Quantum Mechanics and ElectronicStructure Theory

Debasish Koner

Department of ChemistryUniversity of Basel

October 2, 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. Variational method and perturbation method8. Atomic structure (Hydrogen like atoms)9. Atomic units

10. The molecular Hamiltonian11. Born-Oppenheimer approximation

Contents

1. Molecular orbital theory2. Hartree-Fock theory3. Other ab initio methods4. Basis sets5. Molecular orbitals and population analysis6. Density functional theory (DFT)

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

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(n2πx

L2

)

En1,n2 = h2

8m

(n2

1L2

1+ n2

2L2

2

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

Degeneracy: states with same energies.

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

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

),

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 per-turbed 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

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 , 1, ..., 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

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 isassumed that electrons adjust themselves instantaneously to themotion of nuclei and thus separates the nuclear and electronicmotions.

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 theenergies of different electronic states are well separated.

References

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

2. Lectures notes, H. Guo. University of New Mexico