Link to ppt slides for lecture

Reading: Text: Chapter 20. Also TDDFT in Ch. 6,7 and response functions in App. D,E.

Recall the formulation of DFT - the Hohemnberg-Kohn Theorems, and the Kohn-Sham Ansatz (Lecture 5). The theorems say that all properties are determined by the density. The KS formulation was designed to describe the ground state but the formulation so far has done nothing to explicitly address the calculation of excited states. This lecture is devoted to: 1) classification of important excitations; 2) the formal extension of the HK and KS approaches to time dependence; and 3) some results and discussion of the difficulty in construction of functionals that accurately describe excitations.

1. Electronic Excitations - Chapter 2
1. Electron addition or removal: N → N+1 or N → N-1
• Excitation energy: Δ E₊ = E(N+1) - E(N) - μ or Δ E_- = E(N) - E(N-1) - μ
• Minimum Gap for adding - removing electrons: Δ E_gap - Δ E_- = E(N+1) + E(N-1) - 2 E(N)
2. Electron excitation at fixed number N
• Excitation energy: Δ E_j = E_j(N) - E₀(N)
• Minimum excitation energy - always less than Δ E_gap since electrons repel one another
2. Electron addition or removal - meaning of bands in independent particle system
1. In independent-particle system, the ground state is a single determinant with N electrons, with minimum energy E₀(N)
• Determinant of states with eigenvalues less than Fermi energy μ
• Electron addition: Δ E₊ = eigenvalue of empty state - the allowed energies in the empty bands
• In a crystal Δ E₊ = ε_k,i where i denotes an empty band
• Similarly, for removal, Δ E_- = ε_k,i where i denotes a filled band
3. Electron excitation at fixed number N
1. Example: Optical properties of materials
• Absorption of photons to excite electrons
• Perturbation on system -- perturbation theory
• Described by imaginary part of response function
• Examples show strong effects of electron interactions
4. Density Functional Theory: Recall from Lecture 5 - Chapters 6-9
1. The Hohenberg-Kohn Theorems
• Exact theorems that ALL properties of electron systems are functionals of the density
• But no hint of how to accomplish this!
2. Kohn-Sham Ansatz
• Replace interacting-electron problem with a soluble non-interacting particle problem
• In principle, exact density and ground state energy of the interacting-electron system by solving the Kohn-Sham equations
• Key idea: the Exchange-Correlation Functional Exc[n]
• Remarkably good results for the ground state using simple approximations to Exc[n], e.g., LDA and various GGAs.
• No other properties are guaranteed to be given correctly
• Excitations are often found to be very bad in the simple non-interacting electron approach
5. Time Dependent Density Functional Theory - TDDFT
1. Extends the Hohenberg-Kohn Theorems
• Exact theorems that time evolution of system is fully determined by the initial state (wave function) and the time dependent density!
• But no hint of how to accomplish this!
2. Time Dependent Kohn-Sham
• Replace interacting-electron problem with a soluble non-interacting particle problem in a time dependent potential
• Time evolution of the density of Kohn-Sham system is the same as the density of the interacting system!
3. Analysis in terms of time- or frequency-dependent response functions
6. So what is the problem?
1. It is more difficult to make approximate forms for the time-dependent exchange-correlation functionals than for the static functionals!
2. Leads us to explicit many-body methods