Single-Reference ab Initio Methods for the Calculation of...

Single-Reference ab Initio Methods for the Calculation of Excited States ofLarge Molecules

Andreas Dreuw*

Institut für Physikalische und Theoretische Chemie, Johann Wolfgang Goethe-Universität, Marie Curie-Strasse 11,60439 Frankfurt am Main, Germany

Martin Head-Gordon

Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley National Laboratory,Berkeley, California 94720-1470

Received April 8, 2005

Contents1. Introduction 40092. Wave-Function-Based Methods 4012

2.1. Configuration Interaction Singles 40122.1.1. Derivation of the CIS Equations 40122.1.2. Properties and Limitations 4013

2.2. Time-Dependent Hartree−Fock 40142.2.1. Concepts and Derivation of TDHF 40142.2.2. Properties and Limitations 4015

3. Time-Dependent Density Functional Theory 40163.1. Formal Foundations 4016

3.1.1. The Runge−Gross Theorem 40163.1.2. The Action Integral 40183.1.3. The Time-Dependent Kohn−Sham

Equation4018

3.2. Derivation of the Linear-Response TDDFTEquation

4020

3.3. Relation of TDDFT and TDHF, TDDFT/TDA,and CIS

4022

3.4. Properties and Limitations 40233.5. Charge-Transfer Excited States in TDDFT 4024

4. Analysis of Electronic Transitions 40264.1. Molecular Orbitals 40274.2. Transition Density 40274.3. Difference Density 40284.4. Attachment/Detachment Density Plots 4029

5. Illustrative Examples 40295.1. The Initial Steps of the Ultrafast

Photodissociation of CO-Ligated Heme4029

5.2. Charge-Transfer Excited States inZincbacteriochlorin−BacteriochlorinComplexes

4031

6. Brief Summary and Outlook 40347. Acknowledgments 40358. References 4035

1. IntroductionSince the electronic Schrödinger equation (SE) was

first written down, it was clear that it cannot be

solved exactly for real molecular systems due to theelectron-electron interaction and the resulting highdimensionality. Consequently, approximations had tobe introduced, which gave birth to the research fieldof quantum chemistry. The most prominent ap-proximation is the Hartree-Fock approach,1,2 whichtreats each electron independently moving in anaverage field of all other electrons and the nuclei.This leads to an uncoupling of the many-body SE tomany single-particle equations, the so-called Har-tree-Fock (HF) equations, and concomitantly to thefamiliar and in chemistry widely used single-particlepicture of molecular orbitals. However, an inherentapproximation in the HF method is the neglect ofelectron correlation, that is, the explicit electron-electron interactions. Much effort has been under-taken to recover the missing electron correlation andas a consequence a plethora of quantum chemical abinitio methods have emerged. Examples of suchwave-function-based methods are Møller-Plessetperturbation theory (MP),3,4 configuration interaction(CI),5,6 and coupled-cluster approaches (CC).7,8

A conceptually different approach to include elec-tron correlation is represented by density functionaltheory (DFT),9-11 which relies on the electron densityas a fundamental quantity. In DFT, exchange andcorrelation effects are gathered in the so-calledexchange-correlation (xc) functional. Since the exactxc functional is unknown, it is fitted empirically to aset of experimental data or it is modeled on the basisof model systems such as the uniform electron gasand other known properties. Once determined, it isthen employed as a universal xc functional. Depend-ing on the choice of the ansatz, many differentapproximate xc functionals are available today. Ingeneral, xc functionals can be divided into threedifferent classes: local functionals, gradient-correctedfunctionals, and hybrid functionals. In fact, the artof performing a DFT calculation is closely related tochoosing the appropriate xc functional for the systemunder investigation. However, DFT is a formallyexact theory, that is, if the exact xc functional wereused, exact results would be achieved. Density func-tional theory is built on the famous Hohenberg-Kohn* E-mail: [email protected].

4009Chem. Rev. 2005, 105, 4009−4037

10.1021/cr0505627 CCC: $53.50 © 2005 American Chemical SocietyPublished on Web 10/06/2005

theorems, HK I and HK II.12 The first ensures a one-to-one mapping between the electron density and theexternal potential containing the electron-nucleiattraction and any additional magnetic or electricfield, while HK II guarantees the existence of avariational principle for electron densities analogousto the famous Raleigh-Ritz principle for wave func-tions. Together, HK I and HK II, are the necessary

ingredients for the formulation of a many-body theorybased on the electron density alone. Present-day DFTcalculations are almost exclusively done within theso-called Kohn-Sham formalism,13 which corre-sponds to an exact dressed single-particle theory. Inanalogy to HF theory, the electrons are treated asindependent particles moving in the average field ofall others but now with correlation included by virtueof the xc functional. This again gives rise to single-electron molecular orbitals and orbital energies. Allthe methods described so far aim at precise calcula-tion of the electronic ground state of molecularsystems, and in principle, they can be divided intowave-function-based methods and density-based meth-ods. Especially Hartree-Fock and DFT form thebasis on which excited-state calculations are per-formed, as we will show in the following paragraphs.

Knowledge about the energetic position of elec-tronically excited states relative to the ground state,as well as information about geometric and electronicproperties of excited states, is necessary for theexplanation and interpretation of electronic spectraof molecular systems. Furthermore, optically forbid-den (dark) states, which are thus experimentally notor only very poorly accessible, very often play impor-tant roles in determining the dynamics of electroni-cally excited systems. Quantum chemical calculationsof such excited states can, for instance, provide usefulinformation and do indeed often contribute to thefundamental understanding of excited-state dynam-ics.

Sometimes a ground-state method can be tweakedto calculate a particular electronically excited state,if the ground-state formalism can be forced to con-verge onto an energetically higher solution by intro-duction of constraints. These constraints may begiven, for example, as a different spin multiplicity;hence one can use a ground state method to calculatethe ground state of every possible spin multiplicity.Equally well, the excited state of interest can belongto a different irreducible representation of the spatialsymmetry group than the ground state. Then, it ispossible to converge the ground-state calculation forthe lowest solution of each irreducible representationof the point group of the given molecular system.Following this procedure, excited-state propertiessuch as equilibrium geometries and static electricmoments are accessible. The excitation energy isgiven as the difference of the total energies of theelectronic ground state and the excited state obtainedin two independent calculations. This method is alsocommonly referred to as ∆-method. Logically, thismethodology breaks down immediately if one isinterested in excited states of the same spin multi-plicity and same irreducible representation of thespatial symmetry group as the one of the electronicground state. This is particularly often the case if oneseeks to compute optical spectra of large molecularsystems, which mostly do not exhibit any spatialsymmetry. It is also commonly the case that excited-state configurations are not well represented by themodel commonly used for ground-state wave func-tions, for example open-shell singlets.

Andreas Dreuw was born in 1972 in Neuss, Germany. He received hisundergraduate education in chemistry at the universities of Düsseldorfand Heidelberg. In 1997, he joined the theoretical chemistry group ofProf. Lorenz Cederbaum in Heidelberg and received his Ph.D. in 2001.Then he moved to the University of California at Berkeley to join theresearch group of Prof. Martin Head-Gordon as an “Emmy Noether”postdoctoral fellow of the Deutsche Forschungsgemeinschaft. Since 2003,he has been at the University of Frankfurt where he is leading anindependent research group funded within the “Emmy Noether” program.His research interests comprise the development of theoretical methodsfor the calculation of excited states of large molecules as well as theirapplication to biological problems such as electron and energy transferprocesses in phostosynthesis.

Martin Head-Gordon was born in Canberra, Australia (1962), and grewup in Halifax, Canada, and Melbourne, Australia. He received B.Sc. (Hons)and M.Sc. degrees in Chemistry from Monash University, Melbourne,where he worked in the laboratory of Prof. Ron Brown. In 1989, heobtained his Ph.D. in Chemistry from Carnegie-Mellon University, workingwith Prof. John Pople. After postdoctoral research with Dr. John Tully atBell Laboratories, Head-Gordon became an Assistant Professor ofChemistry at the University of California, Berkeley, in 1992. He waspromoted to Full Professor in 2000, and holds a joint appointment in theChemical Sciences Division at Lawrence Berkeley National Laboratory.He was a NSF Young Investigator (1993), a Packard Fellow (1995), anda Sloan Fellow (1995), received the Medal of the International Society ofQuantum Molecular Sciences (1998), and was a Miller Research Professor(2001−2002). His research centers on the development and applicationof electronic structure methods and their realization as practical computeralgorithms, with particular interests in linear scaling, excited states, andelectron correlation.

4010 Chemical Reviews, 2005, Vol. 105, No. 11 Dreuw and Head-Gordon

Today, several quantum chemical approaches forthe calculation of excited states are available that donot require any a priori constraints and that yieldenergies and oscillator strengths of several excitedstates in one single calculation. In analogy to ground-state methods, excited-state methods can also bedivided into wave-function-based methods and elec-tron-density-based methods. Typical wave-function-based methods are CI,6,5 multireference CI (MR-CI)14,15 or multireference MP approaches,16,17 multi-configurational self-consistent field (MCSCF) meth-ods,2 for example, complete active space SCF (CASS-CF),18 and complete active space perturbation theoryof second order (CASPT2).19 These approaches are inprinciple based on the explicit inclusion of excitedstates in the many-body wave function as additionalso-called “excited” Slater determinants constructedfrom the HF ground state by swapping occupied withvirtual orbitals. The expansion coefficients of theSlater determinants are then calculated via theRaleigh-Ritz variation principle, which in the caseof CI corresponds to the diagonalization of theHamiltonian matrix in the basis of the exciteddeterminants.1 In multiconfigurational SCF ap-proaches also the expansion coefficients of the mo-lecular orbitals setting up the Slater determinantsare reoptimized1 making these calculations prohibi-tively computationally expensive for large molecules.

Other prominent wave-function-based approachesare the equation-of-motion20-22 and linear-response23-26coupled cluster theories (EOM-CC and LR-CC, re-spectively), which depending on the level of trunca-tion in the CC expansion can yield very accurateresults. Closely related to these coupled-cluster theo-ries is the symmetry-adapted cluster configurationinteraction (SAC-CI) approach.27 Propagator theoriesemerging from the Green’s function formalism, suchas the algebraic diagrammatic construction (ADC)scheme,28-30 also provide an elegant route to thecalculation of excited state properties. However, allthese wave-function-based methods are limited tofairly small molecules due to their high computa-tional costs.

The cheapest excited-state methods that includecorrelation via the wave function available today arethe CIS(D) approach31,32 and the approximate coupled-cluster scheme of second order (CC2).33,34 The CIS-(D) approach is a perturbative correction to config-uration interaction singles (CIS)35 that approximatelyintroduces effects of double excitations for the excitedstates in a noniterative scheme very similar to MP2,in which doubly excited states are coupled to theground state. It can also be compared to the pertur-bative triples correction (T) to the CCSD scheme.36Similar in spirit is the CC2 method, which is anapproximation to CCSD and of which the linearresponse functions yield excited states and oscillatorstrengths of about MP2 quality. CC2 is also closelyrelated to the ADC(2) scheme mentioned before.37Modern implementations of these wave-function-based approaches allow for the treatment of molec-ular systems of up to about 50 atoms.34

In this review, we want to focus on single-referenceab initio excited state methods, which are applicable

to large molecules and do not explicitly includecorrelation through the ground-state wave function.In quantum chemical calculations, the size of amolecule is defined by the number of basis functionsrather than by the number of atoms, and we areinterested in methods that can handle ca. 5000 basisfunctions in a computation time of maximum a fewdays on a standard personal computer (PC). If oneassumes a typical basis set size of about 15 basisfunctions per second row atom, one can then inves-tigate molecules with up to 300 such atoms. Theserestrictions limit the applicable ab initio methodsessentially to the wave-function-based methods CIS35and time-dependent Hartree-Fock (TDHF), as wellas to the density-based approach time-dependentDFT (TDDFT), which builds upon the electron den-sity obtained from ground-state DFT. CIS and TDHFare fairly old methods that have been widely usednot only in quantum chemistry but also in nuclearphysics, where TDHF is better known as the randomphase approximation (RPA) and CIS as the Tamm-Dancoff approximation (TDA) to it.38 TDDFT, how-ever, is a very modern method that was developedabout 20 years ago39-42 and today has become one ofthe most prominent methods for the calculation ofexcited states of medium-sized to large molecules.Several reviews are available in the literature thatfocus on the theoretical derivation and developmentof the method,41-44 but so far, no review focuses onthe applicability of TDDFT to various systems, itspractical benefits, and its limitations. Also the dif-ferent available techniques for the analysis of com-plicated electronic transitions has never been re-viewed. The aim of this review is thus threefold.First, CIS, TDHF, and TDDFT are rigorously intro-duced by outlining their derivations and theoreticalfooting, where special emphasis is put on theirrelations to each other. Second, different methods forthe analysis of complicated electronically excitedstates are reviewed and comparatively discussed.Third, we focus on the applicability of the presentedmethods and show their limitations and point outtheir strengths and weaknesses.

The review is organized as follows. In the nextsection (section 2), the wave-function-based methodsCIS and TDHF are presented. The derivations andfundamental theoretical concepts are outlined insections 2.1.1 and 2.2.1 for CIS and TDHF, respec-tively, and their properties and limitations are dis-cussed in sections 2.1.2 and 2.2.2. Then we turn totime-dependent DFT (section 3), where we firstreview the theoretical foundations of the theory(section 3.1) and then rederive the basic equationsof linear response TDDFT in a density-matrix for-mulation (section 3.2). The relations between theintroduced methods CIS, TDHF, TDDFT/TDA, andTDDFT (section 3.3) and properties and limitationsof TDDFT are pointed out (section 3.4). In lastsubsection 3.5 of this section, we focus on one of themost severe failures of TDDFT for charge-transferexcited states. In section 4, modern schemes for theanalysis of electronic transitions are reviewed com-prising the analyses of molecular orbitals (section4.1), the transition density (section 4.2), the difference

Calculation of Excited States of Large Molecules Chemical Reviews, 2005, Vol. 105, No. 11 4011

density matrix (section 4.3), and attachment/detach-ment density plots (section 4.4). In the last section5, two illustrative examples of typical theoreticalstudies of large molecules employing TDDFT aregiven, where special emphasis is put onto the ap-plicability, advantages, and limitations of TDDFT.

2. Wave-Function-Based Methods

2.1. Configuration Interaction Singles

2.1.1. Derivation of the CIS EquationsConfiguration interaction singles (CIS) is the com-

putationally as well as conceptually simplest wave-function-based ab initio method for the calculationof electronic excitation energies and excited-stateproperties. The starting point of the derivation of theCIS equations is the Hartree-Fock (HF) groundstate, Φ0(r), which corresponds to the best singleSlater determinant describing the electronic groundstate of the system. It reads

For simplicity, we assume a closed-shell ground-stateelectronic configuration, and thus, the φi(r) cor-respond to doubly occupied spatial orbitals and n )N/2 (N is the number of electrons). Φ0(r) is obtainedby solving the time-independent Hartree-Fock equa-tion, which is given by

with

and

In this equation, ĥ(r) contains the kinetic energy ofthe ith electron and its electron-nuclei attraction,while the Coulomb operator Ĵi(r) and exchangeoperator K̂i(r) describe the averaged electron-elec-tron interactions. They are defined as

Solution of the Hartree-Fock equations for theground-state Slater determinant (eq 1) within a givenbasis set of size K yields n occupied molecularorbitals, φi(r), and v ) K - n virtual orbitals, φa(r).Here and in the following, we use the indices i, j, k,etc. for occupied orbitals, a, b, c, etc. for virtual ones,and p, q, r, etc. for general orbitals. In configurationinteraction, the electronic wave function is then

constructed as a linear combination of the groundstate Slater determinant and so-called “excited”determinants, which are obtained by replacing oc-cupied orbitals of the ground state with virtual ones.If one replaces only one occupied orbital i by onevirtual orbital a and one includes only these “singlyexcited” Slater determinants, Φi

a(r), in the CI wavefunction expansion, one obtains the CIS wave func-tion, ΨCIS, which thus reads

The summation runs over index pairs ia and has thedimension n × v. This ansatz for the many-body wavefunction is substituted into the exact time-indepen-dent electronic Schrödinger equation,

where T̂ has the usual meaning of the kinetic energyoperator

and V̂el-nuc corresponds to the electron-nuclei attrac-tion

where the index i runs over all electrons and K overall nuclei. ZK is the charge of nucleus K. Theelectron-electron interaction, V̂el-el(r) is given as

Projection onto the space of singly excited determi-nants, that is, multiplication of eq 8 from the left with〈Φj

b|, yields

and with

one readily obtains an expression for the excitationenergies ωCIS ) ECIS - E0

�a and �i are the orbital energies of the single-electronorbitals φa and φi, respectively, and (ia||jb) corre-

Φ0(r) ) |φ1(r)φ2(r)...φn(r)| (1)

F̂(r)Φ0(r) ) E0Φ0(r) (2)

F̂(r) ) ∑i

n

f̂i(r) (3)

f̂i(r) ) ĥi(r) + Ĵi(r) - K̂i(r) (4)

Ĵi(r)φi(r) ) [∑j

N ∫dr′ φj/(r′)φj(r′)

|r - r′| ]φi(r) (5)K̂i(r)φi(r) ) [∑

j

N ∫dr′ φj/(r′)φi(r′)

|r - r′| ]φj(r) (6)

ΨCIS ) ∑ia

ciaΦi

a(r) (7)

Ĥ(r)Ψ(r) ) [T̂(r) + V̂el-nuc(r) + V̂el-el(r)]Ψ(r) )EΨ(r) (8)

T̂(r) ) -∑i

1

2∇2i (9)

V̂el-nuc(r) ) -∑i∑K

ZK

|ri - rK|(10)

Vel-el ) ∑i

N

∑j>i

N 1

|ri - rj|(11)

∑ia

〈Φjb|Ĥ|Φia〉cia ) ECIS∑

iaci

aδijδab (12)

〈Φjb|Ĥ|Φia〉 ) (E0 + �a - �i)δijδab + (ia||jb) (13)

∑ia

{(�a - �i)δijδab + (ia||jb)}cia ) ωCIS∑ia

ciaδijδab (14)


sponds to the antisymmetrized two-electron integrals,which are defined as

Equation 14 can be nicely written in matrix nota-tion as an eigenvalue equation

in which we use the unusual symbol A for the matrixrepresentation of the Hamiltonian in the space of thesingly excited determinants to make the connectionto later occurring equations more clear. ω is thediagonal matrix of the excitation energies, and X isthe matrix of the CIS expansion coefficients. Thematrix elements of A are given as

The excitation energies are finally obtained by solv-ing the following secular equation

that is, by diagonalization of the matrix A. Theobtained eigenvalues correspond to the excitationenergies of the excited electronic states, and itseigenvectors to the expansion coefficients accordingto eq 7.

2.1.2. Properties and LimitationsIn the previous section, the derivation of the CIS

working equations has been outlined, and it has beenshown how excitation energies and excited-statewave functions are obtained. Here, we want tocompile some useful properties of CIS:

(1) Since the CIS wave function is determined byvariation of the expansion coefficients of the ansatz(eq 7), its total energy corresponds to an upper boundof the true ground-state energy by virtue of theRaleigh-Ritz principle. Also all excited-state totalenergies are true upper bounds to their exact values.(2) Owing to Brillouin’s theorem, which states that“singly excited” determinants Φi

a(r) do not couple tothe ground state, that is,

the excited state wave functions are Hamiltonianorthogonal to the ground state. (3) In contrast toother truncated configuration interaction methods,for instance, CI with single and double excitations(CISD), CIS is size-consistent, that is, the totalground-state energy of two noninteracting systemsis independent of whether they are computed to-gether in one calculation or are computed indepen-dently. This is immediately plausible having Bril-louin’s theorem in mind, owing to which the ground-state energy within CIS is nothing else but theHartree-Fock energy. And since Hartree-Fock issize-consistent,1 so is CIS. (4) Another useful propertyof CIS is that it is possible to obtain pure singlet and

triplet states for closed-shell molecules by allowingpositive and negative combination of R and â excita-tions from one doubly occupied orbital. Due to itsconceptually simple ansatz (eq 7) and the listedproperties, the CIS excited-state wave functions arewell defined. Hence, their wave functions and corre-sponding energies are directly comparable, which isa particularly necessary prerequisite if one is inter-ested in transitions between excited states.

An analytic expression for the total energy ofexcited states can be obtained from eq 14 by addingE0 and multiplying from the left with the correspond-ing CIS vector. It reads

As a consequence, ECIS is analytically differentiablewith respect to external parameters, for example,nuclear displacements and external fields, whichmakes the application of analytic gradient techniquesfor the calculation of excited-state properties such asequilibrium geometries and vibrational frequenciespossible. Analytic first derivatives for CIS excitedstates have been published,45,46 and also secondderivatives are available.46 They have been imple-mented in various computer codes, for instance,Q-Chem, CADPAC, or TURBOMOLE.47-49

In general, excitation energies computed with theCIS method are usually overestimated, that is, theyare usually too large by about 0.5-2 eV comparedwith their experimental values (see, for instance, refs45, 50, and 51). This is on one hand because the“singly excited” determinants derived from the Har-tree-Fock ground state are only very poor first-orderestimates for the true excitation energies, since thevirtual orbital energies �a are calculated for the (N+ 1)-electron system instead of for the N-electronsystem.1 Consequently, the orbital energy difference(�a - �i), which is the leading term in eq 17, is notrelated to an excitation energy, if the canonicalHartree-Fock orbitals are used as reference. In otherwords, the canonical HF orbitals are not a particu-larly good basis for the expansion of the correlatedwave function, which then needs a high flexibility tocompensate for this disadvantage, that is, doubly andhigher excited determinants are demanded in thewave function. On the other hand, since electroncorrelation is generally neglected within the CISmethod, the error will be the differential correlationenergy, which must be at least on the order of thecorrelation energy of one and sometimes severalelectron pairs. Such energies are typically on theorder of 1 eV per electron pair, and hence, one shouldexpect errors of this magnitude.

Furthermore, CIS does not obey the Thomas-Reiche-Kuhn dipole sum rule, which states that thesum of transition dipole moments must be equal tothe number of electrons.52-54 Thus, transition mo-ments cannot be expected to be more than qualita-tively accurate. While a full presentation of thecomputational algorithms used to evaluate the CISenergy is beyond our present scope, it is still impor-tant to understand the dependence of computationalcost on molecule size. The computational cost for CIS

(ia||jb) ) ∫∫dr dr′[φi(r)φa(r)φj(r′)φb(r′) - φi(r)φj(r)φa(r′)φb(r′)|r - r′| ] (15)

AX ) ωX (16)

Aia,jb ) (�a - �i)δijδab + (ia||jb) (17)

(A - ω)X ) 0 (18)

〈Φia(r)|Ĥ|Φ0(r)〉 ) 0 (19)

ECIS ) EHF + ∑ia

(cia)2(�a - �i) + ∑

ia,jbci

a cjb(ia||jb) (20)


calculations on large molecules is dominated by theevaluation and processing of two-electron integralsin the atomic orbital basis.45 Per state, this cost scaleswith the square of the molecule size for sufficientlylarge systems, because the number of significant two-electron integrals also grows quadratically for largeenough molecules (for very small molecules thegrowth is fourth-order). This is under the assumptionthat an atomic orbital basis of fixed size (per atom)is chosen. Instead, if the atomic orbital basis on agiven atom is increased in size for a given molecule,then the computational cost grows as O(n4), makinglarge basis set calculations very expensive. This canbe reduced to O(n3) by employing auxiliary basisexpansions, as is discussed in section 3.4 for TDDFT.Additionally there are linear algebra operations thecost of which scales with the cube of basis set sizethat are nonetheless relatively small in comparisonto two-electron matrix element contractions.

The small cost of the linear algebra is a result ofthe use of efficient Davidson-type algorithms55 toiteratively obtain a small number of excited-stateeigenvalues and eigenvectors, in comparison to thevery large rank of the CIS matrix, n(occ) × n(virt).Direct diagonalization would scale as the cube of thisrank, n(occ)3 × n(virt)3, which would be with the sixthpower of molecule size. Thus iterative diagonalizationis critical to the applicability of CIS (and the relatedTDHF and TDDFT methods) to large molecules.Additionally, iterative diagonalization allows memoryrequirements to be kept very modest, scaling withthe square of system size. This is because the fullCIS matrix, A, which grows as the square of the rank,n(occ)2 × n(virt)2, or the fourth power of molecule size,is never constructed directly. The iterative diagonal-ization consists of the repeated contraction of the CISmatrix A against a trial vector x for each state toproduce a residual vector r. This step in the AO basisbecomes the contraction of two-electron integralswith a density-like matrix representing x. On today’sstandard computers, this allows for the treatment offairly large molecules of about 300 first row atomsor 5000 or so basis functions.

2.2. Time-Dependent Hartree −FockThe time-dependent Hartree-Fock equations were

written down for the first time by Dirac as early as1930 following a density matrix and equation-of-motion formalism.56 Since then several differentderivations have been given (see, for instance, refs38 and 57-60), most notably the one by Frenkelusing a time-dependent variation principle.57 Thetime-dependent Hartree-Fock equations constitutean approximation to the exact time-dependent Schrö-dinger equation making use of the assumption thatthe system can at all times be represented by a singleSlater determinant composed of time-dependentsingle-particle wave functions. These equations, how-ever, which will be given later in detail, do notcorrespond to the scheme that is generally referredto when today’s quantum chemists speak of time-dependent Hartree-Fock (TDHF). What is meant,though, are the equations that are obtained in first-order time-dependent perturbation theory from Dirac’s

equation, that is, the linear response. In the follow-ing, we will follow this common convention. In mostquantum chemical applications, the TDHF approachis used to calculate electronic excitation spectra andfrequency-dependent polarizabilities of molecularsystems (for the latter see, for instance, ref 61).Furthermore, the linear response TDHF equationsare also well-known in physics under the namerandom phase approximation (RPA) and have beenapplied in various fields.38,59,60

2.2.1. Concepts and Derivation of TDHFThe starting point of the derivation of the TDHF

equation is the general time-dependent electronicSchrödinger equation for molecular systems

where Ĥ is the time-dependent Hamiltonian

where Ĥ(r) is defined as in eq 8 and V̂(r,t) correspondsto an arbitrary single-particle time-dependent opera-tor, for example, time-dependent electric field

With the approximation that Ψ(r,t) can be writtenas a single Slater determinant (which is the well-known Hartree-Fock assumption) of the form

a time-dependent variant of the Hartree-Fock equa-tion is obtained56 that reads

In addition to the definition of the time-indepen-dent Fock operator (eq 3), the operator F̂(r,t) containsthe time-dependent single-particle potential V̂(r,t).Furthermore, the Coulomb and exchange operatorsare analogously defined as in eqs 5 and 6, respec-tively, but acquire a time dependence since thesingle-particle wave functions φi(r,t) are now time-dependent.

Let us assume that at t ) 0 a molecular system isin a stationary state given by a single Slater deter-minant Φ0(r) that obeys the time-independentHartree-Fock equation (eq 2). Now a small time-dependent perturbation is applied, and the un-perturbed orbitals of the Slater determinant willrespond to this perturbation but change only slightly,since the perturbation is weak. The TDHF equationsare obtained via time-dependent perturbation theoryto first order, that is, the linear response of theorbitals and of the time-dependent Fock operator aretaken into account. The latter comprises the time-dependent perturbing potential itself as well as theresponse of the Coulomb and exchange operators dueto the change in the orbitals. One possible derivation

ĤΨ(r,t) ) i ∂∂t

Ψ(r,t) (21)

Ĥ(r,t) ) Ĥ(r) + V̂(r,t) (22)

V̂(r,t) ) ∑i

N

v̂i(r,t) (23)

Φ(r,t) ) |φ1(r,t)φ2(r,t)...φN(r,t)| (24)

F̂(r,t)Φ(r,t) ) i ∂∂t

Φ(r,t) (25)


of the TDHF equations is via a density matrixformulation, which is outlined later in section 3.2 forTDDFT, and as mentioned earlier, several otherroutes can be found in the literature (see, for ex-ample, refs 2, 38, 59, and 60). Therefore we postponea detailed derivation until section 3.2. After somealgebra and Fourier transformation from the time tothe energy domain, one arrives at a non-Hermitianeigenvalue equation, which yields the excitationenergies and transition amplitudes as eigenvaluesand corresponding eigenvectors. It can be writtenconveniently in matrix notation as

where the matrix elements are defined as follows

The leading term on the diagonal of the A matrix isthe difference of the energies of the orbitals i and a,which are the ones from which and to which theelectron is excited, respectively. The second term ofthe A matrix and the elements of the B matrix, theantisymmetrized two-electron integrals (eq 15), stemfrom the linear response of the Coulomb and ex-change operators to the first-order changes in thesingle-particle orbitals. It is worthwhile to note, andit will indeed be important later on, that the responseof the exchange operator corresponds to a Coulomb-like term and vice versa.

Furthermore, comparison of the TDHF eq 26 withthe CIS eigenvalue eq 16 reveals that the latter iscontained in the first one: when the B matrix of theTDHF equation is set to zero, it reduces to the CISscheme. In physics, this approximation is well-knownas the Tamm-Dancoff approximation,38,59 which willalso be discussed later in section 3.2 in the contextof TDDFT. Here, we instantly realize that there existtwo different derivation routes to CIS. One is viaprojection of the Hamiltonian operator Ĥ onto thespace of “singly excited” Slater determinants, aprocedure that we will term CI formalism in thefollowing, and the other is by time-dependent re-sponse theory to first order and subsequent neglectof the B matrix.

2.2.2. Properties and LimitationsIn the previous subsection, the basic concepts and

the working equation of linear response TDHF hasbeen derived, and it has been shown that it yieldsexcitation energies and transition vectors. In com-parison with CIS, TDHF is an extension since itcontains not only “singly excited” states but also“singly de-excited” states, which are constructed byinterchanging the orbital indices i and a. This state-ment ought to reflect the attempt to give a math-ematical procedure a physical meaning, but of course,the “de-excitations” are nonphysical since one cannotde-excite the Hartree-Fock ground state. Historicallyone can justify this statement, since TDHF (RPA) hasbeen constructed to include correlation effects in the

ground state by virtue of some classes of “doublyexcited“ Slater determinants, and in this context, onecan indeed speak about “de-excited” states.59 Inpractice, however, one uses the Hartree-Fock groundstate as reference state, and all necessary expectationvalues are evaluated with respect to it. This ap-proximation is also known as the quasi-boson ap-proximation, which in general is reasonable if corre-lation effects are only small in the correspondingground state. This again is indicated by the magni-tude of the Y amplitudes, which are a measure of theground-state correlation and which, as a conse-quence, should be small compared to the X ampli-tudes.

The time-dependent HF method exhibits similarproperties as the CIS scheme. It is a size-consistentmethod, and one can obtain pure singlet and tripletstates for closed-shell molecules. However, TDHFencounters problems with triplet states, and ingeneral, triplet spectra are only very poorly predictedby TDHF calculations. This is because the HF groundstate is used as the reference, which in many caseseven leads to triplet instabilities.62 In contrast to CIS,TDHF obeys the Thomas-Reiche-Kuhn sum rule ofthe oscillator strengths,63 and thus, one should expectimproved transition moments compared to CIS.

It is worthwhile to note that time-dependent HFhas a close connection to ground-state HF stabilitytheory, in which it is analyzed whether a convergedHF solution is stable in the sense that it correspondsto a minimum in parameter space.2,62,64 The structureof the equations to be solved for a stability analysisare very similar to the ones for TDHF, and they alsocontain the first (gradient) and second derivatives(Hessian) of the energy with respect to variationalparameters.

Since the energy of the excited states in TDHF aregiven by an analytical expression that is similar tothe one for CIS (eq 20), analytical first derivativesare accessible and have indeed been derived andimplemented in connection with TDDFT.65,66 As wewill see later, the analytical derivatives of TDHF arenecessary ingredients for analytical gradients inTDDFT when hybrid functionals are employed (sec-tion 3.4).

In all applications where the orbitals do not exhibittriplet instabilities, the difference matrix (A - B, eq26) becomes positive definite, which allows reductionof the non-Hermitian TDHF eq 26 into a Hermitianeigenvalue equation of half the dimension

with

From a technical point of view, the TDHF eq 26 canthen in analogy to CIS again be solved using theDavidson procedure,55 but the computational cost isroughly twice the one of a CIS calculation since inaddition to the CIS calculation one has to form trialvector-matrix products also for the B matrix.

Although TDHF is in some sense an extension ofCIS or equally well CIS is an approximation to

[A BB* A* ][XY ]) ω[1 00 -1 ][XY ] (26)

Aia,jb ) δijδab(�a - �i) + (ia||jb)Bia,jb ) (ia||bj) (27)

(A - B)1/2(A + B)(A - B)1/2Z ) ω2Z (28)

Z ) (A - B)-1/2(X + Y) (29)


TDHF, TDHF has not been very successful in thequantum chemistry community, that is, it has notbeen applied very often. Probably, this is becauseexcitation energies calculated with TDHF are slightlysmaller than the ones obtained with CIS, but theyare still overestimations. The effect of the additionalB matrix must be only small, since its elements aresupposed to be small. Otherwise the underlying“quasi-boson approximation” is a bad one. But if theelements of the B matrix are small, one can equallywell use the computationally cheaper CIS scheme.More seriously, the problem of the above-mentionedpoor treatment of triplet states does not exist in CIS.In summary, TDHF does usually not constitute asignificant improvement over CIS that would justifyits increased computational cost.

3. Time-Dependent Density Functional Theory

Twenty years after the formulation of the Runge-Gross theorem,39 which laid the theoretical founda-tion for time-dependent density functional theory(TDDFT), it has become one of the most prominentand most widely used approaches for the calculationof excited-state properties of medium to large molec-ular systems, for example, excitation energies, oscil-lator strengths, excited-state geometries, etc. Everyweek a large number of publications containingsuccessful applications of TDDFT appear in theliterature. In this section, first the formal foundationsof TDDFT are reviewed, which comprise the Runge-Gross theorem, the role of the action integral, andthe time-dependent Kohn-Sham equation (section3.1). They represent the necessary ingredients toderive the TDDFT equations in the linear responseformulation (section 3.2). Afterward, relations be-tween CIS, TDHF, TDDFT/TDA, and TDDFT arediscussed in section 3.3, and properties and limita-tions of TDDFT are outlined (section 3.4), wherespecial emphasis is put on the failure of TDDFT forCT excited states, which is one of the most severeproblems of TDDFT (section 3.5).

3.1. Formal Foundations

Traditional ground-state density functional theoryin the Kohn-Sham formulation (KS-DFT) relies onthe Hohenberg-Kohn (HK) theorems and the exist-ence of a noninteracting reference system, the elec-tron density of which equals the electron density ofthe real system (for reviews in the field of ground-state DFT, the reader is referred to refs 9-11, 67,and 68). The first HK theorem (HK I) establishes aone-to-one mapping between the exact electron den-sity, F(r), and the exact external potential, Vext(r), andsince Vext(r) determines the exact ground-state wavefunction Ψ(r), the exact ground-state wave functionis a functional of the electron density, Ψ[F](r). Thesecond HK theorem (HK II) ensures the existence ofa variational principle such that the electronic energyof a system calculated with a trial density is alwayshigher than the total energy obtained with the exactdensity. Both theorems together allow the construc-tion of an exact many-body theory using the electrondensity as the fundamental quantity. The assumption

of the existence of a noninteracting reference system,the ground state of which is a single Slater determi-nant and the electron density of which is by con-struction exactly equal to the electron density of theinteracting real system, leads to the derivation of thewell-known Kohn-Sham (KS) equations.

However, traditional KS-DFT is limited to time-independent systems, that is, ground states, and ifone wants to establish an analogous time-dependenttheory, time-dependent versions of the first andsecond HK theorems must be formulated and provenand a time-dependent KS equation must be derived.In the following three subsections, we present theRunge-Gross theorem, which is a time-dependentanalogue to HK I, we analyze the role of the actionintegral in a time-dependent variational principle,and we will outline the derivation of the time-dependent Kohn-Sham equations. The key steps ofthe mathematical derivations, as well as the associ-ated physical concepts will be outlined.

3.1.1. The Runge−Gross TheoremThe Runge-Gross theorem can be seen as the time-

dependent analogue of the first Hohenberg-Kohntheorem and constitutes the cornerstone of the formalfoundation of the time-dependent Kohn-Sham for-malism.39 It states that the exact time-dependentelectron density, F(r,t), determines the time-depend-ent external potential, V(r,t), up to a spatially con-stant, time-dependent function C(t) and thus thetime-dependent wave function, Ψ(r,t), up to a time-dependent phase factor. The wave function is thus afunctional of the electron density

with (d/dt)R(t) ) C(t). The density, as well as thepotential, has to fulfill certain requirements, whichwe will encounter in detail as we proceed with theproof of the Runge-Gross theorem. The proof startsfrom the general time-dependent Schrödinger equa-tion

where

T̂(r), V̂el-nuc, and V̂el-el(r) correspond to the kineticenergy operator, the electron-nuclei attraction, andelectron-electron repulsion as defined in eqs 9-11,respectively. V̂(t) is a time-dependent external po-tential and is given as a sum of one-particle poten-tials

N is the number of electrons and is constant withtime. The electron density is given as

Ψ(r,t) ) Ψ[F(t)](t) e-iR(t) (30)

i ∂∂t

Ψ(r,t) ) Ĥ(r,t)Ψ(r,t) (31)

Ĥ(r,t) ) T̂(r) + V̂el-el(r) + V̂el-nuc(r) + V̂(t) (32)

V̂(t) ) ∑i

N

v̂(ri,t) (33)

F(r,t) ) ∫|Ψ(r1,r2,r3,...rN,t)|2 dr2 dr3 ... drN (34)


In the following, spin variables will be omitted forclarity. To prove the Runge-Gross theorem, it mustbe demonstrated that two densities FA(r,t) and FB(r,t)evolving from a common initial state Ψ0 under theinfluence of two different potentials vA(r,t) and vB-(r,t) are always different if the two potentials differby more than a purely time-dependent function, thatis

The first assumption to be made is that the potentialscan be expanded in a Taylor series in time around t0according to

Since vA(r,t) and vB(r,t) differ by more than a time-dependent function, some of the expansion coef-ficients, vk

A(r) ) (∂kvA(r,t))/∂tk|t0 and vkB(r) ) (∂kvB(r,t))/∂tk|t0, must differ by more than a constant. Hence,there exists one smallest integer k such that

From here, the proof proceeds in two steps. First itwill be shown that the current densities, jA(r,t) andjB(r,t), corresponding to vA(r,t) and vB(r,t) are alwaysdifferent, and in a second step, it will be derived thatdifferent current densities require different electrondensities. In general, the current density is definedas

By definition, the current density j(r,t) and theelectron density F(r,t) obey the so-called continuityequation69,70

which states that the temporal change of the electrondensity in a certain volume is equal to the flux ofcurrent density through the surface of that volume.Here, at initial time t ) t0, the current densities andthe electron densities are given as

and the time-evolution of the current densities jA(r,t)and jB(r,t) is given by their equation of motion

Subtraction of these equations yields

and evaluation of this expression at t ) t0 gives anexpression that relates the time evolution of thedifferent current densities to the external potentials.It reads

Consequently, if the potentials vA(r,t) and vB(r,t) differat t ) t0, the right-hand side of eq 43 cannot vanishidentically, and hence, the current densities jA(r,t)and jB(r,t) will be different infinitesimally later thant0. Thus we have established a one-to-one mappingbetween time-dependent potentials and current den-sities. It is worthwhile to note that eq 43 holds inthis form only if eq 36 is satisfied for k ) 0. If theinteger k for which eq 36 holds is greater than zero,the (k + 1)th time derivative of (jA(r,t) - jB(r,t)) mustbe evaluated at t ) t0 and the equation of motion isto be applied (k + 1) times. Corresponding math-ematical expressions can be found, for example, inref 41, which have a very similar form as eq 43 andlead to the same conclusion. To be able to make thisargument, the potential v(r,t) must be expandable ina Taylor expansion in time according to eq 36.

Since now a one-to-one mapping between time-dependent external potentials and time-dependentcurrent densities is established, it remains to beproven that different current densities require dif-ferent electron densities. For this objective, thecontinuity equation (eq 39) is applied to FA(r,t) andFB(r,t), and subtraction of the resulting equations anddifferentiation with respect to time yields

Insertion of eq 43 yields

which corresponds to the desired relation betweentime-dependent electron densities and time-depend-ent external potentials. If one can show that theright-hand side of this equation cannot vanish identi-cally, it would be proven that FA(r,t) and FB(r,t) aredifferent if the corresponding external potentials aredifferent. This proof is done by reductio ad absurdumassuming that the right-hand side does vanish.According to Gauss’ theorem, the following equationis valid

vA(r,t) * vB(r,t) + C(t) (35)

v(r,t) ) ∑k)0

∞ 1

k!

∂kv(r,t)

∂tk|t0(t - t0)

k

) ∑k)0

∞ 1

k!vk(r)(t - t0)

k (36)

vkA(r) - vk

B(r) * const (37)

j(r,t) ) 12i

[Ψ*(r,t)∇Ψ(r,t) - ∇Ψ*(r,t)Ψ(r,t)] (38)

∂

∂tF(r,t) ) -∇j(r,t) (39)

jA(r,t0) ) jB(r,t0) ) j0(r)

FA(r,t0) ) FB(r,t0) ) F0(r) (40)

∂

∂tjA(r,t) ) -i〈Ψ(r,t)|[ĵ(r),ĤA(r,t)]|Ψ(r,t)〉

∂

∂tjB(r,t) ) -i〈Ψ(r,t)|[ĵ(r),ĤB(r,t)]|Ψ(r,t)〉 (41)

∂

∂t[jA(r,t) - jB(r,t)] )

-i〈Ψ(r,t)|[ĵ(r),{ĤA(r,t) - ĤB(r,t)}]|Ψ(r,t)〉 (42)

∂

∂t[jA(r,t) - jB(r,t)]t)t0 ) -F0∇(v

A(r,t) - vB(r,t)) (43)

∂2

∂t2[FA(r,t) - FB(r,t)] ) -∇ ∂

∂t[jA(r,t) - jB(r,t)] (44)

∂2

∂t2[FA(r,t) - FB(r,t)] ) ∇[F0∇(vA(r,t) - vB(r,t))] (45)

∫F0(∇(vA - vB))2 d3r )-∫(vA - vB)∇(F0∇(vA - vB)) d3r +

I(vA - vB)F0∇(vA - vB) dS (46)


For real, experimentally realizable potentials, thesurface integral vanishes, because the fall off of theasymptote is at least as 1/r, and the second term ofthe right-hand side vanishes by assumption. Becausethe integrand is nonnegative, it follows that for all r

and since F0 is greater than zero

and thus

which is in contradiction to the assumption (eq 35).Consequently, the right-hand side of eq 45 cannotvanish identically, and for different time-dependentexternal potentials at t ) t0, one obtains differenttime-dependent electron densities infinitesimallylater than t0. With this, the one-to-one mappingbetween time-dependent densities and time-depend-ent potentials is established, and thus, the potentialand the wave function are functionals of the density.

Recently, van Leeuwen presented a generalization ofthe Runge-Gross theorem and proved that a time-dependent density F(r,t) obtained from a many-particle system can under mild restriction on theinitial state always be reproduced by an externalpotential in a many-particle system with differenttwo-particle interaction. For two states with equiva-lent initial state and the same two-particle interac-tion, van Leeuwen’s theorem reduces to the Runge-Gross theorem.71

Furthermore, the expectation value of any quan-tum mechanical operator is a unique functional ofthe density because the phase factor in the wavefunction cancels out according to

Strictly speaking, the expectation value implicitlydepends also on the initial state, Ψ0, that is, it is afunctional of F and Ψ0. For most cases, however,when Ψ0 is a nondegenerate ground state, O[F](t) isa functional of the density alone, because Ψ0 is aunique functional of its density F0 by virtue of thetraditional first Hohenberg-Kohn theorem.

3.1.2. The Action IntegralIn the previous section, the one-to-one mapping

between time-dependent potentials and time-depend-ent functionals has been established, which repre-sents the first step in the development of a time-dependent many-body theory using the density asa fundamental quantity. A second requirement isthe existence of a variational principle in analogy tothe time-independent case, in which it is given bythe above-described second Hohenberg-Kohn theo-rem. In general, if the time-dependent wave function

Ψ(r,t) is a solution of the time-dependent Schrödingereq 31 with the initial condition

then the wave function corresponds to a stationarypoint of the quantum mechanical action integral.

which is a functional of F(r,t) owing to the aboveproven Runge-Gross theorem, that is,

Consequently, the exact electron density F(r,t) can beobtained from the Euler equation

when appropriate boundary conditions are applied.Furthermore from eq 32, the action integral can besplit into two parts, one that is universal (for a givennumber of electrons) and the other dependent on theapplied potential v(r,t) ) Vel-nuc(r) + V(r,t)

The universal functional B[F] is independent of thepotential v(r,t) and is given as

In summary, the variation of the action integral withrespect to the density according to eq 54 is aprescription of how the exact density can be obtained.In the next section, this stationary action principlewill be applied to derive a time-dependent Kohn-Sham equation in analogy to the time-independentcounterpart.

3.1.3. The Time-Dependent Kohn−Sham EquationIn analogy to the derivation of the time-indepen-

dent Kohn-Sham equations, it is assumed that atime-dependent noninteracting reference system ex-ists with external one-particle potential vS(r,t) ofwhich the electron density FS(r,t) is equal to the exactelectron density F(r,t) of the real interacting system.According to the generalization of the Runge-Grosstheorem by van Leeuwen,71 the existence of a time-dependent noninteracting reference system is usuallyensured. The noninteracting system is then repre-sented by a single Slater determinant Φ(r,t) consist-ing of the single-electron orbitals φi(r,t); thus, itsdensity is given by

F0[∇(vA(r,t) - vB(r,t))]2 ) 0 (47)

∇(vA(r,t) - vB(r,t)) ) 0 (48)

vA(r,t) ) vB(r,t) + const (49)

F(r,t) T v[F](r,t) + C(t) T Ψ[F](r,t) e-iR(t)

O(t) ) 〈Ψ[F](r,t)|Ô(t)|Ψ[F](r,t)〉 ) O[F](t) (50)

Ψ(r,t0) ) Ψ0(t) (51)

A ) ∫t0t1 dt 〈Ψ(r,t)|i ∂∂t - Ĥ(r,t)|Ψ(r,t)〉 (52)

A[F] ) ∫t0t1 dt 〈Ψ[F](r,t)|i ∂∂t - Ĥ(r,t)|Ψ[F](r,t)〉 (53)

δA[F]δF(r,t)

) 0 (54)

A[F] ) B[F] - ∫t0t1 dt ∫d3r F(r,t)v(r,t) (55)

B[F] ) ∫t0t1 dt 〈Ψ[F](r,t)|i ∂∂t - T̂(r) -V̂el-el(r)|Ψ[F](r,t)〉 (56)

F(r,t) ) FS(r,t) ) ∑i

N

|φi(r,t)|2 (57)


Provided that the one-particle potential vS(r,t) exists,the single-electron orbitals are then given as thesolution of the time-dependent one-particle Schröd-inger equation

On the other hand, the noninteracting density,which is by assumption equal to the exact density,is also determined by the Euler eq 54, in which theaction integral is varied with respect to the density.For the noninteracting system (V̂el-el ) 0 in eq 56),the action functional takes on the following appear-ance:

where

and v(r,t) corresponds as usual to the time-dependentexternal potential as defined in eq 33. Applying thestationary action principle (eq 54) yields

If now a time-dependent single-particle potential vS-(r,t) exists that allows for the construction of the time-dependent single-particle Schrödinger eq 58, thispotential is a unique functional of the density byvirtue of the Runge-Gross theorem and can accord-ing to eq 61 be expressed as

evaluated at the exact interacting density F(r,t). Toobtain more information about the properties of BS-[F], one considers the action functional (eq 55) of theinteracting system and as a first step rewrites it inthe following form

Axc is the so-called “exchange-correlation” part ofthe action integral and is defined as

If eq 63 is inserted into the stationary action principlecorresponding to the Euler eq 54, one obtains

This equation is only solved by the exact interactingdensity. Comparison with eq 62 gives an expressionfor the time-dependent single-particle potential

Inserting this equation into the time-dependentsingle-particle Schrödinger eq 58 yields the time-dependent Kohn-Sham equations

in which the density is given according to eq 57. Thetime-dependent Kohn-Sham equations are, in anal-ogy to the time-independent case, single-particleequations in which each electron is treated individu-ally in the field of all others. The kinetic energy ofthe electrons is represented by -1/2∇i2; the externaltime-dependent potential v(r,t) and the Coulombinteraction between the charge distribution of allother electrons with the electron under considerationare explicitly contained. In analogy to the traditionaltime-independent Kohn-Sham scheme all exchangeand correlation effects (explicit Coulomb interactionbetween the electrons) are collected in (δAxc[F])/(δF-(r,t)). To this end, no approximation has been intro-duced and consequently the time-dependent Kohn-Sham theory is a formally exact many-body theory.However, the exact time-dependent “exchange-cor-relation” action functional (also called the xc kernel)is not known, and approximations to this functionalhave to be introduced. The first approximation gen-erally made is the so-called adiabatic local densityapproximation (ALDA) in which the originally non-local (in time) time-dependent xc kernel is replacedwith a time-independent local one based on theassumption that the density varies only slowly withtime. This approximation allows the use of a standardlocal ground-state xc potential in the TDDFT frame-work. The available approximate xc functionals willbe discussed in section 3.4.

The time-dependent Kohn-Sham equations can beconveniently expressed in matrix notation in a basisof, say, M time-independent single-particle wavefunctions {øi(r)} such that

Then, the time-dependent KS equation reads

Here, the ith column of the matrix C contains thetime-dependent expansion coefficients of φi(r,t) andFKS is the matrix representation of the time-depend-

i ∂∂t

φi(r,t) ) (- 12∇i2 + vS(r,t))φi(r,t) (58)

AS[F] ) BS[F] - ∫t0t1 dt ∫ d3r F(r,t)vS(r,t) (59)

BS[F] ) ∫t0t1 dt 〈Ψ[F](r,t)|i ∂∂t - T̂(r)|Ψ[F](r,t)〉 (60)

δAS[F]δF(r,t)

) 0 )δBS[F]δF(r,t)

- vS(r,t) (61)

vS(r,t) )δBS[F]δê(r,t)

|ê(r,t))F(r,t) (62)

A[F] ) BS[F] - ∫t0t1 dt ∫d3r F(r,t)v(r,t) -12∫t0t1 dt ∫d3r ∫d3r′

F(r,t)F(r′,t)|r - r′| - Axc[F] (63)

Axc[F] ) BS[F] -12∫t0t1 dt ∫ d3r ∫ d3r′

F(r,t)F(r′,t)|r - r′| -

B[F] (64)

δBS[F]δF(r,t)

) v(r,t) + ∫ d3r′ F(r′,t)|r - r′| +δAxc[F]δF(r,t)

(65)

vS(r,t) ) v(r,t) + ∫ d3r′ F(r′,t)|r - r′| +δAxc[F]δF(r,t)

(66)

i ∂∂t

φi(r,t) )

(- 12∇i2 + v(r,t) + ∫ d3r′ F(r′,t)|r - r′| + δAxc[F]δF(r,t) )φi(r,t)i ∂∂t

φi(r,t) ) F̂KS

φi(r,t) (67)

φp(r,t) ) ∑j

M

cpj(t)øj(r) (68)

i ∂∂t

C ) FKSC (69)


ent Kohn-Sham operator in the given basis. Multi-plication of eq 69 from the right with C† and thensubtraction from the resultant equation of its Her-mitian transpose leads to the Dirac form of the time-dependent Kohn-Sham equation in density matrixform. This equation reads

in which the density matrix Ppr is in general relatedto the electron density via

To obtain excitation energies and oscillator strengthsemploying the time-dependent KS approach, twodifferent strategies can be followed. One possibilityis to propagate the time-dependent KS wave functionin time, which is referred to as real-time TDDFT.72,73This technique still has the status of an expert’smethod but is beginning to be used in chemistry andbiophysics, and some successful applications havebeen reported recently in the literature.74,75 In thefollowing section, however, we want to focus on theanalysis of the linear response of the time-dependentKS equation. This leads to the linear-response TD-DFT equations, which correspond to the widely usedTDDFT scheme implemented in most standard quan-tum chemistry codes.

3.2. Derivation of the Linear-Response TDDFTEquation

In this section, the derivation of the linear responseTDDFT equation is presented. Using a densitymatrix formalism, it is shown how the excitationenergies are obtained from the linear time-dependentresponse of the time-independent ground-state elec-tron density to a time-dependent external electricfield. Before the time-dependent electric field isapplied, the system is assumed to be in its electronicground state, which is determined by the standardtime-independent Kohn-Sham equation, which inthe density matrix formulation takes on the followingappearance:

with the idempotency condition

Fpq(0) and Ppq

(0) correspond to the Kohn-Sham Hamil-tonian and density matrix of the unperturbedground state, respectively. The elements of the time-

independent Kohn-Sham Hamiltonian matrix aregiven as

In the basis of the orthonormal unperturbed single-particle orbitals of the ground state, these matricesare simply given as

and

Again, we follow the convention that indices i, j, etc.correspond to occupied orbitals, a, b, etc. correspondto virtual orbitals and p, q, r, etc. refer to generalorbitals, and �p is the orbital energy of the one-electron orbital p.

Now, an oscillatory time-dependent external fieldis applied, and the first-order (linear) response to thisperturbation is analyzed. In general perturbationtheory, the wave function or in this case the densitymatrix is assumed to be the sum of the unperturbedground state and its first-order time-dependentchange,

The same holds for the time-dependent Kohn-Sham Hamiltonian, which to first order is given asthe sum of the ground-state KS Hamiltonian and thefirst-order change

Substituting eqs 77 and 78 into the time-dependentKohn-Sham eq 70 and collecting all terms of firstorder yield

The first-order change of the Kohn-Sham Hamilto-nian consists of two terms. The first contributioncorresponds to the applied perturbation, the time-dependent electric field itself, and it has been shownthat it is sufficient to consider only a single Fouriercomponent of the perturbation,2 which is given inmatrix notation as

∑q

{FpqPqr - PpqFqr} ) i∂

∂tPpr (70)

F(r,t) ) ∑p,q

M

cp(t)cq/(t)øp(r)øq

/(r)

) ∑p,q

M

Ppqøp(r)øq/(r) (71)

∑q

{Fpq(0) Pqr

(0) - Ppq(0) Fqr

(0)} ) 0 (72)

∑q

Ppq(0) Pqr

(0) ) Ppr(0) (73)

Fpq(0) ) ∫ d3r φp/(r){- 12∇2 - ∑K)1M ZK|r - RK| +

∫ d3r′ F(r′)|r - r′| +δExc

δF(r)}φq(r) (74)

Fpq(0) ) δpq�p (75)

Pij(0) ) δij

Pia(0) ) Pai

(0) ) Pab(0) ) 0 (76)

Ppq ) Ppq(0) + Ppq

(1) (77)

Fpq ) Fpq(0) + Fpq

(1) (78)

∑q

[Fpq(0) Pqr

(1) - Ppq(1) Fqr

(0) + Fpq(1) Pqr

(0) - Ppq(0) Fqr

(1)] )

i∂

∂tPpr

(1) (79)

gpq )12

[fpq e-iωt + fqp

/ eiωt] (80)


In this equation, the matrix fpq is a one-electronoperator and describes the details of the appliedperturbation. Furthermore, the two-electron part ofthe Kohn-Sham Hamiltonian reacts on the changesin the density matrix, on which it explicitly depends.The changes in the KS Hamiltonian due to thechange of the density are given to first order as

such that the first-order change in the KS Hamilto-nian is altogether given as

Turning to the time-dependent change of thedensity matrix induced by the perturbation of the KSHamiltonian, this is to first order given as

where dpq represent perturbation densities. Insertingeqs 80-83 into eq 79 and collecting the terms thatare multiplied by e-iωt yield the following expression

The terms multiplied by eiωt lead to the complexconjugate of the above equation. The idempotencycondition (eq 73) gives an expression for the first-order change of the density matrix of the form

which restricts the form of the matrix dpq in eq 84such that occupied-occupied and virtual-virtualblocks dii and daa are zero, and only the occupied-virtual and virtual-occupied blocks, dia and dai,respectively, contribute and are taken into account.Remembering the diagonal nature of the unperturbedKS Hamiltonian and density matrixes, one obtainsthe following pair of equations:

where we have set xai ) dai and yai ) dia to followconventional nomenclature. In the zero-frequencylimit (fai ) fia ) 0), that is, under the assumption that

the electronic transitions occur for an infinitesimalperturbation, and making use of the fact that in thebasis of the canonical orbitals Fpp

(0) ) �p and Pii(0) ) 1

(eqs 75 and 76), one obtains a non-Hermitian eigen-value equation, the TDDFT equation,

the structure of which is equivalent to the TDHF eq26 introduced in section 2.2. Here, the elements ofthe matrices A and B are given as

where the two-electron integrals are again given inMulliken notation. In comparison with the TDHF eq26, the definitions of the matrix elements differ onlyin their last terms. While in TDHF the last termscorrespond to the response of the nonlocal HF ex-change potential, which yields a Coulomb-like term,they correspond in TDDFT to the response of thechosen xc potential, which replaces the HF exchangepotential in KS-DFT. In the ALDA approximation(see section 3.1.3), the response of the xc potentialcorresponds to the second functional derivative of theexchange-correlation energy, which is also called thexc kernel, and is given as

Explicit expressions for the xc kernel are given, forexample, in ref 76.

An alternative elegant route to the derivation ofthe linear response expressions for TDDFT (eq 88)via the energy-dependent density-density responsefunction ø(r,r′,ω) of the interacting system, whichcontains all physical information about how the exactdensity F(r,ω) changes upon small changes in theexternal potential vext(r,ω), has been presented byMarques and Gross.44 The quantities are energy-dependent since they correspond to the Fouriertransforms of the corresponding time-dependent ones.The change in the density can equally well becalculated using the response of the noninteractingKohn-Sham system, øKS(r,r′,ω) and is given as

From eq 91 a formally exact expression for the exactdensity response function of the interacting systemcan be derived that reads

Knowing that the exact density response functionpossesses poles at the exact excitation energies of thesystem,44 one can starting from eq 92 through a series

∆Fpq(0) ) ∑

st

∂Fpq(0)

∂PstPst

(1) (81)

Fpq(1) ) gpq + ∆Fpq

(0) (82)

Ppq(1) ) 1

2[dpq e

-iωt + dqp/ eiωt] (83)

∑q [Fpq(0)dqr - dpqFqr(0) + (fpq + ∑st ∂Fpq(0)∂Pst dst)Pqr(0) -

Ppq(0)(fqr + ∑

st

∂Fqr(0)

∂Pstdst)] ) ωdpr (84)

∑q

{Ppq(0) Pqr

(1) + Ppq(1) Pqr

(0)} ) Ppr(1) (85)

Faa(0)xai - xaiFii

(0) +

(fai + ∑bj

{∂Fai∂Pbj xbj + ∂Fai∂Pjb ybj})Pii(0) ) ωxai (86)Fii

(0)yai - yaiFaa(0) -

Pii(0)(fia + ∑

bj{∂Fia∂Pbj xbj + ∂Fia∂Pjb ybj}) ) ωyai (87)

[A BB* A* ][XY ]) ω[1 00 -1 ][XY ] (88)

Aia,jb ) δijδab(�a - �i) + (ia|jb) + (ia|fxc|jb)Bia,jb ) (ia|bj) + (ia|fxc|bj) (89)

(ia|fxc|jb) )

∫ d3r d3r′ φi/(r)φa(r) δ2Exc

δF(r)δF(r′)φb/(r′)φj(r′) (90)

δF(r,ω) ) ∫ d3r′ øKS(r,r′,ω)δvS(r′,ω) (91)

ø(r,r′,ω) ) øKS(r,r′,ω) + ∫ d3 r′′ ∫d3 r′′′ ø(r,r′′,ω)[ 1|r′′ - r′′′| + fxc(r′′,r′′′,ω)]øKS(r′′′,r′,ω) (92)


of algebraic manipulations arrive at a pseudo-eigen-value equation similar to eq 88, which yields exactexcitation energies. If furthermore the single-poleapproximation (SPA)77 and the ALDA approximationas described in section 3.1.3 are employed, oneobtains exactly the linear-response TDDFT, eq 88.

In analogy to TDHF and CIS (section 2.2), theTamm-Dancoff approximation (TDA) to TDDFT hasrecently been introduced.78 Again, it corresponds toneglecting the matrix B in eq 88, that is, only theoccupied-virtual block of the initial d matrix (eq 84)is taken into account. This leads to a Hermitianeigenvalue equation

where the definition of the matrix elements of A isstill the same as in eq 89. It is worthwhile to notethat TDA/TDDFT is usually a very good approxima-tion to TDDFT.78,79 A possible reason may be that inDFT correlation is already included in the groundstate by virtue of the xc functional, which is not thecase in HF theory. Since the magnitude of the Yamplitudes and the elements of the B matrix are ameasure for missing correlation in the ground state,they should be even smaller in TDDFT than in TDHFand, thus, be less important. TDDFT is also moreresistant to triplet instabilities than TDHF.

Equations 88 and 89 represent the TDDFT formal-ism, which is solved to obtain excitation energies ωand transition vectors |XY〉 when the unperturbed KSHamiltonian (eq 74), from which the response isderived, contains a so-called pure DFT xc potentialand not also parts of Hartree-Fock exchange. How-ever, today it is very common to include parts ofHartree-Fock exchange in the xc potential, whichcorresponds to using so-called hybrid functionals. Forinstance, the widely used B3LYP functional contains20% HF exchange (cHF ) 0.2 in eq 94).80 In this case,the unperturbed KS Hamiltonian acquires an ad-ditional term and takes on the following appearance:

Using this more general time-independent KS Hamil-tonian and following the same derivation route asabove, one arrives at more general expressions forthe TDDFT equations, which allow for the use ofhybrid functionals. Although the non-Hermitian eigen-value eq 88 remains the same, the elements of itsmatrices A and B are now given as

containing the response of the Hartree-Fock ex-change potential, as well as the one of the chosen xcpotential at a rate determined by the factor cHFdetermined in the hybrid xc functional. Comparingthe definitions of the matrix elements of TDHF (eqs27) and TDDFT (eqs 89) with eq 96, it becomesapparent that the latter equation contains TDHF andTDDFT as limiting cases if cHF ) 1 or cHF ) 0,respectively. In other words, eq 96 corresponds to alinear combination of eqs 27 and 89 thereby combin-ing TDHF and TDDFT in one hybrid scheme.

3.3. Relation of TDDFT and TDHF, TDDFT/TDA,and CIS

In the previous section, we have seen that TDDFTand TDHF are closely related by a hybrid schemethat emerges when the unperturbed Kohn-ShamHamiltonian contains both a nonlocal Hartree-Fockexchange potential and a local xc potential. In Figure1, the relations between the introduced methods CISand TDHF as well as TDDFT/TDA and TDDFT areschematically shown. Starting from the usual time-independent Hartree-Fock scheme, the ground-stateKohn-Sham equations are in principle obtained byexchanging the nonlocal HF exchange potential, vx

HF

with the local Kohn-Sham xc potential vxc. Analysesof the linear response of the ground-state densitycalculated either with HF or with DFT to an externaltime-dependent perturbation lead to the TDHF orTDDFT schemes, respectively, as is indicated by thevertical arrows in Figure 1.

In analogy to the ground-state methods, TDHF andTDDFT are similarly related since the TDHF equa-tions can be converted into the TDDFT equations bysimply replacing the response of the HF exchangepotential, ∂vx

HF/∂F1, by the response of the xc poten-tial from DFT, ∂vxc/∂F1. The Tamm-Dancoff ap-proximation, that is, the neglect of the B matrix inthe TDHF or TDDFT equations, leads in the TDHFcase to CIS, while in the TDDFT case one obtainsthe TDDFT/TDA equations. Of course, CIS andTDDFT/TDA are equally related as TDHF and TD-DFT are. However, in the case of CIS there exists aderivation route other than the Tamm-Dancoff ap-proximation to TDHF, namely, via a CI formalism,that is, direct projection of the molecular Hamiltonianonto the “singly excited” Slater determinants. Thisis only possible because the projection of the Coulomboperator ∑1/(r1 - r2) contained in the molecularHamiltonian yields terms equivalent to the response

AX ) ωX (93)

Fpq(0) ) ∫ d3r φp/(r){- 12∇2 + ∑K)1M -ZK|r - RK| +

∫ d3r′ F(r′)|r - r′| - cHF∫ d3 r′

F(r,r′)

|r - r′|+

(1 - cHF)δExc

δF(r)}φq(r) (94)

Aia,jb ) δijδab(�a - �i) + (ia|jb) - cHF(ij|ab) +(1 - cHF)(ia|fxc|jb) (95)

Bia,jb ) (ia|bj) - cHF(ib|aj) + (1 - cHF)(ia|fxc|bj) (96)

Figure 1. Schematic sketch of the relation betweenHartree-Fock (HF) and density functional theory (DFT),time-dependent Hartree-Fock and time-dependent DFT(TDDFT), and configuration interaction singles (CIS) andthe Tamm-Dancoff approximation to TDDFT (TDA).


of the Hartree-Fock exchange and Coulomb poten-tial. Obviously, this is not the case for TDDFT/TDA, since the response of the xc potential containsthe second derivative of the approximate local xcfunctional. Projection of the KS determinant wouldresult in equations analogous to the CIS equationbut the matrix elements evaluated for the KS orbit-als.

3.4. Properties and LimitationsIn section 3.2, it was presented how the linear

response formulation of time-dependent density func-tional theory allows for the calculation of excitationenergies and transition vectors in the DFT frame-work. At present, TDDFT represents one of the mostprominent approaches for this task, especially whenexcited states of medium-sized or large molecularsystems are under investigation.

Since the exact local xc potential, which is the keyingredient in DFT-based methods, is not known, anapproximate xc functional has to be chosen in anypractical calculation. Many different flavors of xcfunctionals are available today, for example, localfunctionals such as Slater-Vosko-Wilk-Nussair(SVWN),56,81 gradient-corrected ones (GGAs) such asBecke-Lee-Yang-Parr (BLYP),82,83 Perdew-Burke-Enzerhof (PBE),84,85 or Becke-Perdew 1986 (BP86),82,86and hybrid functionals such as Becke3-Lee-Yang-Parr (B3LYP).80 Moreover, the development of im-proved xc functionals is still a very active field ofresearch. At present, the functionals B3LYP and PBEare the most widely used xc functionals in standardground-state DFT applications, and although allthese functionals have been developed with respectto the electronic ground state, they are also employedin TDDFT calculations, which is a consequence of theALDA approximation (section 3.1.3). In many cases,results obtained with TDDFT are quite sensitive tothe choice of the xc functionals, in particular, whenlocal or GGA functionals are compared with hybridfunctionals (see, for instance, refs 87 and 88). Thisaspect is also discussed in section 5.1. Therefore, thereliability of TDDFT calculations should always bechecked by comparison with either wave-function-based benchmark calculations or experimental data,as well as by the sensitivity of the results to thechoices of xc functional.

Although approximate xc functionals are employedin TDDFT, it has been proven to yield accurateresults for valence-excited states the excitation ener-gies of which lie well below the ionization potential.For such states, the typical error of TDDFT lieswithin the range of 0.1-0.5 eV, which is almostcomparable with the error of high-level correlatedapproaches such as EOM-CCSD or CASPT2. How-ever, to reach such high accuracy within the linearresponse formulation of TDDFT, one needs to includelarge sets of virtual orbitals. Still, in TDDFT, theaccuracy is reached at very favorable computationalcost making TDDFT applicable to fairly large mol-ecules.

In general, the hybrid TDDFT scheme is solvedanalogously to the TDHF approach as it has beenoutlined in section 2.2.2. Again, the non-Hermitian

eigenvalue problem (eq 88) can be converted into aHermitian one, since the orbitals are usually real. Itreads

with

The solution of the hybrid TDDFT equations scalesvery similarly to the solution of the CIS and TDHFproblems, which were already discussed in sections2.1.2 and 2.2.2, respectively. However, hybrid TDDFTis slightly more expensive than TDHF since inaddition to the response of the HF exchange, theresponse of the xc potential also needs to be evalu-ated. In common with ground-state DFT calculations,this is usually done numerically on an atom-centeredthree-dimensional quadrature grid.89 With thresh-olding, this step can eventually scale linearly withthe size of the molecule for sufficiently large systems,and thus eventually it becomes insignificant. Theprefactor, however, is quite large, particularly forfiner grids, and thus for medium-sized molecules, thisstep can dominate the computation.

When in addition only pure xc functionals areemployed that do not include parts of HF exchange,the matrix (A - B) becomes diagonal (eq 89) and onecan fully exploit this to avoid its multiplication withthe trial vectors within the iterative Davidson scheme.In this case, the Tamm-Dancoff approximation doesnot imply a decrease in computational cost. On thecontrary, as soon as hybrid functionals are used and(A - B) is thus not diagonal, the matrix multiplica-tion must be performed and then TDDFT/TDA doessave computation time by a factor of approximatelytwo. Another algorithm to solve the TDDFT or TDHFresponse equations has been provided by Olsen etal.,90 which exploits the specific paired structure ofthe non-Hermitian eigenvalue eq 90. This algorithmdoes not require the involved matrices to be explicitlyavailable and thus allows for accurate calculationswith high dimensions.

Additional computation time can be saved by afactor of 3-8 depending on the system of interest ifthe resolution-of-the-identity (RI) approximation91 isused to evaluate the Coulomb-like terms of eq 96.92,93The RI approximation involves computational effortthat scales with the cube of system size for Coulombinteractions (and the fourth power for HF exchange)for fixed basis set size. It is particularly valuable formedium and large atomic orbital basis sets becausethe cost for fixed molecule size grows only as O(n3),while conventional AO integral processing grows asO(n4). The additional error introduced in the excita-tion energies by the RI approximation is negligiblesince the larger errors in the total energies of theground and excited states are subtracted out, makingTDDFT applicable to very large molecular systems.On a modern computer, molecular systems with upto 3000 basis functions (approximately 200 first rowatoms) can be treated by TDDFT in the standardimplementation and systems with up to 4500 or so

(A - B)1/2(A + B)(A - B)1/2Z ) ω2Z (97)

Z ) (A - B)-1/2(X + Y) (98)


basis functions (300 atoms) can be reached by TDDFTwhen the RI approximation is used.

The reason for the accuracy of TDDFT excitationenergies is that the difference of the Kohn-Shamorbital energies, which are the leading term of thediagonal elements of the A matrix in eq 96, areusually excellent approximations for excitation ener-gies.94,95 This is because the virtual KS orbital ener-gies are evaluated for the N-electron system and,thus, correspond more to the single-particle energyof an excited electron than to the energy of anadditional electron as in Hartree-Fock theory, wherethe virtual orbital energies are evaluated for the N+ 1 electron system. Consequently, orbital energydifferences are a much better estimate for valence-excited states in KS-DFT than in HF theory. Al-though TDDFT performs usually very well for valence-excited states, it is now well-known that TDDFT hassevere problems with the correct description of Ry-dberg states, valence states of molecules exhibitingextended π-systems,96,97 doubly excited states,98,99 andcharge-transfer excited states.100-103 For such states,the errors in the excitation energies can be as largeas a few electronvolts, and the potential energysurfaces can exhibit incorrect curvature. The prob-lems with Rydberg states and extended π-systemscan be attributed to the wrong long-range behaviorof current standard xc functionals, since they decayfaster than 1/r, where r is the electron-nucleusdistance. For example, asymptotically corrected func-tionals such as van Leeuwen-Baerends 1994 (LB94)104or statistical averaging of orbital potentials (SAOP)105or local exact exchange potentials106,107 yield substan-tially improved Rydberg state excitation energies.108States with substantial double excitation charactercannot be treated within linear response theory inthe usual ALDA approximation, since only singlyexcited states are contained in the linear responseformalism.98,99 They can however be recovered whenthe xc kernel is allowed to be frequency/energydependent, since the xc kernel is in fact stronglyfrequency-dependent close to a double excitation.98,99In the case of excited charge-transfer (CT) states,which are the topic of section 3.5, the excitationenergies are much too low and the potential energycurves do not exhibit the correct 1/R asymptote whenR corresponds to a distance coordinate between thepositive and negative charges of the CT state.

Since TDDFT and TDHF are very closely relatedlinear response theories, it is not surprising thatTDDFT possesses all the properties of TDHF (section2.2.2). The linear response equations can be derivedvariationally,66 but due to the approximate natureof the employed xc functionals, comparison with theexact total energy is not possible. One can say,though, that TDDFT is variational within the “modelchemistry”109 defined by the approximate xc func-tional. In analogy to TDHF, TDDFT is size-consis-tent, and it can yield pure spin singlet and tripletstates for closed-shell molecules. Furthermore, TD-DFT obeys the Thomas-Reiche-Kuhn sum rule ofthe oscillator strengths.42

Since 2002, analytic first geometric derivatives ofthe excitation energies given by the hybrid expression

of TDDFT (eqs 88 and 96) are available,65,66,110 whichallow for the efficient calculation of first-order prop-erties of excited states, for instance, their equilibriumgeometries and dipole moments. The equilibriumstructures, dipole moments, and harmonic frequen-cies of excited states calculated with TDDFT are ofgenerally high quality, which is comparable to thatof ground-state KS-DFT calculations.66 So far, theanalytic gradients have been implemented in CAD-PAC and TURBOMOLE.48,49

3.5. Charge-Transfer Excited States in TDDFT

As already mentioned in section 3.4, TDDFT yieldssubstantial errors for charge-transfer excitedstates.87,100,101,103,111 The excitation energies for suchstates are usually drastically underestimated, andmoreover, the potential energy curves of CT statesdo not exhibit the correct 1/R dependence along acharge-separation coordinate R,101,102,112 for in realitythe positive and negative charges in a CT stateelectrostatically attract each other and, thus, separa-tion of these charges must result in an attractive 1/Rdependence. The wrong shape of the potential energycurves excludes the CT states from the reliablecalculation of all molecular properties that involvetheir geometric derivative employing TDDFT orTDDFT/TDA.

These failures of TDDFT in the calculation of CTexcited states can be understood by analysis of thebasic eqs 88 and 96 for the case of charge-transferexcited states. In contrast to a valence-excited state,in a long-range, say for example, an intermolecularcharge-transfer state, an electron is transferred froman occupied orbital i of molecule A to a virtual orbitala of another molecule B (Figure 2). For simplicity,let us assume that the overlap between orbitals onmolecule A and orbitals on molecule B is zero.

Figure 2. Schematic sketch of a typical valence-excitedstate, in which the transition occurs on one of the indi-vidual molecules, that is, the orbitals i, j and a, b arelocated on the same molecule in contrast to a charge-transfer excited state in which an electron is transferredfrom orbital i on molecule A to orbital a on molecule B.When the molecules A and B are spatially separated fromeach other the orbitals i and j do not overlap with a and b.


Consequently, for such a state, all terms of eq 96containing products of occupied and virtual orbitalsvanish. This comprises the second and fourth termof the definition of the A matrix (eq 96) being theresponse of the Coulomb potential and the xc poten-tial of the KS operator, respectively. Only the firstterm, which is the difference of the one-particleenergies of the donor orbital i on A and the acceptororbital a on B, and the third term originating fromthe nonlocal HF exchange part of the Kohn-Shamoperator contribute to the A matrix of eq 86. Thisterm contributes to the matrix elements of A, sinceorbitals i and j are both on A and the orbitals a andb are on B. In fact, this term is a Coulomb-like term,since the created holes (orbitals i and j correspondingto the positive charge in the CT state) interact withthe electrons (orbitals a and b reflecting the negativecharge), which relates to the electrostatic attractionwithin the CT state. Consequently, this term isessential for the correct 1/R dependence of thepotential energy curves of CT states along theintermolecular separation coordinate. The same ar-guments are valid for the elements of the matrix B(eq 96), and all terms are zero, that is, this matrixdoes not contribute to CT states at all. For CT states,eq 96 reduces to the following simple expressions

It is now obvious from the definition of the A matrixin eq 100 that the excitation energy of a CT state inTDDFT is simply given by the difference of the orbitalenergies of the electron-accepting and electron-donat-ing molecular orbitals, �a and �i, respectively, whena pure local xc functional (for instance, SVWN,BLYP,82 or LB94) is employed, that is, cHF ) 0.Within HF theory this is already a rough estimatefor the energy of the CT state at large distances, sinceKoopman’s theorem states that -�i and -�a cor-respond to the ionization potential of molecule A andto the electron affinity of molecule B, respectively.This is because the occupied orbitals are calculatedfor the N-electron system, while the virtual orbitalsare formally evaluated for the (N + 1)-electronsystem. This is not the case in density functionaltheory following the Kohn-Sham formalism (DFT),since the same potential is used to calculate theoccupied and virtual orbitals. As a consequence, whilethe HOMO still corresponds to the IP, the LUMO isgenerally more strongly bound in DFT than in HFtheory and cannot be related to the EA. Since thenegative of the LUMO energy is therefore muchlarger than the true EA, the orbital energy differencecorresponding to a CT state is usually a drasticunderestimation of its correct excitation energy.Furthermore, since the excitation energy of a CTstate is simply given by the constant difference of thecorresponding orbital energies, the potential energycurves of CT states do not exhibit the correct 1/Rshape along a distance coordinate but are constant.As already mentioned, the electrostatic attractionbetween the positive charge (the holes i, j on A) andthe negative charge (the electrons a, b on B) is

contained in the second term of eq 100, whichcorresponds to the linear response of HF exchange.Therefore, the correct 1/R long-range behavior of thepotential energy surfaces can in principle be recov-ered by inclusion of nonlocal HF exchange in the xcpotential, which will improve the asymptotic behavioraccording to the factor cHF of the exchange functional(eq 100). This explains the observed asymptoticbehavior of potential energy curves of CT states of atetrafluoroethylene-ethylene complex calculated withthe SVWN, LB94, B3LYP, and “half-and-half” func-tional compared to configuration interaction singles(CIS) (Figure 3).101

The 1/R failure of TDDFT employing pure standardxc functionals has in fact been understood as anelectron-transfer self-interaction error. To clarify this,let us first inspect the case of TDHF (cHF ) 1 in eq90). The excitation energy of a long-range CT state,where an electron is excited from orbital i on moleculeA into orbital a on molecule B, is dominated by theorbital energy difference �a - �i. In general, �acontains the Coulomb repulsion of orbital a with alloccupied orbitals of the ground state including theorbital i, which is no longer occupied in the CT state.In other words, the electrostatic repulsion betweenorbitals a and i, the integral (aa|ii), is contained inthe orbital energy difference although orbital i isempty in the CT state. That means that the trans-ferred electron in orbital a experiences the electro-static repulsion with itself still being in orbital i, thatis, it experiences molecule A as neutral. This electron-transfer self-interaction effect is canceled in TDHFby the response of the HF exchange term, the secondterm of the A matrix (eq 100), being (ii|aa), whichgives rise to the particle-hole attraction. In pureTDDFT employing approximate xc functionals, HFexchange is not present and the electron-transfer self-interaction effect is not exactly canceled leading toan incorrect long-range behavior of their potential

Aiaσ,jbτ ) δστδijδab(�aσ - �iτ) - δστcHF(iσjσ|aτbτ) (99)Biaσ,jbτ ) 0 (100)

Figure 3. Comparison of the long-range behavior of thelowest CT excited state of an ethylene-tetrafluoroethylenedimer along the intermolecular separation coordinatecomputed with TDDFT employing the SVWN (black), LB94(red), B3LYP (green), and “half-and-half” (blue) functionalswith the curve obtained at the CIS (brown) level. Theexcitation energy of the lowest CT state at 4 Å is set tozero.


energy curves.It is also worthwhile to note another peculi

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