# PhD Course and Lecture Series in Density Functional Theory, 7.5 hp

*Course next start: late autumn term 2017 (precise date not set yet)*

Lecturer: Rickard Armiento

Course will be given if there is a large enough interest shown by registration. Register by email to: ricarifm.liu.se

- There will be ten 2-hour lectures with roughly one lecture per week + 1-2 tutorial sessions.
- The course starts on a basic level (e.g., Hohenberg-Kohn and Kohn-Sham theory) and proceeds up to modern trends in density functionals and extensions of density functional theory. A big part of the course is to read and understand some of the most fundamental publications in density functional theory.

## First lecture: Time and place not decided yet.

## Examination

** Examination in summary:** Attend lectures, read papers, and solve 10 problem sets with 5 small problems each. Hand in on time, and you can leave out or be incorrect about 1 problem/set. If you hand in late you need to solve all problems. No limit on re-tries.

*Examination details:*

- Attending the lectures is mandatory (if you have to miss a few, that can be resolved if you talk with me, but usually incur a minor penalty for the hand-in exercises.)
- Read selected articles and handouts (they are mostly parts of review papers, and some of the most well-known and cited research papers in the field).
- There is one set of hand-in problems per lecture = 10 × (4 smallish questions + one larger).
- Correct solutions handed in
**at or before next lecture**: 5 points/problem. - Correct solutions handed in late: 4 points/problem.
- Technically no limit on re-tries for incorrect solutions (but re-tries are scored as late, 4p).
- The larger problems marked with a star (*), must be solved.
- Requirement for passing: 200 points, including all starred problems done.
- Collaboration is ok, but solutions must be written up independently. Please also state who you have collaborated with on your solution.
- The ultimate deadline for the hand-in problems is when the course starts again, unless anything else is said or agreed upon.
- However,
**important:**there is still going to be a deadline after the end of the course,*after which*the student has to accept that hand-ins are not acted on immediately, but waits for the next 'correction aggregation date' which is a few times per term. - You should expect a reply/confirmation from me for hand-ins that are, e.g., sent by email or post. No response does not in any way shape or form impicitly mean passing. I.e., you have to contact me again if you do not hear anything.

*Note: the hand-in exercises are supposed to be easy once you have understood the source material, but that will usually require going through the provided papers + seeking out info in suggested background material/books to the extent required for ones understanding, which does amount to a non-trivial amount of work. You are supposed to be unsuccessful if simply try to knock out the 50 exercises without little or no reading. That said, if you have read the assigned material and still spend significant time solving them, you are strongly encouraged to discuss with me (either at the lectures, or visit my office). To ask your lecturer `what is a good source to learn about X?' is encouraged.*

## Good Books and Reviews

- Walter Kohn's Nobel Lecture Rev. Mod. Phys.
**71**, 1253 (1999). This is a very good first read. The first lectures follows the same structure. - Parr and Yang, Density-Functional Theory of Atoms and Molecules (book, starting to get old, theoretical, but good)
- Carlos Fiolhais, Fernando Nogueira, Miguel A.L. Marques, A Primer in Density Functional Theory (book, theory-oriented)
- Jorge Kohanoff, Electronic Structure Calculations for Solids and Molecules (book, theory-oriented)
- Richard M. Martin, Electronic Structure (book, more practical)
- Klaus Capelle, A Bird's-Eye View of Density-Functional Theory (review)
- Kieron Burke, the ABC of DFT (review)
- Jones and Gunnarsson, The density functional formalism, its applications and prospects (review paper from 1989)

## Lecture outline and course reading

**Lecture 1: Introduction and Overview**** (Lecture notes, pdf)
**

*Modern theory of matter, the many-particle wave-function catastrophe, orbital theories, Thomas-Fermi theory, functionals.*

Reading: Sections 1, 2, and 3 of W. Kohn, *"Nobel Lecture: Electronic structure of matter—wave functions and density functionals"*, Rev. Mod. Phys. **71**, 1253 (1999). [http://dx.doi.org/10.1103/RevModPhys.71.1253],
Recommended to start reading the handout on functionals.

**Lecture 2: Hohenberg-Kohn DFT
**

*From the Schrödinger equation to the universal functional, the two Hohenberg-Kohn theorems, v and N representability, the constrained search formulation.*

Reading: Section IV(a+b) of W. Kohn, *"Nobel Lecture: Electronic structure of matter—wave functions and density functionals"*, Rev. Mod. Phys. **71**, 1253 (1999). [http://dx.doi.org/10.1103/RevModPhys.71.1253],

Handout on functionals,

P. Hohenberg and W. Kohn, *"Inhomogeneous Electron Gas"*, Phys. Rev. **136**, B864 (1964). (15128 APS citations.) [http://dx.doi.org/10.1103/PhysRev.136.B864] - focus on sections I(1-3), III(1) and IV.

**Lecture 3: Kohn-Sham DFT
**

*Electrons vs. Kohn-Sham particles, the Kohn-Sham equations and their solution.*

Reading: Section IVc of W. Kohn, *"Nobel Lecture: Electronic structure of matter—wave functions and density functionals"*, Rev. Mod. Phys. **71**, 1253 (1999). [http://dx.doi.org/10.1103/RevModPhys.71.1253],

W. Kohn and L. J. Sham, *"Self-Consistent Equations Including Exchange and Correlation Effects"*, Phys. Rev. **140**, A1133 (1965) (17289 APS citations) [http://dx.doi.org/10.1103/PhysRev.140.A1133] - focus on sections I, II, and the segment at the very end of the paper that starts 'note added in proof'.

**Lecture 4: The Exchange-Correlation Energy
**

*The xc-hole, the adiabatic connection, scaling relations, spin-polarized DFT.*

Reading: Section V up to Eq. (5.14) of W. Kohn, *"Nobel Lecture: Electronic structure of matter—wave functions and density functionals"*, Rev. Mod. Phys. **71**, 1253 (1999). [http://dx.doi.org/10.1103/RevModPhys.71.1253],

J. Harris *"Adiabatic-connection approach to Kohn-Sham theory"* Phys. Rev. A **29**, 1648 (1984) [http://dx.doi.org/10.1103/PhysRevA.29.1648] - focus on the part of the paper up to and including Eq. (17),

U von Barth and L Hedin, *"A local exchange-correlation potential for the spin polarized case",* C: Solid State Phys.** 5** 1629 (1972) (3409 IOP citations) [http://dx.doi.org/10.1103/PhysRevA.29.1648 - focus on sections 1, 2, and 3.]

**Lecture 5: Exchange-Correlation Functionals I
**

*Locality for functionals, the local-density approximation, gradient expansion, generalized gradient approximations*

Reading:
Section V from Eq. (5.14) up to and including the paragraph with Eqs. (5.18)-(5.20), of W. Kohn, *"Nobel Lecture: Electronic structure of matter—wave functions and density functionals"*, Rev. Mod. Phys. **71**, 1253 (1999). [http://dx.doi.org/10.1103/RevModPhys.71.1253],

L. Kleinman and S. Lee, *"Gradient expansion of the exchange-energy density functional: Effect of taking limits in the wrong order"*, Phys. Rev. B **37**, 4634 (1988). [http://link.aps.org/doi/10.1103/PhysRevB.37.4634] - note: the point is to understand the general discussion of the paper, especially as illustrated in Fig. 1; there is no need to understand the equations in detail

**Lecture 6: Exchange-Correlation Functionals II
**

*Different approaches: empirical, real-space cutoff, constraint satisfaction, compability; jacob's ladder, meta-GGAs, Jacob's ladder.*

Reading:
The remainder (Eq. (5.20) and onwards) of W. Kohn, *"Nobel Lecture: Electronic structure of matter—wave functions and density functionals"*, Rev. Mod. Phys. **71**, 1253 (1999). [http://dx.doi.org/10.1103/RevModPhys.71.1253],

Look over the two papers that started the empirical vs. non-empirical GGA development: J. P. Perdew, *“Accurate Density Functional for the Energy: Real-Space Cutoff of the Gradient Expansion for the Exchange Hole”*, Phys. Rev. Lett. **55**, 1665 (1985) [http://dx.doi.org/10.1103/PhysRevLett.55.1665]; A. D. Becke, *“Density functional calculations of molecular bond energies”*, J. Chem. Phys. 84, 4524 (1986) [http://dx.doi.org/10.1063/1.450025].

Read thoroughly: J.P. Perdew, K. Burke, M. Ernzerhof, *"Generalized Gradient Approximation Made Simple"*, Phys. Rev. Lett. **77**, 3865 (1996) (28355 citations) [http://dx.doi.org/doi/10.1103/PhysRevLett.77.3865].

**Lecture 7: Formal Properties
**

*Size-consistency, Lieb-Oxford bound, open systems with non-integer particle number, Janak's theorem, the derivative discontinuity, step-like structure, "the bandgap problem", koopmans' theorem, properties of the exchange-correlation potential.*

Reading: J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, Jr., *"Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy"*, Phys. Rev. Lett. **49**, 1691 (1982) (1119 citations) [http://dx.doi.org/10.1103/PhysRevLett.49.1691],

Read lightly: J. P. Perdew, and M. Levy *“Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities.”*, Phys. Rev. Lett. 51, 1884 (1983) [http://dx.doi.org/10.1103/PhysRevLett.51.1884].

**Lecture 8: Orbital Functionals and Other Extensions
**

*Exact exchange with OEP, KLI, SIC, Hybrids, Model potentials, DFT+U.*

Reading:
A. D. Becke, *"A new mixing of Hartree–Fock and local density‐functional theories"*, J. Chem. Phys. **98**, 1372 (1993) (6186 citations) [http://link.aip.org/link/doi/10.1063/1.464304] and J. Heyd, G. E. Scuseria, and M. Ernzerhof, *"Hybrid functionals based on a screened Coulomb potential"*, J. Chem. Phys. **118**, 8207 (2003) (1162 citations) [http://dx.doi.org/10.1063/1.1564060] - focus on up to and including section III.

Read lightly: Heyd, G. E. Scuseria, and M. Ernzerhof, *“Hybrid functionals based on a screened Coulomb potential”*, J. Chem. Phys. 118, 8207 (2003). [http://dx.doi.org/10.1063/1.1564060].

Optional reading, a good review paper on the topics covered in this lecture: S. Kummel and L. Kronik, *“Orbital-dependent density functionals: Theory an applications”*, [http://dx.doi.org/10.1103/RevModPhys.80.3].

**Lecture 9: Practical Considerations
**

*Basis sets: plane wave, atomic, real-space; pseudopotentials, periodic DFT, the hartree energy, matrix solvers.*

Reading:
Read lightly: Sections 3 - 6 of A. E. Mattsson, P. A. Schultz, M. P. Desjarlais, T. R. Mattsson, and K. Leung, *“Designing meaningful density functional theory calculations in materials science–a primer”, Modelling Simul. Mater. Sci. Eng. 13 R1 (2005). [http://dx.doi.org/10.1088/0965-0393/13/1/R01) - focus on the issues being discussed rather than the precise numerics,
Optional reading: look up topics from the lecture that interests you in the book “Electronic Structure” by R. M. Martin (isbn: 0521534402),
Optional reading: for a recap of k-space and the reciprocal lattice see either “Electronic Structure” by R. M. Martin (isbn: 0521534402) or "Solid state physics” by Ashcroft and Mermin (isbn: 0030839939),
*

**Lecture 10: Further Extensions and Modern Trends, Summary and Wrap-Up
**

*Orbital-free DFT, relativity, spin-orbit coupling, time-dependent DFT, High-throughput DFT.*

Reading:
Perdew, et al. *“Some Fundamental Issues in Ground-State Density Functional Theory: A Guide for
the Perplexed”* J. Chem. Theory Comput., 5, 902 (2009) [http://dx.doi.org/10.1021/ct800531s) - read the paper thoroughly, but you may skip the abstract (t is quite long and dry, and does not contain anything that is not in the paper.)

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Responsible for this page:
Fei Wang

Last updated:
08/17/17