ISSPI: Time-dependent DFT

1. ISSPI: Time-dependent DFT Kieron Burke and friends UC Irvine Physics and Chemistry Departments http://dft.uci.edu

2. Recent reviews of TDDFT To appear in Reviews of Computational Chemistry

3. Book: TDDFT from Springer

4. TDDFT publications in recent years Search ISI web of Science for topic ‘TDDFT’ • Warning! By 2300, entire mass of universe will be TTDFT papers

5. Road map • TD quantum mechanics->TDDFT • Linear response • Overview of all TDDFT • Does TDDFT really work? • Complications for solids • Currents versus densities • Elastic scattering from TDDFT

6. Basic points • TDDFT • is an addition to DFT, using a different theorem • allows you to convert your KS orbitals into optical excitations of the system • for excitations usually uses ground-state approximations that usually work OK • has not been very useful for strong laser fields • is in its expansion phase: Being extended to whole new areas, not much known about functionals • with present approximations has problems for solids • with currents is more powerful, but harder to follow • yields a new expensive way to get ground-state Exc.

7. TD quantum mechanics

8. Current and continuity • Current operator: • Acting on wavefunction: • Continuity:

9. Runge-Gross theorem (1984) • Any given current density, j(r,t), and initial wavefunction, statistics, and interaction, there’s only one external potential, vext(r,t), that can produce it. • Imposing a surface condition and using continuity, find also true for n(r,t). • Action in RG paper is WRONG • von Leeuwen gave a constructive proof (PRL98?)

10. TD Kohn-Sham equations • Time-dependent KS equations: • Density: • XC potential: Depends on entire history(MEMORY) initial state(s) dependence(MEMORY)

11. Road map • TD quantum mechanics->TDDFT • Linear response • Overview of all TDDFT • Does TDDFT really work? • Complications for solids • Currents versus densities • Elastic scattering from TDDFT

12. Optical response in box

13. Excitations from DFT • Many approaches to excitations in DFT • There is no HK theorem from excited-state density (PRL with Rene Gaudoin) • Would rather have variational approach (ensembles, constrained search, etc.) • TDDFT yields a response approach, i.e, looks at TD perturbations around ground-state

14. In time-dependent external field TDDFT linear response For a given interaction and statistics: HS: KS: RG: KS:

15. Density response where

16. Dyson-like equation Key quantity is susceptibility Dyson-like equation for a susceptibility: Two inputs: KS susceptibility and XC kernel

17. TDDFT linear response • Probe system with AC field of freq w • Ask at what w you find a self-sustaining response • That’s a transition frequency! • Need a new functional, the XC kernel, fxc[r0](r,r’,w) • Almost always ignore w-dependence (called adiabatic approximation) • Can view as corrections to KS response

18. Eigenvalue equations Casida’s matrix formulation (1996) True transition frequencies KS transition frequencies Unoccupied KS orbital Occupied KS orbital

19. Transitions in TDDFT In this equation, fHXC is the Hartree-exchange-correlation kernel, , where fXC is the unknown XC kernel

20. KS susceptibility

21. How good the KS response is

22. Extracting Exc

23. Adiabatic approximation

24. Road map • TD quantum mechanics->TDDFT • Linear response • Overview of all TDDFT • Does TDDFT really work? • Complications for solids • Currents versus densities • Elastic scattering from TDDFT

25. Any e- system subjected to any • Only unknown: • Treat atoms and molecules in INTENSE laser fields • Linear response: • Only unknown: near ground state • Treat electronic excitations in atoms + molecules + solids Basic approximation: ALDA Overview of ALL TDDFT 1. General Time-dependent Density Functional Theory 2. TDDFT linear response to weak fields 3. Ground-state Energy from TDDFT • Fluctuation–dissipation theorem: Exc from susceptibility • Van der Waals; seamless dissociation

26. Methodology for TDDFT • In general: Propagate TDKS equations forward in time, and then transform the dipole moment, eg. Octopus code • Linear response: Convert problem of finding transitions to eigenvalue problem (Casida, 1996).

27. Green fluorescent Protein TDDFT approach for Biological Chromophores, Marques et al, Phys Rev Lett 90, 258101 (2003)

28. Success of TDDFT for excited states • Energies to within about 0.4 eV • Bonds to within about 1% • Dipoles good to about 5% • Vibrational frequencies good to 5% • Cost scales as N2, vs N5 for CCSD • Available now in your favorite quantum chemical code

29. Naphthalene TDDFT results for vertical singlet excitations in Naphthalene Elliot, Furche, KB, Reviews Comp Chem, sub. 07.

30. Road map • TD quantum mechanics->TDDFT • Linear response • Overview of all TDDFT • Does TDDFT really work? • Complications for solids • Currents versus densities • Elastic scattering from TDDFT

31. How good the KS response is

32. Quantum defect of Rydberg series • I=ionization potential, n=principal, l=angular quantum no.s • Due to long-ranged Coulomb potential • Effective one-electron potential decays as -1/r. • Absurdly precise test of excitation theory, and very difficult to get right.

33. Be s quantum defect: expt Top: triplet, bottom: singlet

34. Be s quantum defect: KS

35. Be s quantum defect: RPA KS=triplet fH RPA

36. Be s quantum defect: ALDAX

37. Be s quantum defect: ALDA

38. General notes • Most papers are lin resp, looking at excitations: need gs potential, plus kernel • Rydberg excitations can be bad due to poor potentials (then use OEP, or be clever!). • Simple generalization to current TDDFT • Charge transfer fails, because little oscillator strength in KS response. • Double excitations lost in adiabatic approximation (but we can put them back in by hand) • Typically not useful in strong fields • Exc schemes still under development

39. Road map • TD quantum mechanics->TDDFT • Linear response • Overview of all TDDFT • Does TDDFT really work? • Complications for solids • Currents versus densities • Elastic scattering from TDDFT

40. Complications for solids and long-chain polymers • Locality of XC approximations implies no corrections to (g=0,g’=0) RPA matrix element in thermodynamic limit! • fH (r-r’) =1/|r-r’|, but fxcALDA = d(3)(r-r’) fxcunif(n(r)) • As q->0, need q2 fxc -> constant to get effects. • Consequences for solids with periodic boundary conditions: • Polarization problem in static limit • Optical response: • Don’t get much correction to RPA, missing excitons • To get optical gap right, because we expect fxc to shift all lowest excitations upwards, it must have a branch cut in w starting at EgKS

41. Two ways to think of solids in E fields • A: Apply Esin(qx), and take q->0 • Keeps everything static • Needs great care to take q->0 limit • B: Turn on TD vector potential A(t) • Retains period of unit cell • Need TD current DFT, take w->0.

42. Relationship between q->0 and w->0 • Find terms of type: C/((q+ng)2-w2) • For n finite, no divergence; can interchange q->0 and w->0 limits • For n=0: • if w=0 (static), have to treat q->0 carefully to cancel divergences • if doing q=0 calculation, have to do t-dependent, and take w->0 at end

43. Road map • TD quantum mechanics->TDDFT • Linear response • Overview of all TDDFT • Does TDDFT really work? • Complications for solids • Currents versus densities • Elastic scattering from TDDFT

44. TD current DFT • RG theorem I actually proves functional of j(r,t). • Easily generalized to magnetic fields • Naturally avoids Dobson’s dilemma: Gross-Kohn approximation violates Kohn’s theorem. • Gradient expansion exists, called Vignale-Kohn (VK). • TDDFT is a special case • Gives tensor fxc, simply related to scalar fxc (but only for purely longitudinal case).

45. Currents versus densities • Origin of current formalism: Gross-Kohn approximation violates Kohn’s theorem. • Equations much simpler with n(r,t). • But, j(r,t) more general, and can have B-fields. • No gradient expansion in n(rt). • n(r,t) has problems with periodic boundary conditions – complications for solids, long-chain conjugated polymers

46. Beyond explicit density functionals • Current-density functionals • VK Vignale-Kohn (96): Gradient expansion in current • Various attempts to generalize to strong fields • But is just gradient expansion, so rarely quantitatively accurate • Orbital-dependent functionals • Build in exact exchange, good potentials, no self-interaction error, improved gaps(?),…

47. Basic problem for thermo limit • Uniform gas: • Uniform gas moving with velocity v:

48. Polarization problem • Polarization from current: • Decompose current: where • Continuity: • First, longitudinal case: • Since j0(t) not determined by n(r,t), P is not! • What can happen in 3d case (Vanderbilt picture frame)? • In TDDFT, jT (r,t) not correct in KS system • So, Ps not same as P in general. • Of course, TDCDFT gets right (Maitra, Souza, KB, PRB03).

49. Improvements for solids: currents • Current-dependence: Snijders, de Boeij, et al – improved optical response (excitons) via ‘adjusted’ VK • Also yields improved polarizabilities of long chain conjugated polymers. • But VK not good for finite systems

50. Improvements for solids: orbital-dependence • Reining, Rubio, etc. • Find what terms needed in fxc to reproduce Bethe-Salpeter results. • Reproduces optical response accurately, especially excitons, but not a general functional. • In practice, folks use GW susceptibility as starting point, so don’t need effective fxc to have branch cut