computer program condon to calculate paramagnetic ... · the theory of transition-metal ions by...

Computer program CONDON to calculate

paramagnetic susceptibilities

Helmut Schilder Heiko Lueken

December 21, 2002

Preface

Nearly fifty years ago Tanabe and Sugano published the first ligand fieldenergy diagrams which are applicable to dN electronic configurations, but sixtyyears ago Finkelstein and Van Vleck had already worked on the specific case d3

in connection with magnetochemistry. The case d3 without any ligand field isgiven by Gray and Wills seventy years ago and cited in the book by Condonand Shortley first printed 1935.

The Theory of Transition-Metal Ions by Griffith, first published 1961,gives the state of the art, but to be able to calculate paramagnetic susceptibilitiesnumerically, the theory had to be extended very much, e.g. in the case 5d4, inorder to reduce the size of the basis.

1977 and 1979 two books by Konig and Kremer describe computer pro-grams and give their results: Ligand Field Energy Diagrams and MagnetismDiagrams for Transition Metal Ions, respectively.

Today a computer program for personal computer (PC) should be able togive all these results in shortest time, and for f electrons, too. Such a program,program CONDON, is described and given here! No difference needs to be madebetween strong, intermediate and weak ligand fields, if a full basis of micro-statesis used to cover all these cases, and irreducible tensor operators are not needed.This is the standard in CONDON, but in spite of this it is possible, and may bepreferred for a very few cases of 4f electrons, to use irreducible tensor operatorsin a basis of one specific LS state in order to save time and storage.

Program CONDON is firstly cited in the book by Lueken 1999 and espe-cially written for this book.

All literature cited in this preface here is given at the beginning of thebibliography!

ii

Contents

1 Introduction 1

2 Quantum theory and magnetic susceptibility 2

2.1 Physical quantities and units . . . . . . . . . . . . . . . . . . . . 2

2.2 Basisfunctions: Full micro-state basis . . . . . . . . . . . . . . . 3

2.2.1 Interelectronic repulsion (Hee) and spin-orbit coupling(HSO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.2 Ligand field effect (HLF ) . . . . . . . . . . . . . . . . . . 5

2.2.3 Analysis of eigenfunctions: Russell-Saunders states . . . 6

2.2.4 Change of micro-state basis . . . . . . . . . . . . . . . . . 7

2.3 Basisfunctions: Single 2S+1L term (Russell-Saunders term) . . . 8

2.3.1 Spin-orbit coupling (HSO) and ligand field effect (HLF ) . 8

2.3.2 Analysis of eigenfunctions: Micro-states . . . . . . . . . . 9

2.3.3 Change of Russell-Saunders basis . . . . . . . . . . . . . 9

2.4 Zeeman effect (HM ) . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Paramagnetic susceptibility . . . . . . . . . . . . . . . . . 11

2.4.2 Diamagnetic susceptibility . . . . . . . . . . . . . . . . . 11

2.5 Exchange interactions . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5.1 Molecularfield approximations . . . . . . . . . . . . . . . 12

2.5.2 Exchange coupled dimers: Heisenberg coupling (Hex ) andZeeman effect (HM ) . . . . . . . . . . . . . . . . . . . . . 13

3 Practical guide to program CONDON 15

3.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Source code, dimensions, libraries . . . . . . . . . . . . . . . . . 15

3.3 Files for INPUT and OUTPUT . . . . . . . . . . . . . . . . . . 16

3.4 Data sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

iii

iv Contents

3.5 Test calculation (s1 system) – A first glance at program CONDON 18

3.6 Magnetic susceptibility of Ti3+[3d1] — a worked example . . . . 21

3.6.1 Ti3+ – dialogue for INPUT and OUTPUT, and choice ofparameters . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.6.2 Ti3+ – ‘Interface I’: Start of fitting procedure, ‘0th iteration’ 29

3.6.3 Ti3+ – From a basis of micro-states to ‘interface III’,where magn. field enters . . . . . . . . . . . . . . . . . . . 30

3.6.4 Ti3+ – From ‘interface III’ to ‘interface IV’, where tem-perature enters . . . . . . . . . . . . . . . . . . . . . . . . 36

3.6.5 Ti3+ – From ‘interface IV’ to first susceptibility . . . . . 38

3.6.6 Ti3+ – further susceptibilities . . . . . . . . . . . . . . . 39

3.6.7 Ti3+ – Fitting procedure . . . . . . . . . . . . . . . . . . 40

4 Description of program CONDON 46

4.1 General outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 .MAIN. PROGRAM CONDON . . . . . . . . . . . . . . . . . . 48

4.3 Subroutine SHORTL . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Subroutine SUSFIT . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4.1 Subroutine RESTBL for RESult in form of TaBLes . . . 54

4.5 Subroutine LEVenbergSQX . . . . . . . . . . . . . . . . . . . . . 57

4.5.1 Parameters, which may be fitted . . . . . . . . . . . . . . 59

4.5.2 About fitting in general . . . . . . . . . . . . . . . . . . . 60

4.6 Subroutine LEVenbergMARquardt . . . . . . . . . . . . . . . . . 61

4.7 Subroutine SUSCeptibility . . . . . . . . . . . . . . . . . . . . . 63

4.8 Subroutine STR00 . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.9 Subroutine LEVenbergDERivatives . . . . . . . . . . . . . . . . 67

4.9.1 Special option for analytical derivations . . . . . . . . . . 68

4.10 Subroutine TANABE . . . . . . . . . . . . . . . . . . . . . . . . 68

4.10.1 Three 3d electrons – Interelectronic repulsion . . . . . . . 69

4.10.2 Three 3d electrons – Spin-orbit coupling . . . . . . . . . 73

4.10.3 Three 3d electrons – The crystal field . . . . . . . . . . . 73

4.10.4 Subroutine TANABE continued . . . . . . . . . . . . . . 75

4.11 Subroutine ECKART (Wigner-Eckart) . . . . . . . . . . . . . . 75

4.12 Subroutine BLOCkoUT . . . . . . . . . . . . . . . . . . . . . . . 77

Contents v

4.13 Subroutine RUSSELl . . . . . . . . . . . . . . . . . . . . . . . . 80

4.14 Subroutine REDUCTanabe . . . . . . . . . . . . . . . . . . . . . 82

4.15 Subroutine CHANGE . . . . . . . . . . . . . . . . . . . . . . . . 87

4.16 Subroutine REDUCEckart . . . . . . . . . . . . . . . . . . . . . 88

4.17 Subroutine SAUNDErs . . . . . . . . . . . . . . . . . . . . . . . 91

4.18 Subroutine TANFEL (TANabe for F ELectrons) . . . . . . . . . 91

4.19 Subroutine SELECT . . . . . . . . . . . . . . . . . . . . . . . . 93

4.19.1 CHI-START and its corrections . . . . . . . . . . . . . . 95

4.20 Subroutine SUGANReal . . . . . . . . . . . . . . . . . . . . . . 96

4.21 Subroutine SUGANComplex . . . . . . . . . . . . . . . . . . . . 98

4.22 Subroutine SUGANHeisenberg . . . . . . . . . . . . . . . . . . . 100

4.23 Subroutine SUGANK(omplexH)eisenberg . . . . . . . . . . . . . 102

4.24 Subroutine CHIMAG . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Examples 105

5.1 Single ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.1.1 Test calculation (s1 system) . . . . . . . . . . . . . . . . 105

5.1.2 Ti3+[3d1] (D3d) . . . . . . . . . . . . . . . . . . . . . . . 107

5.1.3 Cr2+[3d4] and Re3+[5d4] (cubic) . . . . . . . . . . . . . . 113

5.1.4 Introduction to Os4+[5d4] . . . . . . . . . . . . . . . . . . 114

5.1.5 Os4+[5d4] (D5h and cubic) . . . . . . . . . . . . . . . . . 115

5.1.6 Os4+[5d4] (cubic – strong field scheme) . . . . . . . . . . 116

5.1.7 U4+[5f2] (cubic) . . . . . . . . . . . . . . . . . . . . . . . 117

5.1.8 Fe2+[3d6] (cubic, spin-crossover) . . . . . . . . . . . . . . 118

5.2 Exchange-coupled systems . . . . . . . . . . . . . . . . . . . . . 120

5.2.1 Nd3+[4f3] (NdPt2, cubic) . . . . . . . . . . . . . . . . . . 120

5.2.2 Gd3+–Cu2+ dimers . . . . . . . . . . . . . . . . . . . . . 120

5.2.3 Dy3+ [4f9] ( [ Dy(C5H5)2(µ−Br) ]2, rhombic) . . . . . . . 122

5.3 Search tactics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.4 Differing results of fitting . . . . . . . . . . . . . . . . . . . . . . 126

6 Bibliography 129

vi Contents

7 Index 133

List of Figures

3.1 Part of general flowchart . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 General flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Flowchart 1 (PROGRAM CONDON) . . . . . . . . . . . . . . . 51

4.3 Flowchart 2 (SUBROUTINE SHORTL) . . . . . . . . . . . . . . 52

4.4 Flowchart 3 (SUBROUTINE SUSFIT) . . . . . . . . . . . . . . . 53

4.5 Flowchart 4 (SUBROUTINE LEVenbergSQX) . . . . . . . . . . 58

4.6 Flowchart 5 (SUBROUTINE LEVenbergMARquardt) . . . . . . 62

4.7 Flowchart 6 (SUBROUTINE SUSCeptibility) . . . . . . . . . . . 64

4.8 Flowchart 7 (SUBROUTINE LEVenbergDERivatives) . . . . . . 67

4.9 Flowchart 8 (SUBROUTINE TANABE) . . . . . . . . . . . . . . 70

4.10 Flowchart 9 (SUBROUTINE ECKART) . . . . . . . . . . . . . . 76

4.11 Flowchart 10 (SUBROUTINE BLOCkoUT) . . . . . . . . . . . . 78

4.12 Flowchart 11 (SUBROUTINE RUSSELl) . . . . . . . . . . . . . 83

4.13 Flowchart 12 (SUBROUTINE REDUCTanabe) . . . . . . . . . . 85

4.14 Flowchart 13 (SUBROUTINE CHANGE) . . . . . . . . . . . . . 87

4.15 Flowchart 14 (SUBROUTINE REDUCEckart) . . . . . . . . . . 89

4.16 Flowchart 15 (SUBROUTINE SAUNDE) . . . . . . . . . . . . . 92

4.17 Flowchart 16 (SUBROUTINE TANFEL) . . . . . . . . . . . . . 93

4.18 Flowchart 17 (SUBROUTINE SELECT) . . . . . . . . . . . . . . 94

4.19 Flowchart 18 (SUBROUTINE SUGANReal) . . . . . . . . . . . . 97

4.20 Flowchart 19 (SUBROUTINE SUGANComplex) . . . . . . . . . 99

4.21 Flowchart 20 (SUBROUTINE SUGANHeisenberg) . . . . . . . . 101

4.22 Flowchart 21 (SUBROUTINE SUGANK(omplexH)eisenberg) . . 103

4.23 Flowchart 22 (SUBROUTINE CHIMAG) . . . . . . . . . . . . . 104

5.1 CHI in SI units of a s1 system at low temperatures . . . . . . . . 107

vii

viii List of Figures

5.2 Mu effective/µB of Ti3+[3d1] (D3d) . . . . . . . . . . . . . . . . . 114

5.3 Mu effective/µB of Cr2+[3d4] and Re3+[5d4] (cubic) . . . . . . . 115

5.4 CHI in SI units of Fe2+[3d6] (cubic, spin-crossover) . . . . . . . . 119

5.5 Reciprocal CHI in SI units of Nd3+[4f3] (NdPt2, cubic) . . . . . . 121

5.6 Mu effective/µB of a Gd3+–Cu2+ dimer . . . . . . . . . . . . . . 123

5.7 CHI in SI units of Dy3+[4f9] ( [ Dy(C5H5)2(µ−Br) ]2, rhombic) . 124

5.8 CHI in SI units of Dy3+[4f9] ( [ Dy(C5H5)2(µ−Br) ]2, rhombic) . 127

List of Tables

3.1 File fort.7 – Magnetic susceptibility of Ti3+[3d1] – a worked example 23

5.1 File fort.7 – s1 system . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2 File fort.7 – Magnetic susceptibility of Ti3+[3d1] (D3d) . . . . . . 108

5.3 File fort.7 – Os4+[5d4] (cubic) . . . . . . . . . . . . . . . . . . . . 117

5.4 File fort.7 – Magnetic susceptibility of a Gd3+–Cu2+ dimer . . . 122

5.5 fort.7, BWYdy.DAT, BWYWdy.DAT – [ Dy(C5H5)2(µ−Br) ]2 . . 125

ix

1 Introduction

The calculation of paramagnetic susceptibilities of ions with ndN or nfN (or ns1

or ns2 or npN ) electronic configuration under the influence of ligand fields bya computer program named CONDON covers three parts, concerning the ions,the ligand field, and the magnetic field.

Susceptibility is calculated as a function of interelectronic repulsion and spin-orbit coupling parameters, of ligand field parameters, of strength and directionof magnetic field, of orbital reduction factors and of temperature. This concernsthe ‘standard’ version. Special routines or options are available in addition.

Interelectronic repulsion is not present in a special routine for 4fN electronicconfiguration. One special option is to use the Van Vleck [9] equation, whichis not considered as a standard any more. Furtheron Heisenberg coupling [8](see Subsection 2.5.2) may be taken into account, if (e.g. symmetrical) dimersare involved. Also there are two options concerning ‘molecular fields’, either‘molecular field theory’ [10] itself or ‘molecular field approach (Curie-Weiss)’[8]. ‘Spin crossover’ [11] may be handled, too.

The calculation of susceptibilities is embedded in a fitting procedure usingthe Levenberg-Marquardt method, which is described in the book ‘NumericalRecipes’ [12], and which varies especially the ligand field parameters.

1

2 Quantum theory and magnetic

susceptibility

2.1 Physical quantities and units

All values inside this program named CONDON are given in SI units, andphysical quantities are defined within this ‘System International (SI)’, e.g. thetemperature is always T measured in Kelvin. An extreme example is the mag-netic susceptibility:

Magnetic susceptibility [SI] = 4π times magnetic susceptibility [CGS], al-though the unit is the number 1 in both cases. This magnetic susceptibilityitself is never used here, but the molar magnetic susceptibility named CHI withits SI unit m3/mol.

As an exception the magnetic susceptibility in CGS unit cm3/mol may betaken for INPUT of measured values, if the filename is CGS. . . or cgs. . . , andfor OUTPUT it appears sometimes in addition to that in SI unit m3/mol.

‘Magnetic field’ means always µ0H and is always given in Tesla, ‘magnetisa-tion’ means molar magnetisation and is defined as CHI∗ (µ0H ) and abbreviatedas CHI∗FIELD here, and its SI unit is Tesla ·m3/mol.

The relation ‘energy = magnetic moment ∗ (µ0H )’ defines the magneticmoment used here, abbreviated as MAGN.MOMENT, with SI unitA ·m2 ≡ J/Tesla, but the unit really used is µB = eh/(4πme)= 9.27402 · 10−24A ·m2, the Bohr magneton in SI units not including µ0.

This magnetic moment times µ0 and times Avogadros constant gives themolar magnetisation defined above!

Some useful definitions that apply here are the following:

µ0 ≡ 4πV · s/(107A ·m) = 4πkg ·m/(107A2 · s2) = 4πTesla ·m/(107A).

The definition of the mol is based on one magnetic centre, except in the caseof unsymmetrical dimers!

As an exception to the use of SI units besides µB , cm−1 is taken for theenergy corresponding to the relation ‘wave number = energy /(Planck constanttimes velocity of light)’. This unit is also used for crystal field parameters asdefined by Wybourne [13], for spin-orbit coupling constants, and for the couplingconstant of the Heisenberg operator.

2

2.2. Basisfunctions: Full micro-state basis 3

2.2 Basisfunctions: Full micro-state basis

Operators of interelectronic repulsion and spin-orbit coupling energy (see 2.2.1)and of ligand field energy (see 2.2.2) are represented as matrices. Each operatoryields matrix elements within state functions forming a basis of it. The basismust be orthonormal and in some sense nearly complete, but instead of real statefunctions linear combinations of these may be chosen, and as such micro-statesare most easily to apply, if there are not too much of them. These micro-statesare defined, besides by its quantum numbers, by some integrals as e.g. spin-orbitcoupling constants, which are taken from literature. Slater-Condon parametersfor interelectronic repulsion are also involved. The operators of ligand fieldenergy include ligand field parameters, e.g. those of Wybourne, and these aredetermined by fitting.

For a number of NREL electrons in a subshell with quantum number l givenas KLWERT there are (2(2 KLWERT+1))!/(NREL!∗(2(2 KLWERT+1))!) or-thonormal, i.e. linearly independent micro-states, which may be used as a basis,e.g. 6!/(2!∗4!) = 15 micro-states for two p electrons. One of the micro-states oftwo p electrons is (1+1−), 1 for each ml value, and the sign of ms written asa superscript to the value of ml (see Condon and Shortley [4]). Nearly alwaysa micro-state is written in such a way, i.e. as a sequence of descending ml/ms

values, and in its conventional order, i.e. the first electron to the left, thenthe second electron, the third etc. The micro-state itself is of course a Slaterdeterminant, and the main diagonal of this determinant is written! Thereforethe micro-state (1−1+) is just the same state times minus one, times minusone because one, i.e. an odd number of interchanges, is necessary to arrive atthe state (1+1−) (the same state remains, if an even number of interchanges isnecessary). In CONDON the micro-state (1+1−) is stored as MSML(1,1,1)=1,MSML(1,1,2)=1, MSML(1,2,1)=1, and MSML(1,2,2)=−1, or written as OUT-PUT in the following way (third sum is the second sum plus two times the firstsum):

No.OF STATE SUM SUM SUM (MSML(II,1,I),I=1,NREL),(MSML(II,2,I),I=1,NREL)

1 2 0 4 1 1 1-1

See subroutine TANABE (Section 4.10) for further examples, especially ford electrons!

Another way of storing (1+1−) is by means of MSMLNW(1,1,−1..+1) andMSMLNW(1,2,−1..+1), using three numbers for p electrons and only ms valueselse (times two, in order to get integer values - the first electron is written tothe right of MSMLNW(II,1,..)):

No.OF ST. SUM SUM SUM MSMLNW(II,1,-KLWERT:+KL..) MSMLNW(II,2,-KL..:+KL..)

1 2 0 4 0 0 1 0 0-1

4 Chapter 2. Quantum theory and magnetic susceptibility

The SUMs are often used, and it is convenient to calculate them only once!

Other sums, namely ms + 1/2 + 2 ∗ (ml+KLWERT) for each electron, serveto calculate all possible linearly independent micro-states, i.e. the full basis ofmicro-states to be used. This is done in subroutine SCHEMA! SubroutineNUMBER makes use of this special ordering of micro states to recover theconsecutive number of an otherwise given micro state.

The case of three 3d electrons is given as an example:

(2+2−1+) will be mapped onto 987 and (−1−−2+−2−) is mapped onto 210.

The numbers are decimal numbers, because d electrons (2(2∗2+1)=10) werechosen, the number of 3 digits correspond to the number of 3 electrons, andeach digit is smaller than the digit left of it. There are exactly 10!/(3!∗7!) =120 such numbers, i.e. one micro-state corresponds to one number, and for thefirst micro-state that with number 987 will be chosen, for the second that with986, and for the last that with 210.

There are two micro-states for one s electron corresponding to the two binarydigits 1 and 0.

In the case of f electrons more than ten digits are required, namely 14, and0123456789ABCD are used, to compare with the hexadecimal system, whichuses furtheron E and F. But the number 13 is used instead of D, and 13 isstored as an INTEGER variable. Up to 3432 micro-states have to be calculatedfor f electrons!

A dimension of the basis of up to 1001 is always accepted by CONDON, or itcannot be run at all. Sufficient storage must be available of course, especially fordimensions of 2002, 3003 or 3432, and appropriate subroutines must be linkedfor these. For the latter cases a much smaller basis of one Russell-Saundersstate is almost always sufficient, however.

2.2.1 Interelectronic repulsion (Hee) and spin-orbit cou-pling (HSO)

The book by Condon and Shortley (The Theory of Atomic Spectra) [4], firstprinted 1935, gives all formulas needed to handle the free ions. Instead of Slater-Condon parameters [4] for interelectronic repulsion mostly Racah parameters [5]are used in the case of d electrons.

For spin-orbit coupling see also the books by Griffith [5], by Eyring, Walterand Kimball [14], and by McWeeny and Sutcliffe [15], the latter especially forstep-up and step-down operators.

As a standard a full basis of micro-states is used, e.g. 252 micro-states for nd5

electronic configuration and 1001 for nf4 electronic configuration, respectively.A special routine for 4fN electronic configuration uses as a basis the LS groundstate only (see Section 2.3). For a full basis of micro-states for Dy3+, Tb3+, and


Gd3+ 1001 becomes 2002, 3003, and 3432, respectively.

Spin-orbit coupling energy and, if there is more than one electron and notthe special routine based on one LS state, interelectronic repulsion energy arecalculated.

The operator for spin-orbit coupling, with ZETA the spin-orbit couplingconstant for one electron, is

HSO = ZETA ∗∑Ni=1(lzi ∗ szi + 1/2 ∗ (l+i ∗ s−i + l−i ∗ s+i)).

For the special routine based on one LS state (see 2.3) the operator, withλSO the spin-orbit coupling constant for the LS ground state, is

HSO = λSO ∗ (Lz ∗ Sz + 1/2 ∗ (L+ ∗ S− + L− ∗ S+)).

There is no interelectronic repulsion to be calculated in the case of oneLS state (see 2.3) as a basis, but here, in the case of a full basis of micro-states, it is to be considered! The operator of interelectronic repulsion is Hee =∑Ni<j=1 e

2/rij , but its application, which leads to Slater-Condon parametersand spherical harmonics working as operators, is difficult: See Subsection 4.10.1for details! The Slater-Condon parameter F 0 is always taken as zero, and theRacah parameter A, too. This is possible, because F 0 and A, respectively, donot contribute to any energy differences, although A = F 0 − F 4/441 and theenergies are not the same therefore. Definitely F 0 is zero, and A is not used.There are no further contributions to Hee in CONDON, e.g. from inner shells,and no further potential and kinetic energies within the ions!

2.2.2 Ligand field effect (HLF )

The book by Griffith (The Theory of Transition-Metal Ions) [5], first printed1961, gives all formulas needed for ligand fields. But the definition of ligand fieldparameters used here is that of Wybourne [13], and the ligand field operatorreads, omitting B0

0 , and with kmax=4 and =6 for d and f electrons, respectively,

HLF =kmax∑

k=2,4

[Bk0 Ck0 (i) +k∑

q=2,3,4,6

(Bkq [Ck−q(i) + (−1)q Ckq (i)] )]

with Ckq = [4π/(2k + 1)]1/2 Y kq and Y kq the spherical harmonics working asoperators on spherical harmonics, i.e. the spherical harmonics times [4π/(2k +1)]1/2 work on the angular part of the wave function (the operator does notwork on the wave function itself). Pay attention to the phase convention forcombining Ck−q(i) and Ckq (i). The number of terms is six for d electrons (k upto 4) and eleven for f electrons (k up to 6), respectively, but the number ofnon-zero terms is always less than six or eleven. Even more terms are often zerobecause of symmetry. This operator is an one-electron operator and written forone electron, i.e. electron i . If there is more then one electron, it works on eachelectron, and the results are summed. See Subsection 4.10.3 for details!


As an example the operator for one f electron in a cubic ligand field withthe threefold axis as the z-axis is given here (see Hutchings[16], error in [17]):

Hc(3)LF = B4

0 [ C40 −

√

10/7 (C4−3 − C4

3 ) ]

+ B60 [ C6

0 + (√

210/24) (C6−3 − C6

3 ) + (√

231/24) (C6−6 + C6

6 ) ] .

Twenty point groups may be chosen, all with at least a twofold rotation axis.Because there is also a ‘test’and a group O(3), and two different orientations arepossible for the cubic group, 23 ligand field operators result for each quantumnumber l , but not all of them are different, e.g. group O(3) means no ligandfield, and s electron means no ligand field, too.

As a standard subroutine TANABE is called, which uses a full basis of micro-states, but one may choose a special routine for 4f block atoms or ions with 4fN

electronic configuration with specific ground term 2S+1L (or with specific Jvalue in addition), namely subroutine (Wigner-)ECKART [18, 19, 20].

Ligand fields of any strength are possible, there is no discrimination betweenweak and strong ligand fields . The matrix elements of the ligand field operatorare calculated and added to the matrix evaluated before.

The resulting real symmetrical matrix is diagonalized, and the eigenvaluesand eigenvectors are taken as a new basis.

This basis may be truncated, and it must be truncated, if its size is too large.Matrix elements of the ‘Zeeman operator’ (see Section 4.14 and Section 4.16)are evaluated in this basis for the three directions of space.

An interface in program CONDON may be seen at this place here, becausea new calculation following changes of magnetic field starts here, and becausethere is no more any reference to micro-state basis beyond this place, if sus-ceptibilities are calculated. This ‘interface (III)’ is the most important ‘general’interface!

Special options allow to use parameters as defined by Stevens [21] and Lea,Leask, Wolf [22], respectively.

‘Symmetry quantum numbers’ [10, 17, 23], but only in a restricted sense,may be obtained, if corresponding options are chosen.

2.2.3 Analysis of eigenfunctions: Russell-Saunders states

The full micro-state basis of the firstly resulting eigenstates may be changedto a basis of Russell-Saunders states [4]. This is regarded as an analysis ofeigenfunctions to get additional information. No susceptibilities are calculated,if this option is selected, and therefore, because the new basis is not appliedanywhere, this basis change is not really a basis change, but the new basismight be applied for a basis change. Subroutines RUSSELl and SAUNDErs are


called to perform this basis change. OUTPUT of new eigenvectors goes to filefort.2, but OUTPUT to the terminal and to file fort.11 is also of importance:See Section 4.13!

If the special routine (Wigner-)ECKART is chosen, one single term 2S+1L(Russell-Saunders term) serves as a basis, definitely the ground term in the caseof ‘real’ calculations (as opposed to ‘tests’). Because such a basis is alwayssmall, it may be chosen for ions with a full micro-state basis greater than 1001(and never for a basis smaller than 253 except for tests). Furtheron this optionshould be chosen for 4f electrons only, because only for 4f electrons the aboveanalysis shows, that the ground state is nearly a pure Russell-Saunders term2S+1L.

The basis of one Russell-Saunders state in subroutine ECKART may bechanged to part of a full micro-state basis with help of subroutine CHANGE(and SAUNDErs), and the eigenvectors in this new basis compared with thoseresulting from the full micro-state basis. But this may serve as a very specialtest only and is not recommended.

To compare the eigenvectors resulting from the full micro-state basis andfrom one Russell-Saunders state, subroutine RUSSELl (and SAUNDErs) shouldbe chosen. Starting with the micro-state basis, one gets the composition of theeigenvectors in terms of Russell-Saunders states. The eigenvectors to comparewith are of course composed of one single Russell-Saunders state!

2.2.4 Change of micro-state basis

Neither matrix elements of the ‘Zeeman operator’ (see Section 4.14) for thethree directions of space are evaluated in the basis of micro-states nor any othermatrix elements not yet calculated up to here. Instead the eigenvectors resultingfrom the last diagonalization serve as a new basis, and this means a change ofbasis. This new basis may be truncated, and it must be truncated, if its sizeis too large. Truncation is possible, because only energetically low-lying statesare relevant in magnetochemistry, and only a very small amount of higher lyingstates would be mixed in by magnetic fields or exchange coupling.

Subroutine REDUCTanabe (see 4.14) calculates the eight matrices FMATand GMAT with dimensions up to the maximum, which may occur for d elec-trons and which is 252, three matrices FMAT representing the three directionsof space of the Zeeman operator, two matrices FMAT for intermediate results,and three matrices GMAT needed for Heisenberg coupling.

If FMAT and GMAT would be evaluated within the full basis of micro-statesas basisfunctions, a change of basis would mean a similarity transformation withhelp of the matrix of the old eigenvectors! This would be very time consuming!


2.3 Basisfunctions: Single 2S+1L term (Russell-Saunders term)

As a standard subroutine TANABE is called, which uses a full basis of micro-states, but an essentially different basis may be used for 4f block atoms orions with 4fN electronic configuration. Subroutine (Wigner-)ECKART [18, 19,20] may be chosen, which is a special routine and uses the specific groundterm 2S+1L as a basis. It works with irreducible tensor operators, and it isalso available for tests with d electrons, if f electrons are selected in .MAIN.PROGRAM CONDON. The dimension of the basis using such a 2S+1L groundterm is always small and cannot exceed 66. In addition tests with some otherspecial terms 2S+1L are possible, but the dimension can never exceed 85.

A single 2S+1L term (or LS state) is called a Russell-Saunders state here.The ground term with a specific J value, i.e. a single 2S+1LJ term, cannot beused as a basis in CONDON, but the results of a calculation with such a basismay be simulated by choosing a nearly ‘infinite’ spin-orbit coupling constant.

The basis is defined in CONDON with help of (ML and MS =) ML and MSvalues, and its definition is given as an example only, namely for the case of the3F state of two d electrons. The consecutively ordered ML/MS values are

−3/−1, −2/−1, −1/−1, 0/−1, 1/−1, 2/−1, 3/−1, −3/0, −2/0, −1/0, 0/0,1/0, 2/0, 3/0, −3/1, −2/1, −1/1, 0/1, 1/1, 2/1, 3/1,

and they define exactly twenty-one basisfunctions.

2.3.1 Spin-orbit coupling (HSO) and ligand field effect(HLF )

There is no interelectronic repulsion to be calculated in this case here of one LSstate as a basis!

The operator for spin-orbit coupling, with λSO the spin-orbit coupling con-stant for the LS ground state, is

HSO = λSO ∗ (Lz ∗ Sz + 1/2 ∗ (L+ ∗ S− + L− ∗ S+)).

Subroutine (Wigner-)ECKART (Section 4.11) [18, 19, 20] evaluates the ma-trix of spin-orbit coupling easily in this Russell-Saunders basis.

Irreducible tensor operators are used to calculate ligand field effects in sub-routine ECKART, applying the Wigner-Eckart theorem. Because to each LSground(!) state belongs one single state, which is identical to one(!) micro-statewith MS=S and ML=L, e.g. the micro-state with MS=2 and ML=6 is one of the65 degenerated states of the quintet-I state, the ligand field operator in subrou-tine ECKART to start with is identical to the ligand field operator in subroutineTANABE (see Subsection 4.10.3) for diagonal elements in a micro-state basis.

2.3. Basisfunctions: Single 2S+1L term (Russell-Saunders term) 9

The specific diagonal element is chosen, because it is surely the simplest!

However neither the product with nor the summation over the Wybourneparameters B20, B40 and B60 is calculated as in TANABE, but each sum overall electrons is stored separately. For reduction according to the Wigner-Eckarttheorem and the further procedure see Subroutine (Wigner-)ECKART (Sec-tion 4.11)!

2.3.2 Analysis of eigenfunctions: Micro-states

If the special routine (Wigner-)ECKART is chosen, one single Russell-Saundersterm 2S+1L serves as a basis. This Russell-Saunders basis of the firstly result-ing eigenstates may be changed to part of a full basis of micro-states [4]. Butthis may serve as a very special test only and is not recommended. No sus-ceptibilities are calculated, if this option is selected, which is performed withhelp of subroutine CHANGE (Section 4.15) (and SAUNDErs). The eigenvectorsin this new basis may of course be compared with those resulting from the fullmicro-state basis, but recommended is to do the reverse, i.e. to compare the sin-gle Russell-Saunders state with the composition, in terms of Russell-Saundersstates, of the eigenvectors resulting from the micro-state basis.

2.3.3 Change of Russell-Saunders basis

The change of Russell-Saunders basis to a basis of eigenfunctions is formallyvery much similar to the corresponding change of micro-state basis, but mucheasier to perform. The following text is essentially a copy!

Neither matrix elements of the ‘Zeeman operator’ (Section 4.16) for thethree directions of space are evaluated in the Russell-Saunders basis nor anyother matrix elements not yet calculated up to here. Instead the eigenvectorsresulting from the last diagonalization serve as a new basis, and this means achange of basis. This new basis may be truncated, and it must be truncated, ifits size is too large. Truncation is possible, because only energetically low-lyingstates are relevant in magnetochemistry, and only a very small amount of higherlying states would be mixed in by magnetic fields or exchange coupling.

Subroutine REDUCEckart (Section 4.16) calculates the eight matrices FMATand GMAT with dimensions up to the maximum, which may occur for d elec-trons and which is 252, three matrices FMAT representing the three directionsof space of the Zeeman operator, two matrices FMAT for intermediate results,and three matrices GMAT needed for Heisenberg coupling.

If FMAT and GMAT would be evaluated within the Russell-Saunders basisas basisfunctions, a change of basis would mean a similarity transformation withhelp of the matrix of the old eigenvectors! This would be an alternative here!


2.4 Zeeman effect (HM )

As to the magnetic field, the fundamental equation for paramagnetic suscepti-bilities [8] is applied as a standard. It uses essentially temperature, directionand strength of magnetic field, and energies and corresponding magnetic mo-ments of eigenstates. The strength of the magnetic field does not enter the VanVleck [9] equation, but this equation is not considered as a standard any more.However, if eigenfunctions are analysed with respect to their magnetic behavier,an advantage is obvious!

The energy due to the magnetic field is given by direction, i.e. directioncosines, and strength of the magnetic field, and its matrix elements are calcu-lated with help of the matrix elements of the ‘Zeeman operator’ [8], which reads,with dimensionless operators and in units of −µB ,

(Lz + 2Sz)~k + (Lx + 2Sx)~i + (Ly + 2Sy)~j .

This operator consists of three components, and its matrix elements arecollected in three matrices named FMAT, namely FMAT( , ,1), FMAT( , ,4),and FMAT( , ,5), the latter one should think as being multiplied by (−1)1/2.

Usually, but not here, the following operator is called ‘Zeeman operator’,namely the operator representing the energy due to the magnetic field, whichreads

HM = µB ((Lz + 2Sz)~k + (Lx + 2Sx)~i + (Ly + 2Sy)~j) · (~kBz +~iBx +~jBy)

= µB ((Lz + 2Sz)Bz/B + (Lx + 2Sx)Bx/B + (Ly + 2Sy)By/B )B.

Because the operator HM/B, which results from the matrices of the ‘Zeemanoperator’ and the direction cosines of the magnetic field, serves as a perturbationoperator (also in the case of ‘VANVLECK’!), the operator HM is called ‘pertur-bation operator times magnetic field’ here (‘magnetic field’ short for strength ofmagnetic field, B = µ0H ).

The matrix elements of the latter operator are added to the diagonal matrixof eigenvalues, and a complex hermitian matrix results, if y-direction is included.Orbital reduction factors [17] may be taken into account here.

A further diagonalization yields new eigenvalues and eigenvectors, i.e. neweigenstates.

CONDON calculates the negative magnetic moment −µn corresponding toone single eigenstate n with help of the matrix of the perturbation operatorbefore. It is calculated in units of µB = eh/(4πme), i.e. µn is a number.

In addition the magnetic susceptibility can be calculated using the Van Vleckequation [9]. This equation does not use the strength, but only the direction

2.4. Zeeman effect (HM ) 11

of the magnetic field. The same matrix of a perturbation operator as beforeis applied in this case. If there are degenerated eigenvalues, the method ofGerloch [24] will be applied to get possibly corrections to the squared Zeemancoefficientsof first order.

An interface (‘IV’) in program CONDON may be seen at this place here,because a new calculation following the change of temperature starts here.

2.4.1 Paramagnetic susceptibility

The partition function is calculated as∑

n exp(−En/kBT ) in Subroutine CHI-MAG (Section 4.24), called to calculate CHI and MAGnetisation (kB = 0.695040the Boltzmann constant in cm−1/Kelvin, if En in cm−1).

The molar magnetisation is defined in CONDONas CHI ∗ (µ0H ) and calcu-lated as µ0µBNA

∑

n µn exp(−En/kBT ) divided by the partition function. It isa function of temperature and is really the component of the molar magnetisa-tion in the direction of H , if the molar magnetisation and H are not parallel.The equation used here is called ‘fundamental equation’ in [8].

The molar paramagnetic susceptibility CHI results after division of the molarmagnetisation by µ0H . It is always the susceptibility in the direction of H , andit is defined by Gerloch [24], if the molar magnetisation and H are not parallel.

In addition the magnetic susceptibility can be calculated using the Van Vleckequation [9]. Zeeman coefficients of first and second order instead of the mag-netic moment are used in this case!

2.4.2 Diamagnetic susceptibility

In general the measured susceptibility is the sum of the diamagnetic and para-magnetic susceptibility. The former has a negative value, the latter has a pos-itive value, but may be zero. The measured molar susceptibility entering theprogram should always be the molar paramagnetic susceptibility and shouldnot include the diamagnetic part. The other way round the calculated molarsusceptibility does not include any diamagnetic contribution.

In spite of this the program handles a quantity CHI-0 as an additive con-stant to the calculated susceptibility. This CHI-0 may be negative or positiveand is indeed mostly zero, it may be, but ought not to be, the diamagneticsusceptibility, it may be a correction, positive or negative, of the diamagneticsusceptibility, or it may play a role in connection with ‘temperature independentparamagnetism (TIP)’ or ‘Pauli-paramagnetism’.

Sometimes a CHI-START value of zero is needed, and to get this value avery first INPUT of 0001, meaning s electrons and possibility of tests, is chosen,and two s electrons corresponding to a closed s shell are selected furtheron. (Seealso 5.1.5.)


CHI values used within the fitting procedure must not become negative. Toavoid this, e.g. if CHI-0 is negative, IOPT3=1 must be sometimes selected, sothat CHI-0 is subtracted from measured CHI instead of being added to CHI-START.

2.5 Exchange interactions

2.5.1 Molecularfield approximations

There are three possibilities in program CONDON to take into account molec-ular fields, two options concerning ‘molecular field approach (Curie-Weiss)’ [8],and one option concerning ‘molecular field theory of Weiss’[8, 10]. Only oneoption can be chosen, i.e. only one of three parameters can be chosen as notzero at the same time. A molecular field parameter for (‘real’) molecular fieldcalculations is strictly to be separated from λMF , the molecular field parame-ter for ‘molecular field approach (Curie-Weiss)’. Instead of the molecular fieldparameter λMF the Weiss constant Θ may be taken.

An ‘infinite’ loop (not a ‘DO’ loop) runs in subroutine SUSC (see 4.7) in thecase of ‘molecular field theory’, until convergence is achieved, and in each runsubroutine TANABE (or ECKART) is called with a different ‘field’ to evaluatea value for CHI. The molecular field in subroutine SUSC is proportional to themagnetisation, it is added to the applied field, and it is the same for all direc-tions, i.e. it is ‘isotropic’ (The sum must not become negative!). Convergence isachieved, if CHI becomes nearly constant.

The ‘molecular field approach’ is based on either the formula (see also 4.19.1)(CHI−calc.)−1 = (CHI−START)−1 − λMF , or on the formula (CHI−calc.)−1 =(T −Θ)/C , with C = molar Curie constant.

If Curie‘s law is valid, Θ = λMF C , both formulas are one and the same, andCHI−START = C/T .

As outlined in Subsection 4.19.1, a correction by means of λMF is alwaysapplied, even if it is zero.

If the Weiss constant Θ is taken as parameter, CHI is calculated not for thetemperature T it belongs to, but for a temperature (T −Θ).

Whereas the ‘molecular field approach’ may be applied in case of param-agnetism only, ‘molecular field theorycovers also ferromagnetism, however in avery restricted sense, i.e. more or less for pedagogical purposes only.

2.5. Exchange interactions 13

2.5.2 Exchange coupled dimers: Heisenberg coupling (Hex)and Zeeman effect (HM )

For (symmetrical) dimers there is the possibility to take Heisenberg coupling [8]into account, and in addition magnetic dipol-dipol interaction may be chosen,too. The coupling constant J is one of the parameters which may be fitted.The size of the basis should be relatively small, and this is possible, because thecorresponding matrices are set up following ‘interface III’. A further reductionof the size of the matrix to be diagonalized is possible, if the ‘Combined Model’[25] is applied.

The coupling energy is given by the following operator, written with dimen-sionless angular momentum operators,

Hex = −2J [ Sz1Sz2 + (S+1S−2 + S−1S+2)/2 ] .

The Heisenberg operator works on a basis of product states, and a matrixresults, which has as its dimension, in the case of a symmetrical dimer, thedimension of the original basis squared. A relatively small basis size is thereforerequired for one of the monomers of a symmetrical dimer.

Each product state is the product of two basis states, i.e. Slater determinants,the first factor of the product a basis state of the first centre and the secondfactor one of the second centre. The first operator of each product of operatorsworks on the first basis state and the second factor on the second basis state.

The basis states are either micro states or LS states. In the case of mi-cro states with more then one electron is each operator a sum of one-electronoperators, and the first one works on the spin of the first electron etc.

The resulting matrix is a sum of three matrices, and each of these matricesis, apart from a common factor for each matrix, a Kronecker product involvingGMAT( , ,1...3), namely GMAT( , ,1)⊗GMAT( , ,1), GMAT( , ,2)⊗GMAT( ,,3), and GMAT( , ,3)⊗GMAT( , ,2), respectively.

A perturbation operator with the same (squared) dimension is also requiredhere, because Heisenberg coupling and Zeeman effect are treated at the sametime. For this operator see Section 2.4 and Section 4.22.

The three matrices GMAT( , ,1...3) with matrix elements of S-operators areevaluated along with the matrices of the ‘Zeeman operator’. The matrices andthe eigenvalues are used to set up a common matrix, and the three matrices ofthe ‘Zeeman operator’ are also used, if magnetic dipol-dipol interaction is takeninto account. If the Van Vleck equation is not used, this matrix and the matrixof the perturbation operator times the magnetic field are added, otherwise thismatrix remains unaltered. See Section 4.22 for details.

The matrix will be diagonalized to get the eigenstates, i.e. eigenvalues andeigenvectors, for the fundamental equation for paramagnetic susceptibilities andthe Van Vleck equation, respectively.


CONDON calculates the magnetic moment µn corresponding to one singleeigenstate n with help of the matrix of the perturbation operator above, andthe same operator is used to evaluate the Zeeman coefficients in the case of theVan Vleck equation.

If a dimer is calculated, but not with ‘Combined Model’, two different sortsof dimers may be specified in subroutine SELECT, either a symmetrical dimeror a dimer with one spin-one-half ion (e.g. with Cu2+, option JACuGd). CHIvalues etc. for dimers are calculated for one(!) ion, except in the case of anunsymmetrical dimer.

As a test (‘JATEST’), and instead of Heisenberg coupling, the Ising modeland the XY model [8], respectively, may be applied, and furtheron any model‘between’ these two models. Two factors, a and b, are required as INPUT, andthe Heisenberg model results, if both factors are one. The Ising model resultsfor a=1 and b=0, and it may be modified by taking b>0, but <1. The XYmodel results for a=0 and b=1, and it may be modified by taking a>0, but <1.(If both a and b are 6= 0, they are not(!) normalized.)

3 Practical guide to program

CONDON

3.1 Outline

Program CONDON is written in STANDARD FORTRAN 77, asks for INPUTin a self-explaining manner and is well commentated. All real variables in pro-gram CONDON are declared DOUBLE PRECISION, and all complex variablesCOMPLEX∗16. The latter is an extension to STANDARD FORTRAN 77, butis offered by most software. The .MAIN. PROGRAM CONDON and nearlyall its subroutines and functions are on file CONDON.f, but on different filesare subroutine SHORTL, which defines EQUIVALENCES and nearly all DI-MENSIONS, subroutines DSPEVW and ZHPEVW, which are links to libraryroutines, and subroutine STR00, which is a very special routine.

Two library routines to diagonalize real symmetrical and complex hermitianmatrices, respectively, must be available. To use the optimized binary routinesof commercial libraries [26] is recommended, but alternatively these routines areavailable on file dspzhp.f, which containes all subroutines and functions neededin source code. The latter were downloaded from WWW.netlib.org [26].

Because all routines are available in source code, program CONDON runs onany computer, which offers a FORTRAN compiler. On a PC sufficient storagemay not be available however for some lanthanide ions, if a full basis of microstates is required. But with a restricted basis, and much less storage is neededthen, results are mostly nearly the same!

3.2 Source code, dimensions, libraries

File CONDON.f is the most important and contains PROGRAM CONDONand nearly all its subroutines and functions in source code for FORTRAN 77.

The matrix to be firstly diagonalized has dimension 1001∗1001 for Ho3+,2002∗2002 for Dy3+, 3003∗3003 for Tb3+ and 3432∗3432 for Gd3+, respectively,and there are several matrices of these dimensions. The matrix to be diagonal-ized in the case of the special routine for 4f block ions however has dimensionup to 66 only (or 85 for a quintet-L state of six f electrons as a test), and thesmallest value of 1001 will work in any case! A full basis of micro-states for any

15

16 Chapter 3. Practical guide to program CONDON

d electronic configuration involves only up to 252 micro-states.

Because there are further matrices with dimensions a little greater than1001, a PARAMETER ND2002 with a minimum value of 1110 is introduced insubroutine SHORTL. Four different subroutines SHORTL, on different files, areavailable. On file shortL.f and on file shortP.f ND2002=1110, but ND2002=3432is set on file shortO.f, and ND2002=2002 on file shortQ.f, and one of these files,i.e. either shortL.f or shortO.f or shortP.f or shortQ.f, has to be linked!

File dspzhp.f contains the relevant part of library LAPACK [26] in sourcecode, but these routines are not optimized! The subroutines which are calledby CONDON are DSPEV and ZHPEV, but to do so two subroutines DSPEVWand ZHPEVW on file pcdszh.f serve as links. On file rzdszh.f are two differ-ent subroutines DSPEVW and ZHPEVW, which are the links to subroutinesDSPEV and ZHPEV, respectively, if IBM library ESSL is used. Instead ofdspzhp.f the libraries liblapack.a and libblas.a (llapack and lblas) may be used!

Library NAG19 is more often available, but it does not include these twosubroutines, but DSPEVD and ZHPEVD instead! To be able to call these, twofurther subroutines DSPEVW and two further subroutines ZHPEVW have beenwritten. On file dszhLP.f are subroutines DSPEVW and ZHPEVW to be linkedwith shortL.f or shortP.f and on file dszhOQ.f are those for shortO.f or shortQ.f!The names DSPEV and ZHPEV may always be substituted by DSPEVD andZHPEVD, respectively!

Subroutine STR00 is separated on file str00.f, because it is a very specialroutine and may be exchanged!

Three examples of COMMAND FILES to compile, with options ‘−g’ (todebug) and ‘−O’ (to optimize), and link program CONDON follow, two forGNU FORTRAN under LINUX, and one for FORTRAN under UNIX on anIBM computer, and the name of the file with the executable program is condon:

g77 −g −O CONDON.f shortL.f str00.f pcdszh.f dspzhp.f −o condon

g77 −g −O CONDON.f shortL.f str00.f pcdszh.f -L. -llapack -lblas −o con-don

f77 −g −O CONDON.f shortL.f str00.f rzdszh.f −lessl −o condon

3.3 Files for INPUT and OUTPUT

Nearly all INPUT enters PROGRAM CONDON, the .MAIN. program, andit comes from file OLDf7.DAT, which may be a copy of file fort.7, and/orfrom the terminal, from file CHIINP.DAT (for ‘measured values’) and fromfile BWYBIN.DAT (for ‘parameters’, which may be fitted), but in special casesCHIINP.DAT and/or BWYBIN.DAT may be missing! Instead of CHIINP.DATand BWYBIN.DAT, any other file names may be used, but a file name beginningwith ‘CGS’ or ‘cgs’, e.g. CGSCHI.DAT, instead of CHIINP.DAT means, that

3.4. Data sources 17

CHI values read in are in CGS units instead of SI units. See Subsection 5.1.2or Section 3.6 for file CHIINP.DAT and Section 4.2 for file BWYBIN.DAT! Theprogram reads from file OLDf7.DAT to the end, i.e. EOF (END OF FILE),after that from the terminal if necessary. If file OLDf7.DAT is not available, anew, empty file is created!

File BWYBIN.DAT or a file with the same FORMAT may be put in oncemore in subroutine LEVenbergMARquardt, if it contains limits for all or partof the ‘fitting parameters’, in order to guarantee such limits when iterating(see 4.6).

Most INPUT is ‘per default’, and many values are stored in some ‘BLOCK-DATA’ subprograms.

OUTPUT is to terminal and to file fort.11 in nearly identical manner, partthereof to file fort.1 to be used for plotting. All INPUT from terminal goes tofile fort.7, easily to be used as INPUT in case of a new start of CONDON. Inthe case of ‘STOP’ a new file fort.8 with a special header is written, to be usedas a ‘new file BWYBIN.DAT’! This file fort.8 is also always written (followinga REWIND(8)), if an iteration is completed, in order to serve as a ‘new fileBWYBIN.DAT’, if the program is interrupted.

If subroutine STR00 is running, the B-values calculated there, but onlythose, which go back to the calling routine, are written to file fort.9. Thisroutine needs special INPUT from a file named e.g. str00.dat, too.

File fort.2 is used for OUTPUT in subroutines RUSSELl and SAUNDErs,but these are special routines for a special purpose. Furtheron file fort.3 servesfor OUTPUT in subroutine CHANGE (and fort.2, if ‘MOROUT’ was chosen),and in very special cases in subroutine SAUNDErs in addition to files fort.1 andfort.2.

For some special cases in connection with option ‘JAFAST’, and becausediagonalization before reaching ‘interface III’ may be time consuming, this in-terface may be written to file fort.13 (‘unformatted’) and read in from this fileinstead of a new calculation, if CONDON is started repeatedly with the sameor a similar BWYBIN.DAT file.

3.4 Data sources

Many data are stored in BLOCKDATA subprograms and should be used ifpossible.

Data for Racah parameters [5] and spin-orbit coupling constants [4] for dblock atoms or ions are preferentially chosen from the book by Griffith [5],otherwise from Konig and Kremer [7]. Most parameters given for d electrons,although derived from experiment, do not take into consideration the ligands,although the influence of individual ligands may be important.


Data for 4f electrons, i.e. Slater-Condon parameters and spin-orbit couplingconstants, are chosen from the book by Dieke [27]. These data should be suffi-ciently good for magnetochemistry, but possibly not for ligand fields in general.

Data for 5f electrons, but only for U5+, U4+ and U3+, are taken from theHandbook on the Physics and Chemistry of the Actinides, Vol. 1 [28]. Thesevalues are theoretical relativistic values. These data should be sufficiently goodin many cases, especially if experimental values are not available. See Subsec-tion 5.1.7 for details!

In subroutine STR00 the expectation values of r2, r4, and, for f electrons, r6

are needed. These may be put in directly, but tables are included for d electrons[30], lanthanides [30], and Uranium (relativistic values [28]).

All these data may be changed according to requirements.

One must choose a definite (point) group out of a given table of 23 possibili-ties (indeed 22, one is a test only), the symmetry not ‘lower’ than correspondingto a remaining twofold rotation axis! All ligand field operators are taken fromthe Handbook on the Physics and Chemistry of Rare Earths, Vol. 23 [17]. Ex-cluding ‘imaginary’ operators and operators involving Bk5 , and excluding theicosahedral group, there are 20 point groups, but the cubic group is involvedwith two different orientations, and the group O(3) is not a point group.

With respect to ‘symmetry quantum numbers’, once more data of [17] areused, and in addition data of ‘Handbuch der Physik’, Bd. 28 [23].

3.5 Test calculation (s1 system) – A first glanceat program CONDON

Firstly part of a more qualitative ‘general’ flowchart is given here to show thegeneral structure of program CONDON.

As a first glance at program CONDON it is shown, how a test calculationfor a s1 system, the very first example of calculations with CONDON given lateron (see Subsection 5.1.1), is managed by CONDON.

Nearly all INPUT enters PROGRAM CONDON, the .MAIN. program, aminimum of INPUT and OUTPUT was chosen, and all INPUT came from theterminal in this case (see file fort.7 in 5.1.1)!

A fitting procedure is started in subroutine SUSFIT, although no fitting waschosen, and subroutine LEVSQX is called to calculate SQX as a measure of thequality of a fit, because not all the CHI values were excluded from fitting on theoutset.

The calculation of CHI starts in subroutine SUSCeptibility, which is calledby subroutine LEVSQX. It is embedded in the Levenberg-Marquardt method,and subroutine SUSCeptibility is also called by subroutine LEVDER, if fitting

3.5. Test calculation (s1 system) – A first glance at program CONDON 19

(‘MAIN’) Program CONDON INPUT from terminal,CHIINP.DAT, BWYBIN.DAT

?SUBROUTINEs SHORTL on shortL.f or shortO.f

or shortP.f or shortQ.f(‘DIMENSIONS/ EQUIVALENCES’)

?Interface I: Start of fitting procedure

?SUSFIT (‘Fitting’) ←→ RESultTaBLes (‘Results’)

? 6

LEVSQX (‘SQX’)

? 6

SUSCeptibility

?Interface II: Start of calculation of CHI

6

TANABE ←→ DSPEVW (‘Diagonalization’)

? 6

REDUCTanabe

?Interface III: Transformation to new basis executed

6

SELECT (here: ‘SUGANR’)

? 6

SUGANReal (‘Magn. field enters’) (↔ DSPEVW (‘Diagonalization’))

?Interface IV: Temperature enters calculation

6

CHIMAG (‘CHI and MAGnetisation’)

Figure 3.1: Part of general flowchart


is required.

Eleven the crystal field defining Wybourne parameters and nine further pa-rameters are all zero except a correction factor ‘F’, the 16th parameter, which be-comes ONE per default. In case of need parameters are converted to Wybourneparameters in subroutine SUSC. Subroutine TANABE will be called for a cal-culation based on a basis of micro-states.

In TANABE subroutine SCHEMA is called to calculate all micro-states.The number of micro-states becomes (KASSUM =) two. The micro-states aregiven in two different ways, as MSML and as MSMLNW, and some specialsums of magnetic quantum numbers defining micro-states are given in MQUER(as ‘SUM SUM SUM’). (MSML(1:2,1,I),I=1,NREL) means ml value(s) of (each)electron in each state, (MSML(1:2,2,I),I=1,NREL) means 2∗ms value(s) of (each)electron in each state, (MSMLNW(1:2,1,KLWERT),KLWERT=0) means 2∗msfor (each) KLWERT in each state, if ms>0, and (MSMLNW(1:2,2,KLWERT),KLWERT =0) means 2∗ms for (each) KLWERT in each state, if ms<0.

The following six lines are available, if MOReOUTput is chosen, which iscurrently however not the case!

No.OF STATE SUM SUM SUM (MSML(II,1,I),I=1),(MSML(II,2,I),I=1)1 0 1 1 0 12 0 -1 -1 0 -1

No.OF ST. SUM SUM SUM MSMLNW(II,1,KLWERT) MSMLNW(II,2,KLWERT)1 0 1 1 1 02 0 -1 -1 0 -1

There is no crystal field, no interelectronic repulsion, and no spin-orbit cou-pling for one s electron!

The 2∗2-null matrix will be ‘diagonalized’ with help of subroutine DSPEVW,and its eigenvectors are calculated! The matrix of eigenvectors needs not nec-essarily be the unit matrix, because the two eigenvalues are both zero anddegenerated therefore, but in practice it is for this trivial case.

In REDUCTanabe the matrices FMAT and GMAT are calculated, and theyare calculated in a new basis, the basis of the above eigenvectors ordered cor-responding to the consecutively ordered eigenvalues, and this new basis has ingeneral a reduced dimension, because high lying eigenvalues will be discarded.Of course is the new basis identical to the old basis in this simple case here!

GMAT is needed only if Heisenberg coupling is included, and it concernsonly spin without any g-factor, i.e the g-factor is formally ONE. FMAT includesorbital and spin, the latter with a g-factor of exactly TWO. In this case herewithout any orbital contribution FMAT is two times GMAT!

FMAT is most important as a ‘general’ interface (‘III’) in program CON-DON! It results from the application of the operator (‘Zeeman operator’) of the

3.6. Magnetic susceptibility of Ti3+[3d1] — a worked example 21

(component of the neg.) magnetic moment in each one of the three directionsof space. For the case of a spherically symmetric atom only the z-direction isneeded, i.e. FMAT( , ,1), and this is evaluated with help of the above eigenvec-tors and the magnetic quantum numbers in MQUER.

No magnetic field is taken into account up to here, i.e. up to this interface!

FMAT is used for the energy levels of an atom or ion in a magnetic field, itis used for the calculation of the (neg.) magnetic moment needed for magneti-sation and CHI out of the appropiate eigenvectors, and its diagonal elementsare the Zeeman coefficients of first order in the Van Vleck formula, if there isno degeneration of eigenvalues. The Zeeman coefficients of second order followfrom the non-diagonal elements of FMAT!

Subroutine SELECT is the next subroutine to run! Here the z-direction isselected, if IOPT1=0 or =1, because there is spherical symmetry, and for asingle ion, corresponding to IOPT2=0, i.e. not a dimer, subroutine SUGANRis called!

In SUGANR, R for ‘Real’, the energy of the magnetic field is added to the twoenergy eigenvalues (both zero) using FMAT( , ,1) and the magnetic field. Theresulting matrix is diagonalized and its eigenvectors are calculated. To the veryend of a computation not only for another temperature, the (neg.) magneticmoments are evaluated, using FMAT( , ,1) again and the newly calculatedeigenvectors.

The paramagnetic susceptibility CHI and the MAGnetisation are evaluated,for the temperature given, in subroutine CHIMAG. CHIMAG returns to SUG-ANR and SUGANR returns to SELECT. If temperature changes (and nothingelse), the new calculation starts in CHIMAG, and therefore another ‘interface(IV)’ may be seen here!

Because only one direction, the z-direction, and not two or three directionsare concerned and no mean value needs to be calculated before, CHI becomesCHISTA dependent on temperature and serves as CHI-START in subroutineSELECT. Out of CHI-START, CALCUL. CHI is evaluated, but in the case ofno corrections as here that makes no difference!

For OUTPUT of the results subroutine RESTBL, that means RESult-TaBLes,is called at the very end! Output corresponding to ‘fitting takes place’ is putout, i.e. columns for MEASURED CHI and CALCUL. CHI, because a minimumof OUTPUT was chosen!

3.6 Magnetic susceptibility of Ti3+[3d1] — aworked example

To demonstrate how CONDON works for an example of interest the calculationof the paramagnetic susceptibility of [Ti(H2O)6]3+ is chosen, a complex with


the form of a distorted octahedron of point group D3d, and the ion involved isTi3+[3d1]. For the fitting procedure see Subsection 5.1.2, where ‘2999’ has beenchosen to get ‘less,less,less’ OUTPUT etc., here ‘2110’ will be chosen instead toget more OUTPUT in the form of some intermediate results (‘MOReOUTput’)and some extra text (‘MOReTEXt’). The last iteration of the fitting procedurethere is once more calculated here as ‘zeroth’ iteration using file fort.8, which isnamed BWYBTi.DAT here, and a ‘first’ iteration is added, but with only B20to be fitted!

File fort.7 will be given here as a table!

File fort.11, somewhat shortened, follows now in its (nearly) original form!Comments are added in lower case.:

3.6.1 Ti3+ – dialogue for INPUT and OUTPUT, andchoice of parameters

PROGRAM CONDON written in STANDARD FORTRAN 77.Copyright 2000 Helmut SCHILDER, Fachhochschule Aachen Germany.PROGRAM CONDON is free software, covered by the GNU GeneralPublic License,and you are welcome to change it and/or distribute copies of itunder certain conditions.Type "show copying" to see the conditions.There is absolutely no warranty for PROGRAM CONDON. Type"show warranty" for details.Type "show more" to see some hints.Scientific use of computational results or program codeshould include the citation:Helmut Schilder: PROGRAM CONDON, Fachhochschule Aachen 2000.RETURN to start calculation, type "9" to stop.

CALCULATION OF (MOLAR) MAGNETIC SUSCEPTIBILITIES OF ATOMS OR IONSWITH NS**N, NP**N, ND**N OR NF**N ELECTRONIC CONFIGURATION,AND WITH 4F**N ELECTRONIC CONFIGURATION WITH SPECIFIC GROUND TERM(2S+1)L, I.E. FOR THE CASE 4F**N THERE ARE TWO ESSENTIALLYDIFFERENT ROUTINES.

Firstly request for l value of electron(s), for MOReOUTput and for MORe-TEXt:

INPUT 4 DIGITS: L VALUE OF 1 EL.(0...3), FOR MORE (OR LESS) OUTPUTOF INTERMEDIATE VALUES (1(9)), OF MORE (OR LESS) TEXT (1(9)),FOR MORE (OR LESS) POSSIBILITIES OF TESTS (1(9))


Table 3.1: File fort.7 – Magnetic susceptibility of Ti3+[3d1] – a worked example

”NO WARRANTY”2110 =KLWERT, . . . , . . . , . . . d el.,MOROUT,MOReTEXt

8 =NGROUP(POINT GROUP) D3d000 =JAVANV,IOPT1,IOPT2 not VANVL,one mean dir.,single ioncgsTi.dat (OF CHI . .) for table of temp.’s, CHI’s & fields(I1,F7.2,3D17.6) FORMAT of cgsTi.DAT0 =JANO(OUTPUT) not all data of cgsTi.dat => fort.11BWYBTin.DAT (BWYBIN) for list of parameters (B20 fitted)0 =JASTRU(cture) no CALL of subroutine STR000 =NOPROZ (IN % YES/NO) fit ‘in %’1 =IOPT4 (CHI,1/CHI, ..) ‘CHI itself’ is fitted1 = No.OF D EL. one d electron0 =JANOC (”RACAHC=4∗B”) ‘DATA from GRIFFITH’41 =LDG1IN,NRSRIN(CHARGE+) charge+1/ no. of trans.-series0 =NEURAH (NEW RACAH) no RACAH’s for 1 el.!0 =NEUKSI (NEW ZETA) no new spin-orb. coupl.-const.0 =NEUORB (ORB.-RED.-F.) orb.-red.-factor(s) = ONE

1 0 =IITMAX,IITMIN one iteration in a fitting proced.0 =JASTOP(later on!) no STOP in advance1 =MOROUU IN TANABE OUTPUT of matrices etc.1 =MOROUU IN SCHEMA ‘MOReOUTput here’

10 =KASSUL(”No.OF ROWS”) (all) 10 micro-states10 =KASSUL(”No.OF ROWS”) (all) 10 micro-states

1 = JABLOC(KING OUT) CALL of subroutine BLOCUT0 =MOROUZ IN BLOCUT/0 little OUTPUT in BLOCUT

0 0.0000 =LIMITI,GRENZD standard OUTPUT eigenvec.0 =TRANSF. IN BLOCUT transform.(!) of d orbitals1 =MOROUU IN REDUCTanabe OUTPUT of matrices etc.10 =NRVALI(No. OF STATES) 10 new states (not micro-states)

1 =MOROUU IN SUGANO (for next line:)110 =JANO1,JANO2,JANO3 always all, but no full matrices0 =JANO OUTPUT IN CHIMAG ‘only first temperature’

5 5 =MOROUU/V IN SELECT 5 temp./ 5 CHI’s0 =NOMORE OUTPUT HERE in RESTBL ‘no first result’

0.00000000E+00 =(B)OMEGA standard (0.1) for OMEGA8 =MOROUZ IN BLOCUT/0 no more OUTPUT in BLOCUT

‘blanc’ (BWYBIN 2nd) ‘blanc’: no ‘limits’ read in!0 =MOROUU IN GAUSS no more OUTPUT in GAUSS0 =JASTOQ(2=STOP) no STOP: get table of OUTPUT0 =MFITIN(FORM OF OUTPUT) no ‘Change’ in RESultTaBLes2 =JASTOR(2=STOP) ‘yes’: STOP


2110 =KLWERT, . . . , . . . , . . . ! ! ! ! ! ! ! ! !

Because more text was chosen (’MORTEX=1’), a few hints are given:

"4 DIGITS" MEANS "FORTRAN FORMAT 4I1".!!!!!!!!! MEANS "ALSO TO FILE fort.7".RESULTING VALUES FOR MOROUT, LESS, JATEST, LTS, MORTEX: 1 0 0 0 1

d electrons have been chosen:

CALCULATION OF (MOLAR) MAGNETIC SUSCEPTIBILITIES OF D BLOCK ATOMSOR IONS WITH ND**N ELECTRONIC CONFIGURATION: THE ATOMS OR IONSWILL BE LISTED, IF ONE CHOOSES DATA FOR RACAH PARAMETERS ANDSPIN-ORBIT COUPLING CONSTANTS FROM THE BOOK BY GRIFFITH LATER ON.

The (point)group is asked for and D3d is selected (because of ’MORTEX’ ahint to C-INF-V and ???, and following INPUT of point group, some informationabout the most important groups is given):

THERE ARE 23 DIFFERENT POSSIBILITIES OF INPUT OF (POINT) GROUPS:D2H,D3H,D4H,D5H,D6H,D-INF-H,D2D,D3D,D4D,D6DD2,D3,D4,D6,C2V,C3V,C4V,C6V,C-INF-V,???,CUB(I),CUB(II),O(3)

INPUT IN I2 THE (CONSECUTIVE) NUMBER(0 => 21 <=> CUB-4FOLD ROT.AXIS)

C-INF-V MEANS: AXIAL WITH VERTICAL PLANES; ??? MEANS:"ALL B-VALUES" ((MORE THAN) "REAL" PART OF C3)

8 =NGROUP (POINT GROUP) ! ! ! ! ! ! ! ! !

SOME USEFUL HINTS FOR THE GROUP CHOSEN: CHOOSE ”MORTEX”

Because of ‘MORTEX’, more information is given!:

B40=21.DQ FOR CUB(I); NEG. B40 FOR CUB(I), IF 4-FOLD COORD.;FOR THE TETRAG. CASE, B44=SQRT(5./14.)*B40 => CUB(I), IF B20=0(FOR D EL.; B64=-SQRT(3.5)*B60);FOR THE TRIG. CASE, B43=-SQRT(10./7.)*B40 => CUB(II), IF B20=0(FOR D EL.; B63=SQRT(210.)*B60/24., B66=SQRT(231.)*B60/24.)(IT IS THEN B40-CUB(II)=-(2/3)*B40-CUB(I) (FOR D EL.;B60-CUB(II)=(16/9)*B60-CUB(I)!)

All but three B values, if any more should be put in, will be discarded:


FROM THE FOLLOWING LIST OF POSSIBLE B VALUES THOSE MARKED WITH 1ARE EXCLUDED:B20, B22, B40, B42, B43, B44, B60, B62, B63, B64, B66

0 1 0 1 0 1 1 1 1 1 1

Now a calculation with a non-zero magnetic field in one ‘mean’ direction fora single ion is chosen (and because of ‘MORTEX’ some further hints are given):

INPUT IN 3I1: "JAVANV(LECK)"=1 FOR FORMULA OF VAN VLECK,IOPT1=0...3 FOR DIRECTION(S) OF FIELD, MOSTLY =0 FOR ONE(MEAN) DIRECTION OR =1 FOR MEAN VALUE OF CHI CALCULATED FOR1,2,3 DIRECTION(S) (DIRECTION AT WILL INCL. Y-DIRECTION: =2,NO Y-DIR.: =3) (IOPT1=0 AND IOPT1=1 ARE NOT ALWAYS REALLYDIFFERENT!) AND IOPT2=0...2 FOR (SINGLE) ION/ (SYMM.)DIMER/COMB.MOD. FOR F EL.


(THERE ARE 12(!) DIFFERENT OPTIONS IOPT12 IN SUBROUTINE SELECT,WHICH RESULT FROM 4*3=12 INPUT-OPTIONS, AND THE LATTER ARE:(IOPT1=:) ONE (MEAN) DIRECTION (=0), MEAN VALUE OF CHI WITHCHI CALCULATED FOR 1,2,3 DIRECTION(S) (=1), DIRECTION AT WILL(WITH Y: =2; WITHOUT Y: =3), - (IOPT1=0 AND IOPT1=1 ARE NOTALWAYS REALLY DIFFERENT!) -,(IOPT2=:) SINGLE ION (=0), (SYMM.)DIMER (=1), DIMER COMB.MOD.FOR F EL. (=2)! (IOPT12=3*IOPT1+IOPT2+1=1...12))

000 =JAVANV,IOPT1,IOPT2 ! ! ! ! ! ! ! ! !

The program confirms one ‘mean’ direction:

DIRECTION(S) CONCERNED AND TO BE CALCULATED: 2 1

A table of temperatures, CHI values, etc. is asked for now:

WHERE ARE THE VALUES FOR: "TO BE FITTED YES(0)/NO(1)",TEMP. IN K, CHI-MEASURED IN SI, SIGMA, FIELD IN TESLA?"RETURN" FOR CHIINP.DAT; H OR h FOR "H ATOM" <=> NO INPUT HERE!INPUT OF "CGS OR cgs" (E.G. CGSCHI.DAT) <=> CHI-MEASURED IN CGSUNITS! INPUT OF 888 FOR "STOP"!

Of course the same file cgsTi.dat is chosen as in the fitting procedure inSubsection 5.1.2, cgs. . . and not e.g. CHIINP.DAT, because the values are incgs units.

NAME READ IN WAS: cgsTi.dat ! ! ! ! ! ! ! ! !


THERE IS NO NEED TO FIT ANY VALUES! THE STANDARD INPUT HEREIS CHI IN SI UNIT M**3/MOL. THE INDIVIDUAL STANDARD DEVIAT-IONS SIGMA CANNOT BE ZERO AND ARE E.G. SET TO ONE. FIRSTLINE MUST BE TEXT!"RETURN", IF FORMAT(I1,F7.2,3D17.6), OR INPUT FORMAT IN A50

(I1,F7.2,3D17.6) ! ! ! ! ! ! ! ! !

HEADER: Magn. susc. of CsTiSH in CGS, 270 - 12 K & 11.5 - 2 K,0.5 T.HERE THE FIRST MAX. 3 ROWS OF VALUES ("NOT", TEMPIK, CHI, SIGMA,FIELD)CHI VALUES PUT IN WILL BE CHANGED TO SUCH IN SI UNITS!

0 270.27 0.183972E-07 0.100000E+01 0.500000E+000 265.28 0.186736E-07 0.100000E+01 0.500000E+000 260.28 0.189627E-07 0.100000E+01 0.500000E+00

NO FURTHER VALUES TO FILE 11 AND/OR TTY (0), VALUES (ONLY) TOFILE 11 (1), TO FILE 11 & TTY (2), (ONLY) TO TTY (3)?

0 JANO(OUTPUT) ! ! ! ! ! ! ! ! !

Not file BWYBIN.DAT, but file BWYBTi.DAT will be chosen now (andbecause of ‘MORTEX’ some information is given about the general content ofsuch a file):

IS THE NAME OF THE NEXT FILE BWYBIN.DAT IN FORMAT(I2,D16.8)?"RETURN" OR GIVE ITS NAME! (8 => STOP, H OR h "NO INPUT FROMFILE")

BWYBTi.DAT (BWYBIN) ! ! ! ! ! ! ! ! !

EACH ROW OF THIS FILE (FIRST ROW MUST BE TEXT!):"TO BE FITTED YES(01)/NO(00)" (THERE IS NO NEED TO FIT ANYCOEFFICIENTS!),AND COEFF. OF LIG. FIELD (WYBOURNE COEFFICIENTSSHOULD BE PREFERRED ,BUT OTHERS ARE POSS. TOO!): 11 SUCH COEFF.PLUS 9 FURTHER ROWS (B20,B22,B40,B42,B43,B44,B60,B62,B63,B64,B66+J,LAM,THET,CHI0,F,MOLF,MIX,...) ARE NEEDED!(09/08 INSTEAD OF 01/00 AS SPECIAL OPTION FOR B43,B44,B63,B64,B66:INPUT MEANS "FACTOR" IN E.G. B44=B40*SQRT(5./14.)*FACTOR)J <=> HEISENBERG COUPLING IN A (SYMMETRICAL) DIMERLAM <=> MOLECULAR FIELD PARAMETERTHET <=> THETA OF CURIE WEISS LAWCHI0 <=> CHI DIAMAGNETICF <=> CORR. OF WEIGHT OF SAMPLE MEASURED CHI COMES FROM


MOLF <=> PARAMETER OF MOLECULAR FIELD THEORY OF WEISS(FIELD WILL BE ITERATED UNTIL CONVERGENCE HAS ACHIEVED)MIX/CROSSOVR <=> A MIXTURE OF SINGLE ION AND DIMER ORDENOMINATOR OF "SPIN CROSSOVER" (ONLY D EL.!),ENTHALPY <=> ENTHALPY FOR "SPIN CROSSOVER",ENTROPY <=> ENTROPY FOR "SPIN CROSSOVER".

The program confirms ‘BWYBTi.DAT’:

CRYSTAL FIELD PARAMETERS ETC. ARE READ IN FROM BWYBTi.DAT

HEADER: BWYBTi.DAT: 1.130%, Restart.THE VALUES READ IN ARE:1 -0.13394885E+05 0 0.00000000E+00 0 -0.12847059E+050 0.00000000E+00 0 0.35591271E+05 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.10000000E+01 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00

Already here(!) to be answered:

INPUT IN I1 YES/NO(0): CALL OF STR00 (1=SOM, 2=PCEM)

0 =JASTRU(cture) ! ! ! ! ! ! ! ! !

The following two questions have to be answered, because fitting is required(and furtheron a statement concerning CHI-0 is given)!:

THERE ARE TWO ESSENTIALLY DIFFERENT POSSIBILITIES FOR A LEASTSQUARE FIT:THE MEASURED VALUES AND THE CALCULATED VALUES ARE BOTH DIVIDED BYTHE MEASURED VALUES (THE RESULT FOR THE QUALITY OF THE FIT IS IN%, AND THIS SHOULD BE PREFERRED HERE: INPUT 0) OR THIS DIVISIONIS NOT DONE (INPUT 1)

THE RESULT FOR THE QUALITY OF THE FIT IS IN % YES(0)/NO(1): 0! ! ! ! ! ! ! ! !

BECAUSE CHI-0 IS NOT CONCERNED, IOPT3 REMAINS 0 <=> FITTINGIS BASED ON UNCORRECTED MEASURED CHI!

CHI ITSELF MAY BE FITTED OR ONE/CHI OR MAGNETIC MOMENT OF 1 ATOMOR EFFECTIVE MAGNETIC MOMENT (BOTH IN BOHR MAGNETONS): INPUT(IOPT4=)1...4 (0=>1)!


1 =IOPT4 (CHI,1/CHI,..) ! ! ! ! ! ! ! ! !

The program resumes about fitting options:

BECAUSE THERE ARE NO SPECIAL FITS, OR "ANALYTICALLY" IS NOTPOSSIBLE, NO INPUT IS REQUIRED FOR "ANALYTICALLY"!

The number of d electrons is asked for:

INPUT IN I1 NUMBER OF D EL.; 0 => 1!

THE NUMBER OF D ELECTRONS IS = 1 ! ! ! ! ! ! ! ! !

Many data are stored in BLOCKDATA subprograms and should be used ifpossible.

INPUT IN I1 YES/NO(0), IF DATA FROM KOENIG/KREMER, I.E.DIFFERENT ZETA, AND RACAHC WILL BECOME =4.*RACAHB! -STANDARD: 0<=>GRIFFITH

0 =JANOC (”RACAHC=4*B”) ! ! ! ! ! ! ! ! ! DATA FROM GRIFFITH INCL.RACAHC AND COUPL.-CONST. HAVE BEEN CHOSEN!

The ion now chosen is Ti3+:

INPUT IN 2I1 CHARGE+1 OF THE ION AND 1ST,2ND,3RD TRANS.-SERIES

CHARGE OF THE ION (0=ATOM) AND No.OF TRANS.-SERIES= 3 1! ! ! ! ! ! ! ! !

TI V CR MN %% %% %% %% %%; ALL WITHOUT S ORB.-OCC.DATA FROM GRIFFITH ARE USED INCL. RACAHC AND COUPL.-CONST.!

RACAHB RESP. RACAHC ARE AND REMAIN ZERO!BECAUSE ANY CHANGE IS NOT POSSIBLE, INPUT NO(0) IN I1!

0 =NEURAH(NEW RACAH) ! ! ! ! ! ! ! ! !

F-INTEG(RAL),I=2,3:0.00000000E+00 0.00000000E+00

THE (POS.!) COUPL.-CONST. ZETA (1(!) EL.) IS = 0.1540E+03

Coupling constant ZETA and orbital reduction factors may be changed:


INPUT IN I1 YES/NO(0): NEW (POS.!) SPIN-ORBIT COUPL.-CONST.FOR 1 ELECTRON.

0 =NEUKSI(NEW ZETA) ! ! ! ! ! ! ! ! !

INPUT IN I1 YES/NO(0): ORB.-RED.-FACTOR(S) OTHER THAN ONE

0 =NEUORB(ORB.-RED.-F.) ! ! ! ! ! ! ! ! !

This is the end of .MAIN. PROGRAM CONDON and some hints out ofshortL.f follow now:

CONTROL OF SOME DIMENSIONS/ EQUIVALENCES:

ND2002, NDS66 = 1110 66, JEQUIV= 0

Subroutine SHORTL on file shortL.f has been linked here! SHORTL standsfor SHORTLEY and for short meaning a short routine – only DIMENSIONS andEQUIVALENCES are defined! ShortL.f is taken as ‘standard’, other possiblechoices are shortO.f, shortP.f and shortQ.f!

Because of ‘MORTEX’, one more hint is given, but it may be ignored here!(It concerns shortO.f and shortL.f as opposite to shortP.f and shortQ.f.)

JEQUIV=0 <=> NO EQUIVALENCE (EIVECV,FMIKR1): NO EL.-EL. REPULSIONAND NO SPIN-ORBIT COUPLING HAVE TO BE (NEW) CALCULATED! THATMEANS: IF "JAFAST" (FOR F EL.!), JEQUIV=1 DOES NOT MATTER!

3.6.2 Ti3+ – ‘Interface I’: Start of fitting procedure, ‘0thiteration’

The fitting procedure starts and ends in subroutine SUSFIT, which calls at thevery end subroutine RESultTaBLes for OUTPUT in form of a table, this tableincluding headers going to file fort.11 and without headers to file fort.1 for plots!

Subroutine SUSFIT calls subroutine LEVSQX, where SQX comes from anda DO LOOP over all rows of the INPUT table (‘CHIINP.DAT’) is started. LEV-SQX calls SUSC, where the calculation of SUSCeptibility starts! But because‘molecular field’, ‘spin crossover and ’subroutine STR00’ are not involved, thecalculation really will start in subroutine TANABE!

Some control of parameters and ‘one iteration’ here in subroutine SUSFITnow!:

MFIT, I.E. No. OF PARAMETERS ACTUALLY BEEN ADJUSTED, = 1


THERE IS No. OF ITERATION = 0INPUT IN I5,I3 MAX. AND MIN. NUMBER OF (FURTHER) ITERATIONS!:(ITER.COUNT 0 0, INPUT MAY BE NEG. FOR "FURTHER" ITER.!)FOR THIS FIT WITH 1 PARAMETER: FOR MAX.=MIN. NUMBER OF ITERATIONSLEVENBERG/MARQUARDT MAY BE DISABLED LATER ON(OMEGA SERVES AS A STEP-WIDE)!

1 0 =IITMAX,IITMIN ! ! ! ! ! ! ! ! !

ESPECIALLY FOR TIME CONSUMING COMPUTATIONS AND/ORFOR COMPUTATIONS IN BATCH MODE(TO GUARANTEE A DEFINITE END):

INPUT IN I1 YES/NO(0): "STOP INSTEAD OF FURTHER INPUT"(FURTHER INPUT FOR:MORE ITER., CHANGE OF PARAMETERS/(PARAMETERS/VALUES TO FIT)OR (FURTHER) TABLES OF OUTPUT)YES =1,3,6/ =2 MEANS STOP FOLLOWING / BEFORE FIRST TABLE OFOUTPUT (=3 / =6 MEANS "NO FITTING / FITTING, BUT 6 COLUMNS")

0 =JASTOP(later on!) ! ! ! ! ! ! ! ! !

In subroutine SUSC control of parameters, which enter subroutine TAN-ABE:

CALCULATED WYBOURNE VALUES (PLUS 9 OTHER VALUES)(1 OR -1 MEANS "FITTED"):

1 -0.13394885E+05 0 0.00000000E+00 0 -0.12847059E+050 0.00000000E+00 0 0.35591271E+05 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.10000000E+01 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00

3.6.3 Ti3+ – From a basis of micro-states to ‘interface III’,where magn. field enters

The calculation of CHI really starts in subroutine TANABE!

Because ‘MOROUT’ was chosen, the following question will be asked, andthe answer given is: ‘Yes’, more OUTPUT please!:

MOROUT.NE.0: INPUT IN I1 MOROUT-HERE-IN-TANABE; =0, 1 OR 2!


MOROUT-HERE-IN-TANABE = 1 ! ! ! ! ! ! ! ! !

Subroutine TANABE calls subroutine SCHEMA!:

HERE IS SUBROUTINE SCHEMA TO PREPARE ALL MICRO STATESINPUT MOROUT-HERE-IN-SCHEMA IN I1!:

1 =MOROUU IN SCHEMA ! ! ! ! ! ! ! ! !

THERE ARE 10 MICRO STATES! OUTPUT OF ALL OR OF .. (IN I4)?

10 =KASSUL(”No.OF ROWS”) ! ! ! ! ! ! ! ! !


1 2 1 5 2 12 2 -1 3 2 -13 1 1 3 1 14 1 -1 1 1 -15 0 1 1 0 16 0 -1 -1 0 -17 -1 1 -1 -1 18 -1 -1 -3 -1 -19 -2 1 -3 -2 1

10 -2 -1 -5 -2 -1

No.OF ST. SUM SUM SUM MSMLNW(II,1,-KLWERT:+KL..) MSMLNW(II,2,-KL..:+KL..)THERE ARE 10 MICRO STATES! OUTPUT OF ALL OR OF .. (IN I4)?

10 =KASSUL(”No.OF ROWS”) ! ! ! ! ! ! ! ! !

1 2 1 5 0 0 0 0 1 0 0 0 0 02 2 -1 3 0 0 0 0 0 0 0 0 0-13 1 1 3 0 0 0 1 0 0 0 0 0 04 1 -1 1 0 0 0 0 0 0 0 0-1 05 0 1 1 0 0 1 0 0 0 0 0 0 06 0 -1 -1 0 0 0 0 0 0 0-1 0 07 -1 1 -1 0 1 0 0 0 0 0 0 0 08 -1 -1 -3 0 0 0 0 0 0-1 0 0 09 -2 1 -3 1 0 0 0 0 0 0 0 0 0

10 -2 -1 -5 0 0 0 0 0 -1 0 0 0 0

Returned to TANABE:


FACTORS TO STEP-UP AND STEP-DOWN OP., STARTING WITH ML=-KLWERT:2.000 2.449 2.449 2.000 0.0000.000 2.000 2.449 2.449 2.000

The ligand field is given by the following (Wybourne-)coefficients!:

BWWOLD(1:11) = -0.1339E+050.0000E+00 -0.1285E+05 0.0000E+00 0.3559E+05 0.0000E+000.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Field in Tesla and in cm−1 (not used here, but later on!):

TESLAS,TESLAC= 0.500000E+00 0.233432E+00

MATRIX FMIKR1 INCL. SPIN-ORB.-COUPL. MATRIX;COLUMNS (IN THE FORM OF ROWS):BECAUSE MOROUU=1, ONLY ONE COLUMN (IN THE FORM OF ONE ROW)!

0.1540E+03 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+000.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

MATRIX FMIKR2 (LIG.-FLD. MATRIX);COLUMNS (IN THE FORM OF ROWS):BECAUSE MOROUU=1, ONLY ONE COLUMN (IN THE FORM OF ONE ROW)!

0.3215E+04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+000.1003E+05 0.0000E+00 0.0000E+00 0.0000E+00

The matrix to be diagonalized is ready (no magnetic field):

MATRIX TO BE DIAGONALIZED:COLUMNS IN THE FORM OF ROWS, STARTING WITH DIAG. ELEM.

BECAUSE MOROUU=1, ONLY ONE COLUMN (IN THE FORM OF ONE ROW)!0.3369E+04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+000.1003E+05 0.0000E+00 0.0000E+00 0.0000E+00

ONE GETS MJ VALUES IN SUBROUTINE BLOCUT: THESE, IN THEFORM "2*MJ MODULO 2*AXIS-No.", MEAN SYMMETRY QUANTUM NUMBERS!ALSO OCCUP. OF SPINORBITALS ARE CALC. IN SUBR. BLOCUT!INPUT IN I1 YES/NO(0) "BLOCK-OUT" ("STANDARD": 0)!

To get occupations of spin orbitals, especially those of real orbitals, thissubroutine will be called, but with option ‘MOROUZ=0’ only (no MJ values):

BLOCKING OUT TAKES PLACE YES/NO(0): 1 ! ! ! ! ! ! ! ! !


SUBROUTINE to BLOCk oUT, if possible, a real symm. matrix DBLMATwith elements abs. greater than PARR and ZERO else.MOROUT.NE.0: INPUT IN I1 MOROUT-HERE-IN-BLOCUT; =0 OR =1!... OR =8 FOR "NO MORE OUTPUT FURTHERMORE HERE IN BLOCUT"!(=0 FOR OUTPUT AT THE END OF BLOCUT ONLY: OCCUP.OF SPINORB.)(=8 CANCELS OUTPUT AT THE END OF BLOCUT AND IN FURTHER ITER.!)

0 =MOROUZ IN BLOCUT/0 ! ! ! ! ! ! ! ! !

NULL-ROW/ -COLUMN IS THERE (YES=T/NO=F): FNUMBER OF BLOCKS AND DIMENSIONS OF THESE= 3

3 4 3INPUT IN I4,F10.4 LIMITS FOR OUTPUT OF EIGENVECTORS;< ZERO => 0 AND 0.0; ZERO => 10 0.1(I.E.: MAX. NUMBER OF EIGENVEC. TO BE TAKEN INTO ACCOUNT ANDLOWEST (ABS.) VALUE OF THESE EIGENVECTORS TO BE TAKEN INTOACCOUNT)

0 0.0000 =LIMITI,GRENZD ! ! ! ! ! ! ! ! !

I, J, EIVEC(I,J) FOR ABS(EIVEC(I,J)).GE. 0.1E+00 UNTILLIMIT= 10(PART OF) THE FIRST ROW CONTAINS (PART OF) THE FIRST EIGENVECTOR

2 1 -0.6628E+00 8 1 0.7487E+00 3 2 0.7487E+009 2 0.6628E+00 4 3 0.7436E+00 5 3 -0.1874E+00

10 3 0.6419E+00 1 4 -0.6419E+00 6 4 -0.1874E+007 4 0.7436E+00 1 5 -0.1273E+00 6 5 0.9823E+007 5 0.1376E+00 4 6 0.1376E+00 5 6 0.9823E+00

10 6 0.1273E+00 3 7 -0.6628E+00 9 7 0.7488E+002 8 -0.7488E+00 8 8 -0.6628E+00 4 9 -0.6543E+00

10 9 0.7562E+00 1 10 0.7562E+00 7 10 0.6543E+00

No. OF VALUES= 24; ON FILE11: 24; MAX. 24 ON TERMINAL

OCCUP. OF SPINORB., 1...TH EIGENVEC., E.G. (ML=)3 WITH(SPIN)UP, 3 DOWN, ...0.000 0.439 0.000 0.000 0.000 0.000 0.000 0.561 0.000 0.0000.000 0.000 0.561 0.000 0.000 0.000 0.000 0.000 0.439 0.0000.000 0.000 0.000 0.553 0.035 0.000 0.000 0.000 0.000 0.4120.412 0.000 0.000 0.000 0.000 0.035 0.553 0.000 0.000 0.0000.016 0.000 0.000 0.000 0.000 0.965 0.019 0.000 0.000 0.0000.000 0.000 0.000 0.019 0.965 0.000 0.000 0.000 0.000 0.0160.000 0.000 0.439 0.000 0.000 0.000 0.000 0.000 0.561 0.0000.000 0.561 0.000 0.000 0.000 0.000 0.000 0.439 0.000 0.0000.000 0.000 0.000 0.428 0.000 0.000 0.000 0.000 0.000 0.572


0.572 0.000 0.000 0.000 0.000 0.000 0.428 0.000 0.000 0.000

D ORBITALS ONLY:OCCUP. OF SPINORB., 1...TH EIGENVEC., DX2-Y2-(SPIN)UP,DX2-Y2-DOWN, DXZ-UP, DXZ-DOWN, DZ**2-UP, DZ**2-DOWN, DYZ-UP,DYZ-DOWN, DXY-UP, DXY-DOWN

D ORBITALS .AND. NGROUP.EQ.8.OR.NGROUP.EQ.22:ORBITALS ARE TRANSFORMED SO AS TO CORRESPOND TO [111]-AXISAS Z-AXIS: PUT IN YES(0)/NO (IN I1)!

0 =JANO ”TRANSF.” IN BLOCUT ! ! ! ! ! ! ! ! !

D ORBITALS .AND. NGROUP.EQ.8.OR.NGROUP.EQ.22:ORBITALS ARE TRANSFORMED SO AS TO CORRESPOND TO [111]-AXISAS Z-AXIS, BUT NOT TO REAL OR PURELY IMAGINARY BASIS ORBITALS:PUT IN YES/NO(0) (IN I1)!

0 =JANO ”TRANSF.” IN BLOCUT ! ! ! ! ! ! ! ! !

HINT: "SPIN UP" MEANS NOW "SPIN IN [111]-DIRECTION"!0.006 0.021 0.067 0.249 0.006 0.021 0.067 0.249 0.067 0.2490.021 0.006 0.249 0.067 0.021 0.006 0.249 0.067 0.249 0.0670.006 0.022 0.073 0.241 0.006 0.022 0.073 0.241 0.073 0.2410.022 0.006 0.241 0.073 0.022 0.006 0.241 0.073 0.241 0.0730.001 0.000 0.077 0.256 0.001 0.000 0.077 0.256 0.077 0.2560.000 0.001 0.256 0.077 0.000 0.001 0.256 0.077 0.256 0.0770.374 0.100 0.014 0.004 0.374 0.100 0.014 0.004 0.014 0.0040.100 0.374 0.004 0.014 0.100 0.374 0.004 0.014 0.004 0.0140.100 0.372 0.004 0.015 0.100 0.372 0.004 0.015 0.004 0.0150.372 0.100 0.015 0.004 0.372 0.100 0.015 0.004 0.015 0.004

OCCUP. OF SPINORB. OF EIGENVEC. 1...LIMIT= 10 WERE LISTED!

The matrix has been diagonalized!:

MATRIX OF ORTHONORMALIZED EIGENVECTORS;COLUMNS = EIGENVECTORS, OUTPUT IN THE FORM OF ROWS:

BECAUSE MOROUU=1, ONLY ONE COLUMN = ONE EIGENV.(IN THE FORM OF ONE ROW)!0.0000E+00-0.6628E+00 0.1150E-01 0.0000E+00 0.0000E+00 0.0000E+000.0000E+00 0.7487E+00 0.0000E+00 0.0000E+00


The eigenvalues are shifted so that the first eigenvalue becomes zero – hereall(!) eigenvalues:

< (ALL+1) EIGENVALUES, IN ASCENDING ORDER (IN ROWS):-0.826655924101E+04... BECOMES:0.0000E+00 0.0000E+00 0.2035E+02 0.2035E+02 0.7953E+03 0.7953E+030.2020E+05 0.2020E+05 0.2031E+05 0.2031E+05

TANABE has called REDUCTanabe, where the basis is changed:

MOROUT.NE.0: INPUT IN I1 MOROUT-HERE-IN-REDUCT; =0, 1, 2 OR 3!(+5, I.E. 5, 6, 7 OR 8, FOR OUTPUT OF G-FACTOR(S) IN ADD.!)

1 =MOROUU IN REDUCT ! ! ! ! ! ! ! ! !

No. OF MICRO STATES = 10INPUT IN I3 THE No. OF STATES TO BE CONSIDERED(0 => "PRE-OCC.", WHICH IS =No.OF STATES OR =ND252 = 252

Of course all 10 states are chosen here, but in principle the basis could betruncated (FMAT and GMAT are evaluated in the new basis):

(Hint: The first of the new basis vectors reads|2,-1/2>(-0.6628E+00)+|-1,-1/2>(0.7487E+00):

0 =NRVALI(No.OF STATES) ! ! ! ! ! ! ! ! !No. OF VALUES (INPUT AND CORR.) = 0 10

The result of the calculation up to now is (‘interface III’ – and this includesthe eigenvalues too!):

MATRIX FMAT(,,1): PERT.-OPER. FOR Z-DIR.(ONLY); COLUMNS(GIVEN AS ROWS):BECAUSE MOROUU=1, ONLY ONE COLUMN (IN THE FORM OF ONE ROW)!

-0.6814E+00 0.3444E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00-0.3048E-01 0.1488E+01 0.0000E+00 0.0000E+00

MATRIX GMAT(,,1): GMAT HERE FOR Z-DIR. (ONLY); COLUMNS(GIVEN AS ROWS):

-0.4999E+00 0.8609E-02 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00-0.7621E-02-0.1171E-03 0.0000E+00 0.0000E+00


3.6.4 Ti3+ – From ‘interface III’ to ‘interface IV’, wheretemperature enters

Following ‘interface III’, the most important ‘general’ interface, which includeseigenvalues and FMAT, subroutine SELECT has been reached!:

(IOPT99=)IOPT(IOPT13,NRSTOR,KOMBIN) = 7( 1 2 1)

‘Selected’ was IOPT99=7 (IOPT13 mostly = IOPT12, NRSTOR = 2 <=>(z,)x-direction, KOMBIN = 1 <=> first loop (if or if not ‘Comb.Model’)), andthis means, that subroutine SUGANR is called for a single ion, and its routinefor ‘more’ OUTPUT is subroutine SUGANO, which will be called repeatedly, if‘1’ is put in now:

INPUT MOROUT-HERE-IN-SUGANO IN I1: 0 => (ALWAYS) RETURN

1 =MOROUU IN SUGANO ! ! ! ! ! ! ! ! !

HERE IS SUBROUTINE SUGANO (FOR OUTPUT ONLY), CALLED BY (1...4 <=>SUGANR, SUGANC, SUGANH, SUGANK): 1, NUMBER OF SORT = 2, 1ST CALL

INPUT YES(0)/NO ONLY MINIMUM OUTPUT, YES(0)/NO "OUTPUT ONLY ONCE"AND YES/NO(0) "FULL MATRICES"

1 1 0 =JANO1,JANO2,JANO3 ! ! ! ! ! ! ! ! !

ALL 10 USED EIGENVALUES OF THE INTERFACE, IN ASCENDING ORDER(IN ROWS):0.0000E+00 0.0000E+00 0.2035E+02 0.2035E+02 0.7953E+03 0.7953E+030.2020E+05 0.2020E+05 0.2031E+05 0.2031E+05

FIELD IN CM**(-1), DIRECTIONS (Z-, X-, Y-) OF THE FIELD,AND MINIMAL AND MAXIMAL DIFFERENCE OF FORMER EIGENVALUES:

0.2334E+00 0.5774E+00 0.8165E+00 0.0000E+00 0.2035E+02 0.2031E+05

SUGANR evaluates the new matrix to be diagonalized using the magneticfield times FMAT for z- and FMAT for x-direction to get small supplements tothe diagonal matrix of eigenvalues!

OUTPUT from SUGANO:

2ND CALL: MATRIX TO BE DIAGONALIZED; COLUMNS(IN THE FORM OF ROWS):


BECAUSE "FULL"=0, ONLY ONE COLUMN (IN THE FORM OF ONE ROW)!

-0.9184E-01 0.4641E-01-0.1221E-02 0.1530E+00 0.2071E+00-0.3716E-02-0.4108E-02 0.2006E+00 0.1891E+00 0.6002E-03

SUGANR calls DSPEVW for diagonalizing the matrix and puts out up to66 eigenvalues:

< (ALL+1) Z,X- EIGENVALUES, IN ASCENDING ORDER (IN ROWS):-0.931476144001E-01... BECOMES: (OUTPUT ONLY ONCE FOR NRROUT = 1)0.0000E+00 0.1839E+00 0.2027E+02 0.2062E+02 0.7951E+03 0.7957E+030.2021E+05 0.2021E+05 0.2031E+05 0.2031E+05

SUGANR evaluates the (neg.) magnetic moments for each level and callsSUGANO (3rd call of SUGANO):

3RD CALL: MATRIX OF ORTHONORMALIZED EIGENVECTORS;COLUMNS = EIGENVECTORS, OUTPUT IN THE FORM OF ROWS:

The elements of the following matrix are not very different from either ±1or 0!:

BECAUSE "FULL"=0, ONLY ONE COLUMN = ONE EIGENV.(IN THE FORM OF ONE ROW)!0.9997E+00-0.2523E-01 0.2307E-03-0.7422E-02-0.2600E-03 0.3204E-050.2218E-07-0.9920E-05-0.9298E-05 0.1334E-06

ALL 10 EIGENVALUES, IN ASCENDING ORDER (IN ROWS):

0.0000E+00 0.1839E+00 0.2027E+02 0.2062E+02 0.7951E+03 0.7957E+030.2021E+05 0.2021E+05 0.2031E+05 0.2031E+05

SUGANR calculates the (components of the neg.) magnetic moments withinthe new basis with help of a linear combination of FMAT for z- and FMAT forx-direction, which is multiplied by one eigenvector from the right and from theleft to get one magnetic moment!

(NEG.) MAGNETIC MOMENTS IN BOHR MAGNETONS:

-0.4041E+00 0.3834E+00-0.7272E+00 0.7464E+00-0.1182E+01 0.1184E+01-0.1870E+00 0.1811E+00-0.9874E+00 0.9934E+00

Subroutine SUGANR calls subroutine CHIMAG, and ‘interface IV’ has beenreached! (New) eigenvalues and (neg.) magnetic moments belonging to theseeigenvalues define this interface!


3.6.5 Ti3+ – From ‘interface IV’ to first susceptibility

Subroutine CHIMAG is called now for the first time, corresponding to ‘interfaceIV’ with its eigenvalues and its (neg.) magnetic moments belonging to theseeigenvalues:

MORE OUTPUT IN SUBROUTINE CHIMAG (FOR CHI/MAGN./MAGN.MOMENT)!FIRST TEMP.: INPUT IN I1: MORE OUTPUT ONLY FOR THIS TEMP.YES(0)/NO

OUTPUT only for one (the first) temperature! This OUTPUT here is onlyof interest, if it is changed later on, e.g. mean values are calculated in subroutineSELECT. It is difficult to read, but relevant and used further on is only CHIIN SI!

0 =JANO OUTPUT IN CHIMAG ! ! ! ! ! ! ! ! !

CHI IN SI AND IN CGS, CHI*FIELD, TEMP.-THETA, FIELD, MAGN.MOMENT,PSEUDO EFF.MAGN.MOM., "VANVLECK", No.OF MICRO ST. CONS.,No. (1...4) OF ROUTINE, No.OF NEW TEMP.=0.1871E-07 0.1489E-02 0.9357E-08 0.2703E+03 0.5000E+000.1333E-02 0.1794E+01 0 10 1 1

Now return to subroutine SUGANR and to subroutine SELECT!:

MORE OUTPUT IN SUBROUTINE SELECT FOR CHI (AND FOR ITS COMPONENTS,IF MORE THEN ONE COMPONENT): FOR FIRST...NNTH TEMP. AND/OR FORFIRST...NRTH VALUES OF CHI! INPUT IN 2I2 NN AND NR (0000 =>0101)!

Not too much OUTPUT is selected here – 4 of 5 lines of output will appearlater on in Subsection 3.6.6!:

5 5 =MOROUU,MOROUV IN SELECT ! ! ! ! ! ! ! ! !

The CHI-value is the same now as that above, but often a mean value and/orextrapolated value in the case of ‘Combined Model’ have been calculated here!

No. OF TEMP., CHI-START CALCULATED = 1 0.18714489E-07

CHI becomes CHISTA, stored as a function of temperature, and serves asCHI-START resp. as CHI(1,1,1) in subroutine SELECT. Out of CHI-START,and out of CHI-meas. = CHI(2,4,1), 54 more CHI( , ,)-values are always eval-uated in SELECT. Two, a calculated and a measured, of all these values areused for fitting. Just the same evaluation is done in subroutine RESTBL oncemore (see Subsection 3.6.6), if this subroutine is called! But in the case of anOUTPUT-‘table’ really not CHISTA is used but CHIFIX, and CHIFIX doesnot contain trial CHI-values as CHISTA may do!


Because of ‘MOROUT’ subroutine SELECT calls RESTBL for more OUT-PUT belonging to this first temperature, but this OUTPUT will be calculatedlater on once more and given in form of a table for all temperatures, and there-fore ‘NO MORE OUTPUT HERE’ will be chosen!:

ATTENTION PLEASE HERE IN SUBROUTINE RESTBL:MOROUT.NE.0! INPUT IN I1 YES(0)/NO: NO MORE OUTPUT HERE

0 =NOMORE OUTPUT HERE ! ! ! ! ! ! ! ! !

The calculation for this temperature is ready now!

3.6.6 Ti3+ – further susceptibilities

Subroutine SELECT returns to subroutine REDUCTanabe, this to TANABE,subroutine TANABE to SUSC and SUSC to LEVSQX! There within the sameDO LOOP a new calculation is started for the second temperature, and allrelevant routines are called again! But at once subroutine SELECT has beenreached once more:

(IOPT99=)IOPT(IOPT13,NRSTOR,KOMBIN) = 7( 1 2 1)

But the calculation continues really in CHIMAG as before, corresponding to‘interface IV’, and CHIMAG returns to SUGANR and this returns to SELECT!:

No. OF TEMP., CHI-START CALCULATED = 2 0.18967977E-07

The DO LOOP continues to run! Now for the 3rd temperature:

(IOPT99=)IOPT(IOPT13,NRSTOR,KOMBIN) = 7( 1 2 1)No. OF TEMP., CHI-START CALCULATED = 3 0.19231170E-07

. . . and so on (5 temperatures have been chosen in Subsection 3.6.5!):

(IOPT99=)IOPT(IOPT13,NRSTOR,KOMBIN) = 7( 1 2 1)No. OF TEMP., CHI-START CALCULATED = 4 0.19504677E-07(IOPT99=)IOPT(IOPT13,NRSTOR,KOMBIN) = 7( 1 2 1)No. OF TEMP., CHI-START CALCULATED = 5 0.19788579E-07(IOPT99=)IOPT(IOPT13,NRSTOR,KOMBIN) = 7( 1 2 1)(IOPT99=)IOPT(IOPT13,NRSTOR,KOMBIN) = 7( 1 2 1)(IOPT99=)IOPT(IOPT13,NRSTOR,KOMBIN) = 7( 1 2 1)(IOPT99=)IOPT(IOPT13,NRSTOR,KOMBIN) = 7( 1 2 1)(IOPT99=)IOPT(IOPT13,NRSTOR,KOMBIN) = 7( 1 2 1)


All temperatures are ready! A SQX has been evaluated in subroutine LEV-SQX. It results from a fit to ‘all values to be fitted’ of file ‘cgsTi.dat’, includingthe last line, an energy!

Subroutine LEVSQX returns to subroutine SUSFITnow!

The SQX-value just evaluated is called ‘SQX-FIX’, because there are noSQX-values before which are lower (It is the first value indeed!). (SXR is aspecial sort of ‘delta’ to be used in connection with convergence test(s)!)

AFTER 0TH ITER.: OMEGA,SQX-FIX,SXR=0.00E+00 0.11297990E-01 0.8680E-02

LIST(I), BWYB(LIST(I))= 1 -0.1339E+05CFMS, "CFOS/ CFOSJ" = 0.203E+02 0.203E+05 0.203E+05

3.6.7 Ti3+ – Fitting procedure

Normally a fitting procedure follows a ‘zeroth’ iteration, and although fittinghad been done in Subsection 5.1.2 before starting this ‘zeroth’ iteration here,one ‘first’ iteration will be added to show how the fitting procedure of CONDONworks. Only one parameter will be fitted however, because this is a demonstra-tion only!

First of all OMEGA, an important parameter ofLEVENBERG/MARQUARDT, is asked for in subroutine LEVMAR:

OMEGA = 0.10000000E+00, RETURN FOR THIS OMEGA OR INPUT OTHERIN E16.8 (IS CHANGED AUTOM., OTHERWISE INPUT A<=>"ERROR"!)(A NEGATIVE OMEGA BECOMES POSITIVE WITH THIS VALUE THE LOWERLIMIT!)(A SPECIAL TEST FOR AN OMEGA>0 AND <1 IS ENABLED, IF999 IS ADDED TO OMEGA)

OMEGA (A PARAMETER OF LEVENBERG/MARQUARDT)=0.10000000E+00 ! ! ! ! ! ! ! ! !

The following OUTPUT consists of two parts, and both represent a calcula-tion not very different from that of the ‘0th iteration’ before. For the first partB20, the (only) parameter to be fitted, was changed in subroutine LEVDER,which was called by subroutine LEVMAR, by a small amount in order to enablenumerical differentiation. This part follows now, for the second part see below!

BWWOLD(1:11) = -0.1340E+050.0000E+00 -0.1285E+05 0.0000E+00 0.3559E+05 0.0000E+000.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

MATRIX FMIKR2 (LIG.-FLD. MATRIX); COLUMNS (IN THE FORM OF ROWS):BECAUSE MOROUU=1, ONLY ONE COLUMN (IN THE FORM OF ONE ROW)!


0.3216E+04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+000.1003E+05 0.0000E+00 0.0000E+00 0.0000E+00

MATRIX TO BE DIAGONALIZED:COLUMNS IN THE FORM OF ROWS, STARTING WITH DIAG. ELEM.

BECAUSE MOROUU=1, ONLY ONE COLUMN (IN THE FORM OF ONE ROW)!0.3370E+04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+000.1003E+05 0.0000E+00 0.0000E+00 0.0000E+00

No (more) OUTPUT in subroutine BLOCUT now or further on:

MOROUT.NE.0: INPUT IN I1 MOROUT-HERE-IN-BLOCUT; =0 OR =1!... OR =8 FOR "NO MORE OUTPUT FURTHERMORE HERE IN BLOCUT"!(=0 FOR OUTPUT AT THE END OF BLOCUT ONLY: OCCUP.OF SPINORB.)(=8 CANCELS OUTPUT AT THE END OF BLOCUT AND IN FURTHER ITER.!)

8 =MOROUZ IN BLOCUT/0 ! ! ! ! ! ! ! ! !

(OUTPUT is continued:)

MATRIX OF ORTHONORMALIZED EIGENVECTORS;COLUMNS = EIGENVECTORS, OUTPUT IN THE FORM OF ROWS:

BECAUSE MOROUU=1, ONLY ONE COLUMN = ONE EIGENV.(IN THE FORM OF ONE ROW)!0.0000E+00 0.6628E+00-0.1150E-01 0.0000E+00 0.0000E+00 0.0000E+000.0000E+00-0.7487E+00 0.4684E-17 0.0000E+00

< (ALL+1) EIGENVALUES, IN ASCENDING ORDER (IN ROWS):-0.826649838411E+04... BECOMES:0.0000E+00 0.1819E-11 0.2033E+02 0.2033E+02 0.7949E+03 0.7949E+030.2021E+05 0.2021E+05 0.2031E+05 0.2031E+05MATRIX FMAT(,,1): PERT.-OPER. FOR Z-DIR.(ONLY); COLUMNS(GIVEN AS ROWS):BECAUSE MOROUU=1, ONLY ONE COLUMN (IN THE FORM OF ONE ROW)!

-0.6815E+00 0.3444E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+000.1479E+01-0.1735E+00 0.0000E+00 0.0000E+00MATRIX GMAT(,,1): GMAT HERE FOR Z-DIR. (ONLY); COLUMNS(GIVEN AS ROWS):( -- removed, because not needed for a single ion -- )ALL 10 USED EIGENVALUES OF THE INTERFACE, IN ASCENDING ORDER(IN ROWS):0.0000E+00 0.1819E-11 0.2033E+02 0.2033E+02 0.7949E+03 0.7949E+030.2021E+05 0.2021E+05 0.2031E+05 0.2031E+05


FIELD IN CM**(-1), DIRECTIONS (Z-, X-, Y-) OF THE FIELD,AND MINIMAL AND MAXIMAL DIFFERENCE OF FORMER EIGENVALUES:0.2334E+00 0.5774E+00 0.8165E+00 0.0000E+00 0.2033E+02 0.2031E+052ND CALL: MATRIX TO BE DIAGONALIZED; COLUMNS(IN THE FORM OF ROWS):BECAUSE "FULL"=0, ONLY ONE COLUMN (IN THE FORM OF ONE ROW)!

-0.9184E-01 0.4641E-02-0.1530E+00 0.1222E-02-0.2071E+00-0.3715E-020.1993E+00-0.2338E-01-0.1891E+00-0.6003E-03

3RD CALL: MATRIX OF ORTHONORMALIZED EIGENVECTORS;COLUMNS = EIGENVECTORS, OUTPUT IN THE FORM OF ROWS:

BECAUSE "FULL"=0, ONLY ONE COLUMN = ONE EIGENV.(IN THE FORM OF ONE ROW)!0.9997E+00-0.2523E-01 0.7429E-02-0.2309E-03 0.2601E-03-0.3207E-05

-0.9829E-05 0.1335E-05 0.9298E-05-0.1333E-06ALL 10 EIGENVALUES, IN ASCENDING ORDER (IN ROWS):0.0000E+00 0.1839E+00 0.2025E+02 0.2059E+02 0.7947E+03 0.7952E+030.2021E+05 0.2021E+05 0.2031E+05 0.2031E+05(NEG.) MAGNETIC MOMENTS IN BOHR MAGNETONS:

-0.4041E+00 0.3834E+00-0.7272E+00 0.7464E+00-0.1182E+01 0.1184E+01-0.1869E+00 0.1811E+00-0.9874E+00 0.9934E+00No. OF TEMP., CHI-START CALCULATED = 1 0.18715027E-07

Now a new B20 will be proposed in subroutine LEVMAR and, although notfor this example here, one would like to restrict this B20 to a given limit, whichmay be incorporated in file BWYBIN.DAT resp. file BWYBTi.DAT:

POSSIBLE INPUT OF LIMITS FOR VALUES TO BE FITTED(WYB.-VAL. HERE!):IS THE NAME OF THE NEXT FILE BWYBIN.DAT IN FORMAT(I2,3D16.8)?"9" OR GIVE ITS NAME! (RETURN => NO INPUT HERE, 8 => STOP,"79"(RESP. 7NAME)<=>IGNORE TEST(<=>START-VAL..NE.WYB.-VAL.)!)

‘blanc’ (BWYBIN)!!LEVMAR!!

Subroutine LEVMAR alters the linearized fitting matrix now and needssubroutine GAUSS to solve a system of linear equations, before it can make theproposal for the new parameter:

MOROUT.NE.0 IN GAUSS: INPUT 1 IN I1 FOR MORE OUTPUT HERE!

0 =MOROUU IN GAUSS ! ! ! ! ! ! ! ! !

The calculation with the newly proposed B20 as parameter leads to thesecond part of OUTPUT, which has exactly the same structure as the first


part and, because the new B20 is nearly identical to that of the ‘0th iteration’,covers virtually the same values as the OUTPUT of that iteration. Thereforethis second part of OUTPUT is discarded here, and the result of fitting followsnow!

AFTER 1ST ITER.: OMEGA,SQX-FIX,SXR=0.30E-01 0.11297989E-01 -0.7362E-09

"FITTED"=+-1; -2/-1<=>8/9 IN INPUT FILE:1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0LIST(I), BWYB(LIST(I))=1 -0.13394924E+05

CFMS, "CFOS/ CFOSJ" = 0.203E+02 0.203E+05 0.203E+05

(LAUFNR.EQ.ITMAX): LAUFNR, ITMAX= 1 1

INPUT 0/RETURN FOR TABLES OF OUTPUT, 1 TO CONTINUE, >1: STOP!

0 =JASTOQ(2=STOP) ! ! ! ! ! ! ! ! !

HINT: OUTPUT CORRESPONDS TO "FITTING TAKES PLACE"INPUT 1, IF FORM OF OUTPUT => "<=> NO FITTING TAKES PLACE"BUT 6 (3) FOR "FITTING (NO FITTING), BUT 6 COLUMNS" (0 ELSE)

0 =MFITIN(FORM OF OUTPUT) ! ! ! ! ! ! ! ! !

HINT: OUTPUT CORRESPONDS TO "FITTING TAKES PLACE"


CHI((1)...(2),1:4,1:7): (1) CALCULATED VALUES, (2) VALUESBASED ON MEASURED ONES

CHI(1,(1)...(4),1:7): (1) CALCULATED VALUES, E.G. FROMFORMULA OF VANVLECK,(2) ... INCLUDING MOLECULAR FIELD PARAMETER,(3) INCLUDING IN ADDITION CHI-0,(4) INCLUDING FURTHERMORE A CORRECTION OF WEIGHT OF SAMPLE.

CHI(2,(1)...(4),1:7):MEASURED VALUES WITH (1) THREE, (2) TWO, (3) ONE, (4) WITHOUT(I.E. AS READ IN) CORRECTIONS (CORRECTIONS IN PRINCIPLE ASABOVE, BUT REVERSED)


CHI(1:2,1:4.(1)...(7)): (1) CHI IN SI UNIT M**3/MOL,(2) RECIPROCAL CHI,(5)! MAGNETISATION IN SI UNIT TESLA*M**3/MOL,(3) MAGNETIC MOMENT OF 1 ATOM IN BOHR MAGNETONS

(E.G. SATURATION MOMENT),(4) EFFECTIVE MAGNETIC MOMENT IN BOHR MAGNETONS,(6) CHI IN CGS UNIT CM**3/MOL,(7) PSEUDO EFFECTIVE MAGNETIC MOMENT

(WITH SQRT(CHI*(T-THETA))!).

Some of the options chosen are indicated now:

CHI ITSELF MAY BE FITTED (1) OR ONE/CHI (2) OR MAGNETICMOMENT OF 1 ATOM (3) OR EFFECTIVE MAGNETIC MOMENT (4)(BOTH IN BOHR MAGNETONS): 1

A (SUM OF) LEAST SQUARE(S) FIT WAS DONE (SQUARED WASCHI-CALC./CHI-MEAS.-ONE), BUT SQRT((THIS SUM)/(No.OF VAL.FITTED)) IS GIVEN HERE AS SQX (*100<=>"%")!:SQX, CRYSTAL FIELD MINIMAL SPLITTING (CFMS) AND THE TWODEGENERACIES BELONGING TO THESE TWO LEVELS, AND THE EIGEN-VALUE OF THE LEVEL "No. OF VAL. (NRVAL)", WHICH MAY BE THECRYSTAL FIELD OVER ALL SPLITTING (CFOS), IF NRVAL HAS BEENCHOSEN IN THE RIGHT WAY! AND: CFOSJ(NRVALJ) <=> NRVALJ=2*J+1,IF F EL. AND <=> NRVALJ=SPIN-MULTIPL.*ORB.-MULTIPL., ELSE.

SQX, CFMS, DEGEN.1, DEGEN.2, CFOS, No. OF VAL., CFOSJ, NRVALJ =0.1130E-01 0.2035E+02 2 2 0.2031E+05 10 0.2031E+05 10

TABLE of OUTPUT follows now!:

"00" MEANS: FITTED, "10": NOT (FROM INPUT),"01": NOT (FROM CALC.)

No. 00 TEMP/K SIGMA B/TESLA MEASURED CHI CALCUL. CHI1 00 270.27 1.000000 0.500 0.18397166E-07 0.18714505E-072 00 265.28 1.000000 0.500 0.18673626E-07 0.18967992E-073 00 260.28 1.000000 0.500 0.18962653E-07 0.19231184E-07

...86 10 2.50 0.000000 0.500 0.73574842E-06 0.43755328E-0687 10 2.00 0.000000 0.500 0.89596964E-06 0.51039543E-0688 00 -10.00 1.000000 0.500 0.20300000E+05 0.20312194E+05

THE TABLE PUT OUT HERE ALSO PUT OUT TO FILE fort.1!


1 ELECTRON(S), 2 =l VALUE, 10 BASIS STATES,10 VALUES USED LATER ON,

0.15400000E+03 =ZETA (1 EL.), -0.13394924E+05 =B20(WYBOURNE),-0.12847059E+05 =B40(WYBOURNE), 0.35591271E+05 =B43(WYBOURNE),

INPUT 1 TO RETURN (TO CALLING ROUTINE), 0 OR >1 TO STOP!

2 =JASTOR(2=STOP) ! ! ! ! ! ! ! ! !

4 Description of program CONDON

4.1 General outline

Program CONDON is written on different files! File CONDON.f consists ofthe .MAIN. PROGRAM CONDON, 32 SUBROUTINEs, 3 FUNCTIONs and 4BLOCKDATA subprograms, but 3 subroutines are text only, concerning ‘WAR-RANTY’, ‘GNU GENERAL PUBLIC LICENSE’, and an ‘OUTLINE’ of CON-DON. Not all these subroutines are explicitely described, i.e. in an own sectionor subsection!

One of four files, with subroutines DSPEVW and ZHPEVW each, serves tolink library routines for diagonalizing real symmetrical and complex hermitianmatrices. These library routines are e.g. on file dspzhp.f, downloaded fromWWW.

Subroutine SHORTL on file shortL.f, shortO.f, shortP.f or shortQ.f definesall important DIMENSIONs and some EQUIVALENCEs, and SHORTL on fileshortL.f differs from SHORTL on file shortO.f etc. in this respect.

Subroutine STR00 is separated on file str00.f, because it is a very specialroutine and may be exchanged!

Standard FORTRAN77 is used and always DOUBLE PRECISION, alsoin case of COMPLEX CONSTANTs and VARIABLEs, although this is notstandard, but is offered by most software.

22 Nassi-Shneiderman flowcharts, consecutively ordered from CHART 1 toCHART 22, decribe program CONDON, and they are given in connectionwith the subroutines concerned, CHART 1 for .MAIN. PROGRAM CONDON.Firstly however a more qualitative flowchart is given here to show the generalstructure of CONDON (part of it is also given in Section 3.5).

There are four interfaces, I to IV, in the program. Calculation starts atinterface I with a fitting procedure, but not before interface II for CHI. Thefirst part of the calculation of CHI, from interface II to interface III, calculateseigenvalues and eigenvectors of the energy matrix (apart from the energy of themagnetic field) as a function of certain parameters, using a full basis of micro-states, and it calculates six matrices out of those eigenvectors, which belong toan in general reduced number of eigenvalues, namely those lying lowest. Inter-face III means this in general reduced number of eigenvalues, the perturbationmatrices FMAT( , ,1), FMAT( , ,4) and FMAT( , ,5) and the matrices GMAT(

46

4.1. General outline 47

(‘MAIN’) Program CONDON INPUT from terminal,CHIINP.DAT, BWYBIN.DAT

?SUBROUTINEs SHORTL on shortL.f or shortO.f

or shortP.f or shortQ.f(‘DIMENSIONS/ EQUIVALENCES’)

?Interface I: Start of fitting procedure

?SUSFIT (‘Fitting’) ↔ RESultTaBLes (‘Results’)

? 6

LEVSQX (‘SQX’)

? 6

?6LEVMAR? 6

?6LEVMAR

LEVDER?6

SUSCeptibility (‘diff. B values enabled’) ↔ STR00 (‘create B values’)

?Interface II: Start of calculation of CHI

6(either:)TANABE (↔ BLOCkoUT) ↔ DSPEVW (‘Diagonalization’)

? 6

REDUCTanabe

?

6(or:)ECKART ↔ DSPEVW (‘Diagonalization’)

? 6

REDUCEckart

?Interface III: Transformation to new basis executed

6

SELECT (‘Magn. field enters in SUGANR, . . . C, . . . H, . . . K’)

? 6 ? 6 ? 6 ? 6SUGANReal SUGANCompl. SUGANHeisenbg. SUGANKompl.H.

? 6 ? 6(↔ DSPEVW) (↔ ZHPEVW)

SUGANReal SUGANCompl.↔ DSPEVW ↔ ZHPEVW? ?

? ?Interface IV: Temperature enters calculation

6

CHIMAG (‘CHI and MAGnetisation’)

Figure 4.1: General flowchart

48 Chapter 4. Description of program CONDON

, ,1...3) needed for Heisenberg coupling.

Part two, following interface III, the most important ‘general’ interface,starts with subroutine SELECT and ends with interface IV at the beginningof subroutine CHIMAG. The ‘Zeeman operator’ works here for up to three di-rections of magnetic field and up to three ‘parts’ of CHI, if Heisenberg couplingand ‘Combined Model’ are taken into account.

Perturbation matrices and energy matrices including the energy of Heisen-berg coupling, if concerned, are set up. If the Van Vleck formula is not used,and not to use it should be the standard method, the energy matrices includethe energy of the magnetic field, too. The energy matrices, if they occur, arediagonalized, and if y-direction of magnetic field is involved, they are complexhermitian matrices. If no energy matrices occur, i.e. ‘Van Vleck’ and no Heisen-berg coupling, an eigenvector matrix identical to the unit matrix enters thefollowing calculation.

Perturbation matrices and eigenvector matrices lead to the evaluation of(neg.) magnetic moments or, in the case of ‘Van Vleck’, of Zeeman coefficients offirst and second order. These and the eigenvalues, the newly evaluated eigenval-ues or, in the case of ‘Van Vleck’ and no Heisenberg coupling the old eigenvalues,are considered as ‘interface IV’.

Part three, following interface IV, consists of subroutine CHIMAG. Tem-perature is taken into account here for the first time. CHI is evaluated, usingeither (neg.) magnetic moments or the Van Vleck formula. Magnetic momentsand molar magnetisation are also determined in the case of ‘Van Vleck’, if mag-netic field is not zero!

Returned to part two, to subroutine SELECT, mean values, extrapolatedvalues etc. are calculated if concerned. Also some ‘corrections’ to the evaluatedCHI are always applied, even if these ‘corrections’ are zero.

Finally, these three parts of program CONDON are embedded in a fittingroutine, which however may be only formally used, i.e. running a ‘zeroth iter-ation’. OUTPUT is to terminal and to file fort.11 in nearly identical manner,part thereof to file fort.1 for plotting, and all INPUT from terminal goes to filefort.7. In the case of ‘STOP’ a new file fort.8 with a special header is written,to be used as a ‘new file BWYBIN.DAT’! This file fort.8 is also always written(following a REWIND(8)), if an iteration is completed, in order to serve as a‘new file BWYBIN.DAT’, if the program is interrupted.

4.2 .MAIN. PROGRAM CONDON

Nearly all INPUT enters PROGRAM CONDON, the .MAIN. program, and itcomes from the terminal or file fort.7 alternatively, from file CHIINP.DAT (for‘measured values’) and from file BWYBIN.DAT (for ‘parameters’), but in specialcases CHIINP.DAT and/or BWYBIN.DAT may be missing! See Subsection 3.6

4.2. .MAIN. PROGRAM CONDON 49

and 5.1.2 for file CHIINP.DAT!

Most INPUT is ‘per default’, and many values are stored in some ‘BLOCK-DATA’ subprograms.

First of all quantum number l of electron(s) is asked for, and it may be 0,1, 2 or 3, and at the same time three digits as a combination of the digits 0(‘standard’), 1 (‘more’) and 9 (‘less’) are asked for. These mean ‘MOROUT’ for(more) OUTPUT of intermediate values (or ‘LESS’), ‘MORTEX’ for (more orless) OUTPUT of text and ‘JATEST’ for the possibility of tests (less possibili-ties: ‘LTS’).

Now the special routine for 4f block atoms or ions with 4fN electronic con-figuration with specific ground term 2S+1L may be chosen, namely subroutineECKART, which uses irreducible tensor operators and the Wigner-Eckart the-orem. To simulate the case of one single specific J value a spin-orbit couplingconstant of e.g. 106 has to be chosen later on!

Subroutine ECKART is also available for tests with d electrons, if f electronsare selected here! The matrix to be diagonalized has a small dimension of up to66 only (or 85 for a quintet-L state of six f electrons as a test), and the smallestND2002=1110 will work in any case! But the ‘standard’ should be not to choosethis special routine.

There are 23 different possibilities of INPUT of (point) groups (indeed only22, because one is a test). They are consecutively ordered and given to theterminal, and the relevant number is required as INPUT, the number 21 or 0(‘RETURN’) for the cubic group with fourfold rotation axis as the main axis.None of the groups has a symmetry ‘lower’ than corresponding to a remainingtwofold rotation axis, i.e. ortho-rhombic!

All possible ‘B-values’ are given now to the terminal, but 0 . . . 11 are ‘marked’with 1 and are excluded (0 for a ‘point group ???’ as a test, 11 for a ‘free ion’):

FROM THE FOLLOWING LIST OF POSSIBLE B-VALUES THOSE MARKED WITH 1ARE EXCLUDED:B20, B22, B40, B42, B43, B44, B60, B62, B63, B64, B66

The Van Vleck formula will be used, if ‘JAVANV’=1 is put in now. Optionsfor fitting being treated separately, two options are required now:

IOPT1 = 0 . . . 3 for direction(s) of field, mostly IOPT1=0 for one (mean)direction, or IOPT1=1 for mean value of CHI calculated for 1, 2 or 3 direction(s).IOPT1=0 and IOPT1=1 are not always really different, e.g. ‘cubic’ correspondsalways to the z-direction only! In the case of a crystal field like tetragonal, forIOPT1=1 a mean value of z- and two times x-direction is calculated, and forIOPT1=0 one direction with direction cosines as 1:2.

IOPT1=2 or IOPT1=3 mean ‘direction at will’ (one! direction), IOPT1=2including y-direction and IOPT1=3 not including it. SUGANR and DSPEVW


are sufficient in the case of IOPT1=3, and only real matrices have to be diago-nalized.

To calculate a mean value of three directions in the case of e.g. ‘cubic’, takeIOPT1=1 and a point group with three directions, and simulate ‘cubic’ with‘cubic B-values’ !

The direction of the magnetic field is asked for in the case of IOPT1 greaterthan 1, and the distance of the two ions in a (symmetrical) dimer in the case of‘magnetic dipol-dipol interaction’ is wanted, i.e. the latter is not computed fora distance of zero.

IOPT2 = 0 . . . 2 for (single) ion, (symmetrical) dimer and ‘Combined Model’for the latter (f electrons and symmetrical dimers only), respectively.

CHI values to be fitted are read in from file CHIINP.DAT as a standard,or from a file with any other name. The special file names ‘h’ or ‘H’ mean‘no file at all’, and CHI values of hydrogen atom are taken as computed fortemperatures stored in BLOCKDATA WWWWWW. A file name beginningwith ‘CGS’ or ‘cgs’, e.g. CGSCHI.DAT, means, that CHI values read in are inCGS units instead of SI units.

Parameters are read in from file BWYBIN.DAT as a standard, or from a filewith any other name, but the FORMAT must be (I2,D16.8). The special filenames ‘h’ or ‘H’ mean ‘no file at all’, and all parameters become zero except a‘correction factor F’, which becomes ‘one’ later on automatically!

An integer ‘yes/no(0)’-parameter belongs to each parameter and determines,if the parameter will be fitted, and it is named INBWWW( ). See Section 4.7for some special cases, where 9 for ‘yes’ and 8 instead of 0 are used.

The number of possible (point) groups (23) times the number of possible lvalues is 92, but there are only 19 different possibilities of INPUT of (max. 11)ligand field parameters, ‘B-values’. B-values corresponding to odd parity of thecrystal field potential (‘imaginary terms’) or 5-fold axis (D5h has no B5-value)are not possible! All ligand field operators are taken from the Handbook on thePhysics and Chemistry of Rare Earths, Vol. 23 [17]. It is not possible to put in‘wrong’ B-values, i.e. such not belonging to the (point) group chosen, becausethey will be ‘masked’, i.e. set to zero! Also an INPUT of e.g. a B44-value for acubic group is not possible, because it will always be generated automatically.

Ligand field parameters as defined by Wybourne [13] as ‘B-values’ should bepreferred as INPUT, and those are always used internally. But other parametersare possible, too: See Section 4.7 for details! Derivatives needed and computedfor fitting are however dependent on which ‘B-values’ are taken.

Now the number of electrons will be asked for and those parameter val-ues, including orbital reduction factors, which cannot be fitted. Most INPUTis ‘per default’, and many values are stored in ‘BLOCKDATA’ subprograms,especially in BLOCKDATA GRIFFI, BLOCKDATA WIGNER and BLOCK-DATA STEVEN. ‘GRIFFI’ contains data from the book by Griffith [5] for

4.3. Subroutine SHORTL 51

d electrons, ‘WIGNER’ F -integrals (Slater-Condon parameters) and spin-orbitcoupling constants for 4f electrons from the book by Dieke [27], and ‘STEVEN’is needed, if ligand field parameters for f electrons as defined by Stevens [21]are read in. Furtheron values from the book by Konig and Kremer [7] for delectrons, with Racah parameter C = 4∗ Racah parameter B, are stored inprogram CONDON.

Most parameters given for d electrons, although derived from experiment,do not take into consideration the ligands, although the influence of individualligands may be important. With 5f electrons this influence must not be ignored!The data from the book by Dieke are sufficiently good for magnetochemistry, butnot for ligand fields in general! Data for 5f electrons in program CONDON, butonly for U5+, U4+ and U3+, are taken from the Handbook on the Physics andChemistry of the Actinides, Vol. 1 [28]. These values are theoretical relativisticvalues. These data should be sufficiently good in many cases, especially ifexperimental values are not available. See Subsection 5.1.7 for details!

CHART 1

PROGRAM CONDON (.MAIN.)

INPUT from TERMINAL or FILE fort.7 and mostlyfrom FILE(S) CHIINP.DAT and/or BWYBIN.DAT

SUBROUTINE SHORTL for DIMENSIONS/ EQUIVALENCES

on EXTRA FILES shortL.f or shortO.f or shortP.f or shortQ.f

(one of these has to be LINKed) (see CHART 2)

END of PROGRAM CONDON (.MAIN.)

Figure 4.2: Flowchart 1 (PROGRAM CONDON)

4.3 Subroutine SHORTL

A Nassi-Shneiderman flowchart for subroutine SHORTL is given (consecutivelyordered as CHART 2). It states, that subroutine SUSFIT is called, and thatsubroutine SHORTL is called by PROGRAM CONDON without RETURNto CONDON. Further information is given by flowCHART 3 and subroutineSUSFIT. Such a structure, i.e. with little information here and more informationlater on, is a main feature of Nassi-Shneiderman flowcharts.

SUBROUTINE SHORTL is for DIMENSIONS and EQUIVALENCES only.All dimensions of importance are defined here for the whole program CONDON,


CHART 2

SUBROUTINE SHORTL for DIMENSIONS/ EQUIVALENCES

SUBROUTINE SUSFIT to FIT SUSceptibilities (and others)by method of LEAST SQUARES (see CHART 3)

END of SUBROUTINE SHORTL, no RETURN (to CONDON)

Figure 4.3: Flowchart 2 (SUBROUTINE SHORTL)

i.e. for all subroutines and functions.

A full micro-state basis for f electrons has the following dimensions: 14, 91,364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, and these dimensionsare the most critical. Corresponding to the value of 1001, dimension ND2002is set to 1110 as a minimum. Two-dimensional matrices of this dimension areFMIKR1, FMIKR2, DBLMAT, EIVECV, BLOMAT and LMAT, and because ofEQUIVALENCE (EIVECV,FMIKR2) and EQUIVALENCE (EIVECV,LMAT)these are four matrices.

Furtheron there are several REAL matrices of dimension 1110∗1110 (=1232100) and COMPLEX∗16 matrices of dimension 784∗784 (= 1229312).

Subroutine SHORTL on file shortL.f differs from SHORTL on files shortO.f,shortP.f and shortQ.f, respectively. ND2002 is set to 3432 as a maximum onfile shortO.f, and to 2002 on file shortQ.f. Subroutine SHORTL on files shortP.fand shortQ.f defines a further EQUIVALENCE (EIVECV,FMIKR1) to savea quarter of the space of above matrices. This EQUIVALENCE means, thatFMIKR1 must be re-calculated if it is needed once more, but in the case of ‘NOFITTING’ and/or option ‘JAFAST’ this needs not to be done anywhere. Ofcourse ND2002=3003 may be set on another file short..f.

An important dimension is ND252 = 252 corresponding to the maximum ofdimension of a full micro-state basis for d electrons. A further PARAMETERis NDS66 = 252 on file shortO.f or = 66 else. NDS66 is the maximum of thedimension, to which the basis is reduced in subroutine TANFEL for f electronsand option ‘JAFAST’.

ND28 = 28 is a further important dimension corresponding to a quartet-Fstate of d electrons. ND28 is the maximal dimension for the monomer in the caseof Heisenberg coupling in a (e.g. symmetrical) dimer. 2 ∗ ND2828 = 2 ∗ ND28 ∗ND28 becomes 1229312 and a little less than ND1110 ∗ ND1110 = 1232100 withND1110 = 1110. Matrices for the case of Heisenberg coupling have dimension(ND1110,ND1110) if they are real and (ND2828,ND2828) if they are complex.Remember please DOUBLE PRECISION for complex quantities! Two of the

4.4. Subroutine SUSFIT 53

real matrices are EQUIVALENCEd to matrices defined before, and real andcomplex matrices are EQUIVALENCEd to save storage.

4.4 Subroutine SUSFIT

Subroutine SUSFIT to FIT SUSceptibilities, called by subroutine SHORTL andwithout RETURN to SHORTL, is the routine where calculation really startsafter the start of CONDON, and where calculation ends or may be continuedin the case of standard calculations. At the very end SUSFIT will mostly callsubroutine RESTBL (‘RESult in form of TaBLes’), but RESTBL returns toSUSFIT and calculation may be continued then. A Nassi-Shneiderman flowchartfor subroutine SUSFIT is given (consecutively ordered as CHART 3).

CHART 3

SUBROUTINE SUSFIT to FIT SUSceptibilities

All iterations, starting with 0th iteration

SUBROUTINE LEVenbergSQX (see CHART 4)to evaluate susceptibilities and SQX, e.g. in %

End of calculation?yes no@@@

��

STOP ?yes no@@@

��

iter. = iter. + 1

STOP

SUBROUTINE RESTBL(RESultTaBLes) to getOUTPUT in form ofTABLES on TERMINALand FILEs fort.11 and fort.1

SUBROUTINE LEVMAR(LEVenbergMARquardt)(see CHART 5) startingwith first iteration(method of Levenberg/Marquardt to propose newtrial parameters)

continue?yes no@@@

��

STOP

END of SUBROUTINE SUSFIT, no RETURN (to SHORTL)

Figure 4.4: Flowchart 3 (SUBROUTINE SUSFIT)

Subroutine SUSFIT and subroutine LEVSQX, which is called by SUSFIT,make use of the method of ‘least squares’ to get a measure ‘SQX’ for the devia-


tion of calculated values and measured values of ‘CHI’. This measure is definedin different ways, mostly a special ‘SQX in %’ is calculated. The existence ofmeasured values is always assumed, and SQX is always calculated!

This calculation of a ‘first’ SQX is called here ‘zeroth iteration of a fittingprocedure’, i.e. fitting itself, if fitting is done, starts with first iteration.

For fitting the Levenberg-Marquardt method is implemented. For a descrip-tion of this method see the book ‘Numerical Recipes’ [12]. Besides subroutineLEVSQX to evaluate SQX two subroutines are involved, LEVMAR, called bySUSFIT, and LEVDER, called by LEVMAR. Subroutine SUSFIT will decidefor convergence of the fitting procedure, or if the iteration count has reached itsmaximally allowed value. Shortly after the start of SUSFIT, a ‘STOP’ for thesetwo cases may be selected in advance to occur before or after OUTPUT of atable, or the program is stopped later on directly, once more with the choice ofOUTPUT of a table before. Alternatively the calculation may be continued.

Any calculation may be continued, including one with only zero iterations,or even one with no parameters to fit in addition (If there are still data to fit,and OMEGA has not reached its maximum value!). There are the followingpossibilities:

i) If there are parameters to fit, the calculation may be continued with moreiterations, with a new OMEGA value or not, or it may be continued with a‘new initialization’, i.e. starting with a ‘zero-th iteration’ (but it is numberedconsecutively, not with zero). In addition the possibilities according to ii) areoffered!

ii) If there are no parameters to fit, a different selection of measured CHIvalues (or energy values) with possibly new individual standard deviations maybe chosen, and different parameters and a different fit status (to be fitted ornot) may be chosen.

In the case of ‘STOP’ a new file fort.8 with a special header is written, tobe used as a ‘new file BWYBIN.DAT’! This file fort.8 is also always written(following a REWIND(8)), if an iteration is completed, in order to serve as a‘new file BWYBIN.DAT’, if the program is interrupted.

4.4.1 Subroutine RESTBL for RESult in form of TaBLes

OUTPUT in form of a table is given to file fort.11, but nearly the same table isgiven to the terminal, and a table without any header, to be used for plots, isgiven to file fort.1. There are different options for such a table, and more thanone table may be called for, if both ‘no STOP in advance’ and ‘non-standard’are selected.

Standard is a relatively small table including ‘CALCUL. CHI’ always and‘MEASURED CHI’ in the case of ‘fitting’ (and if also ‘LESS.NE.0’). Optionallya non-standard table of OUTPUT is possible, i.e. a wide table on files fort.11

4.4. Subroutine SUSFIT 55

and fort.1, and it must be printed in format ‘landscape’ to be readable well!

Standard or non-standard form of tables of OUTPUT, in addition corre-sponding to ‘fitting’ or ‘no fitting’, may be selected at will, and in the caseof ‘non-standard’ there are many different possibilities for the contents of thetables, but default values are always offered. Only default values are possible,if the form of (only one) table is selected in advance in subroutine SUSFIT inorder to guarantee a definite end of computation.

In subroutine RESTBL either, just as in subroutine SELECT and also de-scribed there (see 4.19.1), CHISTA (‘CHI-START’) becomes CHI(1,1,1), or CHI-FIX, the relevant ‘old CHISTA’, becomes CHI(1,1,1), the latter if the fittingroutine had not succeeded in finding a ‘better’ SQX in the last run or the lastrun was the first run. CHISTA, CHIFIX and CHIMEA are of course differentfor each temperature (and magnetic field, if concerned), and a loop runs, if thereis more than one row in the table.

There are three possible corrections in subroutine RESTBL, the same as insubroutine SELECT, and all of them are always applied, even if they are zero:

CHI−calc. = (((CHI(1, 1, 1))−1 − λMF )−1 + CHI−0)/F.

CHI(1,1,1), CHI-calc. and the intermediate results are stored in CHI(1,1...4,1),i.e. CHI-calc. = CHI(1,4,1).

CHIMEA (‘CHI-meas.’) becomes CHI(2,4,1), and ‘MEASURED CHI’ and‘CALCUL. CHI’ are ready for OUTPUT for this temperature. They will begiven as sixth and seventh column of the table of OUTPUT in its standardform in the case of ‘fitting’, in the case of ‘no fitting’, and if ‘LESS’ is zero, only‘CALCUL. CHI’ is given as sixth column!

The first five columns are ‘No.’, ‘00’, ‘TEMP/K’, ‘SIGMA’ and ‘B/TESLA’.

‘00’ means ‘FITTED’, ‘10’means ‘NOT FITTED (FROM INPUT)’, ‘01’means ‘NOT FITTED (FROM CALC.)’.

The individual standard deviations SIGMA must not be zero. If any of themare zero, they are set to one! The sum of ‘1/SIGMA**2’ is equal to the ‘No.OF VALUES’ to be fitted (i.e. ‘not 10’ in INPUT).

As an option and regarded as non-standard, much more and very differentOUTPUT may be chosen, and the calculations needed for this follow now.

Exactly in reversed order compared with CHI-calc., CHI-meas. is alwayscorrected, even if the corrections are zero:

CHI(2, 1, 1) = (((CHI−meas. ∗ F)− CHI−0)−1 + λMF )−1.

The results are stored in CHI(2,1...4,1), i.e. CHI-meas. = CHI(2,4,1).

Up to now CHI stands for any of the eight values CHI(1...2,1...4,1)! Out ofeach of these eight ‘CHI IN SI UNITS’-values six further values are calculatedand stored in CHI( , ,2...7):


ii) 1/CHI, iii) (e.g. saturation)MAGN.MOMENT in µB , iv) µeff=MU EFFEC-TIVE, v) CHI∗(field/Teslas), vi) CHI IN CGS, vii) MU EFF.(THETA) (i.e. afurther µeff ).

The ‘further µeff ’ is calculated with (T−Θ) instead of T , because the Weissconstant Θ may be taken instead of the molecular field parameter λMF whichis zero in this case.

CHI( , , ) stands for any of the 56 values calculated, and altogether there are56 different possible headers for columns of CHI( , , ), but surely fewer differentvalues of CHI( , , ) for one and the same temperature.

One table consists of six columns of CHI( , , ) values. The columns may bechosen freely, but default values are given. Each header of each column con-sists now of three single headers. Firstly the identification of values as ‘CAL-CULATED’ or ‘MEASURED’ is given. Secondly one of the following head-ers follows: CHI IN SI UNITS, 1/CHI, MAGN.MOMENT, MU EFFECTIVE,CHI∗FIELD, CHI IN CGS, MU EFF.(THETA). As a third header, the followinginformation is given:

If the values are calculated values, the calculated CHI(1,J,1) the valuesCHI(1,J,K) are based on is given, and if the values are measured values, thecalculated CHI(1,J,1) the calculated values CHI(1,J,K) are based on is given,with which the measured values CHI(2,J,K) are to be compared.

The four possibilities for this last header, i.e. for the calculated CHI-valuesCHI(1,J,1), are, for J = 1 . . . 4:

CHI-START, CHI-calc.∗F-CHI0, CHI-calc.∗F, and CHI-calc.!

To remember the many possibilities offered here easily: There is one ma-trix of calculated values (‘CALCULATED’) and one matrix of measured values(‘MEASURED’), the columns of each matrix named CHI IN SI UNITS, 1/CHI,etc. The rows of both matrices are named as if they were both ‘CALCULATED’,i.e. CHI-START, CHI-calc.∗F-CHI0, CHI-calc.∗F, and CHI-calc.!

In addition to the table the following OUTPUT is given once more:

No. of electrons, l value, no. of basis states, No. of values used ‘later on’. Ifconcerned, the no. of states used intermediately (if ‘JAFAST’) and the statesused for ‘Combined Model’ are given, too.

To the very end and if concerned, (part of) the following OUTPUT is given:

’DIR. FIELD(Z)’, ’DIR. FIELD(X)’, ’DIR. FIELD(Y)’,‘DISTance (DIMER)’, ‘NEW FIELD’, ‘RACAH-B’,‘RACAH-C’, ‘F-INTEGRAL(2)’, ‘F-INTEGRAL(3)’,‘F-INTEGRAL(4)’, ‘ZETA (1 EL.)’, ‘ORB.RED.F.(Z)’,‘ORB.RED.F.(X)’, ‘ORB.RED.F.(Y)’,‘B20(WYBOURNE)’, ‘B22(WYBOURNE)’, ‘B40(WYBOURNE)’,‘B42(WYBOURNE)’, ‘B43(WYBOURNE)’, ‘B44(WYBOURNE)’,‘B60(WYBOURNE)’, ‘B62(WYBOURNE)’, ‘B63(WYBOURNE)’,

4.5. Subroutine LEVenbergSQX 57

‘B64(WYBOURNE)’, ‘B66(WYBOURNE)’, ‘J(HEISENBERG)’,‘MOLecular FieLD PARAMeter’, ‘THETA of Curie-WEISS’,‘CHI-0 (DIAMagnetic)’,‘CORRection (WEIGHT)’, ‘MOLecular FieLD THEORY’,‘MIXture or CROSSOVeR’,‘ENTHALPY’, ‘ENTROPY’.

Note: the last twenty values correspond to those on file BWYBIN.DAT!

Furthermore, in the case of IOPT1=1 (i.e. may be more than one directionof the magnetic field), CHI for each of the directions as stored in CHIZXY isgiven together with the mean value CHI(1,1,1). But these are the actual values(only), not the fixed ones!

A new file fort.8 with a special header is written, too, to be used as a ‘newfile BWYBIN.DAT’!

4.5 Subroutine LEVenbergSQX

There are three subroutines LEV. . . in connection with fitting, LEVSQX, LEV-MAR, and LEVDER. Least square fits are done always, and the LevenbergMarquardt method [12] is used.

Every fitting method needs a measure for the ‘quality of the fit’, and subrou-tine LEVenbergSQX evaluates such a measure as a value of the quantity SQX.Starting a fitting procedure, firstly a SQX value is evaluated, i.e. in the ‘zerothiteration’. Subroutine LEVSQX evaluates such a SQX value also, if fitting isnot done at all, i.e. there is no ‘first iteration’ etc., so long as measured CHIvalues are given as ‘to be fitted’ (see 4.5.2).

An individual standard deviation SIGMA, which must not be zero, belongsto each CHI value ‘to be fitted’. As a standard all these SIGMAs are ONE.SIGMA may be taken as a kind of weighting factor, but a big value of SIGMAmeans a small weighting factor! The SIGMAs put in will be normalized beforecalculation starts, but they are already normalized if they are all ONE.

There are three special options for fitting in subroutine LEVSQX, namelyNOPROZ, IOPT4 and IOPT3.

The option NOPROZ=0 (default value) means, that CHI(1,J,K)/CHI(2,J,K)is fitted to the number ONE. The sum of (CHI(1,J,K)/CHI(2,J,K) − ONE)-squared is divided by the number of values and the square root thereof is the‘least square deviation SQX’, a pure number, to be multiplied by 100% to getit in % (100%=1). CHI(1,J,K)/CHI(2,J,K) means calculated value divided bymeasured value!

NOPROZ=1 is not possible, if ‘less possibilities of tests’ is chosen, and itmeans, that the sum of (CHI(1,J,K) − CHI(2,J,K))-squared is divided by the


CHART 4

SUBROUTINE LEVenbergSQX

Twice in case of ‘SPIN CROSSOVER’, once else

Temperatures/fields (‘rows’) of CHIINP.DAT

SUBROUTINE SUSCeptibility (see CHART 6)to evaluate susceptibilities etc.

Evaluate SQX, e.g. in %, by method of LEAST SQUARES

SQX has become ‘better’ .OR. 0th iterationyes no@@@

��

Hold new solution (discard any other)

0th iteration?yes no@@@

��

OMEGA

incre-asesOMEGA decreases

END of SUBROUTINE LEVenbergSQX, RETURN to SUSFIT

Figure 4.5: Flowchart 4 (SUBROUTINE LEVenbergSQX)

number of values and the square root thereof times a factor ‘CORR’ is the ‘leastsquare deviation SQX’. SQX is surely not in % now, but because CORR isintroduced, it is comparable in magnitude with SQX in %! CORR is the sum of(1/CHI(2,J,K))-squared divided by the number of values, and it is introducedonly for convenience and has no real meaning. CORR appears sometimes inOUTPUT in addition to SQX. CORR becomes ONE, if at least one of the(1/CHI(2,J,K))-squared values is smaller than a threshold value.

Not only CHI itself, but any of the values CHI( , ,1...4) may be fitted corre-sponding to IOPT4=1. . . 4. IOPT4=1 is the default value, 0 becomes 1! Cor-responding to the seven values i) . . . vii) above, IOPT4=1. . . 4 means fitting ofCHI, 1/CHI, MAGN.MOMENT and MU EFFECTIVE, respectively!

Furtheron an option IOPT3 is available in subroutine LEVSQX to avoid anydivision by zero. If either 1/CHI is fitted or, for NOPROZ=0, 1/CHI-meas. iscalculated, or if CORR is evaluated, a division by zero may occur (CORR is setto ONE in such a case)! To avoid this, an option IOPT3=1 (default value =0) is

4.5. Subroutine LEVenbergSQX 59

possible, which selects CHI( ,2, )instead of CHI( ,4, ) for the fitting procedure,i.e. a shift by CHI-0 is done! IOPT3=1 is not possible, if CHI-0 is zero and/orwill not be fitted!

In subroutine LEVSQX there is a DO LOOP over all rows of CHIINP.DAT,i.e. over all temperatures and/or magnetic fields. Each time subroutine SUS-Ceptibility is called to evaluate CHI etc. In addition there is an outer DOLOOP, running twice in the case of ‘SPIN CROSSOVER’ and once else. Thisouter DO LOOP leads to one single CHI value, because results of the first runare intermediately stored in SUSC and used to evaluate the resulting CHI valuein the second run.

Following the calculation of all CHI values, SQX is evaluated.

After SQX has been evaluated, but not for the zeroth iteration, the questionis asked if SQX has become ‘better’. If the answer is ‘no’, the special parameterof Levenberg-Marquardt method ‘OMEGA’ is decreased (by a factor of 0.3),otherwise it is increased by a factor of three. For zeroth iteration or ‘yes’, thenew solution is stored and the old one discarded, if any. The next iterationstarts with the old solution, if it was not discarded, and with the new solutionelse.

The special parameter ‘OMEGA’ is read in after the zeroth iteration to beused for the first iteration. In connection with ‘OMEGA’ some options areavailable to change it during the fitting procedure. A positive value means‘standard’ as decribed here, a negative OMEGA becomes positive ‘with thisvalue the lower limit’, and an ‘ERROR’-INPUT (e.g. INPUT of ‘A’) means,that ‘OMEGA’ is read in again after each iteration (a further ‘ERROR’-INPUTleads to ‘standard’ again). INPUT of 999 plus an OMEGA greater than zero andlower than one, e.g. 999.1, means a special procedure starting with ‘OMEGA’: Ifan iteration, which is not successful, follows one which was successful, ‘OMEGA’is not increased, but set to a (small) constant value, which must be put in insubroutine LEVMAR (see 4.6).

If one single parameter is fitted and the minimum number of iterations wasset equal to the maximum number (a hint to do so was given there), INPUT ofa negative ‘OMEGA’ may serve to disable the LEVENBERG/MARQUARDTmethod completely! If this second option for negative OMEGA is selected,OMEGA remains constant and the absolute value of OMEGA serves as a step-wide for the parameter to be fitted, where one iteration means one step.

A Nassi-Shneiderman flowchart for subroutine LEVenbergSQX is given (con-secutively ordered as CHART 4).

4.5.1 Parameters, which may be fitted

Altogether 20 parameters may be fitted, but not all of them at the same time.

First of all there is a maximum of 11 ligand-field parameters, and all 11 may


be fitted at the same time, corresponding to a ‘point group ???’ as a test. The‘real’ maximum however is 9, corresponding to an ortho-rhombic point group.

Next the (Heisenberg) coupling constant J may be fitted in the case of a(symmetrical) dimer.

Concerning a ‘molecular field’, one of three parameters may be fitted: Molec-ular field parameter λMF or Weiss constant Θ, or a molecular field parameterfor (‘real’) molecular field calculations.

A formally ‘diamagnetic’ contribution CHI-0 may also be fitted.

Because the weight of a sample and therefore measured CHI (note: CHImeans always ‘molar’ CHI) may be wrong, a correction factor ‘F’ of measuredCHI, with a standard value of ‘one’, may be fitted. The standard ‘MEASUREDCHI’, however, which is measured CHI based on CHI-CALC., is not multipliedby ‘F’, whereas CHI-CALC. (‘CALCUL. CHI’) results following a division by‘F’.

If a dimer contains a little monomer, which cannot be removed (this isnot(!) the spin-one-half monomer in the case of an unsymmetrical dimer!), theproportion of such a monomer may be found by fitting a special ‘mixing’ param-eter. This special parameter, but with a different meaning and in the case of amonomer, is used together with two further parameters for ‘enthalpy’ and ‘en-tropy’ in the case of ‘spin crossover’, and these three parameters may be fittedthen. An example for ‘spin crossover’ [11] will be given in Subsection 5.1.8!

See the end of Subsection 4.4.1 for a list of all 20 parameters, but note, thatparameters other than Wybourne parameters may have been fitted instead ofWybourne parameters! Parameters fitted are marked with 1 or, in special cases,with 9, which becomes −1 later on. See section 4.7 for this latter case!

4.5.2 About fitting in general

There is no need to fit any values, and there is no need to fit all values read infrom file CHIINP.DAT as a standard. Values to be excluded from fitting aremarked with ‘1’ (‘NOT’) in the first column of CHIINP.DAT, and all(!) valuesmay be excluded! ‘0’ is set for all values, if there is no input from file (‘H’ or‘h’ is the ‘name of the file’). ‘1’ is appropiate, if some, but not all, of the CHIvalues to be calculated correspond to ‘measured’ values, which are zero. Each‘NOT’ may be changed into its opposite and reversely after the program halts(and not STOPs, because ‘LESS’ was not zero). If there is at least one ‘YES’,a non-zero SQX value is evaluated (if values to be fitted survive)!

Any fit and indeed any calculation starts with its ‘0th iteration’ to get afirst value of SQX, if not all the CHI values are excluded from fitting on theoutset. If some CHI values are zero and, in the case of ‘NOPROZ=0’, a divisionby these CHI values is impossible, these values are excluded from fitting andmarked with 01 in OUTPUT tables, and if all CHI values are zero in this case,

4.6. Subroutine LEVenbergMARquardt 61

a SQX value cannot be evaluated.

Parameters to be fitted are read in from file BWYBIN.DAT as a standardand are marked there with ‘1’ (or ‘9’ in some special cases). If none have beenmarked, and ‘LESS’ is zero, there is a different form of standard OUTPUTtable, which may be changed however. Each marked parameter may be changedinto ‘not marked’ and reversely after the program halts, except in the case‘LESS.NE.0’.

The number of iterations, i.e. the maximal number if there is no convergenceachieved before, is asked for as INPUT, if there are any parameters marked tobe fitted, but the number put in may be zero! The number may be changedand even made smaller after the program halts.

If a negative CHI-value is involved in fitting, i.e. CHI(1,IOPT31,1) or CHI(2,IOPT31,1) are negative, this will be excluded from fitting automatically andmarked with 01 in OUTPUT tables, whereas a value marked with 10 there isexcluded from fitting because it is marked with ‘NOT’ in CHIINP.DAT. Alsoif (T −Θ) becomes lower than zero, CHI-START becomes zero and is excludedfrom fitting! Furtheron a warning is given, if the number of values fitted changes(but SIGMA values are not corrected)!

To summarize, there are at all three possibilities to manage input in connec-tion with fitting: ‘NOT’ in the first column of CHIINP.DAT, not all parametersmarked to be fitted, and ‘zero iterations’. The standard to do absolutely nofitting should be ‘zero iterations’ !

4.6 Subroutine LEVenbergMARquardt

Subroutine LEVenbergMARquardt starts the first iteration of a fitting pro-cedure using the method of Levenberg and Marquardt. There is always theevaluation of a SQX value in a ‘zeroth iteration’ in subroutine LEVenbergSQXbefore, called by subroutine SUSFIT just as this subroutine here, but this ‘ze-roth iteration’ is now renamed the ‘first iteration’. The Levenberg-Marquardtmethod attempts to reduce this value of SQX. It makes use of subroutine LEV-enbergDERivatives, which evaluates the derivatives of CHI with respect to thefitting parameters.

For theory and some practical matter see W. H. Press, B. P. Flannery, S. A.Teukolsky, W. T. Vetterling, Numerical Recipes (FORTRAN Version), The Artof scientific Computing, Cambridge University Press, Cambridge 1992, page 526[12].

If there is the first iteration, a proposal for new fitting parameters will al-ways be made, and for this first iteration the special parameter of Levenberg-Marquardt method ‘OMEGA’ must be put in (pre-occupied by value 0.1).

Subroutine LEVenbergDERivatives is called to evaluate the above mentioned


CHART 5

SUBROUTINE LEVenbergMARquardt

SQX has become ‘better’ or first iterationyes no@@@

��

First iteration?yes no@@@

��

INPUT of OMEGA as a special parameter ofLEVENBERG/ MARQUARDT method

SUBROUTINE LEVenbergDERivatives (see CHART 7) tocalculate derivatives numerically with respect to all parametersto be fitted (some parameters may be calculated analytically)Propose new trial parameters (part 1): Alter linearized fitting matrixSUBROUTINE GAUSS to solve a system of linear equationsINPUT of limits for new trial parameters is enabled

Propose new trial parameters (part 2)

END of SUBROUTINE LEVenbergMARquardt, RETURN to SUSFIT

Figure 4.6: Flowchart 5 (SUBROUTINE LEVenbergMARquardt)

derivatives numerically, special ones also analytically as a test. The linearizedfitting matrix is altered, subroutine GAUSS is called to solve a system of lin-ear equations, and new trial parameters are proposed. Subroutine LEVMARreturns to SUSFIT, which calls subroutine LEVSQX to evaluate a new SQXvalue.

If there is not the first iteration, there are two possibilities: SQX has becomebetter or not! If SQX has become better, a further iteration follows formallyidentical to the iteration before, but using a new, decreased OMEGA. If SQXhas not become better, the linearized fitting matrix is altered using the oldderivatives and also a new, but increased OMEGA, subroutine GAUSS is calledto solve a system of linear equations, and new trial parameters are proposed.This procedure is also called a (new) iteration! Subroutine LEVMAR returnsto SUSFIT, which calls subroutine LEVSQX to evaluate a new SQX value.

There is no guarantee for SQX to become better, but mostly it becomes! IfSQX remains practically constant during further iterations, or if OMEGA hasbecome very big (1000000000), convergence has achieved.

Very often it happens, that new fitting parameters are proposed, which arenot acceptable. To avoid such parameters, limits for fitting parameters may be

4.7. Subroutine SUSCeptibility 63

put in by giving the name of a file containing such limits. Two zeros or blancsmean ‘no limits for this parameter’ ! As a standard this file is identical to thefile containing the starting parameters, and this is ensured by a check, if not thedigit 7 plus the file name are put in instead of the file name. If such a limit isreached, OMEGA becomes very big (1000000000), a warning message is given,and convergence is formally achieved. But this convergence should be ignoredby discarding the parameter concerned from fitting, and the fitting procedureshould be continued with the remaining parameters.

OMEGA is increased if SQX has not become better. But a special featureis available, which resembles somewhat ‘Simulated annealing’ and works witha decreased OMEGA in this case, if SQX had become better in the iterationbefore (see 4.5). In this case an e.g. very small OMEGA is used and put inhere, and it may happen, that a better SQX is found for this OMEGA or theincreased OMEGAs of the following iterations before that OMEGA is reached,which would have been used else.

A Nassi-Shneiderman flowchart for subroutine LEVenbergMARquardt is given(consecutively ordered as CHART 5).

4.7 Subroutine SUSCeptibility

The calculation of CHI really starts in subroutine SUSCeptibility. This sub-routine is called by subroutine LEVSQX for each row (‘temperature’) of the‘INPUT’, but if no fitting is done, LEVSQX is actually not of great importanceand will merely calculate SQX as a measure of the quality of a fit, if not allthe CHI values are excluded from fitting on the outset. It is also called bysubroutine LEVenbergDERivatives to evaluate derivatives of CHI with respectto parameters to be fitted, if fitting is required.

The calculation of CHI is embedded in the Levenberg-Marquardt method,and subroutine LEVenbergMARquardt calls subroutine LEVenbergDERivativesto call subroutine SUSCeptibility.

The calculation of susceptibilities (‘CHI values’) is of course the purposeof this subroutine, but negative, nearly integer temperatures as input are inter-preted as the negative consecutive numbers of the eigenvalues of the ion handled,e.g. −1.0 means the first eigenvalue, which is always zero. These eigenvalues,but not the first one in the case of ‘SQX in %’, may be fitted just as CHI values,and this is useful to have regard to a measured value of ‘crystal field overallsplitting (CFOS or CFOSJ)’. No magnetic field is needed in these cases.

An ‘infinite’ loop (not a ‘DO’ loop) runs in SUSC in the case of ‘MOLecu-larFieLD’ [10], until convergence is achieved, and in each run subroutine TAN-ABE (or ECKART) is called with a different ‘field’ to evaluate a value for CHI.The molecular field here is proportional to the magnetisation, it is added to theapplied field, and it is the same for all directions. Convergence is achieved, if


CHART 6

SUBROUTINE SUSCeptibility

Ligand field parameters are changed to such of WYBOURNE, if others(Call of STR00 is possible to create WYBOURNE parameters)

In the case of ‘MOLecularFieLD’: until convergence

Neg. temperatures => CHI = eigenvalue(s) and RETURN

Micro-state basis?yes no@@@

��

SUBROUTINE TANABE(see CHART 8) to set up

and diagonalize (real) matrix

SUBROUTINE ECKART(see CHART 9) to set up

and diagonalize (real) matrix

2nd loop in LEV. . . in the case of ‘SPIN CROSSOVER’?yes no@@@

��

Composition of new CHI

END of SUBROUTINE SUSC, RETURN to LEVSQX or LEVDER

Figure 4.7: Flowchart 6 (SUBROUTINE SUSCeptibility)

CHI becomes nearly constant.

Furtheron ‘SPIN CROSSOVER’ [11] is implemented, and subroutine SUSCis called twice by LEVSQX for each row (‘temperature’) of the ‘INPUT’. Inter-mediate results, i.e. those of the first run, are stored in SUSC, and the resultingCHI is evaluated in SUSC after the second run. In the case of fitting, the sameis true for LEVDER.

In the case of a ‘normal’ calculation, virtually nothing is calculated in sub-routine SUSC! Only the decision to use TANABE (‘standard’) or ECKART(‘specific term 2S+1L’) is realized in SUSC.

As ligand field parameters as defined by Wybourne [13] as ‘B-values’, i.e.B2

0 etc. = B20, B22, B40, B42, B43, B44, B60, B62, B63, B64, B66, shouldbe preferred as INPUT, no conversion of parameters is needed as a standard.But other parameters are also possible, and such parameters must be convertedto Wybourne parameters, which are always used internally! This conversion

4.7. Subroutine SUSCeptibility 65

is done in subroutine SUSC! Note, that derivatives needed and computed forfitting are dependent on which ‘B-values’ are taken, and that quite a differentfitting procedure will run with only hopefully the same result!

If the structure of an ion-ligand complex is known, such ‘B-values’ may bederived from orbital overlap (one ‘covalence factor’) and charge factors by callingsubroutine STR00 (see Section 4.8). Only two (or three) parameters have to befitted for one (or two) kind(s) of ligands, and this is far less then the numberof B-values in the case of low symmetry (‘Simple Overlap Model (SOM)’, but‘PCEM’, i.e. PointChargeElectrostaticModel, which uses charges only, is alsoincorporated).

As a special feature, B43, B44, B63, B64 and B66 may be given in fileBWYBIN.DAT as proportional to B43/B40, B44/B40, B63/B60, B64/B60 andB66/B60, respectively, if fitting is chosen by 9 instead of 1 and no fitting by 8instead of 0, respectively (9 and 8 become −1 and −2 later on, respectively).Proportionality is such, that B43/B40 etc. =1.0, if the cubic case(s) would beconcerned. See Subsection 5.1.8 for an example!

Ligand field parameters as defined by Stevens [21] for f electrons as ‘B-values’are converted in SUSC. This is not possible in the case of Ce3+ or Sm3+, if BKQvalues with K=6 are involved, because ‘GAMMA’ is not available for these ions.

For f electrons and the ‘standard’ cubic point group, the work of Lea, Leaskand Wolf is important [22]. The parameters X and W will be converted to B40and B60. This is not possible in the following cases: Eu3+, Gd3+ (both notmentioned in table 1 of [22]), Ce3+ and Sm3+ (both have X=1).

If x- and y-direction are involved and three CHI values are calculated, eachfor one direction, x- and y-direction may be exchanged one to each other, if somesigns (of B-values) are reversed! This is a possibility of internal tests! It is alsopossible to get CHI values for y-direction calculating CHI values for x-direction,but with some signs of B-values reversed.

In subroutine SUSC one is asked for, if any of these options is relevant (if itapplies at all), but not always so: ‘Stevens’ is not possible for ‘less tests’, andthe last option is only possible for ‘more tests’ !

For some special cases, i.e. more than four and less than ten electrons, andnot ‘less output’ or ‘less tests’, and if option ‘JAFAST’ is used, there is a fur-ther option, which is often useful in connection with option ‘JAFAST’. Becausediagonalization before reaching ‘interface III’ is time consuming, this interfacemay be written to file fort.13 (‘unformatted’, and the first iteration must run)and read in from this file instead of a new calculation, if CONDON is startedrepeatedly with the same or a similar BWYBIN.DAT file.

A Nassi-Shneiderman flowchart for subroutine SUSC is given (consecutivelyordered as CHART 6).


4.8 Subroutine STR00

Subroutine STR00 may be called by subroutine SUSC to generate ‘B-values’from orbital overlap (one ‘covalence factor’) and charge factors, if the structureof an ion-ligand complex is known. As input firstly the cartesian coordinatesof the ligands are needed, from file str00.dat or from any other file. This filemust contain, after a first line of text, 5 coordinates in Bohr on each line inFORMAT 5D16.8: Firstly ten x-coordinates of the first sort of ligands, then teny-coordinates and ten z-coordinates, and the same for the second sort of ligand.Zeros serve to cancel ligands!

Covalence factor and one or, if two sorts of ligands are present, two chargefactors are the parameters to be fitted and come from subroutine SUSC, andall ‘B-values’ are evaluated and go back to this subroutine. In the case of lowsymmetry the number of parameters to be fitted is drastically reduced in thisway!

As a standard the ‘Simple Overlap Model (SOM)’ [31, 32] is applied, whichin many cases does not give good ‘B-values’, but only approximations to suchvalues. To calculate the structure factors needed for such models, the expec-tation values of r2, r4, and, for f electrons, r6 are needed. These may be putin directly, but tables are included for d electrons [30], lanthanides [30], andUranium (relativistic values [28]), and may be used, if possible.

The point charge model (‘PCEM’, i.e. PointChargeElectrostaticModel), whichuses charges only, is also incorporated, but ‘SOM’ should be preferred, exceptin simple cases as is e.g. [Ti(H2O)6]3+.

Extra input is required by ‘SOM’ concerning the question of ‘the more cova-lent ligand’. There is a formula ‘β = 2/(1− ρ)’. The minus sign is the ‘normal’sign, and the minus sign should hold for the more covalent ligand, or if only onesort of ligands is present. If there are two sorts of ligands, and if the first ligandis the more covalent one, a plus sign instead of minus should be taken for thesecond ligand. Pre-occupied are a minus sign followed by a plus sign. Input isrequired for ‘times minus one’ yes(1)/no(0) for both these signs. If the secondsort of ligands is chemically identical to the first sort and only the distancesfrom the origin are a little different, twice the same sign should be used!

Eleven B-values ‘Wybourne’ are calculated and put out, namely (20 shortfor B20 etc.) 20/22/40/42/43/44/60/62/63/64/66, but only those of them goback to the calling routine, which are valid for the point group given. Furtheronsixteen B-values 12/12’/22’/41/41’/42’/43’/44’/61/61’/62’/63’/64’/65/65’/66’are calculated and put out, but only to be able to control the point group oncemore. So all possible B-values are calculated, and if any of them, which donot go back to the calling routine, differ remarkably from zero, either the pointgroup is wrong or the setting meets not the requirements of a standard.

Because as a standard the parameters to be fitted are written to file fort.8,i.e. one covalence factor and one or two charge factors, the B-values calculated

4.9. Subroutine LEVenbergDERivatives 67

here, but only those, which go back to the calling routine, are written to filefort.9.

4.9 Subroutine LEVenbergDERivatives

CHART 7

SUBROUTINE LEVenbergDERivatives

Parameters to be fitted

parameter - (1 + DELTA) * parameter

Twice in case of ‘SPIN CROSSOVER’, once else

Temperatures/fields (‘rows’) of CHIINP.DAT

SUBROUTINE SUSCeptibility (see CHART 6)to evaluate susceptibilities etc.

evaluate derivatives= DELTAsusceptibility/DELTAparameter

Set up matrices for SUBROUTINE LEVenbergMARquardt

END of SUBROUTINE LEVDER, RETURN to LEVMAR

Figure 4.8: Flowchart 7 (SUBROUTINE LEVenbergDERivatives)

Subroutine LEVenbergDERivatives is used by subroutine LEVMAR to eval-uate the linearized fitting matrix ALPHA and vector BETA. It calls subroutineSUSCeptibility to evaluate CHI values corresponding to fitting parameters dif-fering by DELTA from the original fitting parameters. With help of the originalCHI the derivatives of CHI with respect to the fitting parameters are calculatedas DELTA susceptibility divided by DELTA parameter.

In subroutine LEVDER there is a DO LOOP over all rows of CHIINP.DAT,i.e. over all temperatures and/or magnetic fields just as in subroutine LEVSQX,and in addition there is the outer DO LOOP just as in LEVSQX, which runstwice in the case of ‘SPIN CROSSOVER’ and once else. But there is one moreDO LOOP including these two DO LOOPs, and this goes over all parametersto be fitted.


If all derivatives are calculated and stored, the linearized fitting matrix AL-PHA and vector BETA are calculated, and subroutine LEVDER returns tosubroutine LEVMAR.

A Nassi-Shneiderman flowchart for subroutine LEVenbergDERivatives isgiven (consecutively ordered as CHART 7).

4.9.1 Special option for analytical derivations

Of all the 16 possible combinations of IOPT3, IOPT4 and NOPROZ the mostimportant is of course ‘000’ identical to ‘010’, i.e. ‘calculated’ CHI, ‘CHI IN SI’and ‘SQX in %’. Four other combinations are to be considered more importantthan the rest, and these are ‘011’, ‘021’, ‘110’ and ‘121’, internally as NOGEW2= (0,)1,−1,2,−2:

NOGEW2 = 0: DEFAULT!

NOGEW2 = 1: DEFAULT, but SQX NOT IN %!

NOGEW2 =−1: Curie straight line (1/CHI), NOT IN %!

NOGEW2 = 2: DEFAULT, but SHIFT BY CHI-0!

NOGEW2 =−2: Curie straight line (1/CHI), SHIFT BY CHI-0, NOT IN %!

The derivatives with respect to F CHI-0 and λMF may be done analyticallyinstead of numerically for these five special cases! Numerically is standard, theoption ‘analytically’ is possible as a test!

4.10 Subroutine TANABE

In subroutine TANABE a full basis of micro-states enters the calculation, i.e.120 micro-states in the case of three 3d electrons. Three 3d electrons are takenas an example for several reasons: This example is sufficient to show the relevantproblems, and the case of three 3d electrons is an example treated in the bookby Condon and Shortley [4] in connection with Russell-Saunders states.

Subroutine SCHEMA is called for these micro-states. The micro-states aregiven in two different ways, as MSML and as MSMLNW, and some special sumsof magnetic quantum numbers defining micro-states are stored in MQUER andgiven here as SUMs (sum of ml values, sum of 2∗ms values, sum of twice first sumand once second sum). (MSML(II,1,I),I=1,NREL) means (ml =) ml values ofeach electron in each state II, (MSML(II,2,I),I=1,NREL) means 2∗ms values ofeach electron in each state II, MSMLNW(II,1,−KLWERT:+KLWERT) means2∗ms for each ml value in each state II, if ms>0, and MSMLNW(II,2,−KLWERT:+KLWERT) means 2∗ms for each ml value in each state II, if ms<0.

The first ten micro-states as stored in MSML are given here:

4.10. Subroutine TANABE 69


1 5 1 11 2 2 1 1-1 12 5 -1 9 2 2 1 1-1-13 4 1 9 2 2 0 1-1 14 4 -1 7 2 2 0 1-1-15 3 1 7 2 2-1 1-1 16 3 -1 5 2 2-1 1-1-17 2 1 5 2 2-2 1-1 18 2 -1 3 2 2-2 1-1-19 4 1 9 2 1 1 1 1-1

10 3 3 9 2 1 0 1 1 1

And the same first ten micro-states as stored in MSMLNW are given here,too, but they are not used in TANABE:


1 5 1 11 0 0 0 1 1 0 0 0 0-12 5 -1 9 0 0 0 0 1 0 0 0-1-13 4 1 9 0 0 1 0 1 0 0 0 0-14 4 -1 7 0 0 0 0 1 0 0-1 0-15 3 1 7 0 1 0 0 1 0 0 0 0-16 3 -1 5 0 0 0 0 1 0-1 0 0-17 2 1 5 1 0 0 0 1 0 0 0 0-18 2 -1 3 0 0 0 0 1 -1 0 0 0-19 4 1 9 0 0 0 1 1 0 0 0-1 0

10 3 3 9 0 0 1 1 1 0 0 0 0 0

A Nassi-Shneiderman flowchart for subroutine TANABE is given (consecu-tively ordered as CHART 8).

4.10.1 Three 3d electrons – Interelectronic repulsion

If one has more than one electron, energy of interelectronic repulsion becomesrelevant. There is however a possibility to ignore this energy completely in thecase of 4f electrons, if only the specific ground term 2S+1L is taken into account,and this may be done in subroutine ECKART with help of irreducible tensoroperators. Using irreducible tensor operators is not considered as a standardmethod any more, but may be useful to save storage and time in very specialcases.

The relevant literature for interelectronic repulsion is once more the bookby Condon and Shortley [4]. The operator of interelectronic repulsion is Hee =∑Ni<j=1 e

2/rij =∑Ni<j=1 e

2/rji, but it is reduced to Slater-Condon parameters


CHART 8

SUBROUTINE TANABE with full micro-state basis

SUBROUTINE SCHEMA to get micro-states

First run .OR. (JEQUIV.NE.0 .AND. ‘new B-values’ .AND.‘TANFEL not selected’)?yes no

@@@

��

Set up matrix of operators of interelectronic repulsionand spin-orbit coupling

First run .OR. (‘new B-values’ .AND.’TANFEL not selected’)?yes no

@@@

��

Set up matrix of ligand field operator

Add the two matrices

Matrix will be blocked out?yes no@@@

��

SUBROUTINE BLOCUT (see10, calls DSPEVW for each block)

SUBROUTINE DSPEVW, alink to DSPEV to diagonalize

Change of basis to one of LS states (Russell-Saunders)?yes no@@@

��

SUBROUTINE RUSSELl, TESTonly, no RETURN (see CHART 11)

��@@

SUBROUTINE REDUCTanabe(see CHART 12)

END of SUBROUTINE TANABE, RETURN to SUSCeptibility

Figure 4.9: Flowchart 8 (SUBROUTINE TANABE)

and a sum of different operators: See below! The Slater-Condon parameter F 0 isalways taken as zero, and the Racah parameter A, too. This is possible, becauseF 0 and A, respectively, do not contribute to any energy differences, althoughA = F 0 − F 4/441 and the energies are not the same therefore. Definitely F 0 iszero, and A is not used. There are no further contributions to Hee in CONDON,e.g. from inner shells!

Subroutine TANABE calculates a matrix FMIKR1 with dimension 120 times120 to manage interelectronic repulsion. This matrix is augmented by the el-ements of spin-orbit coupling and is diagonalized directly or after a matrixFMIKR2 has been added, if the latter is not zero because no ligand fields arepresent. FMIKR1 is never re-calculated, if subroutine SHORTL on file shortL.f(or shortO.f) is linked, but in the case of ‘more’ equivalences, i.e. shortP.f orshortQ.f have been linked, it is new calculated, if it is needed once more! (It is


needed only once, if option JAFAST is chosen!)

The operator Hee works on a micro-state, i.e. a Slater determinant, and theindividual sets are given in conventional order by MSML (see Section 4.10 forthe first 10), e.g. the first micro-state is |a1

1a22a

33 >= |2+2−1+ >. In Condon-

Shortley [4] it is shown, how the reduction from the 3- to the 2-electron problemis made, i.e. from e2/rij to e2/rkt, the latter now working on a product of twospin orbitals (One spin-orbital is e.g. |a1 >= |2+ >). Matrix elements are givenseparately for diagonal and non-diagonal elements.

The calculation of the diagonal matrix elements of FMIKR1 is straightfor-ward. Equation 76(7) gives the summation of ‘direct’ and ‘exchange’ integrals,the latter non-zero only for electrons of like spins. Equation 76(7) reads, for theexample chosen,

3∑

k<t=1

[< akk| < att| e2/rkt |akk > |att > − < akk| < att| e2/rkt |atk > |akt >].

The integrals are given by eq.86(9) and by eq.86(12,13,14) of Condon-Shortley[4]. The first one, < akk| < att| e2/rkt |akk > |att >, is written now as (ab | e2/r12 | ab)= J(a, b), the second integral as (ab | e2/r12 | ba) = K(a, b).

J(a, b) =2l∑

k=2(2)

F k(nl) ck(lmal , lm

al ) ck(lmb

l , lmbl ),

K(a, b) = δ(mas ,m

bs)

2l∑

k=2(2)

F k(nl) [ck(lmal , lm

bl )]

2.

MSML enters a function CKCOE for the ck-coefficients (‘Condon-Shortleycoefficients’), which are defined as ck(lma

l , lmbl ) =< Y lma

l| Ck

mal−mb

l

|Y lmbl

>, with

Ckq = [4π/(2k + 1)]1/2 Y kq and Y kq the spherical harmonics working as opera-tors on spherical harmonics, and which have been programmed on the basis ofGaunt’s formula, eq.86(11). The F k (Slater-Condon parameters) are related tothe Racah parameters and are derived thereof, or they are read in directly. Notethat ck(lma

l , lmbl ) = (−1)m

al −m

bl ck(lmb

l , lmal ), and that if q = ma

l − mbl , then

mbl −ma

l = −q.Much more complicated is it to get the non-diagonal elements! First of all

they are non-zero only, if the two micro-states concerned differ by two individualsets, and eq.76(4) is the relevant formula. In addition there are no matrixelements connecting two states if these differ in regard to their

∑

ms. Eq.76(4)reads here

±[< akk| < att| e2/rkt |bkk > |btt > − < akk| < att| e2/rkt |btk > |bkt >],


the ± sign being determined by the parity of the permutations necessaryto change the conventional order of the second micro-state in such a way, that(N−2) of the sets are matched with those of the first micro-state:

a11 a2

2 . . . akk . . . att . . . aNNa1

1 a22 . . . bkk . . . btt . . . aNN .

The integrals in the latter formula are given by 86(9) of Condon-Shortley[4]. They are written now as (ab | e2/r12 | cd) =

δ(mas ,m

cs)δ(m

bs,m

ds)δ(m

al +mb

l ,mcl+m

dl )

2l∑

k=2(2)

F k(nl) ck(lmal , lm

cl ) c

k(lmdl , lm

bl ).

Subroutine COMPaRE, called with ‘option 2’, compares the two micro-states, tests for the delta-condition, that the sum of spins must remain the same,and for the delta-condition in the latter integral concerning the ml values. If thenon-diagonal element is recognized as zero already here, it is not considered fur-theron anywhere. The plus/minus sign in eq.76(4) is determined in COMPaREby the parity of the permutations needed to change the conventional order ofthe second micro-state into that of the first one, if the non-diagonal element isnon-zero.

In subroutine TANABE three different cases may occur, and the relevantcase is determined with help of two copies of part of MSML, made in COMPaRE,each copy a copy of the two individual sets of one of the two micro-states, whichare named akk, att, b

kk and btt above. Either all (four) spins are parallel or not, and

if they are not parallel, the spins of the respective ‘first’ electron are parallel ornot. For the first case both terms in eq.76(4) are non-zero, for the second caseonly the first term is non-zero and for the third case the second term only! Thetwo copies are also used to compute the ck-coefficients needed!

An example follows: The third and the nineth micro-state rewritten herediffer by two individual sets and both have the SUM’s 4 and 1:


3 4 1 9 2 2 0 1-1 19 4 1 9 2 1 1 1 1-1

A non-zero non-diagonal element occurs therefore. The spins of the respec-tive ‘first’ electron, i.e. actually the second electron of each micro-state, areantiparallel, and no permutation was needed for this comparision to be done.The second term of eq.76(4) reads now−CKCOE(K,L,ML=2,L,ML=1)∗CKCOE(K,L,ML=1,L,ML=0)∗F k,and is actually a sum of terms with K=2 and K=4 (L=2).

Diagonal and non-diagonal elements become the elements of a matrix FMIKR1.


4.10.2 Three 3d electrons – Spin-orbit coupling

The operator of spin-orbit coupling [4, 14, 15], with ZETA the spin-orbit cou-pling constant for one electron, is HSO = ZETA ∗

∑Ni=1(lzi ∗ szi + 1/2 ∗ (l+i ∗

s−i + l−i ∗ s+i)), and consists of a diagonal and a non-diagonal part. As anexample take three d electrons again. Some relevant micro-states as stored inMSML are rewritten here:


2 5 -1 9 2 2 1 1-1-13 4 1 9 2 2 0 1-1 19 4 1 9 2 1 1 1 1-1

The resulting matrix elements from the diagonal part are always a sum overthree electrons, and the result for the second micro-state isZETA/2 ∗ (2 ∗ 1− 2 ∗ 1− 1 ∗ 1) = −ZETA/2. The non-diagonal part workingon the second micro-state gives a factor times the third micro-state plus a fac-tor times the nineth micro-state, i.e. two non-diagonal matrix elements or partsthereof. Subroutine COMPaRE called with option 1 is used to select these twomicro-states out of all (KASSUM−1) micro-states. Corresponding to three elec-trons a maximum of three(!) non-diagonal elements or parts thereof is possible– either s−i or(!) s+i may work!

Diagonal and non-diagonal elements are added to those of matrix FMIKR1.

4.10.3 Three 3d electrons – The crystal field

Please note: ‘Ligand field’ and ‘crystal field’ are used synonymously! The oper-ator of the crystal field, mostly called ‘ligand field operator’, is an one-electronoperator as the operator of spin-orbit coupling and as such by far not so com-plicated as the two-electron operator of interelectronic repulsion. See Subsec-tion 2.2.2 for the ligand field operator HLF ! Again the book by Condon andShortley [4] is relevant, now pp. 169 ,170 and 171!

With Ckq = [4π/(2k + 1)]1/2 Y kq and Y kq the spherical harmonics workingas operators on spherical harmonics, integrals result, which are known as ck-coefficients (‘Condon-Shortley coefficients’), just as in the case of interelectronicrepulsion (see Subsection 4.10.1). The operator HLF works on spin orbitals,and its diagonal elements, but now for more than one electron, following fromeq.66(7) (resp. eq.66(9)), are

∑Nj=1 < ajj |HLF (j) |ajj >.

For each diagonal element a sum over all electrons and over the non-zeroWybourne parameters BWWW(1), BWWW(3) and BWWW(7) is calculated,i.e. B20, B40 and B60 are relevant here, but B60=0 for d electrons. With helpof ML out of MSML for each electron of each micro-state, function CKCOE


calculates the ck-coefficients to be multiplied by the Wybourne parameters toget the diagonal elements as sums of these products.

The non-diagonal elements are a little bit more complicated! Eq.66(6) (resp.eq.66(8)) is relevant. They are non-zero only if the two micro-states concerneddiffer by one individual set and the spins are the same.

The non-diagonal elements of HLF , for more than one electron as before, are± < ajj |HLF (j) |bjj >, the ± sign determined by the parity of the permutationsneeded to change the conventional order of the second micro-state into that ofthe first one.

Subroutine COMPaRE is called with ‘option 0’ to compare the two micro-states, to determine the parity of the permutations needed to change the con-ventional order of the second micro-state into that of the first one, and theabsolute value of the difference of the (ml = ML =) ML values of the differingindividual sets. It makes a copy of each of the two sets to be used to determinethe ck-coefficients needed here.

Because of the special structure of MSML, the sign of the difference of the MLvalues is always the same for those matrix elements, which are actually computed(i.e. those below the main diagonal of the matrix), and for the operator (−C3 +C−3) this means, that always the same part of it (e.g. −C3, and indeed −C3

has actually been chosen) works. (This statement is of course irrelevant for anoperator as (C4 + C−4)!)

For the non-diagonal elements the Wybourne parameters BWWW(1. . . 11except 1,3,7) are relevant, i.e. (BKQ=) B22, B42, B43, B44, B62, B63, B64 andB66, but B6Q=0 for d electrons. For a specific element only non-zero parametersare relevant of course, and the Q-values must be equal plus/minus the differenceof the two ML’s out of MSML for the two individual sets concerned. Each suchparameter is multiplied by a ck-coefficient with the same k = K and by parity,and the sum of these products becomes the non-diagonal element.

Diagonal and non-diagonal elements become the elements of a matrix FMIKR2.

An example follows: The first and the seventh micro-state rewritten herediffer by one individual set, the spins are the same and no permutation is needed.


1 5 1 11 2 2 1 1-1 17 2 1 5 2 2-2 1-1 1

The program determines a non-zero element FMIKR2(7,1) as the sum of thetwo products (‘PARITY’=1, B63=0 for d electrons)−CKCOE(K=4,L,ML=1,L,ML=−2)∗B43∗‘PARITY’ and−CKCOE(K=6,L,ML=1,L,ML=−2)∗B63∗‘PARITY’(and of course the first and the second term are zero, if B43 and B63 respectively

4.11. Subroutine ECKART (Wigner-Eckart) 75

are zero). The minus sign in front of CKCOE corresponds to ‘−C3’, the selectedpart of the operator (−C3 + C−3), and of course is L=2.

Matrix FMIKR2 is very often re-calculated in the case of fitting, whereasmatrix FMIKR1 is (mostly) kept and added to each new matrix FMIKR2.

4.10.4 Subroutine TANABE continued

For the first run two matrices, one for interelectronic repulsion and spin-orbitcoupling and one for ligand field energies, are always added. The resultingmatrix will be diagonalized, either directly via subroutine DSPEVW or viasubroutine BLOCUT to block-out the matrix as far as possible before callingDSPEVW for each block.

As a test, subroutine RUSSELl may be called to change basis to one ofRussell-Saunders states. This results in OUTPUT only, and there is no RE-TURN and continuation of the calculation. Subroutine REDUCTanabe is calledotherwise. The special option ‘SUBROUTINE TANFEL’ (‘WIGNER’) for felectrons (only) may be selected in REDUCTanabe, and if this is done, thissubroutine TANABE here will run without doing any relevant calculations, ifcalled again!

4.11 Subroutine ECKART (Wigner-Eckart)

For the specific case of 4f electrons with specific ground term 2S+1L, the basismay be restricted to this single LS state, or even further to the part 2S+1LJ

of this single LS state. The former possibility is especially easy to apply usingirreducible tensor operators [18, 19, 20], and a dimension of up to only 85, but 66for real examples, may occur. The latter possibility with even lower dimensionsis more complicated and not implemented here, but it might be simulated byapplying a ‘nearly infinite’ spin-orbit coupling constant within the LS state.A subroutine ECKART (‘Wigner-Eckart’), essentially different from subroutineTANABE, is included in CONDON and makes use of the method of irreducibletensor operators.

Interelectronic repulsion needs not to be considered at all in subroutineECKART.

For the ligand field operator in subroutine ECKART the Wigner-Eckarttheorem is applied. To each LS ground(!) state belongs one single state, whichis identical to one(!) micro-state with MS=S and ML=L, e.g. the micro-statewith MS=2 and ML=6 is one of the 65 degenerated states of the quintet-Istate. The ligand field operator in subroutine ECKART works firstly on thissingle micro-state like that mentioned, using function CKCOEfficient, just as inSubsection 4.10.3 for diagonal elements. However neither the product with northe summation over the Wybourne parameters B20, B40 and B60 is calculated,


CHART 9

SUBROUTINE ECKART with basis of one LS stateusing irreducible tensor operators

First run .OR. ‘new B-values’?yes no@@@

��

Set up matrix of ligand field operatorusing irreducible tensor operators

Copy this matrix 2S times, augmenting the main diagonal

Operator of spin-orbit coupling works on the former matrix

Matrix will be blocked out?yes no@@@

��

SUBROUTINE BLOCUT (see10, calls DSPEVW for each block)

SUBROUTINE DSPEVW, alink to DSPEV to diagonalize

Change of basis to one of micro-states?yes no@@@

��

SUBROUTINE CHANGE, TESTonly, no RETURN (see CHART 13)

��@@

SUBROUTINE REDUCEckart(see CHART 14)

END of SUBROUTINE ECKART, RETURN to SUSCeptibility

Figure 4.10: Flowchart 9 (SUBROUTINE ECKART)

but each sum over all electrons is stored separately and reduced according tothe Wigner-Eckart theorem with help of function DREIJ (‘3j-symbols’). Thisspecific diagonal element is chosen, because it is surely the simplest!

The Wigner-Eckart theorem states, that the matrix elements factor into theproduct of two terms, one of which expresses the geometry, symmetry, andselection rules of the system, the other of which containes the dynamics[18, 19,20]. The theorem is applied to the operator Ckq = [4π/(2k + 1)]1/2 Y kq and itsmatrix elements. These matrix elements may be reduced with help of functionDREIJ (‘3j-symbols’), and these reduced matrix elements are common factorsof all matrix elements of Ckq for all q.

Using these reduced matrix elements and the Wigner-Eckart theorem oncemore, but the other way round, the matrix elements are recovered, which leadto the diagonal element of the ligand field operator above.

Starting now with these three reduced matrix elements and using different

4.12. Subroutine BLOCkoUT 77

3j-symbols and specific phase factors, one gets all the other matrix elements ofthe ligand field operator, and a matrix is set up with all these matrix elements.

To apply the spin-orbit coupling operator, this matrix is copied 2S times,augmenting the main diagonal of the matrix, and the spin-orbit coupling oper-ator works on this matrix and modifies it corresponding to spin-orbit coupling.

The operator for spin-orbit coupling [4, 15], with λSO the spin-orbit couplingconstant for the LS state, is

HSO = λSO ∗ (Lz ∗ Sz + 1/2 ∗ (L+ ∗ S− + L− ∗ S+)).

The spin-orbit coupling constant λSO is, for the ground state and for f electrons,simply related to the spin-orbit coupling constant ZETA. The factor is NREL,if the shell is less than half filled, or minus the number of holes else.

Subroutine ECKART calls subroutine DSPEVW, a link to subroutine DSPEV,to diagonalize the matrix to get eigenvalues and eigenvectors. For test purposes(‘JATEST’), especially in connection with symmetry quantum numbers, sub-routine BLOCUT may be called to block-out the matrix before diagonalizing.

Subroutine ECKART calls subroutine REDUCEckart to evaluate FMATsand GMATs for ‘interface III’, with the same or a reduced dimension, e.g. 17 inthe case of the quintet-I state of Holmium3+.

Instead of REDUCEckart, but with no RETURN and only for test purposes(‘JATEST’), subroutine CHANGE may be called to change the basis from LSto micro-state basis.

If ‘JATEST’ was selected and for test purposes only, there is also the possi-bility of calculating other 2S+1L states and 2S+1L states of d electrons.

λSO for the LS state is evaluated, because ZETA is given or read in andnot λSO , and it is called LAMBDA, whereas ZETA is called KSI sometimes. If‘JATEST’ was selected, λSO is evaluated for the state concerned.

A Nassi-Shneiderman flowchart for subroutine ECKART is given (consecu-tively ordered as CHART 9).

4.12 Subroutine BLOCkoUT

SUBROUTINE BLOCUT to BLOCkoUT a real symmetric matrix was writtento block out the matrix to be diagonalized evaluated in subroutine TANABEor subroutine ECKART in order to save computer time. The blocks are diago-nalized one after the other, all eigenvalues are consecutively ordered then, andall eigenvectors are ordered corresponding to eigenvalues. As magnetic field wasnot yet applied, blocking-out is nearly always possible!

As a standard subroutine BLOCUT is called by TANABE for f electrons,but using options ‘JATEST’ or ‘MOROUT’, CALL of BLOCUT always may beselected or may be not selected.


CHART 10

SUBROUTINE BLOCUT to BLOCkoUT a real symmetric matrix(and for symmetry quantum numbers and spinorbital occupations)

Matrix is mapped onto LOGICAL LMAT in such a way, thatLMAT is blocked, and rearrangements are saved (in ‘REIHNF’)

Number and dimensions of blocks are determined,and BLOMAT becomes the blocked matrix

All blocks

Diagonalize the matrix

MOReOUTput selected?yes no@@@

��

All blocks

Eigenvectors and 2 times MJ (‘symm. quantum numbers’)(and, but not for LS basis:) Spinorbital occupations

Ordering of energies and eigenvectors

MOReOUTput selected?yes no@@@

��

(A choice of) eigenvectors

(Not for LS basis:) Spinorbital occupations

(d el. and cub(II)/D3d only:) A different basis?yes no@@@

��

Real spinorbitals?no yes@@@

��

New spinorbitals (d electrons only:) Real spinorbitals

END of SUBROUTINE BLOCUT, RETURN to TANABE / ECKART

Figure 4.11: Flowchart 10 (SUBROUTINE BLOCkoUT)

4.12. Subroutine BLOCkoUT 79

Option ‘MOReOUTput’ serves to select different possibilities of more outputin subroutine BLOCUT, e.g. for single blocks and/or for the whole matrix. As aspecial feature output of two times MJ for each part of an eigenvector is possible,i.e. ‘symmetry quantum numbers’ are available. In addition, but not in the caseof a LS basis, i.e. CALL by subroutine ECKART, spinorbital occupations areavailable.

A hint to ‘symmetry’ or ‘crystal quantum numbers’ [10, 17, 23]: ‘Two timesMJ’ instead of MJ has been chosen in order to have integer values. A ‘symmetryquantum number’ is defined as either MJ modulo ‘axis-No.’ (e.g. modulo 3 in thecase of a 3-fold axis) [10] or as MJ modulo q, and for q the minimum q-value inthe crystal field potential different from zero is taken [17]. The crystal quantumnumber is the number with the lowest absolute value of MJ modulo q. Statescharacterized by a different crystal quantum number cannot be mixed by thecrystal field Hamiltonian [17]. There are also other/ further crystal quantumnumbers, but these are not considered in program CONDON!

Mixing of degenerated states may often, but not always, be avoided by call-ing subroutine BLOCUT. The above treated symmetry quantum number maybe degenerated, e.g. in the case of point group D4 one of the symmetry quan-tum numbers is {±1}. If mixing of degenerated states can be avoided, mixing ofsymmetry quantum numbers as {±1} will be avoided, too, and an eigenvector ischaracterized by +1 then or by −1, and it is not the case, that both eigenvectorsare commonly characterized by {±1}. The result of calling subroutine BLO-CUT may therefore be a change of the resulting composition of eigenvectors insubroutine CHANGE.

Also, in the case of point group D4, a non degenerated symmetry quantumnumber 2 appears two times, and the difference is given by a further symmetryquantum number, which is not available here. It may happen that the basis is byaccident fully adapted to symmetry. In this case an eigenvector is characterizedby 2 and one definite further symmetry quantum number, which may have twodifferent values, but is not available here. In general this is not the case, andeigenvectors with different values of this further symmetry quantum numberwill be mixed.

If option ‘MOReOUTput’ was chosen and the basis is one of micro-states,occupation of spinorbitals may be put out. In addition occupation of real spinor-bitals may be put out for d electrons (similarity transformation to real spinor-bitals, for OUTPUT only).

A special feature applies to d electrons for point groups cubic(II) and D3d,i.e. a real or disturbed octahedron with its threefold axis as its main axis. Inorder to get a more familiar description of the occupation of spinorbitals, theymay be transformed by a similarity transformation in such a way, that the cubic[111]-direction is the direction of the new main axis. The occupation of thesenew spinorbitals in the case of cubic(II) is the same then as the occupation ofspinorbitals resulting from a calculation using cubic(I), i.e. the octahedron with


its fourfold axis as its main axis.

A Nassi-Shneiderman flowchart for subroutine BLOCkoUT is given (consec-utively ordered as CHART 10).

4.13 Subroutine RUSSELl

Subroutine RUSSELl changes the basis of the computed eigenvectors from amicro-state basis to a basis of Russell-Saunders states. All Russell-Saundersstates must be of course determined in a micro-state basis, before this basischange can be performed. Subroutine RUSSELl determines simple Russell-Saunders states and calculates the other, either using subroutine SAUNDErsor with help of subroutine GAUSS, and it correspond to 58, page 226 (chapterVIII5) of the book by Condon and Shortley [4] and to Gray and Wills [3],respectively.

Table 17 of Condon-Shortley reads for the case of three d electrons2PD(2)FGH4PF and is incorporated in and put out by subroutine RUSSELlin the following form (‘empty’ boxes <=> 0 <=> ‘no state’):

H G F D P

Doublets

Quartets

1 1 1 2 1

0 0 1 0 1

Another form of representation is the following (0 <=> ‘no state with S andL’, ‘empty’ boxes <=> neg. values of S and/or L, i.e. ‘not possible’):

L = 5 4 3 2 1 0

S = 3/2

S = 1/2

1 0 1 0

1 1 1 2 1 0

4.13. Subroutine RUSSELl 81

Including all degenerated states of all these states one has the following table(120 states at all):

ML = 5 4 3 2 1 0

MS = 3/2

MS = 1/2

1 1 2 2 2 1 1

1 2 3 5 7 7 7 5 3 2 1

1 1 2 2 2 1 1

1 2 3 5 7 7 7 5 3 2 1

Calculation starts with Russell-Saunders state 4F (3/2, 3), which is ‘simple’,because it corresponds to one micro-state. This micro-state is determined, andsubroutine SAUNDErs is called to evaluate all other (27) degenerated statesof 4F . One of these states is 4F (3/2, 1). The next state to be evaluated is4P (3/2, 1), but (3/2,1) is already occupied by 4F (3/2, 1), which has been cal-culated by SAUNDErs to be (3/5)1/2∗(2 1/2 0 1/2 −1 1/2) + (2/5)1/2∗(2 1/21 1/2 −2 1/2).

An one times one inhomogenious system of linear equations is solved by sub-routine GAUSS (i.e. the single equation (3/5)1/2∗x = −(2/5)1/2) to get the un-normalized composition of 4P (3/2, 1) orthogonal to 4F (3/2, 1) (i.e.−(2/3)1/2∗(21/2 0 1/2 −1 1/2) + 1∗(2 1/2 1 1/2 −2 1/2)), and by normalization it becomesorthonormal to 4F (3/2, 1):4P (3/2, 1) =−(2/5)1/2∗(2 1/2 0 1/2 −1 1/2) + (3/5)1/2∗(2 1/2 1 1/2 −2 1/2).

All other (11) degenerated states of 4P are again evaluated by SAUNDErs.

A new calculation starts with 2H(1/2, 5), which is also ‘simple’, because itcorresponds to one micro-state. Again subroutine SAUNDErs is called to evalu-ate all other (21) degenerated states of 2H, and one of them is 2H(1/2, 4). Nextthe state 2G(1/2, 4) is evaluated, orthonormal to 2H(1/2, 4), etc. Orthonormalto 2F (1/2, 3) are of course the two states 4F (1/2, 3) and 2G(1/2, 3)!

There are two 2D states. One of them is chosen at random, and the secondorthogonal to it. The correct two 2D states are not determined by subroutineRUSSELl.

To the very end a basis transformation is performed and the compositionof eigenvectors in LS , i.e. Russell-Saunders basis is given! OUTPUT of neweigenvectors goes to file fort.2, but OUTPUT to the terminal and to file fort.11is even more important! Especially the portion of each Russell-Saunders state aneigenvector is composed of is given, and one or more parts of each eigenvector,


corresponding to one Russell-Saunders state each, are put out.

The ground state for three free 3d electrons is the quartet-F state, but itis not a pure quartet-F state. For the vanadium atom it consists of 99.94%quartet-F state and 0.06% of both the two doublet-D states! The followingOUTPUT is given:

HINT AND WARNING: MORE THEN ONE RS STATE WITH THE SAME SPIN-AND ORBITAL-MULTIPL.: STATES ARE CHOSEN AT RANDOM!PARTIAL SUMS OF EIGENVECTOR**2, ONE FOR EACH R-S STATE:(SUM OF PARTIAL SUMS = 0.10000000E+01)CRYPTIC TABLE OF THESE R-S STATES, GREATEST (SPIN-)MULTIPL.IN FIRST ROW:IHGFDPS: 0 0 0 0 0 0 0IHGFDPS: 0 0 0 1 0 1 0IHGFDPS: 0 1 1 1 2 1 00.999388E+00 0.310336E-07 0.458771E-31 0.434653E-310.172735E-31 0.188730E-03 0.422710E-03 0.421916E-06

IF "PART OF AN EIGENVECTOR" BECOMES >0.5, OR FOR THE MAX. PARTOF AN EIGENVECTOR, THE FOLLOWING OUTPUT IS GIVEN:(ORDERED AS -MS/-ML - -MS/-ML+1 - ...)PART OF ABOVE EIGENVECTOR, CORRESP. TO R-S STATE No. 1(SUM OF SQUARED PARTS (CONTROL) = 0.99938811E+00)0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+000.000000E+00 0.000000E+00 0.755698E+00 0.000000E+000.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00

-0.534359E+00 0.000000E+00 0.000000E+00 0.000000E+000.000000E+00 0.000000E+00 0.337958E+00 0.000000E+000.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00

-0.168979E+00 0.000000E+00 0.000000E+00 0.000000E+00

Another example is the Er3+-Ion:

0.984 4I + 0.178 2K + 0.019 2L is the composition computed for the groundstate, nearly identical to the citation in [8].

A Nassi-Shneiderman flowchart for subroutine RUSSELl is given (consecu-tively ordered as CHART 11).

4.14 Subroutine REDUCTanabe

The step-up and step-down operators are managed with help of subroutineCOMPaRE, called with option 1, up to now. This subroutine could be used hereto select one micro-state out of all (KASSUM−1) micro-states. In subroutineREDUCTanabe however subroutine NUMBER will be used instead, and step-down operators for spin and orbital magnetic quantum numbers are applied

4.14. Subroutine REDUCTanabe 83

CHART 11

SUBROUTINE RUSSELl for a given number and sort of electrons,for all states of all degenerated LS states and for basis change

All S values, descending

All L values, descending

All once/ repeatedly occuring LS states, taken from given tables

Is the LS state to be created already occupiedby a state of a LS state treated before?

@@@

��yes no

SUBROUTINE GAUSS to get an orthogonal LS state

SUBROUTINE SAUNDE (see CHART 15) to constructall states of the degenerated LS state

Basis change from micro-state basis to a basis of LS states

The eigenvectors, in the new basis, are analysedwith respect to the LS states they are composed of

END of SUBROUTINE RUSSELl, no RETURN (to TANABE)

Figure 4.12: Flowchart 11 (SUBROUTINE RUSSELl)

only. These step-down operators work on spin and orbital magnetic quantumnumbers of each single electron in micro-states as given in MSMLNW, andthe result is zero or a different MSMLNW, which is easily evaluated. Theconsecutive ‘number’ of this MSMLNW is determined by subroutine NUMBERand stored intermediately, and it will be used repeatedly to give the ‘row’ of acolumn, which represents an eigenvector.

The first part of the ‘Zeeman operator’ is applied to calculate the first oneof the matrices FMAT, i.e. that for z-direction. See Section 4.16 for some hintsto ‘basis change’! Application of the second and third part should result in the4th and 5th of the matrices FMAT, respectively, those for x- and y-direction.

But instead of doing so, and because their properties are somewhat moreconvenient, the corresponding step-up and step-down operators are applied toget the second and third one of the matrices FMAT, and indeed only step-downoperators are applied to get the third one of the matrices FMAT, because thesecond one is related to the third one by symmetry.


One gets both the 4th and 5th of the matrices FMAT as linear combinationsof the second and third matrix FMAT. Because orbital reduction factors arepossible, the effectively done calculation is somewhat more complicated, seebelow!

Subroutine REDUCTanabe calculates the eight matrices FMAT and GMATwith dimensions up to the maximum, which may occur for d electrons, andwhich is 252, and a reduction of the number of eigenvalues and eigenvectorsis required for many cases of f electrons with a greater maximum or for casesof Heisenberg coupling in a dimer. High lying eigenvalues and correspondingeigenvectors may be discarded, and a new basis is always introduced consistingof the newly obtained eigenvectors to calculate FMAT and GMAT within thisnew basis.

First of all this possibly new dimension is asked for! If f electrons are con-cerned and valid only in the case of fitting, an option ‘JAFAST’ may be selectedto speed up computation in many but not in all these cases. Subroutine TAN-FEL is called then, and this implies a further change of basis and possibly afurther reduction of basis size, the basis for REDUCTanabe here the larger one.

A little more time is however needed in any case for ‘0th’ iteration, if‘JAFAST’ is selected, and for ‘a few’ electrons or a nearly full shell compu-tation is not faster anywhere! Furtheron the result of fitting has to be verified,at least if ligand field parameters change more or less during fitting.

As an example for the choice of basis sizes take the dimer of the Ho3+-Ion. The dimension of the micro state basis is 1001, to compare with theleading dimension of MSMLNW of ND2002 = 1110. If ‘JAFAST’ is selectedand ‘Combined Model’, too, (2L+ 1) ∗ (2S + 1) = 65 and (2J + 1) = 17 shouldbe taken for the intermediate dimensions, and e.g. 8 later on for ‘CombinedModel’.

A Nassi-Shneiderman flowchart for subroutine REDUCTanabe is given (con-secutively ordered as CHART 12).

The z-direction is considered firstly, i.e. FMAT( , ,1) and GMAT( , ,1),GMAT concerning only spin with a g-factor of formally one and needed forthe case of Heisenberg coupling only. FMAT includes both orbital and spinand a g-factor of two for spin. Both are evaluated with help of MQUER (seeSection 4.10), which containes the (z-components of the neg.) spin- and orbital-magnetic moments within the micro state basis, and with help of the new eigen-vectors from above. An orbital reduction factor is also included in FMAT( ,,1).

If ‘MOROUT’ or ‘JATEST’ were chosen, g-factors (g-factor = FMAT(I,I,1) /(FMAT(I,I,1) − GMAT(I,I,1))) are evaluated and given, but they may be incor-rect, if there are degenerated eigenvalues and non-diagonal elements of FMAT(, ,1) (and any orbital reduction factor should be ONE, too). These g-factors areof course for z-direction, and they may become infinite or indefinite (given as−999.0). For the special case of a twofold degenerated ground state, the correct

4.14. Subroutine REDUCTanabe 85

CHART 12

SUBROUTINE REDUCTanabe to change basis and to reduce(not always) dimension of basis

First run?yes no@@@

��

TANFEL (TANabe for F ELectrons) wanted?yes no@@@

��

INPUT of an ‘intermediate’ basis size forREDUCT and another one for TANFEL

INPUT of basissize for REDUCT

First run .OR. (new B-values AND NOT TANFEL)?yes no@@@

��

calculate FMATs and GMATsTANFEL?

yes no@@@

��

TANFEL?yes no@@@

��

SUBR.TANFEL(see CHART 16)

new B-values?yes no@@@

��

SUBR.TANFEL(see CHART 16)

SUBR.SELECT(see CHART 17)

calculateFMATs,GMATs

SUBR.SELECT(see CHART 17)

END of SUBROUTINE REDUCTanabe, RETURN to TANABE

Figure 4.13: Flowchart 12 (SUBROUTINE REDUCTanabe)

g-factor is evaluated by diagonalization of FMAT( , ,1) and a correspondingsimilarity transformation of (FMAT( , ,1) − GMAT( , ,1)). A supplementarytest on g-factors also for other cases is available in subroutine SUGANR (Sec-tion 4.20). All these g-factors are gJ -factors, i.e. g-factors in the usual senseonly, if a J value is a good quantum number for the state considered! They areindeed calculated as ‘(neg.) magnetic moment divided by angular momentum,both in atomic units’.

Step-down operators are taken from the book by Condon and Shortley [4],page 227 (eq.58(3)).

Step-down operators for spin and orbital magnetic quantum numbers worknow on micro-states as given in MSMLNW and the results are stored interme-diately. Step-up operators are never used here.

Next FMAT( , ,3) for the orbital part (only!) and GMAT( , ,3) for the spinpart within the basis of the above eigenvectors are obtained, and because of


symmetry, FMAT( , ,2) and GMAT( , ,2) have been obtained at the same time,too.

To the very end FMAT( , ,4) and FMAT( , ,5) are obtained with help ofFMAT( , ,2), FMAT( , ,3), orbital reduction factors and GMAT( , ,2) andGMAT( , ,3). One should think FMAT( , ,5), for y-direction of both spin andorbital part, as being multiplied by (−1)1/2.

FMAT( , ,2) and FMAT( , ,3) for both spin and orbital part are computedonly, if OUTPUT is desired, and no orbital reduction factors are taken intoaccount then.

As a test (‘JATEST’), and instead of Heisenberg coupling, the Ising modeland the XY model [8], respectively, may be applied, and furtheron any model‘between’ these two models. Two factors, a and b, are required as INPUT, andthe Heisenberg model results, if both factors are one. The Ising model resultsfor a=1 and b=0, and it may be modified by taking b>0, but <1. The XYmodel results for a=0 and b=1, and it may be modified by taking a>0, but <1.If both a and b are 6= 0, they are not(!) normalized. Factor a multiplies GMAT(, ,1), and factor b multiplies both GMAT( , ,2) and GMAT( , ,3).

A few micro-states of three d electrons are rewritten now (see Section 4.10)as MSMLNW, and it will be shown, how the operators work in practice.


1 5 1 11 0 0 0 1 1 0 0 0 0-12 5 -1 9 0 0 0 0 1 0 0 0-1-13 4 1 9 0 0 1 0 1 0 0 0 0-19 4 1 9 0 0 0 1 1 0 0 0-1 0

-9 4 1 9 0 0 0-1 1 0 0 0 1 0

Step-down operator for spin working on the first micro-state gives the secondmicro-state, and it works on the third electron, that is the electron most left,only. Step-down operator for orbital magnetic quantum number working on thethird electron of the first micro-state gives a factor times the third micro-state,and working on the second electron of the first micro-state, that is the electronmost right, gives a factor times a state ‘−9’, and this is −1 times micro-state 9.

All micro-states possible as result of the work of step-down operators on agiven No.OF micro-STate are evaluated in the form given above, and subroutineNUMBER is called to get the No.OF STate of the resulting micro-states. Thenumber of the micro-states and the No.OF STate of each micro-state, that resultfrom a given micro-state, are stored together with the factors above for eachgiven micro-state, and together with the information about which operator isconcerned, once and for all. Three variables, two integer and one real, are usedto store all this information and to evaluate FMAT( , ,3) and GMAT( , ,3).These three variables, one 1- and two 2-dimensional, do indeed represent thestep-down operators for each micro-state!

4.15. Subroutine CHANGE 87

Because FMAT and GMAT are evaluated not within micro-states as bas-isfunctions, but within the calculated eigenfunctions, this means a change ofbasis! Formally one would multiply the row of micro-states and the matrix ofeigenvectors and get a new row of basisfunctions, each basisfunction of this rowa linear combination of micro-states. The unit matrix would belong to thesenew basisfunctions as the matrix representing the new eigenvectors!

If FMAT and GMAT would be evaluated within the basis of micro-states asbasisfunctions, a change of basis would mean a similarity transformation withhelp of the matrix of the old eigenvectors! This would be very time consuminghowever!

Interface III, consisting of eigenvalues and FMAT( , ,1), GMAT( , ,1),FMAT( , ,4), FMAT( , ,5), GMAT( , ,2) and GMAT( , ,3) has been reached now,if(!) subroutine TANFEL is not called for another basis change. The special op-tion ‘JAFAST’ to call subroutine TANFEL (for f electrons only) may be selectedin this subroutine REDUCTanabe here, and if this is done, REDUCTanabe willrun without doing any relevant calculations, if called again!

4.15 Subroutine CHANGE

CHART 13

SUBROUTINE CHANGE for all states of one degenerated LS state,and for basis change

Comes the state to start with from SUBROUTINE ECKART?yes no@@@ �

��

INPUT the state in form of one or more (weighted) micro-states

SUBROUTINE SAUNDE to compute all states of this LS state(see CHART 15)

Was the state coming from SUBROUTINE ECKART?yes no@@@ �

��

Basis change to micro-state basis OUTPUT and STOP

The eigenvectors, in the new basis, may be analysed with respect toMJ values (‘symm. quantum numbers’)

END of SUBROUTINE CHANGE, no RETURN (to ECKART)

Figure 4.14: Flowchart 13 (SUBROUTINE CHANGE)


Subroutine CHANGE runs for tests only, it changes basis from LS to micro-state basis, and it does not return to the calling routine. Subroutine CHANGEis called by subroutine ECKART with the eigenvectors belonging to a specificterm 2S+1L, especially the ground term of an electronic configuration 4fN . Asd electrons may be treated in ECKART for test purposes, d electrons may betreated here, too.

In ECKART only 2S+1L terms containing a state, which is identical to one(!)micro-state, are possible. Subroutine CHANGE calls subroutine SCHEMA toevaluate all the micro-states of the configuration, and subroutine SAUNDE iscalled to evaluate all degenerated LS states belonging to this one LS state in amicro-state basis.

After subroutine CHANGE has changed the eigenvectors, these may be putout to file fort.2, but only if ‘MOROUT’ was chosen before.

The eigenvectors, in the new basis, may be analysed with respect to the(MJ =) MJ values of their components (‘symmetry quantum numbers’ [10, 17,23]). Corresponding to a possible ‘blocking’ of the matrix to be diagonalizedwith respect to symmetry quantum numbers, the eigenvectors may be blocked,too, and all components (of value zero) belonging to wrong MJ values are dis-carded then. If the matrix is really blocked out in subroutine BLOCUT, block-ing of the eigenvectors beyond that caused by symmetry quantum numbers maytake place. The resulting eigenvectors are written on file fort.3!

To analyse the eigenvectors in the new basis may serve as a very special testand is not recommended. MJ values of the eigenvectors in LS basis are givenin subroutine BLOCUT, if ‘MOROUT’ was chosen.

In addition there is the possibility in subroutine CHANGE to call subroutineSAUNDE to get all LS states belonging to a term containing no state, which isidentical to one(!) micro-state. This may serve as a test to control subroutineRUSSELl, which calls subroutine SAUNDE automatically to get such states.

A Nassi-Shneiderman flowchart for subroutine CHANGE is given (consecu-tively ordered as CHART 13).

4.16 Subroutine REDUCEckart

Subroutine REDUCEckart calculates the eight matrices FMAT and GMAT withdimensions up to the maximum, which may occur for f electrons in subroutineECKART, and which is 85, but for ‘real’ cases only 66. A reduction of the num-ber of eigenvalues and eigenvectors is required for cases of Heisenberg couplingin a dimer. High lying eigenvalues and corresponding eigenvectors may be dis-carded, and a new basis is always introduced consisting of the newly obtainedeigenvectors to calculate FMAT and GMAT within this new basis.

First of all this possibly new dimension is asked for! The maximum for

4.16. Subroutine REDUCEckart 89

CHART 14

SUBROUTINE REDUCEckart to change basis and to reduce(not always) dimension of basis

First run?yes no@@@

��

INPUT of ‘No. of values used later on’: same as before ormaximum for dimers (28), but (2J+1) for ‘Comb.Model’

First run .OR. ‘new B-values’?yes no@@@

��

Evaluate FMAT and GMAT in the basis of those eigenvectors,which belong to ‘No. of values’ lowest eigenvalues.

FMAT (Comp.of Zeeman oper.) & GMAT (for Heisenb.coupl.)are the ‘interface (III)’ with respect to SUBROUTINE SELECT

SUBROUTINE SELECT (see CHART 17)

END of SUBROUTINE REDUCEckart, RETURN to ECKART

Figure 4.15: Flowchart 14 (SUBROUTINE REDUCEckart)

dimers is 28, but (2J+1), if ‘Combined Model’ is selected.

The first part of the ‘Zeeman operator’ is applied to calculate the first oneof the matrices FMAT, i.e. that for z-direction. Application of the second andthird part should result in the 4th and 5th of the matrices FMAT, respectively,those for x- and y-direction.

But instead of doing so, and because their properties are somewhat moreconvenient, the corresponding step-up and step-down operators are applied toget the second and third one of the matrices FMAT, and indeed only step-upoperators are applied to get the second one of the matrices FMAT, because thethird one is related to the second one by symmetry.

One gets both the 4th and 5th of the matrices FMAT as linear combinationsof the second and third matrix FMAT. FMAT( , ,4) for x-direction is the meanvalue of FMAT( , ,2) and FMAT( , ,3). FMAT( , ,5) = one half times (-FMAT(, ,2) + FMAT( , ,3)), but one should think of FMAT( , ,5) as being multipliedby i(maginary).

Because orbital reduction factors are possible (only if ‘JATEST’ was cho-sen!), the effectively done calculation is somewhat more complicated and out-lined following the z-direction.


FMAT and GMAT for z-direction are evaluated with help of the eigenvectorsand (ML and MS =) ML and MS values belonging to these eigenvectors, eacheigenvector given as a column. A ‘real’ eigenvector is the product of a row ofkets and such a column, of course one and the same row of kets for all columns.Each ket is one of the functions of the basis, and it is defined by one ML andone MS value.

As an example take the 3F state of two d electrons: The consecutivelyordered ML/MS values are

−3/−1, −2/−1, −1/−1, 0/−1, 1/−1, 2/−1, 3/−1, −3/0, −2/0, −1/0, 0/0,1/0, 2/0, 3/0, −3/1, −2/1, −1/1, 0/1, 1/1, 2/1, 3/1,

and they define exactly twenty-one basisfunctions.

Because FMAT and GMAT are evaluated not within the just mentionedbasisfunctions, but within the calculated eigenfunctions, this means a change ofbasis! Formally one would multiply the row of kets and the matrix of eigenvec-tors and get a new row of basisfunctions, each basisfunction of this row a linearcombination of kets. The unit matrix would belong to these new basisfunctionsas the matrix representing the new eigenvectors!

If FMAT and GMAT would be evaluated within the basis of simple kets asbasisfunctions, a change of basis would mean a similarity transformation withhelp of the matrix of the old eigenvectors!

If ‘MOROUT’ or ‘JATEST’ were chosen, g-factors (g-factor = FMAT(I,I,1) /(FMAT(I,I,1) − GMAT(I,I,1))) are evaluated and given, but they may be incor-rect, if there are degenerated eigenvalues and non-diagonal elements of FMAT(, ,1) (and any orbital reduction factor should be ONE, too). These g-factors areof course for z-direction, and they may become infinite or indefinite (given as−999.0). For the special case of a twofold degenerated ground state, the correctg-factor is evaluated by diagonalization of FMAT( , ,1) and a correspondingsimilarity transformation of (FMAT( , ,1) − GMAT( , ,1)). A supplementarytest on g-factors also for other cases is available in subroutine SUGANR (Sec-tion 4.20). All these g-factors are gJ -factors, i.e. g-factors in the usual senseonly, if a J value is a good quantum number for the state considered!

For x- and y-direction only step-up operators (L+ and S+ =) L+ and S+are applied to get FMAT( , ,2) for the orbital part (only!) and GMAT( , ,2) forthe spin part, respectively, L+ and S+ working on the above eigenvectors, andbecause of symmetry, FMAT( , ,3) and GMAT( , ,3) have been obtained at thesame time, too.

To the very end FMAT( , ,4) and FMAT( , ,5) are obtained with help ofFMAT( , ,2), FMAT( , ,3), orbital reduction factors and GMAT( , ,2) andGMAT( , ,3). One should think FMAT( , ,5), for y-direction of both spin andorbital part, as being multiplied by (−1)1/2.

FMAT( , ,2) and FMAT( , ,3) for both spin and orbital part are computedonly, if OUTPUT is desired, and no orbital reduction factors are taken into

4.17. Subroutine SAUNDErs 91

account then.

As a test (‘JATEST’), and instead of Heisenberg coupling, the Ising modeland the XY model [8], respectively, may be applied, and furtheron any model‘between’ these two models. Two factors, a and b, are required as INPUT, andthe Heisenberg model results, if both factors are one. The Ising model resultsfor a=1 and b=0, and it may be modified by taking b>0, but <1. The XYmodel results for a=0 and b=1, and it may be modified by taking a>0, but <1.If both a and b are 6= 0, they are not(!) normalized. Factor a multiplies GMAT(, ,1), and factor b multiplies both GMAT( , ,2) and GMAT( , ,3).

A Nassi-Shneiderman flowchart for subroutine REDUCEckart is given (con-secutively ordered as CHART 14).

4.17 Subroutine SAUNDErs

Subroutine SAUNDErs calculates eigenfunctions using angular momentum op-erators, and it corresponds to 58, page 226 (chapter VIII5) of the book byCondon and Shortley [4] and to Gray and Wills [3], respectively.

If the eigenfunctions for (MS =) MS=S, ML=L in a micro-state basis aregiven, this method makes use of the step-down operators for MS and ML andfor (ms =) ms and ml.

Firstly all states with the same ML=L value are computed, i.e. states withMS=S−1, , MS=S−2, etc. Then for each MS value all states with this sameMS value but a different ML value are computed, i.e. states with ML=L−1,ML=L−2, etc. See the Nassi-Shneiderman flowchart for subroutine SAUNDE(consecutively ordered as CHART 15). For details see the book by Condon andShortley!

OUTPUT goes to files fort.1 and fort.2, and in special cases also to file fort.3!

4.18 Subroutine TANFEL (TANabe for F ELec-trons)

Subroutine TANFEL is a supplement to subroutines TANABE (‘TANabe forF ELectrons’) and REDUCTanabe for the cases of f electrons (‘F ELectrons’)in a full basis of very many micro-states, and there is an option ‘JAFAST’ toselect TANFEL in order to save time when iterating in a fitting procedure.In REDUCTanabe the eigenvectors found by TANABE were taken as a newbasis, with the dimension of the basis generally lowered, but each iterationwith different B-values starts in subroutine TANABE again with the number ofmicro-states as dimension of the basis.

Therefore eigenvectors, but generally not all of them, eigenvalues, FMATs


CHART 15

SUBROUTINE SAUNDE for all states of one degenerated LS state

All MS values, descending, starting with MS = S value minus 1

State with the same ML=L value is computed with help ofstep-down operators out of the foregoing state

All MS-values, descending, starting with MS = S value

All ML-values, descending, starting with ML = L value minus 1

State is computed with help of step-down operatorsout of the foregoing state with the same MS value

END of SUBROUTINE SAUNDE, RETURN to RUSSELl / CHANGE

Figure 4.16: Flowchart 15 (SUBROUTINE SAUNDE)

and GMATs of the first run are saved to be used later on in subroutine TANFEL,and TANABE will really run only once! A little drawback is that results haveto be verified in an extra run of CONDON, but that does not matter very much.Furtheron even a little more time is needed for the zeroth iteration, if ‘JAFAST’is selected. Lastly computation speeds up only for the cases of more than a fewand less than nearly all electrons!

As an example take the Ho3+ Ion with a basis of dimension 1001, and nine B-values are fitted. After the ‘0-th’ iteration TANFEL takes into account only thedifferences of the new and the old B-values and evaluates matrices of the partsof the ligand field operator of only dimension 65 (or even 17) in the basis givenin REDUCTanabe (Section 4.14). The matrix elements of the Ckq -operatorsof the ligand field operator in the new basis are determined by subroutinesMATRIV and MATRIW, for q=0 and q 6=0, respectively. The latter operatorsare regarded as (several-)step operators working on micro-states, and with helpof subroutine NUMBER a procedure similar to that in 4.14 is used. The sumof all these matrices and of the diagonal matrix, with the same dimension,of the lowest lying eigenvalues is calculated. Subroutine DSPEVW, a link tosubroutine DSPEV, is called to diagonalize this matrix to get new eigenvaluesand new eigenvectors to serve as a new basis.

A similarity transformation with help of the new eigenvectors is appliedto the FMATs and GMATs by subroutine TRANSF, corresponding to a basischange of the FMATs and GMATs to the new basis of the new eigenvectors. The

4.19. Subroutine SELECT 93

CHART 16

SUBROUTINE TANFEL (TANabe for F ELectrons)to save time when iterating (basis change)

First run?yes no@@@

��

Hold eigenvectors, eigenvalues, FMATs and GMATs

New B-values?yes no@@@

��

All new B-values, using differences only

Set up matrix of reduced size of correspondingparts of the ligand field operator

Add the diagonal matrix of eigenvalues (of reduced size)and the foregoing matrices

SUBROUTINE DSPEVW, a link to DSPEV to diagonalizethe matrix for (new) eigenvalues and (new) eigenvectors

SUBROUTINE TRANSF for a similarity TRANSFormation (basischange) of FMATs and GMATs with help of the foregoing eigenvectors

SUBROUTINE SELECT (see CHART 17)

END of SUBROUTINE TANFEL, RETURN to REDUCTanabe

Figure 4.17: Flowchart 16 (SUBROUTINE TANFEL)

size of the basis may be reduced once more, e.g. a reduction takes place from65 to 17. Subroutine SELECT is called with the new eigenvalues and the newFMATs and GMATs and ‘interface III’ has been reached. A Nassi-Shneidermanflowchart for subroutine TANFEL is given (consecutively ordered as CHART16).

4.19 Subroutine SELECT

Subroutine SELECT selects in the simplest case, i.e. only z-direction and a sin-gle ion (not a dimer), subroutine SUGANReal, and this subroutine only once.If y-direction is involved, subroutine SUGANComplex is used instead of SUG-ANR! For the most complicated case nine (= 3 times 3) calls of subroutines


CHART 17

SUBROUTINE SELECT

Hint: ‘Interface III’ is here: Eigenvalues with correspondingFMATs and GMATs!

CHI-values needed (for mean value, ‘Comb.Model’, . . . )

IOPT99 related to direction(s) of magnetic fieldand to single ion/ dimer/ ‘Comb.Model’

leads to CALL of SUBR.

1 2 3 4 5 6 7 8

XXXXXXXXXXXXXXXXXXXXXXX

��

SUGANSUGANSUGANSUGANSUGANSUGANSUGANSUGANR R H C C K R C

(EIVAL0(1,3))

(EIVAL1(1,3))

(EIVAL2(1,3))

(EIVAL2(1,3))

Hint: ,3) <=> y-direction! EIVAL0/1/2 have different dimens.(For SUGANR/C/H/K see CHARTs 18/19/20/21)

If more than one CHI, calculate mean value, extrapolated value, . . .

CHI serves as CHISTArt to calculate 28 CHI( , , )-values at all,and 28 further values are calculated starting with CHI-measured

END of SUBROUTINE SELECT, RETURN to REDUCT,to REDUCE or to TANFEL

Figure 4.18: Flowchart 17 (SUBROUTINE SELECT)

named SUGAN. . . occur to compute mean values and ‘extrapolated’ valuesfor ‘z- ,x- and y-direction’ and the values needed for ‘Combined Model’ ! Thereare four different subroutines SUGAN. . . , namely SUGANReal, SUGANCom-plex, SUGANHeisenberg and SUGANK(omplexH)eisenberg), and each of themmay be called with a different direction of magnetic field (DI) and a differentnumber of eigenvalues (NR) to be used. They are called with EIVAL0(NR,DI),EIVAL1(NR,DI) and EIVAL2(NR,DI), respectively, in order that the eigenval-ues can be stored there which are evaluated or set equal to the ‘old’ eigenvalues inthe case of ‘VANVLECK’. (DIMENSION EIVAL0(ND28,3), EIVAL1(ND28,3),EIVAL2(ND1110,3))

One CHI value is the result of each CALL. This CHI value becomes theCHI value CHISTA (‘CHI-START’), if there is only one CHI value. Other-wise CHI-START is calculated as a mean value and/or ‘extrapolated’ value,

4.19. Subroutine SELECT 95

the latter in the case of ‘Combined Model’. A special ‘mixing parameter’ (seeSubsection 4.5.1), which may also work here, leads to a further mean value.Furthermore, in the case of more than one direction of the magnetic field, CHIfor each of the directions is stored in CHIZXY.

If a dimer is calculated, but not with ‘Combined Model’, i.e. if IOPT2=1,two different sorts of dimers may be specified in subroutine SELECT, eithera symmetrical dimer or a dimer with one spin-one-half ion (e.g. with Cu2+,gS(pin-Hamiltonian)-factor(s) near 2.1 must be put in for Cu2+). A g-factor oftwo is appropriate for a pure spin-one-half ion, and this is the same as gS(pin-Hamiltonian)=g/(g-1) in this case. If there is some orbital contribution in addi-tion to the spin, this is usually allowed for by a gS-factor different from two, andoften, appropriate to symmetry, by different gS-factors for different directionsof the space. CHI values etc. for dimers are calculated for the sum of both ionsin the case of an unsymmetrical dimer. If a dimer with one spin-one-half ionis specified (option JACuGd), the corresponding calculation is done in subrou-tine SUGANH or in subroutine SUGANK, but the nine matrices FMATCU,GMATCU and WERTCU (WERTCU is twice zero) for the spin-one-half ionare set up here, 3 of the 5 2∗2-matrices FMATCU are multiplied by half therespective gS-factor (2 are not used anywhere), and the 3 matrices GMATCUare multiplied by factors for the Ising model or the XY model.

A Nassi-Shneiderman flowchart for subroutine SELECT is given (consecu-tively ordered as CHART 17).

4.19.1 CHI-START and its corrections

The calculation of each CHI value starts with the calculation of CHISTA (‘CHI-START’). There are three possible corrections in program CONDON, and allof them are always applied, even if they are zero:

CHI−calc. = (((CHI−START)−1 − λMF )−1 + CHI−0)/F.

CHI-START, CHI-calc. and the intermediate results are stored in CHI(1,1...4,1),i.e. CHI-START = CHI(1,1,1) and CHI-calc. = CHI(1,4,1).

Exactly in reversed order CHI-meas. is always corrected, even if the correc-tions are zero:

CHI(2, 1, 1) = (((CHI−meas. ∗ F)− CHI−0)−1 + λMF )−1.

The results are stored in CHI(2,1...4,1), i.e. CHI-meas. = CHI(2,4,1).

If measured and calculated values are to be compared, always CHI(2,J,K) isto be compared with CHI(1,J,K), and if calculated values are fitted to measuredvalues, CHI(1,J,K) is fitted to CHI(2,J,K)!

CHI stands for any of the eight values CHI(1...2,1...4,1)! Out of each of theseeight ‘CHI IN SI UNITS’-values six further values are calculated and stored inCHI( , ,2...7):


ii) 1/CHI, iii) (e.g. saturation)MAGN.MOMENT in µB , iv) µeff=MU EFFEC-TIVE, v) CHI∗(field/Teslas), vi) CHI IN CGS, vii) MU EFF.(THETA) (i.e. afurther µeff ).

CHI( , , ) stands for any of these 56 values!

Instead of the molecular field parameter λMF the Weiss constant Θ may betaken (λMF is then formally zero above!). The ‘further µeff ’ is calculated with(T −Θ) instead of T . If (T −Θ) becomes lower than zero, CHI-START becomeszero and is excluded from fitting!

Since molecular field calculations may be done, the corresponding molecularfield parameter is strictly to be separated from λMF above within programCONDON, and for such calculations λMF , Θ and CHI-0 have to be zero!

Two of the CHI( , , ) values evaluated here, say CHI(1,IOPT31,K) andCHI(2,IOPT31,K), are used for fitting. IOPT31 may be 4 or 2, i.e. IOPT31 = 4− 2∗IOPT3, and K may be 1, 2, 3 or 4, i.e. = IOPT4. IF CHI-0 is not involved,IOPT3 is 0 automatically and IOPT31 is always 4. If CHI(1,IOPT31,1) orCHI(2,IOPT31,1) are lower than zero, they are excluded from fitting and markedwith ‘01’, meaning ‘NOT FITTED (FROM CALC.)’ !

For the ordering of the many possibilities of different CHI values see also 4.4.1!

All and exactly the same CHI( , , ) values are evaluated once more forOUTPUT, if ‘more’ OUTPUT was chosen, and if the OUTPUT routine is calledand the fitting routine had succeeded in finding a ‘better’ SQX value in the lastrun, or this was the first run. Otherwise in the OUTPUT routine ‘RESTBL’(see 4.4.1) ‘old’ CHI( , , ) values are re-calculated for OUTPUT with help ofCHIFIX, which is the relevant ‘old’ CHISTA.

4.20 Subroutine SUGANReal

Subroutine SUGANR, ‘R’ for ‘Real’, takes the eigenvalues and matrix FMAT(, ,1) for z-direction from the ‘interface (III)’ and sets up a perturbation matrixPERTUR equal to the direction cosinus of the magnetic field for z-directiontimes FMAT( , ,1). If x-direction is also involved, FMAT( , ,4) times directioncosinus for x-direction is added.

If formula of Van Vleck was not selected, a matrix will be set up with help ofmagnetic field and eigenvalues, which has to be diagonalized to get eigenvectorsand new eigenvalues. The new matrix is essentially a diagonal matrix of the (old)eigenvalues with small supplements corresponding to PERTUR times magneticfield, and the matrix is diagonalized by subroutine DSPEVW and DSPEV, theformer only being a link to DSPEV.

In the case of ‘Van Vleck’ the old eigenvalues become the new eigenvalues,and the matrix of eigenvectors becomes the unit matrix.

4.20. Subroutine SUGANReal 97

CHART 18

SUBROUTINE SUGANReal for z- and/or x-direction(s)

Only temperature has changed?yes no@@@

��

Called by SUGANH?yes no@@@

��

Using FMAT and direction of magnetic field,set up perturbing matrix PERTUR

Formula of VAN VLECK?no yes@@@

��

Using PERTUR, magnetic fieldand eigenvalues, set up (real)

matrix to be diagonalized Matrix ofeigenvectors becomes

the unit matrixSUBROUTINE DSPEVW, a link toDSPEV to diagonalize the matrix

All eigenvectors

Matrix PERTUR is multiplied by the left and by theright with one eigenvector to get one (neg.) magneticmoment resp. one Zeeman coefficient of first order

Formula of VAN VLECK?yes no@@@

��

2nd order Zeeman coefficients are evaluated and possiblycorrections to the squares of the coefficients of 1st order

SUBROUTINE CHIMAG (see CHART 22) to evaluate CHI

END of SUBROUTINE SUGANR, RETURN to SELECT / SUGANH

Figure 4.19: Flowchart 18 (SUBROUTINE SUGANReal)


Subroutine SUGANR may have been called by subroutine SUGANHeisen-berg, and perturbation matrix PERTUR had been evaluated there already andthe matrix to be diagonalized, too.

A new calculation with only magnetic field changed starts in SUGANR (or inSUGANH), and if only temperature has changed, SUGANR runs only formally,but it does not calculate anything!

SUGANR calculates the (components of the neg.) magnetic moments withinthe new basis with help of matrix PERTUR, which is multiplied by one eigen-vector from the right and from the left to get one magnetic moment or, in thecase of ‘Van Vleck’, one Zeeman coefficient of first order.

In the case of ‘Van Vleck’ also second order Zeeman coefficients have to beevaluated and, if there are degenerated eigenvalues, possibly corrections to thesquared Zeeman coefficients of first order. The method of Gerloch [24] will beapplied to get these corrections. Evaluation of second order Zeeman coefficientsmay be suppressed for special tests, if JATEST was chosen, but the method ofGerloch may become wrong then.

SUGANR calls subroutine CHIMAG to calculate CHI and returns to sub-routine SELECT or to subroutine SUGANHeisenberg.

A Nassi-Shneiderman flowchart for subroutine SUGANReal is given (con-secutively ordered as CHART 18).

A special test on g-factors is available in subroutine SUGANR, if (MO-ROUT.AND.JATEST) are chosen. The g-factor for the ground state is evalu-ated by diagonalization of the correct linear combination, for the chosen direc-tion, of FMAT’s and a corresponding similarity transformation of the correctlinear combination of (FMAT − GMAT)’s, also for a ‘matrix’ of dimension one,but of course subroutine SUGANReal must be reachable, i.e. y-direction is notinvolved. ‘STOP’ follows this test!

4.21 Subroutine SUGANComplex

Subroutine SUGANC, ‘C’ for ‘Complex’, takes the eigenvalues and matricesFMAT( , ,1), FMAT( , ,4) and FMAT( , ,5) for z-direction, x-direction andy-direction, respectively, from the ‘interface (III)’ and sets up a perturbationmatrix CPERTU (‘ComplexPERTU’) equal to the sum of direction cosines ofthe magnetic field times FMAT (FMAT( , ,5) times i(maginary)!).

If formula of Van Vleck was not selected, a complex hermitian matrix will beset up with help of magnetic field and eigenvalues, which has to be diagonalizedto get eigenvectors and new eigenvalues. The hermitian matrix is essentially adiagonal matrix of the (old) eigenvalues with small supplements correspondingto CPERTU times magnetic field, and the matrix is diagonalized by subroutineZHPEVW and ZHPEV, the former only being a link to ZHPEV.

4.21. Subroutine SUGANComplex 99

CHART 19

SUBROUTINE SUGANComplex for a direction including y-direction


��

Called by SUGANK?yes no@@@

��

Using FMAT and direction of magnetic field,set up perturbing matrix ComplexPERTU

Formula of VAN VLECK?no yes@@@

��

Using CPERTU, magnetic fieldand eigenvalues, set up hermitian

matrix to be diagonalized Complex matrix ofeigenvectors becomes

the unit matrixSUBROUTINE ZHPEVW, a link toZHPEV to diagonalize the matrix

All eigenvectors

Matrix CPERTU is multiplied by the left and by theright with one eigenvector to get one (neg.) magneticmoment resp. one Zeeman coefficient of first order


��

2nd order Zeeman coefficients are evaluated and possiblycorrections to the squares of the coefficients of 1st order

SUBROUTINE CHIMAG (see CHART 22) to evaluate CHI

END of SUBROUTINE SUGANC, RETURN to SELECT / SUGANK

Figure 4.20: Flowchart 19 (SUBROUTINE SUGANComplex)


In the case of ‘Van Vleck’ the old eigenvalues become the new eigenvalues,and the (complex) matrix of eigenvectors becomes the unit matrix.

Subroutine SUGANC may have been called by subroutine SUGANK (‘SUG-ANKomplexHeisenberg’), and perturbation matrix CPERTU had been evalu-ated there already and the matrix to be diagonalized, too.

A new calculation with only magnetic field changed starts in SUGANC (or inSUGANK), and if only temperature has changed, SUGANC runs only formally,but it does not calculate anything!

SUGANC calculates the (components of the neg.) magnetic moments withinthe new basis with help of matrix CPERTU, which is multiplied by one eigen-vector from the right and from the left (indeed by the conjugated eigenvectorfrom the left!) to get one magnetic moment or, in the case of ‘Van Vleck’, oneZeeman coefficient of first order.

In the case of ‘Van Vleck’ also second order Zeeman coefficients have to beevaluated and, if there are degenerated eigenvalues, possibly corrections to thesquared Zeeman coefficients of first order. The method of Gerloch [24] will beapplied to get these corrections. Evaluation of second order Zeeman coefficientsmay be suppressed for special tests, if JATEST was chosen, but the method ofGerloch may become wrong then.

SUGANC calls subroutine CHIMAG to calculate CHI and returns to sub-routine SELECT or to subroutine SUGANK.

A Nassi-Shneiderman flowchart for subroutine SUGANComplex is given(consecutively ordered as CHART 19).

4.22 Subroutine SUGANHeisenberg

Subroutine SUGANH, ‘H’ for ‘Heisenberg (real)’, takes the eigenvalues and ma-trices FMAT( , ,1) and FMAT( , ,4) from the ‘interface (III)’ and sets up a(first) perturbation matrix PERTUR just as subroutine SUGANR does. Butthis is not the final matrix PERTUR, which has the squared dimension of thefirst. In the final matrix the first matrix appears along the main diagonal untilthe squared dimension is reached (this is simply the Kronecker product of theunit matrix and the first matrix!). Furtheron the Kronecker product of the firstmatrix and the unit matrix is added to get the final matrix PERTUR. (Notethe importance of the ordering of the two parts of a Kronecker product!)

Of course the dimension of the ‘first’ perturbation matrix must be relativelysmall, and it is limited to ND28 (=28). The final matrix has dimension ND1110(=1110), a little more then ND2828 (=28∗28). Indeed the ‘first’ perturbationmatrix does not really occur here. In the case of ‘Combined Model’ storagelimited to dimension ND28 is used for the single-ion part, too!

A further matrix will be set up with help of magnetic field (formally zero

4.22. Subroutine SUGANHeisenberg 101

CHART 20

SUBROUTINE SUGANHeisenberg for z- and/or x-direction(s)


��

Using FMAT and direction of magnetic field,set up perturbing matrix PERTURUsing GMAT and parameter of Heisenberg coupling and, formagnetic dipol-dipol interaction, FMAT and the distanceof the two ions, and using the eigenvalues, set up (first partof) real matrix to be diagonalized


��

Add PERTUR times magnetic field to the matrixto be diagonalized

SUBROUTINE SUGANReal (see CHART 18)

END of SUBROUTINE SUGANH, RETURN to SELECT

Figure 4.21: Flowchart 20 (SUBROUTINE SUGANHeisenberg)

in the case of ‘VANVLECK’) and eigenvalues, which has to be diagonalizedto get eigenvectors and new eigenvalues. The new matrix is essentially a di-agonal matrix corresponding to the (old) eigenvalues with small supplementscorresponding to PERTUR times magnetic field and to the matrix elements ofHeisenberg coupling (and possibly magnetic dipol-dipol interaction). (‘Small’supplements only, if magnetic field and parameter of Heisenberg coupling aresmall!) The ‘corresponding’ part of the main diagonal is again the sum of twoKronecker products: The Kronecker product of the unit matrix and a matrixwith a main diagonal of the old eigenvalues, and the Kronecker product of amatrix with a main diagonal of the old eigenvalues and the unit matrix.

GMAT( , ,1...3) is used for the matrix elements of Heisenberg coupling (andFMAT( , ,1), FMAT( , ,4) and FMAT( , ,5) for magnetic dipol-dipol interac-tion [10]), see also Subsection 2.5.2 for Heisenberg coupling. GMAT( , ,1...3)may have been multiplied by factors in subroutine REDUCT(anabe) or RE-DUCE(ckart) in order to apply the Ising model or XY model or any model‘between’ these two models instead of Heisenberg coupling.

The Heisenberg operator works on a basis of product states, one factor of the


product for one of the two centres. The resulting matrix is a sum of three matri-ces, and each of these matrices is, apart from a common factor for each matrix,a Kronecker product involving GMAT( , ,1...3), namely GMAT( , ,1)⊗GMAT(, ,1), GMAT( , ,2)⊗GMAT( , ,3), and GMAT( , ,3)⊗GMAT( , ,2), respectively.

If in subroutine SELECT a dimer with one spin-one-half ion (e.g. with Cu2+)was specified, the corresponding calculation is done here in subroutine SUGANHor in subroutine SUGANK, and the dimension of the ‘first’ perturbation matrixis limited to NDS66 (e.g. NDS66=66).

The matrix has to be diagonalized, also of course in the case of ‘Van Vleck’.For this diagonalization subroutine SUGANR is called!

A new calculation with only magnetic field changed starts in SUGANH (or inSUGANR), and if only temperature has changed, SUGANH runs only formally,but it does not calculate anything!

A Nassi-Shneiderman flowchart for subroutine SUGANHeisenberg is given(consecutively ordered as CHART 20).

4.23 Subroutine SUGANK(omplexH)eisenberg

Subroutine SUGANK, ‘K’ for ‘K(omplexH)eisenberg’, takes the eigenvalues andmatrices FMAT( , ,1), FMAT( , ,4) and FMAT( , ,5) from the ‘interface (III)’and sets up firstly a (complex) perturbation matrix C(omplex)PERTU justas subroutine SUGANC does for CPERTU. But this is not the final matrixCPERTU, which has the squared dimension of the first. The procedure to getCPERTU is essentially the same as in subroutine SUGANH for PERTUR andmay be found there!

Of course the dimension of the ‘first’ perturbation matrix must be relativelysmall, and it is limited to ND28 (=28). The final matrix has dimension ND2828(=28∗28), but because it is complex, storage of 2∗28∗28 = 1229312 is needed(1110*1110 = 1232100). COMPLEX*16 (corresponding to ‘DOUBLE PRECI-SION’) is used generally and also for CPERTU, although this is an extension tostandard FORTRAN. Most software offers this extension, however, and ‘SIN-GLE PRECISION’ (corresponding to COMPLEX*8) would produce results tooinaccurate.

Indeed the ‘first’ perturbation matrix does not really occur here. In thecase of ‘Combined Model’ storage limited to dimension ND28 is used for thesingle-ion part, too!

The matrix to be diagonalized is complex hermitian, but in the case of‘Van Vleck’ it is only formally complex, i.e. the imaginary part of the complexmatrix is zero. For the diagonalization of the matrix subroutine SUGANC iscalled, although SUGANR would be possible in the case of ‘Van Vleck’ ! MatrixCPERTU is ‘really’ complex, if y-direction is involved, otherwise the imaginary

4.24. Subroutine CHIMAG 103

CHART 21

SUBROUTINE SUGANK for ‘Heisenberg’, including y-direction


��

Using FMAT and direction of magnetic field,set up perturbing matrix ComplexPERTUUsing GMAT and parameter of Heisenberg coupling and, formagnetic dipol-dipol interaction, FMAT and the distanceof the two ions, and using the eigenvalues, set up (first partof) hermitian matrix to be diagonalized


��

Add CPERTU times magnetic field to the matrixto be diagonalized

SUBROUTINE SUGANComplex (see CHART 19)

END of SUBROUTINE SUGANK, RETURN to SELECT

Figure 4.22: Flowchart 21 (SUBROUTINE SUGANK(omplexH)eisenberg)

part of the complex matrix is zero.

IF in subroutine SELECT a dimer with one spin-one-half ion (e.g. withCu2+) was specified, the corresponding calculation is done here in subroutineSUGANK or in subroutine SUGANH, and the dimension of the ‘first’ pertur-bation matrix is limited to NDS66 (e.g. NDS66=66).

A new calculation with only magnetic field changed starts in SUGANK (or inSUGANC), and if only temperature has changed, SUGANK runs only formally,but it does not calculate anything!

A Nassi-Shneiderman flowchart for subroutine SUGANK(omplexH)eisenbergis given (consecutively ordered as CHART 21).

4.24 Subroutine CHIMAG

Subroutine CHIMAG to calculate CHI (and MAGnetisation) is called by subrou-tine SUGANR or by subroutine SUGANC. A calculation with only temperature


CHART 22

SUBROUTINE CHIMAG to calculate CHI (and MAGnetisation)

Hint: ‘Interface IV’ is here: Eigenvalues with correspondingmagnetic moments!

Calculate CHI for the temperature given,for one direction of the magnetic field

END of SUBROUTINE CHIMAG, RETURN to SUGANR / SUGANC

Figure 4.23: Flowchart 22 (SUBROUTINE CHIMAG)

changed starts here and uses eigenvalues and corresponding (neg.) magnetic mo-ments, and an ‘Interface IV’ may be seen here! In the case of ‘Van Vleck’ thereare Zeeman coefficients of first order instead of magnetic moments and Zeemancoefficients of second order in addition.

One specific direction of the magnetic field is included in each call, also inthe case of ‘Van Vleck’. Up to three different directions may belong to onetemperature, and correspondingly up to three sets of eigenvalues etc. belong to‘Interface IV’, indeed up to nine sets in the case of ‘Combined Model’. BecauseCHIMAG is called by subroutine SUGANR or subroutine SUGANC with thecorrect sets, different storage for these selected in subroutine SELECT, thecorrect case is always selected. If only temperature has changed, regardless ofe.g. a change of the direction of the magnetic field, calculation of CHI startshere in CHIMAG.

Because the definition of the mol is based on one magnetic centre, CHI valuescalculated for dimers are divided by two, except in the case of unsymmetricaldimers.

Because of the importance of the direction of the magnetic field, subroutineCHIMAG calculates not only CHI IN SI values, but also CHI IN CGS val-ues, CHI∗FIELD, MAGN.MOMENT, and (PSEUDO) EFF.MAGN.MOMent.‘MOReOUTput’ may be selected to have a look at all these values. They arealso in principle available in subroutine SELECT, but cannot be seen there ex-cept in the case of debugging. Really used in subroutine SELECT or elsewhereis only CHI IN SI.

If ‘JATEST’ is chosen in addition to ‘MOReOUTput’, extra OUTPUT ofup to 21 values shows the evolution of CHI, if only the first, the first and thesecond, . . . eigenvalue are included in the calculation.

A Nassi-Shneiderman flowchart for subroutine CHIMAG is given (consecu-tively ordered as CHART 22).

5 Examples

5.1 Single ions

5.1.1 Test calculation (s1 system)

To get the molar paramagnetic susceptibilities of a hydrogen atom (a s1 system)is very simple, because there is only one electron spin and no orbital contribu-tion, and this shall serve as a first example of calculations with CONDON!

In Section 3.5 the structure of program CONDON is demonstrated with helpof just this example her!

Not only for the purpose of such a test calculation the program contains atable with a lot of different temperatures and the corresponding paramagneticsusceptibilities of hydrogen, indeed those derived from

µeff = g−factor ∗ (1/2 ∗ (1/2 + 1))1/2µB = 31/2µB

and from Curie’s law CHI = C/Temperature with C = molar Curie constant =µ0NAµ

2eff/(3kB). A magnetic field of 0.1 Tesla enters the calculation by CON-

DON. Because saturation occurs for very low temperatures, there will be adifference between the susceptibilities given in the table and those calculatedby CONDON for such low temperatures. Indeed is µsat = 1µB smaller thanµeff = 31/2µB .

All INPUT comes from terminal, because the just mentioned table is usedand an ‘empty’ list of parameters, both selected by twice INPUT of ‘h’.

Because all INPUT from terminal is written on file fort.7 (and nothing elseis written on file fort.7, except ”NO WARRANTY”), the content of this file orsuch a file may serve as INPUT, is given here as a table and commentated!

OUTPUT is written on terminal and on file fort.11, and the latter OUTPUTis nearly identical to the former. Furthermore there is OUTPUT to files fort.7,fort.8 and fort.1, the latter a copy of a part of fort.11 to be used for plots. The‘,less,less,less’ in the table means less OUTPUT of intermediate values and oftext and less possibilities of tests, and therefore less INPUT is required. Anycombination of the digits 0 (‘standard’), 1 (‘more’) and 9 (‘less’) is possible!

A small part of file fort.11 is given here (‘less OUTPUT’ results in OUTPUTcorresponding to ‘fitting takes place’):

AFTER 0TH ITER.: OMEGA,SQX-FIX,SXR=

105

106 Chapter 5. Examples

Table 5.1: File fort.7 – s1 system

INPUT (= . . . not to put in!) Commenttext at will ”NO WARRANTY”0999 =KLWERT, . . . , . . . , . . . s el.,less,less,less23 =NGROUP (point group) ”point” group O(3), not needed0 =IOPT2 (”DIMER”) single ion or atom/ not a dimerh (OF CHI ..) for table of temp.’s, CHI’s & fieldsh (BWYBIN) for list of parameters (all 0.)0 =JASTOQ(2=STOP) no STOP: get table of OUTPUT

0.00E+00 0.45258159E-01 0.1454E+00

"FITTED"=+-1; -2/-1<=>8/9 IN INPUT FILE:0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

LIST(I), BWYB(LIST(I))=

(LAUFNR.EQ.ITMAX): LAUFNR, ITMAX= 0 0


0 =JASTOQ(2=STOP) ! ! ! ! ! ! ! ! !

HINT: OUTPUT CORRESPONDS TO "FITTING TAKES PLACE"SQX, CFMS, DEGEN.1, DEGEN.2, CFOS, No. OF VAL., CFOSJ, NRVALJ =0.4526E-01 0.0000E+00 2 0 0.0000E+00 2 0.0000E+00 2


No. 00 TEMP/K SIGMA B/TESLA MEASURED CHI CALCUL. CHI1 00 300.00 1.000000 0.100 0.15714051E-07 0.15714038E-072 00 250.00 1.000000 0.100 0.18856861E-07 0.18856845E-073 00 200.00 1.000000 0.100 0.23571076E-07 0.23571056E-074 00 150.00 1.000000 0.100 0.31428101E-07 0.31428074E-075 00 100.00 1.000000 0.100 0.47142152E-07 0.47142107E-076 00 90.00 1.000000 0.100 0.52380169E-07 0.52380117E-07

...57 00 1.00 1.000000 0.100 0.47142152E-05 0.47071342E-0558 00 0.90 1.000000 0.100 0.52380170E-05 0.52283088E-0559 00 0.80 1.000000 0.100 0.58927689E-05 0.58789554E-0560 00 0.70 1.000000 0.100 0.67345933E-05 0.67139931E-0561 00 0.60 1.000000 0.100 0.78570250E-05 0.78243582E-0562 00 0.50 1.000000 0.100 0.94284304E-05 0.93721090E-0563 00 0.45 1.000000 0.100 0.10476034E-04 0.10398907E-04

5.1. Single ions 107

64 00 0.40 1.000000 0.100 0.11785538E-04 0.11675982E-0465 00 0.35 1.000000 0.100 0.13469187E-04 0.13306211E-0466 00 0.30 1.000000 0.100 0.15714050E-04 0.15456604E-0467 00 0.25 1.000000 0.100 0.18856861E-04 0.18415814E-0468 00 0.20 1.000000 0.100 0.23571076E-04 0.22723043E-0469 00 0.15 1.000000 0.100 0.31428100E-04 0.29483177E-0470 00 0.10 1.000000 0.100 0.47142151E-04 0.41134093E-0471 00 0.05 1.000000 0.100 0.94284303E-04 0.61233501E-04

THE TABLE PUT OUT HERE ALSO PUT OUT TO FILE fort.1!1 ELECTRON(S), 0 =l VALUE, 2 BASIS STATES,

2 VALUES USED LATER ON,

"LESS OUTPUT" OR "JASTOP" WAS CHOSEN: STOP

0

10−5

2× 10−5

3× 10−5

4× 10−5

5× 10−5

6× 10−5

7× 10−5

8× 10−5

9× 10−5

0.0001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

χm

ol/

(m3m

ol−

1)

T/Kelvin

’H.plot’ using 3:7’H.plot’ using 3:7

rsrsrsrsrs

rsrs

rsrs

rs

rs

rs

rs

rs

rs

rs

’H.plot’ using 3:6

++++++++++

++

+

+

+

+

Figure 5.1: CHI in SI units of a s1 system at low temperatures

A SQX value of 4.53% for the quality of a ‘fit, i.e. in 0th iteration’, iscalculated!

Using the Van Vleck formula results in a very small value of SQX (0.003%),and no magnetic field enters the calculation (but 0.1 Tesla is taken for themagnetisation).

5.1.2 Ti3+[3d1] (D3d)

Ti3+[3d1] in caesium titanium alum is surrounded by six water molecules inthe form of a distorted octahedron, and the site symmetry, but only down to


12 Kelvin, is D3d, although the alum itself is cubic. Susceptibility data areavailable from [29], and a fitting procedure will be given here. ‘2999’ instead of‘2000’ or ‘2110’ is chosen, ‘2’ for d electron(s) and ‘999’ to get a minimum ofOUTPUT.

A maximum of 99 ‘iterations in a fitting procedure’ is chosen here, and B20,B40 and B43 are marked as to be fitted. To get starting values for the latter, thevalue of ∆O = 20300 cm−1 is used, which is known experimentally ([5]). It is tobe multiplied by 2.1 ∗ (−2/3) to get B40 = −28420 for point group ‘cubic(II)’and the corresponding B43 = 33968. A value for B20 = −20000 is got by trialand error! Furtheron a row with ∆O, marked with 0 as to be fitted and with−10 as the tenth energy niveau, is added to the experimental CHI values of filecgsTi.dat, cgs. . . and not e.g. CHIINP.DAT, because the values are in cgs units.They are taken from [29], but values below 12 Kelvin are excluded from fitting.

File fort.7 will be given here as a table!

Table 5.2: File fort.7 – Magnetic susceptibility of Ti3+[3d1] (D3d)

”NO WARRANTY”2999 =KLWERT, . . . , . . . , . . . d el.,less,less,less

8 =NGROUP(POINT GROUP) D3d0 =IOPT2 (”DIMER”) single ion0 =IOPT1 (1 OR MORE DIR.) one mean direction of magn. fieldcgsTi.dat (OF CHI . .) for table of temp.’s, CHI’s & fields(I1,F7.2,3D17.6) FORMAT of cgsTi.DATBWYBTin.DAT (BWYBIN) parameters B20,B40,B43 to be fitted1 = No.OF D EL. one d electron41 =LDG1IN,NRSRIN(CHARGE+) charge+1/ no. of trans.-series

99 0 =IITMAX,IITMIN 99 iterations in a fitting procedure0 =JASTOQ(2=STOP) no STOP: get table of OUTPUT

File fort.11 follows now in its (nearly) original form!:

PROGRAM CONDON written in STANDARD FORTRAN 77.Copyright 2000 Helmut SCHILDER, Fachhochschule Aachen Germany.

For the GNU General Public License see Section 3.6!

CALCULATION OF (MOLAR) MAGNETIC SUSCEPTIBILITIES OF ATOMS OR IONSWITH NS**N, NP**N, ND**N OR NF**N ELECTRONIC CONFIGURATION,AND WITH 4F**N ELECTRONIC CONFIGURATION WITH SPECIFIC GROUND TERM(2S+1)L, I.E. FOR THE CASE 4F**N THERE ARE TWO ESSENTIALLYDIFFERENT ROUTINES.


INPUT 4 DIGITS: L VALUE OF 1 EL.(0...3), FOR MORE (OR LESS) OUT-PUT OF INTERMEDIATE VALUES (1(9)), OF MORE (OR LESS) TEXT (1(9)),FOR MORE (OR LESS) POSSIBILITIES OF TESTS (1(9))

2999 =KLWERT, . . . , . . . , . . . ! ! ! ! ! ! ! ! !

CALCULATION OF (MOLAR) MAGNETIC SUSCEPTIBILITIES OF D BLOCK ATOMSOR IONS WITH ND**N ELECTRONIC CONFIGURATION: THE ATOMS OR IONSWILL BE LISTED, IF ONE CHOOSES DATA FOR RACAH PARAMETERS ANDSPIN-ORBIT COUPLING CONSTANTS FROM THE BOOK BY GRIFFITH LATER ON.

THERE ARE 23 DIFFERENT POSSIBILITIES OF INPUT OF (POINT) GROUPS:D2H,D3H,D4H,D5H,D6H,D-INF-H,D2D,D3D,D4D,D6DD2,D3,D4,D6,C2V,C3V,C4V,C6V,C-INF-V,???,CUB(I),CUB(II),O(3)INPUT IN I2 THE (CONSECUTIVE) NUMBER(0 => 21 <=> CUB-4FOLD ROT.AXIS)

8 =NGROUP (POINT GROUP) ! ! ! ! ! ! ! ! !

SOME USEFUL HINTS FOR THE GROUP CHOSEN: CHOOSE "MORTEX"FROM THE FOLLOWING LIST OF POSSIBLE B-VALUES THOSE MARKED WITH 1ARE EXCLUDED:B20, B22, B40, B42, B43, B44, B60, B62, B63, B64, B66

0 1 0 1 0 1 1 1 1 1 1

INPUT IN I1 IOPT2=1 FOR (SYMM.)DIMER OR =2 FOR DIMER COMB.MODELFOR F EL. (IOPT2=0 FOR (SINGLE) ION)

0 =IOPT2 (”DIMER”) ! ! ! ! ! ! ! ! !

INPUT IN I1 IOPT1=1 FOR MEAN VALUE OF CHI WITH CHI CALCULATED FOR2 OR 3 DIRECTIONS (IOPT1=0 FOR 1 (MEAN) DIRECTION)

0 =IOPT1 (1 OR MORE DIR.) ! ! ! ! ! ! ! ! !

DIRECTION(S) CONCERNED AND TO BE CALCULATED: 2 1

WHERE ARE THE VALUES FOR: "TO BE FITTED YES(0)/NO(1)",TEMP. IN K, CHI-MEASURED IN SI, SIGMA, FIELD IN TESLA?"RETURN" FOR CHIINP.DAT; H OR h FOR "H ATOM" <=> NO INPUT HERE!INPUT OF "CGS OR cgs" (E.G. CGSCHI.DAT) <=> CHI-MEASURED IN CGSUNITS! INPUT OF 888 FOR "STOP"!

NAME READ IN WAS: cgsTi.dat ! ! ! ! ! ! ! ! !


"RETURN", IF FORMAT(I1,F7.2,3D17.6), OR INPUT "FORMAT" IN A50

(I1,F7.2,3D17.6) ! ! ! ! ! ! ! ! !

HEADER: Magn. susc. of CsTiSH in CGS, 270 - 12 K & 11.5 - 2 K,0.5 T.HERE THE FIRST MAX. 3 ROWS OF VALUES ("NOT", TEMPIK, CHI, SIGMA,FIELD)CHI VALUES PUT IN WILL BE CHANGED TO SUCH IN SI UNITS!0 270.27 0.183972E-07 0.100000E+01 0.500000E+000 265.28 0.186736E-07 0.100000E+01 0.500000E+000 260.28 0.189627E-07 0.100000E+01 0.500000E+00

IS THE NAME OF THE NEXT FILE BWYBIN.DAT IN FORMAT(I2,D16.8)?"RETURN" OR GIVE ITS NAME! (8 => STOP, H OR h "NO INPUT FROMFILE")

BWYBTin.DAT (BWYBIN) ! ! ! ! ! ! ! !

CRYSTAL FIELD PARAMETERS ETC. ARE READ IN FROMBWYBTin.DAT

Start of [Ti(H2O)6](3+)THE VALUES READ IN ARE:1 -0.20000000E+05 0 0.00000000E+00 1 -0.28420000E+050 0.00000000E+00 0 0.33968397E+05 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00

CORR. FACTOR OF CHI-MEAS. IS ZERO, BUT BECOMES ONE!

INPUT IN I1 NUMBER OF D EL.; 0 => 1!

THE NUMBER OF D ELECTRONS IS = 1 ! ! ! ! ! ! ! ! !

DATA FROM GRIFFITH INCL. RACAHC AND COUPL.-CONST. HAVE BEENCHOSEN!INPUT IN 2I1 CHARGE+1 OF THE ION AND 1ST,2ND,3RD TRANS.-SERIES

CHARGE OF THE ION (0=ATOM) AND No.OF TRANS.-SERIES= 3 1! ! ! ! ! ! ! ! !


TI V CR MN %% %% %% %% %%; ALL WITHOUT S ORB.-OCC.

CONTROL OF SOME DIMENSIONS/ EQUIVALENCES:

ND2002, NDS66 = 1110 66, JEQUIV= 0

MFIT, I.E. No. OF PARAMETERS ACTUALLY BEEN ADJUSTED, = 3

THERE IS No. OF ITERATION = 0

INPUT IN I5,I3 MAX. AND MIN. NUMBER OF (FURTHER) ITERATIONS!:(ITER.COUNT 0 0, INPUT MAY BE NEG. FOR "FURTHER" ITER.!)

99 0 = IITMAX,IITMIN ! ! ! ! ! ! ! ! !

CALCULATED WYBOURNE VALUES (PLUS 9 OTHER VALUES):(1 OR -1 MEANS "FITTED"):

1 -0.20000000E+05 0 0.00000000E+00 1 -0.28420000E+050 0.00000000E+00 1 0.33968397E+05 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00 0 0.00000000E+000 0.10000000E+01 0 0.00000000E+00 0 0.00000000E+000 0.00000000E+00 0 0.00000000E+00

No. OF MICRO STATES = 10No. OF TEMP., CHI-START CALCULATED = 1 0.17652499E-07

AFTER 0TH ITER.: OMEGA,SQX-FIX,SXR=0.00E+00 0.36783886E+00 0.9201E+01

AFTER 1TH ITER.: OMEGA,FIX,SQX,SXR0.30E+00 0.36783886E+00 0.43778039E+00 0.3832E+01

AFTER 2ND ITER.: OMEGA,FIX,SQX,SXR0.90E+00 0.36783886E+00 0.40455478E+00 -0.1903E+01

AFTER 3RD ITER.: OMEGA,SQX-FIX,SXR=0.27E+00 0.35666214E+00 -0.2479E+01

etc.

AFTER 32TH ITER.: OMEGA,SQX-FIX,SXR=0.19E-03 0.11375568E-01 -0.2210E-02

AFTER 33TH ITER.: OMEGA,SQX-FIX,SXR=


0.56E-04 0.11298032E-01 -0.1195E-03

AFTER 34TH ITER.: OMEGA,FIX,SQX,SXR0.17E-03 0.11298032E-01 0.11298105E-01 0.1133E-06

AFTER 35TH ITER.: OMEGA,FIX,SQX,SXR0.50E-03 0.11298032E-01 0.11298068E+00 -0.5701E-07

AFTER 36TH ITER.: OMEGA,SQX-FIX,SXR=0.15E-03 0.11297990E-01 -0.1206E-06

"FITTED"=+-1; -2/-1<=>8/9 IN INPUT FILE:1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0LIST(I), BWYB(LIST(I))=1 -0.13394885E+05 -0.12847059E+05 0.35591271E+05

CRYSTAL FIELD MIN. SPLIT. (CFMS) AND EIGENVALUE OF THE LEVEL"No. OF VAL.(NRVAL)", WHICH MAY BE THE CRYSTAL FIELD OVER ALLSPLITTING (CFOS),AND CFOSJ (<=> NRVALJ) = 0.203E+02 0.203E+05 0.023E+05

CONV.MAY BE TOO SLOW: ITER/IT-MIN/IT.-MAX.= 36 0 99SXR (=(SQX**2-(SQX**2-BEFORE))*NRDATA)=-0.12E-06DIFMIN (=(SQX-FIX)/(SQX, IF"BETTER")-ONE, "ELSE"=ZERO)= 0.32E-05


0 = JASTOQ(2=STOP) ! ! ! ! ! ! ! ! !

HINT: OUTPUT CORRESPONDS TO "FITTING TAKES PLACE"SQX, CFMS, DEGEN.1, DEGEN.2, CFOS, No. OF VAL., CFOSJ, NRVALJ =0.1130E-01 0.2035E+02 2 2 0.2031E+05 10 0.2031E+05 10


No. 00 TEMP/K SIGMA B/TESLA MEASURED CHI CALCUL. CHI1 00 270.27 1.000000 0.500 0.18397166E-07 0.18714490E-072 00 265.28 1.000000 0.500 0.18673626E-07 0.18967977E-073 00 260.28 1.000000 0.500 0.18962653E-07 0.19231171E-074 00 255.27 1.000000 0.500 0.19251679E-07 0.19504678E-075 00 250.26 1.000000 0.500 0.19553272E-07 0.19788579E-07

...65 00 13.00 1.000000 0.500 0.19161202E-06 0.18834382E-0666 00 12.50 1.000000 0.500 0.19666370E-06 0.19213036E-0667 00 12.00 1.000000 0.500 0.20268299E-06 0.19606032E-06


68 10 11.50 0.000000 0.500 0.20988352E-06 0.20014732E-0669 10 11.00 0.000000 0.500 0.21782546E-06 0.20440862E-0670 10 10.50 0.000000 0.500 0.22634546E-06 0.20886626E-06

...85 10 3.00 0.000000 0.500 0.62816772E-06 0.38894239E-0686 10 2.50 0.000000 0.500 0.73574842E-06 0.43754886E-0687 10 2.00 0.000000 0.500 0.89596964E-06 0.51039075E-0688 00 -10.00 1.000000 0.500 0.20300000E+05 0.20312191E+05

THE TABLE PUT OUT HERE ALSO PUT OUT TO FILE fort.1!

1 ELECTRON(S), 2 =l VALUE, 10 BASIS STATES,10 VALUES USED LATER ON,

0.15400000E+03 =ZETA (1 EL.), -0.13394885E+05 =B20(WYBOURNE),-0.12847059E+05 =B40(WYBOURNE), 0.35591271E+05 =B43(WYBOURNE),

"LESS OUTPUT" OR "JASTOP" WAS CHOSEN: STOP

To get data for a plot of µeff = MU versus temperature in Kelvin, which willbe given here, another calculation with ‘2000’ or ‘2110’ instead of ‘2999’, likethat in section 3.6, must be done. Following ‘0=JASTOQ(2=STOP)’, INPUTof ‘6=MFITIN(FORM OF OUTPUT)’ is possible, and choosing the ‘standard’form with RETURN, four additional columns are put out, two of them involving‘MU’. File fort.8 is renamed BWYBTi.DAT and used to start a recalculation ofthe last iteration in Section 3.6 in order to demonstrate how CONDON worksusing ‘2110’ for much OUTPUT.

5.1.3 Cr2+[3d4] and Re3+[5d4] (cubic)

All parameters for Cr2+ and Re3+ (in octahedral coordination) are taken fromthe book by Lueken [8]. A possible orbital reduction factor for the cubic caseis set to ONE. All parameters on file BWYBIN.DAT are zero except oneWybourne parameter (F becomes ONE automatically). Curves CrCl and CrCNmean Cr2+, and the ligands are Cl− (or H2O) and CN−, respectively. B40 is21000 (or 27300) and 40950, respectively. Curves ReCl and ReH mean Re3+,and the ligands are Cl− (or H2O) and H−, respectively. B40 is 30975 (or 40320)and 25200, respectively. For K3ReH6 see also [33]!

Note the following. Parameters for Cr2+ besides Wybourne parameters arethose given in the book by Griffith [5]. All parameters given there are derivedfrom experiment, but no hints to ligands are given. The influence of individualligands may be important however, too.


0.8

1

1.2

1.4

1.6

1.8

2

0 50 100 150 200 250 300

µeff/µB

T/Kelvin

’Ti.plot’ using 3:11’Ti.plot’ using 3:10

rsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrs

rsrsrs

rs

rsrs

rs

rs

rs

rsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrs

rsrsrsrsrsrsrsrsrs

rs

Figure 5.2: Mu effective/µB of Ti3+[3d1] (D3d)

5.1.4 Introduction to Os4+[5d4]

In order to keep the size of the basis small, all calculations of paramagneticsusceptibilities cited in the book by Griffith (The Theory of Transition-MetalIons) [5] are either done in the weak field or in the strong field scheme, andonly one Russell-Saunders state without spin-orbit coupling is the basis in theweak field scheme. In the strong field scheme, spin-orbit coupling is takeninto account, but the size of the really used basis is reduced correspondingto and using the pN isomorphism, and ‘high-frequency elements from outsidethe ground configuration’ are added, which are thought to correct for the lostcontributions.

Especially the treatment of the cubic case 5d4 in the strong field schemewith (strong) spin-orbit coupling shows the difficulties involved at that time.Although Tanabe and Sugano a few years ago calculated their ligand field energydiagrams [1] within the strong field scheme (but without spin-orbit coupling!),and this means in principle in a full basis, such a full basis is not applied forthe calculations of paramagnetic susceptibilities of Os4+[5d4] in the book byGriffith [5]. Obviously these calculations are wrong for this reason!

In CONDON the basis used is always a full basis of micro-states, except inthe case 4fN , where alternatively a basis of the LS ground state may be chosen.There is no difference between calculations in a weak and in a strong ligandfield. Calculations in the strong field scheme are not possible, but of course theresults of such calculations, if the strong field scheme may be applied at all, areobtained within the full basis of micro-states! If the strong field scheme is not


0

1

2

3

4

5

6

7

0 50 100 150 200 250 300

µeff/µB

T/Kelvin

’CrCl.plot’ using 3:11’CrCN.plot’ using 3:11’ReCl.plot’ using 3:11’ReH.plot’ using 3:11

Figure 5.3: Mu effective/µB of Cr2+[3d4] and Re3+[5d4] (cubic)

appropriate, but a test calculation in such a scheme is nevertheless wanted, asimulation of calculations in this scheme is possible with a nearly infinite strongligand field.

Up to three orbital reduction factors may be applied, for z-, x- and y-direction of magnetic field, respectively, i.e. only one factor for the cubic case.These factors modify the energy due to the magnetic field and at the same timethe derivatives with respect to the magnetic field, and modify in this way CHI.Further reduction factors, concerning Racah parameters and spin-orbit couplingconstants, respectively, have to be incorporated in these parameters.

5.1.5 Os4+[5d4] (D5h and cubic)

Magnetic susceptibility measurements of Na3OsH7 in the temperature rangebetween 3.5 K and room temperature revealed a weak temperature independentparamagnetism [34]. The coordination polyhedron formed by the seven ligandscan be described as a (distorted) pentagonal bipyramide (point group D5h).

In order to demonstrate the usefullness of a CHI-START value of zero (seeSubsection 2.4.2), three calculations were performed. Firstly (only) CHI-0 wasfitted to the measured CHI values, and CHI-0 became 0.221538E−8. Two fitsof ligand field parameter(s) corresponding to D5h, B20 (not fitted, set to zero)and B40 were performed, one fit to measured CHI values and one fit to CHI-0,and the results were the same, because there is no temperature dependence.B20 was set to zero because of strong correlation between B20 and B40, and


the result for B40 is then 29259 cm−1. Racah parameters were B = 500 andC = 4B, ZETA, the spin-orbit coupling constant for one electron, was 4500,and the magnetic field 3 Tesla.

To compare with K3ReH6 (see 5.1.3), also a calculation in point group Oh,i.e. ‘cubic’ was performed, and the result for B40 was 18153 cm−1.

5.1.6 Os4+[5d4] (cubic – strong field scheme)

The cubic case ‘strong field scheme’ is given in the book by Griffith [5] withZETA, the spin-orbit coupling constant for one electron, = 4500, and (15B +5C + ZETA)/2 = 11000 (i.e. B ≈ 500, if C/B ≈ 4). An average value ofexperimental values for a series of osmihalides M2OsHal6 (M = K,Cs; Hal =F,Cl,Br,I) is given with CHI[CGS]=0.000936, although the individual valuesvary ‘in an apparently haphazard manner’.

Calculated in the strong field scheme were CHI[CGS]=0.000722 with theparameters just given and =0.000848 with reduced parameters (incl. orbitalreduction factor), respectively, each value to be incremented by an estimated‘high frequency part’ between 0.000054 and 0.000068.

A simulation of the ‘strong field scheme’ calculations given in the book byGriffith is possible with a nearly infinite B40 (e.g. B40=2000000) and C/B ≈ 4,and with ‘15 =NRVALI’, i.e. 15 is the size of the basis used ‘later on’, and it re-sults in CHI[CGS]=0.000718. Reduction of 30% results in CHI[CGS]=0.000839,and with ZETA rounded to 3000 in CHI[CGS]=0.000898.

From the estimated ‘high frequency part’ before a more realistic B40 of90000 may be estimated. But the assumption, that calculations based on thisB40 and a full micro-state basis with either 15 or 210 (‘all’) functions used ‘lateron’ result in ‘useful’ CHI values, is apparently wrong. CHI[CGS]=0.000639 andCHI[CGS]=0.000666, respectively are obtained with B = 350, ZETA=3000, andorbital reduction factor =0.7. Apparently the strong field scheme must not beapplied, because the ‘low-lying’ functions are not correctly obtained!

Much better results are obtained with B40=90000, B = 500, ZETA=3000,and without any orbital reduction factor. With 15 and 210 (‘all’) functionsused ‘later on’ CHI[CGS]=0.000844 and CHI[CGS]=0.000901, respectively areobtained. The difference of 0.00057 is a good value for the ‘high frequency part’,and the value 0.000901 is not too much different from 0.000936, the ‘experimen-tal’ value. If ZETA is changed to 2900, CHI[CGS]=0.000951 instead of 0.000901results. Apparently the ‘high frequency part’ was well estimated, but to applythe strong field scheme was not correct, and a full basis is needed to get correctresults!

CHI is definitely constant between 0.05 and 300 Kelvin in all cases. Theformula of Van Vleck is selected, but virtually the same CHI values are obtained,if the fundamental equation with a magnetic field of 0.1 Tesla is chosen. The


ground-state function is not degenerated, and its Zeeman coefficient of 1st orderis always zero. The first three excited functions are degenerated, and they arethe only ones with contribute to the Zeeman coefficient of 2nd order of theground-state function except in the cases of B40 = 90000 and 210 (‘all’) functionsused ‘later on’, where contributions from ‘higher’ functions, i.e. higher than 15,contribute, too. The ground-state function is of course the only one, whichcontributes to CHI.

File fort.7 for the most important calculation will be given now in the formof a table.

Table 5.3: File fort.7 – Os4+[5d4] (cubic)

”NO WARRANTY”2000 =KLWERT, . . . , . . . , . . . d electrons, . . . , . . . , . . .21 =NGROUP (POINT GROUP) 21=cubic group100 =JAVANV,IOPT1,IOPT2 ‘VANVLECK’,z-dir.,single ionh (OF CHI ..) for table of temp. & fields1 =NDATA (OF H-ATOM) first row of table onlyBWYBIN.DAT (BWYBIN) file name – for B40.GT.00 =JASTRU(cture) no CALL of subroutine STR000 =NOPROZ (IN % YES/NO) ‘in %’ – has no real meaning0 =IOPT4 (CHI,1/CHI,..) for fit – has no real meaning4 = No.OF D EL. four electrons1 =JANOC (”RACAHC=4*B”) data from KOENIG/KREMER1 =NEURAH(NEW RACAH) new RACAHParameter(s)0.50000000E+03 =RACAHP new RACAHParameter1 =NEUKSI(NEW ZETA) new spin-orb. coupl.-const.0.3000E+04 =FKSII(=ZETA) new spin-orb. coupl.-const.0 =NEUORB(ORB.-RED.-F.) orb.-red.-factors(s) = ONE210 =NRVALI(No. OF STATES) no. of functions used ‘later on’0 =KASSIN(”2ND ORDER”) all 2nd order Zeeman coeff.0 =JASTOQ(2=STOP) no STOP: get table of OUTPUT3 =MFITIN(FORM OF OUTPUT) ‘Change’ in RESultTaBLes000000000000000000 ‘default values’2 =JASTOR(2=STOP) ‘yes’: STOP

5.1.7 U4+[5f2] (cubic)

Spin-orbit and crystal-field interactions have the same order of magnitude, andsimultanious diagonalization of interelectronic, spin-orbit and crystal-field in-teractions is necessary [35]. There are no distinct groups in the resulting energypattern corresponding to a particular free-ion J value! All this causes no prob-


lems to CONDON, and as a first example Cs2UCl6 [36] is treated (also citedin [35], no susceptibility given there). If all(!) parameters are taken from [35],exactly the same energy values as given there are arrived at (and a constant sus-ceptibility of 0.297E−7 SI units (0.296E−7 at 250 K, 0.295E−7 at 300 K)). Note:The point group is Oh (exactly D3d site symmetry, only slightly distorted), andnot Td!

As a special problem with 5f electrons, there is the ‘problem of expandingthe crystal-field model’, because there are stronger interactions between theelectrons and the crystal field [35]. Especially it appears, that values of theSlater-Condon parameter F 2 are strongly affected by the nature of the ligand,and it was suggested that F 2 might be absorbing some of the effects of the ligandfield. Most parameters given for d electrons, although derived from experiment,do not take into consideration the ligands, although the influence of individualligands may be important. With 5f electrons this influence must not be ignored!

The theoretical relativistic values for U5+, U4+ and U3+ in CONDON, takenfrom the Handbook on the Physics and Chemistry of the Actinides, Vol. 1[28], should be sufficiently good in many cases, if experimental values are notavailable. As an example, the energy values of Cs2UCl6 are 0, 879, 1422, 2328,. . . 59080 compared with 0, 900, 1316, 2239, . . . 41403 in [35] (susceptibility0.305E−7 (0.304E−7 at 200 K, 0.303 at 250 K, 0.300 at 300 K)).

In [35] theoretical & experimental values of F 2 for U4+ and U3+ are com-pared: 75735 & 43770 and 70088 & 40280 (in CONDON from [28]: 76557 and71278). As a hint, experimental values are 40% lower than theoretical values!From some special examples in [35], 30% and 20% are derived for F 4 and F 6,respectively. No such big differences occur for ZETA: For U5+, U4+ and U3+

2050 & 1910 (2174 free ion value), 1846 & 1719 and 1683 & 1624 (in CONDONfrom [28]: 2113, 1926 and 1748).

A second example taken from [35] is UCl4 with symmetry Td isolated in amatrix. Using parameters from [35] or from CONDON (from [28]) to calculatesusceptibility gives very similar (1% diff.) results (Wybourne values from [35]).

A third example taken from [35] is U(BD4)4 with symmetry Td. If parame-ters are taken from [35], there is some similarity of reciprocal CHI as a functionof temperature to a Curie straight line. With parameters from CONDON (from[28]), a totally different reciprocal CHI as a function of temperature results,surely a totally wrong one! With very different Wybourne parameters, however(a fit has to be performed to the ‘Curie straight line’ before), it is possible toget nearly (2.5% diff.) the above ‘Curie straight line’ again (B40 = −3484 =>−7122, B60 = −4240 => −1509)!

5.1.8 Fe2+[3d6] (cubic, spin-crossover)

In the case of spin crossover [11] there is an equilibrium between a low-spin and ahigh-spin complex of Fe2+. The high-spin complex has a different free enthalpy,


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Figure 5.4: CHI in SI units of Fe2+[3d6] (cubic, spin-crossover)

i.e. enthalpy and entropy, and different ligand field parameters. For this examplehere a cubic ligand field with the threefold axis as the z-axis was chosen, and aligand field parameter of −118000 for the low-spin component. All ligand fieldparameters, i.e. two(!) in this case, are divided by the same factor, which is5.0 here and corresponds to the 18th row of parameters of file BWYBIN.DAT,to get the ligand field parameters of the high-spin component. Point groupD3d with the 5th row of parameters of file BWYBIN.DAT 8 1.00000000E+00,meaning B43/B40 exactly as in the cubic group above, gives the same results,but the factor may be fitted, choosing 9 1.00000000E+00, if the point group isreally D3d.

The enthalpy and entropy differences of the high-spin component, com-pared with the low-spin component, which are 2270 and 6.72, respectively,here, correspond to the 19th and 20th row of parameters, respectively, of fileBWYBIN.DAT.

The following formula [37] for the proportion of the high-spin component isused in CONDON:

x =1

1 + exp[∆H/RT] ∗ exp[−∆S/R].

CONDON calculates however

x =exp[−∆H/kT] ∗ exp[∆S/k]

1 + exp[−∆H/kT] ∗ exp[∆S/k],


with k the Boltzmann constant in cm−1/Kelvin, i.e. k = 0.695040, and en-thalpy H and entropy S in cm−1.

CHI computed for the high-spin component is multiplied by this x, CHI forthe low-spin component with (1−x).

A plot for this example is given here!

5.2 Exchange-coupled systems

5.2.1 Nd3+[4f3] (NdPt2, cubic)

Reciprocal CHI values for NdPt2 are known, and a positive molecular fieldparameter λMF of 5.0(4)E+5 indicates dominantly ferromagnetic interactions,consistent with the observed ferromagnetism below 10 Kelvin [38]. Values attemperatures below 10 Kelvin were discarded, and owing to the deviation fromexact stoichiometry, a correction factor F = 1.14 was applied.

Now a molecular field calculation with a molecular field parameter for amolecular field proportional to magnetisation is chosen, no values are discarded,but for values below 10 Kelvin the individual standard deviations SIGMA areset to 10 instead of 1. Starting with the parameters already known [38], a fairlygood fit is possible, and for the values above 10 Kelvin this fit is very goodindeed. At 2 Kelvin the calculated 1/CHI value is 0.92E+04 SI units, and themeasured one, divided by a correction factor F = 1.19, is 1.45E+04, i.e. thecalculated value is 33% too low at 2 Kelvin. The molecular field parameterapplied here is the 17th parameter on file BWYBIN.DAT and is strictly tobe separated from λMF in CONDON, and the result for this molecular fieldparameter is 5.6E+5. X and W corresponding to Lea, Leask, Wolf [22] become0.2560 and 3.8162, respectively.

A plot of reciprocal CHI values versus temperature is given!

5.2.2 Gd3+–Cu2+ dimers

Two Gd3+–Cu2+ dimers have been described by Kahn et al. [39], one of themcontaining perfectly isolated binuclear species. The magnetochemical investi-gation of the latter revealed the expected dependence of the molecular para-magnetic susceptibility on temperature, and an excellent agreement betweenobserved and calculated values was obtained. Because the formula for the lat-ter, not involving any magnetic field, is given, calculated and therefore alsomeasured susceptibilities can be re-calculated by the reader, and this was donehere.

Using exactly as given in [39] g(Spin-Hamiltonian) = 2.10 for copper, thecoupling constant of the Heisenberg operator (J as defined here) = 0.71 cm−1,

5.2. Exchange-coupled systems 121

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’NdPt2.plot’ using 3:7’NdPt2.plot’ using 3:6

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Figure 5.5: Reciprocal CHI in SI units of Nd3+[4f3] (NdPt2, cubic)

and no magnetic field for a calculation with CONDON with a single Russell-Saunders term leads to an excellent agreement between the values obtained andthe re-calculated values.

The INPUT is given here as a table for file fort.7, not including 0.71 on fileBWYBIN.DAT, and the resulting susceptibilities are given as a plot of effec-tive magnetic moment versus temperature. A calculation using a field of 0.1Tesla gives virtually the same result, and taking into account magnetic dipol-dipol interaction (the distance of the two ions is 325 pm) makes no noteworthydifference, too.

If CONDON is linked with file shortO.f, which is possible only, if sufficientstorage is available, a full basis set may be used for Gd3+. The ground stateconsists of 97.66% 8S and 2.30% 6I now, with a g-factor, for a spin-orbit couplingconstant of 1470, of 1.99321 (to be calculated using option ‘JATEST – CHANGEOF BASIS TO RUSSELL SAUNDERS STATES’ and options ‘MOROUT &JATEST – A TEST ON G-FACTORS’, respectively).

To get a quality of a fit of 0.06%, the g(Spin-Hamiltonian) for copper mustbe 2.23 (g-factor = 1.81, g = g(S)/(1 − g(S)) here), and the fitting procedureresults in 0.67 cm−1 for the coupling constant J (only 0.64% for 2.10 and 0.71cm−1, but 0.15% for 2.23 and 0.71 cm−1). The energy difference between nonetand septet is 4.64 cm−1 now instead of (8∗0.71 = 5.68) cm−1 before. See [5] forthe spin-Hamiltonian, especially for Cu2+.

Neither the g-factor of Gd3+ nor the quality of the fit change noteworthy, ifa tetrahedral ligand field for Gd3+, e.g. one corresponding to that calculated for


Table 5.4: File fort.7 – Magnetic susceptibility of a Gd3+–Cu2+ dimer

”NO WARRANTY”3000 =KLWERT, . . . , . . . , . . . f el., . . . , . . . , . . .1 =JAWIGN (SPEC.ROUTINE) ‘small basis’23 =NGROUP (POINT GROUP) O(3)101 =JAVANV,IOPT1,IOPT2 ‘VANVLECK’,z-dir.,dimer0.00 =ABST IN DIMER no magn. dipol-dipol interact.cgsKahn.dat (OF CHI . .) for table of temp.’s, CHI’s & fields(I1,F7.2,3D17.6) FORMAT of cgsKahn.DAT0 =JANO(OUTPUT) not all data of cgsKahn.dat => fort.11BWYBIN.DAT (BWYBIN) for the constant of Heisenberg operator0 =JASTRU(cture) no CALL of subroutine STR000 =NOPROZ (IN % YES/NO) ‘standard’ fit (in %)1 =IOPT4 (CHI,1/CHI,..) CHI itself is fitted

7 = No.OF F EL. seven f electrons0 =NEUKSI(NEW ZETA) no change of spin-orbit coupl. constant0 =JANOS(tevens) Wybourne coefficients remain

8 =NRVALI(No. OF STATES) all functions are used ‘later on’2.1000 =(J)ACUGD(dimer) g(Spin-Hamiltonian) for copper0 =KASSIN(”2ND ORDER”) all 2nd order Zeeman coeff.0 =JASTOQ(2=STOP) no STOP: get table of OUTPUT6 =MFITIN(FORM OF OUTP.) ‘Change’ in RESultTaBLes: 6 columns000000000000000000 ‘default values’2 =JASTOR(2=STOP) ‘yes’: STOP

Dy3+ (see table 5.5), or a magnetic field of 0.1 Tesla or both are introduced. Forthe ligand field, there is a zero-field splitting of 0.002323, 0.002325 and 0.005743cm−1 of the first, second and third excited Kramers doublet, respectively.

Note that the results of the time consuming part of the calculations with afull basis set for Gd3+ may be written (in a first iteration, unformatted) to filefort.13 and repeatedly read in from there to shorten further calculations.

5.2.3 Dy3+ [4f9] ( [ Dy(C5H5)2(µ−Br) ]2, rhombic)

Some results for [ Dy(C5H5)2(µ−Br) ]2 with exchange coupling have been pub-lished [40]. Especially Heisenberg coupling was applied, but also a hint to theXY model is given (based on neutron diffraction and the deuterated compound).A simple cubic (tetrahedral) ligand field was used for a fit, which is surprisinglygood, but a correction factor greater than one was apparently used, and mag-netic dipol-dipol interaction was not taken into account.

The relevant point group C2v is applied now, i.e. NGROUP=15. File shortQ.f

5.2. Exchange-coupled systems 123

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Figure 5.6: Mu effective/µB of a Gd3+–Cu2+ dimer

with ND2002=2002 (and NDS66=66) is linked with the other relevant files tofile condon, because exactly 2002∗2002-matrices are needed, if a full basis isused. One more EQUIVALENCE, which is set in shortQ.f, does not matter,because option JAFAST is selected, at least in the case of iterating! Sufficientstorage may not be available, if a PC is used. Indeed a full basis is not re-ally needed, and option JAWIGN (‘WIGNER’) may be applied along with fileshortL.f and a much smaller basis (66 instead of 2002). Virtually the same plotis obtained after convergence of the fitting procedure, but the ligand field foundis not the same, and the different Wybourne values and coupling constants aregiven below for the example treated here. Magnetic dipol-dipol interaction istaken into account now (J becomes −0.074 and −0.071 cm−1 for ‘full basis’and ‘WIGNER’, respectively, if not).

In principle Heisenberg coupling must not be applied for a dimer like theone treated here, because there is no pure spin state [8], and a good fit overthe entire temperature range is therefore not possible at all. Part of the errorsintroduced may however be absorbed in orbital reduction factors, which havebeen found to be 0.91 and 0.90 for ‘full basis’ and ‘WIGNER’, respectively, anda good fit is possible using these factors, if the lowest temperature (1.72 K)is excluded. The ‘Combined Model’ [25] is used here throughout, with ‘lower’dimension = 8. CHI values at ‘higher’ temperatures are very good now, leadingto a very satisfactory plot of reciprocal CHI versus temperature. A plot ‘CHIin SI versus T/Kelvin’ with temperatures up to 30 Kelvin is given!

For fitting one of the ‘standard’ fitting procedures was applied (see 5.3)! FileCHIINP.DAT with its experimental values is therefore not printed here, but only


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Figure 5.7: CHI in SI units of Dy3+[4f9] ( [ Dy(C5H5)2(µ−Br) ]2, rhombic)

included in the plot given, but an additional line with −16.0 and 650.0 has beenadded in order to fit CFOS = CFOSJ, the ‘Crystal Field Over all Splitting’. Allindividual standard deviations SIGMA were set to one, the magnetic field was0.1 Tesla. Instead of CHIINP.DAT, ‘h’ or ‘H’ (‘H atom’) may be used for valuesof temperatures, if the calculated values given in the plot are recalculated usingthe file BWYdy.DAT, which is printed (in reduced form: 8 lines of zeros must beappended). To be able to use option JAWIGN, a second file BWYWdy.DAT isprinted, too, but input of orbital reduction factors is only possible, if ‘JATEST’is chosen.

One more CHI value of 0.738E−5, measured at 2 Kelvin and 5 Tesla, isnot included in the plot and in the fitting procedure, but was calculated to be0.776E−5.

The XY model instead of the Heisenberg model was also applied, in orderto be able to discuss the results of its application with the ‘hint’ from above.Virtually the same results are obtained!

Files fort.7 (without ‘iter.’), BWYdy.DAT and BWYWdy.DAT are givenhere as one table!

5.3 Search tactics

In general it is not an easy task to get starting parameters for a fit! Valuesfrom literature may be helpful. ‘Trial and error’ searching is often the only

5.3. Search tactics 125

Table 5.5: fort.7, BWYdy.DAT, BWYWdy.DAT – [ Dy(C5H5)2(µ−Br) ]2

”NO WARRANTY”3000 =KLWERT, . . . , . . . , . . . f electrons, . . . , . . . , . . .0 =JAWIGN (SPEC.ROUTINE) ‘full basis’15 =NGROUP (POINT GROUP) C2v – ‘ortho-rhombic’002 =JAVANV,IOPT1,IOPT2 2 = symmetr. dimer and Comb. Model422.50 =ABST IN DIMER distance: for magn. dipol-dipol interact.CHIdy.DAT (OF CHI ..) INPUT of temp.’s, CHI’s & fields(I1,F7.2,3D17.6) FORMAT of CHIINP.DAT2 =JANO(OUTPUT) OUTPUT of CHIINP.DATBWYdy.DAT (BWYBIN) INPUT of parameters0 =JASTRU(cture) no CALL of subroutine STR000 =NOPROZ (IN % YES/NO) ‘standard’ fit (in %)1 =IOPT4 (CHI,1/CHI,..) CHI itself is fitted9 = No.OF F EL. 9 f electrons0 =JANO(FINTEG) no change of Slater-Condon parameters0 =NEUKSI(NEW ZETA) no change of spin-orbit coupl. constant1 =NEUORB(ORB.-RED.-F.) INPUT of (3) orbital reduction factors0.91000000 =RED.FA(CT) . . . for z-direction0.91000000 =RED.FA(CT) . . . for x-direction0.91000000 =RED.FA(CT) . . . for y-direction

0 0 =IITMAX,IITMIN 0 iter. in a fitting procedure0 =JASTOP(later on!) no STOP in advance0 =JANOS(tevens) Wybourne coefficients remain2=INFORM(FORMat free) (unformatted) OUTPUT to file fort.131 =JAFAST(”faster”) option ‘JAFAST’ is chosen0 =NRVALI(for JAFAST) full J-basis (0=>16) for ‘JAFAST’8 =N1728I(Comb.Model) lower dimension for ‘Combined Model’0 =JASTOQ(2=STOP) no STOP: get table of OUTPUT0 =MFITIN(FORM OF OUTP.) no ‘Change’ in RESultTaBLes2 =JASTOR(2=STOP) ‘yes’: STOP

File BWYdy.DAT, SQX=1.07% File BWYWdy.DAT, SQX=1.15%1 +0.49840002E+03 1 +0.53859003E+031 −0.59431378E+03 1 −0.58319885E+031 −0.31217001E+03 1 −0.32466000E+031 +0.67763971E+03 1 +0.70754962E+030 +0.00000000E+00 0 +0.00000000E+001 +0.33339874E+03 1 +0.35709982E+031 −0.16567009E+03 1 −0.17135001E+031 +0.25552971E+01 1 +0.25671730E+010 +0.00000000E+00 0 +0.00000000E+001 +0.95824898E+02 1 +0.99514999E+021 +0.16788675E+03 1 +0.15161797E+031 −0.30689824E−01 1 −0.29295629E−01


possibility! More simple cases are point groups of high symmetry. For lowsymmetries the ‘Simple Overlap Model (SOM)’ [31, 32] may be applied, if thestructure is known (Also the angular overlap model may be helpful).

Searching and fitting in the case of [ Dy(C5H5)2(µ−Br) ]2 was successful,provided that subroutine STR00, which makes use of ‘SOM’, was applied, buthow to prepare file str00.dat with the cartesian coordinates of all ligands of thefirst coordination sphere was an extra task and is not discussed here!

Four starting parameters were chosen purely arbitrary: One covalence factorand two charge factors (for Br and C) were 0.05, −1.0, and −0.2, respectively,and the coupling parameter was −0.1. The carbon atoms were supposed to bethe ‘more covalent’ ligands!

A ‘standard’ fitting procedure, i.e. one with OMEGA =0.1 as pre-occupied,led to a SQX value of (only) 6.5% after 99 iterations, and to starting valuesfor 9 Wybourne parameters and a new starting value of −0.04 for the couplingparameter.

All attempts to fit the measured CHI values starting with these parametersfailed, but introduction of an orbital reduction factor (resp. three equal factors)led to satisfactory fits, the final factor reached by ‘trial and error’ tests, and notexactly the same for two different basis sets.

In all cases an energy value, the ‘Crystal Field Over all Splitting (CFOS)’,was fitted together with the CHI values in order to hold this value in a reasonablerange, and in order to hold the B-values in reasonable ranges.

The fitting procedures led in general to B-values quite different from thestarting values! If one thinks, that this could be incorrect, there is the possi-bility of limiting these differences, e.g. to 30% of a starting value (but mini-mally to the value 60). Such minimum and maximum values were given to fileBWYBIN.DAT, and this file was read in after starting of the fitting procedure.

If such a limit was reached, OMEGA became very big (1000000000), a warn-ing message was given, and convergence was formally achieved. But this con-vergence had to be ignored by discarding the parameter concerned from fitting,and the fitting procedure was continued with the remaining parameters. One B-value remained without reaching its limit! The results were two very satisfactoryfits, for two different basis sets, with the results given in Subsection 5.2.3.

Continuation of the fitting procedures without giving limits led to only littlebetter SQX values (0.96%)!

5.4 Differing results of fitting

A seemingly straightforward fitting procedure with a result looking much betterthan that in 5.2.3 was found for Dy(C5H5)2(µ−Br) later on. It starts with theresult from [40] for the cubic case, B40=1050, B60=−140 and J = −0.18, and

5.4. Differing results of fitting 127

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Figure 5.8: CHI in SI units of Dy3+[4f9] ( [ Dy(C5H5)2(µ−Br) ]2, rhombic)

these parameters and the three special relations (1.0 meaning SQRT(5/14) . . . )B44/B40, B64/B60 and B66/B60 were fitted in 500 iterations using the fullbasis (further conditions as in 5.2.3). The results were used to start a newfitting procedure with all possible (12) parameters, which led to a quality of thefit of 1.56% and J = −0.16 cm−1. The plot is qualitatively much better now inthe range from 0 to 30 Kelvin. Including an orbital reduction factor of 0.96 ina new fit does not change this plot, which is given here, but the quality of thefit to 0.59%.

However, a ‘trial and error’ fitting procedure leads to a nearly identical plotand a quality of the fit of 0.55%, but to different results else, especially toJ = −0.05 cm−1 and an orbital reduction factor of 0.95. An attempt to answerthe question, which result is the more reliable, will be made!

Two calculations with IOPT1=1 to get a partition of the CHI values corre-sponding to the three directions of space show, that the x-direction makes thesmallest contribution to CHI in the first case, whereas in the second case thez-direction makes the smallest one.

A new feature in CONDON makes a calculation possible with magneticdipol-dipol interaction in x-direction. The result for the first case is J = −0.087cm−1, and virtually the same or even a little better plot is obtained. Withoutmagnetic dipol-dipol interaction J = −0.127 cm−1, and with only the isotropicpart J = −0.082 cm−1 is obtained.

For the second case the corresponding J values, for z-direction, are J =−0.052, J = −0.097, and J = −0.052 cm−1, respectively.


Apparently the anisotropic part of magnetic dipol-dipol interaction is nearlyzero, but wrong results are obtained, if the setting of the dimer is incorrect!

The result for the J value of Heisenberg coupling is J = −0.07(2) cm−1, orJ = −0.11(2) cm−1 without magnetic dipol-dipol interaction.

Bibliography

[1] Y. Tanabe, S. Sugano, On the Absorption Spectra of Complex Ions. I +II, J. Phys. Soc. Japan 9 (1954) 753 - 779. pages 114

[2] R. Finkelstein, J. H. Van Vleck, J. Chem. Phys. 8, 790 (1940). pages

[3] Gray and Wills, Phys. Rev. 38, 248 (1931). pages 80, 91

[4] E. U. Condon, G. H. Shortley,The Theory of Atomic Spectra,CambridgeUniversity Press, Cambridge 1970. pages 3, 4, 6, 9, 17, 68, 69, 71, 72, 73,77, 80, 85, 91

[5] J. S. Griffith,The Theory of Transition-Metal Ions, Cambridge UniversityPress, Cambridge 1971. pages 4, 5, 17, 50, 108, 113, 114, 116, 121

[6] E. Konig, S. Kremer, Ligand Field Energy Diagrams, Plenum Press, NewYork 1977. pages

[7] E. Konig, S. Kremer, Magnetism Diagrams for Transition Metal Ions,Plenum Press, New York 1979. pages 17, 51

[8] H. Lueken, Magnetochemie, Teubner, Stuttgart 1999. pages 1, 10, 11, 12,13, 14, 82, 86, 91, 113, 123

[9] J. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities,Oxford University Press, Oxford 1932. pages 1, 10, 11

[10] K.-H. Hellwege, Einfuhrung in die Festkorperphysik, 3. Aufl., Springer,Berlin 1988. pages 1, 6, 12, 63, 79, 88, 101

[11] P. Gutlich, Thermisch und optisch schaltbare Eisen(II)-Komplexe, Angew.Chem. 106 (1994) 2109 - 2141, Angew. Chem. Int. Ed. Engl. 33 (1994)2024 - 2054. pages 1, 60, 64, 118

[12] W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, NumericalRecipes (FORTRAN Version), Cambridge University Press, Cambridge1992. pages 1, 54, 57, 61

[13] B. G. Wybourne, Spectroscopic Properties of Rare Earths, Wiley, NewYork 1965. pages 2, 5, 50, 64

[14] H. Eyring, J. Walter, G. E. Kimball, Quantum Chemistry, Wiley, NewYork 1967. pages 4, 73

129

130 Bibliography

[15] R. McWeeny, B. T. Sutcliffe, Methods of Molecular Quantum Mechanics,Academic Press, London 1969. pages 4, 73, 77

[16] M. T. Hutchings, Point-Charge Calculations of Energy Levels of MagneticIons in Crystalline Electric Fields, Solid State Phys. 16 (1965) 227 - 273.Academic Press, New York. pages 6

[17] C. Gorller-Walrand and K. Binnemans, Rationalization of Crystal-FieldParametrization, Vol. 23 (1996), Chapter 155, in: K. A. Gschneidner, Jr.,L. Eyring (eds.), Handbook on the Physics and Chemistry of Rare Earths,North-Holland Publishing Company, Amsterdam. pages 6, 10, 18, 50, 79,88

[18] B. L. Silver, Irreducible Tensor Methods, An Introduction for Chemists,Academic Press, New York 1976. pages 6, 8, 75, 76

[19] R. N. Zare, Angular Momentum, Understanding Spatial Aspects in Chem-istry and Physics, Wiley, New York 1988. pages 6, 8, 75, 76

[20] E. P. Wigner, Group Theory, Academic Press, New York 1959. pages 6,8, 75, 76

[21] K. W. H. Stevens, Matrix Elements and Operator Equivalents Connectedwith the Magnetic Properties of Rare Earth Ions, Proc. Phys. Soc. A[London] 65 (1952) 209 - 215. pages 6, 51, 65

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Index see next page!

Index

1s electron(s), 4, 6, 11, 20, 1062p electrons, 33d electrons, 4, 21–23, 68, 69, 73,

82, 107, 108, 113–115, 118,119

4f electrons, special routine for, seeWIGNER

5f electrons, 18, 51, 118

actinides, 18, 51, 118, 131alum, 107, 131angular momentum, 13, 85, 91, 130angular overlap model, 126atomic spectra, 4, 129atoms (ions)

4f block, 6, 8, 22, 49, 109carbon, 126cerium, 65charge, 23, 28, 65, 66, 108, 110,

130chromium, 113, 115copper, 14, 95, 102, 103, 120–

122d block, 17, 22, 24, 109dysprosium, see dimer, dyspro-

sium, 4, 15, 122energy levels of, 20, 21, 40, 44,

54, 63, 108, 118, 121, 126,130

erbium, 82europium, 65, 131f block, 22, 109gadolinium, see dimer, GdCu,

5, 15, 65holmium, 15, 77, 84, 92hydrogen, 25, 50, 60, 105, 109,

117, 124iron, 118, 119, 129lanthanide, 15, 18, 66, 130–132neodymium, see NdPt2

osmium, 114, 116, 117, 131p block, 22, 109Rare Earths, 18, 50, 129, 130rhenium, 113, 115, 131s block, 22, 109samarium, 65single, 106spherically symmetric, 21terbium, 4, 15titanium, 21–45, 107–114, 131transition metal, see transition

metal ionsuranium, 18, 66, 117, 118, 131vanadium, 82

axial, 24axis

[111], 34, 79‘axis-No.’, 32, 79fivefold, 50, 115fourfold, 24, 49, 80, 109main, 49, 79, 80threefold, 6, 79, 119twofold, 6, 18, 49z-, 6, 34, 119

B values, see ligand field, parame-ter, Wybourne

basischange of, see change of basisdefinition of, 3, 8, 90functions, 3, 7–9, 87, 90sizes, ii, 3, 4, 6–9, 13, 15, 16, 20,

23, 35, 46, 49, 52, 56, 68,75, 77, 84, 85, 88–93, 114,116, 117, 122, 123, 125

truncation of, 6, 7, 9, 35BATCH MODE, 30BLOCKDATA

GRIFFI, 50STEVEN, 50

133

134 Index

WIGNER, 50WWWWWW, 50

BLOCkoUT, see BLOCUTBLOCUT, see BLOCUT, BLOCk-

oUTBLOCkoUT, 23, 32–34, 41, 47

‘in’ CHANGE, 88‘in’ ECKART, 76, 77‘in’ TANABE, 70, 75

Subroutine, 77–80flowchart, 78

Bohr (unit of length), 66Bohr magneton, 2, 10, 11, 28, 37,

42, 44, 56, 96, 105, 114,115, 123

Boltzmann constant k, 11, 120BWYBIN.DAT, see file

CFMS (Crystal Field Minimal Split-ting), see ligand field

CFOS(J) (Crystal Field Over all Split-ting), see ligand field

cgs units, 2, 16, 25, 26, 38, 44, 50,56, 96, 104, 108–110, 116

CGS. . . .DAT, see fileCHANGE, 7, 9, 17

‘in’ BLOCUT, 79‘in’ ECKART, 76, 77‘in’ SAUNDE, 92Subroutine, 87–88

flowchart, 87change of basis, 6, 7, 9, 19, 20, 23,

35, 37, 47, 70, 75, 76, 80,81, 83–85, 87–93, 121

charge factor, see SOM (SimplifiedOverlap Model)

CHI∗FIELD, see (molar) magneti-sation

CHI-0, see diamagnetismCHI-calc., 12, 21, 44, 45, 54–56, 60,

95, 107, 113CHI-START, 11, 21, 38–40, 42, 55,

56, 61, 94–96, 112, 115corrections, 12, 21, 44, 48, 55,

95

mean value, 25, 38, 48, 49, 57,94, 109

CHI0, see diamagnetismCHIFIX, 38, 55, 96CHIINP.DAT, see fileCHIMAG, 11, 19, 21, 23, 37–39, 47,

48‘in’ SUGANC, 99, 100‘in’ SUGANR, 97, 98Subroutine, 103–104

flowchart, 104CHIZXY, 57, 95CKCOEfficient, see FUNCTIONCOMB.MOD., see Combined ModelCombined Model, 13, 14, 25, 36, 38,

48, 50, 56, 84, 89, 94, 95,100, 102, 104, 109, 123, 125,130

COMMAND FILES, 16COMPaRE, see COMPREcompile(r), 15, 16composition of eigenvectors, 7, 9, 79,

81COMPRE, see COMPRE,

COMPaRECOMPaRE

‘in’ REDUCT, 82‘in’ TANABE, 72–74

CONDON, see also: CONDON, pro-gram

.MAIN., 8, 15, 16, 18, 19, 29,46–52

flowchart, 51‘in’ BLOCkoUT, 79‘in’ ECKART, 75‘in’ SELECT, 95, 96‘in’ SHORTL, 51‘in’ SUSC, 65‘in’ SUSFIT, 53‘in’ TANABE, 70‘in’ TANFEL, 92description of, 46–104

.MAIN. in, 48flowcharts in, 46, 47, 51–53,

58, 62, 64, 67, 70, 76, 78,83, 85, 87, 89, 92–94, 97,

Index 135

99, 101, 103, 104interfaces in, 46–48, 65, 77,

87, 89, 93, 94, 96, 98, 100,102, 104

Subroutines in, 51, 53, 54, 57,61, 63, 66–68, 75, 77, 80,82, 87, 88, 91, 93, 96, 98,100, 102, 103

file condon, 16, 123file CONDON.f, 15, 46general flowchart, 18, 19, 46, 47general outline, 46–48

general flowchart, 47interface, 6, 11, 13, 17, 19–21,

29, 30, 35–39, 42parameter, see file,

BWYBIN.DATpractical guide to, 15–45PROGRAM, see CONDON

.MAIN.program, ii, 1–6, 8, 10–12, 14,

105, 113, 114, 118–121, 127parts of, 48

s1 system, 18–21, 105–107low temp., 105, 107

Subroutine, see SubroutineSubroutines, see CONDON, gen-

eral outlineTi(3+), 21–45, 107–114

Condon, see Condon, Shortley-Shortley coefficients, 71, 73parameter, see Slater-Condon

parameterShortley, ii, 4, 29, 68, 69, 71–

73, 80, 85, 91, 129condon, see CONDON, file condonCONDON.f, see CONDON, file

CONDON.fconfiguration, ii, 1, 4, 6, 8, 16, 22,

24, 49, 88, 109, 114convention

order, 3, 71, 72, 74phase, 5

convergence, 12, 27, 40, 54, 61–64,112, 123, 126

conversion of parameters, 20, 64, 65

Copyright, 22, 108corr. factor F, 20, 27, 44, 50, 56, 57,

60, 68, 110, 120covalence factor, see SOM (Simpli-

fied Overlap Model)crystal field, see ligand fieldcrystal quantum numbers, see sym-

metry quantum numbersCurie

constant, 12, 105straight line, 68, 118Weiss, see molecular field ap-

proachCurie’s law, 12, 105

d electrons, see also: 3d electrons,3–5, 7–9, 16–18, 23, 24, 27,28, 49, 51, 52, 66, 73, 74,77–80, 84, 86, 88, 90, 108,110, 113, 115, 117, 118

d orbitals, 23, 34data sources, see also: BLOCKDATA,

see also: tables included,17–18

diamagnetism, 2, 11, 12, 27, 28, 44,56, 57, 59, 60, 68, 96, 115

Dieke, 18, 51, 130DIMENSION, see PARAMETER,

ND. . .dimension of basis, see basis, sizesDIMENSIONS, see JEQUIVdimensions, see source codedimer

dysprosium, 124, 127, 132GdCu, 120, 122, 123, 132lanthanides, 130, 132symmetrical, 1, 2, 13, 14, 25,

27, 50, 52, 60, 95, 104, 109,125

unsymmetrical, 1, 2, 13, 14, 25,27, 50, 52, 60, 95, 104, 109,122

DIMER / COMB.MOD., see Com-bined Model

direction, see also: magnetic field,direction of

136 Index

[111], see axis, [111]cosinus, see magnetic field, di-

rection ofx-, y-, z-, 6, 7, 9, 10, 12, 21,

25, 36, 37, 42, 48, 49, 63,65, 83, 84, 86, 89, 90, 93–99, 101–103, 109, 115, 125,127

DISTance (DIMER), see magneticdipol-dipol interaction

DSPEV, 16, 70, 76, 77, 92, 93, 96,97

DSPEVD, 16DSPEVW, 15, 16, 19, 20, 37, 46,

47, 49, 70, 75–77, 92, 93,96, 97

dspzhp.f, see filedszhLP.f, see filedszhOQ.f, see file

ECKART, 6–9, 12, 47‘and’ .MAIN., 8‘and’ SUSC, 12‘in’ .MAIN., 49‘in’ BLOCUT, 77–79‘in’ CHANGE, 87, 88‘in’ REDUCE, 88, 89‘in’ SUSC, 63, 64‘in’ TANABE, 69Subroutine, 75–77

flowchart, 76Eckart, see Wigner, Eckarteigenstates, see state functionseigenvalues

degenerated, 11, 21, 44, 84, 90,98, 100

number of, see basis, sizesof atoms (ions), see atoms, en-

ergy levels ofOUTPUT of, 36, 37, 42z,x(,y)-(direction of field-), 37,

96–99eigenvectors

composition of, see compositionof eigenvectors

OUTPUT of, 7, 33–35, 37, 42,78, 79, 81, 82, 88

energies, see eigenvaluesenergy, see . . . , energy of

unit of (recipr. cm), 2, 11, 108,116, 120–123, 127

enthalpy H, see spin crossoverentropy S, see spin crossoverEQUIVALENCE, see JEQUIVexchange interaction, see Heisenberg

coupling, 7, 9, 12, 13, 120,122, 130, 132

expectation values of rn, 18, 66

f electrons, ii, see also: 4f, 5f elec-trons, 4, 5, 8, 15, 18, 49–52, 65, 66, 75, 77, 84, 87,88, 91, 122, 125, 130

F-integrals (F’s), 5, 28, 57, 70–72,118

factor CORR., 58factor F, see corr. factor Fferromagnetism, 12, 120file

BWYBIN.DAT, 16, 17, 19, 26,42, 47, 48, 50, 51, 54, 57,61, 65, 110, 113, 117, 119–122, 126

CHI-0, see diamagnetismcorr. factor F, see corr. factor

FHeisenberg coupling constant,

see Heisenberg coupling con-stant

molecular field parameter, seemolecular field theory pa-rameter

parameter λ, see molecularfield approach parameter λ

parameter Θ, see molecularfield approach Weiss con-stant Θ

parameter MIXture, 27, 57,60, 95

parameters, 16, 17, 20, 22,23, 27, 29, 30, 46, 48, 50,

Index 137

54, 59–63, 66, 67, 105, 106,108, 110, 111, 113, 119, 120,124–126

parameters ‘spin crossover’, seespin crossover

CGS. . . .DAT, 2, 16, 23, 25, 40,50, 108, 109, 122

CHIINP.DAT, 16, 19, 25, 29,47, 48, 50, 51, 58–61, 67,108, 109, 123, 125

condon, see CONDON, file con-don

CONDON.f, see CONDON, fileCONDON.f

dspzhp.f, 15, 16, 46dszhLP.f, 16dszhOQ.f, 16fort.1, 17, 29, 45, 48, 53, 54, 91,

105, 107, 113fort.11, 7, 17, 22, 23, 29, 45, 48,

53, 54, 81, 105, 108, 122fort.13, 17, 65, 122, 125fort.2, 7, 17, 81, 88, 91fort.3, 17, 88, 91fort.7, 16–18, 22–24, 48, 51, 105,

106, 108, 117, 121, 122, 124,125

fort.8, 17, 22, 48, 54, 57, 66,105, 113

fort.9, 17, 67OLDf7.DAT, 16pcdszh.f, 16rzdszh.f, 16shortL.f, 16, 19, 29, 46, 47, 51,

52, 70, 123shortO.f, 16, 19, 29, 46, 47, 51,

52, 70, 121shortP.f, 16, 19, 29, 46, 47, 51,

52, 70shortQ.f, 16, 19, 29, 46, 47, 51,

52, 70, 122str00.dat, 17, 66, 126str00.f, 16, 46

filled shell (more or less), 3, 11, 77,84

first

eigenvalue, 35, 63, 104eigenvector, 34, 35iteration, 22, 40, 43, 53, 54, 57,

59, 61, 62, 65, 122micro-state, 4, 71, 72, 86order Zeeman coefficients, see

Zeeman, coefficientsrun, 55, 59, 64, 70, 75, 76, 85,

89, 92, 93, 96susceptibility, 38temperature, 23, 26, 38, 39, 48,

110value of SQX, 40, 60

fitting, see also: Levenberg-Marquardt

least square, 27, 44, 52, 53, 57,58

matrix, 42, 62, 67procedure, 1, 12, 18, 19, 22, 23,

25, 29, 40, 46–48, 54, 55,57, 59, 61, 63, 65, 91, 96,108, 121, 123–127

flowchart(s), see CONDON, descrip-tion of, see Subroutine

FORMAT, 17, 23, 24, 26, 42, 50, 66,108, 110, 122, 125

FORTRAN, 15, 16, 22, 24, 46, 61,102, 109, 129

compiler, see compile(r)free enthalpy, see spin crossoverfull basis, see basis, sizes(full) micro-state basis, ii, 3–9, 15,

16, 20, 23, 30, 31, 46, 52,64, 68, 70, 77, 79, 80, 83,84, 87, 88, 91, 114, 116,121–123, 125, 127

FUNCTIONCKCOEfficient, 71–75DREIJ(‘3j-symbols’), 76

FUNCTIONS, see CONDON,description of

fundamental equation, 10, 11, 13,116

g-factor, 20, 35, 84, 85, 90, 95, 98,105, 121

138 Index

Gaunt’s formula, 71GAUSS, 23, 42

‘in’ LEVMAR, 62‘in’ RUSSEL, 80, 81, 83

general flowchart, see CONDONGerloch, 11, 130

method of, 11, 98, 100GNU

FORTRAN, 16general public license, 22, 46,

108good quantum number, 85, 90Griffith, ii, 4, 5, 17, 23, 24, 28, 50,

109, 110, 113, 114, 116, 129ground state/term, 4, 6–8, 22, 49,

69, 75, 77, 82, 84, 88, 90,98, 109, 114, 117, 121

group O(3), 6, 18, 24, 106, 109, 122

Henthalpy, see enthalpy Hor h ‘BWYBIN.DAT’, 26, 105,

110or h for ‘H-ATOM’, see atoms,

hydrogenHamiltonian

ligand field, see ligand field, op-erator

spin-, 95, 120–122header, 17, 26, 27, 29, 48, 54, 56,

57, 110Heisenberg

coupling, 1, 7, 9, 13, 14, 20, 48,52, 84, 86, 88, 89, 91, 101,122, 123

constant, 2, 13, 27, 57, 60,101, 103, 120–123, 126, 127

model, 14, 86, 91, 124operator, 2, 13, 101, 120

hermitian matrix, 10, 15, 46, 48, 98,99, 102, 103

high-spin, 118–120

imaginary . . . , 18, 34, 50, 89, 98,102

INBWWW, see parameters to be fit-ted

individual sets, 71, 72, 74infinite

g-factor, 84, 90ligand field, see ligand field, in-

finite strongspin-orbit coupling constant, 8,

75inner shells, 5, 70INPUT, see file, BWYBIN.DAT,

CHIINP.DATinterelectronic repulsion, 1, 4, 5, 8,

20, 29, 69, 70, 73, 75, 117energy of, 3, 5, 69operator of, 3, 5, 69, 70

interface(s), see CONDON, descrip-tion of, see CONDON

intermediate field, see ligand field,intermediate

ions, see atomsirreducible tensor operators, ii, 8,

49, 69, 75, 76, 130Ising model, 14, 86, 91, 95, 101iteration, 17, 33, 41, 48, 53, 54, 59,

62, 63, 91, 93, 123count, 54first, see first, iterationlast, 22, 112, 113, 126more, 30, 54, 62number of, 23, 29, 30, 54, 59,

61, 108, 111, 112, 126zeroth, 22, 29, 30, 40, 43, 48,

53, 54, 57–61, 84, 92, 106,107, 111, 125

J , see Heisenberg, coupling constant3j-symbols, see FUNCTION, DREIJJ value (NRVALJ), 6, 8, 44, 49, 75,

84, 85, 89, 90, 107, 112,113, 117, 125

Jahn-Teller interaction, 131JEQUIV, 15, 19, 29, 46, 47, 51–53,

70, 111, 123

Kronecker product, 13, 100–102

Index 139

KSI, see ZETA

l value (1 el.), see also: s, p, d, f el.,3, 6, 22, 24, 45, 49, 50, 56,72, 75, 83, 107, 109, 113

lanthanides, 15, 18, 66, 130–132LAPACK, see library, LAPACKlast

iteration, see iteration, lastline (‘energy’), 40micro-state, 4

Lea, Leask, Wolf, see ligand field,parameter

LEVDER, see LEVDER, LEVenberg-DERivatives

LEVenbergDERivatives, 18, 40,47

‘in’ LEVMAR, 61, 62‘in’ LEVSQX, 57‘in’ SUSC, 63, 64‘in’ SUSFIT, 54


Levenberg-Marquardt, 1, 18, 30, 40,53, 54, 57, 59, 61–63

LEVenbergDERivatives, see LEVDERLEVenbergMARquardt, see LEVMARLEVenbergSQX, see LEVSQXLEVMAR, see LEVMAR, LEVenberg-

MARquardtLEVenbergMARquardt, 17, 30,

40, 42, 47‘in’ LEVDER, 67, 68‘in’ LEVSQX, 57, 59‘in’ SUSC, 63‘in’ SUSFIT, 53, 54


LEVSQX, see LEVSQX, LEVenberg-SQX

LEVenbergSQX, 18, 19, 29, 39,40, 47

‘in’ LEVDER, 67‘in’ LEVMAR, 61, 62‘in’ SUSC, 63, 64‘in’ SUSFIT, 53, 54


libraryESSL, 16LAPACK, 16, 130NAG19, 16routines, 15, 16, 46

ligand field, ii, 1, 5, 8, 18, 20, 32,70, 73, 117, 130

5f electrons, 118CFMS, 40, 43, 44, 106, 112CFOS(J), 40, 43, 44, 63, 106,

112, 124, 126cubic, 6, 18, 24, 49, 50, 65, 78,

79, 108, 109, 113–122, 126,130

energy diagrams, ii, 114, 129energy of, 3, 75infinite strong, 115, 116intermediate, iioctahedral, 113, 116–119operator, 3, 5, 6, 8, 18, 50, 70,

73–77, 79, 92, 93cubic, 6

parameter, 1, 3, 50, 59, 64, 84,110, 119, 131

Lea, Leask, Wolf, 6, 65, 120,130

Stevens, 6, 51, 65, 130Wybourne, 2, 3, 5, 9, 17, 20,

24, 27, 30, 45, 47, 49, 50,57, 60, 64–66, 70, 73–76,85, 89, 91, 93, 109, 111,113, 118, 122, 123, 125, 126

pentagonal, 50, 115potential, 50, 79rhombic, 49, 60, 122, 125strength of, 6strong, ii, 6, 114strong field scheme, 114, 116tetragonal, 24, 49, 79tetrahedral, 118, 121, 122trigonal, 22–24, 78, 79, 108, 114,

118, 119weak, ii, 6, 114weak field scheme, 114

140 Index

limits for parameters, 17, 23, 42, 62,126

link, 4, 15, 16, 29, 46, 51, 70, 76, 77,92, 93, 96–99, 121, 123

LINUX, 16low / high lying states, 7, 9, 20, 46,

84, 88, 89, 92, 116low symmetry, 18, 49, 65, 66, 126low-spin, 118–120LS, see Russell, Saunders

MAGN.MOMENT (1 ATOM), seemagnetic moment

magnetic centre, 2, 104magnetic dipol-dipol interaction, 13,

50, 101, 103, 121–123, 125,127

distance of 2 ions, 50, 57, 101,103, 121, 122, 125

magnetic field, 1, 2, 7, 9, 10, 13,19, 21, 23, 25, 26, 30, 32,36, 38, 42, 45, 47, 48, 55,58, 59, 63, 67, 77, 96–103,105–110, 113, 117, 122, 125

change of, 6, 19, 30, 98, 100,102, 103

direction of, 1, 10, 11, 23, 25,36, 42, 48–50, 57, 94–99,101, 103, 104, 108, 115

energy of, 10, 46, 48, 115saturation, 105strength of, 1, 10, 11, 105, 116,

121, 122, 124magnetic moment, 2, 28, 38, 44, 56,

58, 96, 104effective, 28, 44, 56, 58, 96, 104,

105, 113–115, 121, 123pseudo effective, 38, 44, 56, 96,

104saturation, 44, 56, 96

magnetic moments, see (neg.) mag-netic moments

magnetic quantum numbers, 3, 20,21, 68, 82, 85, 86

magnetic susceptibility, see (molarpara)magnetic susceptibil-

ity, see also: diamagnetismmagnetisation, see (molar) magneti-

sationmagnetism diagrams, ii, 129Marquardt, see Levenberg-

MarquardtMATRIV

‘in’ TANFEL, 92MATRIW

‘in’ TANFEL, 92matrix

complex, see also: hermitian ma-trix, 53, 99, 100, 102

diagonal, 6, 10, 13, 15, 20, 21,32, 34, 36, 37, 41, 42, 46,48–50, 64, 70, 75, 77, 78,85, 88, 90, 92, 93, 96–103

fitting, see fitting, matrixhermitian, see hermitian matrixreal, 6, 15, 33, 46, 50, 52, 53,

64, 77, 78, 97, 101reduced elements, see reduced

matrix elementsunit, 20, 48, 87, 90, 96, 97, 99–

101micro-state basis, see (full) micro-

state basisMJ values, see symmetry quantum

numbers(molar) magnetisation, 2, 11, 12, 21,

38, 44, 48, 56, 63, 96, 103,104, 107, 120

(molar para)magnetic susceptibility,ii, 1, 2, 6, 9–11, 13, 21–24,26, 29, 39, 52, 53, 58, 63,67, 105, 108–110, 114, 115,118, 120–122, 129

molecular field, 1, 12, 29, 60, 63, 64,96, 120

approach (Curie-Weiss), 1, 12parameter λ, 12, 27, 44, 55–

57, 60, 68, 95, 96, 120Weiss constant Θ, 12, 27, 56,

57, 60, 96theory (Weiss), 1, 12

parameter, 12, 27, 57, 60

Index 141

monomer, 13, 52, 60

Nassi-Shneiderman flowcharts, seealso: flowchart(s), 46, 51,53, 59, 63, 65, 68, 69, 77,80, 82, 84, 88, 91, 93, 95,98, 100, 102–104

NdPt2, 120, 121, 131neg. B40, 24neg. CHI, 11, 61(neg.) magnetic moments, 10, 11,

14, 21, 37, 38, 42, 48, 84,97–100, 104

neg. temperatures, 61, 63, 64, 96ns, np, nd, nf electr. conf., 1, 4, 22,

24, 109NUMBER, 4, 92

‘in’ REDUCT, 82, 83, 86number of electrons (NREL), 3, 28,

50, 56, 77, 83, 110, 117,122, 125

Numerical Recipes, 1, 54, 61, 129

OMEGA, see option(s) (for) OMEGAoption(s) (not all!)

‘more’ opt., see JATEST belowanalytical, 28, 62, 68cgs. . . , see file, CGS. . . .DATINFORM(atfree), see file,

fort.13IOPT1, 21, 23, 25, 49, 50, 57,

108, 109, 117, 122, 125, 127IOPT12, 25, 36IOPT13/IOPT99, 36, 39, 40IOPT2, 21, 23, 25, 50, 95, 106,

108, 109, 117, 122, 125IOPT3, see CHI-0IOPT4, 23, 28, 57, 58, 68, 96,

117, 122, 125JACuGd, 2, 14, 95, 104, 120,

122, 123, 132JAFAST(er), 17, 29, 52, 56, 65,

71, 84, 87, 91, 92, 123, 125JANOL(ea,Leask,Wolf), see

Lea, Leask, WolfJANOS(tevens), see Stevens

JASTOP (in advance), 23, 30,54, 107, 113, 125

JASTRU(cture), see STR00JATEST, 11, 14, 22, 24, 49, 65,

77, 84, 86, 89–91, 98, 100,104, 109, 121, 124

JAVANV(leck), see VanVleck, option

JAWIGN(er), see Wigner, op-tion

LESS (OUTPUT), 21, 22, 24,49, 54, 55, 60, 61, 65, 105–109, 113

LTS (less ‘tests’), 22, 24, 49, 57,65, 105, 106, 108, 109

MFITIN (form of OUTPUT),23, 43, 54, 55, 113, 117,122, 125

MOROUT(put), 17, 20, 22–24,30, 31, 33, 35, 36, 39, 41,42, 49, 77–79, 84, 88, 90,98, 104, 109, 121

MORTEX(t), 22–24, 49, 109NOPROZ (not in %), 23, 57,

58, 60, 68, 117, 122, 125(for) OMEGA, 30, 40, 54, 59,

61–63, 126special for B44,. . . , 27, 50, 60,

65, 119, 127VANVLECK, see Van Vleck,

optionWIGNER, see Wigner, option

orbital reduction factor, 1, 10, 23,28, 50, 84, 86, 89, 90, 113,115, 116, 123–127

OUTPUT, see file, fort.11, . . .routine, see RESTBL, RESult-

TaBLes

p electrons, see 2p electronsparamagnetic susceptibility, see (mo-

lar para)magnetic suscep-tibility

PARAMETERND1110, 52, 100ND2002, 16, 49, 52

142 Index

ND252, 35, 52ND28, 52, 100, 102NDS66, 52, 102, 103

parameters to be fitted, see file,BWYBIN.DAT, see file, CGS. . . .DAT

partition function, 11pcdszh.f, see filePCEM (Point Charge Electrostatic

Model), 27, 65, 66perturbation operator, 10, 11, 13,

14, 35, 42(point) group(s), see also: ligand

field, cubic, . . . , 6, 18, 24,34, 49, 50, 60, 66, 109, 126,130

product states, 13, 101PROGRAM CONDON, see CON-

DON .MAIN.program CONDON, see CONDON

quality of a fit, see also: SQX in %,18, 27, 57, 63, 127

quantum number l, see l value (1electron)

Racah parameter, 4, 5, 17, 23, 24,28, 51, 57, 70, 71, 109, 110,115–117

Rare Earths, see atoms (ions), RareEarths

real spin orbitals, see spin orbitalsREDUCE, see REDUCE, REDUC-

EckartREDUCEckart, 9, 47, 76, 77

‘in’ SELECT, 94‘in’ SUGANH, 101


REDUCEckart, see REDUCEreduced matrix elements, 76REDUCT, see REDUCT, REDUC-

TanabeREDUCTanabe, 7, 19, 20, 23,

35, 39, 47‘in’ SELECT, 94

‘in’ SUGANH, 101‘in’ TANABE, 70, 75‘in’ TANFEL, 91–93


REDUCTanabe, see REDUCTrelativistic values, 18, 51, 66, 118RESTBL, see RESTBL, RESultTa-

BLesRESultTaBLes, 19, 21, 23, 29,

38, 39, 47, 96, 117, 122,125

‘in’ SELECT, 96‘in’ SUSFIT, 53

Subroutine, 54–57RESultTaBLes, see RESTBLREWIND(8), 17, 48, 54RUSSEL, see RUSSEL, RUSSELl

RUSSELl, 6, 7, 17‘in’ CHANGE, 88‘in’ SAUNDE, 92‘in’ TANABE, 70, 75


RUSSELl, see RUSSELRussell, see Russell, Saunders

Saunders (LS), ii, 4–9, 13, 22,49, 64, 68–70, 75–83, 87,88, 92, 109, 114, 121

s electron(s), see 1s electron(s), seealso: s orbital occupation

s orbital occupation, 28, 111SAUNDE, see SAUNDE, SAUNDErs

SAUNDErs, 6, 7, 9, 17‘in’ CHANGE, 87, 88‘in’ RUSSEL, 80, 81, 83

Subroutine, 91flowchart, 92

SAUNDErs, see SAUNDESaunders, see Russell, SaundersSCHEMA, 4, 20, 23, 31

‘in’ CHANGE, 88‘in’ TANABE, 68, 70

SELECT, 14, 19, 21, 23, 25, 36, 38,39, 47, 48

Index 143

‘in’ CHIMAG, 104‘in’ REDUCE, 89‘in’ REDUCT, 85‘in’ SUGANC, 99, 100‘in’ SUGANH, 101, 102‘in’ SUGANK, 103‘in’ SUGANR, 97, 98‘in’ SUSFIT, 55‘in’ TANFEL, 93Subroutine, 93–96

flowchart, 94SHORTL, 15, 16, 19, 29, 46, 47, 51,

53, 70Subroutine, 51–53

flowchart, 52shortL.f, see fileShortley, see Condon, ShortleyshortO.f, see fileshortP.f, see fileshortQ.f, see fileSI units, 2, 17, 25, 26, 38, 44, 50, 55,

56, 95, 104, 107, 109, 110,118–121, 123, 124, 127

similarity transformation, see also:change of basis, 7, 9, 23,34, 79, 85, 87, 90, 92, 93,98

simulatedannealing, 63single J value, 8, 49, 75strong field scheme, 115, 116

site symmetry, 107, 118Slater determinant, 3, 13, 71Slater-Condon parameter, 3–5, 18,

51, 69–71, 118, 125SOM (Simplified Overlap Model), 27,

65, 66, 126, 131source code, 15, 16special fits, see analyticalspherical harmonics, 5, 71, 73spin crossover, 1, 27, 29, 57–60, 64,

67, 118, 119, 129spin orbitals, 32–34, 41, 71, 73, 78,

79spin-orbit coupling, 4, 8, 20, 29, 32,

70, 73, 75, 77, 114, 117

constant of, 1–3, 5, 8, 17, 23,24, 28, 29, 45, 49, 51, 57,73, 75, 77, 109, 110, 113,115–118, 121, 122, 125

energy of, 3, 5operator of, 3, 5, 8, 70, 73, 76,

77SQX in %, 27, 44, 53, 54, 58, 63, 68,

107, 112, 113, 121, 126standard

deviations, 26, 54, 55, 57, 120,124

setting, 66table of OUTPUT, see also: op-

tion(s), MFITIN, 55, 61state functions, 3, 6, 9, 10, 13, 117step operators, 4, 32, 82, 83, 85, 86,

89–92Stevens, see ligand field, parameterstorage (more or less), ii, 4, 15, 53,

69, 100, 102, 104, 121, 123STR00, 15–18, 23, 27, 29, 46, 47,

64, 65, 117, 122, 125, 126Subroutine, 66–67

str00.dat, see filestr00.f, see filestrong field, see ligand field, strongstructure of complex, 65, 66, 126,

131subprograms, 17, 28, 46, 49, 50Subroutine

BLOCUT, see BLOCUTCHANGE, see CHANGECHIMAG, see CHIMAGCOMPRE, see COMPREDSPEV, see DSPEVDSPEVD, see DSPEVDDSPEVW, see DSPEVWECKART, see ECKARTGAUSS, see GAUSSLEVDER, see LEVDERLEVMAR, see LEVMARLEVSQX, see LEVSQXMATRIV, see MATRIVMATRIW, see MATRIWNUMBER, see NUMBER

144 Index

REDUCE, see REDUCEREDUCT, see REDUCTRESTBL, see RESTBLRUSSEL, see RUSSELSAUNDE, see SAUNDESCHEMA, see SCHEMASELECT, see SELECTSHORTL, see SHORTLSTR00, see STR00SUGANC, see SUGANCSUGANH, see SUGANHSUGANK, see SUGANKSUGANO, see SUGANOSUGANR, see SUGANRSUSC, see SUSCSUSFIT, see SUSFITTANABE, see TANABETANFEL, see TANFELTRANSF, see TRANSFZHPEV, see ZHPEVZHPEVD, see ZHPEVDZHPEVW, see ZHPEVW

Subroutines, see CONDON, descrip-tion of

SUGANC, see SUGANC, SUGAN-Complex


SUGANComplex, 36, 47‘in’ CHIMAG, 103, 104‘in’ SELECT, 94‘in’ SUGANK, 102, 103

SUGANComplex, see SUGANCSUGANH, see SUGANH, SUGAN-

HeisenbergSubroutine, 100–102

flowchart, 101SUGANHeisenberg, 36, 47

‘in’ SELECT, 94, 95‘in’ SUGANK, 102, 103‘in’ SUGANR, 97, 98

SUGANHeisenberg, see SUGANHSUGANK, see SUGANK, SUGAN-

K(omplexH)eisenbergSubroutine, 102–103

flowchart, 103

SUGANK(omplexH)eisenberg, 36, 47

‘in’ SELECT, 94, 95‘in’ SUGANC, 99, 100‘in’ SUGANH, 102

SUGANK(omplexH)eisenberg, see SUGANK

SUGANO, 23, 36, 37Sugano, see Tanabe, SuganoSUGANR, see SUGANR, SUGAN-

RealSubroutine, 96–98

flowchart, 97SUGANReal, 19, 21, 36–39, 47

‘in’ .MAIN., 49‘in’ CHIMAG, 103, 104‘in’ REDUCE, 90‘in’ REDUCT, 85‘in’ SELECT, 94‘in’ SUGANH, 100–102‘in’ SUGANK, 102

SUGANReal, see SUGANRSUSC, see SUSC, SUSCeptibility


SUSCeptibility, 12, 18–20, 29,30, 39, 47

‘in’ ECKART, 76‘in’ LEVDER, 67‘in’ LEVSQX, 58, 59‘in’ STR00, 66‘in’ TANABE, 70

SUSCeptibility, see SUSCSUSFIT, 18, 19, 29, 40, 47, 51, 52

‘in’ LEVMAR, 61, 62‘in’ LEVSQX, 58Subroutine, 53–57

flowchart, 53symmetry quantum numbers, 6, 18,

32, 77–79, 87, 88

tables included, see also: datasources, 18, 66, 83

TANABE, 3, 6, 8, 9, 12, 19, 20, 23,29–31, 35, 39, 47

‘and’ .MAIN., 8

Index 145

‘and’ SUSC, 12‘in’ BLOCUT, 77, 78‘in’ ECKART, 75‘in’ REDUCT, 85‘in’ RUSSELl, 83‘in’ SUSC, 63, 64‘in’ TANFEL, 91, 92Subroutine, 68–75

flowchart, 70Tanabe, see Tanabe, Sugano

Sugano, ii, 114, 129TANabe for F ELectrons, see TAN-

FELTANFEL, see TANFEL, TANabe for

F ELectronsSubroutine, 91–93

flowchart, 93TANabe for F ELectrons, 52

‘in’ REDUCT, 84, 85, 87‘in’ SELECT, 94‘in’ TANABE, 70, 75

temperaturechange of, 19, 21, 36, 47, 97–

104first, see first temperatureindependent paramagnetism

(TIP), 11, 115low, see CONDON s1 system,

low temp.minus Θ, 12, 38, 56, 61, 96neg., see neg. temperatures

terminal, 7, 16–19, 47–49, 51, 53,54, 81, 105

test calculation, see CONDON, s1

systemtheory of

atomic spectra, see atomic spec-tra

electric and magnetic suscepti-bilities, see Van Vleck

transition metal ions, see Grif-fith

Weiss, see molecular field the-ory

TRANSF‘in’ TANFEL, 92, 93

transition metal ions, see Griffith,see magnetism diagrams

truncation of basis, see basis, trun-cation of

values to be fitted, 16, 18, 23, 25,28, 30, 38, 40, 43–45, 50,54, 55, 57, 58, 60, 61, 63,95, 96, 107–109, 113, 115,122, 125, 126

Van Vleck, ii, 129equation, 1, 10, 11, 13, 14, 21,

25, 44, 48, 49, 96–99, 101,103, 107, 116

option, 1, 10, 23, 25, 38, 48, 49,94, 96, 98, 100–102, 104,117, 122, 125

VANVLECK (JAVANV), see VanVleck, option

wave functions, see also: state func-tions, 5

weak field, see ligand field, weakweight of sample, 27, 44, 57, 60Wigner, 130

Eckart theorem, 8, 9, 49, 75, 76option, ii, 1, 4, 6–8, 15, 18, 22,

49, 51, 69, 75, 88, 109, 114,122–125

WIGNER (JAWIGN), see Wigner,option

(Wigner-)ECKART, see ECKARTWybourne, see ligand field, param-

eter, 129

XY model, 14, 86, 91, 95, 101, 122,124

z,x(,y)-(direction of field-) eigenval-ues, see eigenvalues

z-axis, see axis, z-Zeeman

coefficients, 11, 14, 21, 48, 97–100, 104, 117, 122

corrections, 11, 97–100effect, 10, 13

146 Index

operator, 6, 7, 9, 10, 13, 20, 48,83, 89

zero-field splitting, 122ZETA, see spin-orbit coupling, con-

stant ofZHPEV, 16, 98, 99ZHPEVD, 16ZHPEVW, 15, 16, 46, 47, 98, 99

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