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


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

Prerequisites• 3210 Quantum Mechanics• 2230 Condensed Matter Physics• Also – 214 Fourier Series, 372 Fourier Transforms, etc.

Resources

Plus many other relevant text books and online references – see the library

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

http://www.abinit.org/


The problem

𝐻=−ℏ2

2𝑚 ∑𝑖

𝛻𝑖2−∑

𝑖∑𝐼

𝑍 𝐼𝑒2

4𝜋 𝜖0|𝒓 𝑖−𝑹𝐼|+12∑𝑖≠ 𝑗

𝑒2

4𝜋 𝜖0|𝒓 𝑖−𝒓 𝑗|−

ℏ2

2𝑚𝐼∑𝐼

𝛻 𝐼2+ 12∑𝐼≠ 𝐽

𝑍 𝐼 𝑍 𝐽 𝑒2

4𝜋 𝜖0∨𝑹𝐼−𝑹 𝐽∨¿¿

electrons and ions is a function of electron co-ordinates, (and spins), and ion co-ordinates, (and spins)

𝐻Ψ=𝐸Ψelectron KE

electron-ion interaction

electron-electron interaction

ion-ion interaction

ion KE

But, so ion KE term is small


Timescales

From CA Ullrich, Time-Dependent Density-Function Theory, Oxford University Press (2012)


The problem

𝐻=−ℏ2

2𝑚 ∑𝑖

𝛻𝑖2−∑

𝑖∑𝐼

𝑍 𝐼𝑒2

4𝜋 𝜖0|𝒓 𝑖−𝑹𝐼|+12∑𝑖≠ 𝑗

𝑒2

4𝜋 𝜖0|𝒓 𝑖−𝒓 𝑗|+𝐸 𝐼𝐼

But, still have electrons is a function of electron co-ordinates, , (and spins).

Born-Oppenheimer approximation- electrons react instantaneously to changes in nuclear positions

𝐻=−12∑𝑖 𝛻𝑖

2−∑𝑖

𝑣 (𝒓 𝑖 )+12∑𝑖 ≠ 𝑗

1

|𝒓 𝑖−𝒓 𝑗|+𝐸 𝐼𝐼

Or, in atomic units,

Need to develop some approximations!

Constant depends on ion positions


The modern world is build upon our understanding of the electronic properties of solids

Why bother?

Solid state (nano) physics, materials physics, space technology etc. etc.

Spectroscopye.g. astrophysics, Earth observation science – ExoMol line lists (TDDFT)Plasma physics…


Chemistry (data from WOK) Density-Functional Thermochemistry 3. The Role of Exact Exchange, AD Becke, J. Chemical Physics 98, 5648 (1993)

46 280

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 629Efficient 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)

Highest cited paper on ADS 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 - data from ADS

Black hole 6765

Particle physics 4164

Metamaterials 2289

Aurora 1785Gamma Ray Burst 1513


3210 revision (see Rae)

Schrödinger equation -

⟨𝑂 ⟩=∫𝑉

𝜓∗�̂�𝜓 𝑑𝒓Expectation values -

Probability density - |

Variational principle -⟨ 𝐸 ⟩= ⟨𝜙|𝐻|𝜙 ⟩⟨𝜙|𝜙 ⟩

≥𝐸0

𝜓=∑𝑛

𝑐𝑛𝜙𝑛 , 𝜕 ⟨𝐸 ⟩𝜕𝑐𝑚

∗ =0=∑𝑛

(𝐻𝑚𝑛❑ −𝐸 𝛿𝑚𝑛 )𝑐𝑛thenIf


Question 1.1

If is normalised and the are orthonormal, show that .

If we wish to minimise subject to the constraint that the are normalised, show that the appropriate Lagrange multiplier,


Full variation and functionals

is a functional of and

A functional maps a function onto a value, For example,

See RM Martin App. A, or e.g. GC Evans, Functionals and their applications, Dover, New York, 1964


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’)

Documents

Quantum Theory of Solids