excitation energies of benzene from kohn-sham theory

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Excitation Energies of Benzene fromKohn]Sham Theory

NICHOLAS C. HANDY, DAVID J. TOZER*Department of Chemistry, University of Cambridge, Cambridge, CB2 1EW, UK

Received 13 July 1998; accepted 20 July 1998

ABSTRACT: Singlet and triplet vertical excitation energies are determined forbenzene using time-dependent Kohn]Sham density functional theory. Thepotential of the HCTH continuum functional is explicitly corrected, using arecently developed procedure, to impose the appropriate y1rr q C asymptotic

Ž U . Ž .behavior. The 48 valence p ª p and Rydberg p ª n s 3 excitation energiescomputed using this corrected potential have a mean absolute error of 0.12 eV.Q 1999 John Wiley & Sons, Inc. J Comput Chem 20: 106]113, 1999

Keywords: benzene; excited states; density functional theory

Introduction

Ž .ohn]Sham density functional theory DFTK is now widely used for the calculation ofground-state properties of molecules. Both local

Ž 1 2 .GGA functionals e.g., BLYP , HCTH and hybridŽ 3.functionals e.g., B3LYP are in common use. The

most successful functionals contain a set of param-eters whose values have been determined in sucha way that the functional gives predictions that are

Žin best agreement with ‘‘experimental’’ data e.g.,.energies, structures for a chosen set of molecules.

However, much of modern chemistry involveslinear response molecular properties, which arisefrom the response of a molecule to an externalelectromagnetic field. In this study we concentrate

Correspondence to: N. Handy*Present address: Department of Chemistry, University of

Durham, South Road, Durham, DH1 3LE, UK

on single-photon vertical excitation energies, whichare determined in DFT through the poles of thefrequency-dependent polarizability. If electric cur-

Ž .rent dependence is ignored, then by now stan-dard time-dependent DFT theory 4 ] 8 leads to thefollowing equation for the determination of theexcitation energies, v, when GGA functionals areused:

1 1 22 2 Ž .H H H y v I c s 0 1Ž .2 1 2

where H is the so-called electric Hessian, given in1a spin]orbital notation by:

Ž . Ž . Ž .H s e y e d q 2 ai ¬ bjai , b j1 a i ai , b j

d vx c Ž . Ž .q ai dr bj 2ž /dr

Thus, orbital energy differences, two-electron inte-Ž .grals of the xo ¬ xo variety, and one-electron inte-

grals involving first derivatives of the exchange-

( )Journal of Computational Chemistry, Vol. 20, 106]113 1999Q 1999 John Wiley & Sons, Inc. CCC 0192-8651 / 99 / 010106-08

EXCITATION ENERGIES OF BENZENE

correlation potential are involved. The magneticHessian, H , is positive definite diagonal, given2by:

Ž . Ž . Ž .H s e y e d 3ai , b j2 a i ai , b j

It has been amply demonstrated, for example,by Casida et al.5 and Bauernschmitt and Ahl-richs,7, 9 that valence excitation energies are calcu-lated to an accuracy of ; 0.3 eV, a great improve-ment over the corresponding time-dependent

Ž .Hartree]Fock or RPA predictions, and also agreat improvement over single configuration inter-

Ž .action SCI predictions. However, it has been alsorecognized that a straightforward implementationwith GGA functionals gives an extremely poordescription of excitations to Rydberg states. Casidaet al.5 were the first to understand that this wasnot due to an absence of electric current depen-dence in the functional, but rather was due to anincorrect asymptotic exchange-correlation poten-tial.

We have recently presented a new technique10

that corrects this deficiency in the asymptotic po-tentials of GGA functionals. Excitation energies ofseveral small molecules, determined using the cor-rected potential, are accurate for both valence andRydberg excitations. The aim of this study is toapply this new approach to the chemically impor-tant molecule, benzene.

We observe that Grimme11 has also presentedaccurate vertical excitation energies using a pa-rameterized single-excitation configuration interac-tion approach. He uses Kohn]Sham eigenvalues,suitably scaled Coulomb contributions, and a pa-rameterized exchange correction to the matrix ele-ments. It is encouraging that such an approach iscapable of giving good results, and we suggestthat the theory in the next section may providepartial understanding of why it is successful.

Asymptotically CorrectedExchange-Correlation Potentials

In this section we outline our new procedure fordetermining excitation energies using an asymp-totically corrected exchange-correlation potential.Full details may be found in ref. 10.

The Kohn]Sham equation, for a GGA func-w xtional, E r , is:x c

1 2 Ž . Ž . Ž . Ž .y = q v r q v r q v r y e f r s 0Ž .e x t J x c p p2

Ž .4

Ž .involving the external potential, v r , thee x tŽ .Coulomb potential, v r , and the exchange-corre-J

Ž . w x Ž .lation potential, v r s dE r rdr r . It is byx c x cnow established that, in the energetically impor-

Ž .tant regions, the form of v r for different GGAsx cŽis similar as is evident from their similar homo.eigenvalues , and also the form of the occupied

molecular orbitals is similar. However, there aresubstantial inadequacies in the asymptotic region.As r ª `, the Kohn]Sham equation for the homoorbital yields:

Ž . Ž .yI q 0 q 0 q v ` y e s 0 5x c hom o

or:

Ž . Ž .v ` s I q e 6x c hom o

where I is the ionization potential. Because it isnow well established, both by calculation and the-ory,12, 13 that for GGA functionals, e ; yIr2, ithom o

Ž .follows that v r must not vanish asymptotically.x cFurthermore, using the property that the exchangehole contains one electron, we can conclude that

Ž .the form of v r in the asymptotic region is givenx cby:

1Ž . Ž .v r ª y q I q e 7x c hom or

There is no GGA in present use that obeys thiscondition; most, if not all, give potentials thatvanish asymptotically. As a result, very few vir-tual orbitals are bound, and this leads to the ob-served deficiencies in Rydberg excitation energiescomputed using regular GGA functionals.

The Kohn]Sham matrix elements involve theŽ < < .evaluation of h v h by quadrature. To resolvek x c l

the error in the asymptotic potential of some GGAfunctional, we define an asymptotically correctedŽ .AC exchange-correlation potential, at a grid pointr as:

Ž . AC Ž . GG A Ž .a If r - as , for any A , v r s v rA A x c x c

Ž .b If r ) bs , ;A ,A A

Ž X .1 r rXAC Ž . Ž .v r s y dr q I q e 8H Xx c hom o< <N r y r

where r is the distance of r from atom A; s isA Athe Bragg]Slater radius of atom A14 ; a, b arenumerical parameters; N is the number of elec-

Ž .trons; and r r is any approximate molecular den-

JOURNAL OF COMPUTATIONAL CHEMISTRY 107

HANDY AND TOZER

Ž .sity we use the Hartree]Fock density . If, for oneACŽ .or more A, as - r - bs , then v r is ob-A A A x c

tained by a linear interpolation of the formulasŽ . Ž .given in eq. 8a and 8b . The expression in eq.

Ž .8b exhibits the appropriate asymptotic behaviorw Ž .xeq. 7 .

Kohn]Sham orbitals and eigenvalues deter-mined using this new potential can then be used in

Ž . Ž . weqs. 1 ] 3 to determine excitation energies thecorrected potential is not used for the term d v rdrx c

Ž .x 10in eq. 2 . Preliminary studies using the recentlydeveloped HCTH functional2 suggest that values

Ž . Ž .of a s 3.5 and b s 4.7 are reasonable for thedetermination of excitation energies and polariz-abilities. We recognize that there are many vari-eties of this recipe, although our investigations10

clearly show that the asymptotic correction is es-sential if Rydberg excitation energies are to bedetermined accurately. It should be noted that inall except the asymptotic region, the potential isthe functional derivative of a GGA functional, andindeed the change to the potential has little effectŽ .less than 0.001 E on total energies, calculated byhthe regular GGA expression.

Before describing the calculations on benzeneusing the asymptotically corrected HCTH poten-tial, we give a little more detail of the implementa-tion. Our program is an unrestricted KS code, but,in this closed-shell case, we slightly amended theHessian construction to give the singlet and triplet

Žexcitation energies separately it is interesting tonote that no transformed two-electron integrals are

.required for the triplet states , the formulas havepreviously been given by Bauernschmitt and Ahl-richs.7 The larger calculations, which involved de-termining the lowest 32 eigenvalues of a matrixexceeding 3000 in dimension, were performed us-ing our own15 version of the iterative Liu]David-son algorithm,16 which we found eminently suitedto this problem. The Hessian matrix was calcu-lated once and stored on disk. The most time-con-suming part of the calculation was the evaluation,by numerical quadrature, of the exchange-correla-tion contributions to the H matrix. We found that1we could use a modest quadrature, approximately4000 points surrounding each atom. However, westress the importance in the Kohn]Sham calcula-tions of not removing radial grid points at largedistances, because, otherwise, diffuse functions be-

Žcome continuum-like orbitals with near-zero in-.stead of positive energies. It appears that the

Euler]Maclaurin radial quadrature scheme17 is ad-vantageous in this regard. For the construction ofthe Hessian, the radial grid can be pruned.

Excitation Energies of Benzene

The principal excited states of benzene of inter-est are those that involve an excitation from thedegenerate homo p orbitals to the degenerate lumo

Ž Up * valence orbitals and also other p ª p va-.lence excitations , together with excitations from

Ž .the homo orbitals to the Rydberg n s 3 orbitalsŽ .3s, 3p, 3d . All of our calculations were performed

˚ ˚at CC s 1.392 A and CH s 1.086 A.Ž .The relevant orbitals in our study are a p ,2 u

Ž . Ž U . Ž . Ž . Ž .e p , e p , a 3s , e 3 p , 3 p , a 3 pp ,1 g 2 u 1 g 1u x y 2 uŽ . Ž . Ž .2 2 2e 3d 3d , e 3d , 3d and a 3d . In2 g x yy x y 1 g z x z y 1 g z

each of the tables the symmetries of the possiblestates are given and their experimental excitationenergies taken from Lorentzon et al.18 Excitedstates of benzene have long been a challengingproblem for the theoretical chemist, stretching asfar back as the days of Parr]Pariser]Pople theory.Great progress has been made in quantum chem-istry, because it has conclusively demonstratedthat both types of states can be calculated reliably,unlike the semiempirical theories, which can onlypredict the valence states. In Table I we give thequantum chemistry prediction for singlet andtriplet excitation energies, using the multiconfigu-ration plus second-order perturbation theoryCASPT2,18 the second-order polarization propaga-tor approximation SOPPA,19 the random-phase

Žapproximation RPA which is time-dependent19. 20Hartree]Fock, and coupled cluster CC2

methodologies. For all of these studies large basissets with additional diffuse functions were used. Itis seen that the correlated methodologies give anaccuracy approaching 0.1 eV, which we under-stand is regarded as an ultimate desirable accu-racy. There are other similar studies in the litera-

Ž .ture, such as the multireference CASSCF pertur-bation theory studies of the p ª p U spectrum byHashimoto et al.21 We shall not comment furtherat this stage on these ab initio studies, except tosay that we think that the correlated methodsCASPT2, SOPPA, and CC2 are incomparably ex-pensive and complicated compared with DFTcalculations! RPA is less expensive than DFT cal-

Ž .culations at present because it does not involveŽnumerical quadrature but of course it is less accu-

.rate .There are several practical questions to be ad-

dressed, for which we concentrated on the singletstates. First, we examined how many occupiedorbitals we could treat as frozen; that is, we did

VOL. 20, NO. 1108


TABLE I.( )Experimental Singlet and Triplet Excitation Energies eV of Benzene Together with ab initio Calculations

from Literature.

18 18 19 19 20State, transition Exp. CASPT2 SOPPA RPA CC2

U1 ( )1 B V, p ª p 4.90 4.84 4.69 5.82 5.232u11 B 6.20 6.30 6.01 5.88 6.461u11 E 6.94 7.03 6.75 7.50 7.071u12 E 7.8? 7.90 } } 8.912 g1 ( )1 E R, p ª 3s 6.334 6.38 6.18 6.54 6.451g1 ( )1 A R, p ª 3p , 3p 6.932 6.86 6.70 6.94 6.972u x y11 E 6.953 6.91 6.76 7.11 7.022u11 A } 6.99 6.83 7.28 7.121u1 ( )2 E R, p ª 3pp 7.41 7.16 7.03 7.16 7.321u1 ( )2 21 B R, p ª 3d , 3d 7.460 7.58 7.35 7.70 7.591g x y y x y11 B 7.460 7.58 7.35 7.68 7.602 g12 E 7.535 7.57 7.34 7.59 7.561g1 ( )2 A R, p ª 3d , 3d 7.81? 7.74 7.56 7.77 7.811g z x z y11 E 7.81 7.77 7.55 7.80 7.802 g11 A 7.81 7.81 7.59 7.85 7.832 g1 ( )23 E R, p ª 3d } 7.57 7.40 7.73 8.881g z

U3 ( )1 B V, p ª p 5.60 5.49 5.50 5.57 }2u31 B 3.94 3.89 3.75 ) }1u31 E 4.76 4.49 4.48 4.70 }1u31 E 7.12 / 7.74 7.12 7.41 7.24 }2 g3 ( )1 E R, p ª 3s } 6.34 6.14 6.44 }1g3 ( )1 A R, p ª 3p , 3p } 6.80 6.64 6.82 }2u x y31 E } 6.90 6.74 7.08 }2u31 A } 7.00 6.84 7.28 }1u3 ( )2 E R, p ª 3pp } 6.98 6.92 7.11 }1u3 ( )2 21 B R, p ª 3d , 3d } 7.53 7.35 7.69 }1g x y y x y31 B } 7.53 7.33 7.65 }2 g32 E } 7.57 7.32 7.51 }1g3 ( )2 E R, p ª 3d , 3d } 7.55 7.57 7.85 }2 g z x y z31 A } 7.62 7.50 7.64 }1g31 A } 7.70 7.59 7.85 }2 g3 ( )23 E R, p ª 3d } 7.56 7.38 7.71 }1g z

not allow excitations from them. We found conclu-Ž .sively that only the six lowest 1s orbitals could

be frozen, with any further freezing affecting someof the excitation energies by more than 0.3 eV.ŽThis is also known to those who perform ab initio

.calculations. All calculations were therefore per-formed with six frozen orbitals. The dimensions of

Ž . Žthe H matrix were therefore m y 21 )15 = m1.y 21 )15, where m is the size of the basis set.

Next we examined the size of the basis set. Theresults of our studies are given in Table II. We

U Ž .found the often-used 6-31 q G basis m s 126totally inadequate. We had great difficulty identi-fying some of the Rydberg states and, furthermorethe valence-state predictions in Table II were sub-stantially different from the results presented in

Table III with a much larger basis. The calculationswith this 6-31 q GU basis is a good demonstrationof how to get things right for the wrong reason,because it is clearly seen in Table II that the va-lence p ª p U predictions are fortuitously excel-lent! Instead, for our investigation, we used the

U Ž .6-31G basis m s 120 together with diffuse func-Žtions at the origin this basis gave much better

.agreement with the final results in Table III . Fol-lowing ref. 22, we used an spd set with exponent

Ž .0.01 m s 130 , and also examined results usingŽ .two such sets, with exponents 0.01, 0.04 , and

Ž . Ž .0.01, 0.004 m s 140 . An examination of theseresults and the individual virtual orbitals indi-

Ž .cated that the set 0.01, 0.04 was the best ‘‘double-zeta’’ quality diffuse set. We then enlarged the


HANDY AND TOZER

TABLE II.( )Singlet Excitation Energies of Benzene eV and Our Density Functional Calculations Using HCTH Functional

a( )both with AC and without Asymptotic Correction.

U U U U631G AC 631G AC 631G AC 631GU ( ) ( ) ( ) ( )State, transition Exp. AC 631 + G + 0.01 + 0.01 + 0.01, 0.04 + 0.01, 0.004

U1 ( )1 B V, p ª p 4.90 5.32 5.42 5.41 5.44 5.422u11 B 6.20 6.06 6.22 6.22 6.24 6.221u11 E 6.94 6.70 7.30 7.01 7.00 7.001u1 ( )1 E R, p ª 3s 6.33 6.53 5.45 6.10 6.15 6.101g1 ( )1 A R, p ª 3p , 3p 6.93 7.17 5.86 6.80 6.81 6.812u x y11 E 6.95 7.18 5.89 6.80 6.79 6.782u11 A } 7.16 5.86 6.83 6.81 6.811u1 ( )2 E R, p ª 3pp 7.41 8.16 6.08 7.35 7.34 7.351u1 ( )2 21 B R, p ª 3d , 3d 7.46 } 6.33 7.48 7.39 7.481g x y y x y11 B 7.46 } 6.34 7.49 7.40 7.492 g12 E 7.54 } 6.34 7.49 7.41 7.491g1 ( )2 A R, p ª 3d , 3d 7.81? } } 7.67 7.63 7.661g z x z y11 E 7.81 } } 7.67 7.66 7.662 g11 A 7.81 } } 7.70 7.63 7.692 g1 ( )23 E R, p ª 3d } } } 7.52 7.38 7.511g z1 ( U)2 E V, p ª p 7.8 / 9.4 8.37 } 8.54 8.52 }2 g

a Various basis sets augmented with spd diffuse functions.

valence basis to TZ2P,23, 24 which together with theŽ .double-zeta diffuse set m s 248 , formed our

definitive basis set. The final matrices with thisbasis were 3405 = 3405.

The results of a calculation using the 6-31GU

Ž .basis and a single spd 0.01 diffuse set, but with-out the asymptotic correction, are given in thefourth column of Table II. It is clearly seen that thepredictions for the Rydberg states are effectivelymeaningless. Further calculations that show theimportance of the AC correction are given forother molecules in ref. 10.

The results using the 6-31GU and TZ2P basissets augmented with the ‘‘double-zeta’’ diffuse setare given in Table III for both the singlet and

Žtriplet states 48 states in all, including the spa-.tial degeneracies, excluding the spin degeneracies .

The ‘‘experimental’’ triplet values are taken as theCASPT2 values from ref. 18.

wWe now comment on some of the DFT TZ2P qŽ . x0.01, 0.04 spd predictions for the individual states,and we compare them with the ab initio predic-tions in Table I.

Ž U .V p ª p B , B , E . When compared with2 u 1u 1uthe experimental values, the mean absolute errorsfor these eight states, for DFT, SOPPA, andCASPT2, are 0.18, 0.20, and 0.13 eV, respectively.

3 ŽRPA fails completely for the B state the excita-1u.tion energy is imaginary , and its prediction for

the 1B state is in error by 0.92 eV. This set of2 u

predictions is typical of RPA, SOPPA, and CASPT2accuracy from many calculations in the literature,and it is becoming clear that the DFT accuracy isalso typical.5, 7 However, we must observe that the

3 ŽDFT prediction for the B state is very poor too2 u.low by 0.62 eV , for which we do not have a

reason. For the E states, the DFT-predicted oscil-1u

lator strength of 0.75 is close to the CASPT2 valueof 0.82. We also observe that the DFT predictionsfor these low-lying valence states do not seem tohave a systematic error pattern; from ab initiocalculations, the 3B state is ionic, and the 1B2 u 2 u

state is covalent, but both have significant errors intheir DFT predictions. On the other hand, the 1B1u

and 1E states are ionic and the DFT errors are1umuch smaller.

Ž Ž . U .V p a ª p E . The poor CC2 prediction2 u 2 g

is explained because of the importance of doubleŽ U .excitations p ª p in wave function calculations

for the singlet state. The experimental value isuncertain, but it must be expected that the CASPT2

Ž . Ž .value for this covalent singlet state 7.90 eV isclose. In the light of these two comments it is

Ž .encouraging that the DFT prediction 8.37 eV isŽwithin 0.5 eV of the CASPT2 prediction however,

.see later .Ž .R p ª 3s E . The DFT prediction for the sing-1 g

let state, 6.24 eV, is close to the observed value of6.33 eV. The CASPT2 prediction is slightly better,

VOL. 20, NO. 1110


TABLE III.U ( )Low-Lying p ª p and p ª n = 3 Singlet and Triplet Excited States of Benzene and Our Density Functional

UCalculations Using Asymptotically Corrected HCTH Functional with 6-31G and TZ2P Basis Sets, Both( )Augmented with Double-Zeta Diffuse spd 0.01, 0.04 Central Set.

U18 UState, transition Exp./ CASPT2 6-31G TZ2P

U1 ( )1 B V, p ª p 4.90 5.44 5.282u11 B 6.20 6.24 6.021u11 E 6.94 7.00 6.941u12 E 7.8? 8.54 8.372 g1 ( )1 E R, p ª 3s 6.334 6.15 6.241g1 ( )1 A R, p ª 3p , 3p 6.932 6.81 6.922u x y11 E 6.953 6.79 6.902u

U11 A 6.99 6.81 6.911u1 ( )2 E R, p ª 3pp 7.41 7.34 7.241u1 ( )2 21 B R, p ª 3d , 3d 7.460 7.39 7.501g x y y x y11 B 7.460 7.41 7.522 g12 E 7.535 7.41 7.511g1 ( )2 A R, p ª 3d , 3d 7.81? 7.63 7.781g z x z y11 E 7.81 7.66 7.762 g11 A 7.81 7.63 7.802 g

U1 ( )23 E R, p ª 3d 7.57 7.38 7.451g zU3 ( )1 B V, p ª p 5.60 5.15 4.982u

31 B 3.94 4.11 4.021u31 E 4.76 4.78 4.661u

U31 E 7.12 7.30 7.202 gU3 ( )1 E R, p ª 3s 6.34 6.11 6.211gU3 ( )1 A R, p ª 3p , 3p 6.80 6.72 6.852u x y

13E 6.90U 6.76 6.882uU31 A 7.00 6.81 6.911uU3 ( )2 E R, p ª 3pp 6.98 7.01 7.131uU3 ( )2 21 B R, p ª 3d , 3d 7.53 7.35 7.461g x y y x yU31 B 7.53 7.38 7.502 gU32 E 7.57 7.37 7.481gU3 ( )2 E R, p ª 3d , 3d 7.55 7.60 7.742 g z x y zU31 A 7.62 7.52 7.681gU31 A 7.70 7.66 7.782 gU3 ( )23 E R, p ª 3d 7.56 7.34 7.401g z

the SOPPA prediction is slightly worse. The RPAprediction is in error by 0.20 eV; indeed, for all theRydberg singlet states, the DFT predictions are farsuperior to the RPA predictions. The predictiveaccuracy of both RPA and DFT is higher for the

Žtriplet states on the assumption that the CASPT2.predictions are within 0.1 eV of the exact values .

Ž .R p ª 3 p , 3 p A , A , E . The DFT predic-x y 1u 2 u 2 utions for the two observed singlet states are within0.05 eV of the observed values, and the spread of

Ž .the DFT predictions 0.02 eV for the singlet statesŽ .is smaller than the CASPT2 value 0.13 eV . The

predictive accuracy of DFT is superior to that ofSOPPA for all these states. The DFT oscillator

Žstrength for the A state is 0.04 for CASPT2 it is2 u.0.06 .

Ž .R p ª 3 pp E . The DFT prediction for the1usinglet state has an error of 0.17 eV, comparedwith the CASPT2 and SOPPA errors of 0.25 and0.38 eV, respectively. Indeed, this state is the mostdifficult for CASPT2. The DFT value for the oscil-lator strength is 0.12, compared with the CASPT2value of 0.06. However, the sum of the oscillator

Žstrengths for the two E states agree DFT: 0.87,1u. 18CASPT2: 0.88 ; Lorentzon et al. stated that ob-

servation suggests that the combined oscillatorstrength should be close to 0.8. The DFT andCASPT2 predictions for the triplet state are veryclose.

Ž .R p ª 3d . For the singlet states for which thereare observed values, the CASPT2 predictions agreewithin 0.12 eV, the DFT predictions are within 0.06


HANDY AND TOZER

eV, whereas the RPA predictions are only within0.24 eV. The CASPT2 and DFT predictions for thetriplet states are within 0.20 eV of each other for allthe states.

ŽIf and this must be questioned in the light-.of the E -state difficulty we assume that the1u

CASPT2 ab initio is the most accurate set availablefor both the triplet and the singlet states that arenot observed, it is then of interest to compare theDFT and experimentrCASPT2 predictions given inTable III, for these 48 states. We find that the meanabsolute difference is only 0.12 eV. For this reasonwe have not undertaken a quantum defect analysisbecause our predictions will be very close to thosealready presented for CASPT2.18

DFT also predicts other low-lying triplet andŽ . U Ž .singlet states. In particular, the s e ª p e2 g 2 u1 Ž . 1transition yields valence states for A 7.21 , A1u 2 u

Ž . 1 Ž . 3 Ž . 3 Ž . 37.34 , E 7.32 , A 7.02 , A 7.07 , and E2 u 1u 2 u 2 uŽ . Ž . Ž .7.05 , and the s e ª 3s a transition yields2 g 1 g

1 Ž . 3 Ž . ŽRydberg states for E 8.29 and E 8.21 exci-2 g 2 g.tation energies in parentheses, in electron volts .

There does not appear to be any comparative stud-ies with which these values can be compared. Thisis surprising because our calculations suggest thatthere are two 1E states that are very close, at 8.292 gand 8.37 eV.

Finally, we note that predictions using the TZ2Pbasis are superior to those using the 6-31GU basis.The mean absolute errors for those states for which

Ž .the observed values are reliable 22 states forŽ U .TZ2P 6-31G are 0.11, 0.14 eV, respectively. This

result is to be expected because the parameters forw Ž .xthe HCTH functional and a and b eq. 8 wereŽoptimized for TZ2P-quality bases. For these 22

states the mean absolute error of CASPT2 was 0.10.eV.

Conclusion

The purpose of this work was to demonstratethat density functional theory can be used to ob-tain single-photon vertical excitation energies ofmolecules of both valence and Rydberg character.Benzene is the classic molecule for such a study.Forty-eight triplet and singlet states are deter-mined to a probable mean absolute accuracy of

Ž0.12 eV. The mean absolute error for the 12 va-lence states was 0.23 eV and the maximum error

.was 0.62 eV. These results were obtained using aGGA functional, HCTH, which has a potential thathas been suitably adjusted in the asymptotic re-

gion. The cost of the method is essentially thesame as any method that involves a single excita-tion matrix. From these and other studies it doesnot appear necessary to include any electric cur-rent dependence in the functional. This may bevery important, because the so-called magneticHessian is strictly diagonal, and therefore first-order changes in the Kohn]Sham orbitals are ob-tained by ‘‘uncoupled theory.’’ It is quite clear thatit is the accuracy of the occupied and virtualorbitals and their energy that are all-important.

It is of much interest that DFT can produce thisaccuracy for these excited states. DFT probably hasits greatest role in applications to larger molecularproblems, which will always be impossible in wavefunction quantum chemistry. Our code for thesecalculations is grossly inefficient; for example, notests for small quantities have been included in thenumerical evaluation of the exchange-correlationcontribution to the electric Hessian. However, thecalculations in this study demonstrate that DFT iscompetitive with wave-function quantum chem-istry for excitation energies. This will be of greatimportance in future studies.

Acknowledgments

The authors are grateful for constructive com-ments from Prof. B. O. Roos, which have beenincluded in this work.

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excitation energies of benzene from kohn-sham theory

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