PA4311 Quantum Theory of SolidsPA4311 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)- Framework (Hohenberg-Kohn, Kohn-Sham)- 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 calculationwww.abinit.org

Course Outline

PA4311 Quantum Theory of Solids

Last time…Determinantal form for N-electron wavefunction (see Rae, Chapter 10)

- electrons are indistinguishable and obey the PEP

Single determinant – Hartree-Fock method (exact inclusion of exchange)

If expand as a sum over determinants – - Full Configuration Interaction method describes both exchange and correlation

Full CI is, in principle, exact

- can converge systematically with number of single particle orbitals in the basis

But size of problem increases exponentially with N - limited to N~10

PA4311 Quantum Theory of Solids

Kohn (1999) ‘In general the -electron wavefunction is not a legitimate scientific concept when

- Nobel prize in Chemistry (1998)

e.g. storage required ~ bytes >> number of atoms in universe when

Nobel Lecture: Electronic Structure of Matter: Wave Functions and Density Functionals. Nobelprize.org. Nobel Media AB, accessed 23 Oct 2014. www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/kohn-lecture.html

Density functional theory - find all the useful properties of a system from without solving for

Electronic Structure, RM Martin (ch.s 4,6,7,12)Time-Dependent Density Functional Theory, CA Ullrich (ch. 2)Various review articles – e.g. U von Barth, Physica Scripta. T109, 9–39 (2004).

PA4311 Quantum Theory of Solids

Hohenberg-Kohn

𝑉 (𝑟 )

Ψ 𝑖 (𝒓 1 ,…,𝒓 𝑁) Ψ 0 (𝒓 1 ,…,𝒓 𝑁)

𝑛0(𝒓 )HK

1. For any system of interacting particles in an external potential , the potential is uniquely determined by the ground state density

- i.e. tells us everything about the system!

2. The energy can be defined as a functional of the density. The density that minimises is the exact ground state density,

PA4311 Quantum Theory of Solids

1. For any system of interacting particles in an external potential , the potential is uniquely determined by the ground state density

Proof proceeds via 2 steps

1. The potential uniquely determines the wavefunction2. No two different ground state wavefunctions can lead to the same ground

state density

RM Martin: ‘…crucial for any practitioner in the field to understand’

PA4311 Quantum Theory of Solids

2. The energy can be defined as a functional of the density. The density that minimises is the exact ground state density,

where is the universal Hohenberg-Kohn functional

Find the exact ground state density by minimising

But… the universal functional, is unknown!

PA4311 Quantum Theory of Solids

Kohn-ShamMap the difficult many particle problem onto an auxiliary system that is easy (…easier…) to solve

Assume the ground state density of the original interacting system is equal to that of some chosen non-interacting system

Net result: independent particle equations with all the difficult many-body terms folded into the ‘exchange and correlation’ potential

𝑉 (𝑟 )

Ψ 𝑖 (𝒓 1 ,…,𝒓 𝑁) Ψ 0 (𝒓 1 ,…,𝒓 𝑁)

𝑛0(𝒓 )HK𝑛0(𝒓 )

𝜓 𝑖=1 ,𝑁 (𝒓 ) 𝜓 𝑖(𝒓 )

𝑉 𝐾𝑆(𝒓)HKKS

PA4311 Quantum Theory of Solids

Kohn-ShamWant to find by minimising

Invent auxiliary non-interacting system with same total energy and density, where is a single determinant and

Non-interacting system KE Hartree interaction

Exchange and correlation energy makes up the difference

External potential

𝐸 𝑋𝐶 [𝑛 ]= (𝑇 [𝑛 ]−𝑇 𝑠 [𝑛 ] )+(𝑊 [𝑛 ]−𝐸𝐻 [𝑛 ] )Trick is that is small

PA4311 Quantum Theory of Solids

Atom

He -2.83 2.77 -6.63 2.00 -0.97

Ne -128.23 127.74 -309.99 65.73 -11.71

Ar -525.95 524.97 -1253.13 231.46 -29.24

Kr -2750.15 2747.81 -6577.87 1171.72 -91.82

Xe -7228.86 7225.10 -17159.16 2880.92 -175.71

Total ground state energy, (Hartrees.), calculated within the LDA – CA Ullrich, Table 2.1

PA4311 Quantum Theory of Solids

Kohn-Sham

where

Single particle equations are ,where,

Exchange and correlation potential