Quantum Theory of Solids

PA4311 Quantum Theory of Solids

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

http://www2.le.ac.uk/departments/physics/people/mervynroy/


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


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?


Last time…

• ‘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 )

Must solve -single electron Schrödinger equations self-consistentlyTotal energy, , is the sum of single particle energies

Hartree Equations Single particle orbitals

Hartree approximation

Problems - PEP not enforced, electrons do not respond to specific configuration of other electrons


Hartree-Fock• Electrons are indistinguishable and obey the Pauli exclusion principle• Exact inclusion of the exchange interaction

The N-electron wavefunction has the form of a Slater determinant


Hartree-Fock

e.g. 2-electrons

Wavefunction is antisymmetric -

Pauli Exclusion Principle: if then


Question 2.3

Show that the 2-electron Slater determinant,

is correctly normalised assuming the single particle orbitals and are orthonormal.


Electron spinSingle particle orbital contains information on the electron spin, Wavefunctions with different spins are orthogonal

Can write this formally in many different ways, e.g. notation of S Raimes, Many Electron Theory, North-Holland publishing company, 1972

spin coordinatespace coordinates spin function, or where ,

wavefunction

∫𝑉

𝑑𝒓→∑𝜁=±1

∫𝑉

𝑑𝑥𝑑𝑦𝑑𝑧 then∫𝑉

𝜓 𝑖∗𝜓 𝑗𝑑𝒓→∑

𝜁=± 1

𝜒 𝑖 (𝜁 ) 𝜒 𝑗 (𝜁 )∫𝑉

𝜙𝑖∗𝜙 𝑗𝑑𝑥𝑑𝑦𝑑𝑧

if and 0 if


Hartree-FockAssume N-electron wavefunction has the form of a Slater determinant

,

then minimise subject to the constraint that each is normalised

(− 12 𝛻𝑖2−𝑣 (𝒓 𝑖 )+𝑣𝐻 (𝒓 𝑖))𝜓 𝑖 (𝒓 𝑖 )−∑

𝑗

𝛿𝜒 𝑖 𝜒 𝑗∫𝜓 𝑗

∗ (𝒓 ′)𝜓 𝑖 (𝒓 ′ )|𝒓 −𝒓 ′|

𝑑𝒓 ′𝜓 𝑗(𝒓 )=𝐸𝑖𝜓 𝑖 (𝒓 𝑖 )

Hartree-Fock Equations

direct term

exchange term (integral operator)


Question 2.4

Starting from the 2-electron wavefunction, derive the Hartree-Fock equation,

for two electrons by minimising subject to the constraint that each is normalised.


Hartree-Fock• Electrons are indistinguishable, and can lower their energy by exchanging (if

spins are the same)• PEP is automatically enforced• But - still an approximation to the full wavefunction (electrons still don’t respond

properly to the particular configuration of the other N-1 electrons)• Calculations are much more difficult than in the Hartree approximation because

of the integral operator

if can maybe use a Hartree-like approach. If might need to use post HF methods…A number of variants of HF are popular in quantum chemistry

Hartree and HF methods are particularly poor for excited states


Post Hartree-Fock• HF Slater determinant is still an approximation to the full electron wavefunction

because we still have an arbitrary constraint• If we solve S.E. exactly we will get a lower energy – the difference between and

is often called the correlation energy• Physically, correlation describes the way that electrons tend to avoid each other

radial distance

𝑔(𝑟

)/𝜌0

Exact – ‘exchange and correlation’ hole

Hartree-Fock – exchange hole

Uniform electron gas

is the pair distribution functionClassically, average density

Hartree


Configuration interactionRemove all constraints from the wavefunction,

Expand as a sum over Slater determinants, each with a different configuration of single particle orbitals (full CI)

Ground state determinant

Single excitationsdouble excitations

triple excitations

electron ’excited’ from th occupied orbital to th unoccupied orbital

Expansion often split configurations by number of ‘excitations’ in each configuration


Configuration interaction

As usual, find the unknown coefficients by minimising the energy subject to the constraint that is normalised.

Then, as before,

Calculation of matrix elements, , is more complicated – but procedure is familiar.


To be sure of the validity of an expansion must run calculation systematically

– check energy is minimised according to variational principle

Configuration interactionNumber of terms in expansion grows very rapidly‘Black art’ to truncate expansion for real systems - see e.g. J. Phys.: Conf. Ser. 242 (2010)

Must set maximum number of unoccupied orbitals If must truncate number of excited state configurations (eg. CISD, CISDT etc.)

2 electrons

5 electrons

⟨𝑣𝑐 ⟩<Δ𝐸𝑖

⟨𝑣𝑐 ⟩<Δ𝐸𝑖

- from Phys. Rev. B 85 205432 (2012)


Quantum Chemistry: acronym zoo

Exact solution

STO-2G, STO-3G, STO-6G, STO-3G*, 3-21G, 3-21++G, 3-21G*, 3-21GSP, 4-31G, 4-22GSP, 6-31G, 6-31G-Blaudeau, 6-31++G, 6-31G*, 6-31G**, 6-31G*-Blaudeau, 6-31+G*, 6-31++G**, 6-31G(3df,3pd), 6-311G, 6-311G*, 6-311G**, 6-311+G*, 6-311++G**, 6-311++G(2d,2p), 6-311G(2df,2pd), 6-311++G(3df,3pd), MINI (Huzinaga), etc. etc.

Complete set of statesSingle particle basis

MethodFull CI

HF

Coup

led

clus

ter,

CCSD

, CC

SDT,

CIS

D, C

ISDT

, M

olle

r-Pl

esse

t, , C

AS

SCF,

MC

SCF

etc.

Limited basis

Full CI wavefunction tells us everything – but do we need it?


The exponential wallKohn (1999) “In general the -electron wavefunction is not a legitimate scientific concept when ”- it contains too much information… Usually we only need to know a

few things – e.g. energy, polarizability, band structure, density etc.

Imagine: store wavefunction on a grid, 10 points per dimension…

No. particles No. grid points Memory required

1 1 Kbyte

4 1Tbyte

7 1 Zettabyte

27 ??

atoms in the universe

DFT - find all the useful properties of a system without solving for

Quantum Theory of Solids

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