computational chemistry (kje-3102) basis...

The Centre for Theoretical and Computational Chemistry


Computational Chemistry (KJE-3102) Basis Sets

Bin Gao ([email protected])

Center for Theoretical and Computational ChemistryDepartment of Chemistry

University of Troms

Feb. 11 and 21, 2011




Outline

1 Generalities about Basis Sets

2 Slater Basis Sets

3 Gaussian Basis Sets

4 Integral Evaluation

5 Pseudopotentials

6 GEN1INT Tool Package

Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 2 / 55




The Schrodinger Equation

Based on Born-Oppenheimer approximation, the motions of electrons can beseparated from nuclei, which results in the electronic Schrodinger equation

Hee = Eee, (1)

with

He = Te + Vne + Vee + Vnn (2)

= N

i=1

122i

N

i=1

M

A=1

ZAriA

+N1

i=1

N

j=i+1

1rij

+ Vnn. (3)





Solving the Schrodinger EquationThe underlying physical laws necessary for the mathematical

theory of a large part of physics and the whole of chemistry are thuscompletely known, and the difficulty is only that the exact applicationof these laws leads to equations much too complicated to be soluble.

P. A. M. Dirac

The equationHee = Eee

is a many-body problem, too complicated to solve directly. Approximate waysof solution

Numerical methods, such as finite-difference, and finite-elementapproach.Expansion method represent e as

e(x1,x2, . . . ,xN)

i

Ci i (x1,x2, . . . ,xN). (4)

J. R. Chelikowsky et al., Phys. Rev. Lett. 72, 1240 (1994).J. R. Chelikowsky et al., Phys. Rev. B 50, 11355 (1994).J. E. Pask et al., Phys. Rev. B 59, 12352 (1999).





Slater Determinants

Pauli exclusion principle the wave function e must be antisymmetricwith respect to the permutation of any two electrons, i.e.

e(x1,x2, . . . ,xi, . . . ,xj, . . . ,xN) = e(x1,x2, . . . ,xj, . . . ,xi, . . . ,xN). (5)

Slater determinant:

SD(x1,x2, . . . ,xN) =1N!

1(x1) 2(x1) N(x1)1(x2) 2(x2) N(x2)

......

. . ....

1(xN) 2(xN) N(xN)

, (6)

where i (x) is the molecular spin orbital, composed of spatial orbital andspin function,

i (x) = i (r)i (s), (i = , ). (7)

Note that it is one-electron wave function (from independent particle model)!Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 5 / 55




Molecular Orbitals and Basis Sets

The spatial part i (r) could in principle be constructed by a linearcombination of basis functions:

i (r) =

j

cijj (r), (8)

where cij is the molecular orbital (MO) coefficient, {j (r)} in definition is acomplete basis set and would require an infinite number of basisfunctions. However, in practice, one has to use a finite number of basisfunctions, for instance the Gaussian Type Orbital (GTOs) function.Linear Combination of Atomic Orbitals (LCAO) strictly speaking,Atomic Orbitals (AO) are solutions for the atom. Atomic Basis Functionshould be more appropriate here.

They should be the basis functions of Hilbert space.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 6 / 55




Hartree-Fock-Roothaan Method

Converged!

Yes

No1 ?n nD D

1Determine nD

0Get initial density matrix D

Compute nF D

Figure: Illustration of SCF procedure.





MOs from DALTON program

i (r) =

j

cijj (r)

Molecular orbitals for symmetry species 1------------------------------------------

Orbital 1 2 3 41 O :1s -1.0004 0.0070 -0.0000 -0.00032 O :1s -0.0013 -0.9097 -0.0000 -0.12773 O :1s 0.0020 0.0270 0.0000 0.22154 O :2px 0.0000 -0.0000 -0.0000 0.00005 O :2py -0.0039 0.2408 0.0000 -0.76866 O :2pz 0.0000 0.0000 -0.9133 0.0000... ... ... ... ... ...





Atomic Basis Functions

Mathematically speaking, the basis set is the set of functions from whichthe wave function is constructed.Requirements of suitable basis functions

The basis should be designed such that it allows for an orderly andsystematic extension towards completeness with respect to one-electronsquare-integrable functions.The basis should allow for a rapid convergence to any atomic or molecularelectronic state, requiring only a few terms for a reasonably accuratedescription of molecular electron distributions.The functions should have an analytical form that allows for easymanipulation. In particular, all molecular integrals over these functionsshould be easy to evaluate. It is also desirable that the basis functions areorthogonal or at least that their nonorthogonality does not present problemsrelated to numerical instability.

Christopher J. Cramer, Essentials of Computational Chemistry: Theories and Models(Second Edition).Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, Molecular Electronic-Structure Theory.





Different kinds of Basis Functions

Slater Type Orbitals (ADF)Gaussian Type Orbitals (DALTON, DIRAC, GAUSSIAN,GAMESS USA, ...)Plane Wave Basis Functions (modelling extended (infinite) systems)Wavelet (MRChem)SIESTA program: Numerical atomic orbitals for linear-scalingcalculations, J. Junquera, O. Paz, D. Sanchez-Portal, and E. Artacho,Phys. Rev. B 64, 235111, (2001).... ...





Slater Type Orbitals (STO)

,n,l,m(r , , ) = NYlm(, )rn1er , (9)

where N is a normalization constant, is called exponent. The r , , and are spherical coordinates, and Ylm is the angular momentum part (sphericalharmonic function describing shape).Pros and Cons:

The exponential dependence on the distance between the nucleus andelectron mirrors the exact orbitals for the hydrogen atom exponentialdecay, s-type Slaters have a nuclear cusp (discontinuous derivative).The STOs do not have any radial nodes; nodes in the radial part areintroduced by making linear combinations of STOs.Very difficult to compute three- and four-centre two-electron integrals.Used in some atomic codes and some molecular codes, for instance,Amsterdam Density Functional (ADF) package.

Frank Jensen, Introduction to Computational Chemistry: Chapter 5.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 11 / 55




Gaussian Type Orbitals (GTO)

Proposed by Boys in 1950,

Spherical-harmonic: ,n,l,m(r , , ) = NYlm(, )r2nl2er2, (10)

Cartesian: ,lx ,ly ,lz (x , y , z) = Nxlx y ly z lz er

2, (11)

where N is a normalization constant, is the exponent, and Ylm is thespherical harmonic function.Cons and Pros:

Instead of cusp, a GTO has a zero slope at the nucleus, so that GTOshave problems representing the proper behaviour near the nucleus.The GTO falls off too rapidly far from the nucleus compared with an STO,and the tail of the wave function is consequently represented poorly.Therefore, more GTOs are necessary for achieving a certain accuracycompared with STOs.However, integrals are more efficient to compute, most commonly used inmolecular codes.

Frank Jensen, Introduction to Computational Chemistry: Chapter 5.Do not be confused by the orbital, they are simply basis functions!





Gaussian Primitives

In practice, all applications take the GTOs to be centred at the nuclei. Forinstance, a primitive Cartesian Gaussian centered at O takes the form:

,lx ,ly ,lz (xO , yO , zO) = N(x Ox )lx (y Oy )ly (z Oz)lz er2O . (12)

Another better candidate when evaluating geometric derivatives HermiteGaussian:

,lx ,ly ,lz (xO , yO , zO) = N(2)lxlylz

(

Ox

)lx ( Oy

)ly ( Oz

)lzer

2O , (13)

as proposed by Reine, Tellgren and Helgaker (Phys. Chem. Chem. Phys. 9(2007), 4771).

The definition of primitive will be given in Contracted Basis Sets part.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 13 / 55




The exponents

How are these Gaussian primitives derived?Gaussian primitives are usually obtained from quantum calculations onatoms, the exponents are typically varied until the lowest total energy ofthe atom is achieved.The exponents could be either optimized independently, or related toeach other by some equation, and parameters in this equation areoptimized, for example, the even-tempered or well-tempered basissets.These primitives from atomic calculations cannot accurately describedeformations of atomic orbitals in the molecule. Augmented functions,such as polarization functions and diffuse functions, are thereforeusually used.In molecular calculations, these Gaussian primitives have to becontracted, i.e., certain linear combinations of them will be used as basisfunctions which are sometimes called Contracted Gaussian TypeOrbitals (CGTO).

Simplified Introduction to Ab Initio Basis Sets. Terms and Notation atwww.ccl.net/cca/documents/basis-sets/basis.html


www.ccl.net/cca/documents/basis-sets/basis.html




Even-Tempered Basis Sets

The optimization of basis function exponents becomes difficult when thebasis set becomes large, since the basis functions start to becomelinearly dependent and the energy becomes a very flat function of theexponents.Notice that the ratio between two successive optimized exponents isapproximately constant, the optimization with fixed ratio thus involvesonly two parameters for each type of basis function, independent of thesize of the basis. Such basis sets are even-tempered basis sets, with thei th exponent given as

i = i . (14)

Pros: easy to generate a sequence of basis sets that are guaranteed toconverge towards a complete basis.Cons: however, the convergence is somewhat slow, and an explicitlyoptimized basis set of a given size will usually give a better answer thanan even-tempered basis of the same size.

It was later discovered that the optimum and constants to a good approximation can bewritten as functions of the size of the basis set, M Frank Jensen, Introduction toComputational Chemistry: Chapter 5.Frank Jensen, Introduction to Computational Chemistry: Chapter 5.





Well-Tempered Basis Sets

The exponents in a well-tempered basis of size M are generatedaccording to

i = i1[

1 + (

iM

)]; i = 1,2, . . . ,M, (15)

the parameters , , and are optimized for each atom.Compared to even-tempered basis set, the well-tempered basis set hasfour parameters, and is thus capable of giving a better result for the samenumber of functions.More general parameterization proposed by Petersson et al., see.From the point of view of users, we need to pick up a suitable basis setduring calculations, in particular, when our interest is in specializedproperties.





Contracted Basis Sets

Contracted GTOs (CGTOs):

Spherical-harmonic: n,l,m(rO , O , O) = Ylm(O , O)r2nl2O

i

wiei r2, (16)

Cartesian: lx ,ly ,lz (xO , yO , zO) = xlxOy

lyOz

lzO

i

wiei r2, (17)

where the normalization constants of individual GTOs (known as the primitiveGTOs, PGTOs) have been adsorbed into the contraction coefficients wi .

(10s4p1d/4s1p)[3s2p1d/2s1p], or (10s4p1d/4s1p)/[3s2p1d/2s1p]: Thebasis in parenthesis is the number of primitives with heavy atoms (firstrow elements) before the slash and hydrogen after. The basis in thesquare brackets is the number of contracted functions.





Segmented Contraction

STO-3G in DALTON:

$ CARBON (6S,3P) -> [2S,1P]$ S-TYPE FUNCTIONS

6 2 071.6168370 0.15432897 0.0000000013.0450960 0.53532814 0.000000003.5305122 0.44463454 0.000000002.9412494 0.00000000 -0.099967230.6834831 0.00000000 0.399512830.2222899 0.00000000 0.70011547

$ P-TYPE FUNCTIONS... ... ... ...





General Contraction

cc-pVDZ in DALTON:

$ CARBON (9s,4p,1d) -> [3s,2p,1d]$ S-TYPE FUNCTIONS

9 3 06665.0000000 0.00069200 -0.00014600 0.000000001000.0000000 0.00532900 -0.00115400 0.00000000228.0000000 0.02707700 -0.00572500 0.0000000064.7100000 0.10171800 -0.02331200 0.0000000021.0600000 0.27474000 -0.06395500 0.000000007.4950000 0.44856400 -0.14998100 0.000000002.7970000 0.28507400 -0.12726200 0.000000000.5215000 0.01520400 0.54452900 0.000000000.1596000 -0.00319100 0.58049600 1.00000000

$ P-TYPE FUNCTIONS... ... ... ...$ D-TYPE FUNCTIONS... ... ... ...





Classification of Basis Sets

STO-3G (minimal basis, single- basis sets): proposed by Hehre,Stewart, and Pople in 1969 mimicking STOs with contracted GTOs.Double- basis sets: two functions for each AO size of basis set doesnot change, but the size of the secular equation would be increased.Triple- basis sets, and higher multiple- basis sets.Split-valence or valence-multiple- basis sets: core orbitals continueto be represented by a single (contracted) basis function, while valenceorbitals are split into arbitrarily many functions, for instance, 3-21G,6-21G, 4-31G, 6-31G, and 6-311G (split-valence basis sets usingsegmented contractions).cc-pVDZ and cc-pVTZ: correlation-consistent polarized Valence(Double/Triple) Zeta basis sets (split-valence basis sets using generalcontractions).

There is one and only one basis function defined for each type of orbital core throughvalence, other minimal basis sets include the MINI sets of Huzinaga and co-workers.Correlation-consistent implies that the exponents and contraction coefficients were

variationally optimized not only for HF calculations, but also for calculations including electroncorrelation.Christopher J. Cramer, Essentials of Computational Chemistry: Theories and Models

(Second Edition).Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 20 / 55




Poples Basis Sets

n-ijG or n-ijkG:n is the number of primitives for the inner shells.ij or ijk are the number of primitives for contractions in the valence shell,and ij describes sets of valence double zeta quality and ijk sets ofvalence triple zeta quality.

Simplified Introduction to Ab Initio Basis Sets. Terms and Notation atwww.ccl.net/cca/documents/basis-sets/basis.html


www.ccl.net/cca/documents/basis-sets/basis.html




Polarization Functions

Adding higher angular momentum basis functions (than any occupiedatomic orbital) to describe the polarization of the charge distribution uponformation of a chemical bond.6-31G* implies a set of d functions are added on first-row atoms.6-31G** implies except for a set of d functions added on first-row atoms,p functions are also added on hydrogen atom.





Diffuse Functions

The highest energy MOs of anions, highly excited electronic states, andloose supermolecular complexes, tend to be much more spatially diffusethan garden-variety MOs.Diffuse functions (functions with small exponents, hence large radialextent) are thus usually included.6-31+G(d): added one s and one set of p functions having smallexponents to heavy atoms.6-311++G(3df,2pd): the second plus indicates the presence of diffuse sfunctions on hydrogen atoms.

Christopher J. Cramer, Essentials of Computational Chemistry: Theories and Models(Second Edition)





Extra Topic: Basis-set Linear Dependence

Basis sets become linearly dependent when the size of the basis and thesize of the molecule grow, in particular if the basis contains diffusefunctions.Orthonormalization of a linearly dependent basis set becomesproblematic.Remove MOs in DALTON:

*ORBITAL INPUT.AO DELETETHROVL

THROVL is the limit for basis set numerical linear dependence[eigenvectors (of AO overlap matrix) with eigenvalue less than THROVLare excluded]. Default is 106.





Extra Topic: Basis-set Superposition Error (BSSE)

Molecular interaction energy EAB (EA + EB).A systematic error would occur when using finite basis sets Basisfunctions from A can help compensate for the basis set incompletenesson B, and vice versa. The energy of AB will therefore be artificiallylowered, and the interaction energy will be overestimated. This is knownas Basis-set Superposition Error (BSSE).Using more basis functions requires very large basis sets, not feasible.Counterpoise-corrected interaction energy EAB (ECPA + ECPB ), whereECPA is the counterpoise-corrected energy of A calculated in the full basisof AB, i.e., by including the ghost basis functions on B. Likewise for ECPB .

Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, Molecular Electronic-Structure TheoryBin Gao (CTCC, UiT) Computational Chemistry KJE-3102 25 / 55




Extra Topic: Crystal Orbitals

k ,(r) =1Nsite

g

eik g(r g), (18)

|k , PBC =1Nsite

g

eik g |g, AO. (19)





Extra Topic: London Atomic Orbitals (LAO)

The London Atomic Orbital is defined by multiplying the conventional AOs bya field-dependent phase factor

(r) = exp[ i2 B (R G)(r P)

]

(r R)(|r R|), (20)

where is centered at the nuclear position R, B is the magnetic field, G isthe gauge origin of the magnetic vector potential, and P the origin of theLondon phase factor exp

[ i2 B (R G)(r P)

]. The second factor

(r) in eqn (20) is the angular part of the AO, typically a solid harmonicfunction Sl,m(r) in the position r of the electron relative to the AO. Finally,(r) is a decaying radial form in r, usually chosen as a contracted Gaussian

(r) =

i

wi exp(air2), (21)

where wi and ai are the radial contraction coefficients and orbitalexponents, respectively.Bast et al., Phys. Chem. Chem. Phys. 13 (2011), 2627.





Recommended Literatures and Websites

Christopher J. Cramer, Essentials of Computational Chemistry: Theoriesand Models (Second Edition): Chapters 4.3 and 6.2.Frank Jensen, Introduction to Computational Chemistry: Chapter 5.Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, MolecularElectronic-Structure Theory: Chapters 6, 8 and 9.Simplified Introduction to Ab Initio Basis Sets. Terms and Notation atwww.ccl.net/cca/documents/basis-sets/basis.html

EMSL Basis Set Exchange at https://bse.pnl.gov/bse/portalSegmented Gaussian Basis Set athttp://setani.sci.hokudai.ac.jp/sapporo/Welcome.do


www.ccl.net/cca/documents/basis-sets/basis.htmlhttps://bse.pnl.gov/bse/portalhttp://setani.sci.hokudai.ac.jp/sapporo/Welcome.do




One- and Two-electron Integrals

O = |O1| =(r)O(r)(r)dr , (22)

O = (1)(1)|O12|(2)(2) (23)

=

(r1)(r1)O(r1, r2)(r2)(r2)dr1dr2.





Gaussian Product Rule

The product of two Gaussians can be written in terms of another Gaussianfunctions, for instance,

exp(ax2A) exp(bx2B) = exp(X 2AB) exp(px2P), (24)

where

p = a + b total exponent, (25)

=ab

a + breduced exponent, (26)

Px =aAx + bBx

pcenter-of-charge coordinate, (27)

XAB = Ax Bx relative coordinate/Gaussian separation, (28)K xab = exp(X 2AB) pre-exponential factor. (29)





Gaussian Overlap Distribution

ab(r) = Gikm(r ,a,A)Gjln(r ,b,B) (30)

= xij (x ,a,b,Ax ,Bx )ykl (y ,a,b,Ay ,By )

zmn(z,a,b,Az ,Bz), (31)

where the x component is

xij (x ,a,b,Ax ,Bx ) = Gi (x ,a,Ax )Gj (x ,b,Bx ) (32)

= K xabxiAx

jB exp(px2P) (33)

= K xab

i+j

k=0

C ijk xkP exp(px2P). (34)





Properties of Gaussian Overlap Distribution

xAxij = xi+1,j , (35)

xBxij = xi,j+1, (36)

xi,j+1 xi+1,j = XABxij , (37)xijAx

= 2axi+1,j ixi1,j , (38)

xijBx

= 2bxi,j+1 jxi,j1, (39)(Px

= Ax

+ Bx

XAB

= bpAx ap

Bx

(40)

xijPx

= 2axi+1,j + 2bxi,j+1 ixi1,j jxi,j1, (41)

xijXAB

= 2(xi+1,j xi,j+1) +1

2p(2ajxi,j1 2bixi1,j ). (42)

Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, Molecular Electronic-Structure Theory:Chapter 9.





Integrals of Primitives Obara-Saika Scheme

Overlap integrals Sab = Ga|Gb = SijSklSmn, where

Sij =Z

xij dx . (43)

Translational recurrence relation

2aSi+1,j iSi1,j + 2bSi,j+1 jSi,j1 = 0, (44)

and horizontal recurrence relation

Si,j+1 Si+1,j = XABSij , (45)

Obara-Saika Scheme

Si+1,j = XPASij +1

2p(iSi1,j + jSi,j1), (46)

Si,j+1 = XPBSij +1

2p(iSi1,j + jSi,j1), (47)

and starting from

S00 =r

pexp(X 2AB). (48)






Integrals of Primitives McMurchie-DavidsonScheme

Hermite Gaussians tuv (r ,p,P) = (/Px )t (/Py )u(/Pz)v exp(pr2P).Overlap distribution expanded in Hermite Gaussians

ij =

i+j

t=0

E ijt t , (49)

t (x ,p,Px ) = (/Px )t exp(pr2P). (50)Auxiliary distributions

tij = Kxabx

iAx

jBt (xP), (51)

and the McMurchie-Davidson recurrence relations

ti+1,j = tt1ij + XPA

tij +

12p

t+1ij , (52)

ti,j+1 = tt1ij + XPB

tij +

12p

t+1ij . (53)






Coulomb Integrals

The complications arise from the presence of the inverse operator 1rC or1

r12.

Using the fact that

1rC

=1

+

exp(r2C t2)dt , (54)

the key-quantity in the evaluation of Coulomb integrals is the so-calledBoys function

Fn(x) = 1

0exp(xt2)t2ndt (55)

and its recurrence relations.






Other Schemes of Evaluating Integrals

Gaussian quadrature see, for instance, Trygve Helgaker, PoulJrgensen, and Jeppe Olsen, Molecular Electronic-Structure Theory:Chapter 9.6.The multiple method for Coulomb integrals see, for instance, TrygveHelgaker, Poul Jrgensen, and Jeppe Olsen, MolecularElectronic-Structure Theory: Chapter 9.13, and Elias Rudberg and PaweSaek, J Chem. Phys. 125 (2006), 084106.Cholesky decomposition for instance, I. Reggen and Tor Johansen,J. Chem. Phys. 128 (2008), 194107, and Linus Boman et al., J. Chem.Phys. 129 (2008), 134107.





Integrals of Contracted GTOs

(r) = (r R)

i

wi exp(air2), (56)

O = |O1| =(r)O(r)(r)dr (57)

=

ij

wiwj(r R) exp(air2)O(r)(r R) exp(ajr2 )dr .





Symmetry

One-electron operator

O = |O1| = |O1| = O (symmetric), (58)

or

O = |O1| = |O1| = O (anti-symmetric). (59)

Two-electron integral O = (1)(1)|O12|(2)(2)

O = O = O = O (60)= O = O = O = O.

From group theory, the integral can only be none zero if the direct productof the bra and ket irreps is equal to (or contains) an irrep correspondingto the operator.

Christopher J. Cramer, Essentials of Computational Chemistry: Theories and Models(Second Edition): Appendix B.





Effective Core Potentials (ECP)

The ECP operator around the core C with L 1 being the largest angular momentum orbital

U(rC) = UL(rC) +L1Xl=0

lXm=l

|Ylm[Ul (rC) UL(rC)]Ylm|, (61)

The functions Ylm(C , C) are real spherical harmonics centered on C

Ylm(C , C) =

8>>>:Y 0l (C , C), if m = 0,

12

hY ml (C , C) + (1)

mYml (C , C)i, if m > 0,

1i

2

hYml (C , C) (1)

mY ml (C , C)i, if m < 0,

(62)

where Y ml is the complex spherical harmonics.The radial functions UL(rC) and Ul (rC) UL(rC) (l = 0, . . . , L 1) are expressed ascombinations of Gaussians

UL(rC) =Ncore

rC+

Xk

dkLrnkLC eckLr2C , (63)

Ul (rC) UL(rC) =X

k

dkl rnklC eckl r2C , (64)

where Ncore is the number of core electrons. nkL and nkl are generally restricted to 0, 1, and 2.Chris-Kriton Skylaris et al. Chem. Phys. Lett. 296 (1998) 445451.Larry E. McMurchie and Ernest R. Davidson, J. Comp. Phys. 44 (1981) 289301.





Model Core Potentials (MCP)

In MCP Version 1, for an atom C with the atomic charge Z and Ncore core electrons,the interaction operator V core between core and valence electrons is approximated as

V core Vmp(rC) = Z Ncore

rC

Xl

Al rnlC el r2C , (65)

where nl = 0 or 1, and the parameters Al and l are adjustable parameters of MCP.Moreover, in order to prevent the valence orbitals from collapsing onto those of coreelectrons, the so-called energy shift operator is usually added into the final Hamiltonian

= X

k

fkk |k (rC) k (rC)| , (66)

where 1 fk 2 are adjustable parameters, and k is the eigenvalue of the k th coreorbital. k (rC) is the k th core orbital, which is represented by real solid-harmonicGaussian functions.

Sigeru Huzinaga, Can. J. Chem. 73 (1995) 619628.Mariusz Klobukowski et al., in Computational Chemistry: Reviews of Current Trends, 3 (1999)

4974.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 40 / 55




MCP (Continue)

In MCP Version 2 (AIMP), the Coulomb potential is represented as

V coreC V coreC =X

j

Aj r nj1 exp(j r 2), (67)

while the exchange operator V coreX is used in the spectral representation VcoreX

V coreX = PVcoreX P (68)

whereP =

Xpq

|p(S1)pqp| (69)

is defined using a finite, non-orthogonal basis {p} with the metric Spq = p|q. Thecore projector operator still has the form

= 2X

k

k |k k |. (70)

Version III of the model potential method is the logical extension of the AIMP method,in which the entire core operator V core = V coreC + V

coreX is used in the spectral

representationV core V core = P(V coreC + V coreX )P. (71)

Sigeru Huzinaga, Can. J. Chem. 73 (1995) 619628.Mariusz Klobukowski et al., in Computational Chemistry: Reviews of Current Trends, 3 (1999)

4974.Bin Gao (CTCC, UiT) Computational Chemistry KJE-3102 41 / 55




Extra Topic: Integrals between Crystal Orbitals

A(k) = k , |A|k , PBC =1

Nsite

g,h

eik (hg)g, |A|h, AO

=

l

eik l0, |A|l , AO. (72)

Challenge: evaluation of Coulomb integrals.See, for example, references at CRYSTAL program homepagehttp://www.crystal.unito.it/.


http://www.crystal.unito.it/




What is GEN1INT?

GEN1INT is a Fortran 90 library (with Python interface) to evaluate thegeometric and magnetic derivatives of one-electron integrals

at zero field (for instance B = 0), andusing contracted London atomic orbitals (LAO)(r ; B) = exp

[ i2 B (R G) rP

](r).

More explicitly, what we evaluate is

N

LR

[K 1B

(r ; B)

]B=0

O({rC} ,nr

) [K 2B (r ; B)

]B=0

dr (73)

wherethe number of differentiated centers N 4, so thatthe number of centers in operator O satisfies N 2,the operator O may also depend on the magnetic field B, which givesnew operator

[

K OB O

]B=0

.





A Generalized One-electron Operator

O({rC} ,nr

)=

Cf ({rC}) nr ,Effective core potential,Model core potential (Version 1).

(74)

where

f ({rC}) =

rmM[LCC r

m0C

], (m0 = 1,2),

rmM[LCC (rC)

],

rmM ,

LC1C1

LC2C2

r1C1 r1C2,

erf(%rC)

rC.

(75)

Multi-index notations

rmM = xmxM y

myM z

mzM , (76)

nr =

(

x

)nx ( y

)ny ( z

)nz. (77)





Geometric Derivatives

Take a molecule with 12 atoms as an example, the number of third orderone-center geometric derivatives is 120, and 2376 for the two-center geometricderivatives, 5940 the three-centers 8436 third order geometric derivatives intotal.What is worse ;-) some guys has special requirement about the order ofgeometric derivatives

1(1)

1(1)

0

2(2)

1

3(2)

1

1(1)

0

2(2)

1

3(2)

1

2(2)

0

3(3)

1

3(2)

0

1(1)

0

2(2)

1

3(2)

1

2(2)

0

3(3)

1

3(2)

0

2(2)

0

3(3)

1

3(3)

0

3(2)

0

H = 3L H + 1 = 4

1 1 |L1|=2

2 2 |L2|=2

(|L1|+ 22

)

X1

X1

X1

Y1

X1

Z1

Y1

Y1

Y1

Z1

Z1

Z1

(|L2|+ 22

)

X2

X2

X2

Y2

X2

Z2

Y2

Y2

Y2

Z2

Z2

Z2

K=2

k=1

(|Lk|+ 22

)

X1

X1

X2

X2

X1

X1

X2

Y2

Z1

Z1

Z2

Z2





Geometric Derivatives (Continue)

X1

R1(r)O ({rC} ,nr )1(r)dr (78)

=R h

X1

1(r)i

O1(r)dr +R1(r)O

hX1

1(r)i

dr .

0 (translational invariance).

Table: Possible geometric derivatives for R = R and the operator with the number ofcenters N 2.

Coincide centers Possible geometric derivativesN = 0 R = R 0N = 1 R = R = C1 0

R = R 6= C1 (1)|L|LC1 +LC1

N = 2 R = R = C1 = C2 0

R = R = C1 6= C2 (1)|L|LC2 +LC2

(R = R) 6= (C1 = C2) (1)|L|PL+LC1

l1=0

`L+LC1l1

l1C1

L+LC1l1C2

R = R 6= C1 6= C2PL

l=0

`Ll

lR

LlR

LC1C1

LC2C2





Magnetic Derivatives

In order to evaluate

LRLR

[K 1B

(r ; B)

]B=0

O({rC} ,nr

) [K 2B (r ; B)

]B=0

dr (79)

to any order K 1 and K 2, we introduce the following auxiliary integral

LRLR

rN1P

[K 1B

(r ; B)

]B=0

O({rC} ,nr

)rN2P

[K 2B (r ; B)

]B=0

dr , (80)

and we have the recurrence relations

{K 1+e,LN1}K 0L0 = i2[(RG)+1{K 1L,N1+e1}K 0L0 (81) (RG)1{K 1L,N1+e+1}K 0L0+ (L)+1{K 1,Le+1,N1+e1}K 0L0 (L)1{K 1,Le1,N1+e+1}K 0L0

],

likewise for K 2.





Magnetic Derivatives (Continue)

(a) the order of functions in shells

do nz = 0, Kmaxdo ny = 0, Kmaxnz

(Kmax(ny+nz), ny, nz)end do

end do

0 y

x

z

yy

xy

xx

xz

yz

zz

xxz

yyy

xyy

xxy

xxx

xyz

yyz

xzz

yzz

zzz

xyyz

xxyz

xxxz

yyyy

xyyy

xxyy

xxxy

xxxx

yyyz

xxzz

xyzz

yyzz

xzzz

yzzz

zzzz

Kmax = 5

yyyyz

xyyyz

xxyyz

xxxyz

xxxxz

yyyyy

xyyyy

xxyyy

xxxyy

xxxxy

xxxxx

xxxzz

xxyzz

xyyzz

yyyzz

xxzzz

xyzzz

yyzzz

xzzzz

yzzzz

zzzzz

{x}{y}{z}

{x}

{x}{y}

{x}{y}{z}

{x}

{x}

{x}

{y}{x}

{x}{y}

{x}{y}

{z}

{x}

{x}

{x}

{x}

{y}{x}

{x}

{x}{y}

{x}

{x}{y}

{x}{y}

{z}

{x}

{x}

{x}

{x}

{x}

{y}{x}

{x}

{x}

{x}

{y}{x}

{x}

{x}{y}

{x}

{x}{y}

{x}{y}

{z}

(b) recurrence relations

{K1+e,LN1}K0= i2

[(RG)+1{K1L,N1+e1}K0

(RG)1{K1L,N1+e+1}K0+ (L)+1{K1,Le+1,N1+e1}K0 (L)1{K1,Le1,N1+e+1}K0

],

do k = Kmax, 1, 1do nx = 1, k

recurrence relation along x directionend dorecurrence relation along y direction

end dorecurrence relation along z direction

This is also the order of basisfunctions (in shells), and geo-metric derivatives in GEN1INT.





Class Organization of GEN1INT

+contracted_cgto()+contracted_sgto()+contracted_lcgto()+contracted_lsgto()

ContrInt

-Center of mass-Gauge origin-Dipole origin-Origin of London phase factor

+__init__()+set_center_of_mass()+set_gauge_origina()+set_dipole_origin()+set_london_orgin()+overlap()+diplen()+potenergy()

PropCGTO

-Center of mass-Gauge origin-Dipole origin-Origin of London phase factor

+__init__()+set_center_of_mass()+set_gauge_origina()+set_dipole_origin()+set_london_orgin()+overlap()+diplen()+potenergy()

PropSGTO

>>> import Gen1Int.PropCGTO>>> S = Gen1Int.PropCGTO.overlap \. . . (coord_bra, exp_bra, coeff_bra, \. . . coord_ket, exp_ket, coeff_ket)

User





ContrInt

contractedCGTO

contractedSGTO

geometricderivatives

contractedLondon SGTO

contractedLondon CGTO

contractedHGTO

contractedLondon HGTO

rmM

[LCC r

2C

]nrr

mM

[LCC r

1C

]nrr

mM

[LCC (rC)

]nrr

mM

nr

erf(%rC)rC

nrMCP

(Version 1) ECP LC1C1

LC2C2

r1C1 r1C2

nr

recurrencerelations quadrature





Parallelization

1(1)

1(1)

0

2(2)

1

3(2)

1

1(1)

0

2(2)

1

3(2)

1

2(2)

0

3(3)

1

3(2)

0

1(1)

0

2(2)

1

3(2)

1

2(2)

0

3(3)

1

3(2)

0

2(2)

0

3(3)

1

3(3)

0

3(2)

0

H = 3L H + 1 = 4

1 1 |L1|=2

2 2 |L2|=2

(|L1|+ 22

)

X1

X1

X1

Y1

X1

Z1

Y1

Y1

Y1

Z1

Z1

Z1

(|L2|+ 22

)

X2

X2

X2

Y2

X2

Z2

Y2

Y2

Y2

Z2

Z2

Z2

K=2

k=1

(|Lk|+ 22

)

X1

X1

X2

X2

X1

X1

X2

Y2

Z1

Z1

Z2

Z2

basis functions of ket

basi

sfu

ncti

ons

ofbr

a

a processor

mpi4py (MPI for Python - Python bindings for MPI) for parallelization.Parallel IO using HDF5 (h5py in Python).





Status of GEN1INT

Not available for the time being :-(http://sourceforge.net/projects/gen1int (stable tar ball),http://repo.ctcc.no/projects/gen1int (developing version).Up to four-center geometric derivatives.The results of high order derivatives and/or large orbital quantumnumbers may not be reliable for, such as nuclear-attraction integrals,which is due to the evaluation of Boys functions.





Recommended Literatures and Websites

Christopher J. Cramer, Essentials of Computational Chemistry: Theoriesand Models (Second Edition): Chapters 4.3 and 6.2.Frank Jensen, Introduction to Computational Chemistry: Chapter 5.Trygve Helgaker, Poul Jrgensen, and Jeppe Olsen, MolecularElectronic-Structure Theory: Chapters 6, 8 and 9.EMSL Basis Set Exchange at https://bse.pnl.gov/bse/portalSegmented Gaussian Basis Set athttp://setani.sci.hokudai.ac.jp/sapporo/Welcome.do

Pseudopotentials of the Stuttgart/Cologne group athttp://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.html

The Ab Initio Model Potential Library athttp://www.uam.es/departamentos/ciencias/quimica//aimp/Data/AIMPLibs.html

Two-electron integral library Libint at http://www.files.chem.vt.edu/chem-dept/valeev/software/libint/libint.html


https://bse.pnl.gov/bse/portalhttp://setani.sci.hokudai.ac.jp/sapporo/Welcome.dohttp://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.htmlhttp://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.htmlhttp://www.uam.es/departamentos/ciencias/quimica//aimp/Data/AIMPLibs.htmlhttp://www.uam.es/departamentos/ciencias/quimica//aimp/Data/AIMPLibs.htmlhttp://www.files.chem.vt.edu/chem-dept/valeev/software/libint/libint.htmlhttp://www.files.chem.vt.edu/chem-dept/valeev/software/libint/libint.html




Thank You for Your Attention !!




Exercise 7 of KJE-3102: Basis Sets 2

Please use any program language you are familiar with, implement theObara-Saika scheme of overlap integrals between two Cartesian Gaussians(along x direction)

Si+1,j = XPASij +1

2p(iSi1,j + jSi,j1), (82)

Si,j+1 = XPBSij +1

2p(iSi1,j + jSi,j1), (83)

starting from

S00 =

pexp(X 2AB). (84)

Please report your code and plot the overlap integrals using i = j = 2,a = b = 1.0, and XAB = 0.2,0.4,0.6,0.8,1.0,1.5,2.0,3.0,4.0,6.0,8.0,10.0.What conclusion could you draw from this picture?


Generalities about Basis SetsSlater Basis SetsGaussian Basis SetsIntegral EvaluationPseudopotentialsGen1Int Tool Package

computational chemistry (kje-3102) basis...

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