J . Phys. Chem. 1993,97, 12678-12684
12678
Time-Dependent Theoretical Treatment of Intervalence Absorption Spectra. Exact Calculations in a One-Dimensional Model Eric Simoni, Christian Reber,? David Talaga, and Jeffrey I. Zink' Department of Chemistry and Biochemistry, University of California, bs Angeles, California 90024 Received: February 4 , 1993; I n Final Form: September 24, 1993"
Intervalence absorption spectra are calculated and interpreted by using the time-dependent theory of spectroscopy and the Feit and Fleck method of numerically integrating the time-dependent Schrodinger equation. These methods give exact eigenvalues and eigenfunctions (within the constraints of the model and the numerical accuracy of thecomputer implementation). The results provide a new physical picture of the absorption spectrum and emphasize that Born-Oppenheimer separation of electronic and vibrational wave functions does not apply to the problem. The details of the calculation are discussed and interpreted. The exact calculation is compared to results obtained in the adiabatic limit. The effect of the temperature on intervalence absorption spectra is discussed. Localization and delocalization are interpreted in terms of the eigenfunctions.
1. Introduction
In this paper we report the results of exact quantum-mechanical calculations of intervalence absorption spectra in a one-dimensional model. The eigenvalues and eigenfunctions are calculated and interpreted. The exact results are compared to the results obtained in the adiabatic limit. The physical meaning of the results, the effects of temperature on the spectra, and the concepts of "localized and delocalized" mixed valence complexes are discussed. 2. Theory In this section the theoretical foundation underlying the calculation of intervalence band spectra in the framework of the time-dependent theory of molecular spectroscopy is briefly presented. The time-dependent approach is very powerful from both the technical and conceptual points of view. The physical meaning will be discussed in the following sections. The fundamental equation for the calculation of an absorption spectrum in the time-dependent theory i ~ ~ 9 - 2 1
The problem of calculating the absorption spectra of mixedvalance complexes has proven to be an interesting challenge.I-l8 Two extremes, the diabatic limit and the adiabatic limit, are intuitively appealing because they can be thought about in terms of Born-Oppenheimer separability of electronic and vibrational wave functions. In the diabatic limit, each part of the coupled system is treated independently. For example, in a typical mixedvalence metal compound containing two metals M, one with oxidation state n and the other with n 1, one diabatic potential surface represents the M"1-M" molecule and the second represents the Mn-Mn+I molecule. The intervalence absorption band, often a low-energy transition in the near-IR region of the absorption spectrum that is absent when the two metal centers have the same oxidation states ( n and n or n 1 and n + l), is treated in terms of a one-electron transition from the Mn to the Mn+I. In the adiabatic limit, the two potential surfaces are coupled giving rise to two new surfaces, the upper and lower adiabatic Z (.w ,) = surfaces, with an energy separation at the coordinate of the "avoided crossing" of twice the coupling term. The intervalence absorption band is then treated in terms of a transition from the lower electronic surface to the upper electronic surface. In both of these pictures, the absorption band envelope (or the vibronic with I ( w ) the absorption at frequency o,EO the energy of the structure, if any is resolved) is interpreted in terms of Franckelectronic origin transition, and r a phenomenological damping Condon factors, Le., the overlaps between the vibrational wave factor. The most important part of eq 1 is ( $ I $ ( t ) ) , the functions of the initial surface and the final surface. autocorrelation function of the initial wavepacket $ and the The diabatic and adiabatic limits may have some approximate wavepacket d ( t )that evolves with time. For twocoupled diabatic validity in certain limits of the coupling, but they are neither potential surfaces, the situation that is used to treat intervalence general nor completely correct. The problem arises because the absorption bands, two wavepackets, $1 and $2, moving on the two Born-Oppenheimer separation of the electronic and vibrational coupled diabatic potential surfaces are needed.25-27*3c38 The total wave functions is incorrect when two coupled surfaces cross. The overlap ( # ~ ( t ) ) is concept of potential surfaces loses its meaning. The intervalence absorption band cannot be interpreted as a purely electronic transition between two potential surfaces.3~9 The initial wavepackets are The time-dependent theory of electronic s p e c t r o s ~ o p y ~ ~ - ~ l provides a powerful method of carrying out exact quantum mechanical calculations of the intervalence absorption spectrum. where p t is the electronic transition moment for the transition The time-dependent Schradinger equation is solved numerically and x ( Q ) is the wave function of the initial state. (The initial by the split operator method of Feit and F l e ~ k . 2 * - The ~~ state's eigenfunctions are calculated by using the method shown calculations in the time domain also provide a physical picture at the end of this section. Eigenfunctions of all of the thermally of the effects of the coupling of electronic and nuclear motions populated eigenstates, weighted by their thermal populations, because the time development of the wavepacket can be followed are needed.) For simplicity, the transition momentspi werechosen and interpreted.25-31 to be coordinate independent, Le., constants, in all of the calculations carried out in this paper. t Current address: Department of Chemistry, University of Montreal, To calculate the autocorrelation function in eq 1, we need Montreal, Canada. Abstract published in Advance ACS Abstracts, November 1, 1993. 4i(t), the propagating wave function in the two coupled diabatic
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Theoretical Treatment of Intervalence Absorption Spectra
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12679
potential surfaces. It is given by the time-dependent SchrMinger equation for two coupled states
where Hi denotes the Hamiltonian (Hi = -l/2MV2 + vi(Q)), V,(Q) is the potential energy as a function of the configurational coordinate Q, -l/2MV2 the nuclear kinetic energy, and V12 = V21 = e is the coupling between the two diabatic potentials (chosen to be coordinate independent.) For simplicity, harmonic diabatic potentials are used in all of the following examples, although the theoretical method is not restricted by the functional form of the diabatic potentials. The diabatic potentials are given by Y(Q) = '/&j(QAQjI2 + Ej (5) with ki = 4 ~ ~ M ( hthe w force ~ ) ~constant, AQi the position of the potential minimum along Q, and Ei the energy of the potential minimum for the diabatic potential surface chosen such that the energy of the lowest eigenvalue is zero. The split operator method developed by Feit and Fleck is used to calculate 4i(t).22-24 Both the configurational coordinate Q and the time are represented by a grid with points separated by AQ and At, respectively. For one diabatic potential surface, the time-dependent wave function q5(Q,t+At) is obtained from q5(Q,r) by using the equation
4(r+At) = exp((E)V') exp(iAtV) exp((E)V2)q5(Q,t)
+
~[(Ao~I
= bvF#(Q,t) + O[(At)3] (6) For the case of t y o coupled diabatic potentials, the exponential operators Pand Vin eq 6 are replaced by 2 X 2 matrices operating simultaneously on 41(Q,t) and 42(Q,t):
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Figure 1. Diabatic potential surfaces that form the basis for the timedependent calculation of the intervalence absorption band of the CreutzTaube compound (solid lines). The adiabatic surfaces are shown by the broken lines.
is a Hanning window function. For coupled diabatic potentials, each eigenfunction qi is an array with two components corresponding to the two diabatic potentials which form the basis in all of our calculations.
3. Calculations A. Diabatic and Adiabatic Potential Surfaces and the Coupled Vibronic System. The diabatic potential surfaces that are used in the following discussion are shown in Figure 1. The first step, and one of the most important ones in relating the calculations to a physical problem, is to define the configurational coordinate In this paper thecoordinate is chosen to represent a metal-ligand stretching mode. It is a non-totally-symmetric mode for the entire molecule, but it can be thought of as a totallysymmetric metal-ligand stretch on each fragment. The motions can be schematically represented as L---M"--B-M"+l-L L-M"+'-B--M+"---L where L is a ligand bonded to the metal M and B is a bridging ligand. At the origin of this coordinate, all of the symmetry-related M-L bond lengths in the entire molecule are equal and the two metals are identical. The positive direction represents increasing M-L and M-B bond lengths on one of the metals and decreasing lengths on the other, corresponding to decreasing and increasing positive charges on the metals, respectively. The negative direction along this coordinate represents the situation opposite to that described above. The diabatic potential surfaces used in this calculation are chosen to be harmonic. Any functional form can be used in the calculation, but in the absence of additional information the harmonic potential is used for simplicity. The diabatic potentials used here are defined in terms of one force constant. To simplify the physical interpretation, dimensioned units of the vibrational frequency, 450 cm-l, and mass, 17 amu, are used to determine the force constant. The minimum of diabatic potential surface 1 is displaced by 0.1 50 A to the negative side of the origin, and that of surface 2 is displaced by an equal amount to the positive side of the origin. The coupling e between the two surfaces is 2800 cm-l. The adiabatic surfaces resulting from coupling the diabatic surfaces defined above are also shown in Figure 1. The lower adiabatic surface has a broad minimum centered at zero when the coupling is 2800 cm-1. If the coupling is decreased, the lower adiabatic surface develops two minima. The upper adiabatic surface is narrower and always has a minimum at Q = 0. B. Eigenvaluesand Eigenfunctionsof theCoupledSystem. The eigenvalues of the coupled system are calculated by using eqs 1 Q.'s3
-
~ [ ( A o ~(7) I Details of the computer implementation of eq 7 are given in the literature.2s-2'92-34 It is interesting to note the different roles of B and 8, the momentum operator, transfers wave function amplitude 4iamong grid points along Q a t each time step but does not transfer amplitude between thediabaticsurfaces. Thesechangesare easily monitored by looking a t the wavepacket 4i(t) after every time step. On theother hand, the potential energyoperator, changes the momentum and transfers amplitude between the diabatic surfaces at each time step but does not couple grid points along Q. The amplitude transfer between the diabatic potentials can be followed by calculating the norms (&(t)l$i(t)) for & ( t ) and &(t) after every time step. The norms are a quantitative measurement for the amount of population change between the two states. In the case of the symmetric intervalence transition, no net population change occurs. Both operators affect the total overlap ( & ( t ) ) in eq 1 and therefore the absorption spectrum. For this reason the adiabatic approximation is incorrect. Only the calculation which simultaneously involves both coupled diabatic surfaces according to eq 7 gives the correct total overlap. The eigenfunction \ki corresponding to the eigenvalue Ei is calculated by using eq 8.22
v.
v,
*,(Ei) = JoT4(t)w ( t ) exp(ft)
dt
$ ( t ) is the time-dependent (propagating) wave function, and w ( t )
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12680 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993
TABLE I: Calculated Eigenvalues (in cm-1) for the Coupled System adiabatic eigenvaluesb exact eigenvalues' lowest surface upper surface 0 240 504 793 1093 1393 1717 2027 2358 2698 3029 3369 3720 4070 4410 4771 5121 5481
0 238 506 791 I os9 1396 1711 2032 2359 269 1 3027 3367 371 1 4057 4406 4758 5112 5469f 5193
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5788 5842 6188~
6392 655W
6202 6390
6562 6923
6984 728W 7568
6980 7293 7571 7603
aow
a034 8150
8148 8404
838Y 8722 9128~ 9291
8726 9299
Exact eigenvalues for the coupled diabatic surfaces. Eigenvalues calculated by first calculatingthe adiabatic surfacesand then calculating the eigenvalues of each of the two surfaces individually. They are exact results for each of the individualsurfacesbut are not exact for the coupled vibronic problem. Theeigenfunctionscorrespondingto theseeigenvalues have 17 or more nodes; see text.
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CONFIGURATIONALCOORMNATE (A)
a
,
~
,
I
5793
cm"
Y l
cm,
- 1
f
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and 7 and are listed in Table I. In practice, a simple way to calculate the eigenvalues is to propagate a displaced wavepacket on both surfaces and calculate the spectrum. In cases where the energy difference between the eigenvalues is small, long propagation times and small values of the damping must be used. The intensities of the lines in the calculated "spectrum" (the Fourier transform of the autocorrelation function) depend on the magnitude of the displacement of the initial wavepacket and the functional form of the wavepacket. To find the eigenvalues listed in Table I, a Gaussian wavepacket displaced by 0.13 A was propagated on the surfaces shown in Figure 1. The eigenfunction corresponding to a given eigenvalue is calculated by using eq 8. The eigenfunctions of seven of the energy levels that are most important in calculating the intervalenceelectron-transfer spectrum are shown in Figure 2. These eigenfunctionsare the exact eigenfunctionsof the coupledpotential surfaces. They consist of two parts (from surfaces 1 and 2 in the diabatic basis set) and are labeled S1 and S2, respectively. The square of the eigenfunctions is the probability density. The maximum of the probability density of the eigenfunction corresponding to the lowest-energy eigenvalue is close to A Q = 0 A (4.02 A and +0.02 A for the components corresponding to surfaces 1 and 2, respectively). As the eigenvalue increases, the number of nodes in the corresponding eigenfunction increases. As is the case for a harmonic potential, there are no nodes for the lowest level, one for the first excited level (at A Q = 0 A for
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b
Figure 2. Eigenfunctions of the coupled system. The eigenfunctions corresponding to a given eigenvalue (given in the figure) have two components labeled S1 and S2 in the figures. (a) The eigenfunctions corresponding to the three lowest eigenvalues. (b) Eigenfunctions
corresponding to higher-energy eigenvalues that are important in the intervalence absorption spectrum (see text for details.)
both surface 1 and 2), and two for the second excited level (at A Q = k0.06 A). Eigenfunctions for four of the higher-energy levels are shown in Figure 2b. Three of them (5793,6392, and 6984 cm-*) are those to which electronic transitions are the most intense in the
Theoretical Treatment of Intervalence Absorption Spectra a
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12681 1.0 I
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Figure 4. Magnitude of the overlap as a function of time. The insert shows the short-time behavior.
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absorption band envelopes were calculated by using r = 350 cm-I. The spectra containing the individual lines were calculated by using r = 10 cm-1. The top spectrum correspondsto room temperature and the bottom spectrum to low temperature. intervalence absorption spectrum (vide infra). It is interesting to note that they correspond to the eigenvalues just above the minimum energy of the upper adiabatic potential surface (although they are the exact levels for the coupled system, not the levels of the upper adiabatic surface.) The eigenfunctions corresponding to these higher-energy levels have the same succession in the number of nodes as the lowest three levels. However, the signs of the components of the eigenfunction associated with S1 and S2 are the same, in contrast to those of the lowest levels. In addition, the maximum of the probability is in the range of the positive value of A Q for the function of surface 1 and negative value for the function of surface 2 in contrast to that of the lowest levels. Interspersed with these eigenfunctions are multinode functions such as that shown in Figure 2b (5469 cm-I). These functions, having more than 17 nodes, alternate with the functions having 0, 1, 2, etc., nodes discussed above. The eigenvalues of these functions are labeled with the letter c in Table I. C. Procedures for Calculating Spectra. The procedure for calculating a spectrum follows three steps. First, the lowestenergy eigenvalue of the coupled surfaces defined by one set of parameters (k,AQ,and e) is calculated by propagating a Gaussian wavepacket on the two coupled surfaces. Second, the eigenfunction corresponding to this eigenvalue is calculated. This eigenfunctionhas two components, one from each of the diabatic surfaces. Third, the absorption spectrum is calculated by multiplying the component from surface one by the transition dipole and propagating it on surface two, and vice versa. The transition promotes diabatic state 1’s component onto surface 2 and diabaticstate 2’scomponent onto surface 1. This graphically intuitive description should not give the false impression that a simple diabatic surface picture is being used; it is only the most convenient way to express the intervalencetransition in thediabatic basis. The spectrum is calculated exactly (to the numerical precision of eq 7), as opposed to the majority of time-independent methods, where the effects of coupling and therefore the spectra
are approximated.’ The spectrum calculated with a large r (I’ = 350 cm-l) shows a broad envelope,and the spectrum calculated with a small r (I’ = 10 cm-l) shows the individual peaks. The spectrum calculated by propagating the eigenfunctions from the lowest-energy level correspondsto the ‘low-temperature limit”. This spectrum is shown in Figure 3b. In general, other eigenfunctions must also be propagated because higher-energy levels are thermally populated. (At 298 K, the populations of the sixlowet-energylevelsforthiscalculationare69.8%,22.1%,6.1%, 1.5%,0.4%, and O.l%, in order of increasing energy as listed in Table I.) In order to calculate the 298 K spectrum, each of these sixeigenfunctionsis used tocalculate sixspectra that are multiplied by their thermal weights and added to obtain the calculated room temperature spectrum. This spectrum is shown in Figure 3a. The spectrum calculated by using r = 10 cm-l clearly shows the components contributing to the increased width. Note that the majority of these components appear on the high-energy side of the higher-energy band in the spectrum leading to an apparent blue shift. D. Wavepacket Dynamics and ( # l # ( t ) ) . The magnitude of the overlap ( # l # ( t ) ) calculated as discussed above with a damping factor of 1Ocm-1 is shown in Figure4. At zero time the magnitude is one. As the wavepackets propagate, the overlap at short time decreases sharply. This fast decrease in the time domain corresponds to a broad spectrum in the frequency domain. (In the example shown, the spectrum spans the energy range from 0 to 10 000 cm-I.) This fast decrease in the overlap is caused both by the motion of the wavepacket away from its starting position and by the transfer of amplitude between diabatic surfaces. In the symmetric example in this paper the net change in the population is zero. However, the time-dependent wavepacket # ( t ) develop a complicated shape that decreasesits overlap with the initial wavepacket. These factors cause a much faster decrease in the overlap at short times than is calculated in either the adiabatic or the diabatic limit. As time develops, the magnitude of the overlap undergoes a complicated series of oscillations. The recurrences in the time domain are related to the separations between the peaks in the frequency domain. One of the recurrences that has an important physical meaning and a simple interpretation is the first one at about 5 fs. When transformed to the frequency domain, the recurrence at 5 fs corresponds to a separation between bands of clt,,,,e-l = 6670 cm-1. This separation divides the spectrum into two components, one at very low frequency (in the IR region of the spectrum) and the other centered at 6400 cm-l (the intervalence band). E. Intensitiesof the Absorption Bands. The absorption spectra that are observed in mixed-valence complexes cannot be interpreted as arising from transitions between potential surfaces, nor
Simoni et al.
12682 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993
can the interpretation beseparated into electronic andvibrational parts. The coupled system has one set of eigenvalues. The absorption spectrum arises from transitions from the thermally populated energy levels to higher-energy levels. The most striking feature of the calculated spectrum in Figure 3 is the separation into two distinct regions, one at low energy and the other at higher energy that is commonly called the “intervalence absorption band”. Transitions to energy levels between these two regions have very low intensity. The bimodal pattern of observed intensities can be understood in a semiquantitative physical picture by examination of the eigenfunctions. The transition intensities can be related semiquantitatively to the nodal structure of the eigenfunctions and quantitatively to the signs of the eigenfunctions. For simplicity, the analysis will first be focused on transitions from only the lowest-energy level to the other energy levels. As discussed in section B, the number of nodes in the eigenfunctions increases with increasing energy of the levels. The overlap of the lowest-energy eigenfunction with the others will decrease as the number of nodes increases. Thus, the intensities of the transitions decrease with increasing energy of the levels involved in this transition. However, this trend does not continue indefinitely because at a certain energy (5793 cm-1 in the example presented here) a new set of eigenfunctions appears that begins with zero nodes and then increases (one node for the level at 6392 cm-I, etc.) For these particular levels with the smaller number of nodes, the overlap again is large and the transitions again have large intensities. As the number of nodes in the eigenfunctionsincrease, the transition intensities again decrease. The net result is that the transitions to the lowest-energy levels give a band at very low energy, the transitions to the upper levels whose eigenfunctions have a small number of nodes give another intense band at higher energy, and in between these two bands there is a gap containing very weak transitions. The intensities of the calculated transitions within the highenergy “intervalenceband” have a distinct pattern: the transitions from a given populated energy level to higher-energy levels alternate between zero and nonzero in intensity. This pattern can also be interpreted easily from the overlap between the eigenfunctions. Appropriate eigenfunctions and their corresponding eigenvaluesare shown in Figure 2. The overlap between the 0-cm-’ function of surface 1 and the 5793-cm-I function of surface 2 has the same magnitude but the opposite sign as the overlapbetween the 0-cm-*function of surface 2 and the 5793-cm-I function of surface 1. Thus the total overlap is zero, and a transition between these levels is not allowed. The opposite situation is found for the 0- and 6392-cm-l functions. In this case the part of the 6392-cm-I eigenfunction on surfaces 1 and 2 have opposite signs, and the total overlap is nonzero. The transition between these two levels is allowed. The intensities of the hot bands, Le., transitions from energy levels higher than the lowest that are thermally populated at room temperature, can be explained in the same way. For example, the 238-cm-I eigenfunctions have the same sign; i.e., the part of the function on surface 1 has a positive sign in the region of the minimum of surface 1, and the part of the function on surface 2 has a positive sign in the region of the minimum of surface 2. Thus, the total overlap with the 5793-cm-1 function will be nonzero, and this transition will be allowed. The total overlap with the 6392-cm-1 function (that changes sign) will be zero, and the transition will be forbidden. F. Comparison with the Adiabatic Limit. It is instructive to compare the results of the calculation that uses two adiabatic surfacesand the Born-Oppenheimer approximation to the results of the exact calculation. The numerical procedures that are used to calculate the spectrum in the adiabatic limit are similar to those used in the exact calculation except that the two adiabatic surfacesare calculated and the eigenvalues of the lowest adiabatic
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Figure 5. Eigenfunctions of the adiabatic potential surfaces. Three eigenfunctionsfrom thelowest adiabaticsurfaceandthree from the highest adiabatic surface are shown and labeled by the eigenvalues.
surface are found by propagating a Gaussian on this surface. The eigenfunctions for the levels that have appreciable thermal population are calculated (in this example, the lowest three), and the rmm temperature spectrum is calculated by propagating these functions on the upper adiabatic surface and adding the three spectra obtained with their respective thermal weights. In order tocompare the results of the two methods in detail, the eigenvalues of the upper surface are calculated by propagating a Gaussian on this surface. The eigenfunctions for the three lowest levels of the upper surface are also calculated. Selected eigenfunctions are shown in Figure 5 , and the eigenvalues are tabulated in Table I. The 17 lowest eigenvalues and the lowest eigenvalues of the upper surface, calculated by using the adiabatic surfaces, are very similar to those found by using the exact calculation. The eigenvaluesare compared in Table I. This aspect of the calculation is not surprising; with a large value of the coupling such as that used in this example, the adiabatic approximation is expected to be relatively good. It is interesting to note that the adiabatic calculation fails badly in one respect: there are too many adiabatic eigenvalues at energies above 5788 cm-1. Thecalculated energies of the levels for the upper adiabatic surface are almost the same as those in the exact calculation that have large transition probabilities in the spectrum. The “extra” levels belong to the lowest adiabatic surface and do not contribute to the electronic absorption spectrum. Thus, the energies of the “vibronic structure” in the adiabatic limit calculation correspond closely to the transition energies in the exact calculation. The spectrum calculated from the adiabatic surfaces (Figure 6) differs from the exact spectrum in two important features: the band shape is more symmetric, and the position of the maximum is shifted to lower energy. These differences result from the fact that even the exact calculation of the transition probabilities in the adiabatic limit is wrong because the starting point is wrong. The transition probabilities can be understood by examination of the shape of the eigenfunctions (Figure 5 ) and their overlaps.
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12683
Theoretical Treatment of Intervalence Absorption Spectra
vibrational infrared band involving only vibrations of the atoms. It is a coupled electroniovibtational transition between the eigenstates of the coupled system. It is important to point out that the vibrational wavenumber that was used to define the diabatic potential surfaces has no physical meaning by itself in the coupled system. The energy separationsbetween the eigenvaluesof the coupled system depend not only on the vibrational wavenumber but also on the displacement of thediabatic surfacesand the value of the coupling, i.e., on the coupled nuclear and electronic motions. H. Localizationvs Delocalization. Another calculated result that has a direct physical meaning is the square of the eigenfunctions shown in Figure 2. The square of the eigenfunctions correspondingto the lowest-enery eigenvaluehave maximum probability at the origin of the configurational coordinate. The physical meaning of this result is that the complex is delocalized, i.e., the M-L bond lengths are the same on both metal atoms. This interpretation corresponds to the classification of a class 3 complex.9 It is interesting to note that the square of the eigenfunctions corresponding to the higher-energy eigenvalues has probabilitiesthat maximize away from the origin, suggesting that there is some "localization" at higher temperatures. Decreasing the coupling between the diabatic surfacesincreases the localization. For example, the eigenfunctions, the diabatic and adiabatic surfaces, and the spectra calculated by using e = 280,1400, and 2800 cm-l are shown in Figure 7. All of the other input parametersare the same as those in the previous calculations. The components of the eigenfunction for e 3 280 cm-' have their maximum probability densities near the minimum of the corresponding diabatic surfaces corresponding to a localized system. As E increases,the maximum in the probability moves toward the origin. The effect on the spectra of increasing e are shown in the bottom panels of Figure 7. In the localized case, the spectrum consists of one band. As e increases, two bands are observed and the intensity of the lowest-energy band increases relative to that of the higher-energy band. The width of the higher-energy band decreases.
0
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8
10
WAVENUMBER (t"')x10-'
Flgure 6. Calculated room temperature absorption spectrum for the adiabatic approximation. The absorption band envelope was calculated by using 'I = 350 cm-I. The individual lines were calculated by using r = 10 cm-1.
The eigenfunctions of the upper surface look almost like the functions of a harmonic potential. Those of the lowest surface are very different because the surface is very flat around A Q = 0 A. The m a t important error in the spectrum resulting from using "exact" calculationsin the adiabatic limit is the calculated transition intensities. Althoughthe energiesin a resolved spectrum are similar, the "adiabatic spectrum" is significantly different from the exact spectrum. G. Pbysical Meaningof the Results and Interpretation of the Parameters. Two of the results of the calculations can be compared with experimentally measured values: the intervalence absorption band and the low-energy absorption of electromagnetic radiation. The former was discussed extensivelyabove. The latter is of interest because the time-dependent theoretical interpretation offers new insight into these transitions. The lowest-energy absorptionband is calculated to occur at 506 cm-I, corresponding to an allowed transition from the lowest eigenstate of the coupled system to the third lowest eigenstate. (The transition to the 238cm-I eigenstate is forbidden.) The 506-cm-1 absorption band is in the infrared region of the spectrum, but it is not a purely
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Figure 7. Effect of changing the coupling c on the diabatic and adiabatic surfaces (top three panels), the eigenfunctions (middle three panels), and the spectra (bottom three panels). The values o f t are 280 (left), 1400 (center), and 2800 cm-1 (right). The left panel corresponds to a localized mixed valence complex and the right panel to a delocalized complex.
12684 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 4. c o n c l ~ i o l l s
A simple method for calculating and interpreting intervalence absorption spectra was presented. The method is based on the time-dependent theory of electronic spectroscopy and the Feit and Fleck numerical integration of the time-dependent Schra dinger equation. The time-dependent method is very versatile because any form of the diabatic potential surfaces (not limited to harmonic potentials), any relative energy separation between the diabatic surfaces, and any form of the coupling can be used. The starting point of the model presented in this paper is the diabatic basis set representing the two valence forms. The transition is interpreted in terms of the part of the wavepacket on one of thediabaticsurfaces being transferred toand propagated on the other diabatic surface. The off-diagonal coupling transfers amplitudebetween the surfaces. Thecalculation emphasizes that the Bom-Oppenheimer separation of electronic and vibrational wave functions does not apply and that the interpretation is not confined to either the adiabatic or the diabatic limit. Acknowledgment. This work was made possible by a grant from the National Science Foundation (CHE91-06471). E.S. was supported in part by the Universit6 Paris 11. We thank Dr. J. Gauss and Prof. E. J. Heller for their implementation of the propagation algorithm.
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