Modeling of the Time-Resolved Vibronic Spectra of Polyatomic Molecules: the Formulation of the Problem and Analysis of Kinetic Equations S

MOLECULAR SPECTROSCOPY

Modeling of the Time-Resolved Vibronic Spectra of Polyatomic Molecules: The Formulation of the Problem and Analysis of Kinetic Equations S. A. Astakhov and V. I. Baranov Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, Moscow, 117975 Russia e-mail: [email protected] Received March 28, 2000

Abstract—A semiempirical parametric method is proposed for modeling three-dimensional (time-resolved) vibronic spectra of polyatomic molecules. The method is based on the use of the fragment approach in the formation of molecular models for excited electronic states and parametrization of these molecular fragments by modeling conventional (one-dimensional) absorption and ﬂuorescence spectra of polyatomic molecules. All matrix elements that are required for calculations of the spectra can be found by the methods developed. The time dependences of the populations of a great number (>103) of vibronic levels can be most conveniently found by using the iterative numerical method of integration of kinetic equations. Convenient numerical algorithms and specialized software for PC are developed. Computer experiments showed the possibility of the real-time modeling three-dimensional spectra of polyatomic molecules containing several tens of atoms. © 2001 MAIK “Nauka/Interperiodica”.

INTRODUCTION tions [14, 15]). Within the framework of the parametric theory of vibronic spectra, calculation methods were Recently, considerable advances have been made in developed which allow one to construct the spectral the experimental methods of molecular vibronic spec- ∆ν ≈ representation of the molecular model specified by a set troscopy. On the one hand, a very high resolution ( of parameters [16Ð19]. The second, no less important 1 cm–1) of the vibrational structure in the spectra was problem, has also been solved, namely, the determina- achieved in supersonic jets, and, on the other, modern tion of parameters of the potential surfaces of mole- laser methods provide the observation of dispersive cules in excited states [20]. spectra (fluorescence from single excited vibronic levels) and time-resolved dynamic spectra of molecules This provided the basis for the formulation and excited by short (pico- and femtosecond) light pulses solution of the problems of modeling and prediction of (see, for example, [1Ð11]). one-quantum, absorption, and fluorescence vibronic spectra. It seems reasonable to develop this parametric The modern vibronic spectroscopy is developing in approach for modeling the time-resolved spectra, the direction of studies of the time-resolved line spec- which obviously can give new information compared to tra. This necessitates the development of the corre- conventional vibronic spectra. However, at first glance, sponding theory, computational methods, and the con- to calculate the time-resolved spectrum of a complex struction of molecular models for the calculation, inter- molecule, it is sufficient to know all the probabilities wij pretation, and prediction of such spectra and obtaining of vibronic transitions, which can be easily calculated from them information on the properties of molecules. by the parametric method; nevertheless, the modeling The conventional absorption and fluorescence spec- of such spectra involves a number of problems in tra of polyatomic molecules have long been analyzed practice. theoretically. The theory was developed from the sim- First, it is not clear to what degree the parameters of plest methods, which used various generalized spectral a molecular model that were obtained (calibrated) for parameters (see, for example, [12]), to the methods the spectra of a certain type will be appropriate for a based on the fundamental molecular model and its quantitative modeling of the spectra of another type parameters, which were not directly related to the spec- (time-resolved spectra). tral experiment (potential surfaces of the ground and excited states in natural molecular coordinates) [13]. Second, the influence of a medium (intermolecular The advantage of this model is obvious, because it interactions) can substantially change the emission of allows one to describe not only spectral but also other excited molecules. The problem arises as to how to sep- physicochemical properties of polyatomic molecules arate the contributions from nonradiative and radiative (even, for example, the development of chemical reac- transitions to the time-resolved spectra.

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Third, processes of absorption and radiative and molecules and is based on the fragmentary method of nonradiative relaxation in excited molecules can occur formation of molecular models and a special system of not only inside the same molecular isomer but can parameters possessing all the required properties [13, result in the formation of other isomers (which is, in 20, 25]. The essence of the fragmentary approach is particular, of great interest for photochemistry). There- that the calibration of the values of parameters of the fore, it is necessary to determine the probabilities of molecular fragments can be performed for the simplest optical and nonradiative transitions between molecular molecules and then these fragments can be used for isomers. constructing models of more complex compounds and Fourth, a purely computational problem arises. In predicting their spectra. It is important that both the prediction of spectral properties and the development addition to the calculation of many probabilities wij of vibrational and vibronic transitions in the determina- of a system of parameters of model fragments are pre- formed in the same experiments (absorption or fluores- tion of time dependences of level populations ni(t), it is necessary to solve a system of linear differential equa- cence). tions of a very high dimensionality (>103) even for mol- The main idea of this approach is to calibrate the ecules of a moderate size containing ≈20–30 atoms. values of parameters of the molecular model (or frag- Although the general methods are well known, the ment) in the simplest experiment (one-dimensional applicability of the corresponding computational pro- absorption or fluorescence spectrum) and then to pre- cedures for modeling spectra is not obvious and dict the results of another, more complex, time- requires a special analysis (because of a high dimen- resolved experiment, which describes in detail the sionality of the problem and the fast computations dynamics of excited states and relaxation processes in required). polyatomic molecules. This will allow one to use data At present, all these questions have no clear answer, bases for molecular fragments [20, 25] in computer although studies are being performed in some fields experiments for direct calculations of multidimensional (for example, nonradiative transitions [21], probabili- spectra, which can be directly compared with the corre- ties of optical transitions between isomers [13, 14], and sponding real experiments, and to refine the parameters generalized inverse vibronic problems [22]). This paper of the molecular model. The computer experiment is devoted to the first part of the general problem, devoted to the study of the effect of various factors on namely, the solution of the direct problem of calcula- the shape of the time-resolved spectrum can be also tion of the time-resolved vibronic spectrum for real used instead of real experiments, which often require large molecular systems. We restrict ourselves to the more complex equipment and are time-consuming. case of an isolated molecule and transitions inside only It is clear that such an approach for modeling multi- one molecular isomer. dimensional vibronic spectra can be realized in princi- ple. The main problem, which determines the predictive ability of this method, is the degree of transferabil- PARAMETRIC APPROACH ity of the parameters of molecular fragments used. As An adequate choice of the parameters of a molecu- for any parametric method, the answer to this question lar model is very important in the solution of a direct can be obtained from model calculations of sufficiently spectral problem. representative series of molecules and comparison of Note that the excited-state lifetimes, decay rates for the results with experiments (this question will be dis- individual lines in the spectrum, and the quantum yield cussed elsewhere). Nevertheless, one can already of fluorescence are the key parameters (along with fre- expect the high efficiency of these models, taking into quencies and intensities of vibronic lines) in the analy- account their good predictive ability in modeling con- sis and modeling of time-resolved spectra [23, 24]. ventional IR and UV spectra of polyatomic molecules These quantities, which determine the time dependence [20, 25Ð27]. of the fluorescence intensity and can be obtained Along with a choice of the molecular model, it is directly from experiments, are in fact the parameters of necessary to solve the computational problems a specific experiment rather than the parameters of the involved. The time-resolved vibronic fluorescence molecular model. Knowing these parameters, one can spectrum can be represented as a three-dimensional reproduce the experiment numerically, but it is impos- surface—the dependence of the intensity on the emis- sible to predict the behavior of another or the same mol- sion frequency and the observation time. To construct ecule under different conditions. Such an approach is such a surface, one should calculate the time depen- incapable of predicting physicochemical properties of dence of the intensity of each spectral line, which is molecules and is not constructive in this sense for the proportional to the probability wij of the corresponding development of methods of modeling time-resolved vibronic transition from the state i to the state j and the vibronic spectra of polyatomic molecules. population ni of the level from which this transition We suggest using a parametric method, which was occurs: developed earlier for calculations of the conventional (ν , ) ν ( ) (not time-resolved) vibronic spectra of polyatomic Iij ij t = h ijwijni t . (1)

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Therefore, to calculate a three-dimensional spectrum The system (2) can be written in the matrix form for || || for a specified molecular model, it is necessary, first, to the population vector n(t) with components n1(t), …, determine the probabilities of vibrational and vibronic nN(t) as transitions between all energy levels of the molecule d and, second, by integrating kinetic equations, to obtain ---- n(t) = P n(t) , (3) the time dependences of populations of all levels and dt intensities of spectral lines. Note that the number of vibronic states, which affect the intensity I (ν , t) of where P is the upper triangle probability matrix with ij ij elements spectral lines, can be very large for polyatomic mole- 3 cules (>10 ). Ðw , i = j  i The methods for calculating the probabilities w of < ij pij = w ji, i j vibrational and vibronic transitions are developed in  > detail, including the corresponding software [13, 16Ð 0, i j 19, 26, 28]. The second problem requires the develop- (its eigenvalues coincide with diagonal elements w ). ment of the efficient methods for solving kinetic equa- i tions, because preliminary estimates showed that the In the most general case, the solution of the system simplest methods could not be applied to real time of equations (3) is expressed in terms of the exponential computer experiments with polyatomic molecules due of the matrix P and initial conditions ||n(0)|| [29, 30]: to a great dimensionality of such systems. Let us dis- Pt cuss this question in more detail. n(t) = e n(0) . (4) To determine ||n(t)|| with the required accuracy by Pt ANALYSIS OF KINETIC EQUATIONS calculating the exponential in the form of a series e = m ∞ (Pt) ∑ ------, one should perform a great number of Let all the excited states of a molecule be numbered m = 0 m! from 1 to N in accordance with their increasing energy. operations of matrix multiplication for each value of t, Then, the kinetic equations for populations ni(t) of the which represents a cumbersome computational prob- energy levels, which represent linear first-order differ- lem and cannot be realized in modeling of the multidi- ential equations with constant coefficients wij equal to mensional spectra of polyatomic molecules with N > the probabilities of transitions from the state j to the 103. state i, will have the form However, one can solve kinetic equations analyti- cally, which substantially simplifies calculations. First N dn we consider the case when the matrix P does not have ------i = Ð w n + ∑ w n , i = 1, 2, …N, (2) dt i i ji j multiple eigenvalues. j = i + 1 Let us successively exclude unknowns and represent the solutions of equation (2) in the form i Ð 1 where wi = ∑j = 1 wij + wi0 are the total probabilities of N N   Ðw t Ðw t transitions from the state i to all low-lying excited states n (t) = n (0) + ∑ b  e i Ð ∑ b e k . (5) i  i ki ki and the ground state (wi0). The initial conditions {ni(0), k = i + 1 k = i + 1 i = 1, 2, …, N} can be different depending on the type of excitation of a molecule (resonance, broadband exci- Expressions for coefficients bki can be obtained by sub- tation, etc.). stituting in the equation for an arbitrary ith level in system (2) the solutions (5) for higher-lying levels (nj(t), Consider the possible ways of solving this problem, j > i): taking into account that the system of kinetic equations (2) is written for all vibronic states of the molecule, dni Ðwi + 1t ------= Ð w n + w A e whose number is large (N > 103), and in modeling mul- dt i i i + 1i i + 1 tidimensional spectra, especially in solving inverse N  k Ð 1  problems, the calculation should be performed repeat- Ðwkt + ∑ w A Ð ∑ w b  e , edly for different parameters of the molecular model,  ki k ji kj k = i + 2 j = i + 1 probabilities wji, initial conditions, time intervals, etc. Therefore, the method of solving kinetic equations where A = n (0) + N b . This yields the recur- should be capable of performing fast calculations of m m ∑l = m + 1 lm quite complicated systems. rence relations for bki

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This method of integration of kinetic equations (2) is wi + 1i Ai + 1 bi + 1i = ------general and allows one to obtain solutions for any prob- wi + 1 Ð wi ability matrix P. In speciﬁc calculations, two operations k Ð 1 of matrix multiplication should be performed for each

wki Ak Ð ∑ w jibkj (6) value of t, and the required calculation time, as can be shown, is proportional to N3, which excludes the possi- b = ------j---=---i--+---1------ki w Ð w bility of the real time modeling of the spectra of poly- k i atomic molecules with N > 103. In addition, note that k = i + 2, i + 3, …, N, the computer calculation of the transformation matrix å for such large values of N is a nontrivial problem due where calculations begin from bN N – 1, for which AN = to the errors in rounding off real numbers and the nN(0). necessity of storage of large data bases. Expressions (5) and (6) are convenient for the anal- The use of the approximate numerical methods for ysis and calculation of ni(t) and time dependences of integrating kinetic equations (2), for example, based on the spectral line intensities; however, they are substan- the Eulerian piecewise linear functions method [29], ≠ tially restricted by the condition wk wi, which is unac- seems the most promising. In this case, we obtain the ceptable for the methods of modeling the spectra of point representation for the solution on the discrete grid polyatomic molecules (see below). The solution in the tk in the form presence of multiple eigenvalues of the matrix P, i.e., ( ) ( ) ∆ ( ) for the same total probabilities of transitions from some n tk + 1 = n tk + tP n tk , ∆ levels (wk = wi), can be similarly represented as a sum tk + 1 = tk + t. of the products of exponentials and polynomials in t with the corresponding recurrent relations for the coef- Indeed, the rate of this iterative process is proportional ficients; however, it is cumbersome. This solution can to N2 and the algorithm does not require the storage of be more easily obtained directly in the matrix form by additional matrices except the matrix P itself. In addi- the transformation of the probability matrix P to the tion, unlike the methods considered above, this method canonical Jordan form [31]. does not involve rather slow computer calculations of λ λ an exponential. Moreover, as the computer experiments Let the eigenvalues 1, …, q be different among N showed, the numerical algorithm can be additionally eigenvalues of the matrix P and the rest s eigenvalues optimized in rate by approximately two orders of mag- λ q + i (i = 1, 2, ..., s) be multiple with the multiplicity ri. nitude due to the direct use of the matrix P in the trian- Then, the similarity transformation M–1PM = J reduces gle form and also automatic exclusion of kinetic equa- the matrix P to the canonical Jordan form J = diag{J0, tions with zero solutions for the specified initial condi- J1, …, Js}, where J0 is the diagonal matrix with ele- tions. For this reason, the efficiency of this method λ λ ments 1, …, q and Ji are Jordan cells of the order ri proves to be satisfactory in practice. However, the most important advantage of this method is the fact that it λ … yields the approximate solution of the system of kinetic q + i 1 0 0 0 equations with any matrix P, in particular, in the case of λ … 0 q + i 1 0 0 multiple eigenvalues, which, as we show below, is quite real. Ji = . . . … . . . Let us analyze the solutions of kinetic equations for … λ 0 0 0 q + i 1 some specific cases. … λ 0 0 0 0 q + i Consider first the simplest nontrivial example of the three-level system that has the ground state S0 and two || || Therefore, the solution will have the form n(t) = lowest excited electronic states S1 and S2 (without J0t J1t vibrational sublevels). In this case, the kinetic equa- ePt||n(0)|| = MeJtM–1||n(0)||, where eJt = diag{e , e , λ λ λ tions have the form Jst J0t 1t 2t qt …, e }, e = diag{e , e , …, e }, and dn ------2 = Ðw n , dt 2 2 2 ri Ð 1 (7) t … t dn 1 t ---- -(------)------1 = Ð w n + w n . 2! ri Ð 1 ! dt 1 1 21 2 λ Jit q + it ri Ð 2 e = e … t . A simple analytic solution of the system (7) upon 0 1 t -(------)-- resonance excitation of the third level {n (0) = n , ri Ð 2 ! 2 20 n (0) = 0} can be easily obtained: . . . … . 1 … ( ) Ðw2t 0 0 0 1 n2 t = n20e , (8)

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w Ðw t Ðw t ( ) 21 ( 1 2 ) n1 t = n20------e Ð e . (9) w2 Ð w1 n2 S2 If w1 = w2 (the multiple eigenvalue of the probability w21 Ðw1t matrix), then n1(t) = n20w21 e t. S1 n1 Expression (8) describes a simple exponential law w20 w10 of the decrease in the population of the upper level S 2 S0 (Fig. 1). Therefore, the decay rate of fluorescence I20(t) w+ w– and I21(t) is characterized, according to (1) and (8), by a single quantity, namely, the total probability w2 = t w20 + w21 of transitions from the S2 state. The depen- Fig. 1. Kinetic curves for a three-level system. dence I10(t) ~ n1(t) is biexponential (9) and is already determined by two quantities, namely, the probabilities w = w + w and w = w of transitions from the S 2 20 21 1 10 2 sitions from all other levels lying above this level and S1 levels. Similar results were obtained in papers (k > i). [23, 32]. The total energy level diagram of a molecule can be Let us call attention to an important feature. The rate separated into groups of vibrational levels belonging to of the exponential increase in population n1(t) of the S1 different electronic states. Consider the ground 0 and level and, hence, the intensity I10(t) of the S1 S0 the first excited 1 states of the molecule and their vibra- transition is w+ = max(w1, w2), and the subsequent pop- tional sublevels numerated by subscripts f and i, respec- ulation decay occurs with the rate w– = min(w1, w2). tively (these subscripts denote sets of vibrational quan- This follows from the fact that the sign of the difference tum numbers). Let us calculate the total probabilities of w2 – w1 in expression (9) depends on the relation vibronic transitions from the vibrational sublevels of between the probabilities w and w . One can see that 1 2 the excited state to the ground state w = ∑ w , → , = already in this simplest three-level case the time depen- 1, i f 1 i 0 f dence of the fluorescence intensity is determined not π4ν 3 64 1, i → 0, f µ2 ν only by the probability of the given transition, but also ∑ ------1, i → 0, f , where 1, i → 0, f and f ε 3 by the probabilities of all the other transitions. For 3h 0c µ example, for w1 > w2, the rate of the exponential decay 1, i → 0, f are frequency and dipole moment of the of fluorescence intensity I10(t) will be determined by vibronic transition, respectively. the total probability w = w + w of transitions from µ 2 20 21 In the FranckÐCondon approximation, 1, i → 0, f = the initially excited S2 state rather than by the probabil- µ1 Ψ ' (Q')Ψ (Q)dQ (µ is the dipole moment of the ity of the S1 S0 transition. In the multi-level system 0∫ i f 10 containing vibrational levels as well, the situation is purely electronic transition), and the probabilities are even more complicated (see below). Therefore, it is determined by the overlap integrals for the wave func- clear that a detailed and correct interpretation of the tions Ψ ' and Ψ : experiment, the determination of constants characteriz- i f ing the lifetimes of energy levels, etc., can only be per- 4 formed after the preliminary and detailed calculations 64π 2 3 2 w , = ------µ ∑ν , → , 〈i| f 〉 . of the spectra. 1 i ε 3 10 1 i 0 f 3h 0c f In the general case, when a system has many excited states, the time dependences of the populations are The terms containing the largest overlap integrals 〈i| f 〉 determined by expression (5) (or by a similar expres- will make the main contribution to the total probability. sion containing additional polynomial factors of t in When the molecular geometry does not change signifi- terms with identical total probabilities wi). The first cantly upon excitation, which is typical for polyatomic term is responsible for the transitions from the ith level molecules, the overlap integrals containing functions with coinciding sets of quantum numbers i and f (or dif- to all the low-lying levels with the total probability wi = i Ð 1 fering in the value of only one of them by unity) will be ∑j = 1 wij + wi0, which characterizes the rate of deple- maximal. All the rest of the integrals will be substan- tion of this level. The rest of the terms contain in expo- tially smaller. The frequencies of transitions with i ≈ f nents the total probabilities of transitions from the lev- will differ from the purely electronic transition fre- ν els lying above the ith level to this level. Therefore, the quency 10 by no more than the value of the order of a time dependence of the population of the ith level and, vibrational quantum (i.e., by several percent of the ν hence, of the intensity of transitions from this level is value of 10). Therefore, we can assume on average ν ≈ ν determined by the total probabilities wi of transitions with sufficient accuracy that 1, i → 0, f 10. Taking this both from this level and by the probabilities wk of tran- into account and also that the vibrational wave func-

OPTICS AND SPECTROSCOPY Vol. 90 No. 2 2001 204 ASTAKHOV, BARANOV

→ → S2 S1 S1 S0 I I 1.0 1.0 0.5 0.5 0 0 0 3000 500 500 400 1000 400 λ –2000 300 λ –1000 300 , cm , cm 200 –1 200 , ps –1 t, ps –3000 t –4000 100 100 0 0

Fig. 2. Fluorescence spectra of the hexatriene model upon excitation of the purely electronic S2 state.

(‡) (b)

I I 1.0 1.0 0.5 0.5 0 0 0 200 1000 200 0 –2000 150 150 λ , cm 100 λ 100 – , ps , cm , ps 1 –4000 50 t –1 –2000 50 t 0 0

Fig. 3. The S S ﬂuorescence spectra of the octatetraene model upon excitation (a) of the purely electronic S state and (b) the –1 1 0 1 1620-cm vibrational sublevel of the S1 state.

tions form total orthonormal systems, we obtain for the because the population of the levels in the S1 state total probability increases only due to vibronic transitions from the S2 state (the probabilities of vibrational transitions within π4 π4 64 3 2 2 64 3 2 the S1 state are very small and neglected). Because the w , = ------ν µ ∑〈i| f 〉 = ------ν µ . (10) 1 i ε 3 10 10 ε 3 10 10 total probabilities for all the sublevels of the given elec- 3h 0c f 3h 0c tronic state are the same (w1, j = w1, w2, i = w2) (10), by One can see from this expression that the total prob- neglecting vibrational transitions, the system of kinetic abilities (inverse lifetimes of the sublevels) for all equations is separated into blocks, which correspond to vibrational sublevels of an electronic state are the same different electronic states. The total probabilities are µ ν and are determined by the parameters 10 and 10 of the the same inside each block, and the equations are not electronic transition. This conclusion is important and connected with each other and are solved separately has been conﬁrmed experimentally [23]. (but together with equations for other blocks). The Consider now a system of three electronic states S , solutions of these equations are proportional to the 0 same sum of exponentials. Therefore, we obtain for the S1, and S2 with vibrational sublevels and vibronic transitions between them (the probabilities of vibrational vibrational sublevels of the S1 state: transitions within the electronic states are assumed   zero). Let the system be initially excited to the S . Then, Ðw1t Ðw2t 2 n , = ∑b  (e Ð e ), (11) the population of the sublevels of the intermediate state 1 j  ij i S1 is described, according to (5), by the expression and the time dependences prove to be identical with an   Ðw1, jt Ðw2, it n , = ∑b  e Ð ∑b e , accuracy to the factor ∑ bij for all j, i.e., for all vibra- 1 j  ij ij i i i tional sublevels of the S1 state.

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From this, a result follows that directly affects the which were developed earlier for the calculation and shape of the time-resolved spectrum: The time dynam- prediction of conventional absorption and fluorescence ics of all vibrational lines of the S1 S0 electronic spectra of polyatomic molecules. All the required transition is the same, it is biexponential, and is deter- matrix elements can be calculated by the methods mined by the total probabilities w1 and w2, while the developed. The method of numerical integration of a intensities of the vibrational lines are determined by the system of many (N > 103) kinetic equations is the most efficient for determining the time dependences of the coefficients ∑ b and probabilities w → of the i ij 1, j 0, k populations of vibronic levels. vibronic transitions. We realized the parametric method of modeling the When the probabilities of vibrational transitions time-resolved vibronic spectra of polyatomic mole- (within the given electronic state) are not zero, expres- cules in the form of special software for a PC. The sions (10) and (11) are not valid in the general case and three-dimensional spectra were calculated for mole- the dynamics of vibrational sublevels of the same elec- cules of polyenes and diphenylpolyenes under different tronic state becomes different. Therefore, the intensi- excitation conditions and for different parameters of ties of the vibrational lines of the S1 S0 transition molecular models. The calculations showed the possi- will change in time differently. The difference in the bility of performing real time computer experiments time dependences of the intensities of vibrational lines with polyatomic molecules containing several tens of in the electronic spectrum is the criterion of a substan- atoms. tial role of vibrational transitions in the dynamics of the The analysis of kinetic equations showed, in partic- excited vibronic states of molecules and shows that the ular, that the time dependence of the intensity of vibra- probabilities of vibrational transitions are comparable tional lines (which is many-exponential in the general with those of vibronic transitions. case) can be used for estimating the role of vibrational Upon population of high vibrational levels (over- transitions in the dynamics of excited vibronic states. tones and combination frequencies), a situation can The direct calculation and modeling of these spectra arise in which the relaxation will occur via the vibra- allow one to separate radiative and nonradiative contri- tional states, with the probabilities of transitions butions to the probabilities of the corresponding transi- between them being close or identical (for example, for tions, which is important in the development of the the most intense vibrational transitions accompanied models of intermolecular interactions and the theory of by a change in one of the quantum numbers by unity). nonradiative transitions [21]. In this case, the total probabilities (including also the We restricted ourselves to the processes caused by electronic component) will be the same for the nonzero the vibronic transitions occurring within the same probabilities of vibrational transitions within electronic molecular isomer. In the general case, processes of states as well. In this situation, the system of kinetic optical excitation and relaxation can result in the tran- equations cannot be separated into blocks, as in the sitions to other molecular isomers, which is of great case of negligibly small probabilities of vibrational interest, in particular, for photochemistry. We intend to transitions, and its matrix will have multiple eigenval- study these processes in the future. After the develop- ues, which requires the use of general methods for solv- ment of the methods for calculating probabilities of ing kinetic equations in calculations of the spectra. vibronic transitions between isomers [14], this method One can expect that this will be most typical in the can be easily extended to the general case. Note that the analysis of the time-resolved fluorescence spectra of simulation of three-dimensional spectra neglecting the polyatomic molecules upon selective (resonance) exci- transitions between isomers and their comparison with tation of vibronic states. Therefore, the iterative method experiments can already give the information that will of numerical integration of kinetic equations (2) be useful for the development of the theory of optically described above appears the most adequate. We devel- induced transitions between isomers of polyatomic oped a convenient calculation algorithm and a software molecules. for the construction of three-dimensional spectra and performing computer experiments with real molecules. REFERENCES Figures 2 and 3 illustrate the spectra calculated upon 1. J. C. Brown, J. M. Hayes, J. A. Warren, and G. J. Small, selective excitation of different vibronic states of mod- Laser in Chemical Analysis, Ed. by G. M. Hieftje, els of hexatriene and octatetraene molecules. The J. M.Travis, and F. E. Lytie (The Humana Press, results of a series of computer experiments and their New York, 1981). detailed discussion will be reported elsewhere. 2. J. A. Warren, J. M. Hayes, and G. J. Small, Anal. Chem. 54, 138 (1982). CONCLUSIONS 3. D. H. Levy, Annu. Rev. Phys. Chem. 31, 197 (1980). 4. A. Amirav, U. Even, and J. Jortner, Chem. Phys. 51 Thus, the problem of modeling time-resolved (1−2), 31 (1980). vibronic spectra of polyatomic molecules can be solved 5. 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