# Lecture 1

```Quantum Theory of Solids
Mervyn Roy (S6)
www2.le.ac.uk/departments/physics/people/mervynroy
PA4311 Quantum Theory of Solids
Course Outline
1. Introduction and background
2. 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 orbitals
5. Effective mass theory
6. 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)
PA4311 Quantum Theory of Solids
Resources
Lecture notes www2.le.ac.uk/departments/physics/people/mervynroy
Abinit website - including comprehensive help and tutorials
• www.abinit.org
Books
• Electronic Structure, RM Martin
• Solid State Physics, Ashcroft and Mermin
• Solid State Physics, Hook and Hall
Plus many other relevant
text books and online
references – see the library
Prerequisites
• 3210 Quantum Mechanics
• 2230 Condensed Matter Physics
• Also – 214 Fourier Series, 372 Fourier Transforms, etc.
PA4311 Quantum Theory of Solids
The problem
electron-ion
interaction
electron KE
Ψ = Ψ

−ℏ2
=
2
ℏ2
−
ion 2
KE
2 −

1
2
+
2
≠

2
1
+
40  −
2
2
40 | −  |
electronelectron
interaction
2
≠
40  −
ion-ion interaction
> 1023 electrons and ions
Ψ is a function of  electron co-ordinates,  , (and spins), and  ion
co-ordinates,  (and spins)
But, mI ≫  so ion KE term is small
PA4311 Quantum Theory of Solids
Timescales
From CA Ullrich, TimeDependent DensityFunction Theory, Oxford
University Press (2012)
PA4311 Quantum Theory of Solids
The problem
Born-Oppenheimer approximation
- electrons react instantaneously to changes in nuclear positions
−ℏ2
=
2
2 −

2
1
+
40  −
2
2
≠
40  −
Or, in atomic units,
1
=−
2
2

−

1
+
2
1
≠
−
+
But, still have N > 1023 electrons
Ψ is a function of  electron co-ordinates,  , (and spins).
Need to develop some approximations!
PA4311 Quantum Theory of Solids
+
Constant depends on
ion positions
Why bother?
The modern world is build upon our understanding of the electronic properties of solids
Solid state (nano) physics, materials physics, space technology etc. etc.
Spectroscopy
e.g. astrophysics, Earth observation science
– ExoMol line lists (TDDFT)
Plasma physics…
PA4311 Quantum Theory of Solids
Highest cited papers in Physical Review suite of journals (2014)
Citations
Generalized Gradient Approximation Made Simple, JP Perdew, K Burke, and M Ernzerhof, Phys. Rev.
Lett. 77, 3865 (1996)
25 083
Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density,
C Lee, W Yang, and RG Parr, Phys. Rev. B 37, 785 (1988)
24 292
Self-Consistent Equations Including Exchange and Correlation Effects, W Kohn and LJ Sham, Phys. Rev.
140, A1133 (1965)
18 399
Inhomogeneous Electron Gas, P Hohenberg and W Kohn, Phys. Rev. 136, B864 (1964)
15 629
Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, G
Kresse and J Furthmüller, Phys. Rev. B 54, 11169 (1996)
14 495
Density-functional exchange-energy approximation with correct asymptotic behavior, AD Becke, Phys.
Rev. A 38, 3098 (1988)
14 191
Special points for Brillouin-zone integrations, HJ Monkhorst, JD Pack, Phys. Rev. B 13, 5188 (1976)
12 938
From ultrasoft pseudopotentials to the projector augmented-wave method, G Kresse and D Joubert,
Phys. Rev. B 59, 1758 (1999)
10 351
Helical
microtubules of
graphitic carbon,
S. Ijima, Nature
354 , 56 (1991)
(24 225)
Electric field
effect in
atomically thin
carbon films, K.S.
Novoselov, A.K.
Geim, et al.
Science 306
(2004) (15 139)
astrophysics (2014)
Maps of Dust Infrared Emission for Use in Estimation of Reddening and Cosmic
Microwave Background Radiation Foregrounds, D.J. Schlegel et al, APJ 500, 525 (1998)
8920
Comparison with other areas
Black hole
6765
Particle physics
4164
Metamaterials
2289
Aurora
1785
Gamma Ray Burst
1513
Density-Functional Thermochemistry 3. The Role of Exact Exchange, AD Becke, J.
Chemical Physics 98, 5648 (1993)
46 280
Chemistry (data from WOK)
PA4311 Quantum Theory of Solids
3210 revision (see Rae)
Schrödinger equation - () = ()
Probability density - |()|2
∗
Expectation values -  =

Variational principle -  =
≥ 0

If  =
, then

PA4311 Quantum Theory of Solids

∗ =0=

−

Question 1.1
If  =    is normalised and the  are orthonormal,
show that   2 = 1.
∗     subject
If we wish to minimise  =

to the constraint that the  are normalised, show that the
appropriate Lagrange multiplier,  = .
PA4311 Quantum Theory of Solids
Full variation and functionals
is a functional of  ∗ and
∗,  =    −    − 1
∗ ,  =  ( − )  = ∫  ∗  −   = 0
−  =0
A functional maps a function onto a value,

=
2
= . For example,

1
=   +  −   =
2

1
See RM Martin App. A, or e.g. GC Evans, Functionals and their applications, Dover, New York, 1964
PA4311 Quantum Theory of Solids
The N-electron wavefunction
The -electron wavefunction depends on N spatial coordinates (and spins)
Ψ(1 , 2 , … , N )
Electrons are indistinguishable: Ψ 1 , 2 , … , N 2 = Ψ 2 , 1 , … , N 2
Fermions are anti-symmetric: Ψ 1 , 2 , … , N = −Ψ 2 , 1 , … , N - they obey
the Pauli exclusion principle
See Tipler (4th Ed Sec. 36.6 on ‘The
Schrödinger equation for 2 identical
particles’)
Expectation values
= ΨΨ =

Ψ ∗ 1 , 2 , … ,  Ψ(1 , 2 , … ,  )  2 …
PA4311 Quantum Theory of Solids
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