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
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
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
PA4311 Quantum Theory of Solids
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
PA4311 Quantum Theory of Solids
Hartree-Fock
e.g. 2-electrons
Wavefunction is antisymmetric -
Pauli Exclusion Principle: if then
PA4311 Quantum Theory of Solids
Question 2.3
Show that the 2-electron Slater determinant,
is correctly normalised assuming the single particle orbitals and are orthonormal.
PA4311 Quantum Theory of Solids
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
PA4311 Quantum Theory of Solids
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)
PA4311 Quantum Theory of Solids
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.
PA4311 Quantum Theory of Solids
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
PA4311 Quantum Theory of Solids
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
PA4311 Quantum Theory of Solids
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
PA4311 Quantum Theory of Solids
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.
PA4311 Quantum Theory of Solids
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)
PA4311 Quantum Theory of Solids
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?
PA4311 Quantum Theory of Solids
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