% ---------- ours @ARTICLE{Segev:2000:PSD, author = "B[ilha] Segev and E[ric] J. Heller", title = "Phase-space derivation of propensity rules for energy transfer processes between {Born-Oppenheimer} surfaces", year = "2000", journal = "J. Chem. Phys.", volume = "112", pages = "4004", abstract = "We consider a simple method for calculating weak Franck-Condon factors. The Wigner transform of the wave function on an initial Born-Oppenheimer state is calculated for the donor potential surface and projected onto the acceptor energy surface energy shell. The integrated projection yields an approximation for the relevant Franck-Condon factors, while phase-space integrand shows where in phase space the leakage occurs between the donor and acceptor states. This in turn determines the initial conditions on the acceptor surface for subsequent IVR and energy flow. Propensity rules are obtained by recognizing phase-space points of closest approach of the initial-state Wigner function and the final-state energy surface. The example of two coupled harmonic oscillators is explicitly solved to demonstrate the power of this phase-space approach." } @ARTICLE{Sergeev:2002:MPP, author = "Alexey V. Sergeev and Bilha Segev", title = "Most probable path on phase space for a radiationless transition in a molecule", year = "2002", journal = "J. Phys. A: Math. Gen.", volume = "35", pages = "1769", abstract = "We study a radiationless transition in a polyatomic molecule when the electronic energy of an excited electronic state is transferred to the vibrational degrees of freedom of the nuclei, and when some nuclear coordinates change abruptly. This jump between the donor energy surface and the acceptor one gives the initial conditions for the subsequent dynamics on the acceptor surface, and the partition of energy between competing accepting modes. In the Wigner representation, the physical problem of recognizing the accepting modes for a radiationless vibronic relaxation reduces to the mathematical problem of finding the maximum of a function of many variables under a constraint. The function is the initial Wigner function of the nuclei and the constraint is energy conservation. In a harmonic approximation for the potential surfaces, the problem is equivalent to finding the distance from a given point to a multidimensional ellipsoid. This geometrical problem is solved in closed form. For nonharmonic potentials, the optimization problem is solved perturbatively." } @ARTICLE{Kallush:2002:SJF, author = "S[himshon] Kallush and B[ilha] Segev and A[lexey] V. Sergeev and E[ric] J. Heller", title = "Surface jumping: {Franck - Condon} factors and Condon points in phase space", year = "2002", journal = "J. Phys. Chem. A", volume = "106", pages = "6006", abstract = "We generalize the concept of Tully-Preston surface hopping to include larger jumps in the case that the surfaces do not cross. Instead of identifying a complex hopping point, we specify a jump between two locales in phase space. This concept is used here to find propensity rules for the accepting vibrational mode(s) in a radiationless vibronic relaxation transition. A model inspired by the S-2 --> S-0 vibronic relaxation transition of the benzene molecule in which 30 modes of vibration compete for the electronic energy is studied within this approach. For this model, we show that almost all of the energy must go to a single C-H local stretching. The initial conditions for vibrations of this mode are a coordinate jump of the hydrogen atom toward the ring. All of the other modes undergo an almost vertical transition, in which the energy that they take is determined by their equilibrium displacement between the two surfaces. We observe that for a large energy gap the masses and frequencies become the defining parameters for choosing the accepting mode. Anharmonicities are very important when a competition between degenerate modes occurs. These conclusions are demonstrated by the specific model considered here but apply in general to any weak internal conversion process." } @ARTICLE{Heller:2002:HJB, author = "E[ric] J. Heller and B[ilha] Segev and A[lexey] V. Sergeev", title = "Hopping and jumping between potential energy surfaces", year = "2002", journal = "J. Phys. Chem. B", volume = "106", pages = "8471", abstract = "Novel approaches to surface hopping (in the case in which surfaces cross in classically allowed regions) and surface ''jumping'' (in cases in which they never cross or they cross in classically forbidden regions) are discussed. Classically forbidden transitions necessarily involve discontinuous ''jumps'' in position or momentum or both (but so as to preserve energy). In general, the jumps are discontinuous changes in nuclear positions or momenta on the time scale of the electronic transition. After reviewing various approaches in one dimension, a phase-space approach is applied to multidimensional systems with large energy gaps, in which the traditional semiclassical approaches are more difficult to apply. The concept of jumps extends the spirit of surface hopping into new regimes." } @ARTICLE{Segev:2002:DCV, author = "B[ilha] Segev and A[lexey] V. Sergeev", title = "Dominant channels of vibronic transitions in moleculs with several identical modes", year = "2002", journal = "Chem. Phys. Lett.", volume = "365", pages = "(no. 1, in press)", abstract = "Weak­coupling radiationless transitions (internal conversion or inter system crossing) are studied assuming separability and symmetry over N identical modes. Franck­Condon factors control the branching ratios between exciting just one of the equivalent modes, or equally distributing the available energy. The dominant process can be predicted by an exact quantum mechanical solution if the wavefunctions are known (Gaussian initial distributions and accepting Morse or Poeschl ­ Teller oscillators, for example); or more generally by a Wigner phase space surface­jumping analysis based on a classical limit of the Wigner function, using only the donor distribution and the acceptor potential surface." } @ARTICLE{Sergeev:2002:SEF, author = "A[lexey] V. Sergeev and B[ilha] Segev", title = "Semiclassical estimation of {Franck - Condon} factors and transition rates for vertical and nonvertical transitions", year = "2002", journal = "J. Chem. Phys.", pages = "(in preparation)", abstract = "We develop a systematic way for estimating multidimensional Franck - Condon factors and transition rates for vertical and nonvertical transitions. We first recognize the dominant accepting modes, or more accurately the dominant accepting region in phase space: the most probable positions and momenta of the nuclei during the electronic transition. In particular, we are able to find these positions and momenta during non vertical as well as during vertical transitions. After recognizing the nuclear transition path(s) in phase space we calculate the transition rate by treating the region in phase space found in the first step as a funnel for the transition. We calculate the flow of probability through this funnel. We use the Wigner representation and its semiclassical limit and find that the transition occurs through a point(s) on the final surface of constant energy where the initial Wigner function is maximal. This dominant contribution is estimated analytically. Results are illustrated for Harmonic, Morse and Poeschl - Teller oscillators." } % ---------- Founding Fathers @ARTICLE{Franck:1925, author = "J. Franck", title = "", year = "1925", journal = "Trans. Faraday Soc.", volume = "21", pages = "536" } @ARTICLE{Condon:1926:TID, author = "Edward Condon", title = "A theory of intensity distribution in band systems", year = "1926", journal = "Phys. Rev.", volume = "28", pages = "1182", abstract = "A theory of the relative intensity of the various bands in a system of electronic bands is developed by an extension of an idea used by Franck in discussing the dissociation of molecules by light absorption. The theory predicts the existence of two especially favored values of the change in the vibrational quantum numbers, in accord with the empirical facts as discussed by Birge. A means of calculating the intensity distribution from the known constants of the molecule is presented and shown to be in semi-quantitative agreement with the facts in the case of the following band systems: SiN, AlO, CO (fourth positive group of carbon), I2 (absorption), O2 (Schumann-Runge system), CN (violet system), CO (first negative group of carbon), N2 (second positive group of nitrogen), and N2 (first negative group of nitrogen). In the case of I2 there is a discrepancy, if Loomis' assignment of n'' values is used, which does not appear if Mecke's original assignment is used. It is suggested that at least some of the lower levels postulated by Mecke are real but that absorption from them always results in dissociation of the molecule and so they are not represented in the quantized absorption spectrum." } @ARTICLE{Condon:1928:NMA, author = "Edward U. Condon", title = "Nuclear motions associated with electron transitions in diatomic molecules", year = "1928", journal = "Phys. Rev.", volume = "32", pages = "858", abstract = "The question of nuclear motions associated with electron transitions is discussed from the standpoint of quantum mechanics. It appears that Heisenberg's indetermination principle gives the clue to the inexactness of the earlier method based on Franck's postulate since its strict application calls for a violation of the principle. The existence of an entirely new type of band spectrum due to the wave nature of matter is predicted and the interpretation of Rayleigh's mercury band at 2476-2482 A.U. as of this type is suggested. Finally it is shown that while Franck's postulate is also true for electron jumps in atoms, it is of but trivial interest because its inexactness is much greater for the electrons than for heavy nuclei." } @ARTICLE{Wigner:1932:QCT, author = "E. {P.} Wigner", title = "On the quantum correction for thermodynamic equilibrium", year = "1932", journal = "Phys. Rev.", volume = "40", pages = "749", abstract = "The probability of a configuration is given in classical theory by the Boltzmann formula exp \[-V / hT\] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed." } @ARTICLE{Morse:1929:DMA, author = "Philip M. Morse", title = "Diatomic molecules according to the wave mechanics. II. vibrational levels", year = "1929", journal = "Phys. Rev.", volume = "34", pages = "57", abstract = "An exact solution is obtained for the Schroedinger equation representing the motions of the nuclei in a diatomic molecule, when the potential energy function is assumed to be of a form similar to those required by Heitler and London and others. The allowed vibrational energy levels are found to be given by the formula E(n)=Ee+h omega 0(n+1 / 2)-h omega 0x(n+1 / 2)2, which is known to express the experimental values quite accurately. The empirical law relating the normal molecular separation r0 and the classical vibration frequency omega 0 is shown to be r03 omega 0=K to within a probable error of 4 percent, where K is the same constant for all diatomic molecules and for all electronic levels. By means of this law, and by means of the solution above, the experimental data for many of the electronic levels of various molecules are analyzed and a table of constants is obtained from which the potential energy curves can be plotted. The changes in the above mentioned vibrational levels due to molecular rotation are found to agree with the Kratzer formula to the first approximation." } @ARTICLE{Poeschl:1933, author = "G. P{\"{o}}schl and E. Teller", year = "1933", journal = "Z. Phys.", volume = "83", pages = "143" } @BOOK{Landau:1958:QMN, author = "L. D. Landau and E. M. Lifshitz", title = "Quantum mechanics. Non-relativistic theory", publisher = "Pergamon", address = "London", year = "1958", } % ---------- Heller @ARTICLE{Heller:1978:QCC, author = "Eric J. Heller", title = "Quantum corrections to quasiclassical photodissociation models", year = "1978", journal = "J. Chem. Phys.", volume = "68", pages = "2066", abstract = "A Wigner phase space treatment is applied to a formula for the total photodissociation cross section to obtain an expansion in powers of h(cross). The lowest order term yields an improvement on the reflection approximation, and a polyatomic generalisation of that approximation is proposed. Numerical examples show the improvement gained by including terms of higher order in h(cross)." } @ARTICLE{Heller:1983:RTN, author = "Eric J. Heller and R. C. Brown", title = "Radiationless transitions in a new light", year = "1983", journal = "Journal of Chem. Phys.", volume = "79", pages = "3336", abstract = "Radiationless transitions in polyatomic molecules prove to be quite amendable to a semiclassical treatment both below and above crossings between the potential surfaces involved in the transition. Below such crossings, tunneling integrals are easily performed which give good estimates of the dependence of the nonradiative rate on the energy gap and excess energy in the electronic state. Above the surface crossing, the transitions become classically allowed and a Tully-Preston surface hopping model suffices. The authors find that a nonlinear dependence of ln(k/sub nr/) vs. E plots is the rule rather than the exception. The ln(k/sub nr/) vs. E plots tend to flatten out with increasing energy. This effect can occur below surface crossings, but is most dramatic when a surface crossing is reached. The recent beam results of Smalley and co-workers (1982-3) on pyrazine and pyrimidine are seen to be a possible case of this simple behavior." } @ARTICLE{Heller:1993:NFC, author = "Eric J. Heller and D. Beck", title = "Nonclassical {Franck-Condon} processes", year = "1993", journal = "Chem. Phys. Lett.", volume = "202", pages = "350", abstract = "Franck-Condon factors govern many important molecular processes, including electronic spectroscopy and radiationless transitions. A better understanding of the propensities for exciting various kinds of motion in nonclassical, nonvertical transitions is necessary to attach the right significance to certain experiments. Picosecond and femtosecond laser pulse experiments in the wings of absorption bands are possible which may reveal some of the new effects discussed by the authors." } @INCOLLECTION{Heller:1995:TME, editor = "Hai-Lung Dai and Robert Field", booktitle = "Molecular Dynamics and Spectroscopy by Stimulated Emission Pumping", year = "1995", publisher = "World Scientific", address = "Singapore", author = "Eric J. Heller", title = "Theoretical Methods for Extracting Vibrational Dynamics: Spectroscopy and Dynamics in the Wings" } % ---------- Medvedev @ARTICLE{Medvedev:1982:FCP, author = "E. S. Medvedev", title = "The {Franck-Condon} principle for radiationless transitions in molecules and a selection rule for Morse oscillators", year = "1982", journal = "Chem. Phys.", volume = "73", pages = "243", abstract = "The Franck-Condon (FC) principle for the tunnel radiationless transition (RT) is formulated. It reads that the RT occurs at constant values of the nuclear coordinates q* and of the classical momenta p*. However, unlike the optical transitions, q* and p* take non-physical values since the tunnel RT is a classically forbidden process. As a result of energy conservation, the potential surfaces of two given electronic states cross with one another at the nuclear configuration q*. It is concluded that the electronic-orbital selection rules and the numerical values of the purely electronic matrix elements are governed by the configuration q* rather than by the equilibrium nuclear configuration as was supposed previously. The configuration q* for the T/sub 1/ approximately to S/sub 0/ intersystem crossing in aromatic hydrocarbons is described in terms of a large displacement of only one H atom from its equilibrium position along the C-H bond (perhaps, there is also some out-of-plane displacement). Using the FC principle, it is found that the anharmonicity of the local C-H bond vibrations results in a strong dependence of the T/sub 1/ approximately to S/sub 0/ intersystem crossing rates upon the sign of the C-H bond-length change between two given electronic states, Delta q=R/sub T1/-R/sub S0/. Namely, the following 'selection rule' holds: these RTs are allowed at Delta q<0 and they are forbidden at Delta q>0, the prohibition factor being of the order of 10/sup 2/-10/sup 4/. Finally, an oscillatory dependence of the FC factors upon Delta q is explained, using the FC principle, in terms of quantum interference in the total transition probability, of which the amplitude is a sum of the transition amplitudes due to different crossing points q*. The interference effects are believed to be insignificant for RTs in molecules with large energy gaps and so they are eliminated in the usual manner by adding up the partial probabilities rather than the partial amplitudes. The classical FC factor thus obtained smoothly depends upon Delta q (and upon other parameters as well). This procedure also provides an analytical continuation of the FC factor to non-integral quantum numbers." } @ARTICLE{Osherov:1989:ICT, author = "V. I. Osherov and E. S. Medvedev", title = "Intercombination conversion T_1 -> S_0 in propioaldehyde", year = "1989", journal = "Opt. Spectrosc. (USSR)", volume = "67", pages = "53", abstract = "The authors have calculated the activity of various modes as well as the rates of transition and of the deuteration effect by using the method of direct summation of Franck-Condon factors for different models of potential curves. The mode with the largest frequency change is not a good acceptor if its anharmonicity it taken into account. For a small number (1-2) of good acceptors, a substantial contribution is made by the more numerous group of moderate acceptors which receive small portions of energy with an appreciable probability. The calculated transition rate is in agreement with the observed rate, the deuteration effect is almost on order of magnitude too high, and this discrepancy cannot be eliminated by changing the parameters of the potential curves within physically allowable limits." } @ARTICLE{Medvedev:1992:LLF, author = "E. S. Medvedev", title = "The {Landau-Lifshitz} formula and the correspondence principle for semiclassical matrix elements", year = "1992", journal = "Theor. Math. Phys.", volume = "90", pages = "146", abstract = "The semiclassical Landau-Lifshitz formula with pre-exponential factor is derived and its connection with the correspondence principle is traced. Illustrations are given of its use for the Morse potential and the modified Poschl-Teller potential, and also for a very simple potential with first-order poles." } @ARTICLE{Medvedev:1994:AQL, author = "E. S. Medvedev", title = "The accuracy of the quasiclassical {Landau-Lifshitz} formula for matrix elements and its application to the analysis of the intensities of vibrational overtone transitions", year = "1994", journal = "J. Chem. Phys.", volume = "100", pages = "7192", abstract = "Intensities of vibrational overtone transitions have been analyzed on the basis of the quasiclassical Landau-Lifshitz formula including the pre-exponential factor derived previously. The accuracy of this formula is shown to be very high, on the order of a few percent. An unusual feature of the Landau-Lifshitz formula is the prediction that the overall rapid decrease in intensity with overtone number is highly sensitive to the behavior of the analytical continuation of the molecular potential into regions where the amplitudes of the wave functions are negligible. The dipole moment enters the slowly varying prefactor, which in general superimposes only small modulations on the above steep intensity falloff. However, this prefactor may go through zero, in which case the intensities of one or two transitions nearest the prefactor's zero are anomalously low. These results imply that knowledge of the analytical continuation of the potential is crucial for extracting meaningful information about the dipole moment from overtone intensities." } @BOOK{Medvedev:1995:RTP, author = "E. S. Medvedev and V. I. Osherov", title = "Radiationless transitions in polyatomic molecules", publisher = "Springer", address = "Berlin", year = "1995", } % ---------- Nikitin @ARTICLE{Nikitin:1993:ITL, author = "E. E. Nikitin and L. P. Pitaevskii", title = "Imaginary time and the Landau method of calculating quasiclassical matrix elements", year = "1993", journal = "Physics - Uspekhi", volume = "36", pages = "851", abstract = "In a number of problems of quantum mechanics, particularly in problems of quantum chemistry, the problem arises of calculating matrix elements of a physics quantity f between states which can be described quasiclassically. (For simplicity we shall have in mind a one dimensional case and consider that f is simply a function of the coordinates f(x>)." } @ARTICLE{Nikitin:1993:QCF, author = "E. E. Nikitin and C. Noda and R. N. Zare", title = "On the quasiclassical calculation of fundamental and overtone intensities", year = "1993", journal = "J. Chem. Phys.", volume = "98", pages = "46", abstract = "Various approximations to the transition dipole moment matrix element (n' mod M mod n) are compared with each other and to exact (numerical) values of this overlap integral for different n to n' transitions in a Morse potential with linear dipole moment function. By partitioning the numerical integral into different contributions that involve the classically allowed and forbidden regions of each wave function, the authors have learned what conditions must be satisfied for validity of the different approximations. In particular, they consider the Landau approximation to the quasiclassical matrix element in which the exact wave function for the upper state is replaced by the Wentzel-Kramers-Brillouin (WKB) wave function in the classically allowed region of that state. The authors find that the Landau approximation is more accurate than might have been expected because of the compensation of the neglected tunneling contribution by the singular behavior of the WKB wave function in the classically allowed neighborhood of the turning point. Based on this study, the authors suggest an improved semiclassical approximation for transition dipole matrix elements that involve an arbitrary dipole moment function. This method is applied to the n'-0 transition of a Morse oscillator using a linear dipole moment function; it can reproduce the exact values of the transition dipole moment matrix element to better than 5% for n'=1 to n'=15. Under the condition the dipole moment function is slowly varying or decreases monotonically with increasing internuclear separation, a simple expression is presented for estimating relative strengths of neighboring high overtone transitions." } @ARTICLE{Nikitin:1994:CLQ, author = "E. E. Nikitin and L. P. Pitaevskii", title = "Calculation of the Landau quasiclassical exponent from the Fourier components of classical functions", year = "1994", journal = "Phys. Rev. A", volume = "49", pages = "695", abstract = "In this paper, we show that the Landau quasiclassical exponent, which determines an exponentially small transition-amplitude, can be recovered from the Fourier analysis of classical functions for both one-dimensional and three-dimensional cases. This finding renders unnecessary the analytical continuation of the potential-energy function into the classically inaccessible region of the configuration space. In addition, we present a criterion for the applicability of the Landau method on the basis of the correspondence-principle matrix elements and discuss the relation between this method and the phase-integral approach." } @ARTICLE{Karni:1994:RLM, author = "Y. Karni and E. E. Nikitin", title = "Recovery of the Landau matrix elements from the classical Fourier components: the one-dimensional dissociating oscillator", year = "1994", journal = "J. Chem. Phys.", volume = "100", pages = "2027", abstract = "The recently suggested method of recovering the Landau exponent of the quasiclassical matrix elements from the attributes of classical motion is illustrated by way of an example of dissociating anharmonic oscillators. For a Morse oscillator, in which case the exact analytical results are available, the so-called improved semiclassical approximation that incorporates the Landau exponential yields quite accurate matrix elements for classically strongly forbidden events. This provides a firm support for the method of estimation of quasiclassical matrix elements between distant states from the information on the motion of system in the classically allowed region." } @ARTICLE{Karni:1994:VPR, author = "Y. Karni and E. E. Nikitin", title = "Vibrational predissociation rate from dynamics of a full collision: a test of the Landau method against the exact results", year = "1994", journal = "J. Chem. Phys.", volume = "100", pages = "8065", abstract = "We applied the Landau method of calculating the coupling matrix element for a bound-free transition to the evaluation of,the vibrational predissociation rate constant of the van der Waals complex ICI*-Ne. Use of the adiabatic separation of the radial and angular motion of the van der Waals modes complemented with the angular averaging with the help of the dominant natural expansion wave function of the bending mode, calculated by Lipkin, Moiseyev, and Leforestier (J. Chem. Phys. 98, 1888 (1993)), yields the rate constant which is about 10%-15% lower compared to the best theoretical estimates. This disagreement is ascribed to the true dynamical interaction between the stretching and bending modes which cannot be-incorporated into the adiabatic separation of motion of these modes. Different approximations for treating bending motion are discussed and, the sensitivity of the rate constant to the variation of principle parameters of the potential energy surface is studied." } @ARTICLE{Karni:1995:ACQ, author = "Y. Karni and E. E. Nikitin", title = "Adiabatically corrected quasiclassical model for the vibrational predissociation of van der Waals complexes", year = "1995", journal = "Chem. Phys.", volume = "191", pages = "235", abstract = "For a collinear model of a van der Waals complex A-BC, we present a quasiclassical adiabatically corrected theory of vibrational predissociation. The approach is based on the recovery of the quasiclassical Landau exponent from the attributes of the classical motion of the system (E.E. Nikitin and L. Pitaevskii, Phys. Rev. A 49 (1994) 695). The theory is applied to the generalized Beswick-Jortner Hamiltonian for a prototype system Ne-ICl and is shown to reproduce the exact level widths quite well. Approximate formulae are suggested which correct the widths, calculated within the original Beswick-Jortner theory, for the main adiabatic effects not included in the first-order perturbation treatment: stiffening of the BC oscillator, softening of the A-BC repulsive interaction, and the increase of the interaction range of A and BC with the growth of the vibrational quantum number of BC." } % ---------- Jortner @ARTICLE{Bixon:1968:IRT, author = "Mordechai Bixon and Joshua Jortner", title = "Intramolecular radiationless transitions", year = "1968", journal = "J. Chem. Phys.", volume = "48", pages = "715" } @ARTICLE{Rice:1968:SCF, author = "Stuart A. Rice and Ian McLaughlin and Joshua Jortner", title = "Some comments on a formal theory of photochemical dissociation reactions", year = "1968", journal = "J. Chem. Phys.", volume = "49", pages = "2756", abstract = "Presents a formal theory of photochemical fragmentation reactions where intramolecular energy transfer is of importance." } @ARTICLE{Bixon:1969:ERL, author = "Mordechai Bixon and Joshua Jortner", title = "Electronic relaxation in large molecules", year = "1969", journal = "J. Chem. Phys.", volume = "50", pages = "4061", abstract = "Considers the problem of the radiative decay of electronically excited states of a large molecule. Both intramolecular vibronic coupling and the interaction with the radiation field are considered. Compound states for a system of decaying indistinguishable levels are constructed using the Fano method. General expressions for the radiative decay rate are derived and applied for the statistical limit of intramolecular vibronic coupling. On a time scale shorter than a typical intramolecular recurrence time the radiative decay is exponential, and the reciprocal lifetime consists of independent contributions of radiative and nonradiative components. The experimental implications of these results for large and medium-size molecules are discussed." } @ARTICLE{Jortner:1969:RTP, author = "Joshua Jortner and Stuart A. Rice and Robin M. Hochstrasser", title = "Radiationless transitions in photochemistry", year = "1969", journal = "Adv. of Photochem.", volume = "7", pages = "149" } @ARTICLE{Jortner:1969:TSR, author = "Joshua Jortner", title = "Theoretical studies of radiationless transitions in polyatomic molecules", year = "1969", journal = "Journal de Chimie Physique et de Physico-Chimie Biologique, spec. issue", pages = "9", abstract = "A theory of electronic relaxation processes in large molecules is considered. The following problems are treated in some detail: (a) The nature of the dissipative channel and nonradiative decay in isolated molecules. (b) The nature of the irreversibility in the electronic relaxation processes. (c) Line broadening of an isolated resonance. (d) Interference effect between resonances and Fano-type antiresonances in molecular spectrum. (e) The coupling between radiative and nonradiative processes in large molecules. (f) Nonradiative decay probabilities and the energy gap law." } % ---------- R. S. Berry @ARTICLE{Nielsen:1968:VAP, author = "Svend Erik Nielsen and R. S. Berry", title = "Vibronic autoionization and predissociation in hydrogen", year = "1968", journal = "Chem. Phys. Lett.", volume = "2", pages = "503", abstract = "High-resolution photoionization experiments on H/sub 2/, HD and D/sub 2/ have shown that (1) near the threshold for ionization, the dominant mechanism for ionization is autoionization; (2) the lifetimes for autoionization are 0.3*10/sup -12/ sec or longer; (3) although the vibrational propensity rule ( tau ( Delta v=-1) [left angle bracket] tau ( Delta v=-2) [left angle bracket]...) is frequently obeyed, there are significant violations of this rule; and (4) predissociation is as fast as or faster than autoionization in some cases for which both are allowed. A theoretical treatment which accounts for these phenomena is reported. The effect of isotopic substitution on the autoionization and predissociation rates is also derived. The method is applied to associative ionization and dissociative recombination." } @ARTICLE{Berry:1968:RSS, author = "R. S. Berry and Svend Erik Nielsen", title = "Rydberg states and scattering states of molecular electrons: e-H/sub 2//sup +/", year = "1968", journal = "J. Chem. Phys.", volume = "49", pages = "116", abstract = "A general method is presented for obtaining a model potential for calculation of Rydberg and ionized states of an electron in a diatomic (or more complex) molecule. The potential is derived from a known wavefunction for the ionic molecule core by expanding the charge distribution, including the fixed-position nuclei, in spherical harmonics. Exchange effects are included in the form of an effective local rho /sup 1/3/ potential. The method is applied to the Rydberg states of H/sub 2/ to examine the various contributions to the level splittings and the dependence of these on internuclear distance R. It is also applied to the determination of phase shifts in e-H/sub 2//sup +/ scattering and to the dependence of these phase shifts on R. The particular utility of the method is the ease and efficiency with which one can explore the behavior of electronic contributions to a wide variety of matrix elements, as functions of all the parameters n(or E), l, m, and R." } @ARTICLE{Berry:1970:DCP1, author = "R. S. Berry and Svend Erik Nielsen", title = "Dynamic coupling phenomena in molecular excited states. I. General formulation and vibronic coupling H/sub 2/", year = "1970", journal = "Phys. Rev. A", volume = "1", pages = "383", abstract = "A general formalism is described for treating diabatic coupling processes in highly excited molecular states. The method treats electronic and nuclear motion quantum mechanically, and uses the adiabatic Born-Oppenheimer states as basis functions. The present paper concentrates on diagnoses of vibronic coupling matrix elements, with Rydberg and continuum states of H/sub 2/ as the test cases. The principal contributions come from the excited electron's interactions with the oscillating finite monopole of the ion-molecule core. The electronic factors in the transition amplitudes are definitely dependent on internuclear distance, particularly in the cases of p and d states. The transition amplitudes accumulate their magnitudes over the full classically allowed range of internuclear distance, especially in the cases of s and p states. Specific application is made to vibronic coupling perturbations in p sigma and p pi Rydberg states of H/sub 2/." } @ARTICLE{Berry:1970:DCP2, author = "R. S. Berry and Svend Erik Nielsen", title = "Dynamic coupling phenomena in molecular excited states. II. Autoionization and predissociation in H/sub 2/, HD, and D/sub 2/.", year = "1970", journal = "Phys. Rev. A", volume = "1", pages = "395", abstract = "For Pt. I see ibid., vol. 1, 383 (1970). Vibrationally induced autoionization and one class of predissociation of electronically excited H/sub 2/ and its isotopes are treated by a perturbed-stationary- state theory. Autoionization and predissociation rates are given for a number of states of H/sub 2/, HD, and D/sub 2/. In addition to direct bound- continuum coupling, consideration is given to effects of higher-order coupling; these effects cause order-of-magnitude changes in isolated cases. The auto-ionization rates vary with principal quantum number n as n/sup -3/, directly with vibrational energy vh nu , and decrease sharply with vibrational quantum change Delta v. As the principal quantum number increases, transitions of successively smaller Delta v become possible; the net effect of this is to override the n/sup -3/ dependence in total autoionization rates. Competition between predissociation and autoionization is examined; the two processes show very different dependence on n and v, with the consequence that decay in most regions of (n,v) space is dominated by one or the other process; but the two mechanisms are competitive for some (n,v) states. The isotope effect also is rather different for the two decay processes, enough so that, in effect, the isotope effect amounts to a qualitative change from one mechanism of decay to the other, with mass change." } % Miscellaneous @ARTICLE{Hupper:1998:USE, author = "Bruno H{\"u}pper and Bruno Eckardt", title = "Uniform semiclassical expansions for the direct part of {Franck-Condon} transitions", year = "1998", journal = "Phys. Rev. A", volume = "57", pages = "1536", abstract = "Semiclassical expansions for traces involving Green's functions receive two contributions, one from the periodic or recurrent orbits of the classical system and one from the phase space volume, i.e., the paths of infinitesimal length. Quantitative calculations require the control of both terms. Here we discuss the contribution from paths of zero length with an emphasis on the application to Franck-Condon transitions. The expansion in the energy representation is asymptotic and a critical parameter is identified. In the time domain, a series expansion of the logarithm of the propagator gives very good results. The expansions are illustrated for transitions onto a linear potential and onto a harmonic oscillator." } @ARTICLE{Hupper:1999:USC, author = "Bruno H{\"u}pper and Bruno Eckardt", title = "Uniform semiclassical calculation of the direct part of the photodissociation cross section of water", year = "1999", journal = "J. Chem. Phys.", volume = "110", pages = "11749", abstract = "The photodissociation cross section of water in the first absorption band naturally splits into two parts, a rather dominant and broad background, and a series of isolated resonances. We focus here on the semiclassical calculation of the background term. We compare various semiclassical methods of calculation, including the reflection approximation, the Wigner-Weyl series, Airy function approximations, and higher-order approximations in the time domain, and find that only the latter gives quantitative agreement. They are numerically efficient as they can be reduced to a 2-D spatial integration together with a Fourier transform." } @BOOK{Beveridge:1970:OTP, editor = "Gordon S. G. Beveridge and Robert S. Schechter", title = "Optimization: Theory and Practice", publisher = "Mc{\-}Graw-Hill", address = "New York", year = "1970", } @BOOK{Bender:1978:AMM, author = "C[arl] M. Bender and S. A. Orszag", title = "Advanced Mathematical Methods for Scientists and Engineers", publisher = "Mc{\-}Graw-Hill", address = "New York", year = "1978", } @ARTICLE{Siebrand:1967:RTP, author = "W. Siebrand", title = "Radiationless transitions in polyatomic molecules. I. Calculation of {Franck-Condon} factors", year = "1967", journal = "J. Chem. Phys.", volume = "46", pages = "440", abstract = "... It is shown that for radiationless transition involving large energy gaps (E) F is dominated by distortions, whereas for optical transitions it is dominated by displacements of the oscillators... " } @ARTICLE{Popov:1994:LOE, author = "Vladimir S. Popov and Alexey V. Sergeev", title = "Large orders of 1/n-expansion for multidijensional problems", year = "1994", journal = "Physics Letters A", volume = "193", pages = "165", abstract = "The asymptotics of large orders of the 1/n-expansion is investigated for multidimensional problems of quantum mechanics and atomic physics, including those with separable variables (the hydrogen molecular ion H/sub 2//sup +/), and those where separation of variables is impossible (a hydrogen atom in electric and magnetic fields). It is shown that the parameters of the asymptotics can be found by means of calculating sub-barrier trajectories with the help of the ''imaginary time'' method, as well as by solution of the eikonal equation." } @ARTICLE{Mills:1985:RNM, author = "I. M. Mills and A. G. Robiette", title = "On the relationship of normal modes to local modes in molecular vibrations", year = "1985", journal = "Molecular Physics", volume = "56", pages = "743", abstract = "A simple model for the effective vibrational hamiltonian of the XH stretching vibrations in H/sub 2/O, NH/sub 3/ and CH/sub 4/ is considered, based on a Morse potential function for the bond stretches plus potential and kinetic energy coupling between pairs of bond oscillators. It is shown that this model can be set up as a matrix in local mode basis functions, or as a matrix in normal mode basis functions, leading to identical results. The energy levels obtained exhibit normal mode patterns at low vibrational excitation, and local mode patterns at high excitation. When the hamiltonian is set up in the normal mode basis it is shown that Darling-Dennison resonances must be included, and simple relations are found to exist between the x/sub rs/, g/sub tt/, and K/sub rrss/ anharmonic constants (where the Darling-Dennison coefficients are denoted K) due to their contributions from morse anharmonicity in the bond stretches. The importance of the Darling-Dennison resonances is stressed. The relationship of the two alternative representations of this local mode/normal mode model are investigated, and the potential uses and limitations of the model are discussed." } @ARTICLE{Freed:1971:BOS, author = "K. F. Freed and W. M. Gelbart", title = "On the Born - Oppenheimer separation and the calculation of the nonradiative transition rates", year = "1971", journal = "Chem. Phys. Lett.", volume = "10", pages = "187", abstract = "The dependence of calculated nonradiative transition rates on the vibronic wavefunctions chosen to describe excited molecular state is discussed. Several fallacies and ambiguities in the literature are commented on and a simple resolution to the problem suggested." } @ARTICLE{Spath:1996:SCF, author = "Bruce W. Spath and William H. Miller", title = "Semiclassical calculation of Franck - Condon intensities for reactive systems", year = "1995", journal = "Chem. Phys. Lett.", volume = "262", pages = "486", abstract = "A Filinov transformation of the initial value representation of the semiclassical propagator has been used to obtain Franck-Condon intensities for a model of the photodetachment of H/sub 3//sup -/ to H+H/sub 2/+e/sup -/. Resonance peak widths and heights are obtained which are in good agreement with quantum results." } @BOOK{Englman:1972:JTE, author = "R. Englman", title = "The {Jahn - Teller} effect in molecules and crystals", publisher = "Wiley Interscience", address = "London - New York", year = "1972", } @BOOK{Englman:1979:NRD, author = "R. Englman", title = "Non-radiative decay of ions and molecules in solids", publisher = "North Holland Pub. Co.", address = "Amsterdam", year = "1979", } @BOOK{Fischer:1984:VC, author = "G. Fischer", title = "Vibronic coupling", publisher = "Academic Press", address = "London", year = "1984", } @ARTICLE{Sharp:1964, author = "T. E. Sharp and H. M. Rosenstock", title = "", year = "1964", journal = "J. Chem. Phys.", volume = "41", pages = "3453" } @ARTICLE{Toniolo:2001:ECF, author = "A. Toniolo and M. Persico", title = "Efficient Calculation of {Franck-Condon} factors and vibronic couplings in polyatomics", year = "2001", journal = "J. Comput. Chem.", volume = "22", pages = "968", abstract = "We present a technique for the calculation of Franck-Condon factors and other integrals between vibronic wave functions belonging to different electronic states. The technique is well suited for the determination of the nonadiabatic or spin-orbit couplings related to radiationless decays in polyatomics. Rigorous or approximate partitions of the internal coordinate space are exploited to achieve better efficiency and/or to go beyond the harmonic approximation. The technique is tested by computing the Internal Conversion and InterSystem Crossing rates of (CH3)(3)CNO in its (1)(n --> pi*) state." } @ARTICLE{Burland:1969, author = "D. M. Burland and G. W. Robinson", title = "Calculated radiationless transition rates for benzene and deuterobenzene", year = "1969", journal = "J. Chem. Phys.", volume = "51", pages = "4548", abstract = "Radiationless rates have been calculated for the /sup 3/B/sub 1u/ to /sup 1/A/sub 1g/ transitions of benzene and perdeuterobenzene as well as for the /sup 1/B/sub 2u/ to /sup 1/A/sub 1g/ transition of benzene. The rates were calculated using a model that considers the radiationless transition as a tunneling process between two multidimensional potential surfaces and assuming both harmonic and anharmonic vibrational potentials. The importance of anharmonicity for the radiationless transition rate provides a possible explanation of environmental effects on these rates. Experimental values for the anharmonicities of the carbon- carbon and carbon-hydrogen vibrations and the size of the lowest triplet state of benzene were used in the calculation." } @ARTICLE{Preston:1971:ESC, author = "R. K. Preston and J. C. Tully", title = "Effects of surface crossing in chemical reactions: The H/sub 3//sup +/ system", year = "1971", journal = "J. Chem. Phys.", volume = "54", pages = "4297", abstract = "Approximate potential-energy surfaces for the two lowest singlet states of H/sub 3//sup +/ are calculated using the diatomics-in-molecules approach. The nonadiabatic terms which couple these surfaces can be directly computed in this approximation. From the magnitudes of these coupling terms it is apparent that, for excitation energies below about 10 eV, nonadiabatic transitions must be confied almost entirely to a region localized at the avoided crossing of the two surfaces. This fact suggests the following simplified picture of the dynamics of the H/sup +/+H/sub 2/ reaction: As the H/sup +/ and H/sub 2/ approach, they remain on the lower potential surface (there is no initial electron jump). They continue to follow adiabatically the lower surface in the close-collision region, so the probability of a nonadiabatic transition does not appear to be related to the lifetime of the collision complex. It is while the products are receding that electronic transitions become important. Consequences of this model on the threshold for formation of H/sub 2//sup +/ and on the partition of vibrational energy in the products are discussed, and a comparison with recent experiments of Krenos and Wolfgang is included." } @ARTICLE{Herman:1995:SSH, author = "M. F. Herman", title = "A semiclassical surface hopping propagator for nonadiabatic problems", year = "1995", journal = "J. Chem. Phys.", volume = "103", pages = "8081", abstract = "A semiclassical propagator is developed for general multisurface, multidimensional nonadiabatic problems. It is demonstrated that this propagator satisfies the time-dependent Schrodinger equation through order h(cross). This is the same order satisfied by the usual semiclassical propagator in single surface problems. The zeroth-order term (in the nonadiabatic coupling) for the propagator is just the well-known single surface adiabatic propagator. The first-order terms involve single hops from the initial adiabatic state to other states. Energy is conserved in these hops and the momentum change accompanying each hop occurs in the direction parallel to the nonadiabatic coupling for the transition. Both transmitted and reflected contributions are included after a hop. The propagator expression has the zeroth-order (single surface) semiclassical form before and after the hop. The complete propagator includes terms with any number of hops and all possible hopping points. These multihop terms are defined analogously to the first-order (single hop) terms. An alternative formulation of the semiclassical propagator, which includes contributions from a broader range of hopping trajectories, is also developed." } @ARTICLE{Herman:1999:ORS, author = "M. F. Herman", title = "Optimal representation for semiclassical surface hopping methods", year = "1999", journal = "J. Chem. Phys.", volume = "110", pages = "4141", abstract = "A semiclassical surface hopping expansion of the propagator is developed for a general representation of the ''fast'' variable quantum states. The representation can be the adiabatic or diabatic representation or any representation between these two. A particular representation is defined, which is optimal in the sense that it minimizes the integrated interstate coupling. The coupling is integrated over a suitable classical trajectory in this definition. Calculations for a simple one-dimensional curve crossing model problem show that the use of this optimal representation can significantly reduce the importance of multihop terms in the expansion. An approximation to this optimal representation is proposed, which is much simpler to implement numerically. Calculations for the model curve crossing problem demonstrate that this approximate optimal representation provides integrated couplings that are very close to those obtained for the optimal representation. These results suggest that this approximate optimal representation provides a computationally attractive representation for use with semiclassical surface hopping methods, when studying problems with curve crossings." } @ARTICLE{Herman:1999:CGR, author = "M. F. Herman", title = "Choosing a good representation of the quantum state wave functions for semiclassical surface hopping calculations", year = "1999", journal = "J. Chem. Phys.", volume = "111", pages = "10427", abstract = "Simple curve crossing problems have shown that the probability of transition between quantum states for the fast degrees of freedom becomes small at low energies, if the adiabatic representation is employed for the quantum state wave functions. Conversely, the transition probability approaches unity at high energies in the adiabatic representation, but it is small if the diabatic representation is employed. Methods are developed and numerically tested for choosing a representation, which is intermediate between the adiabatic and diabatic representations, so as to approximately minimize the interstate coupling in semiclassical surface hopping methods. The reduction in the coupling should diminish the importance of multihop trajectories in surface hopping procedures, resulting in a corresponding reduction in the computational effort required for accurate calculations." } @ARTICLE{BenNun:2000:AIM, author = "M. Ben-Nun and J. Quenneville and T. J. Martinez", title = "Ab initio multiple spawning: Photochemistry from first principles quantum molecular dynamics", year = "2000", journal = "J. Phys. Chem. A", volume = "104", pages = "5161", abstract = "The ab initio multiple spawning (AIMS) method is a time-dependent formulation of quantum chemistry, whereby the nuclear dynamics and electronic structure problems are solved simultaneously. Quantum mechanical effects in the nuclear dynamics are included, especially the nonadiabatic effects which are crucial in modeling dynamics on multiple electronic states. The AIMS method makes it possible to describe photochemistry from first principles molecular dynamics, with no empirical parameters. We describe the method and present the application to two molecules of interest in organic photochemistry-ethylene and cyclobutene. We show that the photodynamics of ethylene involves both covalent and ionic electronic excited states and the return to the ground state proceeds through a pyramidalized geometry. For the photoinduced ring opening of cyclobutene, we show that the disrotatory motion predicted by the Woodward-Hoffmann rules is established within the first 50 fs after optical excitation." } @ARTICLE{Hack:2001:CFM, author = "M. D. Hack and A. M. Wensmann and D. G. Truhlar and M. Ben-Nun and T. J. Martinez", title = "Comparison of full multiple spawning, trajectory surface hopping, and converged quantum mechanics for electronically nonadiabatic dynamics", year = "2001", journal = "J. Chem. Phys.", volume = "115", pages = "1172", abstract = "We present calculations employing the simplest version of the full multiple spawning method, FMS-M or minimal FMS, for electronically nonadiabatic quantum dynamics using three model potential energy matrices with different strengths and ranges for the diabatic coupling. We first demonstrate stability of the branching probabilities and final energy distributions with respect to the parameters in the FMS-M method. We then compare the method to a variety of other semiclassical methods, as well as to accurate quantum mechanical results for three-dimensional atom-diatom reactions and quenching processes; the deviations of the semiclassical results from the accurate quantum mechanical ones are averaged over nine cases. In the adiabatic electronic representation, the FMS-M method provides some improvement over Tully's fewest switches trajectory surface hopping method. However, both methods, irrespective of electronic representation, systematically overpredict the extent of reaction in comparison to the exact quantum mechanical results. The present work provides a baseline for understanding the simplest member of the hierarchy of FMS methods and its relationship to established surface hopping methods." } @ARTICLE{Lee:1995:TAQ, author = "H. W. Lee", title = "Theory and application of the quantum phase-space distribution-functions", year = "1995", journal = "Phys. Rep.", volume = "259", pages = "147", abstract = "A review is given of the quantum phase-space distribution functions with emphasis on both the fundamental characteristics and practical applications of the distribution functions. The distribution functions, such as the Wigner distribution function, the Glauber-Sudarshan P and Q functions, the Kirkwood distribution function and the Husimi distribution function, are treated in a unified fashion based on the classification scheme of Cohen. The fundamental relations of the distribution functions are discussed both in (q, p) phase space and in (alpha, alpha*) complex space, the properties of the distribution functions are compared and relations between them derived. Also discussed is the dynamical equations that govern the time development of the distribution functions. Applications of the distribution functions are illustrated, with particular attention to the Wigner distribution function in studies of collision systems and to the Husimi distribution function in studies of classically chaotic nonlinear systems." } @ARTICLE{Frank:2000:WFM, author = "M. D. Hack and A. M. Wensmann and D. G. Truhlar and M. Ben-Nun and T. J. Martinez", title = "Wigner function of Morse potential eigenstates", year = "2000", journal = "Phys. Rev. A", volume = "61", pages = "4102", abstract = "We find the analytic expressions and obtain plots of the Wigner distribution function for the hound eigenstates of the Morse potential on quantum phase space." } @ARTICLE{Berry:1977:SCM, author = "M[ichael] V. Berry", title = "Semi-classical mechanics in phase space: a study of {Wigner's} function", year = "1977", journal = "Philos. Trans. R. Soc. London A", volume = "287", pages = "237", abstract = "The semiclassical structure of the Wigner functions Psi representing bound energy eigenstates mod psi ) for systems with f degrees of freedom is explored. If the classical motion is integrable, the classical limit of Psi is a delta function on the f-dimensional torus to which classical trajectories corresponding to mod psi ) are confined in the 2f-dimensional phase space. In the semi-classical limit of Psi (h(cross) small but not zero) the delta function softens to a peak of order h(cross)/sup -2/3f/ and the torus develops fringes of a characteristic 'airy' form. Away from the torus, Psi can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f=1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when Psi is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta." } % ---------- added references about methods of calculation of FCF @ARTICLE{Malmqvist:1998:FCF, author = "Per-{\AA}ke Malmqvist and Niclas Forsberg", title = "Franck-Condon factors for multidimensional harmonic oscillators", year = "1998", journal = "Chem. Phys.", volume = "228", pages = "227", abstract = "We present a simple formula for the overlap integrals of two sets of multi-dimensional harmonic oscillators. The oscillators have in general different equilibrium points, force constants, and natural vibration modes. The formula expresses the overlap matrix in the one-dimensional case, [m'\n ''], as a so-called LU decomposition, [m'\n ''] = [0'\0 '']Sigma LmtUtn, where the summation index has a range 0 less than or equal to t less than or equal to min(m,n), i.e., it is the matrix product of a lower-triangular matrix L with an upper-triangular Li. These matrices are obtained from simple recursion formulae. This form is essentially retained in the multi-dimensional case. General matrix elements are obtained by exact and finite expressions, relating them to matrix elements over a single set of harmonic oscillator wave functions. We present test calculations with error estimates, also comparing with literature examples." } @ARTICLE{Baraille:2002:VAC, author = "I. Baraille and C. Larrieu and A. Dargelos and M. Chaillet", title = "A variational approach of the calculation of Franck-Condon factors: The F2BO emission spectrum", year = "2002", journal = "Chem. Phys.", volume = "282", pages = "9", abstract = "We present a new variational method to calculate the Franck-Condon factors beyond the harmonic approximation. This approach is successfully applied to the 172130 emission spectrum, rationalising the experimental data available on this molecular system. The natures of the two l(2)A(1) --> (X) over tilde (l(2)B(2)) and l(2)A(1) --> l(2)B(1) emission bands have been investigated. It appears that only the latter transition has a vibronic structure due, essentially, to the first three vibrational states." } @ARTICLE{Lopez:2002:SEF, author = "J. C. L{\'{o}}pez and A. L. Rivera and Yu. F. Smirnov and A. Frank", title = "Simple evaluation of Franck-Condon factors and non-Condon effects in the Morse potential", year = "2002", journal = "Internat. J. Quant. Chem.", volume = "88", pages = "280", abstract = "The calculation of Franck-Condon factors between different 1-D Morse potential eigenstates using a formula derived from the Wigner function is discussed. Our numerical calculations using a simple program written in Mathematica are compared with other calculations. We show that our results have a similar accuracy as those calculations performed with more sophisticated methods. We discuss the extension of our method to include non-Condon effects in the calculation." } @ARTICLE{Muller:2000:AAC, author = "T. M{\"{u}}ller and P. H. Vaccaro and F. P{\'{e}}rez-Bernal and F. Iachello", title = "Algebraic approach for the calculation of polyatomic Franck-Condon factors: application to the vibronically resolved absorption spectrum of disulfur monoxide (S2O)", year = "2000", journal = "Chem. Phys. Lett.", volume = "329", pages = "271", abstract = "Cavity ring-down spectroscopy (CRDS) has been employed to probe the (C) over tilde (1)A' <-- <(X)over tilde>1A' (pi* <-- ) absorption system of jet-cooled disulfur monoxide (S2O) molecules. Vibronically resolved features possessing up to 8 quanta of excitation in the v(2)' S-S stretching mode of the (C) over tilde state have been characterized, with the onset of rapid predissociation for v(2)' greater than or equal to 4 being manifest in the widths of band contours. Measured vibronic intensities are in good accord with predictions derived from a Lie algebraic (or vibron) treatment of Franck-Condon factors for the corresponding S2O emission spectrum." } @ARTICLE{Mok:2000:NMC, author = "Daniel K. W. Mok and Edmond P. F. Lee and Foo-Tim Chau and DeChao Wang and John M. Dyke", title = "A new method of calculation of Franck-Condon factors which includes allowance for anharmonicity and the Duschinsky effect: Simulation of the HeI photoelectron spectrum of ClO2", year = "2000", journal = "J. Chem. Phys.", volume = "113", pages = "5791", abstract = "A new method of Franck-Condon (FC) factor calculation for nonlinear polyatomics, which includes anharmonicity and Duschinsky rotation, is reported. Watson's Hamiltonian is employed in this method with multidimensional ab initio potential energy functions. The anharmonic vibrational wave functions are expressed as linear combinations of the products of harmonic oscillator functions. The Duschinsky effect, which arises from the rotation of the normal modes of the two electronic states involved in the electronic transition, is formulated in Cartesian coordinates, as was done previously in an earlier harmonic FC model. This new anharmonic FC method was applied to the simulation of the bands in the He I photoelectron (PE) spectrum of ClO2. For the first band, the harmonic FC model was shown to be inadequate but the anharmonic FC simulation gave a much-improved agreement with the observed spectrum. The experimentally derived geometry of the (X) over tilde (1)A(1) state of ClO2+ was obtained, for the first time, via the iterative FC analysis procedure {R(Cl-O)=1.414 +/- 0.002 Angstrom, angle O-Cl-O=121.8 +/- 0.1 degrees}. The heavily overlapped second PE band of ClO2, corresponding to ionization to five cationic states, was simulated using the anharmonic FC code. The main vibrational features observed in the experimental spectrum were adequately accounted for in the simulated spectrum. The spectral simulation reported here supports one of the two sets of published assignments for this band, which was based on multireference configuration interaction (MRCI) calculations. In addition, with the aid of the simulated envelopes, a set of adiabatic (and vertical) ionization energies to all five cationic states involved in this PE band, more reliable than previously reported, has been derived. This led also to a reanalysis of the photoabsorption spectrum of ClO2." } @ARTICLE{Muller:1999:VRE, author = "T. M{\"{u}}ller and P. H. Vaccaro and F. P{\'{e}}rez-Bernal and F. Iachello", title = "The vibronically-resolved emission spectrum of disulfur monoxide (S2O): An algebraic calculation and quantitative interpretation of Franck-Condon transition intensities", year = "1999", journal = "J. Chem. Phys.", volume = "111", pages = "5038", abstract = "Emission spectra obtained from jet-cooled disulfur monoxide (S2O) molecules have been interpreted by means of a novel Lie algebraic formalism that makes possible the facile evaluation of multidimensional Franck-Condon factors. Fluorescence accompanying selective excitation of isolated vibronic bands in the S2O (C) over tilde (1)A(')<--(X) over tilde (1)A(')(pi(*)<--pi) absorption system has been dispersed under moderate spectral resolution, allowing assignment of ground state levels possessing up to 20 quanta of vibration in the nu(2) S-S stretching mode [E-vib((X) over tilde)less than or equal to 13 900 cm(-1)]. Aside from providing a rigorous and economical description for the inherently anharmonic nature of highly-excited polyatomic species, our algebraic approach enables quantitative information on molecular wavefunctions to be extracted directly from spectroscopic data. The emerging picture of S2O vibrational dynamics suggests that the X (1)A(') potential surface is substantially more ''local'' in character than the C (1)A(') manifold. While the observed pattern of X (1)A(') vibrational energies could be reproduced well through use of model Hamiltonians that include only diagonal anharmonicities in the local algebraic basis, successful treatment of the C (1)A(') state necessitated explicit incorporation of off-diagonal anharmonicities that lead to pervasive mixing of local vibrational character. This disparate behavior is manifest strongly in measured C-X transition strengths, thereby allowing detailed investigations of Franck-Condon intensities to discern the underlying dynamics. Structural parameters deduced from algebraic analyses are in good accord with previous predictions of the change in S2O geometry accompanying pi(*)<--pi excitation." } @ARTICLE{Hornburger:1998:DCA, author = "H. Hornburger and G. H. E. Diercksen", title = "Derivation of closed analytical expressions for Rosen-Morse Franck-Condon factors", year = "1998", journal = "J. Math. Chem.", volume = "24", pages = "39", abstract = "The expressions for Rosen-Morse Franck-Condon factors derived previously yield a double sum with alternating terms. For higher values of the quantum number the numerical calculation of the Franck-Condon factors by electronic computers using these expressions leads to numerical overflow inspite of the use of double-precision (32 digits) arithmetic. High values for the quantum number in the final ground state of the Rosen-Morse potential occur in molecular nonradiative rate calculations. Furthermore, the expressions show a lack of clearness with respect to the parameters of the potential. For out-of-plane modes exact closed form expressions and exact recurrence relations are derived. Asymptotic expressions for the matrix elements are calculated. Exact closed form expressions for matrix elements with quantum numbers which correspond to regions close to the dissociation barrier are given." } @ARTICLE{Morales:1995:IRF, author = "J. Morales and V. Gaftoi and J. Lopezbonilla and J. J. Pena", title = "Improved recursion formulas for the calculation of 2-center central potential integrals", year = "1995", journal = "Intern. J. Quant. Chem., Suppl.", volume = "29", pages = "339", abstract = "A commutator algebra procedure is used to get improved recurrence relations for the calculation of any f(r) off-diagonal two-center matrix elements in the general case of displaced arbitrary central potential wave functions. As expected, the proposed formulas reduce properly to the generalized recursion equations for the calculation of one-center integrals. Besides, when f(r) = r(k), one obtains the equivalent of the Kramer rule for two-center multipolar matrix elements for arbitrary potentials, and when f(r) is constant, the Wu formula (for the calculation of Franck-Condon factors) is ameliorated in the sense that the new relation not necessarily considers equal mass. Furthermore, all diagonal matrix elements as well as all off-diagonal integrals between nondisplaced potentials appear, in our treatment, as particular cases in good agreement with already published results. As a useful application, the corresponding recurrence relations for the calculation of one-center hydrogenic matrix elements and two-center Kratzer potential integrals are given as examples. However, our approach is general and can be easily extended to obtain recursion formulae for other potential wave functions." } @ARTICLE{Fernandez:1990:FCF, author = "Francisco M. Fern{\'{a}}ndez and J. F. Ogilvie", title = "Franck-Condon factors for diatomic molecules", year = "1990", journal = "Chem. Phys. Lett.", volume = "169", pages = "292", abstract = "We have investigated the validity of the harmonic-oscillator approximation for the calculation of the Franck-Condon factors (FCF) of diatomic molecules. Our results show that, although in general the FCF obtained with this approximation exhibit the correct order of magnitude, they fail to predict the relative magnitudes of off-diagonal elements of the FCF-array obtained by more accurate methods. We show that this deficiency of the harmonic-oscillator approximation is overcome if the cubic term in the expansion of the potential-energy function about equilibrium is taken properly into account. Second-order perturbation theory, i.e. incorporating the cubic and quartic terms of the Dunham expansion, corrects this transposition of the FCF array caused by the harmonic-oscillator approximation. We also show that the FCF array obtained through the harmonic approximation is almost the transpose of the array obtained by more accurate means. For this reason the relative FCF claculated by means of the harmonic approximation agree remarkably well with the relative intensities obtained from electron impact experiments." } @ARTICLE{Krantzman:1988:ENS, author = "Kristin Krantzman and David Farrelly", title = "Evaluation of non-separable bound-bound Franck-Condon factors using the self-consistent field and adiabatic approximations", year = "1988", journal = "Chem. Phys. Lett.", volume = "152", pages = "196", abstract = "The self-consistent field and adiabatic approximations are used to evaluate non-separable bound-bound Franck-Condon overlap integrals. The two methods rely on different, approximate separations of the nuclear wavefunctions. Both approaches provide good agreement with accurate calculations with less computational complexity than alternative, semiclassical approaches." } @ARTICLE{Domcke:1977:CDA, author = "W. Domcke and L. S. Cederbaum and H. K{\"{o}}ppel and W. Von{\ }Niessen", title = "A comparison of different approaches to the calculation of Franck-Condon factors for polyatomic molecules", year = "1977", journal = "Molec. Phys.", volume = "34", pages = "1759", abstract = "In the customary calculation of vibrational intensity distribution in electronic spectra of polyatomic molecules, the Franck-Condon factors are expressed in terms of the difference of the equilibrium geometries of the initial and the final state. It is more advantageous to express the Franck-Condon factors in terms of the gradient of the final state potential energy surface at the equilibrium geometry of the initial electronic state. The ionization and excitation of the NH/sub 3/ molecule are considered to illustrate this fact. Although the change in geometry is known accurately in this case, rather poor Franck-Condon factors are obtained with the customary harmonic method. The use of anharmonic potential energy functions does not improve the situation. The reason for this failure is discussed. Expanding the final state potential energy around the initial state equilibrium geometry, satisfactory Franck-Condon factors are obtained already in the lowest order of the expansion." } % ---------- Phenylpyrrole molecule @ARTICLE{Okuyama:1998:ESJ, author = "Katsuhiko Okuyama and Yasushi Numata and Shino Odawara and Isamu Suzuka", title = "Electronic spectra of jet-cooled 1-phenylpyrrole: Large-amplitude torsional motion and twisted intramolecular charge-transfer phenomenon", year = "1998", journal = "J. Chem. Phys.", volume = "109", pages = "7185", abstract = "The fluorescence excitation spectrum and the single vibronic level dispersed fluorescence spectra in the region of the S0 reversible arrow S-1 transition were measured for jet-cooled 1-phenylpyrrole. The 0-0 band was observed at 35 493 cm(-1). Long and low-frequency progressions with somewhat irregular intensity distributions appeared on both spectra, and were assigned to torsional motion. The torsional energy levels in the S-0 and S-1 states were obtained up to 25 and 16 quanta, respectively. The torsional potentials in both states could be determined from the sufficient number of energy levels observed. In the So state the most stable conformation was determined to be a twisted form with a dihedral angle of 38.7 degrees, where the planar barrier height was calculated to be 457 cm(-1), and the perpendicular to be 748 cm(-1). On the other hand, it was discovered that 1-phenylpyrrole in the S1 state also had a twisted form with a somewhat smaller dihedral angle of 19.8 degrees, and that the barrier to planarity was 105 cm(-1) and to perpendicularity, 1526 cm(-1). These facts indicated that the electronic excitation caused 1-phenylpyrrole to be rigid to twist. 1-Phenylpyrrole and its derivatives have been reported as a group of twisted intramolecular charge-transfer (TICT) molecules. No indication of TICT appeared on the shape of the S-1-state torsional potential determined. The relation between torsional potential and TICT is discussed based on the results of this study." } @ARTICLE{Manz:2002:TRD, author = "J{\"{o}}rn Manz and Boris Proppe and Burkhard Schmidt", title = "Time-resolved dual fluorescence of 1-phenylpyrrole in acetonitrile: molecular dynamics simulations of solvent response to twisted intramolecular charge transfer", year = "2002", journal = "Phys. Chem. Chem. Phys.", volume = "4", pages = "1876", abstract = "The real-time dynamics of solvation of 1-phenylpyrrole (PhPy) in acetonitrile (ACN) upon electronic excitation is investigated by means of non-equilibrium molecular dynamics simulations. The interaction is modeled by empirical intermolecular pair potentials using partial charges and intramolecular torsional potentials from high level ab initio calculations of ground and excited states of PhPy. The intramolecular torsional motion following sudden excitation from the twisted ground state to the 2 B-1 charge transfer state is strongly damped by the viscous ACN solvent, leading to a near-exponential approach of the perpendicular conformation on a timescale of about 5-10 ps. The intermolecular dynamics is characterized by rapid reorientation of the solvent molecules on a timescale of 100 fs followed by weak quasi-coherent librations. The solvatochromic red-shift of the charge transfer state with respect to the locally excited 1 B-1 state results in dual fluorescence, thus supporting the twisted intramolecular charge transfer (TICT) mechanism for PhPy in a polar solvent." } @ARTICLE{Proppe:2000:TST, author = "Boris Proppe and Manuela Merch{\'{a}}n and Luis Serrano-Andr{\'{e}}s", title = "Theoretical study of the twisted intramolecular charge transfer in 1-phenylpyrrole", year = "2000", journal = "J. Phys. Chem. A", volume = "104", pages = "1608", abstract = "Ab initio results for the electronic spectra of 1-phenylpyrrole are presented. Vertical, emission, and nonvertical excitation energies have been computed using multiconfigurational second-order perturbation theory by means of the CASPT2 method. In the S0 and S1 states the most stable conformation was determined to be a twisted structure with a dihedral angle between the planes of the rings of 41.4 and 25.5, respectively. The most intense feature of the absorption spectrum is predicted to be the S0 (11A) S1 (21A) transition placed in vacuo at 5.07 eV. The lowest-excited state, S1 (11B), identified as the ''locally excited'' (LE) state, is calculated in vacuo to lie vertically 4.39 eV above the ground state, with the position of the 0-0 transition at 4.16 eV and a fluorescence maximum of 4.07 eV. An additional fluorescence feature is found at 3.72 eV in acetonitrile from a ''twisted intramolecular charge transfer'' (TICT) state in a perpendicular conformation. The absence of the low-energy emission in vacuo is rationalized in terms of the different shapes of the S0 and S1 hypersurfaces. From the overall study it is concluded that the 1-phenylpyrrole system can be considered as an example where the TICT occurs." } @PHDTHESIS{Proppe:2001:MDF, author = "Boris Proppe", title = "Modelling the dual fluorescence of 1-phenylpyrrole due to torsion and electron transfer in polar solution", year = "2001", school = "Freie Universit{\"{a}}t Berlin", url = "http://darwin.inf.fu-berlin.de/2001/51/indexe.html", abstract = "The results of quantum chemical, quantum mechanical and classical molecular dynamics simulations of 1-phenylpyrrole are presented and compared to experimental data from the literature. In the first part ab initio results for the geometry of 1-phenylpyrrole are obtained. Vertical, emission, and non-vertical excitation energies have been computed using multiconfigurational second-order perturbation theory by means of the CASPT2 method. In the S0 and S1 states the most stable conformations were determined to be twisted structures with dihedral angles between the planes of the rings of 41.4oand 25.5o, respectively. The lowest-excited state, S1 (11B) is identified as the ``locally excited'' (LE) state; the charge transfer state (CT) S3 (21B) has a higher energy. Based on these ab initio points, parameters for analytic potential functions are fitted in the second part. Vibrational eigenfunctions are computed for these torsional potentials. The Franck-Condon Spectra calculated for the S0 (11A)--> S1(11B) transition compare well with the experimental fluorescence excitation spectrum with a fluorescence maximum of 4.07 eV in the gasphase. An additional fluorescence feature is found at 3.72 eV in acetonitrile from a ``twisted intramolecular charge transfer'' (TICT) state in a perpendicular conformation. In the third part the molecular potentials and partial charges derived from the ab initio calculation are used as parameters in a classical molecular dynamics simulation. The effect of a solvent (acetonitrile) on the torsion and on the stabilization of the charge transfer state is investigated. As a consequence of the large dipole of the CT state in comparison with LE state, the energy of the higher excited CT state in the gas phase drops below the energy of the LE state in the solvent, giving rise to near-degenerate dual fluorescence for both CT and LE states in good agreement with the experimental fluorescence spectrum. Different time scales are found for the relaxation with respect to the torsion and the reorientation of the solvent molecules after charge transfer. From the overall study it is concluded that the 1-phenylpyrrole system can be considered as an example where the TICT occurs in the solvent but not in the gas phase." } @ARTICLE{Fabian:1987:CBP, author = "W. Fabian", title = "Conformational behaviour of phenylpyrroles-a semiempirical molecular orbital study", year = "1987", journal = "Zeitschrift fur Naturforschung, Teil a (Physik, Physikalische Chemie, Kosmophysik)", volume = "42A", pages = "641", abstract = "Equilibrium twist angles and rotational barriers for the three isomeric phenylpyrroles were calculated by means of MNDO and AM1. In each case MNDO incorrectly predicts the perpendicular conformation as the most stable one. In agreement with experimental evidence AM1 predicts only slight deviations from planarity and very low 0 degrees -barriers (1-phenylpyrrole: theta =28 degrees , E=1.6 kJ mol/sup -1/; 2-phenylpyrrole: theta =27 degrees , E=0.8 kJ mol/sup -1/; 3-phenylpyrrole: theta =19 degrees , E=0.2 kJ mol/sup -1/). The reasons for the complete failure of MNDO are analyzed by partitioning the total energy into one- and two-center terms. The most significant improvement of AM1 over MNDO is found to be the much better description of repulsive interactions between nonbonded atoms. Possible further improvements of AM1 are briefly discussed." } @ARTICLE{Rettig:1987:PEC, author = "W. Rettig and F. Marschner", title = "Population of excited charge transfer states and molecular conformation in n-phenylpyrroles", year = "1987", journal = "Nouveau Journal de Chimie", volume = "7", pages = "425", abstract = "N-phenylpyrrole and N-(4-cyanophenyl)-pyrrole as well as several sterically hindered derivatives show a long wavelength fluorescence in polar solvents assigned to arise from a Twisted Intramolecular Charge Transfer (TICT) state. The ground state twist angles of these compounds are determined using photoelectron spectroscopy. The experimental values are in reasonable agreement with quantum chemical force field calculations and demonstrate the importance of the effect of broad angular distribution functions." } @UNPUBLISHED{Zilberg:2002:Aug07, author = "Shmuel Zilberg", title = "Four calculations", month = aug, year = 2002, note = "Files with the data sent by e-mail", abstract = "I send you 4 calculations: 1. HF/DZV result for PB in optimal geometry (PBgroundstate1AHFDZV) - torsion 43. 2. HF/DZV result for PB in perpendicular conformation, which is conformational Transition State ( PBgroundstatePerpConfHFDZV) 3. CIS/DZV result for the first excited state of PB in optimal geometry (PBexsitedstate1BCISDZV) - torsion 19.7 4. CIS/DZV result for the first excited state of PB in perpendicular geometry (PBexsitedstatePerpConf1B1CISDZV) Calculated frequencies in all cases must be correct by factor 0.91. " } @MANUAL{Ochterski:1999:VAG, author = "Joseph W. Ochterski", title = "Vibrational Analysis in Gaussian", organization = "Gaussian, Inc.", email = "help@gaussian.com", year = 1999, url = "http://www.lct.jussieu.fr/manuals/Programmes/Gaussian98/vib.htm", abstract = "One of the most commonly asked questions about Gaussian is ``What is the definition of reduced mass that Gaussian uses, and why is is different than what I calculate for diatomics by hand?'' The purpose of this document is to describe how Gaussian calculates the reduced mass, frequencies, force constants, and normal coordinates which are printed out at the end of a frequency calculation." }