Quantum Mechanics and ElectronicStructure Theory

Debasish Koner

Department of ChemistryUniversity of Basel

October 9, 2019

Goal

I Understanding the basic principles of quantum mechanicsI Understanding the basic principles of electronic structure

theory

Contents

1. Introduction2. The postulates of quantum mechanics3. Translational motion (free particle, particle in 1D/2D box)4. Vibrational motion (harmonic oscillator)5. Rotational motion and Angular momentum (rigid rotor)6. Spin7. Atomic structure (Hydrogen like atoms)8. Atomic units9. Variational method and perturbation method

10. The molecular Hamiltonian11. Born-Oppenheimer approximation12. Molecular orbital theory

Contents

1. Hartree-Fock theory2. Other ab initio methods3. Basis sets4. Molecular orbitals and population analysis

Molecular properties are governed by interaction between electronsand nuclei.

The dynamics of electrons and nuclei are governed by quantummechanics.

Quantum mechanics provides accurate description of microscopicsystems.

Application of quantum mechanics to different chemical problemsand find out an accurate description of molecular properties.

Organic chemistry: Reaction intermediate, transition state, reac-tion mechanism, solvent effects, stereo selectivity, hydrogen bond,hydrophobic interactions

Inorganic chemistry: Metal ligand interactionsPhysical chemistry: Potential energy surface, reaction dynamics,

photochemistryBiological chemistry: reaction mechanism, electrostatic interactionMaterial Chemistry: Band gap, adsorption, designing novel struc-

tures

Classical Mechanics

E = p2

2m + V (q)

F = md2qdt2

F = −dV (q)dq

Translational :Energy

E = p2

2mMomentum

p = mv

Classical MechanicsRotational : Angular Momentum

L = Iω

Moment of inertiaI = mr2

EnergyE = L2

2IVibration : Force F = −kq

Potential energyV = 1

2kq2

Positionq = asin(ωt), ω =

√k/m

Total Energy

E = mω2a2

2

Black-body radiation : Energy can be transferred only in discreteamounts. A black-body emitter is composed of of a set of oscilla-tors which can have the energies 0, hν, 2hν, ..., and no other energy,where h is the Planck’s constant. This is called quantization ofenergy. However the Classical physics allowed a continuous variationin energy

Heat capacities : The Einstein and Debye molar heat capacities.Einstein showed that matter is quantized too and Debye stated thatthe oscillator can have different frequencies.

The photoelectric effects :12mev2

e = hν − Φ

Φ is work function.Atomic spectra : The radiation emitted by atoms consisted of

discrete spectral lines. In classical mechanics, all energies are per-missible. Bohr Model,

En = − µe4

8h2ε20.

1n2 , n = 1, 2, 3...

The duality of matter

Any moving body there is associated with a wave, and the momen-tum of the body and the wavelength are related by the de Broglierelation. For photon,

E = mc2 = hν

mc2 = hcλ, (c = νλ)

p = hλ, (p = mc)

‘Experiments are the only means of knowledge at our disposal.The rest is poetry, imagination. It is time for that imagination tounfold.’

- Planck

Postulates of quantum mechanics

Postulate I: The state of a quantum system is fully described by awavefunction Ψ(q, t), where q = q1, q2, ... are the spatial and spincoordinates of the particles in the system. All the observables canbe determined by Ψ(q, t).

Postulate II: Every observables are represented by Hermitian oper-ators.

position, q → q×momentum, pq → ~

i∂∂q

angular momentum, lx = (ypz − zpy ) = −i~( ∂∂z −

∂∂y )

Hermitian operators :∫φ∗(Ωψ)dτ = [

∫ψ∗(Ωφ)dτ ]∗ =

∫(Ωφ)∗ψdτ

The eigenvalues of hermitian operators are real.Ωψ = ωψ

Eigenfunctions corresponding to different eigenvalues of an hermi-tian operator are orthogonal and can be normalized.∫ψ∗nψmdτ = 0, and

∫ψ∗nψndτ =

∫|ψn|2dτ = 1

Eigenfunctions of a Hermitian operator form a complete set, andan arbitrary wavefunction can be expressed as a linear combinationof the eigenfunctionsΨ =

∑cnψn, with Ωψn = ωnψn

Two operators commute if, [Ω1, Ω2] = Ω1Ω2 − Ω2Ω1 = 0q and pq do not commute. [q, pq]φ = i~φ

Bra and ket notation:bra: 〈φ| = φ∗, ket: |ψ〉 = ψ〈φ|ψ〉 =

∫φ∗ψdτ

〈φ|Ω|ψ〉 =∫φ∗Ωψdτ

Postulates of quantum mechanics

Postulate III: When a system is described by a wavefunction ψ,the mean value of the observable Ω in a series of measurements isequal to the expectation value of the corresponding operator.

〈Ω〉 =∫ψ∗Ωψdτ∫ψ∗ψdτ = 〈ψ|Ω|ψ〉

〈ψ|ψ〉

If the wavefunction is chosen to be normalized to 1,

〈Ω〉 =∫ψ∗Ωψdτ = 〈ψ|Ω|ψ〉

Now, if ψ is an eigen function of Ω, with eigenvalue ω,

〈Ω〉 = ω

Postulates of quantum mechanics

If ψ is not an eigen function of Ω, the wavefunction can be expressedas a linear combination of eigenfunctions of Ωψ =

∑cnψn, with Ωψn = ωnψn

As the eigenfunctions from an orthonormal set, 〈Ω〉 =∑|cn|2ωn

Postulate IV: Born interpretation: The probability that a particlewill be found in the volume element dτ at the point q is proportionalto |Ψ(q)|2dτ (probability density).

The wavefunction itself is a probability amplitude, and has no directphysical meaning. The probability density is real and non- negative,the wavefunction may be complex and negative.

Postulates of quantum mechanics

Postulate V: The Schrodinger equation: The wavefunction evolvesin time according to the equation

i~∂Ψ∂t = HΨ

It can be reduced to a time independent form

Ψ(q, t) = ψ(q)φ(t)

i~ψ(q)∂φ(t)∂t = φ(t)Hψ(q)

divide both side by Ψ

i~φ(t)−1∂φ(t)∂t = ψ(q)−1Hψ(q)

Since both sides depend on different variables, they have to be aconstant value (E )

Time dependent and time independent Schrodingerequations

i~∂φ(t)∂t = Eφ(t), φ(t) ∝ e−iEt/~

Hψ(q) = Eψ(q)

Heisenberg’s uncertainty principle : It is impossible to specify simul-taneously, with arbitrary precision, both the momentum and positionof a particle, as [q, p] = i~.

∆p∆q ≥ ~/2

Momentum p and position q are complementary observables. Timet and energy E is another pair of complementary observables.

∆E∆t ≥ ~/2

Time evolution

Differentiation of an operator 〈Ω〉 with respect to time

d〈Ω〉dt = d

dt 〈ψ|Ω|ψ〉

d〈Ω〉dt = i

~〈[H, Ω]〉, i~∂Ψ

∂t = HΨ

[H, px ] = −~i

dVdx , H = − ~2

2m∂2

∂x2 + V

d〈px 〉dt = i

~〈[H, px ]〉 = −

⟨dVdx

⟩= 〈F 〉

d〈x〉dt = 〈px 〉

m = 〈v〉

Ehrenfest’s theorem: The relation between classical and quantummechanics. classical mechanics deals with average values (expec-tation values) while quantum mechanics deals with the underlyingdetails.

Simple quantum systems

Free particle:

Ψ(x) = e±ikx , k =(2mE

~2

)1/2

E = (~k)2

2m , p = ±k~

Energy is not quantizedParticle in a 1D box : Infinite square well V (x) = 0, 0 ≤ x ≤ L

V (x) =∞, x < 0 and x > L

Particle in a 1D boxAcceptable solution:

Ψn =

√2Lsin

(nπxL

), n = 1, 2, ...

E = n2h2

8mL2

Energy is quantized

Particle in a 2D box

Acceptable solution:

Ψn1,n2(x , y) = 2√L1L2

sin(n1πx

L1

)sin(n2πy

L2

)

En1,n2 = h2

8m

(n2

1L2

1+ n2

2L2

2

), n1 = 1, 2, ...andn2 = 1, 2, ...

Degeneracy: states with same energies. (if L1 = L2 = L)

Simple quantum systemsThe harmonic oscillator: The Schrodinger equation,

− ~2

2m∂2ψ

∂x2 + 12kx2ψ = Eψ

This can be solved by Ladder operator method or polynomialmethod.Energy: Eν = (ν + 1

2)~ω, ν = 0, 1, 2, ... where, ω =√

k/m Thewavefunction:

ψν(x) = NνHν(αx)e−α2x2/2, α =(mk~2

)1/4

Hν(z) are Hermite polynomials

Nν =(

α

2νν!√π

)1/2

The harmonic oscillator

Selection rule: ∆ν = ±1

The Morse potential (anhormonic oscillator)

Simple quantum systems

2D rigid rotor (particle on a ring): The Schrodinger equation,

−~2

2I∂2ψ

∂φ2 = Eψ, I = mr2

Solution:ψml (φ) +

√1/2πeimlφ,ml =

√2IE/~2

Energy:

Eml = m2l ~2

2I ,ml = 0,±1,±2, ...

Angular momentum:lz = ml~

Simple quantum systems3D rigid rotor (particle on a sphere): The Schrodinger equation,

−~2

2I

[1

sin2θ

∂2

∂φ2 + 1sinθ

∂

∂θ

(sinθ ∂

∂θ

)]ψ = Eψ

The solution can be separable

ψ(θ, φ) = Θ(θ)Φ(φ) = Pmll (cosθ)

√1/2πeimlφ

Associated Legendre function : Pmll

Spherical harmonics: Yl ,ml (θ, φ) = Θml (θ)Φl (φ)where l = 0, 1, 2... and ml = l , l − 1, ...− lEnergy:

El ,ml = l(l + 1)~2

2IEl ,ml is independent of the value of ml . As, for a given value of l

there are 2l + 1 values of ml , each energy level is 2l + 1 folddegenerate.

Angular momentum:lx = ypz − zpy , ly = zpx − xpz , lz = xpy − ypx

l2 = l2x + l2

y + l2z

l2|l ,ml〉 = l(l + 1)~2|l ,ml〉, lz |l ,ml〉 = ml~|l ,ml〉

Only certain directions of l are allowed

Spin

Particles (electrons, nucleus) have internal angular momentum iscalled as spinFermion: (half-integer spin) electron, proton, etc.Boson: (integer spin) photon, deuterium, etc.Spin is a purely quantum mechanical phenomenon. Spin is arelativistic effect, has no functional basis.

S2|s,ms〉 = s(s + 1)~2|s,ms〉, Sz |s,ms〉 = ms~|s,ms〉

For electron: s = 1/2, ms = 1/2 (α electron), ms = −1/2 (βelectron)Matrix representation:

|α〉 =(

10

), |β〉 =

(01

),

Atomic structure (Hydrogen like atoms)

Hydrogenic atoms H, He+, Li2+

Motion in a Coulombic field : The Hamiltonian for the two-particleelectron-nucleus system:

H = − ~2

2me∇2

e −~2

2mN∇2

N −Ze2

4πε0r

Converting to center-of-mass and relative coordinate

H = − ~2

2m∇2cm−

~2

2µ∇2− Ze2

4πε0r , m = me+mN , µ = memN/(me+mN)

Removing the center-of-mass, the reduced Hamiltonian can be writ-ten as

H = − ~2

2µ∇2 − Ze2

4πε0r = − ~2

2µ1r∂2

∂r2 r + l2

2µr2 −Ze2

4πε0r

Atomic structure (Hydrogen like atoms)

The separation of the relative coordinates leads to

ψ(r , θ, φ) = R(r)Y (θ, φ)

The Angular equation:

l2Yl ,ml (θ, φ) = l(l + 1)~2Yl ,ml (θ, φ)

The radial equation:[− ~2

2µ1r∂2

∂r2 r + l(l + 1)~2

2µr2 − Ze2

4πε0r

]R(r) = ER(r)

The acceptable solutions are the associated Laguerre functions

Rn,l = ρlLn,l (ρ)e−ρ/2, ρ = (2Z/na)r , with a = 4πε0~2/µe2

for an infinitely heavy nucleus µ = me and a = a0 (bohr radius)

Radial wavefunction

The complete wavefunction

ψnlml = RnlYlml

The Rnl are related to the (real) associated Laguerre functions andthe Ylml are the (in general, complex) spherical harmonics.The wavefunction gives the probability of finding an electron at spec-ified location. Radial distribution function the probability of find-ing the particle at a given radius regardless of the direction.

P(r)dr =∫

surface|ψnlml |

2dτ =∫ π

0

∫ 2π

0R2

nl |Ylml |2r2sinθdrdθdφ

(1)Spherical harmonics are normalized to 1∫ π

0

∫ 2π

0|Ylml |

2sinθdθdφ = 1 (2)

Thus,P(r)dr = R(r)2r2dr (3)

Radial distribution function

Atomic orbitals

Quantum numbersPrinciple quantum numbers (n), n = 1, 2, 3The orbital angular momentum quantum number(l), l = 0, 1, 2, ..., n − 1The magnetic quantum number, (ml ),ml = −l , ..., 0, ..., l

Energy :

En = −Z 2µe4

32π2ε20~21n2 = −hcRH

n2

RH is Rydberg constantThe wavenumber for emission for n2 → n1 transition

ν =(

1n2

1− 1

n22

)RH

Lyman series, (ultraviolet) n1 = 1, n2 = 2, 3, ..Balmer series, (visible) n1 = 2, n2 = 3, 4, ..Paschen series, (infrared) n1 = 3, n2 = 4, 5, ..Brackett series, (far infrared) n1 = 4, n2 = 5, 6, ..

Atomic units

action: ~ = 1mass: me = 1charge: e = 1length: a0 = 4πε0~2

mee2 = 1 bohrenergy: Eh = ~2/(mea2

0) = −2× E1s(H) = 1 hartree = 27.21138eVtime: 1 a.u. = ~/Eh = 2.4189× 10−17 s = 0.024189 fsH atom Hamiltonian in atomic unit : H = −(1/2)∇2 − Z/r

Approximate method

Time-independent perturbation theory : H = H(0) + H(1)

H(1) is the perturbation. The wavefunctions and energy of the per-turbed system can be computed from a knowledge of the unper-turbed system and a procedure for taking into account the presenceof the perturbation.Many-level systems H(0)|n〉 = E (0)

n |n〉The Hamiltonian of a perturbed system :

H = H(0) + λH(1) + λ2H(2) + ...

The perturbed wavefunction of the system :

ψ0 = ψ(0)0 + λψ

(1)0 + λ2ψ

(2)0 + ...

The energy of the perturbed state :

E0 = E (0)0 + λE (1)

0 + λ2E (2)0 + ...

First order correction to the energy:

E (1)0 = 〈0|H(1)|0〉 = H(1)

00

First order correction to the wave function :

ψ0 ≈ ψ(0)0 +

∑k 6=0

H(1)k0

E (0)0 − E (0)

kψ

(0)k

Second order correction to energy :

E (2)0 = H(2)

00 +∑n 6=0

H(1)0n H(1)

n0

E (0)0 − E (0)

n

Higher order corrections are much more complicated.

Variation theory

The Rayleigh ratio :

ε =∫ψ∗trialHψtrialdτ∫ψ∗trialψtrialdτ

According to variation theorem, for any ψtrial, ε ≥ E0, where E0 isthe lowest eigenvalue of the Hamiltonian.Linear variation

ψtrial =∑

iciψi

ε =∫ψ∗trialHψtrialdτ∫ψ∗trialψtrialdτ

=∑

i ,j cicj∫ψiHψjdτ∑

i ,j cicj∫ψiψjdτ

=∑

i ,j cicjHij∑i ,j cicjSij

For minumum of ε, ∂ε/∂ck = 0, this leads to the secular equations,∑i

ci (Hik − εSik) = 0

Solution:det|Hik − εSik | = 0

Many electrons system

The He atom : Hamiltonian

H = −12(∇2

1 +∇22)− Z

r1− Z

r2+ 1

r12

No analytical solution exists for the TISEVariational treatment: trial wavefunction for the ground state

|ψ〉 =

√λ3

πe−λr1

√λ3

πe−λr2

where, λ = Z − σ is effective charge, with σ is shielding factor

E = 〈ψ|H|ψ〉〈ψ|ψ〉

= −λ2 − 2(Z − λ)λ+ 5λ8

∂E/∂λ = 0 condition gives λ = Z − 5/16E = −2.85 Hartree and the exact result is -2.904 Hartree

Many electrons system

The Pauli principle: Electrons are indistinguishable Fermions thatcarry spins. The total wavefunction (including spin) must be anti-symmetric with respect to the interchange of any pair of electrons.

Pauli exclusion principle: No two electrons can occupy the samestate.Exchange operator: X12|φ(x1, x2)〉 = |φ(x2, x1)〉According to Pauli principle,X12|φ(x1, x2)〉 = |φ(x2, x1)〉 = −|φ(x1, x2)〉Four possible He ground state wavefunctions,

|ψ1〉 = |1s(1)〉|1s(2)〉|α(1)〉|α(2)〉 = |1s1s|αα〉

|ψ2〉 = |1s(1)〉|1s(2)〉|α(1)〉|β(2)〉 = |1s1s|αβ〉

|ψ3〉 = |1s(1)〉|1s(2)〉|β(1)〉|α(2)〉 = |1s1s|βα〉

|ψ4〉 = |1s(1)〉|1s(2)〉|β(1)〉|β(2)〉 = |1s1s|ββ〉

Many electrons system

|ψ1〉 and |ψ4〉 do not follow Pauli principle.So the allowed wavefunction will be linear combination of |ψ2〉 and|ψ3〉.Again |ψ2〉 + |ψ3〉 does not follow Pauli principle. So the finalwavefunction will be (1/

√2)(|ψ2〉 - |ψ3〉)

Overall wavefunctions that satisfy the Pauli principle can be writtenas a Slater determinant.

ψ(1, 2) = 1√2

∣∣∣∣∣|1s(1)〉|α(1)〉 |1s(1)〉|β(1)〉|1s(2)〉|α(2)〉 |1s(2)〉|β(2)〉

∣∣∣∣∣

Many electrons system

For N electrons system.

ψ(1, 2, ...,N) = 1√N!

∣∣∣∣∣∣∣∣∣∣∣∣∣

|χ1(1)〉 |χ2(1)〉 ... |χN(1)〉|χ1(2)〉 |χ2(2)〉 ... |χN(2)〉

. . . .

. . . .

. . . .|χ1(N)〉 |χ2(N)〉 ... |χN(N)〉

∣∣∣∣∣∣∣∣∣∣∣∣∣Here, |χ〉 contain both spin and orbital parts and is called a spin-

orbital.

The Molecular Hamiltonian

H = −M∑

K=1

12MK

∇2K −

N∑i=1

12∇

2i −

M∑K=1

N∑i=1

ZKRKi

+N∑

i>j

1rij

+M∑

K>L

ZK ZLRKL

H = Tnuc + Hel + Enuc

Tnuc = −M∑

K=1

12MK

∇2K

Enuc =M∑

K>L

ZK ZLRKL

Tnuc and Enuc act/depend only on nuclear coordinates while, Helacts on the electronic coordinates and also depends on the nuclearcoordinates.

The Born-Oppenheimer approximation

The nuclei are much heavier than the electron and thus move veryslowly compare to electrons. In the BO approximation, it is assumedthat electrons adjust themselves instantaneously to the motion ofnuclei and thus separates the nuclear and electronic motions.

Ψ(R, r) = ψe(r ; R)χn(R)

Heψe(r ; R) = Ee(R)ψe(r ; R)

He = Hel + Enuc

[Tnuc + Ee(R)]χn(R) = Enχn(R)

In this approximation the coupling between nuclear and electronicmotions are neglected and this is a good approximation if the ener-gies of different electronic states are well separated.

The molecular orbital (MO) theory

The electronic Hamiltonian for H+2 :

He = −12∇

2e −

1rA− 1

rB+ 1

RAB

Linear combinations of atomic orbitals (LCAO):ψMO =

∑i

ciψAO

∑i

ci (Hik − ESik) = 0

The secular determinant: det|Hik − ESik | = 0For a basis set consisting of two atomic orbitals, one on atom A andthe other one on identical atom B, it can be expressed as∣∣∣∣∣HAA − ESAA HAB − ESAB

HBA − ESBA HBB − ESBB

∣∣∣∣∣ = 0

Now, HAA = HBB = α, HAB = HBA = βand SAA = SBB = 1, SAB = SBA = S

The molecular orbital (MO) theory

Homonuclear∣∣∣∣∣ α− E β − ESβ − ES α− E

∣∣∣∣∣ = 0

Solution:

E± = α± β1± S

if S = 0

Bonding and antibonding MO

Heteronuclear∣∣∣∣∣αA − E β − ESβ − ES αB − E

∣∣∣∣∣ = 0

Solution: if |αA − αB| >> βand S = 0

E+ = αA −β2

αB − αA

E− = αB + β2

αB − αA

Huckel MO theoryFor conjugated molecules: Only π MOs are treated and molecular

frame is fixed by σ bonds. α are set equal and S are set to zero. βare equal for neighbors, but zero for non-neighbors.Example for ethene: ∣∣∣∣∣α− E β

β α− E

∣∣∣∣∣ = 0

Solution: E = α± β, Total energy = 2(α + β)Example for benzene:∣∣∣∣∣∣∣∣∣∣∣∣∣

α− E β 0 0 0 ββ α− E β 0 0 00 β α− E β 0 00 0 β α− E β 00 0 0 β α− E ββ 0 0 0 β α− E

∣∣∣∣∣∣∣∣∣∣∣∣∣= 0

Solution: E = α± 2β, α± β, α± β

Huckel MO theory

For benzene:

Total energy = 2(α + 2β) + 4(α + β) = 6α + 8βDelocalization energy (aromatic stability) = 6α+ 8β − 3(2α+ 2β)= 2β

Hartree Products

The wavefunction of an electron can be described by a spin orbital(contains the spatial orbital and spin)Now for N noninteracting electrons the Hamiltonian can be writtenas

H =N∑

i=1h(i)

where h(i) is the operator describing the kinetic and potential energyof electron i .

h(i)χj(xi ) = εjχj(xi )As H is sum of one electron Hamiltonians the total wavefunction

will be,

ΨHP(x1, x2, ...xN) = χi (x1)χj(x2)...χk(xN)

HΨHP = EΨHP

with eigenvalue E = εi + εj + ...+ εkDoes not satisfy the antisymmetry and Pauli exclusion principle

Slater Determinants

For N electrons system.

ψ(1, 2, ...,N) = 1√N!

∣∣∣∣∣∣∣∣∣∣∣∣∣

|χ1(x1)〉 |χ2(x1)〉 ... |χN(x1)〉|χ1(x2)〉 |χ2(x2)〉 ... |χN(x2)〉

. . . .

. . . .

. . . .|χ1(xN)〉 |χ2(xN)〉 ... |χN(xN)〉

∣∣∣∣∣∣∣∣∣∣∣∣∣where 1√

N! is the normalization factor.Rows are labelled by electrons and columns are labelled by spinorbitals.Satisfies the antisymmetry and Pauli exclusion principleInterchanging the coordinates of two electrons = interchanging tworows (changes the sign of the determinant, antisymmetry) Two elec-trons occupying the same spin orbital → two identical columns ofthe determinant (determinant is zero, Pauli exclusion)

Slater Determinants and Hartree-Fock Equations

Slater determinant introduces exchange effects, |Ψ|2 is invariant tothe exchange of the space and spin coordinates of any two electron

The motion of electrons with parallel spins is correlated but themotion electrons with opposite spins remain uncorrelated.

The simplest antisymmetric wavefunction used to describe theground state of an N electronic system is a single Slater determinant

|Ψ0〉 = |χ1χ2...χN〉

From the variation principle

E0 = 〈Ψ0|H|Ψ0〉

where H is the full electronic Hamiltonian.To minimize E0 with respect to the choice of spin orbital the Hartree-Fock equation can be derived, which determines the optimal spinorbitals.

Slater Determinants and Hartree-Fock Equations

Hartree-Fock equation is an eigen value equation of the formf (i)χ(xi ) = εχ(xi )

where f (i) is an one electron operator (Fock operator)

f (i) = −12∇

2i −

M∑K=1

ZKriK

+ νHF(i)

where νHF(i) is the average potential experienced by the ith elec-tron due to the presence of other electrons.

f (1) = h(1) + νHF(1) = h(1) +∑

bJb(1)−Kb(1)

Many body electron problem ≈ one electron problem where electron-electron repulsion is treated as an average way.The procedure of solving the Hartree-Fock equation is called self-consistent-field (SCF) method.

The Coulomb and Exchange Operator

J is a two electron coulomb term and K is the exchange termarises due to the antisymmetric nature of the Slater determinant.Hartree-Fock integro-differential equation

h(1)χa(1) +∑b 6=a

[∫dx2|χb(2)|2r−1

12

]χa(1)

−∑b 6=a

[∫dx2χ

∗b(2)χa(2)r−1

12

]χb(1) = εaχa(1)

Now,Jb(1)χa(1) =

[∫dx2χ

∗b(2)r−1

12 χb(2)]χa(1)

andKb(1)χa(1) =

[∫dx2χ

∗b(2)r−1

12 χa(2)]χb(1)

The Coulomb and Exchange Operator

Expectation value of J and K :

〈χa(1)|Jb(1)|χa(1)〉 =∫

dx1dx2χ∗a(1)χa(1)r−1

12 χ∗b(2)χb(2) = [aa|bb]

and

〈χa(1)|Kb(1)|χa(1)〉 =∫

dx1dx2χ∗a(1)χb(1)r−1

12 χ∗b(2)χa(2) = [ab|ba]

The sum of coulomb and exchange potential is the average potentialenergy of electron 1 due to the presence of the other N−1 electrons.Each spin-orbital must be obtained by solving the Hartree-Fock

equation with the corresponding Fock operator f (i). However, f (i)depends on the spin-orbitals of all the other N − 1 electrons.One has to follow an iterative style of solution, and stopping when

the solutions are self-consistent.

Restricted Hartree-Fock

Restricted Hartree-Fock (RHF): In the restricted open-shell for-malism, all electrons except those occupying open-shell orbitals areforced to occupy doubly occupied spatial orbitals. The restrictedopen-shell wavefunction for Li atom

Ψ0 = 1√6

det|ψα1s(1)ψβ1s(2)ψα2s(3)|

This is a real constraint as the 1sα electron has an exchangeinteraction with the 2sα electron, whereas the 1sβ electron doesnot.

Unrestricted open-shell Hartree-Fock

The 2sα electron spin polarizes the 1s orbital, thus 1sα and 1sβexperience different effective potential.Unrestricted open-shell Hartree-Fock (UHF): In this formalism two

1s electrons are not constrained to the same spatial orbital. Theunrestricted open-shell wavefunction for Li atom

Ψ0 = 1/√

6 det|ψα1s(1)ψβ1s(2)ψα2s(3)|

The RHF wavefunction is an eigen function of S2 operator but theUHF wavefunction is not.

Self-Consistent Field (SCF)

Roothaan-Hall equations: A known set of basis functions used toexpand the spinorbitals (the spatial parts of the spinorbitals). Thistransforms the coupled HF equations into a matrix problem whichcan be solved by using matrix manipulations.Spatial wavefunction ψa(1) occupied by electron 1

f1ψa(1) = Eaψa(1)

In terms of spatial wavefunctions

f1 = h1 +∑

u2Ju(1)−Ku(1)

Each spatial wavefunction ψi can be expanded as

ψi =M∑

j=1cjiθj

Now, the problem of calculating the wavefunctions has been trans-formed to one of computing the coefficients cji .

HF-SCF

f1M∑

j=1cjaθj(1) = Ea

M∑j=1

cjaθj(1)

M∑j=1

cja

∫θ∗i (1)f1θj(1)dr1 = Ea

M∑j=1

cja

∫θ∗i (1)θj(1)dr1

Overlap matrix (S):Sij =

∫θ∗i (1)θj(1)dr1 (9.17)

Fock matrix (S):Fij =

∫θ∗i (1)f1θj(1)dr1 (9.18)

M∑j=1

Fijcja = Ea

M∑j=1

Sijcja

The entire set of equations can be written as single matrix equationFc = ScE (Roothaan-Hall equations)

HF-SCF

A non-trivial solution of the Roothaan-Hall equations:det|F − EaS| = 0 (9.21)

Cannot be solved directly as the matrix elements Fi involve inte-grals over the coulomb and exchange operators and those dependon the spatial wavefunctions.

Self-consistent field approach. Each iteration produce a new setof coefficients cja. Continue until a convergence criterion has beenreached.

HF-SCF

A trial set of spin-orbitals is formulated and used to construct theFock operator, then the HF equations are solved to obtain a newset of spin-orbitals.

Those are then used to construct a new Fock operator, and so on.The cycle of calculation and reformulation is repeated until a con-

vergence criterion is satisfied.

HF-SCF orbitals

Excited determinants

//Singly excited determinant

//Doubly excited determinant

Configuration State Function (CSF)

Hartree-Fock ground state determinant:

|Φ0〉 = |φ1φ2...φaφb...φn〉

Only one of 2MCn combination of determi-nant

Singly excited determinant:

|Φpa〉 = |φ1φ2...φpφb...φn〉

One electron has been promoted from φa toa virtual spin orbital φp

Doubly excited determinant:

|Φpqab 〉 = |φ1φ2...φpφq...φn〉

Two electrons have been promoted from φaand φb to virtual spin orbitals φp and φq re-spectively

Configuration Interaction

The Hartree-Fock method relies on averages and does not considerthe instantaneous coulombic interactions between electrons. ThusHF method ignores electron correlations.

Form of the exact wave function for ground or excited states:

|Ψ〉 = C0|Φ0〉+∑a,p

Cpa |Φp

a〉+∑

a < bp < q

Cpqab |Φ

pqab 〉+

∑a < b < cp < q < r

Cpqrabc |Φ

pqrabc〉+ · · · (4)

Configuration Interaction

An ab initio method in which the wavefunction is expressed as Eq.4 is called configuration interaction (CI).

The energy associated with the exact ground-state wavefunctionis the exact non-relativistic ground-state energy (within the Born-Oppenheimer approximation).

The difference between the exact energy E0 and the HF limit E0 iscalled the correlation energy.

Ecorr = E0 − E0

Configuration interaction accounts for the electron correlation ne-glected in the Hartree-Fock method.

Exact energy of a system, HF limit and full CI

A full CI (FCI) calculation includes all CSFs of the appropriate sym-metry for a given finite basis set.

Basis sets

Evaluation of the two-electron integrals is very difficult and timeconsuming with Slater basis functions. As the product of two Gaus-sian functions is also a Gaussian function computing the two-electronintegrals is relatively easy.

Basis sets

References

1. Molecular quantum mechanics, P. Atkins and R. Friedman,Oxford University Press

2. Modern Quantum Chemistry Introduction to AdvancedElectronic Structure Theory A. Szabo and N. S. Ostlund,Dover Publications INC. New York

3. Lectures notes, H. Guo, University of New Mexico4. https://www.wikipedia.org/