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A globally uniform time-independent semiclassical wave function for nonadiabatic scattering is presented. This wave function, which takes the form of a surface-hopping expansion, is motivated by the globally uniform semiclassical wave function of Kay and co-workers for the single-surface case. The surface-hopping expansion is similar to a previously presented primitive semiclassical wave function for nonadiabatic problems. This earlier wave function has the important feature that it correctly incorporates all phase terms, allowing for an accurate treatment of quantum interference effects. The globally uniform expression has important numerical advantages over the primitive formulation. The globally uniform wave function does not have caustic singularities, and the globally uniform calculation avoids a root search for trajectories obeying double-ended boundary conditions that is required by the primitive semiclassical calculation.

A time independent semiclassical surface hopping model was previously derived for the numerical evaluation of the transition amplitudes for multisurface problems. This analysis is revisited and a new method incorporating phase information disregarded by the previous model is derived. The coordinate axis is partitioned into a finite number of intervals. The propagation of the wavefront across an interval is again described by a flux conserving transformation. For the one dimensional case transition amplitudes across many intervals can be evaluated via matrix multiplication of the individual interval transforms. Comparison with exact quantum mechanical calculations show that the results generated by this new model are highly accurate. Further this new approach offers a significant increase in computational efficiency. Accurate results can be obtained in calculations employing larger and hence fewer intervals to partition the system where the original model fails. This increase in efficiency has important implications for multidimensional surface hopping problems.

The role of the relative motions of protons on different chain segments in the relaxation of the dipolar coupling measured in nuclear magnetic resonance experiments is considered. The relaxation of the coupling for a single intersegmental proton–proton pair is evaluated numerically, assuming that the decay of this coupling is dominated by the translational motion of the monomers. It is found that the relaxation of this coupling scales as gr(t)−5/2, where gr(t) is the mean squared relative displacement of two monomers in the melt. This scaling applies for times at which gr(t) is greater than the square of the initial separation of the two coupled protons. The effect of the short range equilibrium structure of the melt on the intersegmental relaxation is qualitatively considered by separating out, from the summation over all interactions, the interactions between the protons corresponding to the first intersegmental peak in the hydrogen–hydrogen radial distribution function. This analysis indicates that the...

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.

The relationship between the diverging of classical trajectories in chaotic many-body systems, the spreading of quantum wave packets, and the validity and use of classical molecular dynamics is explored. This analysis, which is based on the semiclassical description of wave function propagation in terms of a weighted integration over a traveling fixed width coherent state basis, suggests that the exponential divergence of nearby classical trajectories in chaotic many-body systems should result in the rapid delocalization of an initially localized quantum wave packet describing the state of the system. Thus the justification for the use of classical molecular dynamics procedures for these supposedly classical systems cannot be based on the picture of the system wave function remaining localized as its center follows a nearly classical trajectory. The quantum evolution of the system density, on the other hand, requires two propagators, and each of these propagators is represented as an integration over traj...

An improved treatment of the coupling between the lateral and reptative motions in the lateral motion model of polymer melt dynamics is presented. Approximate analytical formulas are also provided for the monomer and center of mass mean squared displacements and for the diffusion constant. Scalings are given for all predicted regions, and expressions are presented for the corresponding crossover times and chain lengths. Results from the full calculations and the analytical expressions are compared with published data from simulations and experimental systems. The theoretical predictions are in excellent agreement with the simulation and experimental data. The results also demonstrate that the approximation analytical formulas agree extremely well with the full calculations. Given the simplicity and accuracy of these analytical expressions, they should facilitate the implementation of the theory and its comparison to experimental data.

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.

Results are presented for the calculated probability for resonant transfer of vibrational excitation energy in clusters of identical molecules. The calculations are performed for two-dimensional clusters in order to allow for calculations on larger clusters. Clusters of 5, 10, and 20 molecules are considered. The probability of resonant transfer is calculated by quantum mechanically propagating the wave function for the vibrational degrees of freedom of the molecules in the cluster, while the rotational and translational degrees of freedom evolve along classical trajectories. The transition probabilities are averaged over a canonical distribution of initial phase space points for the trajectories. If the probability for the transition of a vibrational quantum of energy from one molecule to some other molecule in an N molecule cluster is evaluated as N−1 independent two molecule vibrational problems, the results are found to be in excellent agreement with those obtained from the propagation of the full N m...

The results of a recently presented theory for polymer dynamics are compared with experimental diffusion constant data for polyethylene, poly (ethylene oxide), and poly (dimethyl siloxane) polymer melts. The molecular weight dependence of the self-diffusion constants for these systems is investigated. The various constants that appear in the theory are optimized using theoretical analysis and published results from computer simulations. The only adjustable parameters for each system are the friction coefficients for lateral and reptative motions. It is found that the same ratio of the monomer friction coefficients for lateral and reptative motions can be employed for all three experimental systems and for the two computer simulation systems considered. This is reasonable, since all these systems have fairly simple monomer units without large side groups. This leaves the monomer friction coefficient for lateral motions as the only system-dependent adjustable parameter in these calculations. Excellent agree...

The problem of resonant vibration–vibration (V–V) energy transfer in liquids is explored within a simple model in order to compare two calculational methods. Two bromine diatomics are constrained to move between two fixed argon solvent atoms in one dimension. The time-dependent probability for the transfer of a vibrational excitation between the bromine diatomics is computed semiclassically. The results of single-trajectory molecular dynamics methods are compared with those of multiple-trajectory surface hopping methods. It is found that the two methods give similar results, indicating that the simpler single-trajectory method adequately describes the resonant V to V transfer process. The proximity of the nearly degenerate potential energy surfaces leads to a phase coherence time that exceeds the time required for the transition probability to saturate for this model. As a result, the transition probability remains a nonlinear function of time, and this precludes the extraction of a rate constant from the slope of the resonant V–V transition probability curve for this simple one-dimensional model.

Estimates are evaluated for all constants in the theory of polymer dynamics developed in the preceding paper. These theoretical estimates are employed in calculations on the many chain systems simulated previously by Kremer and Grest and by Paul and co‐workers. The agreement between the theoretical calculations and the simulations of Kremer and Grest is found to be very good for the bead mean squared displacement. The theoretically evaluated center of mass mean squared displacement is in good qualitative agreement with the Kremer–Grest simulations, although there are quantitative differences. Good qualitative agreement with the simulations of Paul and co‐workers is also obtained. It is shown that quantitative agreement can be obtained with modest changes in the theoretical estimates of a few of the constants in the theory.

A theory of melt polymer dynamics for linear chain systems is developed. This theory generalizes recent work, which considers the lateral motion of the chains. A description is provided of the short time dynamics and of the crossover from this early time regime to a highly entangled dynamics. In both of these regimes, an effective friction coefficient for the lateral motion is evaluated by considering the extent of correlation between the displacements of the beads. This correlation is required due to the chain connectivity and the noncrossability of the chain backbones. The crossover time between these two regimes is found to be independent of chain length. In the early time regime, the bead mean squared displacement is found to have a time dependence between g∼t0.4 and g∼t0.5. In the highly entangled regime, g has a t2/7 dependence. The reptative motion of the chains along their own backbones and the coupling between this motion and the lateral chain motion is also included. It is found that the inclusi...

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 ℏ. 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.

A variation of velocity‐aligned Doppler spectroscopy is used to study F=0 and F=1 nuclear hyperfine populations in the ground electronic state of atomic hydrogen as the result of a chemical reaction. In this case, the reaction is the 193 nm photolysis of HBr. Not only can the nuclear hyperfine states be studied, but the method allows unequivocal correlation with the spin–orbit states of the bromine atom, i.e., Br(2P1/2) and Br(2P3/2). For H atoms correlated with Br(2P3/2), the F=0/F=1 is 1.2±0.2. Future directions involving the study of this fundamental chemical product state distribution are also discussed.

A theory of correlated many chain motions is developed for linear chain polymer melts. The center of mass diffusion constant of the chains is shown to be inversely proportional to the number of monomers that have their motion correlated with the motion of any given monomer. This quantity is expressed as the integral over the normalized correlation function for the displacements of different monomers in the melt. Two limiting cases are considered for this correlation function in monodisperse melts. The loss of correlation in the motion of monomers occurs because of the relative motion of the chains. The first limiting case assumes that the relative chain motions are correlated over the entire range of the correlated many chain motion. This case leads to the largest single contribution to the diffusion constant in the long chain limit. The second limiting case employs the shortest possible correlation length for the relative displacements of the chains. Since this case allows for the greatest randomness in the motion of the chains, it is expected that the contribution from this type of correlated motion should dominate the chain diffusion constant. The diffusion constant based on the correlated motions in the melt is consistent with the experimentally observed chain length scaling of both the terminal time and the chain diffusion constant. The case of the diffusion of a single probe chain in a melt of host chains of different length is also discussed. 1995 American Institute of Physics.

A model for stress relaxation of linear chain polymer melts in the linear viscoelastic regime is proposed. This model is developed in the context of the lateral motion model of polymer dynamics, which assumes that polymers diffuse in the melt by sliding laterally along the curvilinear contours of neighboring chains. It is assumed that, in the post‐plateau region of the time dependent stress relaxation, the stress is supported by interchain contacts. When each contact breaks, a new contact is formed, since the number of interchain contacts is an equilibrium property. It is also assumed in this model that the new contacts, formed after the application of the strain, support partial stress because the polymer system is not isotropic at the time of the formation of these new contacts. The stress relaxation is evaluated within this model for monodisperse and bidisperse melts using a computer simulation procedure. The results are compared with published experimental data. Very good agreement is found with the e...

The time independent semiclassical treatment of barrier tunneling has been understood for a very long time. Several semiclassical approaches to time dependent tunneling through barriers have also been presented. These typically involve trajectories for which the position variable is a complex function of time. In this paper, a method is presented that uses only real valued trajectories, thus avoiding the complications that can arise when complex trajectories are employed. This is accomplished by expressing the time dependent wave packet as an integration over momentum. The action function in the exponent in this expression is expanded to second order in the momentum. The expansion is around the momentum, p0(*) , at which the derivative of the real part of the action is zero. The resulting Gaussian integral is then taken. The stationary phase approximation requires that the derivative of the full action is zero at the expansion point, and this leads to a complex initial momentum and complex tunneling trajectories. The "pseudo-stationary phase" approximation employed in this work results in real values for the initial momentum and real valued trajectories. The transmission probabilities obtained are found to be in good agreement with exact quantum results.

We provide an in-depth investigation of transmission coefficients as computed using the augmented-fewest switches surface hopping algorithm in the low energy regime. Empirically, microscopic reversibility is shown to hold approximately. Furthermore, we show that, in some circumstances, including decoherence on top of surface hopping calculations can help recover (as opposed to destroy) oscillations in the transmission coefficient as a function of energy; these oscillations can be studied analytically with semiclassical scattering theory. Finally, in the spirit of transition state theory, we also show that transmission coefficients can be calculated rather accurately starting from the curve crossing point and running trajectories forwards and backwards.

An analysis is provided of the statistical errors in the Monte Carlo evaluation of the conditioned real time discretized path integral propagator. The analysis considers the case of a harmonic potential. For this case, all the required integrals can be performed analytically. This analysis is also relevant to a semiclassical evaluation of the integrals in more general problems. It is found (in the simplest case) that the optimal relative statistical error per independent sampling is proportional to DD/2, where D is the dimensionality of the integrand. Therefore, the number of Monte Carlo samplings must scale as DD in order to achieve a desired level of accuracy. Since D is proportional to the number of time steps in the discretized path integral, this analysis demonstrates that the length of the calculations required increases very rapidly as the number of time steps is increased.

A new approximate technique for real time discretized path integral simulations is introduced. The technique transforms the oscillatory integrands into nonoscillatory functions for accurate Metropolis Monte Carlo evaluation. The method is applied to a system containing a Br2 diatomic in two Ar atoms. The vibrational transition probability of finding the diatomic in its ground vibrational state is calculated at time t. The diatomic is initially in its first vibrational state. Computations are carried out for two different times and the results are compared to the previously published results of full discretized path integral calculation. The convergence of the technique is tested as a function of the number of points in the discretized solvent path.

A discretized real‐time Feynman path integral is applied to a system containing a diatomic in a small cluster of solvent molecules. The system considered consists of Br2 in Ar. An adiabatic separation of variables is assumed. The solvent as well as the Br2 center of mass are gathered into a vector called bath or solvent coordinates. The forced oscillator approximation is used to analytically obtain the vibrational contribution to the transition amplitude. The discretized real‐time propagators (bath dependent only, since the vibrational part is carried out analytically) are highly oscillatory and, therefore, not suitable for Monte Carlo calculations. The coarse‐graining technique introduced by Filinov and developed by Freeman and Doll, and Miller is employed to make the integrals more suitable for Monte Carlo calculations. The computations are carried out for five different times. For each time, we study the convergence of the technique for a range of Gaussian widths used as conditioning functions. We also examine the convergence as a function of the number of points in the discretized description of the solvent path.

A model for the motion of a dense multichain polymer system involving linear chains is presented. The model is based on the assumption that chain motion perpendicular to the direction of the chain backbone dominates the dynamics over motion along the backbone. This model is motivated by the results of recent Monte Carlo computer simulations on multichain systems for chain lengths ranging up to 800 segments. It is found in these simulations that motion perpendicular to the backbone dominates. Since many chains are highly intertwined with any given chain, we assume that the chain motion at an interchain contact point proceeds by sliding along the contour of each entangling chain until the exterior of the ball of density formed by the entangling chain is reached. At this point the contact is broken. This requirement that a chain move by sliding along the random flight contour of another chain at a contact point results in an effective bead friction coefficient for three‐dimensional motion which is an increas...

A version of a surface hopping wave function for nonadiabatic multistate problems, which is free of turning point singularities, is derived and tested. The primitive semiclassical form of the particular surface hopping method considered has been shown to be highly accurate, even for classically forbidden processes. However, this semiclassical wave function displays the usual singular behavior at turning points and caustics in the classical motion. Numerical data has shown that this somewhat reduces its accuracy when the energy is near the crossing energy of the diabatic electronic surfaces. The singularity free version of this surface hopping wave function is derived by partitioning the x-axis into a large number of small steps for one dimensional problems. The adiabatic electronic energy surfaces are approximated to be linear functions within each step. The matching conditions required by the continuity of the wave function and its derivative at each step boundary provide the needed conditions to obtain the amplitudes for changes in electronic state and/or reflection of the trajectory for the motion of the nuclei. This leads to a form of the surface hopping wave function that is free of turning point singularities. The method is tested for a one dimensional model problem, and it is found to be highly accurate at all energies considered, even when the energy is near the crossing energy.

It is shown that a surface hopping expansion of the semiclassical wave function formally satisfies the time independent Schrodinger equation for many-state, multidimensional problems. This wave function includes terms involving hops between different adiabatic quantum states as well as momentum changes without change of state at each point along classical trajectories. The single-state momentum changes correct for the order ℏ2 errors due to the semiclassical approximation that are present even in single surface problems. A prescription is provided for the direction of this momentum change and the amplitude associated with it. The direction of the momentum change for energy conserving hops between adiabatic states is required to be in the direction of the nonadiabatic coupling vector. The magnitude of the posthop momentum in this direction is determined by the energy, but the sign is not. Hops with both signs of this momentum component are required in order for the wave function to formally satisfy the Sch...

A previously developed nonadiabatic semiclassical surface hopping propagator [M. F. Herman J. Chem. Phys. 103, 8081 (1995)] is further studied. The propagator has been shown to satisfy the time-dependent Schrodinger equation (TDSE) through order h, and the O(h2) terms are treated as small errors, consistent with standard semiclassical analysis. Energy is conserved at each hopping point and the change in momentum accompanying each hop is parallel to the direction of the nonadiabatic coupling vector resulting in both transmission and reflection types of hops. Quantum mechanical analysis and numerical calculations presented in this paper show that the h2 terms involving the interstate coupling functions have significant effects on the quantum transition probabilities. Motivated by these data, the h2 terms are analyzed for the nonadiabatic semiclassical propagator. It is shown that the propagator can satisfy the TDSE for multidimensional systems by including another type of nonclassical trajectories that reflect on the same surfaces. This h2 analysis gives three conditions for these three types of trajectories so that their coefficients are uniquely determined. Besides the nonadiabatic semiclassical propagator, a numerically useful quantum propagator in the adiabatic representation is developed to describe nonadiabatic transitions.

The globally uniform semiclassical wave function expresses the solution to the time independent Schrodinger equation in terms of fixed width Gaussian wave packets traveling along a set of trajectories. There is a globally uniform wave function (GUWF) for each value of the Gaussian width parameter gamma. Numerical data show that a small Gaussian width is needed in some regions to obtain accurate results, while a broad Gaussian width provides better results in other regions. Since there is a semiclassically valid GUWF for every positive value of gamma, it is reasonable to employ the GUWF corresponding to a Gaussian width that provides good results at each value of r. A criterion for the r dependent choice of gamma is proposed and tested on one and two dimensional model problems. The results show that the use of an r dependent gamma in the GUWF results in improved accuracy for the model problems considered.

The nonadiabatic surface hopping Herman-Kluk (HK) semiclassical initial value representation (SC-IVR) method for nonadiabatic problems is reformulated. The method has the same spirit as Tully's surface hopping technique [J. Chem. Phys. 93, 1061 (1990)] and almost keeps the same structure as the original single-surface HK SC-IVR method except that trajectories can hop to other surfaces according to the hopping probabilities and phases, which can be easily integrated along the paths. The method is based on a rather general nonadiabatic semiclassical surface hopping theory developed by Herman [J. Chem. Phys. 103, 8081 (1995)], which has been shown to be accurate to the first order in h and through all the orders of the nonadiabatic coupling amplitude. Our simulation studies on the three model systems suggested by Tully demonstrate that this method is practical and capable of describing nonadiabatic quantum dynamics for various coupling situations in very good agreement with benchmark calculations.

A rigorous and practical approach for simulations of nonadiabatic quantum dynamics is introduced. The algorithm involves a natural extension of the matching-pursuit∕split-operator Fourier-transform (MP∕SOFT) method [Y. Wu and V. S. Batista, J. Chem. Phys. 121, 1676 (2004)] recently developed for simulations of adiabatic quantum dynamics in multidimensional systems. The MP∕SOFT propagation scheme, extended to nonadiabatic dynamics, recursively applies the time-evolution operator as defined by the standard perturbation expansion to first-, or second-order, accuracy. The expansion is implemented in dynamically adaptive coherent-state representations, generated by an approach that combines the matching-pursuit algorithm with a gradient-based optimization method. The accuracy and efficiency of the resulting propagation method are demonstrated as applied to the canonical model systems introduced by Tully for testing simulations of dual curve-crossing nonadiabatic dynamics.

A one-dimensional, two-state model problem with two well-separated avoided crossing points is employed to test the efficiency and accuracy of a semiclassical surface hopping technique. The use of a one-dimensional model allows for the accurate numerical evaluation of both fully quantum-mechanical and semiclassical transition probabilities. The calculations demonstrate that the surface hopping procedure employed accounts for the interference between different hopping trajectories very well and provides highly accurate transition probabilities. It is, in general, not computationally feasible to completely sum over all hopping trajectories in the semiclassical calculations for multidimensional problems. In this case, a Monte Carlo procedure for selecting important trajectories can be employed. However, the cancellation due to the different phases associated with different trajectories limits the accuracy and efficiency of the Monte Carlo procedure. Various approaches for improving the accuracy and efficiency of Monte Carlo surface hopping procedures are investigated. These methods are found to significantly reduce the statistical sampling errors in the calculations, thereby increasing the accuracy of the transition probabilities obtained with a fixed number of trajectories sampled.

The equation‐of‐motion–Green’s function method for calculating ionization potentials is analyzed within the framework of a linear matrix eigenvalue representation, and an extended form of the theory is developed. The utility of the modifications presented in this paper is strongly suggested by recent numerical studies which successfully employ a generalized definition of the primary operator space in analogy with configuration selection procedures that have proven useful in configuration interaction calculations. The basic theoretical questions are associated with the choice of the basis operators for the primary space and the approximations to be employed in the evaluation of the individual matrix elements. This extended form of the theory incorporates the lowest order effects of ground state correlation on matrix elements between the shakeup basis operators in the primary operator space. A first approximation to the contributions of basis operators involving ionization and double excitation or ionization and double de‐excitation is incorporated. These terms can contribute in second order to the generalized EOM primary matrix. The possible importance of yet higher order contributions are analyzed in light of the modified primary space. The effect of these generalizations of the theory are studied numerically in the following paper and comparison is made with accurate configuration interaction results on the same systems using identical basis sets.

Solvent effects on vibrational states are treated in the Born–Oppenheimer approximation. A Monte Carlo procedure is devised for simulating a system in which the vibrations are treated as quantum degrees of freedom and all other degrees of freedom are treated classically. The key to this method is the use of a restructured perturbation theory for solving the vibrational Schrodinger equation at each of the solvent configurations sampled. This method also is used to compute Franck–Condon factors. The methods developed are applied to two different problems. First, solvent effects on vibrational transitions in the ground electronic state are considered. It is found that the frequency shift from the gas phase frequency is to the red at low solvent densities, but to blue at high solvent densities. Secondly, the heterogeneous electronic absorption and emission line shapes are simulated in a variety of model solute–solvent systems and the results are discussed in terms of a simple theoretical model.

A detailed comparison is presented between calculated equation‐of‐motion (EOM) ionization potentials and electron affinities and highly converged configuration interaction (Cl) results for a variety of atomic and molecular systems. Since an exact EOM calculation and a full Cl calculation within the same orbital basis set must yield identical results, this type of study allows for the separation of errors due to the approximations employed in solving the EOM equations from those errors arising from the use of an incomplete orbital basis set. The convergence of the EOM calculations at different levels of approximation is also investigated for these same systems. Important EOM basis operators, involving ionization and excitation (shakeup operators) or ionization and de‐excitation, are numerically identified by configuration selection routines and are diagonalized rather than treated perturbatively. Terms involving second order couplings (arising from ground state correlation) between these shakeup states are calculated, as are the lowest order contributions from ionization and double excitation and from ionization and double de‐excitation operators. These terms, which are normally neglected in EOM calculations, are found to be significant in certain cases. The agreement between the most sophisticated EOM and CI results is excellent for the molecular systems studied. The results justify the need for retention of higher order couplings, addition of double shakeup operators, and the adoption of a nonperturbative approach as described in paper I. The agreement is not as good for the atomic systems and possible reasons for these discrepancies are discussed.

A Monte Carlo procedure based on a discrete point representation of the path integral for the density matrix is explored. It is found that the variance of the estimator used to evaluate the energy grows as the square root of the number of discrete points used, and is therefore to be avoided in highly quantum mechanical systems, where the number of discrete points must be large. A new energy estimator based on the virial theorem is proposed and shown to be well behaved. The main points of the paper are illustrated, using the harmonic oscillator as an example.

A time dependent surface hopping formalism is developed for nonadiabatic scattering. Standard semiclasical propagation on single adiabatic surfaces constitutes the zeroth order approximation. Higher order terms include reflections and/or transitions between the adiabatic surfaces with zeroth order propagation occurring between the times when these nonclassical reflections and/or transitions occur. In one dimension this expansion formally satisfies the exact quantum mechanical Schrodinger equation if all terms are retained. In many dimensional problems, the analysis is restricted to the special case involving two adiabatic surfaces and terms which are identified as corrections to the single surface semiclassical propagation are ignored. The resulting nonadiabatic expansion is inherently semiclassical, as opposed to the formally exact one‐dimensional case. This time dependent surface hopping formalism is utilized to derive a nonadiabatic generalization of the frozen Gaussian approximation.

Three recently proposed semiclassical methods for nonadiabatic scattering are numerically compared for a one‐dimensional curve crossing model. The generalized surface hopping approach has been found to be in excellent agreement with quantum results over a wide range of energies. The simpler variant of the self‐consistent eikonal method follows the quantum results well at low energies. At higher energies it still follows the correct overall trend of decreasing transition probability, but it does not quantitatively reproduce the oscillations in the transition probability. The classical analog technique provides relatively good agreement with the quantum results over the entire range of energies examined. It is found in the application of this last method that there are typically four to eight stationary phase contributions at each energy and it is crucial to correctly describe the interference between these terms. It is also necessary to include uniform and analytically continued contributions at many energ...

The generalized surface‐hopping procedure for semiclassical nonadiabatic scattering is applied to a two dimensional model for comparison with quantum results. The model consists of two adiabatic electronic potential energy surfaces with a radial nonadiabatic coupling between the electronic states. Both adiabatic potential surfaces are taken to be constant. The coupling is assumed to be weak. This model is sufficient to explore many of the important features of the generalized surface‐hopping procedure. Comparison of the semiclasssical and quantum results demonstrates good agreement, and the first order quantum scattering amplitude is analytically found to be the ℏ→0 limit of the first order semiclassical scattering amplitude. The behavior of the wave function near singularities in the transition amplitudes is explored in a general context, independent of these simplified models. These singularities are found to be necessary to produce the correct behavior in the wave function.

A local rate is derived for the vibrational population relaxation of a diatomic in a simple liquid or dense gas. The total relaxation rate of the system is obtained as the canonical ensemble average of the local rate. The rate expression is amenable to computer simulations in which the canonical average is performed by a Monte Carlo procedure. The vibrational motion is separated from the other degrees of freedom by an adiabatic approximation which treats the vibration as fast compared with the other motions. The adiabatic vibrational energies and the nonadiabatic couplings between vibrational states depend on the solvent configuration. These vibrational energies and couplings are obtained from quantum perturbation theory. The transitions between vibrational states are described semiclassically and the canonical averaging uses the classical canonical density for solvent configurations. The resulting procedure is a mixed quantum–semiclassical–classical simulation technique.

Results are presented from a computer simulation of the population relaxation of the lowest (1→0) vibrational transition for a system of Br2 in a dense Ar fluid at 300 K. The calculate relaxation time is 253 ps. The method of calculation is a mixed quantum–semiclassical–classical simulation procedure. The vibrational state energies and wave functions are obtained from perturbation theory for fixed values of the rotational and translational variables. The relaxation rate for the vibrational transition is evaluated using a semiclassical surface hopping theory of nonadiabatic processes. The configurations of rotational and vibrational variables are sampled from a classical canonical ensemble density using standard Monte Carlo sampling. The relative efficiency of the rotations and translations in promoting the vibrational population relaxation is examined, and the use of perturbation theory and some assumptions of the model are tested numerically.

A procedure for the evaluation of one‐dimensional uniform semiclassical wave functions is explored. This form of the approximate wave function is tested for the case of an exponential potential and is compared to the exact quantum mechanical and WKB wave functions. It is found that the uniform approximation provides a very accurate representation of the exact wave function, even at the turning points of the classical motion, where the WKB approximation diverges. It is demonstrated how correction terms can be constructed, leading to a formal expansion for the exact wave function. The individual terms of this expansion each involve a finite number of reflection points with uniform semiclassical propagation between these reflections. The analysis is extended to multisurface (nonadiabatic) problems, removing the turning point singularities which are generally present in semiclassical formulations of nonadiabatic problems.

We have developed an improvement formalism for carrying out electronic structure calculations for surfaces, interfaces, and organic solids using muffin‐tin‐orbital (MTO) basis sets. We will first show how the variational flexibility of MTO tail functions can be improved by introducing generalized MTO tail functions composed of modified spherical Bessel functions Kl(κ, r) and their derivatives with respect to κ. Next, we will show how energy dependent MTOs can be constructed by using double basis sets, the first partners being keyed to the radial solutions of the wave equation inside the MT spheres, and the second partners to the energy derivatives of these radial solutions. The energy dependence of these double MTO basis sets introduces additional variational flexibility. Since the first partners will normally be considerably larger than the second partners, the double‐dimension secular equations can be solved economically by using the Lowdin partitioning scheme. Finally, we will indicate how the calculat...

The geometric optical series approach of Bremmer is generalized for multisurface nonadiabatic scattering problems. This method yields the formal solution of the Schrodinger equation as an infinite series of multiple integrals. The zeroth order term corresponds to WKB propagation on a single adiabatic surface, while the general Nth order term involves N reflections and/or transitions between surfaces accompanied by ’’free,’’ single surface semiclassical propagation between the points of reflection and transition. Each term is integrated over all possible transition and reflection points. The adiabatic and diabatic limits of this expression are discussed. Numerical results, in which all reflections are ignored, are presented for curve crossing and noncrossing problems. These results are compared to exact quantum results and are shown to be highly accurate.

The self‐consistent matrix propagator method of Laing and Freed is extended to treat semiclassical nonadiabatic scattering in the collinear atom–diatom system. Applications are made to a model system in which diabatic surfaces are parallel, so the nonadiabatic transitions are not well localized in space, thereby introducing difficulties in some previous nonadiabatic semiclassical methods. In the self‐consistent matrix propagator method nonadiabatic transitions occur at the boundaries of Magnus regions, and the relative phases, associated with trajectories undergoing transitions at different boundaries, must accurately be determined. This necessitates the determination of the absolute phases of the uniformized classical S matrix, a phase which is unnecessary in single potential surface semiclassical scattering. Semiclassical calculations are compared with full close coupled quantum calculations of Schmalz. The agreement is very good even at relatively low energies. The largest errors enter, as anticipated, for highly classically forbidden transitions whose overall probabilities are, however, rather small. The self‐consistent matrix propagator method becomes simpler to apply and more accurate as the total energy increases, i.e., as the fully quantum calculations become prohibitively large. The method has the physical appeal that the self‐consistent trajectories follow essentially adiabatic surfaces in strongly interacting regions and diabatic surfaces in weakly interacting regions, with a self‐consistent interpolation between these regions.The self‐consistent matrix propagator method of Laing and Freed is extended to treat semiclassical nonadiabatic scattering in the collinear atom–diatom system. Applications are made to a model system in which diabatic surfaces are parallel, so the nonadiabatic transitions are not well localized in space, thereby introducing difficulties in some previous nonadiabatic semiclassical methods. In the self‐consistent matrix propagator method nonadiabatic transitions occur at the boundaries of Magnus regions, and the relative phases, associated with trajectories undergoing transitions at different boundaries, must accurately be determined. This necessitates the determination of the absolute phases of the uniformized classical S matrix, a phase which is unnecessary in single potential surface semiclassical scattering. Semiclassical calculations are compared with full close coupled quantum calculations of Schmalz. The agreement is very good even at relatively low energies. The largest errors enter, as anticipated,...

The time reversal and unitarity properties of the semiclassical frozen Gaussian approximation for the propagation of time dependent wave functions are explored. It is found that propagation backwards in time is the adjoint operation to propagation forward in time. This basic property of exact quantum propagation holds exactly in the frozen Gaussian approximation. Unitarity is also demonstrated for frozen Gaussian propagation within the stationary phase approximation. This latter result holds only as long as a prefactor is included in the frozen Gaussian approximation. This point is significant since this prefactor is neglected in many frozen Gaussian variations, and its inclusion can require additional computational effort. The time reversal and unitarity results can be combined to prove the very useful, but nonobvious result that time reversed propagation is the inverse of propagation forward in time in the frozen Gaussian approximation, under the same conditions for which the unitarity property holds.

The effective valence shell Hamiltonian, Hv, which acts within a finite valence space and exactly describes all the valence state energies, is applied to several atomic systems. The n=2 (L shell) Hv of the first row atoms, N, O, and F and n=3 (M shell) s and p orbital Hv of the second row atoms, Si, P, and S, are evaluated through second order using STO 5s4p2d and 6s5p3d basis sets, respectively. The calculations are equivalent to a (perturbative) Bk approximation which incorporates all excited configurations and which chooses the primary (valence) space as all the valence K2(2s)m(2p)n and K2L6(3s)m(3p)n configurations, respectively. Using the calculated matrix elements of Hv, the energies of all the valence states of the neutrals and ions are simultaneously determined from a single ab initio calculation on only one charge state of each of these atomic systems. To understand the dependence of Hv on the choice of core and valence orbitals, several sets of orbitals, obtained within the same primitive orbita...

The propagation of an initially highly excited localized wave packet in an anharmonic oscillator potential is studied within the frozen Gaussian approximation. Comparison is made to quantum mechanical basis set calculations. The frozen Gaussian approximation involves the expansion of the initial wave function in terms of an overcomplete Gaussian basis set. The wave function evolution is evaluated by allowing each Gaussian to travel along a classical trajectory with its shape held rigid. A Monte Carlo algorithm is employed in the selection of the initial Gaussian basis functions. The frozen Gaussian results are very good for times on the order of a few vibrational periods of the oscillator and remain qualitatively correct for the entire length of the calculations which is 12 vibrational periods. The dependence of the calculations on the width of the Gaussian basis functions is investigated and the effect of a simplifying approximation for the prefactor of the Gaussians is tested.