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(LDA) and gradient-corrected functional such as the B88x-|-P86c functional. ... Since the Hellmann-Feynman theorem holds in DFT [17], these are equiva...
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Chapter 8 Optical Properties from Density-Functional Theory Mark E . Casida, Christine Jamorski, Fréderic Bohr , Jingang Guan, and Dennis R. Salahub

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1

Département de chimie, Université de Montréal, C.P. 6128, Succursale centre-ville, Montréal, Québec H3C 3J7, Canada

Density-functional theory (DFT) is a promising method for the calculation of molecular optical properties, since it is less compu­ tationally demanding than other ab initio methods, yet typically yields results of a quality comparable to or better than those from the Hartree-Fock approximation. The calculation of static molec­ ular response properties via DFT has now been studied for several years, whereas work on the corresponding dynamic properties is only just beginning, since none of the previously existing molecular DFT codes were capable of treating them. The present article gives a brief summary of some of our work in this area. This includes an illustration of the quality of results that can be expected from DFT for static molecular response properties (dipole moments, polariz­ abilities, and first hyperpolarizabilities), as well as illustrative early results (dynamic polarizabilities and excitation spectra) from our code deMon-DynaRho, the first molecular time-dependent densityfunctional response theory program.

T h e search for stable materials with enhanced nonlinear optical properties for use i n telecommunications and computer information transmission and storage has spurred a renewed [1,2] interest by chemists in recent years [3-7] i n the nonlinear optical properties of molecules. It is hoped that quantum chemical calculations w i l l help i n the design and preselection of candidate materials. However, several requirements will have to be met if the results of quantum chemical calculations are to find direct application to problems currently of interest i n materials science. These requirements include the ability to handle some reasonably large molecules, the ability to treat the response to a dynamic field, and proper consideration of 1

Current address: Laboratoire de Chimie Physique, Universitéde Reims, Faculté des Sciences, Moulin de la Housse, B.P. 347, 51062 Reims Cedex, France

0097-6156/96/0628-0145$15.00/0 © 1996 American Chemical Society

In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

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solvent or m a t r i x effects. To this list, we can add that good ab initio calculations require large basis sets and an adequate treatment of correlation, and that ac­ curate comparisons w i t h experiment can be complicated by the need to include vibrational and orientational contributions. Because of its scaling properties w i t h respect to calculations on increasingly larger molecules and its ability to treat cor­ relation i n a simple way, density-functional theory ( D F T ) is a promising method for quantitative calculations of the optical properties of molecules i n a size range of practical interest. T h i s article is a report of where we stood i n the F a l l of 1994 i n generating and calibrating essential D F T machinery for treating optical problems. G i v e n that linear as well as nonlinear optical properties are a topic which has long been of interest to chemists and is likely to remain so for some time, we w i l l not restrict the topic only to nonlinear properties but will also discuss the use of D F T for calculating simple polarizabilities and excitation spectra. O u r current work on vibrational contributions [8] and solvent effects [9] w i l l be discussed elsewhere. Let us first situate D F T among the variety of quantum chemical methods available for calculating the optical properties of molecules. A t one extreme, i m ­ pressively quantitative ab initio methods, based upon, for example, M0ller-Plesset perturbation theory [10], equation-of-motion [11], or coupled cluster [12] tech­ niques, have been developed whose application tends to be l i m i t e d to very small molecules. A t the other extreme, semiempirical methods allow the consideration of m u c h larger molecules, but the reliance on parameterizations limits the type of molecules and variety of properties to which any given semiempirical method can be applied w i t h confidence. D F T offers the advantages of an ab initio method, yielding results for a variety of properties that are typically better than those ob­ tained from the Hartree-Fock ( H F ) approximation, but w i t h less computational effort. A l t h o u g h D F T is more computationally demanding than semi-empirical methods, it gives much more reliable results when a broad range of molecular types and properties is considered. Thus D F T represents a promising approach to the quantitative treatment of the optical properties of molecules for systems complex enough to be of interest to bench chemists and materials scientists. A t present, the potential of D F T remains largely untapped i n this respect. A l ­ though there have been numerous applications of D F T to the calculation of elec­ tric response properties of atoms and solids (see Ref. [13] for a review), much less work has been done on molecular systems. Studies assessing D F T for calculation of molecular electric response properties have been for static properties, p r i m a r i l y dipole-polarizabilities and hyperpolarizabilities [9,14-19] of small molecules. W o r k on D F T calculations of dynamic molecular response properties has been l i m i t e d to a few calculations using either spherically-averaged pseudopotentials [20,21] or single-center expansions [22,23], i n order to make use of atomic-like algorithms, but which are not of any general utility for molecular calculations. After a brief look at how well D F T works for static molecular response properties, we focus on the first implementation of time-dependent density functional response theory us­ ing an algorithm appropriate for general molecular calculations, giving a summary of our method and results for N 2 . T h e present results are at the level of the ran­ dom phase approximation. Implementation of the fully-coupled time-dependent local density approximation is i n progress. A more complete description of our methodology w i l l be published elsewhere [24].

In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

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METHODOLOGY

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Since excellent reviews of density-functional theory ( D F T ) are readily accessible [25,26], we w i l l restrict our attention to what is needed for a discussion of the current status of D F T for the calculation of molecular optical properties. S t a t i c P r o p e r t i e s . W i t h few exceptions, molecular applications of D F T are based upon the K o h n - S h a m formalism, i n which the exact ground state energy and charge density of a system of TV electrons i n an external local potential are ob­ tained using the exact exchange-correlation functional. In practice this exchangecorrelation functional must be approximated. T h e terms "local potential" and "ground state" are important. T h e former excludes a full, rigorous treatment of magnetic effects, though useful results can be obtained i n practice [27-29]. T h e latter, together w i t h the fact that the K o h n - S h a m formalism is time-independent, excludes the treatment of dynamic response properties i n the traditional the­ ory. Extensions of the formalism to the time-dependent domain have been made, and dynamic response properties w i l l be discussed i n the next subsection. T h e standard K o h n - S h a m formalism is, however, exact for static electronic electric response properties, i n the l i m i t of the exact exchange-correlation functional. T h e charge density is obtained i n K o h n - S h a m theory as the sum of the charge densities of K o h n - S h a m orbitals i/)f with occupation numbers / f . T h a t is p(r) = , l ( r ) + p*(r),

(1)

where P »

= £/nV>f(r)| .

(Hartree atomic units are used throughout.) the self-consistent K o h n - S h a m equations 4v

2 +

(2)

2

T h e orbitals are found by solving

< (r) ^?(r) = ^ f ( r ) , ff

(3)

where the effective potential v° is the sum of an external potential w h i c h , i n molecular applications, is the sum of nuclear attraction terms and any applied potential v £ , and a self-consistent field ( S C F ) t e r m , a

ppl

ǤOF(r) = / 7 ^ 7 dr' + v'Jp\

,'](r),

(4)

which differs from the corresponding quantity i n the Hartree-Fock approxima­ tion i n that the Hartree-Fock exchange operator has been replaced w i t h the density-functional exchange-correlation potential v° . N o practical exact form of the exchange-correlation potential is known, so it is approximated i n prac­ tice. Popular approximations include the widespread local density approximation ( L D A ) and gradient-corrected f u n c t i o n a l such as the B88x-|-P86c functional. c

In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

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NONLINEAR OPTICAL MATERIALS

If the applied potential corresponds to a uniform electric field,



( )

r

16

when 8v \((jj) = rjF(u) . Since the K o h n - S h a m equations have the form of one-particle, orbital equations, we can rewrite E q . (15) as AVV

Sp"(r, « ) = £ / X K ' s ( r , r ' ; a , ) * , ^ , w) dv', T

(17)

J

where x£ (r,r';u,) = s

T

jL [WITi*)]

k ^ r M ] '

(18)

has the form of the generalized susceptibility for a system of independent particles, and the response of the effective potential 8v° is the sum of the perturbation ) = j

-y^T & + £ / AT(r, r'; u , ) ^ ( r ' , ) dv'. |r-r'|

6

u

(19)

T h e exchange-correlation kernel is given by it-f)

K c (

^ )

r

/-(r,r'; ) = / e ' ^ - ' ' ) g ^ M ^ - 0 . W

(20)

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It reduces to

/

-

(

r

'

r

'

)

=

W )

(

2

1

)

in the static l i m i t (u> = 0), i n which case Eqs. (17)-(21) become the C P K S equa­ tions. W h e n u) ^ 0, solving the dynamic coupled equations allows the dynamic po­ larizability a(cj) to be calculated. T h e method can also be extended to other dynamic properties, including higher-order polarizabilities and excitation spec­ tra. In practice, we obtain excitation spectra by noting that the exact dynamic dipole-polarizability can be expanded i n a sum-over-states representation as r

excited states

« H =

£

j r

1

^

(22)

where the a;/ are vertical excitation energies and the / / are the corresponding oscil­ lator strengths. Since practical calculations use approximate exchange-correlation functionals, the calculated dynamic polarizability w i l l also be approximate. N e v ­ ertheless, it still has the same analytic form as the exact dynamic polarizability, so the poles and residues of the calculated dynamic polarizability can be identi­ fied as (approximate) excitation energies and oscillator strengths. Note that the T h o m a s - R e i c h e - K u h n ( T R K ) sum rule [35] £ / /

= W

(23)

should also be satisfied i n the limit of the exact (time-dependent) exchangecorrelation functional. T h e problem of finding good time-dependent exchange-correlation functionals is still i n its infancy. This problem does not arise at the level of the independent particle approximation ( I P A ) , which consists of taking SVSCF — 0- T h e next level of approximation is the random phase approximation ( R P A ) , where the response of the exchange-correlation potential (second term i n E q . (19)) is taken to be zero, which turns out to be a reasonably good approximation for some purposes [vide infra). Note that the R P A includes some exchange-correlation effects, namely those which enter through the orbitals and orbital energies of E q . (18). A notation such as R P A / L D A gives a more complete description of the level of approximation (i.e. approximation used for the response / approximation used for the unperturbed orbitals a n d orbital energies.) A problem with the R P A is that i t does not re­ duce to the C P K S equations i n the static l i m i t . This requirement is met by the

In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

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adiabatic approximation ( A A ) i n which the reaction of the exchange-correlation potential to changes i n the charge density is assumed to be instantaneous,

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6pT{r',V)

~ 6pr{T>,t)

b{t

t

}

-

(

M

)

T h i s assumption is rigorous i n the static case and is at least reasonable i n the low frequency l i m i t . W h e n the exchange-correlation functional is local, the A A is usually referred to as the time-dependent local density approximation ( T D L D A ) . Since the orbitals and orbital energies used are also at the L D A level, the notation T D L D A / L D A gives a more complete description of this A A . A n approximation w h i c h goes beyond the A A has also been suggested [34]. T h e dynamic results reported here were calculated at the R P A / L D A level. Implementation of the T D L D A is i n progress. COMPUTATIONAL DETAILS T h e calculations reported here were carried out using two programs w r i t t e n at the University of Montreal. T h e first program, deMon (for "densite de MontreaF) [36-38], is a general purpose density-functional program which uses the Gaussiantype o r b i t a l basis sets common i n quantum chemistry. T h e second program, DynaRho (for " D y n a m i c Response of /p"), is a post-deMon program w h i c h we are developing to calculate properties which depend on the dynamic response of the charge density. In DynaRho, the formal equations of the previous section are solved i n a finite basis set representation. A l t h o u g h a full description of how this is accomplished is beyond the scope of the present paper, some insight into the operational aspects of DynaRho can be obtained by considering the p a r t i c u l a r l y simple case of the H molecule oriented along the 2-axis and described using a m i n i m a l basis set. There are only two molecular orbitals i n this case. T h e occu­ pied tr-bonding combination w i l l be denoted by the index i , while the unoccupied and gives results comparable to those obtained from the Hartree-Fock approximation. T h e fully coupled T D L D A includes both the coulomb and exchange-correlation contributions to SVSCF and would be equiv­ alent to the finite field L D A results shown here. T h e fact that the R P A results are far more similar to the finite field results than to the I P A indicates that, as would be expected on physical grounds, the response of the coulomb part of V S C F to an applied electric field is an important part of the polarizability, whereas the response of the exchange-correlation potential is a relatively small contribution. Table II shows the convergence of the mean polarizability values for N w i t h re­ spect to basis set. T h e discrepancy between the calculated value and theoretical l i m i t of the T R K sum arises from the limitations of the basis sets used here, which are oriented towards a good description of the 10-electron valence space of the ground state molecule, but not necessarily of the core. These basis sets are expected to describe only the low lying excited states reasonably well. 2

Less data is available to judge the quality of D F T calculations of molecular hyperpolarizabilties, but indications to date [9,16-19] are that mean first hyper­ polarizabilities are pretty good at the L D A level. T h e L D A value of /? i n Table I is i n much better agreement w i t h experiment than is the H F value. Neverthe-

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EXPERIMENT (a.u.) F I G . 1. Comparison of theoretical and experimental mean polarizabilities for N , C O , C H , H 0 , N H , and H F : independent particle approximation, solid squares; random phase approximation, solid diamonds; finite field, open squares; coupled Hartree-Fock, open triangles. T h e density-functional calcu­ lations used the local density approximation and the T Z V P - f - basis set. T h e coupled Hartree-Fock and experimental values are taken from Ref. [17]. See text for additional details. 2

4

2

3

> 00

o LLI

0

1

2

3

4

5

EXPERIMENT (a.u.) F I G . 2. Comparison of theoretical a n d experimental polarizability anisotropics for N , C O , C H , H 0 , N H , and H F : independent particle ap­ proximation, solid squares; random phase approximation, solid diamonds; L D A finite field, open squares; coupled Hartree-Fock, open triangles. T h e density-functional calculations used the local density approximation and the T Z V P + basis set. T h e coupled Hartree-Fock and experimental values are taken from Ref. [17]. See text for additional details. 2

4

2

3

In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

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T A B L E III. Sensitivity of calculated dipole moment, mean polarizability, polarizability anisotropy, and mean hyperpolarizability of H 2 O (in a.u.) to geometry and choice of functional. A l l calculations use the Sadlej basis set (see text).

Functional

Geometry

LDA

Optimized

0.728

10.80

0.46

P -20.0

B88x+P86c

Optimized

0.713

10.68

0.54

-18.4

LDA

Experimental

0.732

10.56

0.27

-19.1

B88x+P86c

Experimental

0.708

10.46

0.35

-17.4

a

Aa

less, it should be emphasized that a truly rigorous comparison w i t h experiment would require the inclusion of finite frequency effects and vibrational contribu­ tions. Comparison with the singles, doubles, quadruples fourth-order M 0 l l e r Plesset perturbation theory results of Maroulis [10] (Table I) suggests that the L D A static electronic hyperpolarizability is too large. Table III shows the sensitivity of our water results to the geometry used and choice of functional. Neither the mean polarizability nor the mean first hyperpo­ larizability is very sensitive to small changes i n geometry. Roughly speaking, this is because the mean polarizability is a volume-like quantity and the mean hyper­ polarizability is just its derivative. T h e polarizability anisotropy, being related to molecular shape, is much more sensitive to small changes i n geometry. T h e B 8 8 x + P 8 6 c gradient-corrected functional is expected to yield improvements over the L D A for properties which depend upon the long range behavior of the charge density. However, although the improvements for water (and sodium clusters [46]) are i n the right direction, they are not dramatic. D y n a m i c r e s u l t s . Results are given here at the R P A / L D A level. A treatment including coupling of exchange-correlation effects w i l l be reported i n due course. We now have preliminary R P A / L D A results for a half dozen small molecules. For purposes of the present summary, we focus on N , an important benchmark molecule for calculation of excitation spectra [11,12,52], and one for which the experimental dynamic polarizability [51] and experimental excitation energies [53] are readily available. Figure 3 shows our calculated dynamic mean polarizability i n comparison w i t h the experimental quantity. T h e frequency dependence is calculated at the R P A / L D A level, and is combined w i t h the finite field L D A static value to give 2

a(u>) = ( a

R P A

(u,) - a

R P A

(0)) + a

F F

(0).

(30)

A similar procedure is sometimes adopted to graft the dynamic behavior from the time-dependent Hartree-Fock approximation ( T D H F A ) calculations onto better post-Hartree-Fock static calculations. T h e agreement w i t h experiment is reason­ ably good. E x c i t a t i o n spectra represent a considerably more challenging test of the R P A / L D A . We restrict our attention to the singlet-singlet transitions since, as was noted earlier, the singlet-triplet transitions are uncoupled at the R P A level. T h e y are also " d a r k " states i n the sense of having oscillator strengths which are

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16

15

co • m

14

3 Downloaded by KUNGLIGA TEKNISKA HOGSKOLAN on October 15, 2014 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0628.ch008

E

5

13

o Q. Z < UJ

2

12

11 I 0

2

4

I 8

6

PHOTON ENERGY (eV)

F I G . 3. Frequency dependence of the mean polarizability of N . T h e the­ oretical curve (dashed) is for the hybrid finite f i e l d - R P A / L D A calculation described i n the text, w i t h the T Z V P + basis set. T h e experimental curve (solid) is constructed from data taken from Ref. [52]. 2

12.0

>

7.0 -

1

1 TDA

1 1 TDHFA MRCCSD

1 1 ' EXPT R P A / L D A

METHOD

F I G . 4. Comparison of the first three singlet-singlet excitation energies of N calculated by various methods w i t h experiment. T h e T a m m - D a n c o f f ap­ p r o x i m a t i o n ( T D A ) , time-dependent Hartree-Fock approximation ( T D H F A ) and singles and doubles multireference coupled cluster values are taken from Ref. [12]. T h e experimental values are taken from Ref. [52]. T h e R P A / L D A values were calculated using the Sadlej basis. T h e excited states and their dominant one-electron contributions are: a n (3cr —• 1TT ), open square; a * E ~ ( l 7 r —> l 7 r ) , open triangle; and w A (lir —• l 7 r ) , solid triangle. 2

1

u

5

1

U

u

f f

u

s

5

In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

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T A B L E I V . Oscillator strengths for the first four vertical transitions of N 2 having nonzero oscillator strength. The R P A / L D A values are calculated with the Sadlej basis set and do not include a degeneracy factor of 2 for the U states. The time-dependent Hartree-Fock approximation ( T D H F A ) and second-order equations-of-motion ( E O M 2 ) oscillator strengths are taken from Ref. [11]. 1

Excitation —•

3(7 2a

-*

u

TDHFA

u

r'

3cr„

G

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States 2?r

Y+

1

l-Kg b '

1

U

^

EOM2

RPA/LDA

0.091

0.12

0.02

0.65

0.094

0.11

0.32

0.49

0.07

0.15

0.19

0.15

zero by symmetry. Figure 4 shows a comparison of the first 3 singlet-singlet verti­ cal excitation energies for N , calculated by various methods, w i t h the experimen­ tal values. B o t h the Tamm-DancofF approximation ( T D A ) , which is equivalent to a singles configuration interaction treatment of the excited states, and the T D H F A give the wrong ordering of these states. T h e R P A / L D A gives the cor­ rect ordering but, not surprisingly, does not do as well as multireference coupled cluster ( M R C S D ) calculations. These excitations are to spectroscopically "dark states". T h e excitation energies of the first four "bright states" calculated at the R P A / L D A level are compared i n Figure 5 w i t h excitation energies calculated using the T D H F A and using a second-order equation-of-motion ( E O M 2 ) method and w i t h experimental transition energies. Calculated vertical transition energies for these states are strongly influenced by the presence of nearby avoided crossings of the excited state potential energy surfaces. Nevertheless, the R P A / L D A exci­ tation energies are quite reasonable and all within about 1 e V of the experimental results. A comparison of oscillator strengths is given i n Table I V . E x p e r i m e n ­ tal values are difficult to extract with precision and so have been omitted. O u r R P A / L D A oscillator strengths do not seem to be fully converged w i t h respect to basis set saturation, and should be viewed w i t h caution. 2

T h e good quality of the results for N are particularly noteworthy i n view of the fact that conventional (time-independent) K o h n - S h a m theory is a fundamentally single-determinantal theory. One of the important advantages of the present t i m e dependent density-functional response theory approach is that it provides a m u l t i determinantal treatment of the excitations. A l l of the excited states of N treated here have an important multideterminantal character. This is especially true of the S ~ and A states each of which requires a m i n i m u m of four determinants simply to obtain a wavefunction of the correct symmetry. Our R P A / L D A calculation automatically includes not only those determinants required by symmetry, but also contributions from other determinants as well. For the half dozen molecules studied so far, the singlet-singlet excitation ener­ gies obtained at the R P A / L D A level are generally w i t h i n l e V of the experimental values. It is interesting to note that the sum-over-states expression (22) implies a relationship between the quality of the excitation spectrum and the quality of the polarizability. Thus, for a molecule such as N , an absolute error of < 1 e V i n the excitation energies translates into a reasonably small error i n the polarizability, yet for a molecule such as N a with extraordinarily low excitation energies (first 2

2

1

1

U

2

2

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>

16.0-j

>£*

>-

o

DC LU z UJ z

15.0-^ 14.0

o

13.0


open square; b ' *£+ (lw —» I71-3), open triangle; c Ii (3 2 7 r ) , solid square; c ' (3 3 < 7 ) , solid triangle. 2

1

u

g

3

1

u

U

u

g

u

U

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bright state at about 1.8 e V [54]), the R P A / L D A polarizability is considerably worse.

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CONCLUSION T h i s paper has given a summary of where we stand (as of fall 1994) i n gener­ ating and calibrating essential D F T methodology for optical problems. In some ways the methods used here bear a close resemblence to Hartree-Fock-based tech­ niques. However, whereas the Hartree-Fock method is an approximation, D F T electronic electrical response properties are formally exact i n the l i m i t of the exact exchange-correlation functional. T h i s , together w i t h efficiencies arising from the use of only local potentials i n D F T , makes D F T a promising method for quanti­ tative calculations of optical (and other) properties of molecules i n a size range comparable to or greater than that now attainable w i t h the Hartree-Fock method, provided, of course, that the approximate exchange-correlation functionals used i n practical calculations are sufficiently accurate. T h a t this is the case has been illustrated by the quality of static dipole moments, dipole polarizabilities, and first dipole polarizabilities of small molecules. Since optical measurements are made w i t h finite frequency electric fields, the extension to the time-dependent regime is important. Thus dynamic p o l a r i z a b i l ­ ities and excitation spectra, calculated at the R P A level, using the first general molecular implementation of time-dependent D F T (the DynaRho program), have been reported here for the first time. The results to date are quite encouraging, and a full treatment, including the response of the exchange-correlation potential, is already underway. T h i s approach promises to become a powerful technique, applicable to a wide range of complex molecules and materials models. ACKNOWLEDGMENTS F B would like to thank the French M i n i s t r y of Foreign Affairs for financial support. Financial support from the Canadian Centre of Excellence i n Molecular and Interfacial Dynamics ( C E M A I D ) , from the N a t u r a l Sciences and Engineering Research C o u n c i l ( N S E R C ) of Canada, and from the Fonds pour l a formation de chercheurs et l'aide a l a recherche ( F C A R ) of Quebec is gratefully acknowledged. We thank the Services informatiques de l'Universite de M o n t r e a l for computing resources.

LITERATURE CITED [1] Böttcher, C.J.F. Theory of Electric Polarization. Volume I: Dielectrics in static fields; Elsevier Scientific Publishing Company: Amsterdam, Holland, 1973. [2] Böttcher, C.J.F.; Bordewijk, P. Theory of Electric Polarization. VolumeII:Dielectrics in time-dependent fields; Elsevier Scientific Publishing Company: Amster­ dam, Holland, 1978.

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