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Chapter XX Timedependent density functional theory in quantum mis ry Filipp Furche Institut fur Physikalische Chemie Universitat Karlsruhe Kaiserstrarse 12 76128 Karlsruhe Germany Kieron Burke Department of Chemistry and Chemical Biology Rutgers University 610 Taylor Rd Piscataway NJ 08854 USA Timedependent density functional theory TDDFT is increasingly popular for predicting excited state and response properties of molecule and clusters We review the present stateof theart focusing on recent developments for excited states We cover the formalism computational and algorithmic aspects and the limitations of present technology We close with some promising developments Extensive reviews on many aspects of TDDFT exist 1 2 3 4 5 and no pretence at comprehensive coverage is made here instead we rely heavily on our own work and that of our collaborators BACKGROUND Standard ie groundstate density functional theory DFT is derived from traditional wavefunctionbased quantum mechanics The HohenbergKohn HK theorem is a simple rewriting of the RayleighRitz variational principle Time dependent DFT is based on a different theorem 7 which is a simple of th M N p quotN quot quot 39 equation For a given initial wavefunction and particle interaction a timedependent oneelectron density pr t can be generated by at most one timedependent external ie onebody potential By starting in a n n d g n r gr quotnquot t t t e initial wavefunction can be absorbed into the density dependence by virture of HK We define a set of timedependent Kohn Sham TDKS equations that reproduce pr t from a unique timedependent KohnSham KS potential This consists of the external potential the Hartree potential and the unknown time dependent exchangecorrelation XC potential chPr t This is a much more sophisticated object than the groundstate vXcpr as it encapsulates all the quantum mechanics of all electronic systems subjected to all possible time dependent perturbations ELECTRONIC EXCITATIONS To extract electronic excitations apply a weak electric field and ask how the system responds as in standard perturbation theory We don39t need the entire vXcpr t but only its value close to the ground state This is captured in the X0 kernel fxcr r39 t t395chPr t5pr39 t39 This is a new functional introduced by the timedependence Its Fourier transform fxcr r39 co reduces to the groundstate value as a 0 The standard adiabatic approximation ignores the frequency dependence and uses the secondderivative of the groundstate XC energy functional Typical exa ples are the local density approximation LDA generalized gradient approximation GGA and hybrids such as B3LYP Several practical routes have been adopted for extracting excitation energies from TDDFT response theory In 1995 Casida converted the optical response problem into the solution of an eigenvalue problem EVP 2 whose indices are the singleparticle transitions of the groundstate Kohn Sham potential l2 ilQl lm The matrices A and B are the Hessians of the electronic energy The dominant contributions to the A matrix are the Kohn Sham transition frequencies along the diagonal The transition vectors Xn Yn correspond to collective eigenmodes of the TDKS density matrix with eigenfrequencies Qn The Hartree and XC kernels produce both diagonal and offdiagonal contributions to A and B correcting the transitions between occupied and unoccupied levels of the groundstate KS potential into the true transitions of the system If the different KohnSham transitions do not couple strongly to oneanother a useful approximation is to take only the diagonal elements of A One can view the KS transition as being corrected by an integral over fxcr r39 m on the transition matrix elements and the KS oscillator strengths will be good approximations to the true ones Alternatively many physicists propagate the TDKS equations in real time usually n a realspace grid inside a large sphere They calculate the timedependent dipole moment of their system whose Fouriertransform yields the optical response COM PUTATIONAL ASPECTS The response theory outlined above can be recast in variational form 9 To this end one defines a Lagrangian L which is stationary with respect to all its parameters at the excited state energies L depends on the ground state KS molecular orbitals MOs on the excitation vector and three Lagrange multipliers This is convenient for excited state property calculations because the HellmannFeynman theorem holds for L The LCAOlinear combination of atomic orbitals MO expansion reduces the computation of excited state energies and properties to a finitedimensional optimization problem for L which can be handled algebraically The stationarity conditions for L lead to the following problems which have to be solved subsequently in an excited state calculation i Ground state KS equations in a finite basis results are the ground state KS MOs and their eigenvalues Computational strategies to solve this problem have been developed over decades eg direct SCF selfconsistent field RI resolution of the identity and linear scaling methods Efficient excited state methods take advantage of this technology as much as ssible ii The finitedimensional TDKS EVP Casida39s equations 0 M results are the excitation energies and transition vectors They are used a to compute transition moments and b to analyze the character of a transition in terms of occupied and virtual MOs Complete solution of the TDKS EVP for all excited states leads to a prohibitive ON6 scaling of CPU time and to ON5 lO N is the dimension of the oneparticle basis In most applications only the lowest states are of interest iterative diagonalization methods such as the Davidson method are therefore the first choice 12 13 M In these iterative procedures the timedetermining step is a single matrixvector operation per excited state and iteration which can be cast into a form closely resembling a ground state Fock matrix construction 15 In this way a single point excitation energy can be computed with similar effort as a singlepoint ground state energy Block algorithms lead to additional savings if several states are computed at the same time 16 Sometimes the TammDancoff approximation is used 3 variation of L iii The Zvector equation results are the TDKS relaxed excited state density and energy weighted density matrices Excited state properties such as dipole moments and atomic populations can be computed from the excited state density matrix analytical gradients of the excited state energy with respect to the nuclear positions require the energy weighted density matrix as well Using iterative methods similar to those above the cost for computing the Z vector is again in the range of the cost for a singlepoint ground state which amounts to constraining Y to zero in the energy Geometry optimizations for excited states are therefore not significantly more expensive than for ground states Flexible Gaussian basis sets developed for ground states are usually suited for excited state calculations The smallest recommendable basis sets are of split valence quality and have polarization functions on all atoms except H eg SVP or 631Gquot Especially in larger systems these basis sets can give useful accuracy eg for simulating UV spectra see below However excitation energies are typically overestimated by 0205 eV and individual oscillator strengths may be qualitatively correct only A useful but not sufficient indicator of the quality is the deviation between the oscillator strengths computed in the length and in the velocity gauge which approaches zero in the basis set limit Triplezeta valence basis sets with two sets of polarization functions eg cc pVTZ or TZVPP usually lead to basis set errors well below the functional error larger basis sets are used to benchmark Higher excitations and Rydberg states may require additional diffuse functions PERFORMANCE Vertical excitation and CD spectr So far simulation and assignment of vertical electronic absorption spectra has been the main task of TDDFT calculations in chemistry Most benchmark studies agree that lowlying valence excitations are predicted with errors of ca 04 eV by LDA and GGA functionals w W 18 Hybrid functionals can be more accurate but display a less systematic error pattern Traditional methods such as timedependent HartreeFock TDHF or configuration interaction singles CIS often produce errors of 12 eV at comparable or higher computational cost Bearing in mind that UVVIS spectra of larger molecules are mostly lowresolution spectra recorded in solution and in view of the relatively low cost of a TDDFT calculation errors in the range of 04 eV are acceptable for many purposes Calculated oscillator strengths may be severely in error for individual states but the global shape of the calculated spectra is often accurate Because semi local functionals often predict the onset of the continuum to be 12 eV too low due to the lack of derivative discontinuity this is especially true for excitations in the continuum excitation energy gt HOMO energy 19 Rotatory strengths which determine electronic circular dichroism CD spectra can be computed from magnetic transition moments in the density matrix based approach to TDDF response theory 20 The simulated CD spectra predict the absolute configuration of chiral compounds in a simple and mostly reliable way In particular TDDFT also works well for inherently chiral chromophores 2 and transition metal compounds where semiempirical methods tend to fail Successful applications of TDDFT vertical excitation and CD spectra have been reported in various areas of chemistry including metal clusters fullerenes aromatic compounds porphyrins and corrins and many other organic chromophores As an example we show the simulated and measured CD spectra of the chiral fullerene C76 in Fig 1 16 We used the BeckePerdew86 GGA together with the RlJ approximation and a SVP basis set augmented with diffuse 5 functions a uniform blueshift of 04 eV was applied to all excitation energies to correct systematic errors of the calculation and solvent effects The computed spectrum reproduces the main features of the experimental spectrum even the intensities are in the right range The absolute configuration of C76 can be determined in this way because the measured spectrum can be assigned to one of the two enantiomers whose CD spectra differ by their sign only The simulated spectrum involves 240 excited states its calculation took 30h on a single processor 12 Ghz Athlon PC using TURBOMOLE V54 23 300 300 200 7 200 A 100 7 100 A T I E 9 7 0 7 0 9 a e 4 E 7100 7 7100 7200 7 7200 7300 7 7300 AE nm Figure 1 The simulated CD spectrum of fullerene C76 compared to experiment Excited state structure and dynamic An adequate description of most photophysical and photochemical properties requires information on excited potential energy surfaces beyond vertical excitation energies Early benchmark studies indicated at least qualitative agreement of excited potential surfaces calculated using TDDFI39 and correlated wavefunction ds 24 25 An increasing number of excited state reaction path calculations using TDDFT have been reported A limitation of most studies is that the reaction paths do not correspond to minimum energy paths MEPs ie the internal degrees of freedom other than the reaction coordinate are no optimized Analytical gradients of the excited state energy with respect to the nuclear positions are a basic prerequisite for systematic studies of excited state potential energy surfaces even in small systems Implementations have become available only recently 26 2 27 While errors in adiabatic excitation energies are similar to errors in vertical excitation energies the calculated excited state structures dipole moments and vibrational frequencies are relatively accurate with errors in the range of those observed in ground state calculations The traditional CIS method which has almost exclusively been used for excited state optimizations in larger systems is comparable in cost but significantly less accurate Moreover the KS reference is much less sensitive to stability problems than the HF reference which is an important advantage especially if the ground and excited state structures differ strongly Individual excited states of larger molecules can be selectively investigated pumpprobe experiments The resulting timedependent absorption fluorescence IR and resonance Raman spectra can be assigned by TDDFT excited state calculations First applications show that calculated vibrational frequencies are accurate enough to determine the excited state structure by comparison with experiment 2 The combination of TDDFT and transient spectroscopy methods a promising strategy for excited state structure elucidation in larger systems Computed normal modes of excited states can be used to study the vibronic structure of UV spectra within the FranckCondon and HerzbergTeller approximation 29 For a detailed understanding of photochemical reactions beyond MEPs excited state nuclear dynamics simulations including nonadiabatic couplings are necessary The first steps towards this ambitious goal have already been made 30 31 QUALITATIVE LIMITATIONS OF PRESENT FUNCTIONALS Next we discuss situations where today s approximations in TDDFT produce much larger errors or entirely miss important aspects of the optical response Inaccurate groundstate KS potentials It had been wellknown for many years that the X0 potentials of LDA and GGA are inaccurate At large distances they decay exponentially rather than as the correct 1r This can be a severe problem for TDDFT since the orbital energies can be very sensitive to the details of the potential This is not a problem if only lowlying valence excitations of large molecules are required but the energy of lowlying diffuse states is often considerably underestimated while higher Rydberg states are completely missing in the bound spectrum 32 There now exist several schemes for imposing the correct asymptotic decay of the X0 potential 33 But such potentials are not the functional derivative of any XC energy While this has no direct effect on vertical excitation energies other excited state properties are not welldefined Exact exchange DFT methodology is developing rapidly see next section which does not sufferfrom this problem Furthermore when correctly interpreted even the physicists39 TDLDA calculations recoverthe correct oscillator strength despite these difficulties Adiabatic approximation The frequencydependence of the X0 kernel is ignored in most calculations A simple approximation is to use the mdependent XC kernel of the uniform gas 3 However any collective motion of the electrons that does not deform the density eg an overall boost should not excite the electrons but a frequencydependent kernel violates this exact condition whereas adiabatic approximations do not 35 Multiple excitations In principle the exact electronic response functions contain all levels of excitation But Casida s equations span the space of KS singleparticle excitations only and this is unchanged by a frequency independent XC kernel ie within the adiabatic approximation Extended systems Unlike groundstate DFT there are nontrivial complications when TDDFT is applied to bulk systems These arise because the X0 kernel has longrange contributions comparable to the Hartree 1rr39 However our usual local and semilocal approximations yield XC kernels that are of the form 53rr39 or derivatives thereof Thus they have le effect on the calculated optical response of extended systems Charge transfer problems Charge transfer CT excitations are notoriously predicted too low in energy by up to 1 eV or more 36 ln chainlike systems such as polyenes polyacenes or other conjugated joolymers the error in CT excitation energies increases with the chain length 37 8 In the limit of complete charge separation this can be related to the lack of derivative discontinuities in semi local functionals To correct CT excitation energies methods have been suggested that estimate the derivative discontinuity from a ASCF calculation The validity of this approach depends on assumptions such as complete charge separations that may rarely be justified in real systems PROMISING DEVELOPMENTS Here we discuss several promising paths to overcome present limitations Exact exchange Many problems are related to spurious selfinteraction which affects energies and potentials computed with semilocal functionals The self interaction free exact exchange functional leads to a potential with the correct 1r tail greatly improving the description of Rydberg states 1 Moreover the absence of selfinteraction is a prerequisite for a correct derivative discontinuity as has been demonstrated numerically The use of exact exchange potentials improves the description of optical properties of conjugate olymers Unfortunately exchange alone is not enough So far calculations employing the full frequencydependent exchange kernel have been reported for solids only 42 Excitation energies of valence states obtained with exchangeonly potentials plus ALDA kernel are not systematically better than those from GGA calculations Moreover the neglect of correlation effects generally leads to an overestimation of the energy of ionic states as is well known eg from TDHF Adding an LDA or GGA correlation potential to the x only potential leads to marginal exchange and correlation is lost In practice one often resorts to hybrid functionals which contain a relatively small fraction of exact exchange only Thus moderately diffuse states and certain CT excitations can still be handled 4 A more fundamental solution may require correlation functionals compatible with exact exchange Beyond the adiabatic approximation Higherorder excitations are accounted for by dramatic frequencydependence in fxc and building it into the kernel allows one to recover eg a double excitation close to a single In fact the usual adiabatic approximation simply combines both into one peak which will be a good approximation to the total oscillator strength 44 45 Over the last year it has been shown that incorporation of the essential terms of the polarizability from the BetheSalpeter equation ie an orbitaldependent functional recovers excellent excitonic peak shifts in semiconductors 46 47 Chemists with long molecules should be aware of this as the standard methodology misses these effects TD current DFT The RungeGross theorem in fact establishes that the potential is a functional of the current density jr This approach allowed Vignale and Kohn 48 to construct a gradient expansion in jr that goes beyond the adiabatic approximation without violating exact conditions for boosts This formulation leads naturally to ultra nonlocal functionals that can shift exciton peaks and correct polarizability problems 4 but no accurate approximation is yet available OUTLOOK TDDl l39 in its present incarnation works remarkably well for many systems and properties The number of papers is growing exponentially While most are focused on extracting electronic transitions there are many other promising applications For example atoms and molecules in intense laser fields can be handled with this formalism Recently it has been shown that scattering cross sections can also be extracted 0 This is a goldenage of TDDl l39 in quantum chemistry in which we are right now discovering which systems and properties can be handled routinely where our favorite approximations fail and how to fix these failures We anticipate several more exciting years Acknowledgments This work was supported by the Center for Functional Nanostructures CFN of the Deutsche Forschungsgemeinschaft DFG within project C21 and by NSF under grant number CHE9875091 and DOE under grant number DEFGOZ01ER45928 1Gross E K U Dobson J F Petersilka M Density functional theory of timedependent phenomena Top Curr Chem 1996 181 81172 2Van Leeuwen R Key concepts in timedependent density functional theory Int J Mod Phys B 2001 15 19692023 BOnida G Reining L Rubio A Electronic excitations density functional versus manybody Green39 a unction approaches Rev Mod Phys 2002 74 601659 AMaitra N T Wasserman A Burke K What istimedependent density functional theory Successes and Challenges In Electron correlations and materials properties 2 Gonis A Kioussis N Ciftan M Eds KluwerlPlenum 5Furche F Rappoport D Density functional methods for excited states equilibrium structure and electronic spectra In Computational Photochemistry Olivucci M Ed Elsevier Amsterdam to appear 6Hohenberg P Kohn W Inhomogeneous electron gas Phys Rev 1964 136 88648871 7Runge E Gross E K U Densityfunctional theory for timedependent systems Phys Rev Lett 1984 52 9971000 8Appel H Gross EKU Burke K Excitations in TimeDependent DensityFunctional Theory Phys Rev Lett 2003 90 043005 gFurche F Ahlrichs R Adiabatic timedependent density functional methods for excited state properties J Chem Phys 2002 117 7433 7447 mCasida M E TimeDependent Density Functional Response Theory for Molecules In Recent advances in density functional methods Chong D P Ed World Scientific Singapore 1995 pp 155193 Bauernschmitt R Ahlrichs R Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory Chem Phys Lett 1996 256 454464 UOIsen J Jensen H J A Jizrgensen P Solution of the large matrix equations which occur in response theory J Comput Phys 1988 74 265282 138tratmann R E 8cuseria G E Frisch M J An efficient implementation of timedependent densityfunctional theory for the calculation of excitation energies of large molecules J Chem Phys 1998 109 82188224 Chernyak V 8chulz M F Mukamel 8 Tretiak 8 Tsiper E V Krylovspace algorithms for timedependent HaitreeFock and density functional computations J Chem Phys 2000 113 3643 Weiss H Ahlrichs R Haser M A direct algorithm for selfconsistent field linear response theory and application to 060 Excitation energies oscillator strengths and frequencydependent polarizabilities J Chem Phys 1993 99 12621270 Furche F Dissertation Universitat Karlsruhe 2002 Hirata 8 HeadGordon M Timedependent density functional theory within the TammDancoff approximation Chem Phys Lett 1999 314 291299 Parac M Grimme 8 Comparison of multireference MollerPlesset theory and timedependent methods for the calculation of vertical excitation energies of molecules J Chem Phys A 2002 101 6844 6850 WQWasserman A Maitra N T Burke K Accurate Rydberg excitations from the local density approximation Phys Rev Lett 2003 91 263001 2 Furche F On the density matrix based approach to timedependent density functional theory J Chem Phys 2001 114 59825992 2lFurche F Ahlrichs R Wachsmann C Weber E 80banski A VOgtIe F Grimme 8 Circular dichroism of helicenes investigated by timedependent density functional theory J Am Chem Soc 2000 122 17171724 22Autschbach J Jorge F E Ziegler T Density functional calculations on electronic circular dichroism spectra of chiral transition metal complexes Inorg Chem 2003 42 28672877 23Ahlrichs R Bar M Haser M Horn H Kolmel C Electronic 15 16 18 structure calculations on workstation computers The program system Turbomole Chem Phys Lett 1989 162 165169 See also httpwwwturbomolecom 24Casida M E Casida K C Salahub D R Excitedstate potential energy curves from timedependent densityfunctional theory a cross section of formaldehyde39 sAi manifold Int J Quantum Chem 1998 1 25Sobolewski A L Domcke W Ab initio potentialenergy functions for excited state intramolecular proton transfer a comparative study of o hydroxybenzaldehyde salicylic acid and 7hydroxy1indanone Phys Chem Chem Phys 1999 1 30653072 26Van Caillie C Amos R D Geometric derivatives of density functional theory excitation energies using gradientcorrected functionals Chem Phys Lett 2000 317 159164 27Hutter J Excited state nuclear forces from the TammDancoff approximation to timedependent density functional theory within the plane wave basis set framework J Chem Phys 2003 118 3928 3934 28Rappoport D Furche F Photoinduced intramolecular charge transfer in 4dimethylaminobenzonitrile A theoretical perspective J Am Chem Soc 2004 126 12771284 29Dierksen M Grimme 8 Density functional calculations ofthe vibronic structure of electronic absorption spectra J Chem Phys 2004 120 35443554 30Chernyak V Mukamel S Densitymatrix representation of nonadiabatic couplings in timedependent density functional TDDFT theories J Chem Phys 2000 112 35723579 3lRohrig U E Frank Hutter J A L J VandeVondele Rothlisberger U QMMM CarParrinello molecular dynamics study of the solvent effects on the ground state and on the first excited singlet state of acetone in water Chem Phys Chem 2003 4 11771182 32Casida M E Jamorski C Casida K C Salahub D R Molecular excitation energies to highlying bound states from timedependent densityfunctional response theory Characterization and correction of the timedependent local density approximation ionization threshold J Chem Phys 1998 108 44394449 33Casida M E Salahub D R Asymptotic correction approach to improving approximate exchangecorrelation potentials time dependent densityfunctional theory calculations of molecular excitation spectra J Chem Phys 2000 113 89188935 Gross EKU Kohn W Local densityfunctional theory of frequency dependent linear response Phys Rev Lett 1985 55 28502852 1986 57 923 E 35Dobson JF Harmonic potential theorem Phys Rev Lett 1994 73 7 34 36Dreuw A Weisman J L HeadGordon M Longrange charge transfer excited states in timedependent density functional theory require nonlocal exchange J Chem Phys 2003 119 29432946 37Pogantsch A Heimel G Zojer E Quantitative prediction of optical excitations in conjugated organic oligomers A density functional theory study J Chem Phys 2002 117 59215928 Parac M Grimme S Substantial errors from timedependent density functional theory for the calculation of excited states of large TE systems Chem Phys Chem 2003 4 292295 Tozer D J Relationship between longrange chargetransfer excitation 38 39 1 H H III VI Excited states from timedependent density functional theory Peter Elliott Department of Physics and Astronomy University of California Irvine CA 92697 USA Kieron Burke Department of Chemistry University of California Irvine CA 92697 USA Filipp Furche Institut fur Physikalische Chemie Universitat Karlsruhe Kaiserstra e 12 76128 Karlsruhe Germany Dated March 21 2007 Timedependent density functional theory TDDFT is presently enjoying enormous popularity in quantum chemistry as a useful tool for extracting electronic excited state energies This article explains What TDDFT is and how it differs from groundstate DFT We show the basic formalism and illustrate With simple examples We discuss its implementation and possible sources of error e discuss many of the major successes and challenges of the theory including Weak elds strong elds continuum states double excitations charge transfer high harmonic generation multipho ton ionization electronic quantum control van der Waals interactions transport through single molecules currents quantum defects and elastic electronatom scattering Contents B Testing TDDFT C Saving standard functionals Introduction 2 D Electron scattering A ovemew 3 VII Beyond standard functionals Groundstate review 3 A DOUble imitations A Formalism 3 B Poiymers B Approximate Functionals 5 0 011615 0 Basis Sets 6 D Charge transfer Timedependent theory 7 VIII other toplcs A RungeGross theomm 7 A Groundstate XC energy B Kohn Sham equations 8 girong elds C Linear response 9 ansport D Approximations 10 IX Summary Implementation and basis sets 11 References A Density matrix approach 11 B Basis Sets 11 C Naphthalene converged 12 D Double zeta basis sets 12 E Polarization functions 12 F Triple zeta basis sets 12 G Diffuse functions 13 H Resolution of the identity 13 l Summary 13 Performance 13 A Example Napthalene Results 14 B In uence of the groundstate potential 15 1 N2 a very small molecule 15 2 Napthalene a small molecule 16 C Analyzing the in uence of the KC kernel 16 D Errors in potential vs kerne 17 E Understanding linear response TDDFT 17 Atoms as a test case 18 A Quantum defect 19 I INTRODUCTION Groundstate density functional theory 173 has be come the method of choice for calculating groundstate properties of large molecules because it replaces the in teracting manyelectron problem with an effective single particle problem that can be solved much more quickly It is based on rigorous theorems1 2 4 and a hierar chy of increasingly accurate approximations such as the local density approximation LDA generalized gradient approximations GGA7s577 and hybrids of exact ex change with GGA8i For example a recent groundstate calculation9 for crambin C203H317N55OB4S5 a small protein using TURBOMOLE10 on a 15 GHZ HP ita nium workstation took just 6h52m extraordinarily fast for 2528 electronsr But formally groundstate density functional theory predicts only groundstate properties not electronic excitations On the other hand timedependent density functional theory TDDFT11715 applies the same philosophy to timedependent problems We replace the complicated manybody timedependent Schrodinger equation by a set of timedependent singleparticle equations whose or bitals yield the same timedependent densityi We can do this because the RungeGross theorem16 proves that for a given initial wavefunction particle statistics and in teraction a given timedependent density can arise from at most one timedependent external potential This means that the timedependent potential and all other properties is a functional of the timedependent densityr Armed with a formal theorem we can then de ne time dependent KohnSham TDKS equations that describe noninteracting electrons that evolve in a timedependent Kohn Sham potential but produce the same density as that of the interacting system of interest Thus just as in the groundstate case the demanding interacting time dependent Schrodinger equation is replaced by a much simpler set of equations to propagaterThe price of this enormous simpli cation is that the exchangecorrelation piece of the Kohn Sham potential has to be approxi mate i The most common timedependent perturbation is a longwavelength electric eld oscillating with frequency an In the usual situation this eld is a weak perturbation on the molecule and one can perform a linear response analysisr From this we can extract the optical absorption spectrum of the molecule due to electronic excitations Thus linear response TDDFT predicts the transition fre quencies to electronic excited states and many other properties This has been the primary use of TDDFT so far with lots of applications to large molecules Figure 1 compares TDDFT and experiment for the electronic CD spectrum of the chiral fullerene C751 A total of 240 optically allowed transitions were required to simulate the spectrumi he accuracy is clearly good enough to assign the absolute con guration of C75 TDDFT calculations of this size typically take less than a day on lowend personal computers r 300 200 A 100 A a m g 9 v 0 Sr 3 2 El 100 c 200 r 300 AE nm FIG 1 TDDFT calculation and experiment for the electronic CD spectrum of fullerene VDC76 TDDFT calculations were performed with the BP86 functional and an augmented SVP basis set 17 The RI J approximation together with TZVP auxiliary basis sets 18 was used Experimental data in CH2C12 are from 19 A random walk through some recent papers using TDDFT gives some feeling for the breadth of ap plications Most are in the linear response regime In inorganic chemistry the optical response of many transition metal complexes20735 has been calculated an even some Xray absorption36 37i In or ganic chemistry heterocycles3843 among others4446 have been examined Other examples include the re sponse of thiouracil47 stetrazine48 and annulated porphyrins49r We also see TDDFT7s use in study ing various fullerenes50755i 1n biochemistry TDDFT is nding many uses56766i DNA bases are under examination and an overview of their study may be found in Ref 67r 1n photobiology potential energy curves for the transcis photoisomerization of proto nated Schiff base of retinal68 have been calculated Large calculations for green and blue uorescent proteins have also been performed69 70r Doing photochem istry with TDDFT71 properties of chromophores727 76 and dyes77783 have been computed For these and other systems there is great interest in chargetransfer excitations84792 but as we later discuss intermolecu lar charge transfer is a demanding problem for TDDFTr Another major area of application is clusters large and small covalent and metallic and everything inbetween937112 including MetCars113i Several studies include solvent effects1147122 one example be ing the behavior of metal ions in explicit water123r TDDFT in linear response can also be used to examine chirality1247127 including calculating both electric and magnetic circular dichroism26 1287132 and has been applied to both helical aromatics133 and to artemisinin complexes in solution134i There have also been applica tions in materials135 136 and quantum dots137 but as discussed below the optical response of bulk solids requires some nonlocal approximations138i Beyond the linear regime there is also growing interest in second and thirdorder response1397l42 in all these elds In particular the eld of nonlinear optics has been heavily investigated1437l45 especially the phenomenon of two photon absorption146 1467153 In fact TDDFT yields predictions for a huge variety of phenomena that can largely be classi ed into three groups the nonperturbative regime with systems in laser elds so intense that perturbation theory fails ii the linear and higherorder regime which yields the usual optical response and electronic transitions and iii back to the groundstate where the uctuation dissipation theorem produces groundstate approxima tions from TDDFT treatments of excitations A Overview This work focuses primarily on the linear response regimer Throughout we emphasize the difference be tween small molecules atoms diatomics etc and the larger molecules that are of greater practical interest where TDDFT is often the only practical rstprinciples method We use napthalene Clng as an example to show how the selection of the basis set and of the KC functional affects excitation energies and oscillator strengths computed using TDDFTr Small molecules are somewhat exceptional because they usually exhibit high symmetry which prevents strong mixing of the KS states due to con guration interaction also basis set require ments are often exacerbated for small systemsr Naphtha lene is large enough to avoid these effects yet reasonably accurate gas phase experiments and correlated wavefunc tion calculations are still available We use atomic units throughout 52 Ft me 1 so that all energies are in Hartrees 1 H2272 eV2627r5 kcalmol and distances in Bohr 20529 unless oth erwise noted For brevity we drop comma7s between ar guments wherever the meaning is clear In DFT and TDDFT there is a confusing wealth of acronyms and abbreviations Table l is designed to aid the readers nav igation through this mazer The content of this review is organized as follows Sec tions H and Ill cover the basic formalism of the the ory that is needed to understand where it comes from why it works and where it can be expected to fail Sec tion IV is all about details of implementation especially basisset selectionr On the other hand section V is de voted to performance and analyzing the sources of er ror in the basisset limit In section VI we then look at a few atoms in microscopic detail this is because we know the exact groundstate Kohn Sham potential in such cases and so can analyze TDDFT performance in great depth Section VII is devoted to the many at tempts to go beyond standard functional approximations and especially discusses where such attempts are needed The last substantial section section Vlll covers topics outside the usual linear response approach to excitations TABLE I Table of acronyms and abbre 39 AC corrected ALDA Adiabatic LDA B88 Becke GGA BSLYP Hybrid functional using Becke exchange and LYP corre1ation CASPT2 Comp1ete active space 2quot order perturbation theory Coupled c1uster Con gurationsinteraction sing1ets electronrelectron e 00 55350 externa EXX Exact exchange Generalized gradient approximation Hohenberngohn Hartr Hartr 0 mo E73gt ee ch ee p1us exchangercorrelation Kohanham LB94 van LeeuwensBaerend L 73 m s asymptotica11y corrected functional ation U gt r o r 9L a m 39 m 5 ltlt n w w n o E E LSDA Loca1 spin density approximation L Localized HanreesFock accurate approximation to EXX LeesYangrParr corre1ation Mean aoso1ute error optimised effective potentia1 PBE PerdewsBurkesEmzerhof GGA PBEO Hybrid based on PBE RungerGross R o mgt5m wmww pa quotUPU gt0 3 TDKS Time ExchangecorreIation including groundstate functionals derived from TDDFT challenges for strong elds and transport through single molecules Section TX is a summary 11 GROUND STATE REVIEW In this section we review ground state DFT rather quickly For a more comprehensive review we recom mend 154 Many of the results discussed here are re ferred to in later sections A Formalism Groundstate DFT is a completely different approach to solving the manyelectron problem than the traditional solution of the Schrodinger equation The Hohenberg Kohn theoreml of 1964 states that for a given non degenerate groundstate density nr of Fermions with a given interaction the external potential 175 r that pro duced it is unique up to an additive constantr Hence if the density is known then 175 r is known and so P the Hamiltonian is known From this and the number of particles determined by the integral of the density all properties of the system may be determined In partic ular the groundstate energy of the system E would be known This is what we mean when we say these prop erties are functionals of the density ergr It was later shown that this holds even for degenerate ground states4 and modern DFT calculations use an analo gous theorem applied to the spin densities 71a r 715 Where 016 i respectfullyr The total energy for N electrons consists of three parts the kinetic energy TN the electronelectron interaction Vee 11 and the external potential energy V5 11 each of which is de ned below 2 NMH N Zv m 7 lt1 1 PM 1 7t new ngZ r FV W lt2 1 7 N mm m varmm 3 By the RayleighRitz principle E mIin IJlHl IJ 4 LIIVee111VemIJ 7 mIinT If we simply rewrite the minimization as a two step process155 E min min Vee 11 71007113 IIgt71 71 where the inner search is over all interacting wavefunc tions yielding spin densities 71a 71 We may pull the last term out of the inner minimization E ltFnan5 Z dsf vexwr 71000 113 Elnmmal 7 s where Finmng W i rgwltT1vee1gt lt6 Tlnavn l Veelna77wl 7 7 is a universal functional independent of vexw Minimizing the total energy density functional Eq 5 for both spin densities by taking the functional derivative 66710 and using the EulerLagrange multi plier technique leads to the equation 6Fnan5 6n 7 vext0r M 7 8 where M is the chemical potential of the system Next we imagine a system of noninteracting electrons with the same spin densities Applying the HK theo rem to this noninteracting system the potentials vsar that give densities no r as the groundstate spin den sities for this system are unique This is the ctitious Kohn Sham KS system2 and the fully interacting problem is mapped to a noninteracting one which gives the exact same density Solving the KS equations which is computationally simple at least compared to the fully interacting problem which becomes intractable for large particle numbers then yields the groundstate density The KS equations are 1 7V2 1 1371 97711377 1 7 9 with spin densites N0 RAF Zl jarl2 a 10 j1 where vsmvsg are the KS potentials and C7 is the number of spin a electrons NCZ N5 N In Fig 2 we plot the exact density for the He atom from a highly accurate wavefunction calculation and below we plot the exact KS potential for this system One can see that the KS potential is very different from the external potential This is due to the fact that the KS single effective potential for the noninteracting system must give the correct interacting electron density Because the coulomb repulsion between the electrons shields the nucleus and makes the charge density decay less rapidly than 5 the KS potential is shallower than 175 To derive an expression for 1730 r we note that the Euler equation that yields the KS equations is 6T a WNW 11 11gt Here Ts is the kinetic energy of the KS electrons N a l 7 3 V 2 Ts 7 d r ElV Wrl If we rewrite Fnan5 in terms of the KS system Fna7n3 Twang UM Excna7m3l 7 12 where UM is the Hartree energy given by 7 l 3 3 0rn0r UM 7 2 61 rd r r7 M 13 and the exchangecorrelation XC energy is de ned by Eq 7 and Eq 12 Exclnmnbl Tlnmn13Tslnmn l elnmnbliUlnl 14 lnserting this into Eq 8 and comparing to Eq 11 gives a de nition of the KS potential vsar vextr vHF vxcar 7 15 where the Hartree potential is the functional derivative of U 5Un 7 3 57Lr 7 d7 lrir while the KC potential is given by the functional derivar tive of the KC energy vur 16 We ma Lxcg 7 W 17 This then closes the relationship between the KS sysr tem and the original physical problem Once Exc 72a mg is known exactly or approximated 11mg 1 is determined by di erentiation e KS equations can be solved energy functl nately Exc 72a mg is not known exactly and m approximated There exists a functlonal soup of many di erent approximations of varying accuracy and compur tational cost Many of these are discussed in Section II B In Fig 2 when the KS equation is solved with this exact potential the HOMO level is at 724592 eV This 39 39 39 In exact FT is exactly true157 only energy level of the ctitious KS system that has an immediate physical interpretation Before leaving the groundrstate review we mention the optimized e ective potential OEP method158 159 Here 39 is written as a ction 0 th KS orbitals which in turn are functionals of the den sity The exchange energy is then given by the familiar HF de nition a 1 N far ar 7ar wrl Exe ZZdrdr T cy1 o 18 However in contrast to HF a single e ective potential 39 39 39 Accurate orbital E tinuities and the correct asymptotic decay of the KS potential160 As we will see in Section VB these are important for TDDFT linear response B Approximate Functional n groundrstate DFT calculation we must use approximations for the functional dependence of the 47FT2 n0 Top panel 7 found via the QMO method156 Bottorn panel 7 n e atorn Th exact radial density for the He atorn The ex ternal and KS potentials for t e H e KS potential is found by inverting the KS equations using the exact KS orbitals easily found for He if exact density is known XC energy on the spin densities There now exists a hierarchy of such approximations The simplest of these is the local density approximation LDA where the KC energy at a point r is calculated as if it were a uniform electron gas with the spin densities n0 gr in a constant positive background The exchange energy for the uniform gas can be deduced analytically but the correlation contribution is found using a combination of man rbody theory and highly accurate Monte Carlo simulations for the electron gas of di erent densities1617164 LDA works remarkably well given the vast di erent between homogeneous electron gases and atoms or molecules However total energies are generally underesr timated Typically the KC energy is underestimated by about 7 When the performance of LDA is examined carefully this comes about via a nice but not completely accidental cancellation of errors between the exchange part underestimated by about 10 and correlation overestimated by 200 7 300 which is typically 4 times smaller than exchange n obvious improvement to LDA would be to include information about how rapidly the density is changing via its gradient This leads to the generalized gradient approximation GGA 1n the original Kohn Sham paper of 1965 the simplest of these was suggested The gradient expansion approximation GEA is found by examining the slowly varying limit of the electron gas2 165 However it was soon found that GEA failed to improve on the accuracy of LDA and sometimes made things worse It was not until the late 80s that accurate GGA7s were constructed The most popular of these are BLYP B885 for exchange and LYP6 for correlation and PBEW These generally reduce atomization errors of LDA by a factor of 2 7 5 FEB is a functional designed to improve upon the performance of LDA without losing the features of LDA which are correct As such it reduces to LDA for the uniform electron gas A GGA should also satisfy as many exact conditions as possible such as the Lieb Oxford bound or the form of the exchange energy in the slowly varying limit of the electron gas In this regard PBE is a nonempirical functional where all parameters are determined by exact conditions Because of its ability to treat bulk metals it is the functional of choice in solidstate calculations and increasingly so in quantum chemistry When choosing a GGA using PBE or not PBE should no longer be a question Although the crambin calculation of the introduction used BP86 for reasons explained in Finally hybrid functionals mix in some fraction of exact exchange with a GGA This is the Hartree Fock ex change integral Eq 18 evaluated with the KS orbitals which are functionals of the density Only a small fraction of exact exchange 20 7 25 is mixed in in order to preserve the cancellation of errors which GGAls make use of166 The most widely used functional in chemistry is the hybrid function BSLYP which contains 3 experimentally tted parameters6 8 167 although the parameter in B88 has recently been derived 168 Other hybrid functionals include PBEO where 25 of exact exchange is mixed in with the PBE functional169 A less wellknown feature to users of ground state DFT is that while their favourite approximations yield very good energies and therefore structures vibrations ther mochemistry etc and rather good densities they have poorly behaved potentials at least far from nuclei Fig ure 3 illustrates this for the He atom showing the LDA potential compared to the exact KS potential While the A exact 7 FIG 3 Eb act and LDA KS potentials for the He atom While the exact potential falls OH as 71r the LDA decays much too quickly This is common for nearly all present functionals and has major consequences for TDDFT potential is generally good in the region r lt 2 it decays much too fast far from the nucleus The true KS poten tial falls off as 71r whereas LDA decays exponentially Hence the KS eigenvalues and eigenvalues will be poor for the higher levels To understand why poor potentials do not imply poor energies and why these potentials are not as bad as they look see Ref 170 But as we shall see in section V this has major consequences for TDDFT Over the past decade the technology for treating orbitaldependent functionals has developed and such functionals help cure this problem158 This is called the optimized effective potential OEP1717173 The rst useful orbital functional was the selfinteraction cor rected LDA of Perdew and Zunger162 More gener ally the OEP method can handle any orbitaldependent functional including treating exchange exactly Orbital dependent functionals naturally avoid the selfinteraction error that is common in density functionals An al most exact implementation of the OE equations is localized Hartree Fock LHF174 175 available in TURBOMOLEHO C Basis Sets To actually solve the KS equations the KS orbitals p0r are expanded in a nite set of basis functions XAF part Zcpw Xur 19 The most common choice by far for the basis functions in quantum chemistry are atomcentered contracted Carte sian Gaussians176 Mm 0 x1wywgtlevgte7cwrimgt2 20 ll 1 1y 1 and lzl are positive integers or zero and V ll 1 1y 1 l 1 is somewhat loosely called quantum number of Xv l 01 2 3 corresponds to TABLE II Singlepoint calculations using PBE functional for the reaction energy for naphthalene combustion using PBETZVPRI geometries Reference value computed us ing standard enthalpies of formationfrom NIST178 using thermal and ZVPE corrections at the PBETZVPRI level important because CH bonds are broken and 0H bonds are formed Augmentation aug with diffuse functions somewhat improves the smaller basisset results but is not economical in this case Using the resolution of the identity for the Coulomb operator R1 1 1 1 r time with no loss of accuracy results are found with SVP but convergence Basis set Negative reaction energy kcalmol 1 1 9168 SVP 10600 631G 10471 SVP 11086 augSVP 11155 TZVP 11245 TZVPP 11312 ccpVTZ 11290 augTZVP 11302 augTZVPRI 11302 ZVP 11403 Reference Value 12163 5pdf type Cartesian Gaussians The exponents CW and the contraction coef cients on are optimized in atomic calculations Other common basis functions in use are Slater type orbitals plane waves or piecewise de ned functions on a numerical grid The approximation of the orbitals p0r by a nite linear combination of basis functions also called LCAO linear combination of atomic orbitals Eq 19 leads to a nite number of MOs Thus the KS equations and all derived equations are approximated by nite dimensional matrix equations These equations can be treated by established numerical linear and nonlinear al gebra methods When the basis set size is systematically increased the computed properties converge to their ba sis set limit In a nite basis set all operators become nite matri ces the matrix elements are integrals eg wad3rxrltrgtHan1xpltrgt lt21gt The calculation and processing of such integrals is the main effort in virtually all DFT calculations Gaussian basis functions have the distinct advantage that most in tegrals can be evaluated analytically and that they are spatially local The latter implies that many integrals vanish and need not be calculated Whether a certain integral vanishes or not can be decided in advance by so called prescreening techniques177 To illustrate the effect of choosing various basis sets we show in Fig 11 the reaction energy for naphthalene combustion in the gas phase 01ng 1202 5 10002 4H20 22 The basis sets are listed in order of increasing size and are wellknow in quantum chemistry and are described in detail in section IV We see that hydrogen polarization functions basis sets ending in P are improves all the way to TZVP We can see after the TZVPP result the basis set error is below the functional error and the result is effectively converged We have reached the stage where adding more orbitals which increases the computational cost is no longer going to drastically improve the result On the other hand the crambin calculation mentioned in the introduction is very large and so only an SVP basis set could be used On rst impression comparison to the reference value indicates quite a large error AB 76 kcalmol However given that 48 electron pair bonds are broken and formed the error per carbon atom 76 kcalmol is typical for this functional III TIME DEPENDENT THEORY In this section we introduce all the basis elements of TDDFT and how it differs from the groundstate case A Runge Gross theorem The analog of the Hohenberg Kohn theorem for time dependent problems is the Runge Gross theorem16 which we state here Consider N nonrelativistic elec trons mutually interacting via the Coulomb repulsion in a timedependent external potential The Runge Gross theorem states that the densities nrt and n rt evolv ing from a common initial state 110 11t 0 under the in uence of two external potentials v5 rt and vng rt both Taylor expandable about the initial time 0 are al ways different provided that the potentials differ by more than a purely timedependent rindependent function Avextrt W 7 23 where Avem rt v5 rt 7 17ng rt 24 Thus there is a onetoone mapping between densities and potentials and we say that the timedependent potential is a functional of the timedependent density and the initial state The theorem was proven in two distinct parts In the rst RG1 one shows that the corresponding current densities differ The current density is given by jrt WW 3r WM 25 where A 1 N Kr Z VJNF U 5F i Fjo 26 is the current density operator The equation of motion for the difference of the two current densities givesl6 W 7noltrgtVAvextltrogt 27gt If the Taylorexpansion about t 0 of the difference of the two potentials is not spatially uniform for some order then the Taylorexpansion of the current density differ ence will be nonzero at a nite order This establishes that the external potential is a functional of the current density van 1 Mr t In the second part of the theorem RGH continuity is used 7v jrt 28 which leads to 82 Anrt t 70 v movmext r 0 29 Now suppose Ave r 0 is not uniform everywhere Might not the lefthand side still vanish Apparently not for real systems because it is easy to showl79 dgr Ave r 0V no rVAven r 0 d3r V Avext r0n0 rVAven r0 inolVAvext r 0l2 l 30 Using Greenls theorem the rst term on the right van ishes for physically realistic potentials i e potentials arising from normalizable external charge densities because for such potentials Ave r falls off at least as lr But the second term is de nitely negative so if Avex r0 is nonuniform the integral must be nite causing the densities to differ in 2nd order in t This argument applies to each order and the densities nrt and nrt will become different in nitesimally later than t Thus by imposing this boundary conditions we have shown that 175 71 10 rt Notes 0 The difference between nrt and 71 rt is non vanishing already in rst order of Ave rt ensur ing the invertibility of the linear response operators of section 111C 0 Since the density determines the potential up to a timedependent constant the wavefunction is in turn determined up to a timedependent phase which cancels out of the expectation value of any operator 0 We write vext 71 1 0l Ft because it depends on both the history of the den sity and the initial wavefunction This functional is a very complex one much more so than the ground state case Knowledge of it implies solution of all timedependent Coulomb interacting problems 0 If we always begin in a nondegenerate ground state180 181 the initialstate dependence can be subsumed by the Hohenberg Kohn theoreml and then 175 rt is a functional of nrt alone 175 rt 0 A spindependent generalization exists so that 175 rt will be a functional of the spin densities nan5182 B Kohn Sham equations Once we have a proof that the potential is a functional of the timedependent density it is simple to write the TD Kohn Sham TDKS equations as iv vsanrtgt jart 31 whose potential is uniquely chosen via the RG theorem to reproduce the exact spin densities No WW l jartl2 7 32 of the interacting system We de ne the exchange correlation potential via rt venom czgw M rt 33gt lr M where the second term is the familiar Hartree potential Notes 0 The exchangecorrelation potential vxc0rt is in general a functional of the entire history of the densities no rt the initial interacting wavefunc tion 10 and the initial Kohn Sham wavefunc tion 0181 But if both the KS and interacting initial wavefunctions are nondegenerate ground states it becomes a simple functional of 710rt alone 0 By inverting the single doublyoccupied KS equa tion for a spinunpolarized twoelectron system it is quite straightforward but technically demand ing to nd the TDKS potential from an exact timedependent density and has been done several times1837185 0 In practical calculations some approximation is used for vxc rt as a functional of the density and so modi cations of traditional TDSE schemes are needed for the propagation186 0 Unlike the groundstate case there is no self consistency merely forward propagation in time of a density dependent Hamiltonian 0 Again contrary to the groundstate there is no central role played by a singlenumber functional such as the groundstate energy In fact an action was written down in the RG paper but extrem izing it was later shown not to yield the TDKS equations187 C Linear response The most common application is the response to a weak longwavevlength optical eld 6175 rt 7 expio2tz 34 In the general case of the response of the groundstate to an arbitrary weak external eld the systems rstorder response is characterized by the nonlocal susceptibility mm 2dt dsr XWnorr t7t 6vexWr t 35 This susceptibility X is a functional of the groundstate density 710r A similar equation describes the density response in the KS system 6mm 2dt dgr XSWnolrr t7t 6vsar t Here X3 is the KuhnSham response function constructed om KS energies and orbita s XSWrr w 600 Z q w7wqi0 wwq7i0 37gt where q is a double index representing a transition from occupied KS orbital i to unoccupied KS orbital a raw 6W 7 em and q0r Ur wr 0 means the limit as 0 goes to zero from above ie along the positive real axis Thus X3 is completely determined by the groundstate KS potential It is the susceptibility of the noninteracting electrons sitting in the KS groundstate potential To relate the KS response to the true response we examine how the KS potential in 33 changes 61730047 t Memo r7 75 JV 617ch0 rt 38 lt1gtqar EEOV 7 lt1gtZar EgoV Since 6vgxcart is due to an in nitesimal change in the density it may be written in terms of its functional derivative ie rt 7 Z dgr cw fHXmrra H gmm 7 39gt where MEXCU rt focaa no IT7 t i 75 lo y 40 no The Hartree contribution is found by differentiating 16 6vHrt inar t 6t 7 t 41 lrir l fHn0rr t 7 t while the remainder fXCWnolrrt 7 t is known as the KC kernel By the de nition of the KS potential 6710 rt is the same in both Eq 35 and 6 e can then insert Eq 39 into 36 equate with Eq 35 and solve for a general relation for any 6710 rt After Fourier transforming in time the central equation of TDDFT linear response188 is a Dysonlike equation for the true X of the system XWrr w XSWrr w Zdgrldgrg Xswlrr1w 0102 l X lt7 fxc0102r1r2wgt XUQUI Qrw7 lrl F2l 42 Notes 0 The XC kernel is a much simpler quantity than chgn rt since the kernel is a functional of only the groundstate density 0 The kernel is nonlocal in both space and time The nonlocality in time manifests itself as a frequency dependence in the Fourier transform fxcaa Nb 0 If fxc is set to zero in Eq 42 physicists call it the Random Phase Approximation RPA The inclusion of fxc is an exacti cation of RPA in the same way the inclusion of vxc r in groundstate DFT was an exacti cation of Hartree theory 0 The Hartree kernel is instantaneous ie local in time ie has no memory ie given exactly by an adiabatic approximation ie is frequency indepen ent O The frequencydependent kernel is a very sophisti cated object since its frequencydependence makes the solution of an RPAtype equation yield the exact X including all vertex corrections at every higher order term It de es physical intuition and arguments based on the structure of the TDDFT equations are at best misleading If any argument cannot be given in terms of manybody quantum mechanics Eq 42 cannot help 0 The kernel is in general complex with real and imaginary parts related via Kramers Kronig189 Next Casida190 used ancient RPA technology to produce equations in which the poles of X are found as the solution to an eigenvalue problem The key is to ex pand in the basis of KS transitions We write 6710 rt as 5710 w Z anwlt1gt ar anwlt1gtqar 43 where q ai if q i a This representation is used to solve 36 selfconsistently using Eq 39 and yields two coupled matrix equations191 6v 7 6v 44 A B B A 711 0 X 7 J 0 11 Y where AqaqU qq aawqaanq0 BquU QMW an PW an PW and anqUw drdr qar fHXCWrr w ago 45 with 5qu 02 dr 39i39qa r 6175 rw 46 At an excitation energy 617 0 and choosing real KS orbitals and since A 7 B is positive de nite we get ZQquUw aqU 2 qu 47 WW where squot A 7 B12A BA 7 B12 or QquUw ga qq IUW JV 2WKtzoq0 48 Oscillator strengths fq may be calculated190 from the normalized eigenvectors using 2 fqa g Elsa207 gTsilQaq Z gTsilZajq 2 49 N u 11 Excitation Energies eV N N N singlet txiplet Exact KS Exact TDDPT FIG 4 Transitions for the Helium atom using in groundstate DFT on the left and TDDFT on the right In both cases the exact functionals have been used The results for employing the exact XC kernel in TDDFT linear response are known from calculations using Ref 192 In each pair of lines on the right the triplet is the lower where Sqq qq w uq Figure 4 shows the results of exact DFT calculations for the He atom On the left we consider just transitions between the exact groundstate KS occupied Is to un occupied orbitals These are not the true excitations of the system nor are they supposed to be However ap plying TDDFT linear response theory using the exact kernel on the exact orbitals yields the exact excitations of the He atom Spindecomposing produces both singlet and triplet excitations D Approximations As in the groundstate case while all the equations above are formally exact a practical TDDFT calculation requires an approximation for the unknown XC potential The most common approximation in TDDFT is the adi abatic approximation in whic vi nkrt aminoltrgtinornm so ie the KC potential at any time is simply the ground state XC potential at that instant This obviously be comes exact for suf ciently slow perturbations in time in which the system always stays in its instantaneous groundstate Most applications however are not in this slowly varying regime Nevertheless results obtained within the adiabatic approximation are remarkably accu rate in many cases Any groundstate approximation LDA GGA hybrid automatically provides an adiabatic approximation for use in TDDFTi The most famous is the adiabatic lo cal density approximation ALDA It employs the func tional form of the static LDA with a timedependent den sity deg cln7 7133 A 71 Ft 7422 71a rt 715 M nonort 51 Here 5 na n is the accurately known exchange correlation energy density of the uniform electron gas of spin densities npnli For the timedependent exchange correlation kernel of Eq 40 Eq 51 leads to 2 unif d exc ALDAn0rtrt 63r7r 6t7t d d m7 na xcaa lt52gt The time Fouriertransform of the kernel has no frequencydependence at all in any adiabatic approxima tion Via a Kramers Kronig relation this implies that it is purely real189i Thus any TDDFT linear response calculation can be considered as occuring in two steps 1iAn 39 dttDFT quot is done and a selfconsistent KS potential foundi Transitions from occupied to unoccupied KS or bitals provide zeroorder approximations to the op tical excitations to An approximate TDDFT linear response calcula tion is done on the orbitals of the groundstate cal culation This corrects the KS transitions into the true optical transitionsi In practice both these steps have errors built into themi IV IMPLEMENTATION AND BASIS SETS In this section we discuss how TDDFT is implemented numericallyi TDDFT has the ability to calculate many different quantities and different techniques are some times favored for each type For some purposes eg if strong elds are present it can be better to propa gate forward in time the KS orbitals using a real space grid193 194 or with plane waves195i For niteorder response Fourier transforming to frequency space with localized basis functions may be preferable196i Below we discuss in detail how this approach works emphasiz ing the importance of basisset convergencei A Density matrix approach Instead of using orbitals we can write the dynamics of the TDKS systems in terms of the oneparticle density n0noor 11 matrix 70 rr t of the TDKS determinanti 70rr t has the spectral representation N Ya Frt ja We 7 53 ie the C7 TDKS orbitals are the eigenfunctions of 70 The eigenvalue of all TDKS orbitals which is their occu pation number is always 1 which re ects the fact that the TDKS system is noninteractingi Equivalently ya satis es the idempotency constraint yamt dxlvaltrr1tgtvaltrlm lt54gt The normalization of the TDKS orbitals implies that the trace of 70 be Nor Using the TDKS equations 31 one nds that the timeevolution of 70 is governed by the vonNeumann equation 706 lHalnl t VOW 55 where PL7 rt 7V22 v50rt is the TDKS one particle Hamiltoniani Although 70 has no direct physical meaning it provides the interacting density and current density The density is simply MW VAT 56 and the current density can be obtained from j7 rt Vr 7 Vry7 rr t 57 rr Thus one can either propagate the TDKS orbitals using the TDKS equations 31 or equivalently one can propa gate the TDKS oneparticle density matrix 70 using the vonNeumann equation 55 subject to the idempotency contstraint 54 and normalized to Nor n practice it is often preferable to use 70 instead of the TDKS orbitalsi 70 is unique up to a gauge transfor mation while the orbitals can be mixed arbitrarily by unitary transformationsi Both n7 andj7 are linear in 70 while they are quadratic in the orbitals also the TDKS equations are inhomogeneous in the orbitals due to the density dependence of H0 while they are homogeneous in 70 A response theory based on the TDKS density matrix is therefore considerably simpler than one based on the orbitalsi Finally the use of 70 is computationally more ef cient than using orbitals196i B Basis Sets In response theory the basis functions XL r are usu ally chosen to be timeindependent for strong elds or coupled electronnuclear dynamics timedependent basis functions can be more appropriate C Naphthalene converged Table III shows the basis set convergence of the rst six singlet excitation energies of naphthalene computed us ing the PBE XC functional the corresponding oscillator strengths for some of the transitions are also given Simi lar basisset convergence studies on small model systems should precede applications to large systems In prac tice the systems and states of interest the tar et accu racy the methods used and the computational resources available will determine which basis set is appropriate With a model small molecule we can nd the basisset convergence limit of a method Both excitation energies and oscillator strengths are essentially converged within the aug QZVP basis set QZVP stands for a quadruple zeta valence basis set with polarization functions199 and the pre x aug denotes additional diffuse functions on nonhydrogen atoms which were taken from Dun ning7s aug ccpVQZ basis set200 For C and H this cor responds to 855p4d3f2g and 453p2dlf respectively where the numbers in brackets denote shells of contracted Gaussian type orbitals CGTOs as usual We will take the aug QZVP results as a reference to assess the effect of smaller basis sets D Double zeta basis sets The smallest basis in Table III is of split valence SV or double zeta valence quality201 without polarization functions This basis set consists of two CGTOs per va lence orbital and one per core orbital ie 352p for C and 25 for H Another popular double zeta valence ba sis set is 6 31G 202 The SV basis set can be used to obtain a very rough qualitative description of the low est valence excited states only eg l lBgu and 1 132 Higher and diffuse excitations such as l 1A are much too high in energy or can be missed completely in the spectrum Since unpolarized basis sets also give poor results for other properties such as bond lengths and en ergies their use is generally discuraged nowadays E Polarization functions By adding a single set of polarization functions to non hydrogen atoms the SV results for valence excitations can be considerably improved at still moderate compu tational cost The resulting basis set is termed SVP and consists of 35219161 for C and 25 for H201 The basis set errors in the rst two valence excitation ener gies is reduced by about 50 There is also a dramatic improvement in the oscillator strength of the dipole al lowed transitions This is expected from the limiting case of a single atom where the rst dipole allowed transition from a valence shell of l quantum number l generally involves orbitals with lquantum number l 1 Basis sets of SVP or similar quality are often the rst choice 12 for TDDFT applications to large systems especially if only the lowest states are of interest andor diffuse exci tations are quenched eg due to a polar environment The popular 631G basis set 202 203 has essentially the same size as SVP but performs slightly poorer in our example Adding a single set of 19 type polarization functions to hydrogen atoms produces the SVP basis set201 These functions mainly describe CH 0 type excitation in molecules which usually occur in the far UV and are rarely studied in applications In our example going from SVP to SVP has no signi cant effect This may be different for molecules containing strongly polarized hydrogenelement or hydrogen bridge bonds Next aug SVP is an SVP basis set augmented by a lslpld set of primitive Gaussians with small expo nents from Dunning s aug ccpVDZ 200 often called diffuse functions As shown in Table III the effect of diffuse augmentation is a moderate downshift of less than 01 eV for the rst two singlet excitation energies This behavior is typical of lower valence excited states having a similar extent as the groundstate Our example also shows that diffuse functions can have a signi cant effect on higher excitations An extreme case is the 1 Au state which is an excitation into the 106m orbital having the character of a 35 Rydberg state of the entire molecule The excitation energy of this state is lowered by more than 1 eV upon diffuse augmentation While polarization functions are necessary for a quali tatively correct description of transition dipole moments additional diffuse polarization functions can account for radial nodes in the rstorder KS orbitals which fur ther improves computed transition moments and oscilla tor strengths These bene ts have to be contrasted with a signi cant increase of the computational cost In our example using the aug SVP basis increases the com putation time by about a factor of 4 ln molecules with more than 3040 atoms most excitations of interest are valence excitations and the use of diffuse augmentation may become prohibitively expensive because the large ex tent of these functions confounds integral prescreening F Triple zeta basis sets In such cases triple zeta valence TZV basis sets can be a better alternative The TZVP def2TZVP Ref 204 basis set corresponds to 553p2dlf on C and 35119 on H It provides a description of the valence electrons that is quite accurate for many purposes when density functionals are used At the same time there is a sec ond set of polarization functions on nonhydrogen atoms The excitation energies of valence states are essentially converged in this basis set see Table III However diffuse states are too high in energy There is very little change going to the TZVPP basis which differs from TZVP only by an additional set of polarization functions on H Dun ning7s ccpVTZ basis set 205 performs similar to TZVP 13 TABLE 111 Basis set convergence of the rst six singlet excitation energies in eV and oscillator strengths length gauge of naphthalene The basis set acronyms are de ned in the text Nbf denotes the number of Cartesian basis functions7 and PU denotes the CPU time seconds on a single processor of a 24 GHz Opteron Linux workstation The PBE functional was used for both the groundstate and TDDFT calculations The groundstate structure was optimized at the PBETZVPRl level Ebrperimental results were taken from Ref 197 Basis set 1 lBgu 1 lBgu Osc Str 2 1Ag 1 1B19 2 lBgu Osc Str 1 1144 Nbf CPU SV 4352 4246 00517 6084 5254 5985 11933 6 566 106 24 SVP 4272 4132 00461 5974 5 149 5869 11415 6 494 166 40 6 31G 4293 15 6021 5 185 90 7 013 166 40 SVP 4262 4125 00466 5960 5 136 5852 11402 6 505 190 48 augSVP 4213 4056 00417 5793 4 993 5666 11628 5 338 266 168 TZVP 4209 4051 00424 5834 5030 5715 11455 6 215 408 408 TZVPP 4208 4050 00425 5830 5027 5711 11464 6 231 480 568 CCpVTZ 4222 4064 00427 5870 5061 5 747 11355 6062 470 528 augTZVP 4193 4031 00407 5753 4 957 5622 11402 5141 608 2000 aug TZVPRl 4193 4031 00407 5752 4 957 5621 11401 5 142 608 400 QZVP 4197 4036 00416 5788 4 989 5667 11569 5 672 1000 6104 augQZVP 192 4029 00406 748 54 5616 11330 5 071 1350 28216 expt 397 40 445 47 0102 0109 550 552 528 522 563 555 589 12 13 5 6 and TZVPP However Dunning basis sets are based on a generalized contraction scheme for valence orbitals as opposed to the segmented contracted SV TZV and QZV basis sets The latter are more efficient for larger sys terns7 because more integrals vanish G Diffuse functions Adding a 151p1d1f set of diffuse functions to TZVP we obtain the aug TZVP basis set The aug TZVP ex citation energies of all states except the 1 1Au Rydberg state are within 001 eV of the reference aug QZVP re sults and can be considered essentially converged for the purposes of present TDDFT A similar observation can be made for the oscillator strengths in Table 111 Going to the even larger quadruple zeta valence QZV basis sets the results change only marginally but the computation times increase substantially ln density functional theory7 these basis sets are mainly used for bechmarks and calibration H Resolution of the identity For comparison we have included results that were ob tained using the resolution of the identity approximation for the Coulomb energy RIJ 206 207 It is obvious that the error introduced by the RlJ approximation is much smaller than the basis set error while the com putation time is reduced by a factor of 5 The RlJ approximation is so effective because the computation of the Coulomb Hartree energy and its respose is the bottleneck in conventional TDDFT calculations RlJ replaces the fourindex Coulomb integrals by threeindex and twoindex integrals which considerably lowers the algorithmic pre factor208 It is generally safe to use with the appropriate auxiliary basis sets As soon as hy brid functionals are used however the computation of the exact exchange becomes ratedetermining I Summary To summarize7 for larger molecules SVP or similar basis sets are often appropriate due to their good cost to performance ratio We recommend to check SVP results by a TZVP calculation whenever possible Dif fuse functions should be used sparingly for molecules with more than about 20 atoms V PERFORMANCE This chapter is devoted to studying and analyzing the performance of TDDFT7 assuming basisset convergence We dissect many of the sources of error in typical TDDFT calculations To get an overall impression a small survey is given by Furche and Ahlrichs209 Typical chemical calculations are done with the B3LYP167 functional and typical results are transition frequencies within 04 eV of exper iment and structural properties of excited states are al most as good as those of groundstate calculations bond lengths to within 1 dipole moments to within 5 vi brational frequencies to within 5 Most importantly7 this level of accuracy appears suf cient in most cases to qualitatively identify the nature of the most intense tran sitions often debunking cruder models that have been used for interpretation for decades This is proving espe cially useful for the photochemistry of biologically rele vant molecules69 TABLE IV Performance of various density functionals for the rst six singlet excitation energies in eV of naphthalene An augTZVP basis set and the PBETZVPRI groundstate structure was used The best estimates of the true excitar tions were from experiment and calculations as described in 14 TABLE VI Performance of various density functionals and correlated wavefunction methods for the oscillator strengths of the rst three dipoleallowed transitions of naphthalene A augTZVP basis set and the PBETZVPRl groundstate structure was used for all except the CASPT2 results which were taken from Ref 197 Method 1 133 1 132 2 149 1 131 2133 1117 Method 1 133 1132 2 1337 Pure density functionals LSDA 00000 00405 11517 LSDA 4191 4940 5623 5332 Bp86 00000 00411 11552 BP86 4193 4027 5770 4974 5627 5337 pBE 00000 00407 11402 PBE 4193 4031 5753 4957 5622 5141 BgLYP 00000 00539 12413 Hybrids PBEO 00000 00574 12719 B3LYP 4393 4282 6062 5422 5794 5311 LHFLSDA 00000 00406 12089 PBEO 4474 4379 6205 5611 5889 5603 LHFpBE 00000 00403 12008 best 40 45 55 55 55 57 015 00002 00743 18908 CC2 00000 00773 14262 CASPT2 00004 00496 13365 TABLE V Performance of various wavefunction methods for expt 0002 0102 0 109 1 27 1 3 the excitations of Table l The augTZVP basis set and the PBETZVPRl groundstate structure was used for all ex cept the CASPT2 results which were taken from Ref 197 Ebrperimental results are also from Ref 197 Method 1 13 1 132 2 149 1 1319 2 133 1 14 013 5139 4934 7033 5251 5770 5352 002 4375 4753 5053 5333 5013 5735 CASPT2 403 4 5 553 554 554 expt 397 40 445 47 550 552 523 522 55533351555 best 40 45 55 55 55 57 A Example Napthalene Results s an illustration compare the performance of var ious density functionals and wavefunction methods for the rst singlet excited states of naphthalene in Tables 1V V and V1 All calculations were performed using the aug TZVP basis set the complete active space SCF with secondorder perturbation theory CASPT2 results from Ref 197 were obtained in a smaller double zeta valence basis set with some diffuse augmentation The experimental results correspond to band maxima from gasphase experiments however the position of the band maximum does not necessarily coincide with the vertical excitation energy7 especially if the excited state structure differs signi cantly from the groundstate structure For the lower valence states7 the CASPT2 results can there fore be expected to be at least as accurate as the experi mental numbers For higher excited states7 the basis set used in the CASPT2 calculations appears rather small7 and the approximate secondorder coupled cluster values denoted RlCC2 2107212 might be a better reference Thus our best guess denoted best in the Tables is from experiment for the rst 4 transitions CASPT2 for the 5th7 and RlCC2 for the 6th We begin with some general observations 0 The excitation energies predicted by the GGA func tionals BP86 and PBE differ only marginally from the LSDA results an exception being the 1 1Au Ry dberg state whose PBE excitation energy is sub stantially lower than those of all other methods Note however that GGA functionals generally im prove over LSDA results for other excited state properties such as structures or vibrational frequen cies 0 Hybrid mixing leads to systematically higher exci tation energies On average this is an improvement over the GGA results which are systematically too low However while diffuse excitations bene t from hybrid mixing due to a reduction of selfinteraction error7 valence excitation energies are not always im proved as is obvious for the 1 lBgu and 2 lBgu valence states 0 The 1 IBM state is erroneously predicted below the 1 lBgu state by all density functionals which is a potentially serious problem for applications in pho tochemistry this is not corrected by hybrid mixing 0 The con gurationinteraction singles ClS method which uses a Hartree Fock reference that is compu tationally as expensive as hybrid TDDFT produces errors that are substantially larger7 especially for valence states The coupled cluster and CASPT2 methods are far more expensive and scale pro hibitively as the system size grows The 1 13 excitation is polarized along the short axis of the naphthalene molecule In Platt s nomencla ture of excited states of polycyclic aromatic hydrocar bons PAHs7 1 IBM corresponds to the 1La state This state is of more ionic character than the 1 lBgu or 1L1 state Parac and Grimme have pointed out 213 that GGA functionals considerably underestimate the excita tion energy of the 1La state in PAHs This agrees with the observation that the 1 13 excitation of naphtha lene is computed 0405 eV too low in energy by LSDA and GGA functionals leading to an incorrect ordering of the rst two singlet excited states B In uence of the ground state potential From the very earliest calculations of transition frequencies188 190 it was recognized that the inac curacy of standard density functional approximations LDA GGA hybrids for the groundstate XC potential leads to inaccurate KS eigenvalues Because the approx imate KS potentials have incorrect asymptotic behavior they decay exponentially instead of as 717 as seen in Fig 3 the KS orbital eigenvalues are insufficiently nega tive the ionization threshold is far too low and Rydberg states are often unbound Given this disastrous behavior many methods ave been developed to asymptotically correct potentials214 215 Any corrections to the groundstate potential are dissatisfying however as the resulting potential is not a functional derivative of an energy func tional Even mixing one approximation for vxc r and another for fxc has become popular A more satisfying route is to use the optimized effective potential OEP method159 173 and include exact exchange or other selfinteraction free functionals216 This produces a far more accurate KS potential with the correct asymptotic behavior The chief remaining error is simply the correlation contribution to the position of the HOMO ie a small shift All the main features below and just above I are retained 1 N2 a very small molecule A simple system to see the effect of the various ground state potentials is the N2 molecule In all the cases discussed below a SCF step was carried out using the groundstate potential to nd the KS levels These are then used as input to Eq 47 with the ALDA XC kernel In Table Vll the KS energy levels for the LDA functional are shown It is very clear to see that the eigenvalues for the higher unoccupied states are positive As mentioned this is due to the LDA potential being too shallow and not having the correct asymptotic behavior Comparing the basis set calculation with the basisset free calculation the occupied orbitals are in good agreement However for the unoccupied states that are unbounded in LDA basis sets cannot describe these correctly and give a positive energy value which can vary greatly from one basis set to another In Table Vlll the bare KS transition frequencies between these levels are shown Note that they are in rough agreement with the experimental values and that they lie inbetween the singletsinglet and singlettriplet 15 TABLE Vll Orbital energies of the KS energy levels for N2 at separation R 20744au Orbitals calculated using the LDA potential are shown for two diHerent numerical methods The rst is fully numerical basis set free while the other uses the Sadlej 52 orbitals basis set217 The OEP method uses the EX KLl approximation and is also calculated basis set free Energies in eV Orbital LDA A OEP basis set free Sadlejb Occupied orbitals 17g 38005 38082 39111 1017 38002 38078 39107 27g 2824 2852 3554 2044 1344 1340 2029 17 1189 1186 1853 79 1041 1038 17 15 Unoccupied orbitals 17 22 223 844 40g 004 066 505 27m gt 0 193 404 3042 gt 0 135 354 log gt 0 276 50g gt 0 320 249 609 gt 0 233 27Fg gt 0 389 217 37 gt 0 204 From Ref218 bfrom Ref 219 TABLE Vlll Comparison of the vertical excitation energies for the rst twelve excited states of N2 calculated using dif ferent methods for the SCF step In all cases the KS or bitals from the SCF step are inputed into Casida s equations with the ALDA XC kernel For the LDA calculated with the Sadlej basis set the bare KS transition frequencies are given to demonstrate how they are corrected towards their true val ues using Casida s equations Also given are the mean abso lute errors for each method errors in backets are calculated for the lowest eight transitions only Excitation energy eV 3 ALDA ALDAquot LB94 OEPcl Expt State Excitation BARE K Singlet a singlet transitions wlAn 17m a 1wg 953 1020 1027 932 1055 1027 alz 1 a 1wg 953 953 9 53 913 1009 992 g 309 a 1wg 315 904 923 353 9 75 9 31 aHIE 3ag a 4057 e e 1043 7 1247 12 20 oilla 2 a 1m 7 e 1337 7 1432 1353 sing 30 a 27Fu e e 35 e 1307 1290 Singlet a triplet transitions 03H 20 a hr 11 1005 1105 11 19 B82 17m a 1wg 953 9 53 953 9 13 1009 9 57 nga 17m a 1wg 953 3 30 391 3 32 934 3 33 B3111 309 a 1wg 315 7 50 752 7 14 3 12 3 04 My 1 a 1wg 953 734 307 729 351 774 1332 30 H40 12 32 1195 1200 g g 1033 Mean Absolute Error 0 051 027 54 053 034 Using Sadlej basis set From Ref219 bBasis set free From Ref218 CF rom Ref220 dUsing KLl approximation From Ref218 EComputed in 221 from the spectroscopic constants of Huber and Herzberg 222 transitions223 The ALDA XC kernel f g DA then shifts the KS transitions towards their correct values For the eight lowest transitions LDA does remarkably well the mean absolute error MAE being 027eV for the Sadlej basis set For higher transitions it fails drastically the MAE increases to 054eV when the next four transitions are included This increase in the MAE is attributed to a cancellation of errors that lead to good frequencies for the lower transitions218 Since LDA binds only two unoccupied orbitals it cannot accurately describe transitions to higher orbitals 1n basis set calculations the energies of the unbound orbitals which have converged will vary wildly and cannot give trusted transition frequencies One class of XC functionals that would not have this problem are the asymptotically corrected AC functionals 214 215 2247226 LB94227 is one such of these and its performance is shown in Table V111 AC XC potentials tend to be too shallow in the core region so the KS energy levels will be too low while the AC piece will force the higher KS states to be bound and their energies will cluster below zero Thus it is expected that using AC functionals will consistently underestimate the transitions frequencies A much better approach is using the OE method The KS orbitals found using this method are self interaction free and are usually better approximations to the true KS orbitals OEP will also have the correct asymptotic behavior and as we can see in Table V11 all orbital energies are negative In Table V111 the MAE for CE is 034eV much lower than LDA Since OEP binds all orbitals it allows many more transitions to be calculated A common OEP functional is exact exchange or the KLl approximation228 to it which neglects cor relation effects but these are generally small contribu tions to the KS orbitals Using these with ALDA for fxc which does contain correlation leads to good transition frequencies as shown in Table V111 Although LDA is sometimes closer to the experimental values for the lower transitions the value of CE lies in it ability to describe both the higher and lower transisions 2 Napthalene a small molecule Returning to our benchmark case of Naphthalene us ing more accurate LHF exchangeonly potentials from Sec 1113 together with an LSDA or PBE kernel produces excitation energies in between the GGA and the hybrid results except for the 1 1Au Rydberg state whose excita tion energy is signi cantly improved Whether the LSDA kernel or the PBE GGA kernel is used together with an LHF potential does not change the results signi cantly The 1 1319 and especially the 1 1Au states are dif fuse and it is not surprising that their excitation energy is considerably underestimated in the LSDA and GGA 16 TABLE 1X Naphthalene Effect of groundstate potential on the excitations of Table IV A groundstate calculation using exact exchange OEP LHF is performed and the excitations are found using a LDAPBE kernel respectfully The result is then compared to that found if the LDAPBE functional had been used for both steps Method 11334 11324 214g 113w 21334 1114 LSDA 4191 4026 5751 4940 5623 5332 LHFLSDA 4317 4143 5898 5097 5752 5686 PBE 4193 4031 5753 4957 5622 5141 LHFPBE 4295 4121 5876 5091 5741 5693 best 40 45 55 55 55 57 treatment Using the asymptotically correct LHF poten tial corrects the excitation energy of the 1 1A which is a pure oneparticle excitation out of the 16 valence into the 10ag Rydberg orbital the latter may be viewed as a 35 orbital of the Clngr ion On the other hand a strong mixture of valence and Rydberg excitations oc curs in 1 1319 The LHF potential improves the GGA results only marginally here suggesting that more ac curate XC kernels are necessary to properly account for valenceRydberg mixing C Analyzing the in uence of the XC kernel In this section we discuss the importance of the KC kernel in TDDFT calculations As mentioned earlier the kernels used in practical TDDFT are local or semilocal in both space and time Even hybrids are largely semi local as they only mix in 20 7 25 exact exchange 25 Continuum N u Excita on Energies eV 2 a Exact KS ALDAxc Exact TDDFT 19 FIG 5 The spectrum of Helium calculated using the ALDA XC kernel229 with the exact KS orbitals 1n realistic calculations both the groundstate XC po tential and TDDFT XC kernel are approximated A sim ple way to separate the error in the KC kernel is to look at a test case where the exact KS potential is known Figure 5 shows the spectrum of He using the exact KS potential but with the ALDA XC kerneli It does rather well 229 very well as shall see later in section VI when we examine atoms in more detail Very similar results are obtained with standard GGA7sr 25 Continuum Z4 N u Excitation Energies eV N N E 19 Exact KS ALDAx EXX FIG 6 The spectrum of Helium calculated using the ALDAx kernel and the exact exchange kernel229 Again the exact KS orbitals were used The importance of nonlocality for the XC kernel can be seen as the exchange part of ALDA gives a noticeable error compared to the exchange part of the true functional the AEXX kernel for He The errors in such approximate kernels come from the locality in space and time We can test one of these sep arately for the He atom by studying the exchange limit for the KC kerneli For two spinunpolarized electrons fX iler 7 r l ie it exactly cancels half the Hartree termi Most importantly it is frequencyindependent so that there is no memory ie the adiabatic approxima tion is exact In Fig 6 we compare ALDAx ie the ALDA for just exchange to the exact exchange result for Her Clearly ALDA makes noticeable errors relative to exact exchange showing that nonlocality in space can be important Thus the hybrid functionals by virtue of mixing some fraction of exact exchange with GGA will have only slightly different potentials mostly in the asymptotic region but noticeably different kernelsr D Errors in potential vs kernel In this section we examine the relative importance of the potential and kernel errors It has long been be lieved that fixing the defects in the potential especially 17 its asymptotic behavior has been the major challenge to improving TDDFT results2247226r We argue here that this is overly simplistic and is due to tests being carried out on atoms and small molecules In large molecules where the interest is in the many lowlying transitions the potential can be sufficiently accurate while the kernel may play a larger role In fact our analysis of the general failure of TDDFT in underestimating the 1La transitions in PAH7s sheds some light on its origin Using the selfinteraction free LHF potential does not cure this problem as is obvious from Tabr lXi To the best of our knowledge the cause of this shortcoming of TDDFT is not well understood We note however that the same incorrect ordering of 1La and 1L1 occurs in the CIS approximation which is f 39 t 39 free The analysis here shows that this is a failure of our approximations to the KC kernel rather than to the groundstate potentialr E Understanding linear response TDDFT Several simple methods have evolved for qualitatively understanding TDDFT results The most basic is the singlepole approximation SPA which originated188 in including only one pole of the response function The easiest way to see this here is to truncate Eq 47 to a lgtlt1 matrix yielding an often excellent approximation to the change in transition frequency away from its KS value193 230 022 m wig quaanqa SPA 58 The original SPA was on the unsymmetric system yield ing 02 m wm7 anqa which for a spinsaturated system becomes 0 m mg Zqu 229 This can also provide a quick and dirty estimate since only KS transitions and one integral over fxc are needed While it allows an es timate of the shift of transitions from their KS energy eigenvalue differences it says nothing about oscillator strengths which are unchanged in SPA from their KS values In fact a careful analysis of the TDDFT equa tion shows that oscillator strengths are particularly sensi tive to even small offdiagonal matrix elements whereas transition frequencies are less so231r A more advanced analysis is the double pole approximation232 DPA which applies when two tran sition are strongly coupled to one another but not strongly to the rest of the transitions Then one can show explicitly the very strong effect that offdiagonal elements have on oscillator strengths showing that sometimes an entire peak can have almost no contributionr One also sees polerepulsion in the positions of the transitions a phenomenon again missing from SPAr The DPA was used recently and very successfully to explain Xray edge spectroscopy results for Soltransition metal solids as one moves across the periodic table233i These transitions form a perfect test case for DPA as the only difference between them is caused by the spinorbit TABLE X Transition frequencies and oscillator strengths OS calculated using the double pole approximation DPA for the lowest lBgu transitions in Naphthalene The PBE functional was used with a aug TZVP basis set on top of a PBETZVPRI ground state structure KS DPA Full TDDF w OS w OS w OS lBgu 4117 102 4245 0001 4191 0 ZBgu 4131 100 6748 202 5633 114 splitting several eV of the 2p12 and 2p32 levels In a ground state KS calculation this leads to a 21 branching ratio for the two peaks based simply on degenearcy as all matrix elements are identical for the two transitions Experimentally while this ratio is observed for Fe large deviations occur for other elements These deviations could be seen in full TDDFT calcu lations and were attributed to strong core hole correla tions The SPA while it nicely accounts for the shifts in transition frequencies relative to bare KS transitions but yields only the ideal 21 branching ratio However the DPA model gives a much simpler and more benign interpretation The sensitivity of oscillator strengths to off diagonal matrix elements means that even when the off diagonal elements are much smaller than diagonal el ements of order 1 eV they cause rotations in the 2 level space and greatly alter the branching ratio Thus a KS branching ratio occurs even with strong diagonal corre lation7 so long as off diagonal XC contributions are truly negligible But even small off diagonal correlation7 can lead to large deviations from KS branching ratios We can use DPA to understand the lowest lBgu tran sitions in our naphthalene case In Table X we list the TDDFT matrix elements for the PBE calculation for the two nearly degenerate KS transitions 1au gt 2Z93g and 2b1u gt 2bgg along with their corresponding KS transi tion frequencies Contour plots of the four orbitals in volved are shown in Fig 7 We note rst that these two KS transitions are essentially degenerate so that there is no way to treat them within SPA The degen eracy is lifted by the off diagonal elements which cause the transitions to repel each other and strongly rotate the oscillator strength between the levels removing al most all the oscillator strength from the lower peak232 The DPA yields almost the correct frequency and oscil lator strength ie none for the lower transition but the higher one is overestimated with too much oscil lator strength This must be due to coupling to other higher transitions In the DPA in fact the higher transi tion lands right on top of the third transition so strong coupling occurs there too This example illustrates i that solution of the full TDDFT equations is typically necessary for large molecules which have many coupled transitions but also ii that simple models can aid the interpretation of such results All of which shows that while models developed for well separated transi 533 FIG 7 The four orbitals involved in the rst two lBgu contour value i 007 au The PBE functionals and an aug TZVP basis set were used tions might provide some insight for speci c transitions in large molecules the number and density of transitions make such models only semi quantitative at best VI ATOMS AS A TEST CASE In this section we look more closely at how well TDDFT performs for a few noble gas atoms As ex plained above this is far from representative of its be havior for large molecules but this does allow careful study of the electronic spectra without other complica tions Most importantly for the He Be and Ne atoms we have essentially exact ground state KS potentials from Umrigar and coworkers156 234 This allows us to dis sect the sources of error in TDDFT calculations 02 continuum 00 2p o2 25 393 2 gt4 O4 O7 25 LE O6 O8 15 10 FIG 8 Singlet energy level diagram for the helium atom The Rydberg series of transition frequencies clustered below the ionization threshold can be seen The frequencies clus ter together making it dif cult to assess the quality of the TDDFT calculated spectra As discussed in the text the quantum defect is preferable for this purpose A Quantum defect In Fig 8 we show the KS orbital energy level diagram of the helium atom The zero is set at the onset of the continuum and is marked with a dotted line For closed shell atoms and for any spherical oneelectron potential that decays as 1r at large distances the boundstate transitions form a Rydberg series with frequencies 1 can I 2n 7102 59 where I is the ionization potential and pan is called the quantum defect Quantum defect theory was de veloped by Ham 235 and Seaton 236 before even the Hohenberg Kohn theorem1 The great value of the quantum defect is its ability to capture all the information about the entire Rydberg se ries of transitions in a single slowlyvarying function the quantum defect as a function of energy MAE w I which can often be t by a straight line or parabola In Table XI we report extremely accurate results from wavefunction calculations for the helium atom We show singlet and triplet values that have been obtained by Drake 198 We also give results from the exact ground state KS potential shown in Fig 2 156 For each col umn on the left are the transition frequencies while on the right are the corresponding quantum defects Note how small the differences between transitions become as one climbs up the ladder and yet the quantum defect remains nite and converges to a de nite value 19 TABLE XI Transition energies w and quantum defects QD for He atom sRydberg seriesau The ionization en ergy is 090372 au Transition Singlet a Triplet a KS5 w QD w QD w QT 18 gt 28 07578 01493 07285 03108 07459 02196 18 gt 38 08425 01434 08350 03020 08391 02169 18 gt 48 08701 01417 08672 02994 08688 02149 18 gt 58 08825 01409 08883 02984 08818 02146 18 gt 68 08892 01405 08926 02978 08888 02144 18 gt 78 08931 01403 08926 02975 08929 02143 C Accurate non relativistic calculations from Ref 198 bThe differences between the KS eigenvalues obtained with the exact potential from Ref 156 l l l 025 6 gt gt lt yelt 93 020 D 390 E 3 015 M C U 3 0 010 He atom O Singlet 005 A T met X Exact KS 000 39 39 015 010 005 000 Energy au FIG 9 The exact 9 KS quantum defect and the exact singlet and triplet quantum defects of He and their parabolic ts The quantum defect may clearly be described as a smooth function of energy in this case a linear t Thus knowing the quantum defect for a few transitions allows us to nd it for all transitions and hence their frequencies All the information of the levels of Fig 8 and of Ta ble XI is contained in Fig 9 This clearly illustrates that the quantum defect is a smooth function of energy and is well approximated in these cases as a straight line The quantum defect is thus an extremely compact and sensi tive test of approximations to transition frequencies Any approximate groundstate KS potential suggested for use in TDDFT should have its quantum defect compared with the exact KS quantum defect while any approxi mate XCkernel should produce accurate corrections to the groundstate KS quantum defect on the scale of Fig 9 To demonstrate the power of this analysis we test two common approximations to the groundstate poten tial both of which produce asymptotically correct po tentials These are exact exchange 237 see Sec II B Quantum defect 01 7 En ergy a u FIG 10 The Be p quantum defect of LB94 exact exchange OEP7 and KS and their best ts While both functionals give the correct asymptotic behavior of the KS potential7 by calculating the quantum defect we can learn more about their performance and LB94 227 Exact exchange calculations are more demanding than traditional DFT calculations7 but are becoming popular because of the high quality of the po tential 238 239 On the other hand LB94 provides an asymptotically correct potential at little extra cost beyond traditional DFT 215 226 240 In Fig 10 we show the 19 Be quantum defect obtained with LB94 OEP and exact KS potentials Fig 10 immediately shows the high quality of the exact exchange potential The quan tum defect curve is almost identical to the exact one apart from being offset by about 01 On the other hand the quantum defect of LB94 was poor for all cases studied24l 242 This shows that just having a poten tial that is asymptotically correct is not enough to get a good quantum defect B Testing TDDFT To see how well TDDFT really does we plot quantum defects for atoms We take the He atom as our proto type as usual in this section In Fig 11 we plot rst the KS quantum defect and the exact singlet and triplet lines as before in Fig 9 Then we consider the Hartree approximation This is equivalent to setting the KC ker nel to zero This changes the postion of the singlet curve7 but leaves the triplet unchanged from its KS value7 be cause the direct term includes no spin ipping lt de nitely improves over the KS for the singlet Lastly we include ALDA XC corrections Only if these signi cantly improve over the Hartree curves can we say TDDFT is really working here Clearly it does reducing the Hartree error enormously These results are also typical of He P transitions and Be S and P transitions For reasons as yet unclear7 the 5 Quantum defect M 010 005 000 Energy au 015 FIG 11 The corrections due to using the Hartree or ALDA kernel on the exact KS 3 quantum defect of He Using the Hartree kernel only effects the singlet values7 shifting them too low If a good XC kernel is then used it should move both the triplet and singlet quantum defects from the Hartree kernel towards the exact ones242 In this case7 ALDA does a good job and is performing well 5 gt d transitions fail badly for both these systems2417 243 C Saving standard functionals We have a problem with the incorrect longrange be havior of the potential from standard density functionals only when Rydberg excitations are needed But it would be unsatisfactory to have to perform a completely differ ent type of calculation eg OE7 in order to include such excitations when desired especially if the cost of that calculation is signi cantly greater However it is possible with some thought and care and using quantum defect theory7 to extract the Ryd berg series from the shortranged LDA potential To see this consider Fig 12 which shows both the bare KS response and the TDDFT corrected response of the He atom The 6function absorptions at the discrete transi tions have been replaced by straightlines7 whose height represents the oscillator strength of the absorption mul tiplied by the appropriate density of states247 In the top panel7 just the KS transitions are shown7 for both the KS potential and the LDA potential of Fig 3 from section HR The exact curve has a Rydberg series con verging to 09037 the exact ionization threshold for He The LDA curve on the other hand has a threshold at just below 06 But clearly its optical absorption mimics 15 exact KS 7 E 1 LDA a 7 T3 05 7 EL o 0 y 0 6 8 1 12 14 16 1g 15 a I r x exact calculatlons 7 E 17 l experiment ltgt E J 39035 LDAALDAWH 00015 11 E 05 7 1 090065555265666 2 r O 0 l 1 06 8 1 12 14 16 18 0 FIG 12 He atom The top panel shows the bare exact KS and LDA spectra and the lower panel shows the TDDFT corrected spectra LDAALDA results are from 244 but unshifted the exact calculations are from 245 multiplied by the density of states factor see text and the experimental results are from 246 that of the exact system even in the Rydberg series re gion and is accurate to about 20 The TDDFT ALDA corrections are small and overcorrect the bare LDA re sults but clearly are consistent with our observations for the bare spectra Why is this the case ls this a coincidence Returning to Fig 3 of the introduction we notice that the LDA or GGA potential runs almost exactly parallel to the true potential for 7 S 2 ie where the density is Thus the scattering orbitals of the LDA potential with transi tion energies between 06 and 09 almost exactly match the Rydberg orbitals of the exact KS potential with the same energy When carefully de ned ie phase space factors for the continuum relative to bound states the oscillator strength is about the same This is no coinci dence but due to the lack of derivative discontinuity of LDA its potential differs from the exact one by roughly a constant The fruit yl of TDDFT benchmarks is the 77 gt 77 transition in benzene This occurs at about 5 eV in a groundstate LDA calculation and ALDA shifts it cor rectly to about 7 eV230 Unfortunately this is in the LDA continuum which starts at about 65 eVl This is an example of the same general phenomenon where LDA has pushed some oscillator strength into the continuum but its overall contribution remains about right We can go one step further and even deduce the en ergies of individual transitions While the existence of a quantum defect requires a longranged potential its value is determined by the phaseshift caused by the deviation from 717 in the interior of the atom The quantum defect ezt7act07 248 is a formula for extracting exact FIG 13 He atom solution of Eq60 for 71 as a function of 7 The 71 2O orbital was used for the exact case and the scattering orbital or energy E I 1LSDA was used for the LDA the effective quantum defect from a scattering orbital of a shortranged KS potential such as that of LDA The QDE is dln 7 1 71 1 U77139 2 60gt d7 71 7 Here k 2lEl is written as k 71 1 with 71 71 7 7171 where 71 numbers the bound state and 7171 is the quantum defect U is the con uent hypergeometric function 249 lfthe extractor is applied to an orbital of a longranged potential it rapidly approaches its quantum defect In Fig 13 we plot the results of the QDE for the He atom applied to both the exact KS potential and the LDA potential The LDA potential runs almost paral lel to the exact one in the region 1 lt 7 lt 2 where 7100 can already be extracted accurately and orbitals corresponding to the same f7eque710y exact and LDA are therefore very close in that region In the spirit of Ref250 we compare the exact energynormalized 20s orbital which is essentially identical to the zeroenergy state in the region 0 lt 7 lt 6 and the LDA orbital of energy I e DA 0904 7 0571 0333 Notice how good the LDA orbital is in the region 1 lt 7 lt 2 We show in Fig13 the solution of Eq60 when this scat tering LDA orbital is employed Clearly the plateau of the LDA curve in the 1 lt 7 lt 2 region is an accurate estimate of the quantum defect The value of 11 on this plateau is 0205 an underestimation of less than 4 with respect to the exact value Thus given the ionization potential of the system LDA gives a very accurate prediction of the asymptotic quantum defect The ionization potential is needed to choose the appropriate LDA scattering orbital but the results are not terribly sensitive to it We repeated the same procedure with the LDA ionization potential de ned as ELDAHe7ELDAHe0974 instead of the exact one and foun 713A 0216 overestimating the exact 7100 by just 1 D Electron scattering Lastly in this section we mention recent progress in de veloping a theory for lowenergy electron scattering from molecules This was one of the original motivations for developing TDDFTi One approach would be to evolve a wavepacket using the TDKS equations but a more di rect approach has been developed251 252 in terms of the response function X of the N l electron system assuming it is bound This uses similar technology to the discrete transition casei Initial results for the simplest case electron scat tering from He suggest a level of accuracy compara ble to boundbound transitions at least for low energies the most dif cult case for traditional methods due to boundfree correlation253i TDDFT using the exact groundstate potential and ALDA produces more accu rate answers than static exchange251 a traditional low cost method that is used for larger molecules254 255 However that TDDFT method fails when applied to electron scattering from Hydrogen the true prototype as the approximate solution of the TDDFT equations very similar to the single pole approximation of Sec V E fails due to stronger correlationsi To overcome this a much simpler method has been developed that uses an old scattering trick256 to deduce scattering phase shifts from boundstate energies when the system is placed in a box yielding excellent results for a very demanding casei VII BEYOND STANDARD FUNCTIONALS We have surveyed and illustrated some of the many present successful applications of TDDFT in the previous section In these applications standard approximations local gradientcorrected and hybrid see sect H B are used both for the groundstate calculation and the exci tations via the adiabatic approximation seci lll D In this section we survey several important areas in which this approach has been found to fail and what might be done about it The errors are due to locality in both space and time and these are intimately related In fact all memory effects ie dependence on the history of the density184 implying a frequencydependence in the KC kernel can be subsumed into an initialstate dependence180 but probably not viceversai Several groups are attempting to build such effects into new H 39 t L 39 2577266 but none have shown universal applicability yet The failure of the adiabatic approximation is most no ticeable when higherorder excitations are considered and found to be missing in the usual linear response treatment190i The failure of the local approximation in space is seen when TDDFT is applied to extended systems eg polymers or solidsi Local approxima tions yield shortranged XC kernels which become ir relevant compared to Hartree contributions in the long 22 wavelength limit The Coulomb repulsion between elec trons generally requires long ranged ie lr exchange ef fects when longwavelength response is being calculated Thus several approaches have been developed and ap plied in places where the standard formulation has failed These approaches fall into two distinct categories On the one hand where approximations that are local in the density fail approximations that are local or semi local in the currentdensity might work In fact for TDDFT the gradient expansion producing the leading corrections to ALDA only works if the current is the ba sic variable267i Using the gradient expansion itself is called the Vignale Kohn approximation268 269 and it has been tried on a variety of problems The alternative approach is to construct orbital dependent approximations with explicit frequency dependence270 271 This can work well for speci c cases but it is then hard to see how to construct general density functional approximations from these examples More importantly solution of the OE equations is typ ically far more expensive than the simple KS equations making OEP impractical for large molecules A Double excitations As rst pointed out by Casida190 double excita tions appear to be missing from TDDFT linear response within any adiabatic approximationi Experience272 273 shows that like in naphthalene sometimes adia batic TDDFT will produce a single excitation in about the right region in place of two lines where a double has mixed strongly with a Single In fact when a double excitation lies close to a single excitation elementary quantum mechanics shows that fxc must have a strong frequency dependence 270 Re cently postadiabatic TDDFT methodologies have been developed270 274 275 for including a double excita tion when it is close to an opticallyactive single excita tion and works well for small dienes270 276 It might be hoped that by going beyond linear response non trivial double excitations would be naturally included in eg TDLDA but it has recently been proven that in the higherorder response in TDLDA the double excita tions occur simply at the sum of singleexcitations277i Thus we do not currently know how best to approximate these 39 39 This problem is particularly severe for quantum wells where the external potential is parabolic leading to multiple near degeneracies between levels of excitation 2 74 Returning to our naphthalene example based on a HF reference the 2 1A9 state has according to the RlCC2 results a considerable admixture of double excitationsi This is consistent with the fact that the CIS method yields an excitation energy that is too high by 15 eV compared to experiment The TDDFT results are much closer yet too high by several tenths of eVi B Polymers s a N o m o o o o l l l mu polarizability per ongomer uml a u 3 l 10 number ofoligoma units FIG 14 ALDA and VK static axial polarizability of poly acetylene compared with RHF and MP2 results from Refs 278r280 ALDA severely overestimates the polarizability compared to the accurate MP2 calculation Hartree Fock is also incorrect However using the VK functional gives almost exact agreement at least in this case An early triumph of the VK functional was the static polarizabilities of longchain conjugated polymers These polarizabilites are greatly underestimated by LDA or GGA7 with the error growing rapidly with the number of units281 On the other hand HF does rather well and does not overpolarize The VK correction to LDA yields excellent results in many but not all cases show ing that a currentdependent functional can correct the overpolarization problem Naturally orbitaldependent functionals also account for this effect282 but at much higher computational cost C Solids Again7 in trying to use TDDFT to calculate the opti cal response of insulators7 local approximations has been shown to fail badly Most noticeably they do not describe excitonic effects283 or the exciton spectrum within the band gap On top of this7 the gap is usually much smaller than experiment because adiabatic appoximations can not change the gap size from its KS value One approach is using the VK approximation in TD CDFT This has proven rather successful although a single empirical factor was needed to get agreement with experiment2847286 An alternative is to study the manybody problem287l and ask which expressions must the KC kernel include in order to yield an accurate absorption spectrum2887 289 However the presently available schemes require an expensive GW calculation in the rst place290l A recent review can be found in Ref 291 D Charge transfer As is usually the case whenever a method is shown to work well it starts being applied to many cases and spe ci c failures appear Charge transfer excitations are of great importance in photochemistry especially of biolog ical systems7 but many workers have now found abysmal results with TDDFT for these cases This can be understood from the fact that TDDFT is a linear response theory When an excitation moves charge from one area in a molecule to another both ends will relax ln fact7 charge transfer between molecules can be wellapproximated by groundstate density functional calculations of the total energies of the species involved But TDDFT must deduce the correct transitions by in nitesimal perturbations around the groundstate with out an relaxation Thus it seems a poor problem to tackle with linear response Many researchers are studying this problem to understand it and nd practical solutions around it88 2927295 VIII OTHER TOPICS In this chapter we discuss several topics of specialized interest where TDDFT is being applied and developed in ways other than simple extraction of excitations from linear response In the first of these we show how TDDFT can be used to construct entirely new approximations to the ground state XC energy This method is particularly useful for capturing the longrange uctuations that produce dis persion forces between molecules which are notoriously absent from most groundstate approximations In the second we brie y survey strong eld applica tions7 in which TDDFT is being used to model atoms and molecules in strong laser elds We nd that it works well and easily for some properties but less so for others In the last we discuss the more recent hot area of molecular electronics Here many workers are using groundstate DFT to calculate transport characteristics7 but a more careful formulation can be done only within and beyond TDDFT We review recent progress toward a more rigorous formulation of this problem A Ground state XC energy TDDFT offers a methocl2967 2987301 to nd more sophisticated groundstate approximate energy func tional using the frequencydependent response function Below we introduce the basic formula and discuss some of the exciting systems this method is being used to study This adiabatic connection procedure uses the 10 I l v I I 0 MM A BP86 0 FDTBP36 A x PBEO 3978 FDTPBEU c E t 3 0 39 I D KP gt3 00 3quot 51 a I u 5 DJ on a 5 E g 10 o a m 5 20 I a l r I r I 25 3 35 4 He He Dlstance Aug FIG 15 Binding energy for the Helium dimer interacting via Van der Waals VdW forces from Ref 296 Using the uctuation dissipation theorem FDT new XC energy functionals may be constructed using any ground state func tional The curves from the standard ground state functionals BP865 297 and PBE7 are given as well as the FDT curves with these as input Clearly the FDT is needed to accurately describe VdW interaction uctuation dissipation formula Exon0 Ol dA d339r d339r M 61 lr r l where the pair density is PMM Z 0 X 0n0rrgiwgt n0r63r r and the coupling constant A is de ned to multiply the electron electron repulsion in the Hamiltonian but the external potential is adjusted to keep the density xed302 303 XQU is given by Eq 42 with the KC kernel A Any approximation to the KC kernel X000 yields a sophisticated XC energy EXC It is interesting that if we set XC effects to zero in conventional DFT we end up with the highly inaccurate Hartree method of 1928 However when calculating the linear response if the KC kernel is zero ie within the random phase approximation the KC energy calculated using Eq 61 still gives useful results Computationally this procedure is far more demand ing than conventional DFT but as the above example has shown even poor approximations to the KC kernel can still lead to good results Using this method to nd the KC energy has the ability to capture effects such as dy namical correlation or Van der Waal interactions which 24 are missing from conventional ground state DFT approx imations and are thought to be important in biological systems In particular the coef cient in the decay of the energy between two such pieces C5 in E gt C5R6 where R is their separation can be accurately within about 20 evaluated using a local approximation to the frequency dependent polarizability299 304 306 In Fig 15 the binding energy curve for two Helium atoms interacting via Van der Waals is shown Using the uctuation dissipation formula Eq 61 and the PBEO XC kernel clearly gives more accurate results than semi local functionals Recently the frequency integral in Eq 61 has been performed explicitly but approximately yielding an explicit non local density functional299 301 304 307 312 applicable at all sepa rations TDDFT response functions have also been used in the framework of symmetry adapted perturbation theory to generate accurate binding energy curves of Van der Waals molecules 313 One can go the other way and try using Eq 61 for all bond lengths314 315In fact Eq 61 provides a KS density functional that allows bond breaking without arti cal symmetry breaking300 In the paradigm case of the H2 molecule the binding energy curve has no Coulson Fischer point and the dissociation occurs correctly to two isolated H atoms Unfortunately simple approximations while yielding correct results near equilibrium and at in nity produce an unphysical repulsion at large but nite separations This can be traced back300 to the lack of double excitations in any adiabatic fXC Study of the convergence of EXC with basis sets has also led to an obvious aw in the ALDA kernel at short distances296 Further work is needed to nd accurate XC kernels One method298 to test these is by examining the uni form electron gas as the frequency dependend susceptibil ity can be found easily and uses the well known Lindhard function Hence different approximate XC kernels may be tested and their results compared to highly accurate Monte carlo simulations B Strong elds Next we turn our attention to the non perturbative regime Due to advances in laser technology over the past decade many experiments are now possible in regimes where the laser eld is stronger than the nuclear attraction11 The time dependent eld cannot be treated perturbatively and even solving the time dependent Schrodinger equation in three dimensions for the evolution of two interacting electrons is barely feasible with present day computer technology316 For more electrons in a time dependent eld wavefunc tion methods are prohibitive and in the regime of not too high laser intensities where the electron electron in teraction is still of importance TDDFT is essentially the only practical scheme available3177323 There are a whole host of phenomena that TDDFT might be able to predict high harmonic generation multi photon ion ization above threshold ionization above threshold dis sociation etc but only if accurate approximations are available With the recent advent of atto second laser pulses the electronic time scale has become accessible Theoretical tools to analyze the dynamics of excitation processes on the attosecond time scale will become more and more important An example of such a tool is the time dependent electron localization function TDELF 325 326 This quantity allows the time resolved observation of the formation modulation and breaking of chemical bonds thus providing a visual understanding of the dynamics of excited electrons for an example see Ref 324 The natural way of calculating the TDELF is from the TDKS orbitals High harmonic generation is the production from medium intensity lasers of very many harmon ics sometimes hundreds of the input intensity Here TDDFT calculations have been rather succesful for atoms 3277329 and molecules330 331 Recent experiments have used the HHG response of molecules to determine their vibrational modes 332 Calculations have been per formed using traditional scattering theory333 If this method grows to be a new spectroscopy perhaps the elec tron scattering theory of Sec VlD will be used to treat large molecules Multi photon ionization occurs when an atom or molecule loses more than one electron in an intense field About a decade ago this was discovered to be a non sequential process ie the probability of double ioniza tion can be much greater than the product of two inde pendent ionization events leading to a 7knee7 in the dou ble ionization probability as a function of intensity3347 336 TDDFT calculations have so far been unable to accurately reproduce this knee and it has recently been shown that a correlation induced derivative discontinuity is needed in the time dependent KS potential185 Above threshold ionization AT1 refers to the proba bility of ionization when the laser frequency is less than the ionization potential ie it does not occur in lin ear response337 338 Again this is not well given by TDDFT calculations but both this and MP1 require knowledge of the correlated wavefunction which is not directly available in a KS calculation Since the ionization threshold plays a crucial role in most strong field phenomena Koopmans theorem relat ing the energy level of the KS HOMO to the ionization energy must be well satisfied This suggests the use of self interaction free methods such as OEP159 173 or LDA SIC rather than the usual DFT approximations LDAGGAetc with their poor potentials see Fig 3 25 in Sec 1113 The field of quantum control has mainly concentrated on the motion of the nuclear wave packet on a given set of precalculated potential energy surfaces the ultimate goal being the femto second control of chemical reactions 339 With the advent of atto second pulses control of electronic dynamics has come within reach A marriage of optimal control theory with TDDFT appears to be the ideal theoretical tool to tackle these problems340 341 Recent work3427344 has shown the ability of TDDFT to predict the coherent control of quantum wells using Terahertz lasers However they remains many dif culties and challenges including the coupling between nuclei and electrons3457347 in order to develop a general purpose theory C Transport There is enormous interest in transport through single molecules as a key component in future nanotechnology348 Present formulations use ground state density functionals to describe the stationary non equilibrium current carrying state349 But several recent suggestions consider this as a time dependent problem3507353 and use TDCDFT for a full descrip tion of the situation Only time will tell if TDDFT is really needed for an accurate description of these devices FIG 16 Schematic representation of a benzene 14 di thiol molecule between two gold contacts The molecule plus gold pyramids 55 atoms each constitute the contended molecule as used in the DFT calculations for the Landauer approach Imagine the setup shown in Fig 16 where a conducting molecule is sandwiched between two contacts which are connected to semi infinity leads The Landauer formula for the current is 1 00 I g dE TltEgtULltEgt 7 ME 62gt where TE is the transmission probability for a given energy and fLRE is the Fermi distribution function for the left right lead The transmission probability can be written using the non equilibrium Green s functions NEGF of the system Ground state DFT is used to nd the KS orbitals and energies of the extended molecule and used to nd the self energies of the leads These are then fed into the NEGF method which will determine T and hence the current The NEGF scheme has had a number of successes most notably for atomic scale point contacts and metal lic wires Generally it does well for systems where the conductance it high However it was found that for molecular wires the conductance is overestimated by 1 3 orders of magnitude Various explanations for this and the problems with DFT combined with NEGF in general have been suggested Firstly the use of the KS orbitals and energy levels has no theoretical basis The KS orbitals are those orbitals for the non interacting problem that reproduce the correct ground state density They should not be thought of as the true single particle excitations of the true system However as we have seen they often repro duce these excitations qualitatively so it is not clear to what extent this problem affects the conductance The geometry of the molecules was also suggested as a source of error DFT rst relaxes the molecule to nd its geometry whereas in the experiments the molecule may be subject to various stresses that could rotate parts of it and or squash parts together However calculations have shown that the geometry corrections are small354 The approximation that the non equilibrium XC functional is the same as for the static case has been suggested as a major source of error In fact neither the HK theorem nor the RG theorem are strictly valid for current carrying systems in homogeneous electric elds A dynamical correction to the LDA functional for the static case has been derived using the Vignale Kohn functional T DCDFT but were found to yield only small corrections to ALDA355 In a similar vein the lack of the derivative dis continuity and self interacting errors SIE in the approximations to the XC functional may be the source of the problem354 In Hartree Fock calculations and also in 0PM calculations282 with EXX exact exchange which have no SIE the conductances come out a lot lower in most regions356 Also calculations have been done using a simple model357 with a KS potential with a derivative discontinuity The I V curves for this system are signi cantly different from those predicted by LDA This problem is most severe when the molecule is not strongly coupled to the leads but goes away when it is covalently bonded Recent OEP calculations of the transmission along a H atom chain verify these features356 26 amaze oo 3 00000390209o30 o o go ogo cgo ogo39 Goggogo 0000 FIG 17 Ring geometry for gauge transformation of electric elds Despite these problems quantitative results can be found for molecular devices By looking at what bias a KS energy level gets moved between the two chemical potentials of the leads and hence by Eq 62 there should be a conductance peak one can qualitatively predict postitions of these peaks358 although the magnitude of the conductance may be incorrect by orders of magnitude Since transport is a non equilibrium process we should expect that using static DFT will not be able to accu rately predict all the features Recently a number of methods have been suggested to use TDDFT to calculate transport In Ref 359 the authors present a practical scheme using T DDFT to calculate current The basic idea is to pump the system into a non equilibrium ini tial state by some external bias and then allow the KS orbitals to evolve in time via the TDKS equations The RG theorem then allows one to extract the longitudinal current using the continuity equation Using transparent boundary conditions in the leads these solve problems with propagating KS in the semi in nite leads and us ing an iterative procedure to get the correct initial state they are able to nd the steady state current An alternative formulation uses periodic bound ary conditions and includes dissipation360 In the Landauer Buttiker formulism dissipation effects due to electron electron interaction and electron phonon interaction are neglected as the molecule is smaller than the scattering length However there would be scattering in the leads Imagine a molecule in the ring geometry with a spatially constant electric eld Via a gauge transformation this can be replaced by a constant time dependent magnetic eld through the center of the ring If there is no dissipation the electrons would keep accelerating inde nitely and never reach a steady state 1n the classical Boltzmann equation for transport scat tering is included via a dissipation term using 739 the av erage collision time A master equation approach is ba sically a generalization of the Boltzmann equation to a fully quantum mechanical system The master equation is based on the Louville equation in quantum mechanics and for a quantum mechanical density coupled to a heat bath or reservoir it is written as 296 7211166 0139 63 where C is a superoperator acting on the density whose elements are calculated using Fermils Golden rule with Veg h in a certain approximation weak coupling and instantaneous processes A KS master equation353 can be set up modifying C for single particle reduced density matrices so that it will give the correct steady state The TDKS equations are then used to prop agate forward in time until the correct steady state density is found The current in then extracted from this Recent calculations have shown it can give cor rect behaviour such as hysteresis in lV curves352 361 IX SUMMARY We hope we have conveyed some of the spirit and ex citement of TDDFT in this noncomprehensive review We have explained what TDDFT is and where it comes from We have shown that it is being used and often works well for many molecular excitations lts useful ness lies neither in high accuracy nor reliability but in its qualitative ability to yield roughly correct absorption spectra for molecules of perhaps several hundred atoms Thus we emphasize that usually there are many exci tations of the same symmetry all coupled together and that these are the circumstances under which the theory should be tested For many molecular systems TDDFT 27 is now a routine tool that produces useful accuracy with reasonable confidence That said we have discussed some of the areas where TDDFT in its current incarnation is not working such as double excitations charge transfer and extended sys tems But there has been significant progress in two out of three of these both in understanding the origin of the problem and finding alternative approaches that may ultimately yield a practical solution We also stud ied how well TDDFT works for a few cases where the exact groundstate solution is known describing the ac curacy of different functionals We also surveyed some applications beyond simple linear response for optical ab sorption such as groundstate functionals from the adi abatic connection strong elds and transport In each of these areas more development work seems needed be fore TDDFT calculations can become a routine tool with useful accuracy y wonder how long DFT s preemminence in elec tronic structure can last For sure Kohn Sham DFT is a poor player that struts and frets his hour upon the stage of electronic structure and then is heard no more After all its predecessor ThomasFermi theory is now obso lete being too inaccurate for modern needs Many al ternatives for electronic excitations such as GW are be coming computationally feasible for interesting systems But we believe DFT and TDDFT should dominate for a few decades yet We thank Michael Vitarelli for early work and Dr Meta van Faassen and Dr ax Koentopp for pro viding figures KB gratefully acknowledges support of the US Department of Energy under grant number DE FG0201ER45928 and the NSF under grant CHE 0355405 This work was supported in part by the Cen ter for Functional Nanostructures CFN of 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aromatic hydrocarbons DL Kokkin and TW Schmidt J Phys Chem A 110 6173 2006 Electronic transitions of thiouracils in the gas phase and in solutions Timeedependent density functional theory TDeDFT study MK Shukla and J Leszczynski J Phys Chem A 108 10367 2004 setetrazine in aqueous solution A density functional study of hydrogen bonding and electronic ezcitations M Odelius B Kirchner and J Hutter J Phys Chem A 108 2044 2004 Observation and interpretation of annulated pore phyrins Studies on the photophysical properties of mesortetraphenylmetalloporphyrins JE Rogers KA Nguyen DC Hufnagle DG McLean WJ Su KM Gossett AR Burke SA VinogradoV R Pachter and PA Fleitz J Phys Chem A 107 11331 2003 BET and TDDFT study related to electron transfer in nonbonded porphine center dot center dot center dot Os 60 complezes TLJ ToiVonen Tl Hukka O Cramar iuC TT Rantala and H Lemmetyinen J Phys Chem A 110 12213 2006 Optical absorption and electron energy loss spectra of carbon and boron nitride nanotubes a rsteprinciples approach AG Marinopoulos L Wirtz A Marini V OleVano A Rubio and L Reining Appl Phys A 78 1157 2004 Realetime Ab initio simulations of excited carrier dy namics in carbon nanotubes Y Mi amoto A Rubio and D Tomanek Phys ReV Lett 97 126104 2006 Photoelectron spectroscopy of 0784 dianions T Ehrler JM Weber F Furche and MM Kappes Phys Rev Lett 91 113006 2003 Photoelectron spectroscopy offullerene dianions Os 762 07782 and 07842 OT Ehrler F Furche JM Weber and MM Kappes J Chem Phys 122 094321 2005 Optical and loss spectra of carbon nanotubes Dee polarization e ects and intertube interactions AG 29 Marinopoulos L Reining A Rubio and N Vast Phys Rev Lett 91 046402 2003 Structures and electronic absorption spectra of a recently F 2 o and E Sicilia ChemEur J 12 6797 2006 Are conical intersections responsible for the ultrafast processes of adenine protonated adenine and the cor responding nucleosides SB Nielsen and Tl Solling ChemPhysChem 6 1276 2005 Molecular dynamics in electronically excited states using timeedependent density functional theory 1 Tavernelli UF Rohrig and U Rothlisberger Mol Phys 103 963 2005 A TDDFT study of the optical response of DNA bases base pairs and their tautomers in the gas phase A Tso lakidis and E Kaxiras J Phys Chem A 109 2373 20 Calculating absorption shifts for retinal proteins Come putational challenges M Wanko M Ho mann P Strodet A KosloWski W Thiel F Neese T Frauen heim and M Elstner J Phys Chem B 109 3606 2005 A timeedependent density functional theory investigae tion of the spectroscopic properties of the betaesubunit in Orphycocyanin YL Ren J Wan X Xu QY Zhang and GF Yang J Phys Chem B 110 18665 2006 Linear and nonlinear optical response of aromatic amino acids A timeedependent density functional in vestigation J Guthmuller and D Simon J Phys Chem A 110 9967 2006 Time dependent density functional theory modeling of chiroptical properties of small amino acids in solution MD Kundrat and J Autschbach J Phys Chem A 110 4115 2006 Theoretical study on photophysical and photosensitive properties of aloe emodin L Shen HF Ji and HY Zhang TheochemJ Mol Struct 758 221 2006 Electronically ezcited states of tryptamine and its microe hydrated complez M Schmitt R Brause CM Marian S Salzmann and WL Meerts J Chem Phys 125 124309 2006 Photoinduced processes in protonated tryptamine H Kang C JouVet C DedonderLardeux S Martren Chard C Charriere G Gregoire C Desfrancois JP Schermann M Barat and JA Fayeton J Chem Phys 122 084307 2005 A TDDFT study of the excited states of DNA bases and their assemblies D Varsano R Di Felice MAL Mar ques and A Rubio J Phys Chem B 110 7129 2006 TDeDFT calculations of the potential energy curves for the transecis photoeisomerization of protonated Schi base of retinal H Tachikawa and T lyama J Pho tochem Photobiol B Biol 76 55 2004 Timeedependent densityefunctional approach for biologi cal chromophores The case of the green uorescent pro tein M A L Marques X Lopez D Varsano A Cas tro 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2005 NLO properties of l h p A timesdependent density functional study A Karton MA Iron ME Van der Boom and JML Martin J Phys Chem A 190 5454 2005 A time dependent density functional theory study of alphar84 phycocyanobilin in Ciphycocyanin J Wan X Xu YL Ren and GF Yang J Phys Chem B 109 11088 2005 Photoionization cross section and angular distribution calculations of carbon tetra uoride B T011011 M Stener G Fronzoni and P DecleVa J Chem Phys 124 214313 2006 Timerdependent density functional theory determinas tion of the absorption spectra of naphthoquinones D Jacquemin J Preat V Wathelet and EA Perpete Chem Phys 328 324 2006 A Theoretical Investigation of the Ground and Ezcited States of Coumarin 151 and Coumarin 120 RJ CaVe K Burke and EW Castner Jr J Phys Chem A 106 9294 2002 A TDPDFT study on triplet ezcitedsstate properties of curcumin and its implications in elucidating the photo sensitizing mechanisms of the pigment L Shen HF J1 and HY Zhang Chem Phys Lett 409 300 2005 Times 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in RuX MeCO2MesDAB XCl I DAB1 7diaza7137butadiene e T DFT and CASSCFCASPTQ methods N Ben Amor S Zalis and C Daniel Int J Quant Chem 106 2458 2006 Chargestransfer pi pi ezcited state in the 7sazaindole dimer A hybrid con guration interactions singlestimes dependent density functional theory description R Gelabert M Moreno and JM Lluch J Phys Chem A 110 1145 2006 HartreerFoch ezchange in time dependent density func tional theory application to charge transfer ezcitations in solvated molecular systems L Bernasconi M Sprik and RHutter Chem Phys Lett 394 141 2004 Timesdependent densityrfunctional theory in vestigation of the uorescence behavior as a function of alhyl chain size for the 47NNr k E E e i e p sy s tems 4sNNsdiethylaminobenzonitrile and 47NNr quot p F 39 l 39 ile CJ Jamorski and ME Casida J Phys Chem B 108 7132 2004 89 Failure of timesdependent density functional theory for longsrange char estransfer cit states The zincbacteriochlorins bacterlochlorin and 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