J . Phys. Chem. 1994, 98, 5817-5823
5817
Calculation of Dispersion Energy Shifts in Molecular Electronic Spectra Notker Rosch and Michael C. Zerner'yt Lehrstuhl fur Theoretische Chemie, Technische Universitat Miinchen, 0-85747 Garching, Germany Received: December 9, 1993; In Final Form: April 3, 1994"
An efficient quantum chemical method for calculating dispersion energy shifts in molecular electronic spectra is described. The method makes use of separate calculations for solvent and solute molecules using a perturbation theory formula which averages over solvent configurations. To maintain the balance between the dispersion depression of the ground state and that of an excited state in a finite configuration interaction (CI) treatment, the Thomas-Reiche-Kuhn sum rule is invoked twice, correcting for the missing higher excited and continuum states in both the solute and the solvent. The resulting modified perturbation expression for the dispersion energy is remarkably stable with respect to the size of the employed CI. The model is implemented and applied in the framework of the intermediate neglect of diatomic overlap method using a previously presented selfconsistent reaction field description for the solvent and the solute molecules. Applications to acetone, benzene, naphthalene, and chrysene in cyclohexane clearly demonstrate the success of the method.
I. Introduction This paper is concerned with the disDersion contribution to molecular electronic spectra that arises when a chromophore is placed in solution. This energy shift is the difference in the dispersion energies between the excited state of the chromophore and the ground state. We define the dispersion energy here as the sum of all induced-dipole-induced-dipole interactions between solute and solvent. The appropriate interaction Hamiltonian between a solute "M" and N solvent molecules "S" is written as the sum of dipolar terms N
V= xRL3pMBMSpS p= 1
where pM and pS are the dipole moments for solute and solvent molecules. The angular term
contains the dyadic product of the unit vector npwith itself which points along the direction Rp = RM- Rs,between the point dipoles ps and pM. Then the dispersion term may be considered to be "what remains of the second-order perturbation energy when all terms which can be associated with permanent and induced dipoles of the individual molecules are removed".l This kind of language is, of course, based on consideration of solute and solvent molecules with wave functions that do not overlap. There is a large amount of literature in this area, and the calculation of the dispersion energy is not easy,24 let alone that of the difference in dispersion energies between two states. Most often approximate formulas involving molecular ionization potentials and mean polarizabilities are derived that have very little accuracy. Yet the influence of dispersion on spectroscopic shifts is believed to be large, often as large as 2000 cm-I. Dispersion generally leads to red shift^,^ as discussed later in this text. Often the entire solvent shift of a nonpolar molecule in a nonpolar solvent is associated with dispersi~n.'*~-~ In the context of modern quantum chemistry a supermolecule calculation might be contemplated as reference. Such a calculation between solute and solvent is not really feasible, although cluster calculations in the presence of a reaction fields-13 or in the t Permanent address: Quantum Theory Project, University of Florida, Gainesville, FL 3261 1 . 0 Abstract published in Advance ACS Absrracts, May 15, 1994.
presence of other solvent molecules treated classically are.1416 The Hartree-Fock calculations would produce supermolecular orbitals that might be localized to either solute or solvent molecules and would include the major interactions between solute and solvent. Single excitations localized on the solute generally make up the major features of the low-energy spectrum of the ~ o l u t e l ~ - ~ ~ and, supposedly at higher energies so that the solute spectrum is not obscured, the solvent. The interaction of such local excitations with one another gives rise to excitonic interactions,7J which become of importance when solvent and solute are of a very similar nature. In addition, there are charge-transfer excitations, from solvent to solute and solute to solvent, and their interactions with one another and with the localized excitations that would be included in any CIS (configuration interaction singles) treatment. The transition energies of charge-transfer excitations are generally of higher energy than the local excitations and can be estimated byl7-I9
where the excitation is from orbital 9i localized on the solvent (solute) to orbital $ Jlocalized ~ on solute (solvent), EA is an electron affinity, IP is an ionization potential, and R Mis~a mean distance between solvent and solute. Equation 3a is the frozen orbital approximation for any excitation 4p-$a,Jiathecoulomb integral, and Kia the corresponding exchange integra1.17J9.22J3 When charge-transfer excitations of the above type are low lying, the effects can be dramatic and can lead to many interesting phenomena, in both absorption and emission. There are a variety of double excitations that might be catalogued in a similar fashion. Those localized to orbitals on the solute or solvent are those of importance in correlating the ground state of solute or solvent, r e s p e c t i ~ e l y . ~Those ~ - ~ ~doubles which consist of two localized single excitations, one on the solute and one on the solvent, give rise to the dispersion interactions of interest here. Then there are the various double excitations, generally of higher energy, that involve at least one charge-transfer term. These double excitations also interact with one another and with the single excitations as well as the ground state in any CISD (CI singles and doubles) treatment. In general, to restore the balance that a CIS treatment shows in the calculations of electronic spectra, once doubles are included, triples must also be included.26~27 The triples of greatest
0022-365419412098-58 17%04.50/0 0 1994 American Chemical Society
5818 The Journal of Physical Chemistry, Vol. 98, No. 23, 1994
importance are those doubly excited relative to the singles that make up the principal description of the low-lying states. In other words, once the ground state is correlated, the excited states must also be. Although a useful way of cataloguing thevarious contributions in a supermolecule calculation, such a scheme is, in general, not of great practical utility unless the supermolecular orbitals are truly local or can be made ~0.28-3O If they are local, a CIS calculation on a cluster in a reaction field or in the presence of solvent molecules modeled classically is useful and will include electrostatic interactions and induced interactions but, of course, will not include the dispersion effects, which are a true correlation (many body) effect. Dispersion effects require double excitations of the type previously discussed. If the molecular orbitals ( M O s ) are strictly local, these excitations could be separated. If not, all doubles would need to be included, a large computational task, even in a small cluster-one that would need to be repeated as different conformational relations between solute and solvent were explored via a Monte Carlo or molecular dynamics simulation. Even so, such a calculation would be of dubious accuracy due to the fact that general CISD (CI singles and doubles) procedures suffer from deterioration with system size (size extensivity problem~).~2--25J1Modern supermolecule quantum chemistry is probably not the way yet to approach this problem except in very specific cases.
Rosch and Zerner treatment will not predict a nearly constant red shift in the T-T* spectra of naphthalene in different solvents when compared with the gas phase, nor a red shift in the n-r* excitation of acetone in nonpolar solvents relative to the gas phase and, at the same time, an increasing blue shift with solvent dielectric strength. In both cases these red shifts are believed to be due to dispersion, and reproducing this shift requires explicit and detailed knowledge of the electronic states of the solvent. In the former case, selfconsistent reaction field (SCRF) theory applied to naphthalene in cyclohexane at the dipole level leads to no shift (no dipoles!) and a t the quadrupole moment level to a small blue shift of 4 cm-1 for the lBlu state and 5 cm-1 for the lBzu state, compared to experimental red shifts of 300 and 950 cm-l, r e ~ p e c t i v e l y .A~ * ~ ~ simulation of acetone in cyclohexane in which acetone is treated quantum mechanically and cyclohexane classically leads to a small average blue shift of 20 cm-I.l6 This result does not change even when the acetone molecule and three nearest cyclohexanes are treated quantum mechanically at the CIS level.I6 A SCRF treatment of acetone leads to a small red shift (20 cm-I) to a good sized blue shift (170 cm-I), depending on which theory is used, as discussed below. Experimentally, a red shift of 400 cm-I is reported between acetone in the gas phase and in cyclohexane solution.39 We start with the second-order perturbation theory expression for Ed(I), the dispersion energy of state I, as given by1-2-40941
4pSKoeMS@)pMI JpMI JeMs@)
11. Theory
We begin our treatment of the dispersion energy with the consideration of a general wave function +( 1...N) =
+
A p t ( 1...p)+Slj(p 1...p iJ,k,...
+ g) x
(7)
where the superscript S(p) refers to thepth solvent molecule and the superscript M to thesolute molecule. N is Avogadro's number. In the above
in which the molecular orbitals that make up are assumed to be local on moiety "A" and are obtained from separate calculations of solute I+Mi) (or from a small supermolecule of solute and a few tightly bound solvent molecules) and solvent IJ.slj). Fixing the number of electrons in each fragment (p, q, etc.) and ignoring the antisymmetrizer A between moieties allows a separation of the Hamiltonian into
in which
can be solved for the solute (or small supermolecule) separately. Hs can be expanded as the sum of solvent Hamiltonians and their interactions
H , = H,,
+ H,, + ... + HSN+ V,,,,, + ...
+
E,, - Po EM,- EMI
(6b)
p'Ij pSOK
= (+'It~I~Mj)
(8)
($s&lrc/sK)
(9)
are the transitiondipoles. Theangular termOMS@) and thedistance Rp are defined as in eq 2. In eq 7 ESK and EM^ are the energies of the Kth state of the solvent, I ~ K ) , and Jth state of the solute, I+MJ), respectively. The brackets of eq 7 suggest that time or ensemble averaging should be performed. Averaging of interacting dipoles assuming a Boltzmann distribution and fixed Rpis well studied.1.5 Several techniques are available for averaging over ( l / R $ ) . Assuming a solvent of constant density p s distributed outside a sphere of radius a M surrounding the solute leads to68
with
and After both angular and ( R p 4 ) averaging, eq 7 becomes can be solved separately. The interactions between solvent molecules can be treated classically (by simulations, or, as in the next section, by "thermal" averaging), and the problem of interest is to evaluate the solute-solvent interaction terms between wave functions of the type given in eq 4. Several recent studies have shown how this might be done in the context of a reaction field a p p r o a ~ h , ~and , ~ such ~ - ~techniques ~ do successfully pick out the major interactions between solute and solvent. Extending these ideas for electronic spectroscopy reproduces fairly well observed shifts for a chromophore in different solvents,8-l3~35-37for often the differences in dispersion energies are small. But such a
with C M S s being the van der Waals coefficient for two interacting molecules. There are many ways in which a M can be estimated. The simplest is from the mass density, itself, and leads to9
Calculation of Dispersion Energy Shifts
The Journal of Physical Chemistry, Vol. 98, No. 23, 1994 5819
aM3= r~~ = ~ M M / ~ T N P M
(1 3a)
where MMis the molar mass in daltons. Most studies with the reaction field method in which a similar volume appears suggest larger values of a ~such , as93
and
If the electrons of the solvent are considered in equilibrium with those of the solute during the excitation process,8
or
uM = rM+ 0.5A
(13d)
Assuming a distribution of distances Rp= r M + rs, r M + 3rs, rM Srs, etc., and the number of solvent molecules at these distances as being proportional to 4 7 r ( r ~+ r ~ ) ~ / ( 2 r s4T(rM )~, + 3r~)2/(2rs)~, etc., yields42
+
2
Rp" = 4 ~ ( 2 r ~ ) - ~[rM+ (2n - l)rSl4
p= 1
(14a)
n= 1
Comparing eq 10 with the n = 1 term of eq 14a (by far the largest term) yields42
For rs = rM, this expression yields a~ = 1 . 7 r ~which , is larger than most empirical estimates of this value. For the moment we will use eq 14b, a posteriori, in that our best results are obtained with this choice. We leave a final choice of ansatz for later study, along with a study of more realistic shapes of molecule^.^'^^ The implementation of eq 12 is reasonably straightforward. We calculate the spectrum of a given solvent IVo) I@K), and the transition dipoles, ( $ S o ( p l p ~in ) , the presence of a reaction field8J&13 and store this information for all further calculation using this solvent. The spectrum of a given chromophore (solute) is then generated in the presence of a reaction field, and the final transition energy is obtained through
-
AE,(A) =
- (rclOlm0)l
[($IlfwI)
+
'/2g(WolPl~o)[($,lPl$,) - ( ~ 0 l P l ~ O + )l 1/2g(?2)($11~I$1)[($oI~I$o) - ($IIPI~~II)I
+
[Ed(1)
(15)
where AEI(A)is the estimated absorption energy to state I $ I ) from theory A of ref 8, t the dielectric constant, and 9 the diffraction index. The first term on the right-hand side is calculated directly with
H = H o = H'
(16a)
H ' = -g(4($olPl$o)P
(16b)
g(e) = 2(c - 1)/(2€
+ qaM3
(16c)
The next term is the solvent cost term, which together with the first yield the total system energy8
E1 = ($Ilfl$I)
+ 1/2g(wolPl$o)($IIPI$I)
(164
The third term of eq 15 represents the relaxation of the solvent upon absorption due to electron polarization (but not reorganization relaxation), and the last term is the dispersion correction of eq 7. In model B of ref 8 H'is defined as -g(t) ($olpl$o)p/2 to yield the total energy directly from the quantum chemical calculation,*
Because of the open-ended sums, the evaluation of eq 7 is tedious, and the calculation of ,!&(I) is sensitive to the details of the CI over the solute and solvent. For example, the simplest treatment for excited states is CIS (CI singles), and this is, in fact, the level of theory invoked to parametrize most semiempirical methods utilized to reproduce spectra. Such a treatment here might be expected to yield a reasonable result for the dispersion energy of the ground state of the solute, &(O), but a relatively poorer description for excited states. The reason for this is that the CIS procedure only generates some of the single excitations relative to 1$1);i.e., if $1
=P
i
( 1 / d ) tI41$1+*.4i&m***I+ I$l$1***4m$,**-I1
then CIS will generate I@r) and I$*j), but not 1qmaij), which is a double from I$o). This implies that, if there are n occupied and m virtual orbitals, that n X m states enter the evaluation of & ( o ) but only ( n + m) states enter the evaluation of &(I) in a meaningful way. Evaluating eq 12 from CIS is not useful and will lead to misleading results. CISD (CI singles and doubles) includes all single excitations with respect to both 13.0) and and would appear to be a balanced theory for the calculation of relative &(I) - ?./d(o) dispersion energies. For large chromophores this involves a considerable amount of work, roughly proportional to n2m2, and suffers from size extensity problems. Perhaps more important, however, is that CISD is not a balanced theory for spectroscopy, as mentioned in the Introduction. A more direct, and perhaps simpler way, is to calculate &(I) - & ( o ) in a series of calculations, one for each I, in which [,bo) and I$I)are used to generate single excitations, i.e., a two-reference CIS. The dimension of each separate CI is then roughly nm 2nm = 3nm, nm being the number of singles generated from I$o) and 2nm the number of singles generated from I$[). (Most of these singles have four open shells and generate two singlets.) We will examine this technique, similar in outlook to RPA calculations24~43 in the Results section, but we still wish to make the evaluation of eq 12 somewhat less sensitive to the nature of the CI. One way to proceed is to invoke the Thomas-Reiche-Kuhn sum r ~ l ewhich ~ ~ states , ~ ~that
+
In eq 18, W is the number of electrons in the solute molecule. From eq 12
5820
The Journal of Physical Chemistry, Vol. 98, No. 23, 1994
Y
Y
I
= ETlC(1) 2
1 ( P ~ K I > ~ ( P ~ L ~- )~~ ( ~E ~ 1K a2aM3L* K>M(ES~ - ESo EM, - EMI)(EMK - EM,)
--qz
+
)
Rosch and Zerner positive for all low-lying EM1 of interest, &(J) -&(I) is negative for E M ,C EM,. Thus, dispersion should red shift all absorption. This was demonstrated originally by Liptay,' but in a somewhat different fashion, and as discussed in the Introduction, the observation of this red shift prompted this study. Equation 21 will be stable with respect to the size of the solute CI, but not necessarily with respect to the CI treatment of the solvent. However, the sum rule correction can also be applied to the solvent, estimating the effects of missing solvent excitations as was done for the solute. This leads to
(19) EdCBIC(I)is the term calculated from eq 12 with sums from K = 0 to M. The second term in eq 19 is the remainder not included in the finite sum. Similarly, eq 18 can be split up yielding
$0
where SumM(1) is the sum rule calculated for state I when the sum is restricted to states from $0 to $M. We now assume that all the lower-lying states, I $ K) , K IM, have been included in &"lC((I) and thus that EM^ - E'1) in eq 19 is nearly constant and approximated by ( E M ) -EM]. We then pull this term from the denominator and derive
Ed(1) = EPlC(I) -
The quantity (EM) - E M [that appears twice in eq 21 can be written as ( ( E M )- EMo)- ( E M ,- EMo).The latter term (EM1 - EMo)is calculated directly from the CI. The former ( ( E M ) -E'o) wewillapproximate by thenegativeof theaveragevalence orbital energy, which under Koopmans' approximation is an estimate for the ionization limit of excitations from each orbital, averaged over the important excitations. The assumption here is that most of the states missing from the sums will have transition energies clustering around the ionization limits and that states that lie in the continuum have negligibly small transition moments. We will see that eq 21 is very stable in estimating &(I) from different sizes of the solute CI, and so we examine this equation further. Suppose BI = j/2NM,Le., that no excited states are included explicitly. Then
NM
Ed(1) = -
X
(12aM3((EM)- EM,)
(PsLo)z
T
L*
--
X
aiaM3( (EM) - EM])
[
(2Lo)2
+
. .I
(22) L#O(EL- Eo ( EM))2 The second term in brackets is positive, and in practice about 10% or less of ~ W Ma quantity which, in turn, is like a solvent polarizability, reduced in value by ( E M ) .Since ( E M )- EM, is "M
+ E'Ix
+
( 2 -)ESo (EM)- E M ,
[ ( E s )-Po + (EM) -EM,]-' (23)
W i t h s o = O,eq23reducestoeq21 asitshould. Theadditional terms proportional to BSo increase the dispersion energy. Evaluation of eq 23 shows the same stability as does eq 21 to the CI of the solute but is also stable with respect to the size of the solvent CI. The actual role of these individual terms, however, is difficult to evaluate as the total dispersion energy calculated from either eq 21 or eq 23 is dependent on the solvent parameter us3. Bso and ( E s ) are properties of the solvent alone (as is as) and need to be evaluated only once. Finally, we note a simple formula can be derived when the sums over excited states are empty and BM1= J2P and bo=
3/2p.
Although this formula is not very accurate, it is qualitatively correct and strongly suggests that higher excited states show a systematic increase in dispersion stabilization, as discussed earlier. Our examples show this to be generally true, although exceptions occur when using the more accurate representations of ,!$(I) (for example, eq 21 or 23) for the value of the Ed""(1) term depends on a numerator somewhat sensitive to the nature of the state I$,), as, for example, its symmetry. The details do matter, and for this reason we believe that generic formulas such as eq 24 will not be universally useful. 111. Sample Applications
(EL-Eo+ ( E M ) ) ( l - E M , / ( E L - E o + (EM)))
P'
B,'
We first examine the stability of eqs 21 and 23 with respect to the size and nature of the CI using naphthalene in cyclohexane as an example and using the INDO/CI model Hamiltonianl9.46 as implemented in the program ZIND0.47 This is a minimum basis set of valence orbitals only m e t h ~ d parametrized ~~J~ directly on low-lying electronic spectra at the CIS l e ~ e l . ' ~The . ~ spectrum ~ of cyclohexane is calculated49in the presence of the reaction field of cyclohexane (e = 2 . 2 2 0 , ~= 1.446), and the excitation energies and transition dipoles of these states are stored. The cavity radius for cyclohexane, as = 3.22 A, is from eq 13b with p = 0.7785 g/cm3. The spectrum of naphthalene is then calculated in the reaction field of cyclohexane; a cavity radius of 6.00 A is derived
The Journal of Physical Chemistry, Vol. 98, No. 23, 1994 5821
Calculation of Dispersion Energy Shifts
TABLE 1: Examination of Calculated Dispersion Energies for Naphthalene in Cyclohexane. 240 Solvent States (Sum Rule = 10.10 e)b CIS 457 (1 to 24/25 to 43) EO El E2
374 264
-7 8 -244 -559 -210 -616 -595 CIS 305 (9 to 24/25 to 43)
EO El E2
356 232
-72 -224
-227 -655
E(ABS)b 21.20 1.65 1.15
-559 -61 1
18.63 1.24 0.77
CIS 101 (15 to 24/25 to 34) EO El E2
206 66
8.38 1.23 0.72
-99 -279
-309 -559 -820 -638 CISD 606 (1 5 to 24/25 to 34)
EO 6.61 El 5 -1 53 -477 -470 6.87 E2 -164 -407 -1223 -718 6.99 101 Solvent States (Sum Rule = 5.48 e)CIS 457 (1 to 24/25 to 43) EO 21.20 El 583 -120 -26 1 -559 1.65 E2 422 -316 -661 -595 1.15
CIS 305 (9 to 24/25 to 43) EO El E2
554 373
-1 12 -337
-243 -703
-559 -61 1
18.63 1.24 0.77
CIS 101 (15 to 24/25 to 34) EO El E2
319 114
-153 -420
-331 -878
8.38 1.23 0.72
-559 -638
E(ca1c) refers to the first term in eq 21 or 23. Energy values are in cm-I. The sum rule values are in electrons (e). The ground state is designated E O the first and second excited states are designated E l and E2. Cyclohexanehas 36valenceelectrons. Naphthalene has48 valence electrons. (I
from eq 14b. This study of the dispersion shifts is summarized in Table 1. Recall that naphthalene has 48 valence electrons. The first three calculations are for different size CIS calculations, the easy “10 up, 10 down” (the highest 10 occupied MO’s and the lowest 10 unoccupied, 100 states plus the ground-state reference), then 19 up and 15 down, and then 19 up and 24 down, in the presence of 240 excited states of the solvent.49 As discussed, such a treatment is far better for estimating the dispersion of the ground state than that of the excited states, as seen from the sum rule and the fact that Emlcgenerates blue shifts for the two lowest T- T* transitions reported. Examining the sum rule, the largest of these calculations accounts for 21.2 electrons of the ground state, but for only 1.65 electrons of the first T T* and 1.15 electrons of the second ?r T * excitation. In spite of this unbalance in the sum rules eqs 21 and 23 yield reasonably consistent results. The dispersion shift of E2 is about 2.5 times that of E l . Equation 23, along with a cavity radius chosen from “Abe’s formula”,42 eq 14b, yields results in best agreement with experiment. The results from eq 21 are proportional to those of eq 23, roughly 2.1 times smaller and could be made toequal those of eq 23 by simply choosing a radius scaled by the third root of 2.1, in this case closely corresponding to the radius calculated from the mass density eq 13b. Equation 23, however, should be insensitive to the size of the CI and the number of states generated for the solvent. This idea is checked in the next three calculations in which 101 solvent states are generated by the 10 u p 1 0 down ansatz plus the ground state. This generates a sum rule for the solvent of 5.48 electrons vs 10.10 electrons for the 240 solvent state calculation. (Cyclohexane has 30 valence electrons.) As anticipated, the results from eq 23 are relatively insensitive to this change, less than lo%, whereas those from eq 21 are about 33% reduced. For this reason we choose to exploit eq 23 with solute cavity chosen from eq 14b and solvent cavity from eq 13b.
-
-
TABLE 2 n-u* Excitation Energy of Acetone in Cyclohexane (in cm-’). model A B A1 28 252 27 807 -190
28 368 27 924 -7 3
27 990 27 552 -445
B1 28 116 27 678 -319
E(D1SP)b shift The reference excitation energy (gas phase) for this CIS = 121 is 27 997 cm-l. See ref 8 for the definition of models A, B, A l , and B1. The experimentalshift is reported at -400 cm-I; see text. The calculated spectrum of acetone is somewhat sensitive to geometry. E(ABS) is the value calculated of the absorption before corrected for the dispersion energies and E(DISP) after corrections. The shift is relative to the energy calculated for acetone in the gas phase. The last calculation in this table is a CIS from four references, the ground state, and the three single excitations that make up The sum rule the lowest two excited ?r T * excitations. calculations for the three states involved, ground state, first excited, and second excited, yield 6.61, 6.87, and 6.99 electrons, respectively, demonstrating the more balanced nature of this CI. The sum rule for excitations from the ground state is somewhat worse than for the other CI calculations reported, although it is clearly better for excitations from the two lowest excited states. Equation 23 yields dispersion shifts somewhat larger than the other results, although the E2 shift is still 2.5 times that of E l . Note that this is the only one of these pilot calculations in which Ecalcfor E2 has even the correct sign, indicating the difficulty in obtaining the dispersion shifts without approximating the effects of the missing states. All results in this table obtained from eq 23 are reasonably close to the observed values. The observed spectroscopic shifts for naphthalene in cyclohexane are -300 and -950 cm-1 for E l and E2, respectively.50 The approximate eq 24, free of the specific details of the calculation, is no more successful than other approximate equations of this nature introduced in the literature.l,51-52 It does, however, seem to confirm the general conclusion that higher excited states have a larger red shift than lower ones, although the prediction would be a shift of E2 some 20% larger than that of E l . Experimentally the shift of E2 is 3 times that of E l . Table 2 summarizes results for then T * excitation of acetone using a 121 CIS c a l c ~ l a t i o nand ~ ~the different models of ref 8. Experimentally, a 400-cm-1 red shift is observed, and this would seem to be in better accord with those theories that use a mean field for electronic dielectric relaxation, with both ground and excited states responding to electron dielectric relaxation (models A1 and B1) rather than those that assume a “sudden approximation’’, with electronic dielectric relaxation only relaxing the excited state.8 In Table 3 we examine the calculated spectrum of benzene, naphthalene, chrysene, and azulene in cyclohexane. The calculations proceed as described before. The solute cavity radius is calculated from eq 14b. The CIS includes 10 occuppied and 10 virtual orbitals, but the results, as before, are reasonably stable with respect to the size of the CI. All models,8 A, B, A l , and A2, give a very good description of the observed red shifts for the first three molecules in this table. Azulene has a permanent dipole moment and differs from the other molecules in this study in that permanent moments contribute to the spectroscopic shifts. Thus, the ‘Lb state is blueshifted and the ‘LA, lKb, and states are red-shifted. This trend is clearly reproduced in our calculations. The ‘Kb state is found red-shifted less than the lower-lying ‘LA state, an observation not reproduced by these calculations. The reasons for this are not apparent. All models, A, B, A l , and B1 again produce similar results, although, due to the rather large permanent dipole moment, they need not.
-
-
5822 The Journal of Physical Chemistry, Vol. 98, No. 23, 1994
TABLE 3: Some Calculated Solvent Shifts in Cyclohexane (in cm-') shift spectra in vapor calc
benzene 'Bzu
'Blu 'Ey
obs
-316 (-209)' -598 -1209
calc
obsd
27 798 (0) 38 620 (0.002) 47 885 (0) 52 308 (1 -99)
naphthalene
ILb (X) 'La b) chrysene
-332 -3OoE (-275)' -879 -95W (-902)'
-243 (-486)' (-252)' 'La ('BJ -733 (-1030)b 'La ('Ag) -296 IBb ('BU) -1666 (-1620)b azu Ien e 'Lbb) +162 (+164)' 'LA(x) -288 (-333)' 'Kbb) -446 (-285)' 'Bb (X) -1475 (-1650)' ILb
('Bu)
31 300 (0.006) 32 460 (0.002) 34 655 (0.200) 37 330 (0.18) 28 556 (0.002) 28 250 (0.005) 29 390 (0.005) 32 231 (0.300) 32 430 (0.36) 34 039 (0) 38 087 (2.39) 39 110 (1.29) 16 940 (0.018) 28 230 (0.045) 34 784 (0.183) 38 505 (1.691)
14 27oC 29 76oC 34 164e 37 382c
These calculated values are from "10 up, 10 down" CIS, 101 spin states and for model B1. However (see ref 8),all models are very similar as the first term that differentiatesthem is the quadrupole term which is small. The exception is azulene; see text. A total of 240 solvent states are included; reducing this to 101 solvent states lowers the calculated shifts by about 6%. The shifts calculated by eq 24 are -547, -634,and -725 cm-' for benzene, -560 and -611 cm-I for naphthalene, and -505, -591,-634,and -735 cm-' for chrysene. * The values in parenthesesare for n-pentane as a solvent, from ref 50. The differences between the shifts observed in n-pentane and cyclohexane are expected to be small: see, for example, the case of naphthalene in this table. From ref 6. Thesearetheobservedgas-phasevaluesreportedinref51.Thenumbers in parentheses are oscillator strengths. For chrysene two vibrational levels of ILb are given, whereas the 'A, transition, which is forbidden, is not reported. Fromref 51.ITheobservedshift is for 2-chloropropane, from ref 5 1. The error in these values compared to cyclohexane could be large.
IV. Conclusions The calculation of thedispersion contribution to the molecular electronic spectra of a chromophore in solution is difficult for any molecule of size even with the most modern computers. This is due to the prohibitive number of excitations that must be included in the CI of a supermolecule consisting of solute and several solvents. In this paper we describe a method that allows the calculation of the dispersion contribution from separate calculations of solvent and solute, using a formula which averages over the solvent conformations.'.*,40 Even so, it is difficult to maintain a balance between calculating the dispersion depression on the ground state and that of the excited states of interest. Most finite configuration interaction treatments that are reasonable for calculating transition energies and transition probabilities yield a blue shift for the dispersion contribution. By invoking the Thomas-Reiche-Kuhn sum rule twice, we correct for the missing higher excited and continuum states in both solute and solvent and derive a formula that is reasonably stable with respect to the details of the CI we employ. This model successfully yields the observed red shift in the cases we examine here: that of acetone, benzene, naphthalene, and chrysene in cyclohexane. The first excited state of azulene is blue-shifted in aprotic solvents, while the second third and fourth are red-shifted. These trends are easily picked up by these calculations. We have assumed for this work a fixed spherical shape of both solvent and solute molecules. Although most molecules appear to be effectively spherical in a solvent environment (tumbling and so forth, apparently add to this observation 5 9 , we have recently worked out formulas for elliptical cavities.53 Other more realistic shaped cavities, such as those that surround atoms with intersecting van der Waals spheres are available,35-37,54 but we have not yet evaluated this formulation for a direct calculation
Rosch and Zerner of dispersion. The results we report here are sensitive to the effective cavity radii, as pointed out previously (see eqs 10-14), but there are fortuitous cancellations when looking at energy differences that reduce this sensitivity. We have implemented this formulation within the INDO/CI approximation, although there are no restrictions to this model Hamiltonian. Our development is general. In practical applications, dispersion energies calculated within the INDO approximation will likely be underestimated for excited states that show Rydberg character (smaller molecules and higher excited states). The reason for this is that the INDO model used here does not have a basis set that represents Rydberg states very well. On the other hand, the valence states which carry most of the oscillator strength are well represented in the INDO/CI model, better than all but the best ab-initio calculations. A theory like the one presented here, with the reaction field and with a model for the cavitation energy, should be able to yield absoluteenergiesof solvation.52 This is a different typeof problem in the sense that systematic errors (like the cavitation energy itself and even the cavity shape and size) will not cancel in the subtraction of ground- and excited-state energies.
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