( )
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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Theory
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-
In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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NONLINEAR OPTICAL MATERIALS
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
In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
158
NONLINEAR OPTICAL MATERIALS
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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CASIDA ET AL.
Optical Properties from Density-Functional
159
Theory
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
In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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160
NONLINEAR OPTICAL MATERIALS
>
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
In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
4
8.
CASIDA ET AL.
Optical Properties from Density-Functional
Theory
161
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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