J. Phys. Chem. 1991, 95, 3539-3545
3539
Theory of Solvent Effects on the Vlslble Absorption Spectrum of &Carotene by a Lattice-Filled Cavity Model Jon Applequist Department of Biochemistry and Biophysics, Iowa State University, Ames, Iowa SO01 1 (Received: September 27, 1990)
The visible absorption spectra of al/-trans-&carotene in 21 solvents reported by Myers and Birge are analyzed in terms of a cavity model for solvent effects on strongly allowed electronic transitions. The chromophore is treated as a classical point dipole oscillator at the center of a spherical cavity in a continuum having the optical dielectric constant of the solvent. A simple cubic lattice of polarizable points filling the cavity space outside a cylindrical region approximating the size of the @-carotenemolecule represents the local solvent structure. The theory reproduces the observed dependence of oscillator strength, absorption peak wavenumber, and root-mean-square band wavenumber on solvent polarizability. Hypothetical vapor spectra are calculated from the observed spectra in three solvents, using a relationship between the complex polarizabilities in solution and vapor phase for arbitrary band shapes. The predicted vapor spectra show a large change in band shape having diminished vibronic peaks, an rms wavenumber of 24 890 f 240 cm-I, and oscillator strength 3.63 f 0.04. Predicted shifts in band shape from one solvent to another, using the same theory, show distortionsof vibrational structure that are not seen experimentally. It is concluded that the theory is satisfactory for predicting solvent effects on oscillator strength and wavenumber of absorption spectra of this type but that it contains an inherent artifact in the treatment of vibrational structure.
Introduction The visible absorption spectrum of 8-carotene (Figure 1) in solution offers an important test case for theories of solvent effects on electronic absorption spectra. The lowest energy transition, a T-T* transition near 450 nm (22000 cm-I), has a large oscillator strength and is well separated from higher energy transitions.' The extensive experimental study of all-trans-@-carotenein a wide range of solvents by Myers and Birge' showed that both the absorption wavenumber and the oscillator strength are strongly affected by solvent and that the effects are well correlated with the refractive index of the solvent, but apparently unrelated to solvent polarity. (Their study included 21 solvents with refractive indexes from 1.33 to 1.62 and dipole moments from 0 to 3.9 D.) The absorption band shows some partially resolved vibrational structure that varies only slightly among different solvents. Myers and Birge presented a theory that focused on the effect of solvent on the oscillator strength and showed that the observed dependence of oscillator strength on solvent polarizability is fairly well accounted for by a model in which the chromophore is treated as a point dipole transition moment at the center of a cylindrical cavity having approximately the dimensions of the all-trans-@carotene molecule. The effect of the solvent is to decrease the oscillator strength, the magnitude of the decrease being greater the larger the solvent polarizability. Myers and Birge showed that this is a consequence of the prolate cavity geometry and the polarization of the transition parallel to the long axis of the cavity. Their theory treated the interactions between solvent and chromophore by a first-order perturbation method, which does not predict a shift in absorption wavenumber from this interaction. Some of the limitations of perturbation theory were overcome recently in a classical cavity treatment of solvent effects on optical properties? The theory permits calculation of the effects of solvent polarizability on oscillator strength and wavenumber for an arbitrary assembly of interacting chromophores and nonchromophoric groups. The purpose of the present study is to test the theory on the relatively simple Bcarotene system, where only one electronic transition need be considered and where the effects exhibited by the experimental data are related to solutesolvent interactions and not to intramolecular interactions in the solute. A spectral feature that has been almost entirely neglected in previous applications of the classical theory is the presence of vibrational structure in the absorption band. It will be shown here that the effect of solvent on band shape is easily incorporated into ( I ) Myers, A. B.; Birge, R. R. J . Chem. Phys. 1980, 73, 5314. (2) Applequist, J. J . Phys. Chem. 1990, 94, 6564.
the theory, and the 8-carotene spectra offer a test of this aspect to the theory. The present study examines the model in an oversimplified form, using a point dipole approximation for the chromophore and a simple cubic lattice structure for the solvation shell. It would be possible to treat the chromophore by a more realistic distributed transition charge density and a more realistic solvation structure without adding greatly to the complexity of the formalism; but such calculations will be much more demanding computationally, and it is useful to gain some insight into the behavior of the model by a simpler means before this is done. It will be seen that solvent effects on oscillator strength and wavenumber are reproduced adequately by this simple version of the model. Certain features of the solvent effect on band shape, involving wavenumber shifts of vibronic components, are not predicted well by the model, and this fact presents a problem that will have to be addressed in future studies. The representation of the solvent environment of a chromophore by a simple cubic lattice of polarizable points was employed previously by Warshe13 in a variety of problems, including the calculation of environmental effects on the electronic transition energy of the protonated Schiff base of retinal. Warshel and Lappicirella4treated the effects of protein environment of the heme chromophore of hemoglobin by a related model that assigned fixed charges and induced dipoles to sites determined by the known protein structure; results were obtained for shifts in both wavenumber and oscillator strength of the transitions. Blair et ale5 examined a more realistic approach to solvent effects by a dynamic modeling of the water environment of formaldehyde, giving a prediction of the blue shift of the lowest energy electronic transition. The cited studies are all based on quantum mechanical treatments of the transitions, which make it possible to include effects of both fixed and induced moments on the chromophore. The classical model used here neglects permanent moments in the solvent molecules, an approximation that appears to be reasonable in the case of @-carotene. The simplicity of the formalism and the computations make the model attractive for the further exploration undertaken here. From the results on @-carotenewe hope to learn something of the validity of the model for more complex solutes where the classical approach has a substantial advantage in computational economy. (3) (a) Warshel, A. Proc. Narl. Acad. Sci. U.S.A. 1978, 75, 2558. (b) Warshel, A. J . Phys. Chem. 1979,83, 1640. (4) Warshel, A.; Lappicirella, A. J . Am. Chem. Soc. 1981, 103, 4664. (5) Blair, J. T.; Krogh-Jespersen,K.; Levy,R. M. J. Am. Chem. Soc. 1989, 111,6948.
0022-3654/91/2095-3539$02.50/00 1991 American Chemical Society
3540 The Journal of Physical Chemistry, Vol. 95, No. 9, 1991
Here v is a vector antiparallel to the dipole moment induced in the solvent lattice by that of the chromophore, FL is the lattice reaction field coefficient at the chromophore (Appendix B), and Fc is the continuum reaction field coefficient at the chromophore, given by7
Figure 1. Structure of all-trans-@-carotene.
It should be mentioned that Abe and Iweibo6 used a different cavity theory to calculate solvent effects on the oscillator strength of &carotene and found a trend comparable to that observed by Myers and Birge. However, the effect in their theory is unrelated to the geometry of the molecule or the cavity, and their theory must be regarded as inconsistent with that of both Myers and Birge and the present work.
Theory Our model consists of an electric dipole oscillator representing the solute chromophore located at the center of a spherical cavity in an otherwise continuous dielectric medium. The cavity also contains a lattice of polarizable point particles representing solvent molecules within the region bounded by the spherical cavity wall and the outer contact limit of the solute molecule. The chromophore and the lattice solvent molecules interact with each other by way of the fields of their induced dipole moments, including both the direct fields and the reaction fields resulting from the interaction with the surrounding continuum. The polarizability of the solvent in both the continuum and the lattice is taken to be that which is characteristic of optical frequencies; thus, orientation of permanent dipoles under the influence of the fields of either induced or permanent charge distributions is neglected, and the solvent is treated in all respects as a nonpolar medium of dielectric constant e. This has the effect of neglecting solute-solvent interactions arising from permanent-moment contributions which could affect the transition energies. It is assumed further that electronic transitions in the solvent occur only at frequencies much greater than that of the solute absorption band, so that the solvent polarizability is essentially nondispersive in the frequency range of interest. The theory that gives the absorption spectrum of the solution from the complex polarizability of the isolated chromophore has been given in detail elsewheree2 The results needed for the present calculations will be given here. They take a simpler form than those for more general solutes. The steps needed to obtain them from previous results are given in the appendixes. The chromophore is represented by a dipole oscillator polarized along unit vector i and located at the center of a spherical cavity of radius a. The polarizability tensor of the isolated chromophore is a0 = aoQQ
(1)
where a. is a complex function of wavenumber V. We consider two cases for comparison with the experimental data: (i) a. is given by a Lorentzian function of D, and (ii) a. is an arbitrary function of v that may be chosen to reproduce experimental band shapes. The Lorentzian form of a. is a. = Do/(ro2- p2
+ ir?)
(2)
where Dois the dipole strength (in units of cm), p0 is the resonance wavenumber, and r is the half-peak bandwidth. The imaginary part of eq 2 gives the absorption spectrum of the isolated molecule? The theory of the cavity model predicts that the absorption spectrum of the solution is likewise given by a Lorentzian function with the same bandwidth r and with dipole strength (Appendix A)
DI
Doli - vI2
Applequist
F~ = 2(c - 1)/(2€
+ i)a3
(5)
For the more general band shape the form of the polarizability function is altered by the solvent interaction according to (Appendix C) fio-l
- p - v12fi1-1 = 3 v - c + FL)
(6)
where fiI is the complex polarizability of the chromophore in solution and overbars denote isotropic averages (e.g., suo = a0/3). In this case the dipole strength in solution is again given by eq 3 (Appendix A). In place of eq 4 for the wavenumber shift, one finds a corresponding formula for the shift in mean-square wavenumber of the absorption band (Appendix D) ( V 2 ) I = ( P2)o - Do(Fc
+ FL)
(7)
where subscripts 0 and 1 denote isolated molecule and solution values, respectively, and
where t, is the molar absorption coefficient at wavenumber 9. It is worth noting that eq 7 is not meaningful for a Lorentzian band because ( 9 )is infinite for that case whenever r > 0. On the other hand, eq 4 is strictly applicable only to a Lorentzian band. The remaining formulas needed for the present calculations are the following. The imaginary part of the chromophore polarizability in solution is obtained from e, using2
where N A is Avogadro's number. The real part is obtained from the Kronig-Kramers transform*
where the Cauchy principal value of the integral is indicated and the limits xIand x2 represent the known range of the absorption spectrum. (It will be noted that contributions to Re fi, from higher energy transitions are omitted.) The oscillator strength of the chromophore in solution is obtained from the conventional relationg
st,
f = (2303m,c2/NAae2)
dii
(1 1)
where meand e are respectively the mass and charge of the electron and c is the vacuum velocity of light. From eqs 2 and 9, the f defined by eq 11 is related to the dipole strength D by
f=
+
[ 3 ~ ' / ~ / ( 2 t1)](4n2m,$/3e2)D
(12)
The oscillator strength thus contains medium effects arising from the reaction field effects present in D,the cavity field factor2 3e/(2c + I), and a factor arising in the calculation oft, from the complex refractive index. (In previous calculations2 I omitted the coefficient in e in eq 12, and the values off so calculated are not directly comparable to experimental values calculated by eq 1 1.)
(3)
Computational Methods The chromophore is located at the origin of a coordinate system with its axis i parallel to one coordinate axis. The boundary of the @-carotenemolecule is regarded as a cylinder with half-length
(4)
Electric Polarization, 2nd ed.; Elsevier: Amsterdam, 1973; Vol. 1, p 134.
and resonance wavenumber (Appendix B) VI2
= Po2 - Do(Fc
+ FL)
(6) Abe, T.; Iweibo, 1. Bull. Chem. SOC.Jpn. 1986, 59, 2381.
(7) BBttcher, C. J. F.; van Belle, 0. C.; Bordewijk, P.; Rip, A. Theory of
(8) Moscowitz, A. Ado. Chem. Phys. 1962,4, 67. (9) Kauzman, W. Quantum Chemistry; Academic Press: New York, 1957; p 581.
The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 3541
Solvent Effects on 8-Carotene Absorption Spectrum 1.8
f
1.6
'1.41
I
I
I
I
I
I
h
I
\
25
4.0
I
!-
I
I
I
I
I
I
I
I
I
I
I
-1 3.5
\
43.0
I
( E -1) / ( E + 2 )
0.1
Figure 2. Oscillator strength of visible absorption spectrum of @-carotene as a function of solvent polarizability: left ordinate, relative oscillator strength referred to benzene solution;right ordinate, absolute oscillator strength; (0)experimental data of Myers and Birge;' (-) theoretical curve for cavity model.
14 A and radius 4.42 A, as given by Myers and Birge.' The cylinder is centered at the origin with its axis parallel to the chromophore axis. The spherical cavity in the continuum is likewise centered at the origin and given a radius equal to the half-height of the cylinder. The region between the cylinder and the cavity wall is filled with points on a simple cubic lattice whose unit cell edges, of length d , are parallel to the coordinate axes. The coordinate origin serves as translation origin for the lattice. A lattice point is excluded if its distance from the cylinder wall is less than a contact distance c, or if its distance from the cavity wall is less than a contact distance c,. The values c, = c, = 1.O A were chosen somewhat arbitrarily. Each lattice point is assigned a polarizability aL such that a bulk medium composed of such a lattice would have the dielectric constant t in accordance with the Lorentz-Lorenz relation, aL = (3&/4n)(e - l)/(t 2). An effort was made to find a value of d that would apply to all solvents, so that the only variable quantity from one solvent to another is e. The major computational step is the calculation of v and FL by eqs A3 and B3 for use in eqs 3,4,6, and 7. The computation of the matrices involved has been described previously.* For treatment of the detailed band shape, the experimental molar absorption coefficients of @-carotenein isopentane, benzene, and carbon disulfide were read from the published graphs of Myers and Birge,' covering a range of approximately 18OW28 000 cm-I. Data were read at intervals of 100 cm-l with an estimated accuracy of 2% of the peak values. Each spectrum was fitted by a set of five log-normal bands by the procedure of Siano and Metzler,lo and the analytical log-normal expression was used in the subsequent calculation of solvent effects. The integrals in eqs 8 and 10 were evaluated numerically over the range 16000-34000 cm-I, using an increment of 50 cm-' in 8. The calculations were carried out on a Hitachi Data Systems AS/9180 computer. Double precision was used in all of the matrix calculations. Single precision was used in the log-normal curve fitting.
I
I
0.3
0.2
(e -1)/ ( e +2)
Figure 3. Absorption wavenumber of visible spectrum of @-caroteneas a function of solvent polarizability: (0)experimental data of Myers and Birge' for wavenumber of strongest vibronic peak; ( 0 )root-mean-square wavenumber of absorption band; (-1 theoretical wavenumber for a single Lorentzian peak in the cavity model; (---) theoretical rms wavenumber in the cavity model. 151
I
I
I
I
I
I
I
+
Results Figures 2 and 3 show the experimental data of Myers and Birge' for the oscillator strength and absorption wavenumber of @-carotene as a function of solvent polarizability measured by (t - l)/(e + 2). The oscillator strengths were given by these authors as relative valuesf,/fb, wherefb is the oscillator strength in benzene andf, is that in the given solvent. The absorption wavenumber pI reported was that of the most intense vibronic peak (see Figure 4a). Theoretical values offi/fb were calculated from eq 3, converting D s t o f s by eq 12. The results were sensitive to the choice of the lattice distance d , and it was found by trial and error that a fixed value d = 4.1 5 A for all solvents gave good agreement with experiment, as shown by the solid curve in Figure 2. (The lattice ~
(10) Siano. D.
B.;Metzler. D. E. J . Chem. Phys.
1969, 51, 1856.
-v ,
103cm-1
Figure 4. Visible absorption spectra of @-carotene.(a) Observed spectra of Myers and Birge' in three solvents: (-) isopentane; (--) benzene; (- -) carbon disulfide. The curves are sums of log-normal bands fitted
-
to the experimental spectra and extend the high-wavenumber tails beyond the observed range by 4000 cm-' or more. (b) Hypothetical vapor spectra of @-carotenecalculated from each of the spectra in (a). Dashed and solid curves correspond to the source spectra in (a).
so constructed had 96 lattice points in the allowed region.) The absolute oscillator strength of @-carotenein hexane has been given as 2.4911 and 2.69.12 We assume a midvaluefi = 2.60 in hexane (c = 1.891) and from the theoretical curve in Figure 2 obtain the "vapor"-phase oscillator strength$, = 3.90. This fixes the scale of absolute oscillator strengths on the right side of Figure 2. The solvent effect on absorption wavenumber is treated most simply by eq 4,assuming that the absorption band can be approximated by a Lorentzian with its maximum at pI. The shift term Do(Fc + FL)is fixed by the structure taken for the solid curve in Figure 2. Numerical calculations show that is an approximately linear function of (c - I)/(€ + 2). Thus, a good estimate of p0 is obtained by a least-squares straight-line fit to the experimental data of Figure 2, giving the extrapolated intercept po (11) Bayliss, N . S.J . Chem. Phys. 1948, 16, 287. (12) Mulliken, R.S.;Riecke, C.A. Rep. Prog. Phys. 1941, 8, 231.
3542 The Journal of Physical Chemistry, Vol. 95, No. 9, 1991
Applequist
TABLE I: Root-Mean-Square Wavenumbers and Oscillator Strengths of @-CaroteneSpectra in Solution a d Hypotbetical Vapor State (j2)’/*,
solvent isopentane benzene
6
1.825 2.244 carbon disulfide 2.640 a
f
cm-’
solution
vawr
solution
vawr
23190 22280 21 570
25150 24850 24680 24 890 i 240’
2.50 2.12 1.90
3.68 3.60 3.61 3.63 i 0.04’
Mean values.
= 24400 cm-’. Equation 4 then gives the solid curve in Figure 3, which agrees well with experiment. It is of interest to note that FLis found to be 89-90% of Fc + FL over the experimental range o f t . Thus, the wavenumber shift is dominated by the lattice reaction field, though the continuum reaction field is not negligible even for the rather large cavity adopted here. The assumption of a Lorentzian band shape is avoided by the use of eq 6. aI is determined by the experimental spectra via eqs 9 and 10, and v and FLare fixed for each value of c by the lattice structure assumed for Figure 2. Thus, the absorption spectrum of the isolated chromophore (a hypothetical vapor state) is obtained from Im fi0 calculated from eq 6. Figure 4a shows the observed spectra in three solvents (as represented by a sum of five lognormal bands each), and Figure 4b shows the hypothetical vapor spectra calculated from each observed spectrum. If the theory were fully correct and the experimental data accurate, these vapor spectra should all be the same. The spectra differ in important details, but they agree fairly well in position, intensity, and overall breadth. Table I gives the root-mean-square wavenumbers and oscillator strengths obtained from these spectra, and it is seen that both quantities for the vapor spectra are constant within a standard deviation of 1%. Equation 7 provides another related test of the theory. By use of the mean oscillator strength and rms wavenumber of the vapor from Table I with the same lattice structure as before, eq 7 gives the dashed curve in Figure 3 for the rms wavenumber in solution. The filled circles represent the rms wavenumbers of the observed solution spectra from Table I, and these show the predicted shifts with solvent polarizability. (According to eqs 4 and 7, the two curves in Figure 3 should be parallel; this is not quite the case because slightly different values of fo were used for the two.) An experimental test of the vapor spectrum predicted in Figure 4b is lacking. To create a direct experimental test of the theory for general band shapes, eq 6 may be used to predict the spectrum in solvent B from the observed spectrum in solvent A. This is done by eliminating a,, between the equations for the two solvents, giving (P - VA~’&A-’ ~ ( F c A FLA)= (d - VB(’~B-’ 3(Fce + FLB)
+
+
+
(13) where subscripts A and B denote values for the corresponding solvents. Figures 5-7 show the predicted spectra along with the experimental spectra in each of three solvents. The predictions are consistent with each other and with experiment only on the high-wavenumber side of each absorption band. On the lowwavenumber side the theory fails to predict the band shape, which is strongly affected by vibrational structure. In general, the theory predicts too little shift in the positions of the vibronic bands, so that the wavenumbers of the calculated peaks tend to lie close to those of the spectrum from which they were calculated. In spite of this failing, the theory predicts the overall shift in intensity quite well, as shown by the mean-square wavenumbers. That is, the theory predicts a redistribution of intensity among the vibronic bands with little shift in band positions, while experiment shows a relatively constant distribution of intensity among vibronic bands and a shift in the position of the overall, band system. Discussion The comparison between theory and experiment for the solvent effects on oscillator strength and certain measures of band wavenumber shows that both effects are accounted for within the same formalism for a simple model. While the solvent effects on
V,
103cm-1
Figure 5. Absorption spectrum of @-carotenein isopentane: (0)experimental data of Myers and Birge;’ (--) calculated from spectrum in benzene; (- - -) calculated from spectrum in carbon disulfide.
-V ,
103 cm-1
Figure 6. Absorption spectrum of &carotene in benzene: (0)experi-
mental data of Myers and Birge;’ (-1 calculated from spectrum in isopentane; (- - -) calculated from spectrum in carbon disulfide.
oscillator strength and wavenumber both arise from similar interactions between the chromophore and solvent, the two effects differ qualitatively in their dependence on the solvent structure. According to either eq 4 or eq 7 the absorption band is always red-shifted in the presence of solvent, as the shift term is always positive. (In the case of FL this fact rests on the positive definite property of the matrix of Appendix A.I3) The oscillator strength, however, can either increase or decrease in the presence of solvent according to eq 3, depending on the orientation and magnitude of the lattice moment expressed by the vector v. In the case of a prolate cylindrical molecule with axial P, v is parallel to ii and of the same sign, so that the oscillator strength is decreased by the solvent. In the case of an oblate cylindrical molecule with axial i, v is antiparallel to 0, and the effect of the solvent is reversed. The theory of Myers and Birge’ for a cylindrical cavity produced the same results regarding the direction of change of the oscillator strength. The results in Figures 5-7 show that where the theory deals with the band shape it is only partially successful. The predicted solvent effects on the relatively unstructured high-wavenumber region agree fairly well with experiment, but the predicted wavenumbers and intensities of the vibronic components, where these are resolved, do not agree with experiment. The theoretical results generally show large shifts of intensity among the vibronic components and small shifts in their wavenumbers. The experimental spectra show just the reverse, with little shifting of intensity among vibronic components and a uniform shifting of peak positions, giving an overall shift of the band position with little change in shape. ~~
(13) Applequist, J. J . Chem. Phys. 1979, 71, 1983.
~~~~
The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 3543
Solvent Effects on @-CaroteneAbsorption Spectrum
I
I
I
-v ,
IO'cm-'
Figure 7. Absorption spectrum of @-carotenein carbon disulfide: (0) experimental data of Myers and Birge;) (-) calculated from spectrum in isopentane; (--) calculated from spectrum in benzene. A true vapor spectrum of &carotene to compare with the calculated vapor spectra of Figure 4b would provide an important test of the theory. The three calculated spectra agree in showing a major shift of intensity to the high-wavenumber region, with most of the vibronic structure lost in the low-wavenumber tail. It should be borne in mind that the calculated vapor spectra would not be expected to reproduce observed spectra for at least two reasons: (i) the collision broadening effects present in solution have not been accounted for, and (ii) the populations of vibrational levels, which affect the intensities of the vibronic bands, would most likely differ in vapor and in solution. However, the oscillator strength and mean-square wavenumber of the vapor spectrum would be expected to correspond to the calculated spectra, and it would be of interest to see what correspondence there might be in the band shape. The validity of the classical theory of chromophore interactions for predicting band shapes has been tested in the past, mainly in connection with the spectra of dimeric chromophores and larger aggregates.Iel7 Much attention has been given to the question of validity in the case of strong coupling, where the interaction energy between chromophores is much greater than the bandwidth. To place the &carotene problem in this context, we may take ijo - PI as a measure of energy of coupling of the chromophore with itself by way of the reaction field. The data in Figure 3 show this to be as large as 3700 cm-I, which is comparable to the observed half-peak bandwidths of about 4000 em-'. The strength of coupling would then be regarded as intermediate by the criteria used for interacting chr~mophores.'~ DeVoe,14using a formalism closely related to the present one, calculated the absorption spectrum of a dimer in which the monomer spectrum was taken to be a superposition of Gaussian bands, assuming various coupling energies. In the strong coupling region he noted a tendency for the absorption intensity to become concentrated in a narrow band. This result is clearly associated with the assumed band shape, as a single Lorentzian band treated in the same way undergoes a shift with no change in bandwidth.l* There appears to be experimental support for DeVoe's finding in the strongly shifted and narrowed absorption band in the spectrum of a cyanine dye aggregate.Ig (14) DeVoe, H. J. Chem. Phys. 1964.41, 393. (15) Hemenger, R. P. J. Chem. Phys. 1977.66, 1795. (16) Hemenger, R. P.; Kaplan, T.; Gray, L. J. J . Chem. Phys. 1979, 70, 3324. (17) Lukashin, A. V.; Frank-Kamenetskii, M.D. Chem. Phys. Lett. 1977, 45. 36. __ ,
(18) Applequist, J.; Sundberg, K. R.;Olson, M.L.; Weiss, L. C. J . Chem. Phys. 1979, 70, 1240. (19) Cooper, W. Chem. Phys. Leu. 1970, 7 , 73.
However, HemengeP obtained theoretical results for a dimer similar to those of DeVoe and showed that the narrowing of the band in strong coupling is greatly reduced in an alternative theory in which the vibrational levels of the electronic states are treated more accurately. He concluded that the classical theory is invalid in the region of intermediate or strong coupling. Lukashin and Frank-Kamenet~kiiI~ reached a similar conclusion in treating a dimer spectrum for a model in which the monomer spectrum consisted of a progression of vibronic lines. They compared results from a self-consistent-field (SCF) treatment, which is closely related to the classical theory, with those from an exact solution to the SchrGdinger equation for a case in which the coupling energy was 3 times the spacing of the vibronic lines. The S C F method underestimated the shift of the vibronic frequencies and overestimated the shift of intensity among the lines, tending to concentrate the intensity in the lowest energy line. The calculated solvent shifts for @-carotene show effects analogous to those found by these authors for chromophore interaction in dimers. A tendency for concentration of the band intensity in a narrow range is seen in the calculated vapor spectra of Figure 4b. (In this case the narrowing occurs on removal of the coupling, which is the reverse of the situation found in the dimer. The effect is more strongly evident if one adopts an artificially large value for Fc or FL, which produces a vapor spectrum dominated by a single narrow line.) The shifts of the vibronic bands in Figures 5-7 show discrepancies relative to experiment similar to those noted by Lukashin and Frank-Kamenetskii. Thus, the weight of the theoretical and experimental evidence suggests that an artifact does exist in the classical theory as it is applied to both dimer spectra and solvent effects on monomer spectra. The artifact appears to limit the validity of the theory for predicting changes in band shapes, particularly where vibrational structure is present. Aside from fundamental assumptions of the classical model, there are physical simplifications in the present treatment which could be suspected of producing artifacts. These include (i) the point dipole treatment of the chromophore, (ii) the omission of higher energy transitions of &carotene, (iii) the representation of the local solvent structure by points on a regular lattice, and (iv) the treatment of the solvent polarizability as nondispersive. It seems unlikely that any of these could be responsible for the problems with the band shape because the classical formalism could treat all of these aspects of the model more accurately, again giving eq 6 but with different routes in the calculation of v, Fc, and FL. The refinement of the model to include more realistic structures should give a more reliable theory of the solvent effects on oscillator strengths and band wavenumbers, but the use of the classical theory to predict effects on band shape remains problematical.
Acknowledgment. I am grateful to David A. Rabenold for helpful suggestions during the course of this study and to Carol and David Metzler for providing a computer program for band fitting. This investigation was supported by a research grant from the National Institute of General Medical Sciences (GMl3684). Appendix A. Solvent Effect on Dipole Strength Let the chromophore be composed of n Lorentzian oscillators. In the absence of solvent oscillator i has dipole strength Do(')and polarization direction ai. Let the cavity contain in addition N nondispersive oscillators, which may include both the solvent lattice and any nondispersive polarizability assigned to the solute. The dipole interaction equation takes the partitioned matrix form'*
where p and E are column vectors listing the components of the indued dipole moments and cavity field components, respectively, along the polarization directions of all oscillators; A is the interaction matrix, which includes both dipole and reaction field
3544 The Journal of Physical Chemistry, Vol. 95, No. 9, 1991
tensors; and subscripts 1 and 2 denote the dispersive and nondispersive subsystems, respectively. The normal modes of the system are a set of n Lorentzian oscillators with dipole strengths D,(I)which obey the sum ruleI3 n
i= I
n
= CDo("lii - vil2 i= I
Applequist
The orthogonality condition for the eigenvectors (ref 18, eq 30) gives
(A2)
n
Et 1J(i)fl*k(i)= D,(k)6.Jk
ill
Here
vi = UA22-lA2l.i
Hence, the summation of eq B8 over i gives ('43)
where U is the 3 X N matrix whose columns are the unit vectors of the nondispersive oscillators and A21,iis the ith column of A21. We are interested in the special case in which all dispersive oscillators are polarized in the same direction (all ii= 1) and are located at the same point (all vi = v). Then eq A2 becomes
Equation A4 is equivalent to eq 3 with Do = ED$) and D , = EDl('). This result also applies to the arbitrary band shape of Appendix C by letting n OD.
-
Appendix B. Wavenumber Sum Rules The eigenvalue equation derived from eq A1 is1* (A?, - A12A22-'A21)tl(i) = B?CIItl(i)
(B1)
where B? and tl(O are the ei envalue and eigenvector, respectively, of the ith normal mode; A l l is A l l evaluated at B = 0; and C I I is the diagonal matrix of elements I/Do('). Let the oscillators of Appendix A be located at the same point in the cavity with the same polarization direction i. Oscillator i has resonance wavenumber poi in the absence of interactions. AYl may be written
Q
A?, = CIIA - FCJ
(B2)
where A is the diagonal matrix of elements tr0? and J is the n X n matrix whose elements are all unity. Also
AI2A22-lA21 = FLJ
(B3)
because the elements of this matrix are the lattice reaction field coefficients relating the sites of the dispersive oscillators and are all equal to FL. Thus eq B1 becomes
[A - (Fc + FL)CII-IJ]tl(i) = B?tl(')
(B4)
Equation B4 implies that B? is an eigenvalue of the matrix A (Fc FL)CII-IJ. Hence we obtain the "unweighted wavenumber sum rule" by setting the sum of the eigenvalues equal to the trace of the matrix, giving
+
/=I
- v)( 1 l...l)tl(i)
i= I
With use of eq A4, this becomes the "weighted wavenumber sum rule" n
n
i=l
i=l
C3?DI(')/DI = CPO?Do(')/Do- Do(Fc + FL) (B12)
Like eq B5, this sum rule reduces to eq 4 for a single Lorentzian oscillator; however, the sums in eqs B5 and B12 are quite different in general, and it is noteworthy that the same shift term Do(Fc FL)occurs in both. Equation B12 is extended to an arbitrary band shape in Appendix D.
+
Appendix C. Relationship between Complex Polarizabilities for Arbitrary Band Shape Let q in eq 1 be an arbitrary complex function of wavenumber. Let the electric field represented by the 3 X 1 column vector E be the uniform throughout the cavity. Then eq A1 takes the expanded form
Solving for pl and
p2
AIIM+ I A12~2= aTE
(C1)
AZIC(1 + A2+2 = UTE
(C2)
w2, and using eqs A3 and B3, one finds = (All - FL)-I(8 - v ) ~ E
(C3)
= A22-I[UT - A21(All - FL)-I(i - v ) ~ ] E
(C4)
The total induced moment fit in the system within the cavity is then wt
= fir1+ uw2
(C5)
+
= [(All - FL)-'(i - V)(U- v ) ~ az]E
(C6)
where the polarizability of the nondispersive subsystem is az = UAzz-'UT
(C7)
The coefficient of E in eq C6 is the total polarizability tensor a, of the cavity contents. The polarizability tensor of the chromophore in solution is al = a,- a2.With All = a,,-I - Fc, eq C6 then gives a1 =
This reduces to eq 4 for a single oscillator. A more general form of this sum rule for the case Fc = 0 was given previously.18 To obtain a second wavenumber sum rule that can be generalized to arbitrary band shapes, we make use of the definition of the normal-mode electric dipole moment M(i):
n
+
Ii - v12?80?Do(i) - DoDl(Fc FL)= C B ~ D , ( (~B)l l )
(a0-l
- Fc - FL)-'(i - v ) ( i - v ) ~
(C8)
On taking isotropic averages, eq C8 becomes eq 6. This result is closely related to DeVoe's14 equation for normal-mode polarizabilities of a system of identical coupled oscillators, and the present derivation was prompted by that of DeVoe.
(B6)
Appendix D. Mean-Square Wavenumber Shift for an Arbitrary Band The Lorentzian polarizability of eq 2 takes the following form in the limit r = Ozo
037)
a ( e ) = D/(Po2- P2) - ( ~ D T / ~ B -~e)) ~ ( B( D~l )
where the superscript T denotes the matrix transpose. Multiplying eq B4 by li - v12tl(i)TJ,we obtain
where 6(v0 - e ) is the Dirac delta function. Let the arbitrary polarizability function of Appendix C be regarded as a continuous distribution of such &function oscillators such that the number of oscillators in the range v0 to v0 + dpo is po(v0) dso and the dipole
p(') = (6
which gives the normal-mode dipole strength D,(i)= w(i)Tw(i) = 16 - ~ 1 2 f ~ ( i ) T J f ~ ( i )
li - vJ2tl(i)TJAtl(i) - (Fc + FL)D&fi) = e?Dl(')
(B8)
using JCI1-IJ= D d , The first term on the left may be transformed by summing over i and expanding:
(20) Rabenold,
D.A.; Rhodes, W.J . Chem. Phys. 1984,80, 3866.
3545
J . Phys. Chem. 1991, 95, 3545-3549 strength of each oscillator at eo is Do(ro). Then Im a0@) = -(7r/2)Jmp0(eO)Do(eo)eo-'6(eo
- e) deo
-
(D2)
The integrals in eqs D5 and D6 are equal to the corresponding sums in eq B12 in the limit n -. Furthermore, the system of n Lorentzian oscillators, each regarded in Appendix B as an individual chromophore, is equivalent to a single chromophore whose polarizability is the sum of n Lorentzian functions. Hence, eq B12 applies to the present case and takes the form
= -TP~(V)D0(~)/28 (D3) In the presence of interaction with the solvent the corresponding relation is Im al(e)= - r p l ( e ) D 1 ( e ) / 2 e (D4) The mean-square wavenumbers then become, from eqs 8 and 9 (e2), = x m s 2 p , ( y ) Do@) dij/lmpo(ij) 0 Do@)de (D5)
Registry No. rmns-&Carotene, 7235-40-7.
Matrix Infrared Study of the Nd:YAG Laser Depolymerization Products of Solid P4Ol0 Matthew McCluskey and Lester Andrews* Department of Chemistry, The University of Virginia, Charlottesville, Virginia 22901 (Received: October 2, 1990)
Molecules produced by the depolymerization of P4O10 have been the subject of a matrix isolation infrared study. Nd:YAG laser ablation of the hexagonal modification of phosphorus(V) oxide produced bands at 1408.2, 1025.7, 767.2, 576.8, and 41 2 cm-' due to argon-isolated P4Ol0, while laser ablation of the orthorhombic modification produced intense bands at 1319.4 and 386.3 cm-I due to the PO2 molecule and absorptions for a large number of additional depolymerization species. The above products are compared to those from the thermolytic decomposition of P4O10 vapor at 900-950 OC, which produced strong absorptions due to P409 and PO2and weaker bands due to P40e P205, and P204 Reactive laser ablation of red phosphorus in oxygen produced a spectrum similar to that from the laser ablation of orthorhombic P4O10. In all of the decomposition reactions studied, HOP0 and HOP02 were observed as a result of partial hydrolysis of the P4Olo samples.
Introduction In the past several years, a study of the matrix infrared spectra of less-common phosphorus oxides has been conducted in this laboratory. Several different experimental methods of producing novel species from the reactions of phosphorus and oxygen have been employed, including the reactions of P4 and PH3 with 0 atoms from a microwave discharge, the photolysis reactions of P4 and PH, with O,,and the reaction of discharged P4 from a coaxial microwave discharge tube with O2.I+ In a series of experiments, reactions of P2 with 0,and O3 were e ~ a m i n e d . ~ . ~ These studies identified P20, the phosphorus analogue of nitrous oxide, oxo-bridged P203,several isomers of oxo-bridged P204,and the ring isomers of P 4 0 and P402. They also confirmed earlier reports of PO, PO2, and P2O5.Iq4 While the molecules mentioned above were produced by combination reactions of the elements, they may also be viewed as the thermodynamically stable products of depolymerization of the more familiar phosphorus oxides, P406 and P4010.8 The purpose of the present study was to identify the depolymerization products of P4O10, using Nd:YAG laser ablation techniques combined with matrix-isolation infrared spectroscopy. Two polymorphic modifications of P4OlOwere utilized. The hexagonal modification, P4OI0(H),is the commercially available form, produced by burning P4 in oxygen. It is a volatile white powder, with a vapor pressure of 16 Torr at 250 OC,and is aggressively hydroscopic. The orthorhombic modification, P4Ol0(O), is produced by heating P4OI0(H)above its melting point in a closed container. It is extensively polymerized, with a glassy structure consisting of sheets of interlocking rings, which lowers ( I ) Andrews, L.; Withnall, R. J . Am. Chem. Soc. 1988, 110, 5605. (2) Withnall, R.; Andrews. L. J. Phys. Chem. 1988, 92, 4610. (3) Mielke, 2.;Andrews, L. Inorg. Chem. 1990, 29, 2773. (4) Withnall, R.; Andrews, L. J . Phys. Chem. 1987, 91, 784. ( 5 ) Mielke, 2.;McCluskey. M.; Andrews, L. Chem. Phys. Lerr. 1990,165, 146. (6) McCluskey, M.; Andrews, L. J . Phys. Chem. 1991, 95, 2679. (7) McCluskey, M.; Andrews, L. J . Phys. Chem. 1991, 95, 2988. (8) Lohr Jr., L. L. J . Phys. Chem. 1990, 94, 1807.
0022-3654191 12095-3545$02.50/0
the vapor pressure of P4OlO(O)(34 Torr at 440 "C) and reduces the rate of h y d r a t i ~ n . ~ J ~ In the present work, the products of Nd:YAG laser ablation of P4OI0(H)and P4OlO(O)were isolated in argon matrices and studied with infrared spectroscopy. The spectra show extensive depolymerization of P4OlO(O). The results are compared with the products of reactive laser ablation of red phosphorus and oxygen and to the results of the thermolytic decomposition of P4Ol0 vapor at 900-950 OC. In general, the PxO,product distributions closely resemble those obtained in the reactions of evaporated P2 with O3and 02,lending mechanistic support to the assignments to PO,, P2OS,P20, and the P204 isomers. (Detailed descriptions of the vibrational assignments have already been given and will not be repeated here.'-7) The observation of these molecules with seven different chemical methods is possible because of their thermodynamic stability and suggests that they may also be observed outside of the matrix environment.
Experimental Section The refrigeration unit (CTI Cryogenics Model 22) and the vacuum system have been described previou~ly.~*~ The laser ablation of P4O10 was performed by mounting a pressed pellet of P4O10 (hexagonal or orthorhombic modifications) 2 cm in front of the matrix window. A synthetic quartz lens (1 in diameter, 2411.focal length) mounted onto a rod inside the vacuum vessel focused the laser into a tight spot on the sample surface. The Nd:YAG laser (Quanta Ray DCR-11) was operated in Qswitched mode at a repetition rate of 10 Hz. Typically, the laser was operated at, or slightly above, the lasing threshold with 20-40 mJ/pulse. The rod on which the sample was mounted could be rotated periodically to refocus the laser onto a fresh portion of the sample. The orthorhombic modification of P4O10 was prepared by heating the familiar hexagonal form at 400 OC in an evacuated (9) Mellor, J. W. Comprehensive Treatise on Inorganic and Theoretical Chemistry; Wiley: New York, 1971; Vol. VIII, Supplement 111. (IO) Hill, W. L.; Faust, G. T.; Hendricks, S. B. J. Am. Chem. Soc. 1943, 65. 194.
0 1991 American Chemical Society
