Report

Tutorial: Time-dependent density-functional theory Carsten A. Ullrich University of Missouri XXXVI National Meeting on Condensed Matter Physics Aguas de Lindoia, SP, Brazil May 13, 2013 Outline 2 PART I: ● The many-body problem ● Review of static DFT PART II: ● Formal framework of TDDFT ● Time-dependent Kohn-Sham formalism PART III: ● TDDFT in the linear-response regime ● Calculation of excitation energies Optical spectroscopy 3 ● Uses weak laser as Probe Photoabsorption cross section ● System Response has peaks at electronic excitation energies Na2 Green fluorescent protein Na4 Theory Energy (eV) Vasiliev et al., PRB 65, 115416 (2002) Marques et al., PRL 90, 258101 (2003) Linear response 4 tickle the system observe how the system responds at a later time (r, t ) (r, t ) The formal framework to describe the behavior of a system under weak perturbations is called Linear Response Theory. 5 Linear response theory (I) Consider a quantum mechanical observable ground-state expectation value ˆ with 0 0 ˆ 0 Time-dependent perturbation: Hˆ 1 (t ) F (t )ˆ , The expectation value of the observable time-dependent: ˆ t t0 now becomes (t ) (t ) ˆ (t ) , t t0 The response of the system can be expanded in powers of the field F(t): (t ) 0 1 (t ) 2 (t ) 3 (t ) ... = linear + quadratic + third-order + ... 6 Linear response theory (II) For us, the density-density response will be most important. The perturbation is a scalar potential V1, Hˆ 1 (t ) d 3 r V1 (r, t )nˆ (r) N where the density operator is nˆ (r) (r rl ) l 1 n(r, t ) (t ) nˆ (r) (t ) The linear density response is n1 (r, t ) dt d 3r nn (r, r, t t ) V1 (r, t ) nn (r, r, t t ) i (t t ) 0 [nˆ (r, t t ), nˆ (r)] 0 Linear response theory (III) 7 Fourier transformation with respect to (t t ) gives: n1 (r, ) d 3 r nn (r, r, ) V1 (r, ) 0 nˆ (r) n n nˆ (r) 0 0 nˆ (r) n n nˆ (r) 0 nn (r, r, ) i i n 1 n n where n En E0 is the nth excitation energy. ►The linear response function has poles at the excitation energies of the system. ► Whenever there is a perturbation at such a frequency, the response will diverge (peak in the spectrum) Spectroscopic observables 8 First-order induced dipole polarization: p1 () α()E() In dipole approximation, one defines a scalar potential associated with a monochromatic electric field, linearly polarized along the z direction: V1 (r, t ) E z sin(t ) This gives p1, z ( ) d 3 r z n1 (r, ) And the dynamic dipole polarization becomes 2 3 zz ( ) d r z n1 (r, ) E d 3 r d 3 r zz (r, r, ) From this we obtain the photoabsorption cross section: zz ( ) 4 Im zz ( ) c 9 Spectrum of a cyclometallated complex F. De Angelis, L. Belpassi, S. Fantacci, J. Mol. Struct. THEOCHEM 914, 74 (2009) TDDFT for linear response 10 Gross and Kohn, 1985: n1 (r, t ) d 3 r dt (r, t , r, t ) V1 (r, t ) d 3 r dt s (r, t , r, t ) V1, s (r, t ) Exact density response can be calculated as the response of a noninteracting system to an effective perturbation: (t t ) V1, s (r, t ) V1 (r, t ) dt d r f xc (r, t , r, t ) n1 (r, t ) r r 3 xc kernel: Vxc [n](r, t ) f xc (r, t , r, t ) n(r, t ) n (r ) 0 Frequency-dependent linear response 11 n1 (r, ) d 3 r r, r, V1 r, d r s r, r, V1, s r, 3 many-body response function: r, r, m noninteracting response s function: 0 nˆr m m nˆr 0 Em E0 i c.c.( ) exact excitations Ω k* (r ) j (r ) *j (r) k (r) r, r, f k f j ( j k ) i j ,k KS excitations ωKS 1 V1, s V1 d r f xc r, r, n1 r, r r 3 12 The xc kernel: approximations 1 f xc (r, r, ) (r, r, ) (r, r, ) | r r | 1 s 1 Formally, the xc kernel is frequency-dependent and complex. Random Phase Approximation (RPA): Adiabatic approximation: frequency-independent and real f xcRPA 0 g .s. 2 V [ n ]( r ) Exc[n] adia xc f xc (r, r) n(r) n(r)n(r) 2 unif d exc (n) ALDA (r, r ) Adiabatic LDA (ALDA): f xc dn2 nn 0 (r ) (r r) 13 Analogy: molecular vibrations Molecular vibrations are characterized by the eigenmodes of the system. Using classical mechanics we can find the eigenmodes and their frequencies by solving an equation of the form A 2 jk r m jk jr 0 j dynamical coupling matrix (contains the spring constants) frequency of rth eigenmode eigenvector gives the mode profile mass tensor So, to find the molecular eigenmodes and vibrational frequencies we have to solve an eigenvalue equation whose size depends on the size of the molecule. Electronic excitations 14 An electronic excitation can be viewed as an eigenmode of the electronic many-body system. This means that the electronic density of the system (atom, molecule, or solid) can carry out oscillations, at certain special frequencies, which are self-sustained, and do not need any external driving force. n1 (r, ) d r (r, r, ) V1 (r, ) 3 finite density response: the eigenmode of an excitation diverges when ω equals one of the excitation energies set to zero (no extenal perturbation) Electronic excitations with TDDFT 15 1 n1 (r, ) d r s r, r, d r f xc (r, r, ) n1 (r, ) | r r | 3 3 Find those frequencies ω where the response equation, without external perturbation, has a solution with finite n1. M. Petersilka, U.J. Gossmann, E.K.U. Gross, PRL 76, 1212 (1996) H. Appel, E.K.U. Gross, K. Burke, PRL 90, 043005 (2003) We define the following abbreviation: 1 f Hxc (r, r, ) f xc (r, r, ) | r r | 16 Warm-up exercise: 2-level system Noninteracting response function, where jk j k : k* (r ) j (r ) *j (r) k (r) s r, r, f k f j jk i j ,k We consider the case of a system with 2 real orbitals, the first one occupied and the second one empty. Then, 1 (r ) 2 (r ) 2 (r)1 (r) 1 (r ) 2 (r ) 2 (r)1 (r) s r, r, 21 21 221 1 (r ) 2 (r ) 2 (r)1 (r) 2 212 2-level system 17 n1 (r, ) d 3r s r, r, d 3r f Hxc (r, r, )n1 (r, ) 221 3 3 2 d r ( r ) ( r ) ( r ) ( r ) d r f Hxc (r, r, )n1 (r, ) 1 2 1 2 2 21 Multiply both sides with 1 (r)2 (r) f Hxc (r, r, ) and integrate over r. Then we can cancel terms left and right, and 221 3 3 1 2 d r d r 1 (r)2 (r) f Hxc (r, r, )1 (r)2 (r) 2 21 2 212 221 d 3r d 3r 1 (r ) 2 (r ) f Hxc (r, r, )1 (r) 2 (r) 221 12 f Hxc 12 2 2 21 TDDFT correction to Kohn-Sham excitation The Casida formalism for excitation energies 18 Excitation energies follow from eigenvalue problem (Casida 1995): A * K K X 1 0 X * A Y 0 1 Y Aia ,ia ii aa a i K ia ,ia K ia ,ia 1 d r d r r a r f xc, r, r, i r a r r r 3 3 * i For real orbitals and frequency-independent xc kernel, can rewrite this as ia ii 2 2 2 K Z Zia aa ai ai ai ia ,ia ia 19 The Casida formalism for excitation energies The Casida formalism gives, in principle, the exact excitation energies and oscillator strengths. In practice, three approximations are required: ► KS ground state with approximate xc potential ► The inifinite-dimensional matrix needs to be truncated ► Approximate xc kernel (usually adiabatic): f adia xc V (r ) r, r n(r) stat xc advantage: can use any xc functional from static DFT (“plug and play”) disadvantage: no frequency dependence, no memory → missing physics (see later) 20 How it works: atomic excitation energies LDA + ALDA lowest excitations Exp. full matrix SMA Vasiliev, Ogut, Chelikowsky, PRL 82, 1919 (1999) SPA 21 A comparison of xc functionals Study of various functionals over a set of ~ 500 organic compounds, 700 excited singlet states D. Jacquemin et al., J. Chem. Theor. Comput. 5, 2420 (2009) Mean Absolute Error (eV) 22 Excited states with TDDFT: general trends Energies typically accurate within 0.3-0.4 eV Bonds to within about 1% Dipoles good to about 5% Vibrational frequencies good to 5% Cost scales as N2-N3, vs N5 for wavefunction methods of comparable accuracy (eg CCSD, CASSCF) Available now in many electronic structure codes challenges/open issues: ● complex excitations (multiple, charge-transfer) ● optical response/excitons in bulk insulators Single versus double excitations 23 1 n1 (r, ) d r s r, r, d r f xc (r, r, ) n1 (r, ) | r r | 3 Has poles at KS single excitations. The exact response function has more poles (single, double and multiple excitations). 3 Gives dynamical corrections to the KS excitation spectrum. Shifts the single KS poles to the correct positions, and creates new poles at double and multiple excitations. ► Adiabatic approximation (fxc does not depend on ω): only single excitations! ► ω-dependence of fxc will generate additional solutions of the Casida equations, which corresponds to double/multiple excitations. ► Unfortunately, nonadiabatic approximations are not easy to find. 24 Charge-transfer excitations ZincbacteriochlorinBacteriochlorin complex (light-harvesting in plants and purple bacteria) TDDFT error: 1.4 eV TDDFT predicts CT states energetically well below local fluorescing states. Predicts CT quenching of the fluorescence. Not observed! Dreuw and Head-Gordon, JACS (2004) Charge-transfer excitations: large separation 25 exact ct 1 I d Aa R (ionization potential of donor minus electron affinity of acceptor plus Coulomb energy of the charged fragments) What do we get in TDDFT? Let’s try the single-pole approximation: ctSPA La Hd 2 d 2 r d 3r La (r) Hd (r) f Hxc (r, r, ) La (r) Hd (r) The highest occupied orbital of the donor and the lowest unoccupied orbital of the acceptor have exponentially vanishing overlap! TDDFT ct a L d H For all (semi)local xc approximations, TDDFT significantly underestimates charge-transfer energies! 26 Charge-transfer excitations: exchange a , HF d , HF 2 3 TDHF d r d ct L H r a , HF L d , HF H 1 R La (r ) La (r ) Hd (r) Hd (r) | r r | and use Koopmans theorem! TDHF reproduces charge-transfer energies correctly. Therefore, hybrid functionals (such as B3LYP) will give some improvement over LDA and GGA. Even better are the so-called range-separated hybrids: 1 f ( | r r |) 1 f ( | r r |) | r r | | r r | | r r | SR GGA x Exc E LR HF x E f (x) e x , f (x) erfc(x) E GGA c Excitations in finite and extended systems 27 0 nˆ r j j nˆ r 0 r, r, lim c.c. 0 E j E0 i j j The full many-body response function has poles at the exact excitation energies: Im Im finite x xx x x Re extended Re ► Discrete single-particle excitations merge into a continuum (branch cut in frequency plane) ► New types of collective excitations appear off the real axis (finite lifetimes) 28 Metals: particle-hole continuum and plasmons In ideal metals, all single-particle states inside the Fermi sphere are filled. A particle-hole excitation connects an occupied singleparticle state inside the sphere with an empty state outside. pl From linear response theory, one can show that the plasmon dispersion goes as 2 (q) pl q ... 4ne 2 pl m 29 Plasmon excitations in bulk metals Sc Al Quong and Eguiluz, PRL 70, 3955 (1993) Gurtubay et al., PRB 72, 125114 (2005) ● In general, excitations in (simple) metals very well described by ALDA. ●Time-dependent Hartree (=RPA) already gives the dominant contribution ● fxc typically gives some (minor) corrections (damping!) ●This is also the case for 2DEGs in doped semiconductor heterostructures 30 Plasmon excitations in metal clusters Yabana and Bertsch (1996) Calvayrac et al. (2000) Surface plasmons (“Mie plasmon”) in metal clusters are very well reproduced within ALDA. Plasmonics: mainly using classical electrodynamics, not quantum response Insulators: three different gaps 31 Band gap: Optical gap: E g E g , KS xc E optical g Eg E exciton 0 The Kohn-Sham gap approximates the optical gap (neutral excitation), not the band gap! 32 Elementary view of Excitons Real space: Mott-Wannier exciton: weakly bound, delocalized over many lattice constants Reciprocal space: An exciton is a collective interband excitation: single-particle excitations are coupled by Coulomb interaction 33 Excitonic features in the absorption spectrum ● Sharp peaks below the onset of the single-particle optical gap ● Redistribution of oscillator strength: enhanced absorption close to the onset of the continuum Optical absorption of insulators 34 Silicon RPA and ALDA both bad! ►absorption edge red shifted (electron self-interaction) ►first excitonic peak missing (electron-hole interaction) Why does ALDA fail? G. Onida, L. Reining, A. Rubio, RMP 74, 601 (2002) S. Botti, A. Schindlmayr, R. Del Sole, L. Reining, Rep. Prog. Phys. 70, 357 (2007) Linear response in periodic systems 35 GG (k , ) sGG (k , ) G1G 2 sGG 1 (k , ) VG1 (k ) G1G 2 f xcG1G 2 (k , ) G 2G (k , ) Optical properties are determined by the macroscopic dielectric function: 2 (Complex index of refraction) mac ~ ( ) n For cubic symmetry, one can prove that Therefore, one needs the inverse dielectric matrix: 1 mac ( ) lim GG (k , ) G 0 k 0 G0 1 G1G (k, ) GG VG (k)GG (k, ) The xc kernel for periodic systems 36 f Hxc (r, r, ) e i (q G )r qFBZ G,G f Hxc ,GG (q, )e i (q G)r TDDFT requires the following matrix elements as input: K iaG,0iGa0 qFBZ GG ik i ei (q G )r ak a f Hxc ,GG (q, ) ak a e i (q G)r ik i k a k i q ,G 0 k a k i q ,G0 Most important: long-range ik i e but iqr f (q 0) limit of “head” (G G 0) : ak a q q 0 ALDA xc , 00 f exact xc , 00 (q, ) q const . 0 1 (q, ) q 2 0 q Therefore, no excitons in ALDA! 37 Long-range xc kernels for solids ● LRC (long-range corrected) kernel (with fitting parameter α): f LRC xc ,GG q |qG | 2 GG ● “bootstrap” kernel (S. Sharma et al., PRL 107, 186401 (2011) 1 boot GG (q,0)vG (q ) f xc ,GG (q, ) s 00 (q,0) ● Functionals from many-body theory: (requires matrix inversion) exact exchange excitonic xc kernel from Bethe-Salpeter equation 38 Excitons with TDDFT: “bootstrap” xc kernel S. Sharma et al., PRL 107, 186401 (2011) 39 Extended systems - summary ► TDDFT works well for metallic and quasi-metallic systems already at the level of the ALDA. Successful applications for plasmon modes in bulk metals and low-dimensional semiconductor heterostructures. ► TDDFT for insulators is a much more complicated story: ● ALDA works well for EELS (electron energy loss spectra), but not for optical absorption spectra ● Excitonic binding due to attractive electron-hole interactions, which require long-range contribution to fxc ● At present, the full (but expensive) Bethe-Salpeter equation gives most accurate optical spectra in inorganic and organic materials (extended or nanoscale), but TDDFT is catching up. ● Several long-range XC kernels have become available (bootstrap, meta-GGA), with promising results. Stay tuned! 40 The future of TDDFT: biological applications (TD)DFT can handle big systems (103—106 atoms). Many applications to large organic systems (DNA, light-harvesting complexes, organic solar cells) will become possible. Charge-transfer excitations and van der Waals interactions can be treated from first principles. N. Spallanzani, C. A. Rozzi, D. Varsano, T. Baruah, M. R. Pederson, F. Manghi, and A. Rubio, J. Phys. Chem. (2009) 41 The future of TDDFT: materials science K. Yabana, S. Sugiyama, Y. Shinohara, T. Otobe, and G.F. Bertsch, PRB 85, 045134 (2012) Vacuum Si Si ● Combined solution of TDKS and Maxwell’s equations ● Strong fields acting on crystalline solids: dielectric breakdown, coherent phonons, hot carrier generation ● Coupling of electron and nuclear dynamics allows description of relaxation and dissipation (TDDFT + Molecular Dynamics) 42 The future of TDDFT: open formal problems ► Development of nonadabatic xc functionals (needed for double excitations, dissipation, etc.) ► TDDFT for open systems: nanoscale transport in dissipative environments. Some theory exists, but applications so far restricted to simple model systems ► Strongly correlated systems. Mott-Hubbard insulators, Kondo effect, Coulomb blockade. Requires subtle xc effects (discontinuity upon change of particle number) ► Formal extensions: finite temperature, relativistic effects… TDDFT will remain an exciting field of research for many years to come! 43 Literature Time-dependent Density-Functional Theory: Concepts and Applications (Oxford University Press 2012) “A brief compendium of TDDFT” Carsten A. Ullrich and Zeng-hui Yang arXiv:1305.1388 (Brazilian Journal of Physics, Vol. 43) C.A. Ullrich homepage: http://web.missouri.edu/~ullrichc [email protected]