ISSP I1 ISSPI: Time-dependent DFT Kieron Burke and friends UC Irvine Physics and Chemistry Departments

ISSP I2 Recent reviews of TDDFT To appear in Reviews of Computational Chemistry

ISSP I3 Book: TDDFT from Springer

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

ISSP I5 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

ISSP I6 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.

ISSP I7 TD quantum mechanics

ISSP I8 Current and continuity Current operator: Acting on wavefunction: Continuity:

ISSP I9 Runge-Gross theorem (1984) Any given current density, j(r,t), and initial wavefunction, statistics, and interaction, there’s only one external potential, v ext (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?)

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

ISSP I11 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

ISSP I12 Optical response in box

ISSP I13 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

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

where Density response

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

ISSP I17 TDDFT linear response Probe system with AC field of freq  Ask at what  you find a self-sustaining response That’s a transition frequency! Need a new functional, the XC kernel, f xc [  0 ](r,r’,  ) Almost always ignore  -dependence (called adiabatic approximation) Can view as corrections to KS response

ISSP I18 Eigenvalue equations Casida’s matrix formulation (1996) True transition frequencies KS transition frequencies Occupied KS orbital Unoccupied KS orbital

In this equation, f HXC is the Hartree-exchange- correlation kernel,, where f XC is the unknown XC kernel Transitions in TDDFT

ISSP I20 KS susceptibility

ISSP I21 How good the KS response is

ISSP I22 Extracting Exc

ISSP I23 Adiabatic approximation

ISSP I24 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

ISSP I25 Overview of ALL TDDFT 1. General Time-dependent Density Functional Theory Any e - system subjected to any Only unknown: Treat atoms and molecules in INTENSE laser fields 2. TDDFT linear response to weak fields Linear response: Only unknown: near ground state Treat electronic excitations in atoms + molecules + solids 3. Ground-state Energy from TDDFT Fluctuation–dissipation theorem: Exc from susceptibility Van der Waals; seamless dissociation Basic approximation: ALDA

ISSP I26 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).

ISSP I27 Green fluorescent Protein TDDFT approach for Biological Chromophores, Marques et al, Phys Rev Lett 90, (2003)

ISSP I28 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 N 2, vs N 5 for CCSD Available now in your favorite quantum chemical code

ISSP I29 TDDFT results for vertical singlet excitations in Naphthalene Elliot, Furche, KB, Reviews Comp Chem, sub. 07. Naphthalene

ISSP I30 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

ISSP I31 How good the KS response is

ISSP I32 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.

ISSP I33 Be s quantum defect: expt Top: triplet, bottom: singlet

ISSP I34 Be s quantum defect: KS

ISSP I35 Be s quantum defect: RPA KS=triplet RPA fHfH

ISSP I36 Be s quantum defect: ALDAX

ISSP I37 Be s quantum defect: ALDA

ISSP I38 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

ISSP I39 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

ISSP I40 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! f H (r-r’) =1/|r-r’|, but f xc ALDA =  (3) (r-r’) f xc unif (n(r)) As q->0, need q 2 f xc -> 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

ISSP I41 Two ways to think of solids in  fields A: Apply  sin(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.

ISSP I42 Relationship between q->0 and  ->0 Find terms of type: C/((q+ng) 2 -  2 ) For n finite, no divergence; can interchange q->0 and  ->0 limits For n=0: –if  =0 (static), have to treat q->0 carefully to cancel divergences –if doing q=0 calculation, have to do t-dependent, and take  ->0 at end

ISSP I43 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

ISSP I44 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).

ISSP I45 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

ISSP I46 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(?),…

ISSP I47 Basic problem for thermo limit Uniform gas: Uniform gas moving with velocity v:

ISSP I48 Polarization problem Polarization from current: Decompose current: where Continuity: First, longitudinal case: –Since j 0 (t) not determined by n(r,t), P is not! What can happen in 3d case (Vanderbilt picture frame)? –In TDDFT, j T (r,t) not correct in KS system –So, P s not same as P in general. –Of course, TDCDFT gets right (Maitra, Souza, KB, PRB03).

ISSP I49 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

ISSP I50 Improvements for solids: orbital-dependence Reining, Rubio, etc. Find what terms needed in f xc 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

ISSP I51 Our recent work Floquet theory Double excitations Understanding how it works – Single- and Double-pole approximations X-ray spectra Rydberg series from LDA potential Quantum defects Errors in DFT for transport TDDFT for open systems Elastic electron-atom scattering

ISSP I52 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

ISSP I53 Elastic scattering from TDDFT Huge interest in low energy scattering from biomolecules, since resonances can lead to cleavage of DNA Traditional methods cannot go beyond 13 atoms Can we use TDDFT? Yes!

ISSP I54 Simple scheme for spherical case Eg e- scattering from H. Put H- into spherical box, and consider E>0 states. Old formula due to Fano (1935): Exact for any Rb beyond potential.

ISSP I55 Is KS a good starting place?

ISSP I56 Is the LDA potential good enough?

ISSP I57 TDDFT corrections

ISSP I58 Summary TDDFT is different from DFT Linear response TDDFT turns KS orbital differences into single optical excitations Value is in semi-quantitative spectra –Can help determine geometry –Identify significant excitations Troubles with strong fields Troubles with solids Current- or orbital-dependence are promising alternatives for solids and long-chain polymers