PA4311 Quantum Theory of Solids Quantum Theory of Solids Mervyn Roy (S6) www2.le.ac.uk/departments/physics/people/mervynroy
Jan 02, 2016
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