introduction to electronic structure theory

Introduction to Electronic StructureTheory

Dage Sundholm

Department of ChemistryUniversity of Helsinki

Finland

CSC Spring School, 20.3.2013

What is Electronic Structure Theory

• Electronic structure theory (EST) describes the motion ofelectrons in atoms and molecules.

• The motion of electrons and nuclei of molecules can beseparated into

1. Electronic motion2. Vibrational motion of the nuclei3. Rotational motion of molecules

• In most electronic structure calculations, one is mainlyinterested in (1)

• In this brief introduction, (1) will be discussed, (2) ismentioned, and (3) is omitted.

Born-Oppenheimer Approximation

• Nuclei are much heavier than electrons

• The lightest nucleus i.e., the proton is 1836 times heavierthan the electron.

• The heavy nuclei move much slower than the fast electrons

• The electronic motion can always adjust to the nuclearpositions

• The electronic Schrodinger equation can be solved forgiven fixed positions of the nuclei

• The decoupling of electronic and nuclear motions is calledBorn-Oppenheimer approximation (BOA)

Born-Oppenheimer Approximation

• The decoupling of electronic and nuclear motions is calledBorn-Oppenheimer approximation (BOA)

• BOA is the birth of molecular structure

• Calculating the electronic energy for a number of nuclearpositions yields the potential energy surface of themolecule

• At the non-relativistic Schrodinger level, the Hamiltonian is

H = − ~2

2m

∑i

∇2i −

~2

2MA

∑A

∇2A −

∑i,A

ZAe2

4πε0riA

+∑A>B

ZAZBe2

4πε0RAB+∑i>j

e2

4πε0rij

Hamiltonian

H = − ~2

2me

∑i

∇2i −

~2

2MA

∑A

∇2A −

∑i,A

ZAe2

4πε0riA

+∑A>B

ZAZBe2

4πε0RAB+∑i>j

e2

4πε0rij

where• me is the electron mass,• e is the charge of the electron,• MA is the mass of nucleus A,• ZA is charge of nucleus A,• riA is the distance between electron i and nucleus A,• RAB is internuclear distances,• rij is interelectronic distances.

Hamiltonian• In EST calculations, atomic units (a.u.) are used.

• The electron mass me = 1• The charge of the electron e = 1• Dirac’s constant ~ = 1• Coulomb’s constant 1

4πε0= 1

H = −∑

i

12∇2

i −∑

A

12MA

∇2A −

∑i,A

ZA

riA+∑A>B

ZAZB

RAB+∑i>j

1rij

• The 1st term is the kinetic energy of the electrons• The 2nd term is the kinetic energy of the nuclei• The 3rd term is the electron-nucleus Coulomb attraction• The 4th term is the nucleus-nucleus Coulomb repulsion• The 5th term is the electron-electron Coulomb repulsion

Hamiltonian

H = −∑

i

12∇2

i −∑

A

12MA

∇2A −

∑i,A

ZA

riA+∑A>B

ZAZB

RAB+∑i>j

1rij

The Schrodinger equation reads

HΨ(r ,R) = EΨ(r ,R)

The decoupling of electronic and nuclear motions (BOA) yields

Ψ(r ,R) = Ψ(r ; R)Φ(R)

• Ψ(r ,R) is a function of electron and nuclear coordinates.• Ψ(r ; R) is a function of electron coordinates for given

parameters R.• Φ(R) is a function of the nuclear coordinates.

Electronic and Nuclear Schrodinger Equations

The electronic Schrodinger equation in BOA−∑i

12∇2

i −∑i,A

ZA

riA+∑A>B

ZAZB

RAB+∑i>j

1rij

Ψ(r ; R)

= Ee(R)Ψ(r ; R)

The nuclear Schrodinger equation in BOA(−∑

A

12MA

∇2A + Ee(R)

)Φ(R) = EnΦ(R)

Computational Procedure

• The electronic Schrodinger equation is solved for given R

• The nuclear positions are adjusted to obtain the lowestenergy.

• The molecular structure with the lowest energy is calledequilibrium structure or equilibrium geometry.

• The equilibrium structure is THE molecular structure.

• Check whether it is a minimum or a saddle point!

• For a minimum, the second derivative of the energy withrespect to nuclear displacements should have a positivecurvature in all dimensions.

• A first estimate of the nuclear motion (vibrationalfrequencies) of the molecule is the by-product.

Example of potential energy surfaces

Wave packages on potential energy surfaces

Chemical reactions and other eventsinvolving nuclear motion can be modeledas wave packets moving on the potentialenergy surface.

Computer Exercise

In the Computer exercise this afternoon you optimize themolecular structure of some small molecules.

• Optimize the molecular for your favourite small molecule

• Investigate how the molecular structure depends on thelevel

• Check whether you have found a minimum or a transitionstate

• Search for the transition state of a chemical reaction

• Simulate the vibrational spectrum

Mathematics: Dirac Notation

Electronic structure theory needs appropriate mathematicaltools.A wave function ψ(r) can be represented as a vector

|Ψ〉 =n∑

i=1

ai |Φi〉

where |Ψ〉 is a state vector, ai are expansion coefficients, and|Φi〉 are fixed basis vectors of the Hilbert space.

|Ψ〉 =

a1a2...

an

〈Ψ| =(a∗1a∗2 · · · a∗n

)

Slater determinants

The Slater determinant is a useful but clumsy notation

Ψ =1

N!

∣∣∣∣∣∣∣∣∣χ1(x1) χ2(x1) · · · χN(x1)χ1(x2) χ2(x2) · · · χN(x2)

......

. . ....

χ1(xN) χ2(xN) · · · χN(xN)

∣∣∣∣∣∣∣∣∣The equivalent Dirac notation is simpler and more practical

Ψ = |χ1(x1)χ2(x2) · · ·χN(xN)〉

The notation using occupation number (ni ) vectors is

Ψ = |n1,n2, · · · nN〉

where ni is 0 or 1. Appropriate mathematics is developed tosimplify expressions involving occupation number vectors.

Occupation number vectors

• The abstract occupation number vectors can be used torepresent Slater determinants.

• The main advantage is that an efficient algebra can bedeveloped for them.

|Ψ〉 = |k1, k2, · · · , kN〉 ki = 1 for occupied orbital andki = 0 for unoccupied orbital

Inner product: 〈k |m〉 = ΠNi=1δki mi

|c〉 =∑

k ck |k〉, 〈c|d〉 =∑

k c∗k dk , |k〉〈k | = 1

Second Quantization

The whole machinery of the second quantization is obtain fromthe algebra of the occupation number vectors.

Creation operator:a†P |k1, k2, · · · ,0P , · · · , kN〉 = δkP ,0Γk

P |k1, k2, · · · ,1P , · · · , kN〉

with ΓkP = ΠP−1

Q=1(−1)kQ

Annihilation operator:aP |k1, k2, · · · ,1P , · · · , kN〉 = δkP ,1Γk

P |k1, k2, · · · ,0P , · · · , kN〉

Anticommutation relations:[a†P ,a

†Q]+ = 0 [aP ,aQ]+ = 0 [a†P ,aQ]+ = δPQ

Operators in Second Quantization

Hamiltonian and Energy (E = 〈H〉):

H =∑pq

hpqa†paq +∑pqrs

gpqrsa†pa†r asaq

One-electron interaction (e.g., dipole moment 〈µ〉 withµ = {x , y , z}):

µ =∑pq

µpqa†paq

Two-electron interaction (e.g., Coulomb and exchange energy〈 1

r12〉):

1r12

=∑pqrs

(1

r12)pqrsa†pa†r asaq

The N-body Wave Function

The most simple ansatz for the N-body wave function is theHartree Product

Ψ(x1, x2, · · · , xN) = χ1(x1)χ2(x2) · · ·χN(xN)

where xi are 4N coordinates, i.e., the three Cartesiancoordinats plus the spin.

χi(xi) are spin orbitals. χi(xi) = φi(ri)|α〉 or χi(xi) = φi(ri)|β〉

|α〉 and |β〉 denote spin up and spin down, respectively.

The Hartree product does not fulfil the antisymmetry conditionof the electrons (fermions).

Accurate N-body Wave Functions

John C. Slater introduced determinants as N-bodywave function.

The Slater determinants fulfil the antisymmetrycondition of fermions. How can the wave function beimproved?

The use of large one-particle basis sets makes it possible toapproach the basis-set limit.

By writing the wave function as a linear combination of manySlater determinants improves the quality of the wave function.

By considering all possible Slater determinants in a givenone-particle corresponds to the solution of the Schrodingerequation in that basis.

The Hartree-Fock ModelThe Hartree-Fock or the mean-field model is obtained byassuming that the wave function consists of one single Slaterdeterminant.

In the Hartree-Fock model, the electrons are assumed to movein the average field of all other electrons and nuclei.

The electronic motion is uncorrelated.

The Hartree-Fock Model

The equations of the Hartree-Fock model is obtained byexpressing the energy as expectation value of the Hamiltonianfor a one-Slater-determinant wave function

EHF = 〈ΨHF |H|ΨHF 〉

Differentiating the energy expression with respect to linearparameters of the one-particle functions (orbitals) φi

Ensuring that the orbitals are orthonormal 〈φi |φj〉 = δij

The Hartree-Fock Lagrangian reads

L(φi) = EHF (φi)−∑

ij

εij(〈φi |φj〉 − δij

)

The Fock Equation

The Hartree-Fock energy for closed-shell systems

EHF = 2∑

i

hii +∑

ij

(2giijj − gijji

)hij =

∫φ∗i (1)h1φj(1)dr1

gijkl =∫ ∫

φ∗i (1)φ∗k (2)r−112 φj(1)φl(2)dr1dr2

The molecular orbitals (MO), φi , are expanded in a set ofnon-orthonormal atomic orbitals (AO) χµ

φi =∑µ

χµCµp

Inserting it into the energy expression and differentiation withrespect to Cµp yields the Fock equation

The Fock Equation

The Fock equation for non-orthonormal AOs reads∑ν

F AOµν Cνi = εi

∑ν

SµνCνi

where the Fock matrix has been introduced

F AOµν = hµν +

∑i

(2gµνii − gµiiν

)The AO Fock matrix can be calculated in the AO basis

F AOµν = hµν +

∑στ

DAOστ

(gµνστ −

12

gµστν

)

The Hartree-Fock Iterative Procedure

1. Guess C2. Calculate DAO = CDMOCT

3. Calculate FAO

4. Diagonalize FAOC = SCε5. Determine an error vector e.g., e = FDS− SDF6. Check the convergence7. Introduce damping using e.g., DIIS8. Make a new guess for C and go to 29. End when converged

Hellmann-Feynman TheoremAssume that the Hartree-Fock wave function (Ψ) is perturbedby an external perturbation V :

The perturbed energy can be written as:

E(λ) = 〈Ψ + λ∆Ψ|H + λV |Ψ + λ∆Ψ〉

The first-order correction to the energy is obtained bydifferentiating E(λ) with respect to λ

dE(λ)

dλ= 2Re〈∆Ψ|H + λV |Ψ + λ∆Ψ〉+ 〈Ψ + λ∆Ψ|V |Ψ + λ∆Ψ〉

dE(λ)dλ

∣∣∣λ=0

= 2Re〈∆Ψ|H|Ψ〉︸︷︷︸ +〈Ψ|V |Ψ〉

= 0

Open-Shell Systems

For molecules with unpaired electrons or for general open-shellsystems, different orbitals are used for describing the motion ofthe α (spin-up) and the β (spin-down) electrons.

The unrestricted Hartree-Fock (UHF) model is ageneralization of the restricted Hartree-Fock (RHF) model forclosed-shell molecules.

The matrix sizes of the UHF are twice the size of the RHF ones.

UHF is not unproblematic as it might introduce spincontamination effects, convergence problems, negative gapbetween the highest occupied molecular orbital and the lowestunoccupied molecular orbital (HOMO-LUMO gap).

Correlation effects are often larger for open-shell systems thanfor the closed-shell ones.

Density Functional TheoryHartree-Fock does not consider any electron correlation effectsby definition

ECorrelation = EExact non−relativistic − EHartree−Fock

Density functional theory (DFT) is the most simple model thatconsider electron correlation effects.

DFT is structurally quite similar to Hartree-Fock (HF)

The effctive HF potential

veff = vnuc + vJ + K

The effctive DFT potential

veff = vnuc + vJ + vxc

Density Functional Theory

DFT is formally exact. However vxc is not known.

The art of DFT is to find an accurate approximation to vxc

DFT is based on the electron density, ρ(r), which is theprobability to find an electron in dr .

The electron density is related to the wave function

ρ(r1) = n∫

dr2

∫dr3 · · ·

∫drn|Ψ(r1, r2, . . . , rn)|2

The density uniquely determines the Hamiltonian, because•∫ρ(r)dr = n

• The cusps at the nuclei yield the nuclear positions• The slope of the nuclear cusps yields the nuclear charge

Density functional theory

Simple DFT approach:

• Take the exact ρ(r)

• Use ρ(r) to obtain H

• Construct and solve the Schrodinger equation

• The exact ground-state energy is obtained

The proceduce can be formulated mathematically as

E0 = E [ρ(r)]

which is an energy functional that is variational E [ρ] ≥ E [ρ]

Hohenberg-KohnThe energy can be divided into something that explicitly dependon the electron density and the universal part (the rest)

E [ρ] =

∫drρ(r)(vnuc + velectrostatic) + FHK

The energy functional

E [ρ] = T [ρ] + J[ρ] + E ′xc[ρ] +

∫drρ(r)vnuc

The kinetic contribution is:

T [ρ] = Ts + Tc[ρ] with Ts = −12

∑i∈occ

〈ψi |∇2|ψi〉

The Kohn-Sham energy is:

EKS = Ts + J[ρ] + Exc[ρ] +

∫drρ(r)vnuc

Kohn-Sham Equations

InEKS = Ts + J[ρ] + Exc[ρ] +

∫drρ(r)vnuc

everything is known except Exc[ρ], which is small compared tothe rest.

Different DFT functionals have their own approximation to Exc[ρ]

Minimization of EKS with respect to the orbitals yields theKohn-Sham equations

(−1

2∇2

i + vnuc + vJ + vxc

)ψi = εiψi with vxc =

δExc[ρ]

δρ

Density functionals

LDA: vxc ∝ ρ(r)13

GGA: vxc = f (ρ, ~∇ρ)

Hybrid vxc = f (ρ, ~∇ρ) + EHF,x

Meta GGA vxc = f (ρ, ~∇ρ, τ) where τ=∑

i12 |∇ψi(r)|2

EB3LYPxc [ρ] = (1−a)ES

x [ρ]+aEHFx +bEB88

x [ρ]+cELYPc +(1−c)EVWN

c [ρ]

a = 0.20 b = 0.72 c = 0.81

Stairway to heaven

• The exact functional• · · ·• Stairway to heaven• Orbital dependent functionals• Exact exchange functionals• Parametrized functionals• Meta GGA• Hybrid functionals• GGA functionals molecules• LDA functionals formal• Hartree-Fock Slater, the first• Thomas Fermi

http://www.youtube.com/watch?v=w9TGj2jrJk8

Configuration Interaction

Configuration interaction (CI) is the most simple ab initiomethod to consider electron correlation.

The wave function is expanded as a linear combination ofSlater determinants (configurations).

|ΨCI〉 =∑

I

CI |I〉

The expansion coefficients are determined by minimizing the CIenergy, with the constraint that the states are orthonormal.

ECI = 〈ΨCI |H|ΨCI〉 with 〈ΨCI |ΨCI〉 = 1

LCI = ECI − ε (〈ΨCI |ΨCI〉 − 1)


|ΨS〉 All configurations that are obtained by single excitationsfrom the |ΨHF 〉 reference.

|ΨD〉 All configurations that are obtained by double excitationsfrom the |ΨHF 〉 reference.

|ΨT 〉 All configurations that are obtained by triple excitationsfrom the |ΨHF 〉 reference.

|ΨN〉 All configurations that are obtained by all N excitationsfrom the |ΨHF 〉 reference.

|ΨCI〉 including all terms up to |ΨN〉 is the full configurationinteraction (FCI) or exact diagonalization wave function i.e.,it is the solution of the Schrodinger equation in the given basisset.

Excited Slater Determinants

The Slater determinants of the CI model are obtained by single(S), double (D), triple (T), . . . alternations of the occupation ofthe Fock space.

Matrix Elements of the CI Hamiltonian Matrix

Brillouin’s theorem says that single excited determinants donot directly couple with the Hartree-Fock reference.

The Hamiltonian matrix is banded because only Slaterdeterminants which are conncted via singles and doubleexcitations yield non-vanishing matrix elements.


CI is a straightforward method to consider electron correlationeffects.

Truncated CI wave functions are not size-extensive i.e., theenergy of two non-interacting fragments is not twice the energyof one of the fragments.

The only exception is the FCI wave function.

FCI calculations are computationally expensive. The largestFCI calculation as far as I know had 2500 billion (2.5 1012)Slater determinants.

FCI calculations are indispensable as benchmark, whereastruncated CI has lost its importance as production tool due tothe size problem.

Slater-Condon rules

Most matrix element of the CI Hamiltonian vanish,

because configurations whose occupation number vectorsdiffer in more than two position pairs do not directly couple.

The reason is that the Hamiltonian contains only one- andtwo-body interaction terms

The non-vanishing matrix elements of the CI Hamiltonian canbe identified using second quantization.

The obtained expressions are called Slater-Condon rules

One-electron Slater-Condon rules|k〉 is a occupation number vector containing the occupation ofthe orbitals for a given Slater determinant.

kP = 1 for occupied orbitals

kP = 0 for unoccupied orbitals

fPQ are the matrix elements of the one-electron operator f

Identical bra and ket : 〈k|f |k〉 =∑

P

kP fPP

bra and ket differ in one pair : 〈k|f |l〉 = ΓkI Γl

J fIJ

bra and ket differ in two or more pairs : 〈k|f |l〉 = 0

withΓk

P = ΠP−1Q=1 (−1)kQ

Two-electron Slater-Condon rules

gPQRS are the matrix elements of the two-electron operator g

Identical bra and ket 〈k|g|k〉 = 12∑

PR kPkR (gPPRR − gPRRP)

differ in one pair 〈k|g|l〉 = ΓkI Γl

J∑

R kR (gIJRR − gIRRJ)

differ in two pairs 〈k|g|l〉 = ΓkI Γk

JΓlK Γl

L (gIKJL − gILJK )

differ in more than two pairs 〈k|g|l〉 = 0

withΓk

P = ΠP−1Q=1 (−1)kQ

CAS and RAS Models

The Complete Active Space and Restricted Active Space

Multireference Configuration InteractionMultireference configuration interaction (MRCI) belong to thecategory of ”As Full CI as possible” models.

RAS can be considered as a MRCI with orbital optimization.

The reference can also consist of a few selected importantconfigurations with single and double replacements to thecomplementary space.

Multireference Perturbation Theory

The CASPT2 method is a multireference (MR) perturbationtheory method with a CASSCF wave function as the reference.

MRPT should be formulated such that when |Φ0〉 → |HF 〉MRPT→ Møller-Plesset PT. This is not trivial because the MRFock operator is not unique.

In CASPT2, the spin-averaged first-order density matrix of theCAS function is used to define the Fock operator.

The open-shell and closed-shell configurations are then treatedunequally, which is fixed with semi-empirical level shifts

The NEVPT2 model avoids the problem by adding two-electronterms to H0, but it yields orbital labelling dependent results.


In the single reference case, H0 is the sum of one-electron Fockoperators, which is diagonal in canonical orbital basis.

The CAS reference is not an eigenfunction of the n-electronFock operator, that is, FCAS is not a proper H0. This is fixed byusing projection operators

H0 = PFP + QFQ

where P projects to the CAS reference.

The Fock operator of the MR case is not diagonal. Unitarytransformations in the individual spaces remove off-diagonalelements in each orbital block, but not between the inactive,active and secondary orbital blocks.


The first-order wave function is obtained as in the singlereference case as

(H0 − E0) Ψ1 = (H0 − H) Ψ0

which has to be solved iteratively.

The first order wave function is expanded in the first-orderinteraction space

Ψ1 =∑pqrs

T qspr EpqErsΨ0

where Epq is the spin-averaged excitation operator that movesan electron from the q orbital of the active or inactive spaces toan empty orbital p of the secondary space.


The configurations of Ψ1 comprise those who havenon-vanishing Hamiltonian matrix elements with Ψ0

This is called internal contraction that reduces significantlythe number of Tqs

pr amplitudes

Internal contraction leads to complicated expressions involvinghigher-order density matrices and non-orthogonalconfigurations including linear dependencies.

CASPT2 is able to handle bond breaking, reactions, transitionmetals, and excited states.

CASPT2 yields pure spin states.

CASPT2 Problems

The internal contraction scheme leads to intruder states as forMR coupled-cluster, MRCI and other similar approaches.

CASPT2 is not a black box method. It requires experience anda deep understanding of the electronic structure of theinvestigated system to obtain reliable results.

All orbitals that should be in the active space cannot beincluded due to the huge computational costs.

The borderline between active and secondary orbitals is notobvious.

CASPT2 with the minimal reference based on chemical intuitionoften yield unreliable results.

Coupled-Cluster ModelsThe coupled-cluster ansatz for the wave function is

ΨCC = exp(T )Φ0 where Φ0 = ΦHF in most cases

exp(T ) = 1 + T +T 2

2!+

T 3

3!+ · · · T

n

n!

The cluster operator T consists of single T1, double T2, triple T3replacement operators.

T = T1 + T2 + T3 + · · ·+ Tn

The maximum is n = number of electrons, which is identical tofull CI

When the series is cut at T2, it is the CCSD model and a cut atT3 yields the CCSDT model, etc.

Coupled-Cluster ModelsThe cluster operators Tk operating on the HF referencegenerates all Slater determinants for single (S), double (D),triple (T), ... replacements.

T1 =occ∑

i

vir∑a

tai a†aai

T2 =14

occ∑ij

vir∑ab

tabij a†aa†baiaj

tai and tab

ij are the singles and doubles amplitudes, respectively

exp(T ) = 1 +(

T1

)︸︷︷︸ +

(T2 +

12

T 21

)︸︷︷︸ +

(T3 + T2T1 +

16

T 31

)︸︷︷︸ · · ·

single double triple

Coupled-Cluster Models

The coupled cluster equations for the amplitudes are obtainedby projecting the coupled-cluster Schrodinger equation againstthe excited Slater determinants

〈µ|exp(−T )H exp(T )|Φ0〉 = 0

〈µ| are the Slater determinant that enter the coupled-clusterstate with connected amplitudes T1, T2, T3, · · ·

〈µ| = 〈Φ0|τ †µ; 〈µ| = 〈Φai |, 〈Φ

abij |, 〈Φ

abcijk |, · · ·

where τµ are second-quantization operators that creates allpossible single, double, triple, ... replacements from theHartree-Fock reference (Φ0)

Coupled-cluster energyThe coupled-cluster energy is obtained by projecting againstthe Hartree-Fock reference

E = 〈Φ0|exp(−T )H exp(T )|Φ0〉

The coupled-cluster energy can be expressed as

ECC = EHF +∑

ia

fiatai +

14

∑iajb

〈ij ||ab〉tabij +

12

∑iajb

〈ij ||ab〉tai tb

j

〈pq||rs〉 = 〈pq|rs〉 − 〈pq|sr〉

〈pq|rs〉 =

∫ ∫φ∗p(1)φr (1)r−1

12 φ∗q(2)φs(2)dr1dr2

fia are elements of the Fock matrix in the MO basis

Approximate Coupled-Cluster SD and Related MethodsCoupled-cluster energy:

Second-order Møllet-Plesset perturbation theory

Coupled-cluster equations:

Similarity-transformed Hamiltonian:

Approximate Coupled-Cluster and Related MethodsJacobi matrix:

Jacobi matrix for CC2:

Jacobi matrix for CIS(D∞):

Jacobi matrix for ADC(2):

Approximate Coupled-Cluster and Related Methods

Time-dependent density functional theory

TDDFT response equations:

Van der Waals InteractionThe van der Waals (vdW) interaction originates from fluctuatingdipoles due to the correlated motion of the electrons.

Hartree-Fock and the commonly used DFT functionals are notable to take vdW interaction into account.

Second-order Møller-Plesset perturbation theory (MP2) is thecheapest ab initio method that include vdW interaction.

MP2 overestimate vdW interaction. The spin-component-scaled(SCS) MP2 yields often similar vdW energies as thecomputationally more expensive CCSD(T) method.

A simple semi-empirical patch to consider vdW interactions isGrimme’s D3 method, which consider vdW interaction atmolecular mechanics (MM) level.

B3LYP+D3 is good and pragmatic level when optimizing themolecular structure of large molecules such as biomolecules.

Basis setsBasis sets can be divided into the

• Gaussian-type (GTO) basis functions (in most quantumchemistry programs)

• Slater-type (STO) basis functions (in ADF), efficient forDFT

• Plane-wave basis sets, the favourite in the physicscommunity (VASP)

• Numerical basis functions (GPAW uses grid andatom-centered basis functions)

• Fully numerical (Octopus)

GTOs are better than STOs in ab initio correlation calculations,because GTOs have the wrong asymptotic form at the nucleiand at infinity. (Almlof)

Classes of Gaussian-Type Basis Sets

• Cartesian basis sets χi(r) = x lymzn exp(−αi r2), (6d,10f)

• Spherical basis sets χi(r) = r lY ml (θ, ϕ) exp(−αi r2), (5d, 7f)

• Even-tempered basis sets• Uncontracted primitive basis sets with constant ratio

between the exponents αi−1/αi = αi/αi+1

• General contracted basis sets• All basis functions comprise all primitive basis functions

• Segmented contracted basis functions• As general contracted, but for small blocks of primitive basis

functions• Completeness-optimized uncontracted basis sets

• Optimize the exponents to obtain as complete basis set aspossible in a given range of exponents

Basis setsBasis sets can be downloaded fromhttps://bse.pnl.gov/bse/portal

A few personal principles concerning the choice of basis set.• Use as large basis set as possible

• Use more than one basis set

• The series of Dunning basis sets is very useful whenaiming at benchmarking.

• The Dunning basis sets are large, which put limitations onthe size of the molecule and the level of theory.

• Diffuse basis functions might be needed when themolecule is not very large or when it is negatively charged

• Special basis sets might be need for some properties.

Basis sets

A few personal principles concerning the choice of basis set.• The MOLCAS ANO basis sets should be used only in

combination with MOLCAS and other programs that cantake the advantage of general contracted basis sets.

• I prefer the Karlsruhe basis sets

• the def2-TZVP basis sets is usually enough for mypurposes, def2-SVP in molecular structure optimization oflarge molecules.

• When diffuse basis functions are needed, the def2-TZVPDis much more cost efficient than Dunning’s aug- basis sets

• The Pople basis sets are not very accurate

• Use Stuttgart pseudopotentials instead of LANL2DZ ECPs

Direct Diagonalization

Large eigenvalue problems cannot be solved by constructingthe matrix and diagonalizing it.

The largest matrix that fit into the memory of a moderncomputer is say 64 GB = 2 · 109 double precsion words.

The size of the largest matrix is then less than 105

The largest CI problem that has been solved consisted of2.5 · 1012 Slater determinants

The CI matrix has to be diagonalized using direct methods

Direct methods means that the linear transformation H C = σ isperformed without construction of H.

Davidson-Olsen diagonalizationThe Hamiltonian matrix can be written as

H = H0 + H1 (1)

The eigenvector and the eigenvalue is analogously written as

C = C(0) + C(1) and E = E (0) + E (1) (2)

The approximate eigenvalue E (0) is obtained as

E (0) =C(0)T HC(0)

C(0)T C(0)(3)

inserting Eq. (1) and Eq. (2) into HC = EC and neglectingquadratic terms yields

(H0 − E (0))C(1) = −(H− E (0))C(0) + E (1)C(0) (4)

Davidson-Olsen diagonalization

The correction vector C(1) is the obtained as

C(1) = −(H0 − E (0))−1[(H− E (0))C(0) − E (1)C(0)] (5)

The energy correction E (1) is obtained by multiplying Eq. (5)from the left by C(0)T

and require orthonormality of the C(0) andC(1) vectors

E (1) =C(0)T

(H0 − E (0))−1(H− E (0))C(0)

C(0)T (H0 − E (0))−1C(0)(6)

which can be inserted in Eq. (5)


The correction vector C(1) obtained using

C(1) = −(H0 − E (0))−1×[(H− E (0))C(0) − C(0)T

(H0 − E (0))−1(H− E (0))C(0)

C(0)T (H0 − E (0))−1C(0)C(0)

](7)

is orthogonal against C(0) by construction regardless of howthe partitioning of H into H0 + H1 is done.

• The algorithm requires that the inverse (H0 − E (0))−1 timesa vector must be calculated.

• In the Davidson diagonalization algorithm H0 is thediagonal of the H matrix and the E (1) correction term isomitted.


The Davidson-Olsen formulation, which is correct to first order,has the following advantages as compared to the slightlysimpler Davidson diagonalization procedure

• The update vector C(1) is orthogonal against C(0)

• A better H0 does not introduce any new formal problemsnot even for the case H0 → H.

• Block diagonal H0 or other more accurate approximationsto H can be used.

• An algorithm with only two vectors (C(0) and C(1)) leads toconvergence in only a few iterations.

• Very large CI problems can therefore be solved

The zeroth-order Hamiltonian

The better H0 the better is the convergence.

H0 =

(Hblock LT

L D

)The matrix times vector operation is the

H0x = y

x =

(x1x2

)y =

(y1y2

)Hblockx1 + LT x2 = y1

Lx1 + Dx2 = y2

The zeroth-order Hamiltonian

x2 = D−1 (y2 − Lx1)

Hblockx1 + LT D−1 (y2 − Lx1) = y1(Hblock − LT D−1L

)x1 = y1 − LT D−1y2

x1 =(

Hblock − LT D−1L)−1 (

y1 − LT D−1y2

)x =

(x1x2

)y =

(y1y2

)

Direct Inversion of the Iterative Space

One of the most important developments in electronic structurecalculations is the Direct Inversion of the Iterative Space (DIIS)method for damping the iterative procedure of the solution ofthe Fock and Kohn-Sham equations.

In the update of the new Fock matrix, the Fock matrices areconsidered

F AOn =

n∑i

ωiF AOi

where F AOi are the Fock matrices of the previous iterations and

the DIIS weights ωi are obtained by solving the DIIS equation.


The DIIS equation is obtained by constructing the DIISLagrangian

L = ω†Bω − λ

(1−

m∑i

ωi

),

with

Bij = 〈ei |ej〉.

constructed from the error vector

ei = F AOi DiS − SDiF AO

i

Di is the density matrix of iteration i and S the overlap matrix


The DIIS equation is a matrix equation of the size of theiteration space or of the m last iterations.

B11 B12 · · · B1m −1B21 B22 · · · B2m −1

......

. . ....

...Bm1 Bm2 · · · Bmm −1−1 −1 · · · −1 0

ω1ω2...ωmλ

=

00...0−1

Solution of the DIIS equation yields ωi which combined withF AO

k , k = 1,m yields the new extrapoleted (damped) F AOi

The End

Now this is not the end.

It is not even the beginning of the end.

But it is, perhaps, the end of the beginning.

Sir Winston Churchill, Speech in November 1942British politician (1874 - 1965)

introduction to electronic structure theory

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