Advanced TDDFT II. Double Excitations in TDDFT

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1 Advanced TDDFT II. Double Excitations in TDDFT f xc Neepa T. Maitra Hunter College and the Graduate Center of the City University of New York

2 First, quick recall of how we get excitations in TDDFT: Linear response (Petersilka, Gossmann & Gross (1996), Casida, (1996)) Poles at true excitations Poles at KS excitations adiabatic approx: no ω-dep Need (1) ground-state v S,0 [n 0 ](r), and its bare excitations (2) XC kernel ~ δ(t-t ) Yields exact spectra in principle; in practice, approxs needed in (1) and (2).

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

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

5 Double (Or Multiple) Excitations How do these different types of excitations appear in the TDDFT response functions? Consider: χ poles at true states that are mixtures of singles, doubles, and higher excitations χ S -- poles at single KS excitations only, since one-body operator connect Slater determinants differing by more than one orbital. can t χ has more poles than χ s? How does f xc generate more poles to get states of multiple excitation character?

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

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

8 χ -1 = χ s -1 - f Hxc

9 General case: 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 usual adiabatic matrix element dynamical (non-adiabatic) correction Maitra, Zhang, Cave,& Burke JCP (2004), Casida JCP (2005)

10 Simple Model System: 2 el. in 1d Vext = x 2 /2 Vee = λ δ(x-x ) λ = 0.2 Exact: ½ : ½ ½: ½ Exact: 1/3: 2/3 2/3: 1/3 Dressed TDDFT in SPA, f xc (ω)

11 When are states of double-excitation character important? (i) Some molecules eg short-chain polyenes Lowest-lying excitations notoriously difficult to calculate due to significant doubleexcitation character. Cave, Zhang, Maitra, Burke, CPL (2004) Mazur, Wlodarczyk, J. Comp. Chem. (2008) fuller analysis, implementation in Niedoda

12 When are states of double-excitation character important? (ii) Coupled electron-ion dynamics propensity for curve-crossing means need accurate double-excitation description for global PES Levine, Ko, Quenneville, Martinez, Mol. Phys. 104, 1039 (2006) (iii) Certain long-range charge transfer states! Stay tuned for next lecture! (iv) Electron-induced quantum chaos in quantum wells Wasserman, Maitra, Heller, PRA 77, (2008) (v) Certain autoionizing resonances Next!

13 Before doing this, as we are speaking of frequency-dependent kernels, note more generally some other recent ω-dep kernels developed in the literature (but that don t capture doubles correctly): -- exact-exchange kernel, A. Görling, (PRA,1998) -- VK kernel of TDCDFT, G. Vignale & W. Kohn (PRL,1996) -- Nanoquanta kernel, M. Gatti, V. Olevano, L. Reining, I. Tokatly, arxiv (2008) and citations therein derived from putting many-body theory into TDDFT -- See also some talks next week (Casida, Görling)

14 Autoionizing Resonances When energy of a bound excitation lies in the continuum: KS (or another orbital) picture ω ω bound, localized excitation continuum excitation True system: Electron-interaction mixes these states Fano resonance ATDDFT gets these mixtures of single-ex s

15 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.

16 a i ω = 2(ε a ε i ) ω 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-ex

17 Getting the cross-section in TDDFT Photo-absorption cross section: Where, ^ ^ How does the exact kernel add the resonant bump to χ s? Need to do a degenerate perturbation theory analysis in the continuum, diagonalizing the bound double-excitation with the continnum states. Aha! Luckily this is closely along the lines of what Fano did in 1961

18 Near a Fano resonance Fano s Universal Resonance Formula ^ U. Fano, Phys. Rev. 124, 1866 (1961) ^ ^ Take T = n(r) where ^ ^ ^ ^ ^ Fano line-shapes

19 Consider Fano into TDDFT to be Kohn Sham states Adapt Fano analysis: For near resonance In this 1 st approx, we considered coupling between douby-excited to continuum only, and took Γ as the smallest energy scale in the system.

20 To find we need Use Kramers-Kronig relation.... Assuming

21 So looks like Krueger, Mullady, Maitra (in progress)

22 Summary (so far) ATDDFT fails to describe double-excitations as strong frequencydependence is needed. Diagonalizing in the (small) subspace where double excitations mix with singles, we can derive a frequency-dependent kernel that does the job. Shown to work well for simple model systems, as well as real molecules (although more testing is needed). Likewise, in autoionization, resonances due to double-excitations are missed in ATDDFT.

23 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.

24 References: Double excitations within time-dependent density functional theory linear response, N.T. Maitra, F. Zhang, R.J. Cave, and K. Burke, J. Chem. Phys. 120, 5932 (2004) A dressed time-dependent density functional treatment of the $2^1 A_g$ States of butadiene and hexatriene, R.J. Cave, F. Zhang, N.T. Maitra, and K. Burke, Chem. Phys. Lett. 389, 39 (2004) Exact exchange-correlation kernel for dynamic response properties and excitation energies in density functional theory, A. Goerling, Phys. Rev. A 57, 3433 (1998) Current-dependent exchange-correlation potential for dynamical linear response theory, G. Vignale and W. Kohn, Phys. Rev. Lett. 77, 2037 (1996) Transforming nonlocality into frequency dependence: a shortcut to spectroscopy M. Gatti, V.Olevano, L. Reining and I. V. Tokatly, cond-mat.mtrl-sci: arxiv: v1 Propagator corrections to adiabatic time-dependent density-functional theory linear response theory, M. E. Casida, J. Chem. Phys. 122, (2005). Application of the Dressed Time-Dependent Density Functional Theory for the Excited States of Linear Polyenes, G. Mazur and R. Wlodarczyk, J. Comp. Chem. in press (2008). Conical intersections and double excitations in time-dependent density functional theory, B.G. Levine, C. Ko, J. Quenneville and T. J. Martinez, Mol. Phys. 104, 1039 (2006) Valence and core photoionization dynamics of acetylene by TD-DFT continuum approach, G. Fronzoni, M. Stener, P.Decleva, Chem. Phys. 298, 141 (2004) Effect of configuration interaction on intensities and phase shifts, U. Fano, Phys. Rev. 124, 1866 (1961)

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f xc Advanced TDDFT PDF Created with deskpdf PDF Writer - Trial :: II. Frequency-Dependent Kernels: Double Excitations Advanced TDDFT II. Frequency-Dependent Kernels: Double Excitations f xc Neepa T. Maitra Hunter College and the Graduate Center of the City University of New York First, quick recall of how we get excitations