PA4311 Quantum Theory of Solids

Quantum Theory of SolidsMervyn Roy (S6)www2.le.ac.uk/departments/physics/people/mervynroy

PA4311 Quantum Theory of Solids

1. Introduction and background2. The many-electron wavefunction

- Introduction to quantum chemistry (Hartree, HF, and CI methods)

3. Introduction to density functional theory (DFT)- Periodic solids, plane waves and pseudopotentials

4. Linear combination of atomic orbitals5. Effective mass theory6. ABINIT computer workshop (LDA DFT for periodic solids)

Assessment: 70% final exam 30% coursework – mini ‘project’ report for ABINIT calculation(Set problems are purely formative)

Course Outline

PA4311 Quantum Theory of Solids

Last time…The modern world is build upon our understanding of the electronic properties of solids…

Born-Oppenheimer approximation – electrons respond instantaneously to ion motion

N-electron wavefunction contains all the information about the system

is a function of spatial coordinates, and spins

The variational principle is a useful starting point to find approximations to

Rae 5th Ed. Sec. 7.3 – Variational principle & complete sets of states (q. 1.1)M. L. Boas, 2nd Ed. ,Ch. 4, Sec. 9 – Lagrange multipliers

PA4311 Quantum Theory of Solids

The N-electron wavefunctionThe -electron wavefunction depends on N spatial coordinates (and spins)

Electrons are indistinguishable: Fermions are anti-symmetric: - they obey the Pauli exclusion principle

⟨𝑂 ⟩=⟨Ψ|�̂�|Ψ ⟩=∫𝑉

Ψ∗ (𝒓 1 ,𝒓 2 ,…,𝒓 𝑁 )�̂�Ψ (𝒓 1 ,𝒓 2 ,…,𝒓 𝑁)𝑑𝒓𝟏𝑑𝒓 2…𝑑𝒓 𝑁

Expectation values

See Tipler (4th Ed Sec. 36.6 on ‘The Schrödinger equation for 2 identical particles’)

PA4311 Quantum Theory of Solids

The density operator

We can calculate the electron density by finding the expectation value of

�̂�=∑𝑖=1

𝑁

𝛿(𝒓 −𝒓 𝑖)

PA4311 Quantum Theory of Solids

A hierarchy of methods

• Hartree‘Independent’ particle approximation

• Hartree-FockExact inclusion of the exchange interaction

• Configuration InteractionPost Hartree-Fock methods attempt to include exchange and correlation

• The exponential wallDo we really need to know the full wavefunction?

PA4311 Quantum Theory of Solids

Hartree approximation• ‘Independent’ electron picture – (electrons are distinguishable)

• Electrons interact via mean-field Coulomb potential - (respond to avg. charge density)

(− 12 𝛻𝑖2−𝑣 (𝒓 𝑖 )+∫ 𝑛 (𝒓 ′ )

|𝒓 𝑖−𝒓′|𝑑𝒓 ′)𝜓𝑖 (𝒓 𝑖 )=𝐸𝑖𝜓 𝑖 (𝒓 𝑖 )

Key points Replace interaction term with average potential, -electron wavefunction is separable, Must solve -single electron Schrödinger equations self-consistentlyTotal energy, , is the sum of single particle energies

Hartree Equations Single particle orbitals

PA4311 Quantum Theory of Solids

Question 2.1

If the Schrödinger equation is separable so that

show that the expectation value of the density operator , is

PA4311 Quantum Theory of Solids

Derivation of Hartree equations

(− 12∑𝑖 𝛻𝑖2−∑

𝑖

𝑣 (𝒓 𝑖 )+12∑𝑖≠ 𝑗

1|𝒓 𝑖−𝒓 𝑗|)Ψ (𝒓 1 ,…,𝒓 𝑁 )=𝐸Ψ (𝒓 1 ,..𝒓 𝑁 )

(− 12 𝛻𝑖2−𝑣 (𝒓 𝑖 )+∑

𝑗

𝑁

∫ |𝜓 𝑗 (𝒓′ )|2

|𝒓 𝑖−𝒓′|𝑑 𝒓 ′)𝜓 (𝒓 𝑖 )=𝐸𝑖𝜓 (𝒓 𝑖 )

Ψ (𝒓 1 , ..𝒓 𝑁 )=∏𝑖=1

𝑁

𝜓 𝑖 (𝒓 𝑖 ) ,Assume the independent electron form of the wavefunction,

then minimise subject to the constraint that each is normalised.

⋮

PA4311 Quantum Theory of Solids

12∑𝑖≠ 𝑗

1

|𝒓 𝑖−𝒓 𝑗|→𝑣𝐻 (𝒓 𝑖 )=∫ 𝑛 (𝒓 )

|𝒓 𝑖−𝒓′|𝑑 𝒓 ′ .

Assume that the full electron interaction can be replaced by a mean field term,

Use the method of separation of variables to show that the -electron Schrödinger equation can be separated into single particle equations.

Question 2.2

PA4311 Quantum Theory of Solids

Self consistent field approximationThe single particle equations must be solved self-consistently

Guess

Calculate

Solve Eq.s -

Calculate new

Self consistent?No

Use

new

Yes Calculation finished

PA4311 Quantum Theory of Solids

Hartree approximation• Electrons are distinguishable & wavefunction is not antisymmetric

- Pauli exclusion principle has to be put in by hand

• Electrons do not respond to the particular (as opposed to the average) configuration of the other N-1 electrons

• Self interaction problem

• Calculations are numerically complex

But – Hartree-like calculations are important for modern DFT

PA4311 Quantum Theory of Solids

Hartree approximation

Interaction effects (exchange and correlation) are important when the coulomb interaction energy is large compared to

Hartree-like approximations better when

𝐸1

𝐸2

𝐸3

Infinite square well

Interaction goes like

goes like 𝐸1

𝐸2

𝐿 𝐿