Hartree-Fock Method and Beyond - İYTEnanodft09.iyte.edu.tr/pdf/DT-talk.pdf · 2009-08-28 · Conﬁguration InteractionTheory Möller-Plesset PerturbationTheory Overview of Coupled-Cluster

Configuration Interaction TheoryMöller-Plesset Perturbation Theory

Overview of Coupled-Cluster TheoryTutorial

Hartree-Fock Method and Beyond

Daniele Toffoli

Department of Chemistry, Middle East Technical University, Ankara, Turkey.

August 19, 2009

Daniele Toffoli Hartree-Fock Method and Beyond



Outline

1 Configuration Interaction TheoryCorrelation EnergyThe algebraic approximationA simple example: the H2 moleculeSize Extensivity

2 Möller-Plesset Perturbation TheoryNon-degenerate Rayleigh-Schrödinger Perturbation TheoryPerturbative Expansion of the correlation energySize extensivity of MPn methods

3 Overview of Coupled-Cluster Theory

4 TutorialOverview of GAMESS-USGaussian basis sets: an overview




Correlation EnergyThe algebraic approximationA simple example: the H2 moleculeSize Extensivity









SCF Method: drawbacks

A restricted closed shell HF solution is unable to properlydescribe molecular dissociation (H2 dissociates in closedshell fragments, i.e. H+ and H−)

UHF is sometimes used for open-shell systems.

Lack of correlation in the electronic motion.





SCF Method: drawbacks





Correlation Energy

EC = EExactNR − EHF

correlation energy: energy difference between the energy ofthe exact solution and the energy of the Hartree-Fock solution.

The correlated electron motion can be described in terms ofvirtual excitations, or equivalently through mixing of electronicconfigurations (Slater determinants).





Correlation Energy








Correlation Energy








Overview of CI Theory

Upon solution of the SCF equations in a finite basis of 2Kfunctions:

fχi = ǫiχi

We get(

2KN

)

Slater determinants, of which the Hartree-Fock

one has the lowest energy.

This set of Slater determinants constitutes our N electron basis.







fχi = ǫiχi

We get(

2KN

)










fχi = ǫiχi

We get(

2KN

)








The algebraic approximation

All(

2KN

)

Slater determinants can be classified based on the

orbital differences with respect to the HF reference state:

|Ψ0〉 = |χiχj . . . χa . . . χb . . . χc . . . χk 〉 SCF state

|Ψra〉 = |χiχj . . . χr . . . χb . . . χc . . . χk 〉 singly excited

|Ψrsab〉 = |χiχj . . . χr . . . χs . . . χc . . . χk〉 doubly excited

. . .

|Φ〉 = c0|Ψ0〉 +∑

ar

cra|Ψr

a〉 + (12!

)2∑

abrs

crsab|Ψrs

ab〉 + . . .






All(

2KN

)

Slater determinants can be classified based on the

orbital differences with respect to the HF reference state:

|Ψ0〉 = |χiχj . . . χa . . . χb . . . χc . . . χk 〉 SCF state

|Ψra〉 = |χiχj . . . χr . . . χb . . . χc . . . χk 〉 singly excited

|Ψrsab〉 = |χiχj . . . χr . . . χs . . . χc . . . χk〉 doubly excited

. . .

|Φ〉 = c0|Ψ0〉 +∑

ar

cra|Ψr

a〉 + (12!

)2∑

abrs

crsab|Ψrs

ab〉 + . . .






The variational principle is invoked. We need to solve amatrix eigenvalue problem:

HC = EC

The dimensionality of the CI matrix is equal to the numberof Slater determinants included in the CI expansion, i.e.(

2KN

)

The dimension is prohibitively large even for the smallestsystems (∼ 30 × 106 for Nel = 10 and Nb = 19). Iterativediagonalization techniques.





Sparsity of the CI matrix

|Φ〉 = c0|Ψ0〉 + cS |S〉 + cD|D〉 + cT |T 〉 + . . .

〈Ψ0|H|S〉 = 0 in virtue of the Brillouin Theorem.

〈S|H|Q〉 = 〈Ψ0|H|T 〉 = 0 and so on

〈D|H|Q〉 = 〈Ψrsab|H|Ψtuvw

cdef 〉 6= 0 only if ab ∈ cdef andrs ∈ tuvw

The CI matrix is sparse.





Sparsity of the CI matrix

Further reduction of computational complexity is due to:

Molecular symmetry: the Hamiltonian matrix will be blockdiagonal in a symmetry adapted basis set, and the matrixdimension is reduced compared to complete neglect ofsymmetry.

Spin properties: Spin-adapted basis set (ConfigurationState Functions, CSFs) instead of simple determinants.For singlet states, we discard determinants for whichNα 6= Nβ.

Use of optimal one electron basis: natural spin orbitals.





One electron reduced density matrices

γ1(x1,x′1) = N

∫

Ψ(x1,x2, . . . ,xN)Ψ∗(x′1,x2, . . . ,xN)dx2, . . . ,dxN

The electron density, ρ(x1) is simply the diagonal elementof the one-electron reduced density matrix:

Trγ1(x1,x′1) = N

γ1 is hermitean, i.e. γ†1 = γ1.

〈Ψ|O1|Ψ〉 = tr(hγ1).






Density matrix in HF theory

γ1(x1,x′1) =

∑

i

∑

j

χi(x1)γijχ∗j (x

′1)

from which we have:

γab = δab for occupied orbitals

γrs = 0 for virtual orbitals

The diagonal elements can be interpreted as orbital occupationnumbers.






Density matrix for a CI wave function

γ1(x1,x′1) =

∑

i

∑

j

χi(x1)γijχ∗j (x

′1)

γ is not diagonal.

γ† = γ i.e. can be diagonalised by a unitary change ofbasis: natural spin-orbitals

χ′i =

∑

j

χjUij






Density matrix for a CI wave function

In the natural spin-orbital basis:

γ1(x1,x′1) =

∑

k

χ′k (x1)λkχ

∗k (x′

1)

λk are occupation numbers:0 ≤ λk ≤ 1

In the basis of the natural orbitals the CI expansion is morecompact.





FCI in minimal basis for the H2 molecule

The Hartree-Fock solution in minimal basis is determined solelyby symmetry considerations:

ψ1 ≡ 1 =1

√

2(1 + S)(φ1 + φ2)

ψ2 ≡ 2 =1

√

2(1 − S)(φ1 − φ2)

φ1 and φ1 are 1s orbitals centered on the two nuclei.

S12: extra-diagonal overlap matrix element.

ψ1 is totally symmetric: doubly occupied in the HF groundstate.






Only two closed shell determinants mix in the FCI ground state(intermediate normalization):

|Ψ〉 = 1|11〉 + c|22〉

FCI matrix elements

〈11|H|11〉 = EHF = 2ǫ1 − J11

〈11|H|22〉 = 〈11||22〉 = K12 = 〈22|H|11〉〈22|H|22〉 = ǫ2 − 4J12 + J22 + 2K12






FCI eigenvalue problem

(

2ǫ1 − J11 K12

K12 2ǫ2 − 4J12 + J22 + 2K12

)(

1c

)

= ǫ

(

1c

)

We now subtract from both sides the Hartree-Fock energy:

(

2ǫ1 − J11 00 2ǫ1 − J11

)






CI eigenvalue problem

(

0 K12

K12 2∆

)(

1c

)

= Ec

(

1c

)

Correlation energy in FCI

Ec = ∆ −√

∆2 + K 212

where 2∆ = 2(ǫ2 − ǫ1) + J11 + J22 − 4J12 + 2K12.





Size Extensivity of the CI parameterization

Size extensivity: For N weakly interacting molecules theenergy of the super system should be proportional to thenumber of molecules. In the limit where the separationbetween the molecules is infinite, the energy of thesuper-system is exactly N times the energy of a singlemolecule.

An electronic structure method is size-extensive if theenergy of the super system is predicted to be exactly Ntimes the energy of a single molecule. Otherwise themethod is not size-extensive.





Size Extensivity of the FCI parameterization

FCI wave function of two non interacting H2 molecules

Only four determinants mix in the FCI wave function, due tospatial and spin symmetry:

|Φ0〉 = |Ψ0〉 + c1|11112222〉 + c2|21211212〉 + c3|21212222〉

FCI eigenvalue problem

0 K12 K12 0K12 2∆ 0 K12

K12 0 2∆ K12

0 K12 K12 4∆

1c1

c2

c3

= Ec

1c1

c2

c3






FCI correlation energy

Solving the linear system of equations we obtain:

c1 = c2 = 2K12(Ec−4∆)

c3 = c21

from which:Ec = 2(∆ −

√

∆2 + K 212)

which must be compared with the correlation energy for a

single H2 molecule: Ec = ∆ −√

∆2 + K 212. It therefore follows

that FCI is indeed size consistent.






The condition c3 = c21 is only valid in the limit of non interacting

systems. One could hope that this could represent a good firstorder approximation. This approximation is at the basis of thesuperior accuracy of Coupled-Cluster methods (the socalled disconnected amplitudes).





Size Extensivity of the DCI parameterization

DCI wave function of two non interacting H2 molecules

Only doubly excited determinants mix in the DCI wave function:

|Φ0〉 = |Ψ0〉 + c1|11112222〉 + c2|21211212〉

DCI eigenvalue problem

0 K12 K12

K12 2∆ 0K12 0 2∆

1c1

c2

= Ec

1c1

c2






DCI correlation energy

Solving the linear system of equations we obtain:

c1 = c2 = 2K12(Ec−4∆)

Ec = ∆ −√

∆2 + 2K 212

which is clearly not size consistent.






For N non interacting molecules we need to solve:

0 K12 K12 . . . . . .

K12 2∆ 0 0 . . .

K12 0 2∆ 0 . . .

K12 0 0 2∆ . . .

. . . . . . . . . . . . . . .

1c1

c2

c3

. . .

= Ec

1c1

c2

c3

. . .

to obtain:

c1 = c2 = c3 = . . . =K12

Ec − 2∆

and the correlation energy will be given by:

Ec = NK12c1 = ∆ −√

∆2 + NK 212






Ec ∼√

N for N −→ ∞. DCI is therefore unable to recover anyfraction of the correlation energy for large N, since Ec

N −→ 0.Every truncated CI method is not size consistent.





Weight of the HF wave function

It is easy to demonstrate that the coefficient of the HFdeterminant in the DCI wave function is given by:

c0 =1

√

1 + Nc21

In the limit of large N, the coefficient of the HF solutiongoes to zero, i.e. c0 = 1√

1+Nc21

−→ 0 for N −→ ∞.





Multiconfigurational Self-Consistent Field Theory(MCSCF)

MCSCF ansatz

|ΨMCSCF 〉 = c1|Ψ1〉 + c2|Ψ2〉 + . . .+ cN |ΨN〉

combine the variational optimization of the CI coefficientstogether with the variational optimization of theone-particle spin orbitals.

The MCSCF wave function is very flexible: a large portionof the molecular PES can be explored, away from theequilibrium geometry.

Too few, even if carefully selected, determinants (activeand inactive space selections).





Modern CI: Multireference CI methods

An MCSCF wave function is chosen as a reference:enhanced accuracy into a larger configurational space.

The number of configurations is increased by a factorequal to the number of configurations in the MCSCF wavefunction.

Computationally very expensive: single and doubleexcitations.




Non-degenerate Rayleigh-Schrödinger Perturbation TheoryPerturbative Expansion of the correlation energySize extensivity of MPn methods









PT Equations

Assume our Hamiltonian can be written as:

H = H0 + V

H0 is an Hamiltonian for a problem for which the exacteigenfuntions and eigenvalues can be computed:

H0|Ψ(0)n 〉 = E (0)

n |Ψ(0)n 〉

V : (small) perturbation operator.

Example: Anharmonic part of PESs, correlation energy.





PT Equations


H = H0 + V


H0|Ψ(0)n 〉 = E (0)

n |Ψ(0)n 〉







PT Equations


H = H0 + V


H0|Ψ(0)n 〉 = E (0)

n |Ψ(0)n 〉







PT Equations

We expand both the energy, Ei , and the wave function, |Φi〉 forthe i-th level in Taylor series of the order perturbation:

Ei = E (0)i + λE (1)

i + λ2E (2)i + λ3E (3)

i + . . . ,

and

|Φi〉 = |i〉 + λ|Ψ(1)i 〉 + λ2|Ψ(2)

i 〉 + λ3|Ψ(3)i 〉 + . . . ,

E (1)i and |Ψ(1)

i 〉: first-order corrections for the i-th level.

E (2)i and |Ψ(2)

i 〉: second-order corrections for the i-th level.





PT Equations

Imposing the condition 〈i |Φi〉 = 1 to the wave function:

|Φi〉 = |i〉 + λ|Ψ(1)i 〉 + λ2|Ψ(2)

i 〉 + λ3|Ψ(3)i 〉 + . . . ,

we obtain:

〈i |Ψ(n)i 〉 = 0

valid at each order in the perturbation.





PT Equations

Insert the perturbation expansion for the energy and wavefunction in the Schrödinger Equation:

(H0 + λV )(|i〉 + λ|Ψ(1)i 〉 + λ2|Ψ(2)

i 〉 + . . .) =

(E (0)i + λE (1)

i + λ2E (2)i + . . .)(|i〉 + λ|Ψ(1)

i 〉 + λ2|Ψ(2)i 〉 + . . .)





PT Equations

Group terms of the same order in the perturbation:

H0|i〉 = E (0)i |i〉

(H0 − E (0)i )|Ψ(1)

i 〉 = (E (1)i − V )|i〉

(H0 − E (0)i )|Ψ(2)

i 〉 = (E (1)i − V )|Ψ(1)

i 〉 + E (2)i |i〉

(H0 − E (0)i )|Ψ(3)

i 〉 = (E (1)i − V )|Ψ(2)

i 〉 + E (2)i |Ψ(1)

i 〉 + E (3)i |i〉





PT Equations

If now we multiply each member of the set on the left by 〈i |, weobtain the following equations:

E (0)i = 〈i |H0|i〉

E (1)i = 〈i |V |i〉

E (2)i = 〈i |V |Ψ(1)

i 〉E (3)

i = 〈i |V |Ψ(2)i 〉

. . .

E (n)i = 〈i |V |Ψ(n−1)

i 〉. . .





PT Equations

We can rearrange the first order equations to obtain:

(E (0)i − H0)|Ψ(1)

i 〉 = (V − 〈i |V |i〉)|i〉

|Ψ(1)i 〉 is expanded in the basis of the eigenvectors of H0

|Ψ(1)i 〉 =

∑

j

cj |j〉





PT Equations

First order correction to the wave function

|Ψ(1)i 〉 =

∑

j

′〈j |Ψ(1)i 〉|j〉

=∑

j

′ 〈j |V |i〉|j〉(E (0)

i − E0j ).

First order correction to the energy

E (1)i = 〈i |V |i〉





PT Equations

Second-order correction to the energy

E (2)i =

∑

j

′ 〈i |V |j〉〈j |V |i〉(E (0)

i − E (0)j )

Second-order correction to the wave function

|Ψ(2)i 〉 =

∑

j

′∑

k

′ 〈j |V |k〉〈k |V |i〉(E (0)

i − E0j )(E (0)

i − E0k )

|j〉

− E (1)i

∑

j

′ 〈j |V |i〉(E (0)

i − E (0)j )2

|j〉





PT Equations

Third-order correction to the energy

E (3)i =

∑

j

′∑

k

′ 〈i |V |j〉〈j |V |k〉〈k |V |i〉(E (0)

i − E (0)j )(E (0)

i − E (0)k )

− E (1)i

∑

j

′ |〈j |V |i〉|2

(E (0)i − E (0)

j )2.

or:

E (3)0 = A(3)

0 + B(3)0





The fluctuation potential

The electrostatic Hamiltonian is partitioned as follows:

H = H0 + V

with

H0 =∑N

i=1(−12∇2

i + vHF (i)) (Fockian)

V =∑

i<j1rij−

∑Ni=1 vHF (i) (fluctuation potential).





The unperturbed Hamitonian

Any Slater determinant built from HF spin orbitals is aneigenstate of H0:

H0|n〉 =N

∑

i=1

ǫi |n〉

with:

|Ψ(0)n 〉 ≡ |n〉 = |χiχj . . . χk 〉

We can construct a perturbative expansion of the correlationenergy.





MPn Equations

By using the Slater-Condon rules one obtains:

MP1 Equations

E (1)0 = 〈Ψ(0)

0 |V |Ψ(0)0 〉

= 〈Ψ(0)0 |

∑

i<j

1rij

−N

∑

i=1

vHF (i)|Ψ(0)0 〉

= −12

∑

ab

〈ab||ab〉





MPn Equations

NOTE

E(MP1) = E (0)0 + E (1)

0 =

N∑

a=1

ǫa − 12

∑

ab

〈ab||ab〉

= EHF

A perturbative expansion of the correlation energy must beginwith the second order correction.





MPn Equations

MP2 Equations

E (2)0 = 〈0|V |Ψ(1)

0 〉

=∑

n

′ 〈0|V |n〉〈n|V |0〉(E (0)

0 − E (0)n )

.

|n〉 denotes an excited determinant.

The sum spans all the excited states.

Only doubly excited Slater determinants can contribute(Brillouin theorem).





MPn Equations

MP2 Equations

E (2)0 =

14

∑

abrs

〈ab||rs〉〈rs||ab〉ǫa + ǫb − ǫr − ǫs

=14

∑

ab

∑

rs

|〈ab||rs〉|2ǫa + ǫb − ǫr − ǫs

.





MPn Equations

MP3 Equations

E (3)0 =

∑

m 6=0

∑

n 6=0

〈0|V |m〉〈m|V |n〉〈n|V |0〉(E (0)

0 − E (0)m )(E (0)

0 − E (0)n )

− E (1)0

∑

n 6=0

|〈n|V |0〉|2

(E (0)0 − E (0)

n )2.





Size-consistency order by order

Consider the special case of N non interacting H2 molecules ina minimal basis set:

Hartree-Fock solution

|HF 〉 = |111112121313 . . . 1N 1N〉

1 and 2 denotes the HOMO (bonding orbital σg)and LUMO(anti-bonding orbital σ∗u) orbitals.

Subscripts refer to a particular molecule, from 1 to N.

Overbars denote spatial orbitals occupied by electrons withβ spin.





Size-consistency of Hartree-Fock

Size consistency of E (0)0

E (0)0 = 〈HF |H0|HF 〉 = 〈HF |

2N∑

k=1

f (k)|HF 〉

= 2N

∑

i=1

〈1i |f |1i〉

= 2Nǫ1






Size consistency of E (1)0

E (1)0 = 〈HF |V |HF 〉 = 〈HF |

∑

k<l

1rkl

|HF 〉 − 〈HF |2N∑

k=1

vHF (k)|HF 〉






NOTE

We need to calculate integrals of the type:

〈ab|ab〉 =

∫∫

φa(r1)φb(r2)1

r12φa(r1)φb(r2)dr1dr2

The only bi-electronic integrals which are different fromzero must involve spatial orbitals of the same molecule.






Size consistency of E (1)0 ; Part I

〈HF |∑

k<l

1rkl

|HF 〉 =12

N∑

i=1

{〈1i1i ||1i1i〉 + 〈1i 1i ||1i 1i〉

+ 〈1i1i ||1i1i〉 + 〈1i 1i ||1i 1i〉}

=12

N∑

i=1

{〈1i 1i |1i 1i〉 + 〈1i1i |1i1i〉}

=12

N∑

i=1

{2〈1i1i |1i1i〉}

= NJ11.






Size consistency of E (1)0 ; Part II

〈HF |2N∑

k=1

vHF (k)|HF 〉 =N

∑

i=1

{〈1|vHF |1〉 + 〈1|vHF |1〉}

=N

∑

i=1

{〈1|J1 + J1 − K1 − K1|1〉

+ 〈1|J1 + J1 − K1 − K1|1〉}= 2NJ11






I+II

EHF = E (0)0 + E (1)

0 = 2Nǫ1 − NJ11

= N(2ǫ1 − J11)

The HF energy is indeed size-consistent.

HF and DFT are size-consistent EE methods. They retaintheir accuracy with increasing system size.





Size-consistency of MP2

NOTE

〈1i 1i |V |2i 2i〉 = 〈1i 1i |∑

k<l

1rkl

|2i 2i〉 − 〈1i 1i |2N∑

k=1

vHF (k)|2i 2i〉

= 〈1i 1i |∑

k<l

1rkl

|2i 2i〉

= 〈1i 1i ||2i 2i〉= 〈1i 1i |2i 2i〉 − 〈1i 1i |2i2i〉= 〈1i 1i |2i 2i〉= 〈12|12〉= K12






MP2 Energy

E (2)0 =

N∑

i=1

K 212

2(ǫ2 − ǫ1)

=NK 2

12

2(ǫ2 − ǫ1)

The MP2 energy is size-consistent as well.






MP3 Energy

E (3)0 = A(3)

0 + B(3)0

Let consider the two terms separately.






B(3)0 contribution

B(3)0 = −E (1)

0

∑

n 6=0

|〈n|V |0〉|2

(E (0)0 − E (0)

n )2

= NJ11

N∑

i=1

K 212

4(ǫ2 − ǫ1)2

=N2J11K 2

12

4(ǫ2 − ǫ1)2

The B(3)0 contribution is NOT size consistent!






A(3)0 contribution

A(3)0 =

∑

m 6=0

∑

n 6=0

〈0|V |n〉〈n|V |m〉〈m|V |0〉(E (0)

0 − E (0)n )(E (0)

0 − E (0)m )

=

N∑

i=1

K 212

4(ǫ2 − ǫ1)2 {−NJ11 + J11 + J22 − 4J12 + 2K12}

= − N2K 212J11

4(ǫ2 − ǫ1)2

+NK 2

12

4(ǫ2 − ǫ1)2 (J11 + J22 − 4J12 + 2K12).






MP3 Energy

E (3)0 = A(3)

0 + B(3)0

=NK 2

12

4(ǫ2 − ǫ1)2 (J11 + J22 − 4J12 + 2K12)

The MP3 energy is size-consistent as well.

For higher orders the situation is similar. The MPnexpansion is size-extensive order by order.











The exponential ansatz

Nowadays CC is the method of choice when calculatingground-state molecular properties in the vicinity of themolecular equilibrium geometry.

It is based on the exponential ansatz for the N-electronwave function:

|Ψ〉 = eT |HF 〉|HF 〉 is the HF solution.

T is a multi-electron operator, called cluster operator.




The Cluster operator

It can be written as a sum of strings of creation-annihilationoperators:

The expansion of T contains a finite number of terms.

T = T1 + T2 + T3 + . . .+ TN




The Cluster operator

Cluster operator in second quantization

T1 =∑

ar

t raa†

r aa

T2 =∑

a<br<s

t rsaba†

sa†r abaa

T3 =∑

a<b<cr<s<t

t rstabc a†

t a†sa†

r ac abaa

t ra, t

rsab, t

rstabc , . . .: cluster amplitudes. The parameters of the

CC wave function.Daniele Toffoli Hartree-Fock Method and Beyond



Exponential vs linear ansatz

CI ansatz

|ΨCI〉 = (1 + T )|HF 〉

CC ansatz

|Ψ〉 = eT |HF 〉

The exponential ansatz is at the basis of the CCsize-extensivity

For a non-truncated T operator the Full-CI wave function isrecovered.




The CC equations

Insert the exponential ansatz in the Schrödinger Equation:

HeT |HF 〉 = EeT |HF 〉.Multiply in turn both sides by the HF reference state, 〈HF |,and a determinant in the excitation manifold, Ψrst...

abc...

〈HF |HeT |HF 〉 = E

〈Ψrst...abc...|HeT |HF 〉 = E〈Ψrst...

abc...|eT |HF 〉

The resulting non-linear system must be (iteratively) solvedfor the cluster amplitudes.




CCD wave function

Cluster operator

T ∼ T2 =∑

a<br<s

t rsaba†

sa†r abaa

Wave function

|ΨCCD〉 = (1 + T2 +12

T 22 +

13

T 32 + . . .)|HF 〉

The CCD wave function contains also quadruple, sextuple. . . excitations.The former enter as products of amplitudes correspondingto pair-excitations (disconnected excitations).




CC Theory is NOT variational

The CI coefficients are variationally optimized.

The CC amplitudes are obtained by projection of theSchrödinger equation on the excitation manifold.

The accuracy is high enough for the loss of variationalproperty to be of no practical concern.




Drawbacks and new developments in CC Theory

It is assumed that the system can be (reasonably) welldescribed by a single determinant.

It is not possible to explore large portions of the PES.

A multireference generalization is being developed.

New developments also in vibrational structure theory,VCC.




Overview of GAMESS-USGaussian basis sets: an overview









Gamess QCP

Developed in the group of Mark S. Gordon at AmesLaboratory/Iowa State University:http://www.msg.chem.iastate.edu

Free for academic purposes.

Large selection of wave function parameterizations.

Explicit vibrational wave function methods.





Gamess QCP

Capabilities

RHF, UHF, MCSCF.

DFT, MPn, CI, CC, EOM-CC.

VSCF and post-VSCF methods.

Calculation of static and dynamic molecular properties(pol. hyper-pol.).

Scalar and spin-orbit relativistic effects.

Solvent effects.

Good selection of plug-ins for different types ofcalculations.





Choice of the one-electron basis

Requirements

The basis set should be complete.

The basis elements should approximate at best the atomicorbitals.

The integral evaluation must be fast.






0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 7

func

.

radius (a.u.)

Gaussian typeSlater type






(nodeless) Gaussians vs Slater type functions

GTOs display a wrong behaviour at the nucleus (cuspcondition).

GTOs decay too fast at large distances.

STOs: numerical integral evaluation, feasible but costly(ADF).

GTOs: the integration is analytical.

GTOs are usually preferred over STOs.





Classification of basis sets

split-valence basis set

The core electrons do not participate in the chemical bonding.The main focus is in the description of the valence orbitals(anisotropy in the electron distribution).

STO-nG: n Gaussian are fitted to a Slater type orbital:Minimal basis.

DZV: two basis functions for each valence orbital.

TZV: three basis functions for each valence orbital.

nZV: n basis functions for each valence orbital.






Polarization functions

Polarization functions: functions with higher angularmomentum, l + 1, l + 2, . . .. Used to describe thedirectionality of the chemical bond.

essential in correlated calculations (angular correlation:two electrons are on the opposite side of the nucleus)

Denoted with a P in their name: DZP, TZP etc.






Diffuse functions

Used for describing situations of diffuse electrondistributions (e.g. attachment processes).

Essential for the calculation of excited states.

Essential for properties depending on the "tail" of the wavefunction.





Contracted basis sets, CGTOs

Basis functions are expressed as (fixed) linear combinations ofprimitive gaussian functions (PGTOs). The number ofparameters to be determined variationally is lower.

Convenient for core electrons.

The integral evaluation over molecular orbitals is moreefficient.

Types of contractions

general: each primitive enters in all of the CGTOs. Usedfor correlated calculations.

segmented: each primitive is used in only one CGTO.Contractions are optimized in molecular HF calculations.





Common basis sets

Pople style basis sets

Optimized in atomic Hartree-Fock calculations. Mainly used forSCF calculations (DFT and HF).

3-21G: split-valence basis. 3 PGTOs for the core orbitals.2 PGTOs for the inner part of the valence, 1 PGTO for theouter part. (Es: for C (6s3p)→[3s2p])

6-311G: Triple split-valence basis.

polarization and diffuse (+) functions can be added.





Common basis sets

Correlation-consistent basis sets

Correlating (virtual) atomic orbitals are represented bybasis functions optimized at the CISD level.

Functions that contribute to similar amounts of correlationenergy are included at the same stage.

General contractions.

Standard nomenclature: (aug-)cc-pVXZ for valencecorrelation (X=D,T,Q,5 . . .).





One- and N-electron requirements





References

Attila Szabo, Neil Ostlund: "Modern Quantum Chemistry:Introduction to Advanced Electronic Structure Theory",Dover

Frank Jensen, "Introduction to Computational Chemistry".

Poul Jorgensen, Jeppe Olsen, Trygve Helgaker, "MolecularElectronic-Structure Theory".


Documents

Hartree-Fock Method and Beyond - İYTEnanodft09.iyte.edu.tr/pdf/DT-talk.pdf · 2009-08-28 · Conﬁguration InteractionTheory Möller-Plesset PerturbationTheory Overview of Coupled-Cluster