the generalized active space concept in multiconfigurational self-consistent field methods

THE JOURNAL OF CHEMICAL PHYSICS 135 044128 (2011)

The generalized active space concept in multiconfigurationalself-consistent field methods

Dongxia Ma1 Giovanni Li Manni2 and Laura Gagliardi1a)

1Department of Chemistry and Supercomputing Institute University of Minnesota Minneapolis Minnesota55455-0431 USA2Department of Physical Chemistry University of Geneva 1211 Genegraveve Switzerland

(Received 12 May 2011 accepted 28 June 2011 published online 29 July 2011)

A multiconfigurational self-consistent field method based on the concept of generalized active space(GAS) is presented GAS wave functions are obtained by defining an arbitrary number of activespaces with arbitrary occupation constraints By a suitable choice of the GAS spaces numerousineffective configurations present in a large complete active space (CAS) can be removed whilekeeping the important ones in the CI space As a consequence the GAS self-consistent field approachretains the accuracy of the CAS self-consistent field (CASSCF) ansatz and at the same time can dealwith larger active spaces which would be unaffordable at the CASSCF level Test calculations onthe Gd atom Gd2 molecule and oxoMn(salen) complex are presented They show that GAS wavefunctions achieve the same accuracy as CAS wave functions on systems that would be prohibitive atthe CAS level copy 2011 American Institute of Physics [doi10106313611401]

I INTRODUCTION

Multiconfigurational self-consistent field (MCSCF)methods1ndash3 generate accurate wave functions for chemi-cal problems of strong non-dynamical correlation energysuch as bond breaking and dissociations4 potential energyhypersurface degeneracies (conical intersections)5 sym-metry breaking problems (Cope rearrangement)6 biradicalsituations7 8 organic molecules photophysics9ndash12 transitionmetal bonding13ndash18 and spectroscopy19ndash24 and actinidechemistry25ndash28 MCSCF wave functions are often used asreference wave function for subsequent multireferenceconfiguration interaction or perturbation calculations suchas complete active space second-order perturbation theory(CASPT2)29 30 to include the dynamical correlation

The most commonly used MCSCF approach is the com-plete active space SCF (CASSCF) method31 In CASSCF aset of molecular orbitals is chosen to be active and all pos-sible configurations constructed from this set of active or-bitals with correct space and spin symmetry form a config-uration space A full configuration interaction (FCI) wavefunction is generated in the configuration space and at thesame time the orbitals are optimized via all possible ro-tations between inactive-active active-virtual and inactive-virtual spaces CASSCF has become the most popular MC-SCF method mostly because the wave function is completelydefined by selecting the active orbitals

Since a FCI is performed within the CAS space the ma-jor drawback of CASSCF is that the number of configurationstate functions (CSFs) or Slater determinants (SDs) scalesfactorially with the number of active orbitals and the worstcase is when the number of active electrons is about the sameas the number of active orbitals CASSCF calculations with

a)Author to whom correspondence should be addressed Electronic mailgagliardumnedu

a CAS larger than 16 electrons in 16 orbitals are currentlynot feasible Most of the configurations in the CI space con-tribute only marginally to the total wave function Ivanic andRuedenberg32ndash35 systematically investigated the amount of in-effective configurations (the authors referred to them as ldquothedeadwoodrdquo) present in a FCI wave function and found thatif one is aiming for chemical accuracy (1 mhartree) the dead-wood represents more than 99 of the FCI space Henceone way to reduce the computational cost is to put some con-straints on the active space to remove some of the ineffectiveconfigurations

One can for example partition the active space into sev-eral subspaces and then apply limits on the occupations ofeach subspace Well-known procedures are the generalizedvalence bond method36 constrained CASSCF (CCASSCF)method37 quasi-CASSCF (QCASSCF) method38 restrictedCI (RCI) method39 40 occupation-restricted-multiple-active-space (ORMAS) SCF method41 and restricted active space(RAS) SCF method42 43 In the CCASSCF method the ac-tive space is partitioned into an arbitrary number of or-bital spaces and the electron occupation number of eachspace is kept constant (disconnected spaces) the CI expan-sion is expressed in term of CSFs QCASSCF follows thesame partitioning scheme as CCASSCF but the CI expan-sion is expressed in terms of Slater determinants rather thanCSFs in order to obtain a more efficient direct-CI basedalgorithm

A RAS CI space is obtained by dividing the active spaceinto three subsets RAS1 RAS2 and RAS3 The total num-ber of active electrons in all the three RAS spaces togetherwith the maximum number of holes in RAS1 and the max-imum number of particles in RAS3 are used as restric-tions to define the configuration space Normally the neardoubly occupied orbitals are put in RAS1 near empty or-bitals in RAS3 and most active orbitals in RAS2 The RAS

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044128-2 Ma Li Manni and Gagliardi J Chem Phys 135 044128 (2011)

structure includes many usual CI spaces as special cases Forinstance when there are no orbitals in RAS1 or RAS3 RASreduces to CAS In a sense the RAS1 and RAS3 spaces addsome dynamic correlation to the RAS2 space The RASSCFapproach has been recently employed to generate referencewave functions for subsequent perturbation treatment to sec-ond order the RASPT2 approach44ndash47 We have also recentlyproposed a new approach SplitCAS to determine a suitablezeroth-order wave function for multiconfigurational pertur-bation theory The same ansatz as in complete active spacewave function optimization is split in two parts a princi-pal space (A) and a much larger extended space (B) Parti-tioning technique of Loumlwdin is employed to map the initialeigenvalue problem to a dimensionality equal to that of (A)only48

The concept of generalized active space (GAS) was firstproposed by Olsen42 49 It can be understood as a general-ization of RAS Instead of three active spaces in principleGAS allows an arbitrary number of active spaces Instead of amaximum number of holes in RAS1 and a maximum numberof electrons in RAS3 accumulated minimum and maximumelectron occupation numbers are used in GAS to define thewave function

In this paper we describe a new implementation ofthe concept of GAS self-consistent field (GASSCF) Thereare some similarities between our GASSCF method and theORMAS-SCF method41 In both of them an arbitrary num-ber of orbital spaces can be defined the configuration spacesare both expanded in SDs and inter-space electron excitationsare allowed (connected spaces) However these two methodsdiffer in the way that the electron occupation number con-straints for active spaces are defined The final CI spaces maybe different and as a consequence each method may haveadvantages or limitations according to the systems in exam

The paper is organized as follows In Sec II we for-mulate the GAS wave function from a theoretical point ofview and we describe the algorithm We also compare theGAS wave function to the CAS and RAS wave functions andto ORMAS approach In Sec III we present some test cal-culations on the Gd atom the Gd2 molecule and the Ox-oMn(salen) complex Finally in Sec IV we present someconclusions

II THEORY AND ALGORITHM

A Definition of the GAS wave function

The following input parameters need to be specified (1)number of GAS spaces ngas (2) number of orbitals in eachGAS space per each irreducible representation and (3) accu-mulated minimum and maximum number of electrons occu-pying the GAS spaces minocc(igas) and maxocc(igas) (igasruns from 1 through ngas) In other terms one has to definethe minimum and maximum electron occupation number forthe first space then the minimum and maximum electron oc-cupation number for the first two spaces (GAS1 + GAS2)and so on all the way to the whole active space

An example to demonstrate the accumulated occupationnumbers is as follows

GAS1 GAS2 GAS3 GAS4Minocc 0 4 5 8Maxocc 2 4 6 8

The distribution of these electrons among the GASspaces in this case is the first orbital space contains from zeroup to two electrons the first two spaces together contain fourelectrons in total the first three spaces contain from five upto six electrons the occupation counts for all four spaces to-gether must be same as the total number of active electrons

B Reduction of the GAS wave function to the RASand CAS wave function

It can be easily proved that CAS and RAS are two specialcases of GAS

If a RAS wave function is defined bynactel total number of active electronsorbitals in RAS1 RAS2 and RAS3 spacesnhole1 maximum number of holes in RAS1nelect3 maximum number of electrons in RAS3

The equivalent GAS function is defined as followsngas = 3orbitals in each GAS spaces same as in RAS spacesminocc(1) = number of spin orbitals in RAS1minus nhole1maxocc(1) = number of spin orbitals in RAS1minocc(2) = maxnactel minus nelec3 minocc(1)maxocc(2) = minnactel number of spin orbitals inRAS1 + RAS2minocc(3) = maxocc(3) = nactel

CAS is an even simpler case The equivalent GAS wave func-tion is define by

ngas = 1minocc(1) = maxocc(1) = nactel

A pictorial description of the GAS wave function to-gether with the CAS and FCI wave functions is reported inFig 1 The vertical line defines a single determinant refer-ence state (HF state) All possible distributions of the elec-trons among the reference orbitals generate the FCI space(big circle) If instead the reference orbitals are partitioned

FIG 1 Graphical description of HF FCI CAS and GAS configurationspaces


044128-3 Generalized active space SCF J Chem Phys 135 044128 (2011)

into three groups (inactive active and virtual orbitals) andonly the permutations within the active orbitals are allowedleaving the inactive orbitals doubly occupied and the virtualempty the CAS space is generated (intermediate circle) Themissing correlation is described in the figure as the differencebetween the two circles and arises from the fact that no exci-tations are allowed fromto the inactivevirtual orbitals In theGAS formulation the reference orbitals are partitioned intoan arbitrary number of spaces (little circles) Intra-space exci-tations are allowed (both connected and disconnected circles)as well inter-space excitations (connected circles)

C Direct CI

The CI algorithm for GAS is analogous to the one forRAS described in Ref 42 We describe it only briefly here Itis determinant based and it uses Handyrsquos technique to separatethe determinants into alpha strings and beta strings50 51

|α(Iα)β(Iβ)〉 = α(Iα)β(Iβ)|vac〉 (1)

Graphical representation of alpha strings and beta strings fol-lowing reverse lexical order are used to order the strings

For the CI expansion

|0〉 =sum

IαIβ

C(Iα Iβ )|α(Iα)β(Iβ)〉 (2)

the direct CI σ vector is defined as

σ (Iα Iβ )=sum

JαJβ

〈β(Jβ )α(Jα)|H |α(Iα)β(Iβ)〉C(Jα Jβ)

(3)where H is the non-relativistic electronic Hamiltonian in afinite basis

H =sum

kl

hklEkl + 1

2

sum

ijkl

(ij |kl)(Eij Ekl minus δjkEil) (4)

and Ekl excitation operator

Ekl = adaggerkαalα + a

daggerkβalβ (5)

The two electron contribution can be divided into three termsone term involving two α excitations one involving two β

excitations and the third one involving mixed excitations

D Orbital optimization

The current implementation of GASSCF consists of atwo-step procedure At each iteration a CI is performed firstand then followed by orbital optimization according to thesuper-CI scheme52 with the quasi-Newton update53 as a con-vergence accelerator

At the end of each CI optimization step a reference state|0〉 is obtained A unitary transformation of orbitals is per-formed by an exponential operator

exp(iλ) = 1 + iλ + 1

2(iλ)2 + middot middot middot (6)

where

λ = isum

pgtq

κpq(Epq minus Eqp) = isum

pgtq

κpqEminuspq (7)

By truncating exp(iλ)|0〉 after the first two terms one obtainsa variational wave function

|0〉 = |0〉 minussum

pgtq

κpqEminuspq |0〉 (8)

The linear variational parameters κpq can be obtained bysolving the superconfiguration interaction (super-CI) secularproblem

HX = ESX (9)

where

H0pq = 〈0|H |pq〉 = 〈0|H Eminuspq |0〉 (10)

Hpqrs = 〈pq|H |rs〉 = 〈0|EminuspqH Eminus

rs |0〉 (11)

and S is the overlap matrix of the super-CI states |pq〉Spqrs = 〈0|Eminus

pqEminusrs |0〉 (12)

After solving for the parameters κpq the molecular orbitalsare transformed to a new set of orbitals With the new set oforbitals a new GASSCF iteration is performed until the κpq

are all equal to zero which indicates that the GASSCF con-vergence has been reached As in the RASSCF scheme theorbital rotations within each GAS space are redundant whilethe inter-GAS rotations are included in the orbital optimiza-tion Some of these rotations might be quasi-linear dependentfor some choices of GAS wave function This may cause con-vergence difficulties

E A comparison between GAS and ORMAS

As already mentioned the GAS and ORMAS wave func-tions differ in the definition of the various spaces While OR-MAS requires a minimum and a maximum electron occupa-tion number for each space GAS requires an accumulativeminimum and maximum electron occupation number for eachspace The CI expansions are thus different in the two ansaumltzeWe will now describe two extreme cases (Fig 2) to illustratethe two different approaches

1 Case in which ORMAS is preferable

Let us assume to have a system with K active orbitalspaces and a number of active electrons defined as

N = 1 +Ksum

i=1

Ni

where Ni is the number of electrons in each orbital spaceinvolved only in intra-space excitations and one extra elec-tron that can be added to any of the K spaces This electronguarantees that there are always inter-space excitations (seeFig 2(a))

Since the single electron is involved in inter-space ex-citations the K spaces are all connected By following theORMAS scheme it is possible to specify a minimum elec-tron occupation number equal to Ni and a maximum equalto Ni + 1 per each space As an effect of this choice the CI



(a)

(b)

FIG 2 Pictorial representation of the ORMAS (a) and GAS (b) schemes

expansion will contain all the configurations with the singleelectron located in different orbital spaces

Within the GAS scheme this system could be describedby electron occupation number constraints shown in Table I

This GAS CI space includes the whole ORMAS CI spaceand also some multiple inter-space excitations which aremostly ineffective configurations In this case ORMAS-SCFwould be more efficient than GASSCF For example the an-ion of a long chain conjugated π system

2 Case in which GAS is preferable

Let us consider a system whose active space can bepartitioned into a few orbital spaces some of them areconnected whereas others are not connected (Fig 2(b)) We

TABLE I GAS electron occupation number constraints for case 1

GAS(1) GAS(2) GAS(K)

Minocc N1 N1 + N2 N

Maxocc N1 + 1 N1 + N2 + 1 N

define one set of connected spaces as a group Within eachgroup inter-space electron excitations occur according to theuser specifications however excitations among groups (aliasnot connected orbital spaces) are undesired (they representineffective configurations) Let us consider for example asystem containing two transition metal centers where eachmetal center is a local multiconfigurational region that byitself cannot be described by CASSCF and charge transferbetween these centers does not occur The user could separatethe active orbitals into two groups one per each metal centerThis partitioning will exclude from the CI expansion thoseconfigurations featuring charge transfer Then a furtherpartitioning is done within each group in order to furtherreduce the CI space and make the calculation feasible Byusing ORMAS this kind of constraints cannot be imposedThe ORMAS wave function in this case would contain manyineffective configurations involving electron excitationsamong different groups and the charge transfer between thetwo metal centers cannot be avoided This case shows thesuperiority of GAS over ORMAS scheme

III TEST CALCULATIONS

The GASSCF code has been implemented in the MOL-CAS 77 quantum chemistry software package54 which hasbeen used to perform all the calculations discussed below Wepresent some benchmark results on Gd atom Gd2 moleculeand oxoMn(salen) complex in order to verify the accuracy ofthe GAS approach The initial motivation of the developmentof GASSCF was to be able to describe systems like transitionmetal clusters which in general are highly multiconfigura-tional but with many ineffective configurations A typical casein which the number of configurations increases dramaticallyis in going from Gd atom to the Gd2 molecule The Gd atomhas an electronic configuration [Xe]4f 76s25d1 A reasonableactive space would consist of 10 valence shell electrons dis-tributed in the 4f 5d and 6s orbitals CAS(1013) When con-sidering the Gd2 molecule as we will explain later the 6porbitals also play important roles the corresponding activespace would thus be CAS(2032) which is prohibitively largefor conventional CASSCF and RASSCF The oxoMn(salen)molecule has been used as a benchmark test for ORMAS-SCF and it thus represents an ideal system for a comparisonbetween GASSCF and ORMAS-SCF

A Gd atom

The aim of this set of calculations is to verify thatby using a GASSCF ansatz the CI expansion and conse-quently the computational cost may be reduced with respectto CASSCF without loss in accuracy

We computed three low-lying electronic states of thegadolinium atom namely the ground state [Xe]4f 76s25d1

9Do and two excited states [Xe]4f 76s15d2 11Fo and[Xe]4f 76s25d1 7Do The D2h point group symmetry con-straints were imposed and an ANO-RCC-VTZP type ba-sis set was employed Note that these calculations are notintended to reproduce experimental accuracy Basis set and



TABLE II Comparison between CASSCF and GASSCFa on three low-lying levels of Gd atom All energy values have been shifted by +11 260hartree

Number of Energy Eb

System Method determinants (hartree) (hartree)

9Do 4f 76s25d1 CAS(1013) 1160 minus1692485GAS2(1013) 269 minus1692385 0000100GAS5(1013) 101 minus1692348 0000137

11Fo 4f 76s15d2 CAS(1013) 39 minus1678458GAS2(1013)c 6 minus1676635 0001823

7Do 4f 76s25d1 CAS(1013) 12 577 minus1654021GAS2(1013) 3340 minus1653402 0000619GAS5(1013) 700 minus1651866 0002155

aGAS descriptions please see textbE is the energy difference from the CASSCF energycIn this case GAS2(1013) and GAS5(1013) configuration spaces are identical

active spaces might not be adequate to describe the systemproperly For instance although 6p orbitals are not involvedin the main configurations describing the electronic states an-alyzed they might contribute as correlating orbitals The pur-pose is to compare the GAS method performance versus thecorresponding CAS performance

We initially performed CASSCF calculations with anactive space containing the 10 valence electrons distributedamong the 4f 5d and 6s orbitals CAS(1013) The GAS cal-culations were performed with two different choices of GASspaces We introduce the following notation to define the ac-tive spaces in the GAS calculations GASn(xy) where n indi-cates the number of spaces that we have introduced and (xy)the total number of active electrons and active orbitals re-spectively (a) GAS2(1013) consists of two orbital spacesone with seven electrons in the 4f orbitals and the secondwith three electrons in 5d and 6s orbitals (b) GAS5(1013)consists of five orbital spaces obtained by partitioning the 4forbitals into four different orbital spaces according to symme-try considerations

In Table II the total energies for the three electronicstates obtained with the GAS and CAS approaches are re-ported together with the number of Slater determinants En-ergy differences between the GAS and CAS values are alsopresented With respect to CAS(1013) GAS2(1013) elimi-nates the configurations generated by the excitations betweenthe 4f orbitals and 5d6s orbitals With GAS5(1013) moreconfigurations are eliminated since the 4f orbital space hasbeen further divided up into four subspaces Inspection ofTable II shows that the sizes of the GAS CI spaces are oneor two orders of magnitude smaller than the size of the CASCI space and the energy difference is at most of the order ofthe mhartree if not lower

B Gd2 molecule

Gd2 is a challenging system both theoretically and ex-perimentally It is the highest spin diatomic molecule knownto date with a ground state 19minus

g (σ 1g σ 1

u π2gπ2

uδ2gδ

2uφ

2gφ

2u)4f middot

σ 2g σ 1

g π2uσ 1

u Many attempts both theoretically and experimen-tally have been performed in the past years to determine its

minus0460

minus0440

minus0420

minus0400

minus0380

minus0360

minus0340

minus0320

20 25 30 35 40 45 50 55 60 65

E (+22523 Hartree)

GdminusGd (Aring)

CAS(2026)GASminus2(2026)GASminus5(2032)

FIG 3 Gd219minus

g potential energy curve by using CAS(2026)GAS2(2026) and GAS5(2032)

ground state and spectroscopic constants Lombardi et al55

fitted the Raman spectra into a Morse potential and deter-mined a ground state vibrational constant ωe = 1387 plusmn 04cmminus1 and a spectroscopic dissociation energy of 21 plusmn 07eV From the theoretical side Cao and Dolg performed a sys-tematic investigation on lanthanide dimers including Gd256

The reader should refer to the original reference for moredetails

In this subsection we will investigate the Gd219minus

g

state with several CAS and GAS choices The basis set usedthroughout this section is of ANO-RCC-VDZP type and allcalculations were performed within the D2h point group

The full valence shell active space for this system consistsof 20 valence electrons in 32 molecular orbitals arising from4f 5d 6s and 6p orbitals of each Gd atom A CAS(2032)would generate about 14 billion Slater determinants for the19minus

g state which at present is not feasible Electronic stateswith a lower spin multiplicity would correspond to an evenlarger number of determinants Gd2 is thus presently not treat-able with conventional CASSCF or RASSCF approachesSince all 4f orbitals are always singly occupied in the 19minus

g

state they could be separated from the others and constitute asubspace within the GAS approach This molecule representsan ideal system for the GAS approach

An alternative possibility would be to remove some or-bitals from CAS(2032) to make the CASSCF calculationfeasible We have explored various CAS and GAS choicesIn Table III we report total energies at a fixed bond dis-tance of 300 Aring obtained with the different CAS and GASchoices together with the number of Slater determinants andthe equilibrium bond distance Re In the CAS(2026) calcula-tion six orbitals are moved from CAS to the secondary spaceGAS2(2026) is its analogous but the active space is dividedinto two subspaces the first one contains 14 electrons in the4f orbitals and the second contains 6 electrons in the 12 5dand 6s orbitals Following the same logic by separating the4f orbitals from the others in CAS(2032) the GAS2(2032)is formed In order to further reduce the size of the configu-ration space GAS5(2032) was built by dividing the 4f sub-space of the GAS2(2032) into four different GAS spaces



4 fσ g (100) 4 fπ u (100) 4 fπ u (100) 4 fδ g(100) 4 fδ g(100) 4 fφu (100) 4 fφu (100)

4 fσ u (100) 4 fπ g (100) 4 fπ g (100) 4 fδ u (100) 4 fδ u (100) 4 fφg (100) 4 fφg (100)

6sσ g (181) 5dσ g(092) 5dπ u (094) 5dπ u (094) 5dδ g (002) 5dδ g (002)

6sσ u (098) 5dσ u (007) 5dπ g (005) 5dπ g (005) 5dδ u (000) 5dδ u (000)

6 pσ g (007) 6 pπ u (003) 6 pπ u (003)

6 pσ u (001) 6 pπ g(002) 6 pπ g (002)

FIG 4 The natural orbitals of Gd2 GAS5(2032) at equilibrium bond distance Orbital labels and occupation numbers are listed below each orbital

(4 in 4) (4 in 4) (4 in 4) and (2 in 2) respectively accordingto symmetry considerations

Table III shows that in going from CAS(2026) toGAS2(2026) the number of determinants is reduced by 99and the energy deviation is only 17 mhartree In going fromCAS(2032) not doable to GAS2(2032) the number of

TABLE III Gd219minus

g state Comparison of GASSCF against CASSCFa

All energy values have been shifted by +22 520 hartree

Number of E (R = 300 Aring)Method determinants (hartree) Re (Aring)

CAS(2026) 2 137 560 minus3434520 308GAS2(2026) 23 808 minus3432858 308CAS(2032) sim14 times 109 NAGAS2(2032) 474 016 minus3455416GAS5(2032) 138 304 minus3455380 306

aCAS and GAS spaces description see text For all the results listed in this table theleading configuration is (σ 1

g σ 1u π2

g π2u δ2

gδ2uφ

2gφ2

u)4f middot σ 2g σ 1

g π2uσ 1

u

determinants reduces by three orders of magnitude More-over when we further partition the 4f subspace going fromGAS2(2032) to GAS5(2032) the number of determinants isreduced by about 70 and the energy deviation is of the or-der of 10minus5 hartree

We calculated the potential energy curves for theGd2

19minusg state by using CAS(2026) GAS2(2026) and

GAS5(2032) (Fig 3) See supplementary material57 for thedata used in the plot GAS5(2032) predicts an equilibriumbond distance of 306 Aring while CAS(2026) and GAS(2026)of 308 Aring

The curves obtained with CAS(2026) and GAS(2026)are not smooth throughout the dissociation pathway be-cause some correlating orbitals are missing In other wordsCAS(2026) and GAS(2026) are not big enough spaces to de-scribe the whole dissociation path consistently Along the re-action path the 4f orbitals are always singly occupied The oc-cupation numbers of the other active orbitals for GAS2(2026)and GAS5(2032) are reported in Table IV The natural or-bitals of Gd2 GAS5(2032) at equilibrium bond distance



TABLE IV Occupation numbers of natural active orbitals for GAS2(2026) and GAS5(2032)

R (Aring) 6sσg 6pσg 5dσg 5dδg 5dπu 6pπu 5dπu 6pπu 5dδg

GAS2(2026) 306 185 001 093 ndash 096 ndash 096 ndash 003520 158 004 025 ndash 029 ndash 081 ndash 026540 151 004 099 ndash ndash 003 ndash 003 042600 147 003 099 ndash ndash 003 ndash 003 046

GAS5(2032) 306 181 007 092 002 094 003 094 003 002520 157 005 052 009 058 003 058 003 009540 150 004 098 001 000 003 000 003 042600 146 003 098 001 000 003 000 003 046

R (Aring) 6pσu 5dσu 6sσu 5dδu 5dπg 6pπg 5dπg 6pπg 5dδu


GAS5(2032) 306 001 007 098 000 005 002 005 002 000520 002 031 123 009 033 002 033 002 009540 002 098 132 001 000 002 000 002 060600 002 098 136 001 000 003 000 003 056

along with occupation numbers are given in Fig 4 ForGAS2(2026) in the region R = 520 Aring to 540 Aring the ac-tive orbitals 5dπu and 5dπg are progressively replaced by theorbitals 6pπu and 6pπg The orbital spaces in the bonding re-gion and in the dissociation region are thus different and a(2026) active space cannot describe this change in a smoothway This behavior is cured by using the GAS5(2032) spacewhich includes all the orbitals that change along the dissoci-ation Near dissociation (R gt 550 Aring) the GAS2(2026) andGAS5(2032) curves are very similar because the two wavefunctions become more similar and the extra orbitals presentin GAS5(2032) are nearly empty

Near equilibrium the 5dπ orbitals are active inGAS2(2026) while the four 6pπ orbitals and two of thefour 5dδ orbitals are in the virtual space Inspection of theGAS5(2032) shows that these orbitals are correlating or-bitals and give a non-negligible contribution to the wave func-tion From the above analysis it can be concluded that whileGAS2(2026) and CAS(2026) are satisfactory active spacesat equilibrium they cannot describe the dissociation regionconsistently On the other hand GAS5(2032) contains all thenecessary orbitals to describe the entire curve and determinespectroscopic constants We fitted our GAS5(2032) potentialenergy curve to a Morse potential and obtained De = 21 eVand ωe = 140 cmminus1 These values are in good agreement withLombardirsquos experimental values De = 21plusmn07 eV and ωe

= 1387 plusmn 04 cmminus1

C GAS applied to OxoMn(salen) compound

The OxoMn(salen) (salen = NNprime-bis(salicylidene)-ethylenediamine dianion) system (Fig 5) is used as a prod-uct specific catalyst during the Jacobsen-Katsuki asymmet-ric epoxidation of olefins58ndash61 The importance of this cata-lyst lies in the fact that it guarantees high enantiomeric ex-cess In order to understand the reason for this high selec-tivity many experimental and theoretical studies have been

attempted However there are still many conflicting opinionsconcerning the reaction mechanism and the bare catalyst

Linde et al employed density functional theory (DFT)with the B3LYP exchange-correlation functional to studya simplified cationic model similar to the neutral Ox-oMn(salen) species studied here except that the chlorineligand was removed62 They found that the singlet tripletand quintet states are quasi-degenerate the singlet being theground state and the triplet and quintet 14 and 26 kcalmolrespectively above the ground state They also state that theMn-Oax bond has a triple bond character in the singlet spinstate double bond character in the triplet spin state and sin-gle bond character in the quintet spin state

Cavallo and Jacobsen employed DFT with the Becke-Perdew exchange-correlation functional (BP86) to study theneutral model (chlorine included) and found that the triplet ismore stable than the singlet spin state63

In 2001 Abashkin et al addressed the same issue in amore systematic manner64 They performed DFTBP86 andDFTB3LYP calculations with the DZVP basis set They usedboth cationic and neutral models to compare their resultswith the ones obtained by Linde et al They determined the

FIG 5 The neutral model here used for the OxoMn(salen) compound



TABLE V Number of Slater determinants CASSCF iterations until convergence and absolute energies for allthe CAS and GAS choices here discussed and for the ORMAS calculations by Ivanic66 (S) and (T) stand forsinglet and triplet spin states respectively

Number of SDs MCSCF iterations MCSCF energy (au) CASPT2 energy (au)

CAS(1010) S 31 878 12 minus2260897474 minus2263409690CAS(1211) S 106 953 25 minus2260902407 minus2263412384CAS(1211) T 152 460 42 minus2260902818 minus2263423504CAS(1413) S 1 473 186 17 minus2260927332 minus2263422117CAS(1413) T 2 208 492 20 minus2260935376 minus2263436353ORMAS(1010) S 2424 ndash minus225142305 ndashORMAS(1211) T 8836 ndash minus225142749 ndashGAS(1211) S 14 010 34 minus2260899367 ndashGAS(1211) T 20 410 62 minus2260897926 ndashGAS(1413) S 185 192 13 minus2260923416 ndashGAS(1413) T 282 919 34 minus2260929083 ndashGAS(1817) S 11 313 365 20 minus2260973576 ndashGAS(1817) T 18 436 215 48 minus2260979368 ndash

relative energies of singlet triplet and quintet spin states us-ing the coupled cluster method including up to perturbativelyconnected triple excitations CCSD(T) They found an im-portant discrepancy in the predictions of BP86 and B3LYPfunctionals Using the hybrid B3LYP functional they con-firmed the results obtained by Linde that the triplet is lower inenergy than the singlet for both cationic and neutral modelHowever both their DFTBP86 and CCSD(T) results indi-cated that singlet is more stable than the triplet (6 kcalmolat DFTBP86 level and 145 kcalmol at CCSD(T) level oftheory)

Ivanic et al performed the first multiconfigurationalab initio study of the neutral OxoMn(salen)65 66 They per-formed geometry optimization at CASSCFMRMP2 level oftheory as implemented in the GAMESS package67 on the neu-tral model compound They also used this compound to testthe ORMAS approach66 At the singlet optimized geome-try CASSCF MRMP2 and ORMAS methods predicted thetriplet to be more stable than the singlet by 29 kcalmol23 kcalmol and 28 kcalmol respectively

TABLE VI Occupation numbers for all the CAS choices (S) and (T) standfor singlet and triplet spin states respectively

CAS(1010) CAS(1211) CAS(1413)

S S T S T

3dx2minusy2 ndash 197 100 197 100σ (salen) ndash ndash ndash 197 195σ (Oax) 191 191 189 191 190π1(Oax) 184 184 174 185 176π2(Oax) 184 184 195 184 195π1(L) 196 196 197 196 197π1(R) 196 196 195 196 195σ (salen) ndash ndash ndash 004 006σ (Oax) 009 009 011 009 010π1

(Oax) 016 017 026 016 025π2

(Oax) 016 017 105 017 104π1

(L) 004 004 003 004 003π1

(R) 004 004 005 004 005

We performed a series of CASSCF calculations followedby second order perturbation correction (CASPT2) togetherwith a series of GASSCF calculations on the singlet andtriplet states of the neutral model at the geometry optimizedby Ivanic et al65 We used basis set of the atomic naturalorbital type of double-zeta plus polarization quality (ANO-RCC-VDZP)68ndash72 Scalar relativistic effects were included us-ing the Douglas-Kroll-Hess Hamiltonian The computationalcosts arising from the two-electron integrals were reduced byemploying the Cholesky decomposition technique73 In orderto prevent weak intruder states in the CASPT2 calculationsan imaginary shift of 02 units was added to the external partof the zero-order Hamiltonian At CASPT2 level the 1s or-bitals of C N and O atoms were kept frozen together withorbitals up to and including the 2p for Cl and orbitals up toand including 3s for Mn All calculations were performed inC1 symmetry

Our first choice of complete active space for both sin-glet and triplet spin states consisted of 14 electrons distributedamong 13 orbitals These orbitals are six doubly occupiedbonding orbitals σ (salen) σ (Oax) π1(Oax) π2(Oax) π1(L)and π1(R) one non bonding orbital 3dx2minusy2 (Mn) and sixanti-bonding orbitals σ (salen) σ (Oax) π1

(Oax) π2(Oax)

π1(L) and π1

(R) The symbols L and R indicate the left(L) and right (R) side of the salen ligand respectively Oax

refers to the oxygen in axial position with respect to Mn(salen) refers to orbitals which are linear combinations of the3dxy(Mn) and the 2p atomic orbitals on O and N atoms of thesalen ligand pointing to the metal center This choice reflectsthe fact that to fully probe the multiconfigurational nature ofthe system all the 3d atomic orbitals of the Mn atom alongwith the orbitals of π -type on the ligand must be includedin the active space However in order to compare our re-sults with the ones obtained by Ivanic et al65 66 we also per-formed smaller CASSF calculations (a) CAS(1010) for thesinglet by removing the σ (salen) σ (salen) and 3dx2minusy2 (Mn)orbitals from the (1413) active space and (b) CAS(1211)by adding the 3dx2minusy2 (Mn) to the CAS(1010) for both sin-glet and triplet spin states The number of CASSCF iterationsuntil convergence number of Slater determinants in the CI



TABLE VII Occupation numbers for all the GAS choices and ORMAS (S) and (T) stand for singlet and tripletspin states respectively

GAS3(1211) GAS3(1413) GAS4(1817) ORMAS66

S T S T S T S

3dx2minusy2 198 100 197 100 197 100 ndashσ (salen) ndash ndash 197 195 197 195 ndashσ (Oax) 191 189 191 190 191 190 191π1(Oax) 184 174 185 175 185 175 185π2(Oax) 184 195 184 195 185 195 185π1(L) 197 197 197 197 196 196 197π1(R) 196 195 196 195 196 196 196π2(L) ndash ndash ndash ndash 193 193 ndashπ2(R) ndash ndash ndash ndash 193 193 ndashσ (salen) ndash ndash 004 005 004 005 ndashσ (Oax) 009 011 009 010 009 010 009π1

(Oax) 017 027 016 025 016 025 015π2

(Oax) 017 105 017 104 017 104 015π1

(L) 003 003 003 003 004 004 003π1

(R) 004 005 004 005 004 004 004π2

(L) ndash ndash ndash ndash 007 006 ndashπ2

(R) ndash ndash ndash ndash 007 007 ndash

expansion and absolute CASSCF and CASPT2 energiesfor all the CAS choices are reported in Table V We alsopresent the ORMAS results by Ivanic66 and our GAS re-sults For each MCSCF iteration the orbital optimizationtakes about 6 to 12 iterations to converge and no obvi-ous difference between GAS and CAS calculations has beenobserved

For GAS3(1211) and GAS3(1413) the CI space wasreduced by partitioning the active space into three sub-spaces The first space includes five orbitals for GAS3(1211)π1(Oax) π2(Oax) π1

(Oax) π2(Oax) and 3dx2minusy2 (Mn) and

seven for GAS3(1413) by adding orbitals σ (salen) andσ (salen) into the same subspace the second space includesσ (Oax) and σ (Oax) orbitals and the third space include theremaining four active π -type orbitals of the salen ligand Be-side the GAS3(1211) and GAS3(1413) that can be directlycompared with the equivalent CAS calculations we also com-puted a bigger GAS4(1817) for both singlet and triplet spinstates For GAS4(1817) the active space was partitioned intofour subspaces the first two being identical to the ones ofthe GAS3(1413) and the other two containing an extendedset of π orbitals on ligand spatially separated (left and rightside) The equivalent CAS(1817) would give about 06 bil-lion Slater determinants that at present is not practical Wealso imposed constraints on the electron occupation numberof each space in a way that no inter-space excitations couldoccur In other words subspaces were disconnected for anyGAS choice for both singlet and triplet spin states Goingfrom CAS to the equivalent GAS choices only 13 of deter-minants survive for both spin states with an energy deviationsof the order of mhartree

Unlike the ORMAS calculations the optimized CASSCFand GASSCF orbitals were not further localized Note that theORMAS-SCF calculations were performed with different ba-sis set thus the total energies of ORMAS-SCF are not com-parable with the other methods in this table

In Fig 6 we report the singlet natural orbitals obtainedfor the GAS4(1817) calculation The triplet natural orbitalsare very similar The occupation numbers of the active or-bitals corresponding to the aforementioned CAS calculationsare reported in Table VI while the occupation numbers of theactive orbitals for the GAS choices are reported in Table VII

3dx2 minusy2

σ (salen) σ (Oax ) π1(Oax ) π 2 (Oax )

σ (salen) σ (Oax ) π1(Oax ) π 2

(Oax )

π1 L( ) π 2 L( ) π 2 R( ) π1 R( )

π1 L( ) π 2

L( ) π 2 R( ) π1

R( )

FIG 6 Active natural orbitals for the GAS4(1817) choice



the ORMAS singlet occupation numbers by Ivanic66 are alsolisted as comparison

The CAS(1211) and GAS3(1211) natural orbitals andcorresponding occupation numbers are almost identical to theones obtained by Ivanic et al the only difference being anoccupation number of 197 for the 3dx2minusy2 (Mn) orbital for thesinglet spin state that we included into the active space Wedid not encounter the orbital switching described by Ivanic66

Moving the 3dx2minusy2 (Mn) orbital out of the active spacemdashgoing from CAS(1211) to CAS(1010) respectivelymdashcausesa non-negligible energy deviation of 5 mhartree (3 kcalmol)Within the CAS(1211) or GAS3(1211) choices the two spinstates appear almost degenerate with energy gaps of 025kcalmol and 088 kcalmol respectively

The CAS(1413) GAS3(1413) and GAS4(1817)choices show that the triplet is the ground state Triplet-singlet energy gaps of 5 kcalmol 36 kcalmol and of 36kcalmol were obtained by the CAS(1413) GAS3(1413)and GAS4(1817) choices respectively The perturbative cor-rection (CASPT2) at the CAS(1413) reference wave functionconfirms the CASSCF and GASSCF results the triplet beingthe ground state with the singlet 88 kcalmol above

In including the σ (salen) and σ (salen) orbitals thetriplet σ (salen) orbital has some mixing with one of theπ (Oax) orbitals The natural orbitals for the GAS4(1817) andcorresponding occupation numbers differ from the ones ob-tained by smaller CAS or GAS and by Ivanic More orbitals ofπ -type have been included into the GAS4(1817) active spaceand non-negligible values of occupation numbers for the newadded anti-bonding orbitals were obtained According to ourstudy among all the various active spaces investigated theGAS4(1817) space describes better the multiconfigurationalnature of the system

The singlet spin state is dominated by the closedshell configuration σ (salen)2σ (Oax)2π (Oax)4π (L)4π (R)4

(3dx2minusy2 )2 with a weight of 765 for the CAS(1413) and717 for the bigger GAS4(1817) The triplet spin stateis dominated by the following electronic configurationσ (salen)2σ (Oax)2π (Oax)4π (L)4π (R)4(3dx2minusy2 )1πlowast

1 (Oax)1with a weight of 738 for the CAS(1413) choice and 686for the GAS4(1817)

IV DISCUSSION

We have developed a generalized active space wave func-tion formalism that can be employed to perform MCSCF cal-culations in those cases where a conventional CASSCF ap-proach is not viable The GAS wave function is obtained bydividing the active space into an arbitrary number of sub-spaces requiring accumulated minimum and maximum oc-cupation numbers We have demonstrated that RAS and CASare special cases of GAS

By an appropriate choice of the GAS spaces the userscan eliminate many of the ineffective configurations thatwould be present in a large CAS space but keep the importantones in the CI space

An aspect that should be mentioned is that GASSCF isnot strictly size-extensive The size extensivity of GASSCFdepends on the choice of the active spaces It is strictly

size extensive when all the GAS spaces are all disconnectednamely no inter-space excitation involved but when thespaces are connected it is not size extensive any more

Although in the examples presented in this paper no con-vergence difficulty in the orbital optimization step was en-countered there might be cases where the near linear depen-dency of some orbital rotations could cause such problems

The GASSCF formulation has special advantages for sys-tems where the orbitals can be easily separated into differentgroups For instance (a) lanthanide and actinide complexeswhere the f orbitals can be put in one or more GAS spacesseparately (b) resonance states or Rydberg states where theouter electrons almost do not correlate with the inner elec-trons (c) molecules with several localized conjugated bondsand (d) molecular magnets in which several atoms are cou-pled high-spin and of great interest but forebodingly difficultby direct CASSCF

The method has thus a promising potential for the treat-ment of strongly correlated systems We will employ it tostudy clusters of metals and oligomeric species The GASSCFwave function is also a better reference wave function for sub-sequent perturbative treatment (PT2 for example) and we willexplore this aspect in the future

ACKNOWLEDGMENTS

We gratefully acknowledge financial support from DOE(Grant No DE-SC002183) DM thanks UMN MRSEC forfinancial support

1J Hinze J Chem Phys 59 6424 (1973)2E Dalgaard and P Joslashrgensen J Chem Phys 69 3833 (1978)3M W Schmidt and M S Gordon Annu Rev Phys Chem 49 233 (1998)4K Andersson and B O Roos Int J Quantum Chem 45 591 (1993)5M Merchaacuten and L Serrano-Andreacutes J Am Chem Soc 125 8108 (2003)6D A Hrovat K Morokuma and W T Borden J Am Chem Soc 1161072 (1994)

7R Lindh and B J Persson J Am Chem Soc 116 4963 (1994)8R Lindh and G Karlstrom Chem Phys Lett 289 442 (1998)9L Serrano-Andreacutes M Merchaacuten I Nebotgil R Lindh and B O Roos JChem Phys 98 3151 (1993)

10L Serrano-Andreacutes R Lindh B O Roos and M Merchaacuten J Phys Chem97 9360 (1993)

11L Serrano-Andreacutes and B O Roos J Am Chem Soc 118 185 (1996)12M P Fuumllscher L Serrano-Andreacutes and B O Roos J Am Chem Soc 119

6168 (1997)13G La Macchia G Li Manni T K Todorova M Brynda F Aquilante

B O Roos and L Gagliardi Inorg Chem 49 5216 (2010)14M Radon and K Pierloot J Phys Chem A 112 11824 (2008)15S Creve K Pierloot M T Nguyen and L G Vanquickenborne Eur J

Inorg Chem 1999 107 (1999)16B J Persson B O Roos and K Pierloot J Chem Phys 101 6810 (1994)17K Pierloot B J Persson and B O Roos J Phys Chem 99 3465 (1995)18B O Roos A C Boriacuten and L Gagliardi Angew Chem Int Ed 46 1469

(2007)19L Gagliardi and B O Roos Inorg Chem 42 1599 (2003)20K Pierloot E Van Praet L G Vanquickenborne and B O Roos J Phys

Chem 97 12220 (1993)21K Pierloot E Tsokos and L G Vanquickenborne J Phys Chem 100

16545 (1996)22K Pierloot J O A De Kerpel U Ryde and B O Roos J Am Chem

Soc 119 218 (1997)23K Pierloot J O A De Kerpel U Ryde M Olsson and B O Roos J

Am Chem Soc 120 13156 (1998)24A Delabie K Pierloot M H Groothaert R A Schoonheydt and

L G Vanquickenborne Eur J Inorg Chem 3 515 (2002)25L Gagliardi and B O Roos Chem Soc Rev 36 893 (2007)



26B O Roos P-Aring Malmqvist and L Gagliardi J Am Chem Soc 12817000 (2006)

27L Gagliardi Theor Chem Acc 116 307 (2006)28K Pierloot and E van Besien J Chem Phys 123 204309 (2005)29K Andersson P-Aring Malmqvist B O Roos A J Sadlej and K Wolinski

J Chem Phys 94 5483 (1990)30K Andersson P-Aring Malmqvist and B O Roos J Chem Phys 96 1218

(1992)31B O Roos P R Taylor and P E M Siegbahn Chem Phys 48 157

(1980)32J Ivanic and K Ruedenberg Theor Chem Acc 106 339 (2001)33J Ivanic and K Ruedenberg Theor Chem Acc 107 220 (2002)34L Bytautas J Ivanic and K Ruedenberg J Chem Phys 119 8217

(2003)35L Bytautas and K Ruedenberg Chem Phys 356 64 (2009)36L G Yaffe and W A Goddard Phys Rev A 13 1682 (1976)37S P Walch C W Bauschlicher B O Roos and C J Nelin Chem Phys

Lett 103 175 (1983)38H Nakano and K Hirao Chem Phys Lett 317 90 (2000)39A I Panin and O V Sizova J Comput Chem 17 178 (1996)40A I Panin and K V Simon Int J Quantum Chem 59 471 (1996)41J Ivanic J Chem Phys 119 9364 (2003)42J Olsen B O Roos P Joslashrgensen and H Joslashrgen Aa Jensen J Chem

Phys 89 2185 (1988)43P-Aring Malmqvist A Rendell and B O Roos J Phys Chem 94 5477

(1990)44P-Aring Malmqvist K Pierloot A R M Shahi C J Cramer and

L Gagliardi J Chem Phys 128 204109 (2008)45S M Huber M E Ertem F Aquilante L Gagliardi and C J Cramer

Chem-Eur J 15 4886 (2009)46V Sauri L Serrano-Andreacutes A M R Shahi L Gagliardi S Vancoille

and K Pierloot J Chem Theory Comput 7 153 (2011)47A R M Shahi C J Cramer and L Gagliardi Phys Chem Chem Phys

11 10964 (2009)48G Li Manni F Aquilante and L Gagliardi J Chem Phys 134 034114

(2011)49T Fleig J Olsen and C M Marian J Chem Phys 114 4775 (2001)50N C Handy Chem Phys Lett 74 280 (1980)51P J Knowles and N C Handy Chem Phys Lett 111 315 (1984)

52A Banerjee and F Grein Int J Quantum Chem 10 123 (1976)53J Olsen D L Yeager and P Joslashrgensen Adv Chem Phys 54 1 (1983)54F Aquilante L De Vico N Ferreacute G Ghigo P-Aring Malmqvist

P Neograacutedy T B Pedersen M Pitonak M Reiher B O RoosL Serrano-Andreacutes M Urban V Veryazov and R Lindh J ComputChem 31 224 (2010)

55Y Liu L Fang X Shen X Chen J R Lombardi and D M LindsayChem Phys 262 25 (2000)

56X Cao and M Dolg Mol Phys 101 1967 (2003)57See supplementary material at httpdxdoiorg10106313611401 for the

data used in plotting Gd2 potential energy curves58E N Jacobsen in Catalytic Asymmetric Synthesis edited by I Ojima

(VCH New York 1993) p 15959E N Jacobsen and N S Finney Chem Biol 1 85 (1994)60T Katsuki J Mol Catal A Chem 113 87 (1996)61C Dalton K Ryan V Wall C Bousquet and D Gilheany Top Catal 5

75 (1998)62C Linde B Aringkermark P-O Norrby and M Svensson J Am Chem Soc

121 5083 (1999)63L Cavallo and H Jacobsen Angew Chem Int Ed 39 589 (2000)64Y G Abashkin J R Collins and S K Burt Inorg Chem 40 4040 (2001)65J Ivanic J R Collins and S K Burt J Phys Chem A 108 2314

(2004)66J Ivanic J Chem Phys 119 9377 (2003)67M W Schmidt K K Baldridge J A Boatz S T Elbert M S Gordon

J H Jensen S Koseki N Matsunaga K A Nguyen S Su T L WindusM Dupuis and J A Montgomery J Comput Chem 14 1347 (1993)

68B O Roos R Lindh P-Aring Malmqvist V Veryazov and P-O WidmarkJ Phys Chem A 108 2851 (2004)


70B O Roos V Veryazov and P-O Widmark Theor Chem Acc 111 345(2004)

71B O Roos R Lindh P-Aring Malmqvist V Veryazov P-O Widmark andA C Boriacuten J Phys Chem A 112 11431 (2008)

72B O Roos R Lindh P-Aring Malmqvist V Veryazov and P-O WidmarkChem Phys Lett 409 295 (2005)

73F Aquilante T B Pedersen R Lindh B O Roos A Saacutenchez de Meraacutesand H Koch J Chem Phys 129 024113 (2008)



structure includes many usual CI spaces as special cases Forinstance when there are no orbitals in RAS1 or RAS3 RASreduces to CAS In a sense the RAS1 and RAS3 spaces addsome dynamic correlation to the RAS2 space The RASSCFapproach has been recently employed to generate referencewave functions for subsequent perturbation treatment to sec-ond order the RASPT2 approach44ndash47 We have also recentlyproposed a new approach SplitCAS to determine a suitablezeroth-order wave function for multiconfigurational pertur-bation theory The same ansatz as in complete active spacewave function optimization is split in two parts a princi-pal space (A) and a much larger extended space (B) Parti-tioning technique of Loumlwdin is employed to map the initialeigenvalue problem to a dimensionality equal to that of (A)only48

The concept of generalized active space (GAS) was firstproposed by Olsen42 49 It can be understood as a general-ization of RAS Instead of three active spaces in principleGAS allows an arbitrary number of active spaces Instead of amaximum number of holes in RAS1 and a maximum numberof electrons in RAS3 accumulated minimum and maximumelectron occupation numbers are used in GAS to define thewave function

In this paper we describe a new implementation ofthe concept of GAS self-consistent field (GASSCF) Thereare some similarities between our GASSCF method and theORMAS-SCF method41 In both of them an arbitrary num-ber of orbital spaces can be defined the configuration spacesare both expanded in SDs and inter-space electron excitationsare allowed (connected spaces) However these two methodsdiffer in the way that the electron occupation number con-straints for active spaces are defined The final CI spaces maybe different and as a consequence each method may haveadvantages or limitations according to the systems in exam

The paper is organized as follows In Sec II we for-mulate the GAS wave function from a theoretical point ofview and we describe the algorithm We also compare theGAS wave function to the CAS and RAS wave functions andto ORMAS approach In Sec III we present some test cal-culations on the Gd atom the Gd2 molecule and the Ox-oMn(salen) complex Finally in Sec IV we present someconclusions

II THEORY AND ALGORITHM

A Definition of the GAS wave function

The following input parameters need to be specified (1)number of GAS spaces ngas (2) number of orbitals in eachGAS space per each irreducible representation and (3) accu-mulated minimum and maximum number of electrons occu-pying the GAS spaces minocc(igas) and maxocc(igas) (igasruns from 1 through ngas) In other terms one has to definethe minimum and maximum electron occupation number forthe first space then the minimum and maximum electron oc-cupation number for the first two spaces (GAS1 + GAS2)and so on all the way to the whole active space

An example to demonstrate the accumulated occupationnumbers is as follows

GAS1 GAS2 GAS3 GAS4Minocc 0 4 5 8Maxocc 2 4 6 8

The distribution of these electrons among the GASspaces in this case is the first orbital space contains from zeroup to two electrons the first two spaces together contain fourelectrons in total the first three spaces contain from five upto six electrons the occupation counts for all four spaces to-gether must be same as the total number of active electrons

B Reduction of the GAS wave function to the RASand CAS wave function

It can be easily proved that CAS and RAS are two specialcases of GAS

If a RAS wave function is defined bynactel total number of active electronsorbitals in RAS1 RAS2 and RAS3 spacesnhole1 maximum number of holes in RAS1nelect3 maximum number of electrons in RAS3

The equivalent GAS function is defined as followsngas = 3orbitals in each GAS spaces same as in RAS spacesminocc(1) = number of spin orbitals in RAS1minus nhole1maxocc(1) = number of spin orbitals in RAS1minocc(2) = maxnactel minus nelec3 minocc(1)maxocc(2) = minnactel number of spin orbitals inRAS1 + RAS2minocc(3) = maxocc(3) = nactel

CAS is an even simpler case The equivalent GAS wave func-tion is define by

ngas = 1minocc(1) = maxocc(1) = nactel

A pictorial description of the GAS wave function to-gether with the CAS and FCI wave functions is reported inFig 1 The vertical line defines a single determinant refer-ence state (HF state) All possible distributions of the elec-trons among the reference orbitals generate the FCI space(big circle) If instead the reference orbitals are partitioned

FIG 1 Graphical description of HF FCI CAS and GAS configurationspaces




C Direct CI





|0〉 =sum

IαIβ



σ (Iα Iβ )=sum

JαJβ



H =sum

kl

hklEkl + 1

2

sum

ijkl




daggerkβalβ (5)






exp(iλ) = 1 + iλ + 1


where

λ = isum

pgtq


pgtq

κpqEminuspq (7)



pgtq



HX = ESX (9)

where



rs |0〉 (11)









N = 1 +Ksum

i=1

Ni





(a)

(b)









Minocc N1 N1 + N2 N

Maxocc N1 + 1 N1 + N2 + 1 N




A Gd atom







Number of Energy Eb









B Gd2 molecule


g (σ 1g σ 1

u π2gπ2

uδ2gδ

2uφ

2gφ

2u)4f middot

σ 2g σ 1

g π2uσ 1


minus0460

minus0440

minus0420

minus0400

minus0380

minus0360

minus0340

minus0320

20 25 30 35 40 45 50 55 60 65

E (+22523 Hartree)

GdminusGd (Aring)


FIG 3 Gd219minus






g




g









6 pσ g (007) 6 pπ u (003) 6 pπ u (003)

6 pσ u (001) 6 pπ g(002) 6 pπ g (002)










g σ 1u π2

g π2u δ2

gδ2uφ

2gφ2


g π2uσ 1

u











GAS5(2032) 306 181 007 092 002 094 003 094 003 002520 157 005 052 009 058 003 058 003 009540 150 004 098 001 000 003 000 003 042600 146 003 098 001 000 003 000 003 046



GAS5(2032) 306 001 007 098 000 005 002 005 002 000520 002 031 123 009 033 002 033 002 009540 002 098 132 001 000 002 000 002 060600 002 098 136 001 000 003 000 003 056



















CAS(1010) CAS(1211) CAS(1413)

S S T S T


(Oax) 016 017 026 016 025π2

(Oax) 016 017 105 017 104π1

(L) 004 004 003 004 003π1

(R) 004 004 005 004 005



(Oax) π2(Oax)

π1(L) and π1






GAS3(1211) GAS3(1413) GAS4(1817) ORMAS66

S T S T S T S


(Oax) 017 027 016 025 016 025 015π2

(Oax) 017 105 017 104 017 104 015π1

(L) 003 003 003 003 004 004 003π1

(R) 004 005 004 005 004 004 004π2









3dx2 minusy2



(Oax )

π1 L( ) π 2 L( ) π 2 R( ) π1 R( )

π1 L( ) π 2

L( ) π 2 R( ) π1

R( )












IV DISCUSSION








ACKNOWLEDGMENTS


















































C Direct CI





|0〉 =sum

IαIβ



σ (Iα Iβ )=sum

JαJβ



H =sum

kl

hklEkl + 1

2

sum

ijkl




daggerkβalβ (5)






exp(iλ) = 1 + iλ + 1


where

λ = isum

pgtq


pgtq

κpqEminuspq (7)



pgtq



HX = ESX (9)

where



rs |0〉 (11)









N = 1 +Ksum

i=1

Ni





(a)

(b)









Minocc N1 N1 + N2 N

Maxocc N1 + 1 N1 + N2 + 1 N




A Gd atom







Number of Energy Eb









B Gd2 molecule


g (σ 1g σ 1

u π2gπ2

uδ2gδ

2uφ

2gφ

2u)4f middot

σ 2g σ 1

g π2uσ 1


minus0460

minus0440

minus0420

minus0400

minus0380

minus0360

minus0340

minus0320

20 25 30 35 40 45 50 55 60 65

E (+22523 Hartree)

GdminusGd (Aring)


FIG 3 Gd219minus






g




g









6 pσ g (007) 6 pπ u (003) 6 pπ u (003)

6 pσ u (001) 6 pπ g(002) 6 pπ g (002)










g σ 1u π2

g π2u δ2

gδ2uφ

2gφ2


g π2uσ 1

u











GAS5(2032) 306 181 007 092 002 094 003 094 003 002520 157 005 052 009 058 003 058 003 009540 150 004 098 001 000 003 000 003 042600 146 003 098 001 000 003 000 003 046



GAS5(2032) 306 001 007 098 000 005 002 005 002 000520 002 031 123 009 033 002 033 002 009540 002 098 132 001 000 002 000 002 060600 002 098 136 001 000 003 000 003 056



















CAS(1010) CAS(1211) CAS(1413)

S S T S T


(Oax) 016 017 026 016 025π2

(Oax) 016 017 105 017 104π1

(L) 004 004 003 004 003π1

(R) 004 004 005 004 005



(Oax) π2(Oax)

π1(L) and π1






GAS3(1211) GAS3(1413) GAS4(1817) ORMAS66

S T S T S T S


(Oax) 017 027 016 025 016 025 015π2

(Oax) 017 105 017 104 017 104 015π1

(L) 003 003 003 003 004 004 003π1

(R) 004 005 004 005 004 004 004π2









3dx2 minusy2



(Oax )

π1 L( ) π 2 L( ) π 2 R( ) π1 R( )

π1 L( ) π 2

L( ) π 2 R( ) π1

R( )












IV DISCUSSION








ACKNOWLEDGMENTS

















































(a)

(b)









Minocc N1 N1 + N2 N

Maxocc N1 + 1 N1 + N2 + 1 N




A Gd atom







Number of Energy Eb









B Gd2 molecule


g (σ 1g σ 1

u π2gπ2

uδ2gδ

2uφ

2gφ

2u)4f middot

σ 2g σ 1

g π2uσ 1


minus0460

minus0440

minus0420

minus0400

minus0380

minus0360

minus0340

minus0320

20 25 30 35 40 45 50 55 60 65

E (+22523 Hartree)

GdminusGd (Aring)


FIG 3 Gd219minus






g




g









6 pσ g (007) 6 pπ u (003) 6 pπ u (003)

6 pσ u (001) 6 pπ g(002) 6 pπ g (002)










g σ 1u π2

g π2u δ2

gδ2uφ

2gφ2


g π2uσ 1

u











GAS5(2032) 306 181 007 092 002 094 003 094 003 002520 157 005 052 009 058 003 058 003 009540 150 004 098 001 000 003 000 003 042600 146 003 098 001 000 003 000 003 046



GAS5(2032) 306 001 007 098 000 005 002 005 002 000520 002 031 123 009 033 002 033 002 009540 002 098 132 001 000 002 000 002 060600 002 098 136 001 000 003 000 003 056



















CAS(1010) CAS(1211) CAS(1413)

S S T S T


(Oax) 016 017 026 016 025π2

(Oax) 016 017 105 017 104π1

(L) 004 004 003 004 003π1

(R) 004 004 005 004 005



(Oax) π2(Oax)

π1(L) and π1






GAS3(1211) GAS3(1413) GAS4(1817) ORMAS66

S T S T S T S


(Oax) 017 027 016 025 016 025 015π2

(Oax) 017 105 017 104 017 104 015π1

(L) 003 003 003 003 004 004 003π1

(R) 004 005 004 005 004 004 004π2









3dx2 minusy2



(Oax )

π1 L( ) π 2 L( ) π 2 R( ) π1 R( )

π1 L( ) π 2

L( ) π 2 R( ) π1

R( )












IV DISCUSSION








ACKNOWLEDGMENTS


















































Number of Energy Eb









B Gd2 molecule


g (σ 1g σ 1

u π2gπ2

uδ2gδ

2uφ

2gφ

2u)4f middot

σ 2g σ 1

g π2uσ 1


minus0460

minus0440

minus0420

minus0400

minus0380

minus0360

minus0340

minus0320

20 25 30 35 40 45 50 55 60 65

E (+22523 Hartree)

GdminusGd (Aring)


FIG 3 Gd219minus






g




g









6 pσ g (007) 6 pπ u (003) 6 pπ u (003)

6 pσ u (001) 6 pπ g(002) 6 pπ g (002)










g σ 1u π2

g π2u δ2

gδ2uφ

2gφ2


g π2uσ 1

u











GAS5(2032) 306 181 007 092 002 094 003 094 003 002520 157 005 052 009 058 003 058 003 009540 150 004 098 001 000 003 000 003 042600 146 003 098 001 000 003 000 003 046



GAS5(2032) 306 001 007 098 000 005 002 005 002 000520 002 031 123 009 033 002 033 002 009540 002 098 132 001 000 002 000 002 060600 002 098 136 001 000 003 000 003 056



















CAS(1010) CAS(1211) CAS(1413)

S S T S T


(Oax) 016 017 026 016 025π2

(Oax) 016 017 105 017 104π1

(L) 004 004 003 004 003π1

(R) 004 004 005 004 005



(Oax) π2(Oax)

π1(L) and π1






GAS3(1211) GAS3(1413) GAS4(1817) ORMAS66

S T S T S T S


(Oax) 017 027 016 025 016 025 015π2

(Oax) 017 105 017 104 017 104 015π1

(L) 003 003 003 003 004 004 003π1

(R) 004 005 004 005 004 004 004π2









3dx2 minusy2



(Oax )

π1 L( ) π 2 L( ) π 2 R( ) π1 R( )

π1 L( ) π 2

L( ) π 2 R( ) π1

R( )












IV DISCUSSION








ACKNOWLEDGMENTS





















































6 pσ g (007) 6 pπ u (003) 6 pπ u (003)

6 pσ u (001) 6 pπ g(002) 6 pπ g (002)










g σ 1u π2

g π2u δ2

gδ2uφ

2gφ2


g π2uσ 1

u











GAS5(2032) 306 181 007 092 002 094 003 094 003 002520 157 005 052 009 058 003 058 003 009540 150 004 098 001 000 003 000 003 042600 146 003 098 001 000 003 000 003 046



GAS5(2032) 306 001 007 098 000 005 002 005 002 000520 002 031 123 009 033 002 033 002 009540 002 098 132 001 000 002 000 002 060600 002 098 136 001 000 003 000 003 056



















CAS(1010) CAS(1211) CAS(1413)

S S T S T


(Oax) 016 017 026 016 025π2

(Oax) 016 017 105 017 104π1

(L) 004 004 003 004 003π1

(R) 004 004 005 004 005



(Oax) π2(Oax)

π1(L) and π1






GAS3(1211) GAS3(1413) GAS4(1817) ORMAS66

S T S T S T S


(Oax) 017 027 016 025 016 025 015π2

(Oax) 017 105 017 104 017 104 015π1

(L) 003 003 003 003 004 004 003π1

(R) 004 005 004 005 004 004 004π2









3dx2 minusy2



(Oax )

π1 L( ) π 2 L( ) π 2 R( ) π1 R( )

π1 L( ) π 2

L( ) π 2 R( ) π1

R( )












IV DISCUSSION








ACKNOWLEDGMENTS




















































GAS5(2032) 306 181 007 092 002 094 003 094 003 002520 157 005 052 009 058 003 058 003 009540 150 004 098 001 000 003 000 003 042600 146 003 098 001 000 003 000 003 046



GAS5(2032) 306 001 007 098 000 005 002 005 002 000520 002 031 123 009 033 002 033 002 009540 002 098 132 001 000 002 000 002 060600 002 098 136 001 000 003 000 003 056



















CAS(1010) CAS(1211) CAS(1413)

S S T S T


(Oax) 016 017 026 016 025π2

(Oax) 016 017 105 017 104π1

(L) 004 004 003 004 003π1

(R) 004 004 005 004 005



(Oax) π2(Oax)

π1(L) and π1






GAS3(1211) GAS3(1413) GAS4(1817) ORMAS66

S T S T S T S


(Oax) 017 027 016 025 016 025 015π2

(Oax) 017 105 017 104 017 104 015π1

(L) 003 003 003 003 004 004 003π1

(R) 004 005 004 005 004 004 004π2









3dx2 minusy2



(Oax )

π1 L( ) π 2 L( ) π 2 R( ) π1 R( )

π1 L( ) π 2

L( ) π 2 R( ) π1

R( )












IV DISCUSSION








ACKNOWLEDGMENTS























































CAS(1010) CAS(1211) CAS(1413)

S S T S T


(Oax) 016 017 026 016 025π2

(Oax) 016 017 105 017 104π1

(L) 004 004 003 004 003π1

(R) 004 004 005 004 005



(Oax) π2(Oax)

π1(L) and π1






GAS3(1211) GAS3(1413) GAS4(1817) ORMAS66

S T S T S T S


(Oax) 017 027 016 025 016 025 015π2

(Oax) 017 105 017 104 017 104 015π1

(L) 003 003 003 003 004 004 003π1

(R) 004 005 004 005 004 004 004π2









3dx2 minusy2



(Oax )

π1 L( ) π 2 L( ) π 2 R( ) π1 R( )

π1 L( ) π 2

L( ) π 2 R( ) π1

R( )












IV DISCUSSION








ACKNOWLEDGMENTS


















































GAS3(1211) GAS3(1413) GAS4(1817) ORMAS66

S T S T S T S


(Oax) 017 027 016 025 016 025 015π2

(Oax) 017 105 017 104 017 104 015π1

(L) 003 003 003 003 004 004 003π1

(R) 004 005 004 005 004 004 004π2









3dx2 minusy2



(Oax )

π1 L( ) π 2 L( ) π 2 R( ) π1 R( )

π1 L( ) π 2

L( ) π 2 R( ) π1

R( )












IV DISCUSSION








ACKNOWLEDGMENTS

























































IV DISCUSSION








ACKNOWLEDGMENTS
















































the generalized active space concept in multiconfigurational self-consistent field methods

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