11233
J. Phys. Chem. 1994, 98, 11233-11235
Reply to the Comments on “Time-Dependent Theoretical Treatment of Intervalence Absorption Spectra. Exact Calculations in a One-Dimensional Model’’ David
S. Talaga, Christian Reber,? and
Jeffrey I. Zink*
Department of Chemistry and Biochemistry, University of Califomia, Los Angeles, Los Angeles, California 90024 Received: May 31, 1994 In their comment’ on our paper,2 Schatz and Piepho “conjecture” that the model we present is “precisely equivalent” to that presented in their 1978 paper.3 On the basis of this conjecture, they believe that there are systematic errors in our calculations, specifically “parity” or “interchange symmetry” errors in determining the “vibronic selection rules”. Ondrechen et al. claim that our calculations use an “improper dipole pera at or".^ All of these statements are incorrect. In this reply we show that the models, though similar, are indeed different and give different results. We outline the physical assumptions that we use for our calculations and show how these are different from those discussed by Schatz et al. and Ondrechen et al. When we use the same Hamiltonian and physical assumptions as they, we calculate identical spectra. We show that the model problem that they actually calculate has a physical meaning different from a simple intervalence electron transfer. To clear up this confusion, we include a detailed discussion of the transition dipole moment operator, p i j = J’y&(r)p.e&l(r) dr. The diabatic potential energy surfaces for the model we use are shown in Figure la. The potential energy operator that we used in our paper is
-0.6 -0.4 -0.2 0.0 0.2
A
0.6 -0.6 .0.4
-0.2
0.0
A
0.2
0.4
0.6
Figure 1. (a) Model 1. Diabatic potentials generated from eq 1 using A Q = 0.15 A, i j = 450 cm-’, m = 17 g/mol, and E = 2800 cm-I. (b) Model 2. The diabatic potentials generated from eq 2 using the same parameters as in (a). I
0
2000
4000 cm
0
ZOO0
2000
8000
10000
GOOO
8000
10000
GOOO
8000
10000
I
4000 cm
GOOO I
4000 cm
0
where Q is a non-totally symmetric coordinate that changes sign under permutation of the identical sites, k is the harmonic force constant, and AQ is a bond length change along Q. The diabatic surfaces represent symmetrically displaced harmonic oscillators coupled by a constant interaction term E (model 1). The potential energy operator used by Schatz et al. (model 2) is
0.4
1
Figure 2. (a) Spectrum generated from the potential in Figure l a using the dipole in eq 3. (b) Spectrum generated from the potential in Figure l b using the dipole in eq 3. (c) Spectrum generated from the potential in Figure la using the dipole in eq 6 with A = 0.15 A and K = ijm/ 4 R a J V ~ a awhere ~ i j = 100 cm-I, m = 17 g/mol, R, is the Rydberg constant in cm-’, NA is Avagadro’s number, me is the mass of an electron in grams, and uo2 is the Bohr radius in A.
et al. Both models use the following operator in the diabatic basis: The diabatic surfaces represent harmonic oscillators displaced in energy as shown in Figure lb. The two states are coupled by an interaction term that is linear in Q. In the 2 x 2 matrices, each diagonal element operates on the nuclear part of the wave function from a single diabatic electronic state whereas the offdiagonal elements transfer population from one state to the other. All of the calculations in our paper were based on the assumption that pit, f 0, pi=j = 0 and that p , j is a constant over the coordinate (the Condon approximation). The spectrum shown in Figure 2a is the same as that in Figure 3b of our original paper. Schatz et al. indicate that they also make the Condon approximation. The spectrum that we calculate using model 2 with the Condon approximation is shown in Figure 2b and is the same as that shown in Figure l b from the comment by Schatz et al. and Figure 1 from the comment by Ondrechen
* To whom correspondence should be addressed. Department of Chemistry, University of Montreal, Montreal, Canada.
0022-365419412098-11233$04.50/0
(3) The meaning of this operator is that, upon interaction with the photon, amplitude is transferred from one diabatic electronic state to the other. It is interesting to note that both of the diabatic surfaces in Figure 1 (with coupling) give the same adiabatic surfaces and eigenvalues, but the spectra calculated from these two models are quite different. Both the Schatz et al. basis set expansion calculation and our time-dependent theoretical calculation generate the same spectra for model 2. Schatz et al. choose selection rules, as they discuss in their papers, based on an interpretation of the electronic states represented by the surfaces in model 2. Our intensities are based on our choice of p i j , which prepares the initial excited state wave packet for propagation. Schatz et al. incorrectly believe that the spectrum calculated using model 1 0 1994 American Chemical Society
Comments
11234 J. Phys. Chem., Vol. 98, No. 43, 1994 should be the same as that of model 2. However, as is seen by comparing parts a and b of Figure 2 , the spectra calculated by using time-dependent theory and the Condon approximation are very different. The source of their erroneous conclusion is the conjecture that because the potential in model 1 can be transformed to the potential in model 2 by a coordinateindependent unitary transformation, the spectra must therefore be the same. Schatz et al. overlook the fact that all operators that depend on the electronic basis, including the transition dipole moment operator, must be transformed. The required transformation operators are (4) The transition dipole moment operator from model 2 as represented in the basis of model 1 is
When we calculate the spectrum with this transformed transition dipole operator and the potential defined by eq 1, it is identical to that calculated using model 2. Ondrechen et al. use the potential of model 1 and the dipole moment given by eq 5 with a different formalism to calculate a spectrum identical to that in Figure 2b. Our analysis shows that equivalence between the models can be obtained by the requisite transformation of the transition dipole moment operator. Although the operators of model 2 can be transformed to model 1, the transformation raises the question of the physical meaning of the transition dipole moment of eq 5. The photon produces a wave packet in the excited electronic state, which is propagated according to the Hamilt~nian.~The transition dipole moment operator, /IIJ, must represent this change from the ground electronic state to the excited electronic state. To represent a change of electronic state, it is most intuitive for the matrix representation of /II to have nonzero off-diagonal elements. In contrast, the dipole of eq 5 requires that the projection of the wave packet from state 1 be propagated on the surface for state 1 and that from state 2 change sign and be propagated on the surface for state 2. There is no population transfer between the two states as a result of the photon. We maintain that, in order to represent an intervalence charge transfer, /IiJ must transfer charge. This restriction of /IIJ is tantamount to requiring that the off-diagonal elements of PI 0; the off-diagonal elements transfer probability density from one diabatic electronic basis state to the other. Our choice of model, including the transition dipole moment operator, was kept simple for purposes of illustrating the use of the time-dependent theory in a basis that provides a simple and natural way of representing the intervalence charge transfer. One diabatic potential energy surface in model 1 represents a single valence localized form of the molecule MnBM”+’; the other surface represents the other trapped valence form Mn+’BMn. A vertical transition from one state to the other corresponds to moving an electron from one site to the other without a change in nuclear positions. The functional form of the off-diagonal elements in PIJ depends on the molecule of interest and the polarization of light. The physical meaning of the simple /IIJ that we used is that the light is polarized in the Z direction, which is a vector bisecting the site-bridge-site angle in the symmetric geometry. An advantage of the time-dependent theory is that any appropriate form of /IIJ can be used without increase in computational difficulty or time. For example, another common
*
functional form of P i j that is appropriate for different situations (e.g., light polarized in the X direction) is one that changes sign at the origin. A specific function that has this property and also approaches zero at long bond lengths is
The spectrum calculated by using this form of PIJ is shown in Figure 2c. Note that the “vibronic selection rules” are identical to those of Schatz et al. and Ondrechen et al. but that the intensities of the individual bands are very different. This result graphically illustrates the point that there is not one “correct” calculation, nor is there a unique “selection rule” for this problem. Even when a single Hamiltonian is used, the physical assumptions of the problem (molecular geometry, the polarization of the light, and the symmetry and coordinate dependence of PIJ)can lead to different spectra. Finally, we respond to the minor points raised in the comments by Schatz et al. Our calculations are exact calculations of spectra including coupling of the nuclear and electronic kinetic energies “within the constraints of the model and the numerical accuracy of the computer implementation.” Detailed considerations of the numerical accuracy have been presented previously in the literature.6 The eigenvalues in our original paper were determined to an accuracy of 0.1%, which was sufficient for that work. The eigenvalues that were included in our table were those below 10 000 cm-’ that contributed at least 0.001% of the intensity of the most intense band in the intervalence spectrum. We note that because we use a spectral method these states, though not listed in our table, are automatically included in the calculated spectra regardless of the magnitude of their contribution. The trend in the intensities of the low-energy absorption features in our paper is in agreement with the trends discussed by Schatz et al. for model 2. The “sticks” in our figures illustrate this point. The spectra in our figures are calculated using eq 1 from our original paper. Note that the Gaussian convolution occurs before multiplication by w. When broadening of the spectra is caused by large values of the phenomenological damping factor r and the resulting intensities are multiplied by w , the broadened peaks appear larger. In particular, the zero-energy transition is also given width by this factor. If the band widths are caused by inhomogeneous broadening, then a distribution of the sticks (multiplied by w ) would be summed, and the resulting broad band would not have any contributions from the zero-energy transition. In summary, the spectra calculated by Schatz et al. should not be identical to those we calculate. The sources of this confusion are the transition dipole and the basis that they use for their expansions. Their spectra are calculated from model 2, or equivalently, as done by Ondrechen et al., using the model 1 potential with a transition dipole given by eq 5. (Our timedependent methods using these models numerically reproduce their spectra.) However, spectra using model 2 must be interpreted either in a basis where the physical meaning is unclear or as not resulting from an intervalence charge transfer. The time-dependent methods that we use in our calculations provide new physical insight into intervalence absorption. In our published calculations, we chose to use a simple physical model with the Condon approximation to illustrate the use of and the insight available from time-dependent calculations. However, the equations in our original paper are not limited to harmonic diabatic potential bases and can use any form of coupling and any form of the transition dipole moment. This flexibility does not require an increase in computational effort.
Comments The present analysis demonstratesthat there is not one “correct” calculation nor is there a unique “selection rule” for intervalence charge transfer absorption spectra. The physical aspects of the problem (molecular geometry, the polarization of the light, requirement of charge transfer, and the symmetry and coordinate dependence of P i j ) can lead to different spectra.
Acknowledgment. This work was made possible by a grant from the National Science Foundation (CHE91-06471).
J. Phys. Chem., Vol. 98, No. 43, 1994 11235
References and Notes (1) Schatz, P.N.; Piepho, S. B. J . Phys. Chem., this issue. (2) Simoni, E.; Reber, C.; Talaga, D.; Zink, J. I. J. Phys. Chem. 1993, 97, 12678-12684. (3) Piepho, S. B.; Krausz, E. R.; Schatz, P. N. J . Am. Chem. Soc. 1978, 100,2996. (4) Ondrechen, M. J.; Ferretti, A.; Lami, A.; Villani, G. J. Phys. Chem., this issue. 15) Heller. E. J. Potential Energy Surfaces and Dynamics Calculations; Truhkr, D., Ed.; Plenum Press: New Ybrk, 1981. . (6) Feit, M. D.; Fleck, J. A,; Steiger, A. J. J . Comput. Phys. 1982,47, 412.
