1. Advanced TDDFT II Neepa T. Maitra Hunter College and the Graduate Center of the City University of New York Memory-Dependence in Linear Response a. Double Excitations b. Charge Transfer Excitations f xc

2. Poles at KS excitations Poles at true excitations Need (1) ground-state v S,0 [n 0 ](r), and its bare excitations (2) XC kernel Yields exact spectra in principle; in practice, approxs needed in (1) and (2). adiabatic approx: no  -dep ~  (t-t’) First, quick recall of how we get excitations in TDDFT: Linear response Petersilka, Gossmann & Gross, PRL 76, 1212 (1996) Casida, in Recent Advances in Comput. Chem. 1,155, ed. Chong (1995) n0n0

3. Well-separated single excitations: SMA When shift from bare KS small: SPA Useful tool for analysis Zoom in on a single KS excitation, q = i  a TDDFT linear response in quantum chemistry codes: q =(i  a) labels a single excitation of the KS system, with transition frequency  q =  a -  i, and Eigenvalues  true frequencies of interacting system Eigenvectors  oscillator strengths

4. Interacting systems: generally involve mixtures of (KS) SSD’s that may have 1,2,3…electrons in excited orbitals. single-, double-, triple- excitations Non-interacting systems eg. 4-electron atom Eg. single excitations near-degenerate Eg. double excitations Types of Excitations

5. Double (Or Multiple) Excitations  – poles at true states that are mixtures of singles, doubles, and higher excitations  S -- poles at single KS excitations only, since one-body operator can’t connect Slater determinants differing by more than one orbital.    has more poles than  s ? How does f xc generate more poles to get states of multiple excitation character? Consider: How do these different types of excitations appear in the TDDFT response functions?

6. Exactly solve one KS single (q) mixing with a nearby double (D) Simplest Model:

7. This kernel matrix element, by construction, yields the exact true  ’s when used in the Dressed SPA, strong non- adiabaticity! Invert and insert into Dyson-like eqn for kernel  dressed SPA (i.e.  - dependent): adiabatic

8.  -1 =  s -1 - f Hxc

9. An Exercise! Deduce something about the frequency-dependence required for capturing states of triple excitation character – say, one triple excitation coupled to a single excitation.

10. Diagonalize many-body H in KS subspace near the double-ex of interest, and require reduction to adiabatic TDDFT in the limit of weak coupling of the single to the double: N.T. Maitra, F. Zhang, R. Cave, & K. Burke JCP 120, 5932 (2004) usual adiabatic matrix element dynamical (non-adiabatic) correction Practical Approximation for the Dressed Kernel So: (i) scan KS orbital energies to see if a double lies near a single, (ii)apply this kernel just to that pair (iii)apply usual ATDDFT to all other excitations

11. Alternate Derivations  M.E. Casida, JCP 122, 054111 (2005) M. Huix-Rotllant & M.E. Casida, arXiv: 1008.1478v1 -- from second-order polarization propagator (SOPPA) correction to ATDDFT  P. Romaniello, D. Sangalli, J. A. Berger, F. Sottile, L. G. Molinari, L. Reining, and G. Onida, JCP 130, 044108 (2009) -- from Bethe-Salpeter equation with dynamically screened interaction W(  )  O. Gritsenko & E.J. Baerends, PCCP 11, 4640, (2009). -- use CEDA (Common Energy Denominator Approximation) to account for the effect of the other states on the inverse kernels, and obtain spatial dependence of f xc -kernel as well.

12. Simple Model System: 2 el. in 1d Vext = x 2 /2 Vee =  (x-x’) = 0.2 Dressed TDDFT in SPA, f xc (  ) Exact: 1/3: 2/3 2/3: 1/3 Exact: ½ : ½ ½: ½

13. (i) Some molecules eg short-chain polyenes Lowest-lying excitations notoriously difficult to calculate due to significant double- excitation character. When are states of double-excitation character important? R. Cave, F. Zhang, N.T. Maitra, K. Burke, CPL 389, 39 (2004); G. Mazur, R. Wlodarczyk, J. Comp. Chem. 30, 811, (2008); Mazur, G., M. Makowski, R. Wlodarcyk, Y. Aoki, IJQC 111, 819 (2010);  Grzegorz Mazur talk next week M. Huix-Rotllant, A. Ipatov, A. Rubio, M. E. Casida, Chem. Phys. (2011) – extensive testing on 28 organic molecules, discussion of what’s best for adiabatic part… Other implementations and tests:

14. (ii) Coupled electron-ion dynamics - propensity for curve-crossing means need accurate double-excitation description for global potential energy surfaces Levine, Ko, Quenneville, Martinez, Mol. Phys. 104, 1039 (2006) (iv) Near conical intersections - near-degeneracy with ground-state (static correlation) gives double-excitation character to all excitations (iii) Certain long-range charge transfer states! Stay tuned! When are states of double-excitation character important? (v) Certain autoionizing resonances …

15. Autoionizing Resonances When energy of a bound excitation lies in the continuum: bound, localized excitationcontinuum excitation   Electron-interaction mixes these states  Fano resonance KS (or another orbital) picture ATDDFT gets these – mixtures of single-ex’s True system: M. Hellgren & U. van Barth, JCP 131, 044110 (2009)  Fano parameters directly implied by Adiabatic TDDFT (Also note Wasserman & Moiseyev, PRL 98,093003 (2007), Whitenack & Wasserman, PRL 107,163002 (2011) -- complex-scaled DFT for lowest-energy resonance )

16. Auto-ionizing Resonances in TDDFT Eg. Acetylene: G. Fronzoni, M. Stener, P. Decleva, Chem. Phys. 298, 141 (2004) But here’s a resonance that ATDDFT misses: Why? It is due to a double excitation.

17. bound, localized double excitation with energy in the continuum single excitation to continuum Electron-interaction mixes these states  Fano resonance  ATDDFT does not get these – double-excitation  a  i  a i e.g. the lowest double-excitation in the He atom (1s 2  2s 2 ) A. Krueger & N. T. Maitra, PCCP 11, 4655 (2009); P. Elliott, S. Goldson, C. Canahui, N. T. Maitra, Chem. Phys. 135, 104110 (2011).

18. ATDDFT fundamentally fails to describe double-excitations: strong frequency-dependence is essential. Diagonalizing in the (small) subspace where double excitations mix with singles, we can derive a practical frequency-dependent kernel that does the job. Shown to work well for simple model systems, as well as real molecules. Likewise, in autoionization, resonances due to double-excitations are missed in ATDDFT. Summary on Doubles Next: Long-Range Charge-Transfer Excitations

19. Long-Range Charge-Transfer Excitations Notorious problem for standard functionals Recently developed functionals for CT Simple model system - molecular dissociation: ground-state potential - undoing static correlation Exact form for fxc near CT states

20. Eg. Zincbacteriochlorin-Bacteriochlorin complex (light-harvesting in plants and purple bacteria) Dreuw & Head-Gordon, JACS 126 4007, (2004). TDDFT predicts CT states energetically well below local fluorescing states. Predicts CT quenching of the fluorescence. ! Not observed ! TDDFT error ~ 1.4eV TDDFT typically severely underestimates Long-Range CT energies But also note : excited state properties (eg vibrational freqs) might be quite ok even if absolute energies are off (eg DMABN, Rappoport and Furche, JACS 2005)

21. e First, we know what the exact energy for charge transfer at long range should be: Now to analyse TDDFT, use single-pole approximation (SPA): Why usual TDDFT approx’s fail for long-range CT: -A s,2 -I1-I1 Ionization energy of donor Electron affinity of acceptor Dreuw, J. Weisman, and M. Head-Gordon, JCP 119, 2943 (2003) Tozer, JCP 119, 12697 (2003) Also, usual ground-state approximations underestimate I i.e. get just the bare KS orbital energy difference: missing xc contribution to acceptor’s electron affinity, A xc,2, and -1/R

22. E.g. Tawada, Tsuneda, S. Yanagisawa, T. Yanai, & K. Hirao, J. Chem. Phys. (2004): “Range-separated hybrid” with empirical parameter  Short-ranged, use GGA for exchange Long-ranged, use Hartree-Fock exchange (gives -1/R) E.g. Optimally-tuned range-separated hybrid choose  system-dependent, chosen non-empirically to give closest fit of donor’s HOMO to it’s ionization energy, and acceptor anion’s HOMO to it’s ionization energy., i.e. minimize Stein, Kronik, and Baer, JACS 131, 2818 (2009); Baer, Livshitz, Salzner, Annu. Rev. Phys. Chem. 61, 85 (2010) Gives reliable, robust results. Some issues, e,g. size-consistency Karolweski, Kronik, Kűmmel, JCP 138, 204115 (2013) Functional Development for CT… Correlation treated with GGA, no splitting

23. …Functional Development for CT: Others, are not, e.g. Heßelmann, Ipatov, Görling, PRA 80, 012507 (2009) – using exact-exchange (EXX) kernel. E.g. Gritsenko & Baerends JCP 121, 655, (2004) – model asymptotic kernel to get closed—closed CT correct, switches on when donor-acceptor overlap becomes smaller than a chosen parameter E.g. Hellgren & Gross, PRA 85, 022514 (2012): exact f xc has a  -dep. discontinuity as a function of # electrons; related to a  -dep. spatial step in f xc whose size grows exponentially with separation (latter demonstrated with EXX) E.g. Many others…some extremely empirical, like Zhao & Truhlar (2006) M06-HF – empirical functional with 35 parameters!!!. E.g. Maitra JCP 122, 234104 (2005) – form of exact kernel for open-shell---open- shell CT What has been found out about the exact behavior of the kernel?

24. Let´s look at the simplest model of CT in a molecule  try to deduce the exact f xc to understand what´s needed in the approximations. 2 electrons in 1D

25. Simple Model of a Diatomic Molecule Model a hetero-atomic diatomic molecule composed of open-shell fragments (eg. LiH) with two “one-electron atoms” in 1D: “softening parameters” (choose to reproduce eg. IP’s of different real atoms…) Can simply solve exactly numerically  (r 1,r 2 )  extract  (r)   exact First: find exact gs KS potential (   s )

26. Molecular Dissociation (1d “LiH”) “Peak” and “Step” structures. (step goes back down at large R) Vext Vs n Vext x

27. R=10 peak step asymptotic x V Hxc J.P. Perdew, in Density Functional Methods in Physics, ed. R.M. Dreizler and J. da Providencia (Plenum, NY, 1985), p. 265. C-O Almbladh and U. von Barth, PRB. 31, 3231, (1985) O. V. Gritsenko & E.J. Baerends, PRA 54, 1957 (1996) O.V.Gritsenko & E.J. Baerends, Theor.Chem. Acc. 96 44 (1997). D. G. Tempel, T. J. Martinez, N.T. Maitra, J. Chem. Th. Comp. 5, 770 (2009) & citations within. N. Helbig, I. Tokatly, A. Rubio, JCP 131, 224105 (2009).

28. n(r) v s (r) Step has size  I and aligns the atomic HOMOs Prevents dissociation to unphysical fractional charges. II II bond midpoint peak step, size  I “Li” “H” v H xc at R=10 peak step LDA/GGA – wrong, because no step! asymptotic Vext The Step At which separation is the step onset? Step marks location and sharpness of avoided crossing between ground and lowest CT state..

29. A Useful Exercise! To deduce the step in the potential in the bonding region between two open-shell fragments at large separation: Take a model molecule consisting of two different “one-electron atoms” (1 and 2) at large separation. The KS ground-state is the doubly-occupied bonding orbital: where    r) and n(r ) =  1 2 (r) +  2 2 (r) is the sum of the atomic densities. The KS eigenvalue  0 must =   =  I  where I 1 is the smaller ionization potential of the two atoms. Consider now the KS equation for r near atom 1, where and again for r near atom 2, where Noting that the KS equation must reduce to the respective atomic KS equations in these regions, show that v s, must have a step of size       = I 2 –I 1 between the atoms.

30. So far for our model: Discussed step and peak structures in the ground-state potential of a dissociating molecule : hard to model, spatially non-local Fundamentally, these arise due to the single-Slater-determinant description of KS (one doubly-occupied orbital) – the true wavefunction, requires minimally 2 determinants (Heitler-London form) In practise, could treat ground-state by spin-symmetry breaking  good ground-state energies but wrong spin-densities See Dreissigacker & Lein, Chem. Phys. (2011) - clever way to get good DFT potentials from inverting spin-dft Next: What are the consequences of the peak and step beyond the ground state? Response and Excitations

31. What about TDDFT excitations of the dissociating molecule? Recall the KS excitations are the starting point; these then get corrected via f xc to the true ones. LUMO HOMO  ~ e -cR Near-degenerate in KS energy “Li”“H” Step  KS molecular HOMO and LUMO delocalized and near-degenerate But the true excitations are not! Find: The step induces dramatic structure in the exact TDDFT kernel ! Implications for long-range charge-transfer. Static correlation induced by the step!

32. e First, we know what the exact energy for charge transfer at long range should be: Now to analyse TDDFT, use single-pole approximation (SPA): Recall, why usual TDDFT approx’s fail for long-range CT: -A s,2 -I1-I1 Ionization energy of donor Electron affinity of acceptor Dreuw, J. Weisman, and M. Head-Gordon, JCP 119, 2943 (2003) Tozer, JCP 119, 12697 (2003) Also, usual ground-state approximations underestimate I i.e. get just the bare KS orbital energy difference: missing xc contribution to acceptor’s electron affinity, A xc,2, and -1/R

33. Important difference between (closed-shell) molecules composed of (i)open-shell fragments, and (ii)those composed of closed-shell fragments. HOMO delocalized over both fragments HOMO localized on one or other  Revisit the previous analysis of CT problem for open-shell fragments: Eg. apply SMA (or SPA) to HOMO  LUMO transition But this is now zero ! q = bonding  antibonding Now no longer zero – substantial overlap on both atoms. But still wrong. Wait!! !! We just saw that for dissociating LiH-type molecules, the HOMO and LUMO are delocalized over both Li and H  f xc contribution will not be zero!

34. Undoing KS static correlation… These three KS states are nearly degenerate:  0 LUMO  0 HOMO in this basis to get: The electron-electron interaction splits the degeneracy: Diagonalize true H atomic orbital on atom2 or 1 Heitler-London gs CT states where  ~ e -cR “Li”“H” Extract the xc kernel from:

35. What does the exact fxc looks like? KS density-density response function: Interacting response function: Finite overlap between occ. (bonding) and unocc. (antibonding) Vanishes with separation as e -R Extract the xc kernel from: Vanishing overlap between interacting wavefn on donor and acceptor Finite CT frequencies only single excitations contribute to this sum Diagonalization is (thankfully) NOT TDDFT! Rather, mixing of excitations is done via the f xc kernel...recall double excitations lecture…

36. Exact matrix elt for CT between open-shells Maitra JCP 122, 234104 (2005) ………… Note: strong non-adiabaticity! Interacting CT transition from 2 to 1, (eg in the approx found earlier) KS antibonding transition freq, goes like e -cR  0  0 - nonzero overlap _         Upshot: (i) f xc blows up exponentially with R, fxc ~ exp(cR) (ii) f xc strongly frequency-dependent Within the dressed SMA the exact f xc is:…

37. How about higher excitations of the stretched molecule? Since antibonding KS state is near-degenerate with ground, any single excitation  0   a is near-generate with double excitation (  0   a,  0   a ) Ubiquitous doubles – ubiquitous poles in f xc (  ) Complicated form for kernel for accurate excited molecular dissociation curves Even for local excitations, need strong frequency-dependence. N. T. Maitra and D. G. Tempel, J. Chem. Phys. 125 184111 (2006).

38. Long-range CT excitations are particularly challenging for TDDFT approximations to model, due to vanishing overlap between the occupied and unoccupied states; optimism with non-empirically tuned hybrids Require exponential dependence of the kernel on fragment separation for frequencies near the CT ones (in non-hybrid TDDFT) Strong frequency-dependence in the exact xc kernel enables it to accurately capture long-range CT excitations Origin of complicated  -structure of kernel is the step in the ground- state potential – making the bare KS description a poor one. Static correlation. Static correlation problems also in conical intersections. What about fully non-linear time-resolved CT ?? Non-adiabatic TD steps important in all cases Fuks, Elliott, Rubio, Maitra J. Phys. Chem. Lett. 4, 735 (2013) Summary of CT