1. Lecture 9: Advanced DFT concepts: The Exchange-correlation functional and time-dependent DFT Marie Curie Tutorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science 100 Washington Square East New York University, New York, NY 10003

2. Ψ 0 λ is the ground state wavefunction of a Hamiltonian H=T+λW+V. Note that when λ=0. V=V KS and when λ=1, V=V ext.

3. λ λ λλ

8. Where f xc can be obtained by integrating over r ’ with a Change of variables s=r-r ’.

11. Gradient Corrections Extend the LDA form to include density gradients: Example: Becke, Phys. Rev. A (1988) Functional form chosen to have the correct asymptotic behavior:

12. Motivation for TDDFT Photoexcitation processes Atomic and nuclear scattering Dynamical response of inhomogeneous metallic systems.

13. The time-dependent Hamiltonian Consider an electronic system with a Hamiltonian of the form: Where V(t) is a time-dependent one-body operator. Our interest is in the solution of the time-dependent Schrödinger equation: Let V be the set of time-dependent potentials associated and let N be the set of densities associated with time-dependent solutions of the Schrödinger equation. There exists a map G such that G : V N

14. The Hohenberg-Kohn Theorem Since V(t) is a one-body operator: Assume the potential can be expanded in a Taylor series: Suppose there are two potentials such that Then, there exists some minimum value of k such that

15. The Hohenberg-Kohn Theorem For time-dependent systems, we need to show that both the density n(r,t) and the current density j(r,t) are different for the two different potentials, where the continuity equation is satisfied: For any operator O(t), we can show that:

16. The Hohenberg-Kohn Theorem From equation of motion, we can show that And, in general, for the minimal value of k alluded to above: Hence, even if j and j’ are different initially, they will differ for times just later than t 0.

17. The Hohenberg-Kohn Theorem For the density, since It follows that: Therefore, even if n and n’ are initially the same, they will differ for times just later than t 0. Hence, any observable can be written as a functional of n and a function of t.

18. Actions in quantum mechanics and DFT Consider the action integral: Schrödinger equation results requiring that the action be stationary according to: Hence, if we view A as a functional of the density,

19. Hohenberg-Kohn and KS schemes Hohenberg-Kohn: Kohn-Sham formulation: Introduce a non-interacting system with effective potential V KS (r,t) that gives the same time-dependent density as the interacting system. For a non-interacting system, introduce single-particle orbitals ψ i (r, t ) such that the density is given by KS action:

20. Time-dependent Kohn-Sham equations From : Adiabatic LDA/GGA:

21. Linear response solution for the density Strategy: Solve the Liouville equation for the density matrix to linear order. Quantum Liouville equation for the density operator ρ(t): Time-dependent density:

22. Linear response solution for the density Write the density operator as: To linear order, we have Solution:

23. Linear response solution for the density To linear order: Where the Fourier transform of the response kernel is: Hence, poles of the response kernel are the electronic excitation energies.

24. from Appel, Gross and Burke, PRL 93, 043005 (2003).

25. Lecture Summary Adiabatic connection formula provides a rigorous theory of the exchange-correlation functional and is the starting point of many approximations. Generalization of density functional theory to time-dependent systems is possible through generalization of the Hohenberg-Kohn theorem. In linear response theory, the response kernel (or its poles) is the object of interest as it yields the excitation energies.