Classical and quantum mechanical infrared echoes from resonantly coupled molecular vibrations

The nonlinear response function associated with the infrared vibrational echo is calculated for a quantum mechanical model of resonantly coupled, anharmonic oscillators at zero temperature. The classical mechanical response function is determined from the quantum response function by setting variant Planck's over 2pi-->0, permitting the comparison of the effects of resonant vibrational coupling among an arbitrary number of anharmonic oscillators on quantum and classical vibrational echoes. The quantum response function displays a time dependence that reflects both anharmonicity and resonant coupling, while the classical response function depends on anharmonicity only through a time-independent amplitude, and shows a time dependence controlled only by the resonant coupling. In addition, the classical response function grows without bound in time, a phenomenon associated with the nonlinearity of classical mechanics, and absent in quantum mechanics. This unbounded growth was previously identified in the response function for a system without resonant vibrational energy transfer, and is observed to persist in the presence of resonant coupling among vibrations. Quantitative agreement between classical and quantum response functions is limited to a time scale of duration inversely proportional to the anharmonicity.     

Interpreting nonlinear vibrational spectroscopy with the classical mechanical analogs of double-sided Feynman diagrams

Observables in coherent, multiple-pulse infrared spectroscopy may be computed from a vibrational nonlinear response function. This response function is conventionally calculated quantum-mechanically, but the challenges in applying quantum mechanics to large, anharmonic systems motivate the examination of classical mechanical vibrational nonlinear response functions. We present an approximate formulation of the classical mechanical third-order vibrational response function for an anharmonic solute oscillator interacting with a harmonic solvent, which establishes a clear connection between classical and quantum mechanical treatments. This formalism permits the identification of the classical mechanical analog of the pure dephasing of a quantum mechanical degree of freedom, and suggests the construction of classical mechanical analogs of the double-sided Feynman diagrams of quantum mechanics, which are widely applied to nonlinear spectroscopy. Application of a rotating wave approximation permits the analytic extraction of signals obeying particular spatial phase matching conditions from a classical-mechanical response function. Calculations of the third-order response function for an anharmonic oscillator coupled to a harmonic solvent are compared to numerically correct classical mechanical results.     

Interference and quantization in semiclassical response functions

Application of the Herman-Kluk semiclassical propagator to the calculation of spectroscopic response functions for anharmonic oscillators has demonstrated the quantitative accuracy of these approximate dynamics. In this approach, spectroscopic response functions are expressed as multiple phase-space integrals over pairs of classical trajectories and their associated stability matrices. Here we analyze the Herman-Kluk semiclassical approximation to a linear response function and determine the origin of the capacity of this method to reproduce quantum effects in a response function from classical dynamical information. Our analysis identifies those classical trajectories that contribute most significantly to the response function on different time scales. This finding motivates a procedure for computing the linear response function in which the interference between pairs of classical trajectories is treated approximately, resulting in an integral over a single average trajectory, as in a purely classical calculation.     

Semiclassical eigenvalues for a non-adiabatic system

We have performed a semiclassical quantization of the Meyer-Miller classical analog Hamiltonian for the Jahn-Teller E × e system, in the regime of small linear coupling. The primitive semiclassical energies are in good qualitative and reasonable quantitative agreement with the exact quantum-mechanical solution to this problem. These results demonstrate that such a classical model is able to accurately describe the behavior of non-adiabatic bound state systems, but that uniformization will be necessary to obtain more precise agreement with quantum mechanics.     

Critical comparison of classical and quantum mechanical treatments of the phase equilibria of water

The Gibbs ensemble Monte Carlo simulation technique was used to compare the phase equilibria of the rigid TIP4P water model [Jorgensen et al., J. Chem. Phys. 79, 926 (1983)] utilizing classical and quantum statistical mechanics. The quantum statistical mechanical treatment generally resulted in lower liquid densities and higher vapor densities, narrowing the phase envelope. As a result, the calculated critical temperatures and normal boiling points were lower from the quantum simulations than the classical by 22 and 17 K, respectively, but the critical densities were equal within the estimated uncertainties. When the phase diagram from the quantum statistical mechanical treatment was increased by 22 K, it agreed with the classical results quite well throughout the entire simulated temperature range. A semiclassical treatment, involving a low order expansion in Planck's constant, resulted in good agreement with the path integral results for second virial coefficients, but gave densities and vapor pressures that fluctuated between the values for the classical and quantum statistical mechanics values, with no definite agreement with either.     

Equilibrium atomic structure: Rotating atoms

Exact rigid body solutions for two-electron atoms have been derived within classical mechanics. These solutions describe equilibrium constellations of the electrons, i.e. the electrons do not experience any force within these constellations. Moreover it is shown that exactly two nondegenerate equilibria exist. This new type of strong correlation generates for high total angular momentum a rotation spectrum converging to the threshold of double ionisation. Level positions obtained by semiclassical quantization should be observable for instance in multiphoton absorption spectra.     

Symmetry-adapted correlation function for semiclassical quantization

We study a very simple method to incorporate quantum-mechanical symmetries, including the permutational symmetry on an equal footing with spatial symmetries, into the semiclassical calculation of correlation functions. This method is applied to the calculation of energy spectra to verify its validity by reproducing quantum energy levels for systems of bosons (symmetrized) and fermions (antisymmetrized). The mechanism of how the phase-space structure of classical dynamics is linked with the relevant quantum symmetry is discussed.     

Ambiguities in the semiclassical assignment of the asymmetric rotor rotational quantum numbers

The semiclassical quantization of the rigid asymmetric rotor is revisited in the context of classical inelastic collisions. It is shown that the standard bin histogram method, widely used in quasiclassical trajectory calculations involving linear target molecules, cannot be generalized to the case of asymmetric top molecules owing to ambiguities in the assignment of the final classical action to a particular rotational quantum state. These ambiguities result from pairs of states which are indistinguishable within the bin histogram approach at all the common levels of semiclassical theory. A single value of the classical action can thus correspond to two different quantum states, preventing the distinction between these states in the calculation of rotational cross sections. Our results are illustrated for the rotational states J=1-4 of the water molecule at its equilibrium geometry.     

A comparison between different semiclassical approximations for optical response functions in nonpolar liquid solutions

The temporal behavior of optical response functions (ORFs) reflects the quantum dynamics of an electronic superposition state, and as such lacks a well-defined classical limit. In this paper, we consider the importance of accounting for the quantum nature of the dynamics when calculating ORFs of different types. To this end, we calculated the ORFs associated with the linear absorption spectrum and the nonlinear two-pulse photon-echo experiment, via the following approaches: (1) the semiclassical forward-backward approach; (2) an approach based on linearizing the path-integral forward-backward action in terms of the difference between the forward and backward paths; (3) an approach based on ground state nuclear dynamics. The calculations were performed on a model that consists of a two-state chromophore solvated in a nonpolar liquid. The different methods were found to yield very similar results for the absorption spectrum and "diagonal" two-pulse photon echo (i.e., the homodyne-detected signal at time t=t(0) after the second pulse, where t(0) is the time interval between the two pulses). The different approximations yielded somewhat different results in the case of the time-integrated photon-echo signal. The reasons for the similarity between the predictions of different approximations are also discussed.     

Wave model for conservative bound systems

In the hidden variable theory, Bohm proved a connection between the Schrodinger and Hamilton-Jacobi equations and showed the existence of classical paths, for which the generalized Bohr quantization condition is valid. In this paper we prove similar properties, starting from the equivalence between the Schrodinger and wave equations in the case of the conservative bound systems. Our approach is based on the equations and postulates of quantum mechanics without using any additional postulate. Like in the hidden variable theory, the above properties are proven without using the approximation of geometrical optics or the semiclassical approximation. Since the classical paths have only a mathematical significance in our analysis, our approach is consistent with the postulates of quantum mechanics.     

Time-dependent probability of quantum tunneling in terms of the quasisemiclassical method

In view of the rapid progress in experiments of the tunneling dynamics in the time domain, we develop a quasisemiclassical method that is aimed at a study of the proton-transfer dynamics in a large system such as tropolone and its interesting derivatives, to which not only full quantum mechanics, but even a standard semiclassical theory is never easy to apply. In our very tractable method for multidimensional systems, the tunneling paths are generated in terms of the generalized classical mechanics, but the quantum phases arising from the action integral, the Maslov index, and the semicalssical amplitude factor as well in the semiclassical kernels are entirely neglected. This approach is called the quasisemiclassical method. One of the technical issues involved in the general semiclassical scheme is how to locate points from which a tunneling path emanates. Hence the studies of such tunneling points and the quasisemiclassical method should be examined collectively. We test several ways of determining the tunneling point, including those already proposed in the literature and a newly proposed one. It is shown numerically that the quasisemiclassical method with an appropriate choice of tunneling points reproduces the full quantum mechanical tunneling probability reasonably well. This case study indicates that the present conventional approach is promising to the study of large systems. The role of tunneling points in the initial process of tunneling is also discussed.     

Coherent laser excitation of Rydberg states in a weak static magnetic field

We study one-photon excitation of atomic Rydberg- and continuum states close to a photoionization threshold in the presence of a weak static external magnetic field. A semiclassical closed orbit representation for the atomic transition amplitudes is derived, which exhibits the connection between quantum mechanics and the classical dynamics of the excited electron whose motion under the combined influence of the Coulomb field of the ionic core and the magnetic field is chaotic.     

Multiscale modeling of materials based on force and charge density fidelity

The approximate representation of a quantum solid as an equivalent composite semiclassical solid is considered for insulating materials. The composite is comprised of point ions moving on a potential energy surface. In the classical bulk domain this potential energy is represented by potentials constructed to give the same structure and elastic properties as the underlying quantum solid. In a small local quantum domain the potential is determined from a detailed quantum calculation of the electronic structure. The new features of this well-studied problem are (1) a clearly stated theoretical context in which approximations leading to the model are introduced, (2) the representation of the classical domain by potentials focused on reproducing the specific quantum response being studied, (3) development of "pseudoatoms" for a realistic treatment of charge densities where bonds have been broken to define the environment of the quantum domain, and (4) inclusion of polarization effects on the quantum domain due to its distant bulk environment. This formal structure is illustrated in detail for a SiO(2) nanorod. More importantly, each component of the proposed modeling is tested quantitatively for this case, verifying its accuracy as a faithful multiscale model of the original quantum solid. To do so, the charge density of the entire nanorod is calculated quantum mechanically to provide the reference by which to judge the accuracy of the modeling. The construction of the classical potentials, the rod, the pseudoatoms, and the multipoles is discussed and tested in detail. It is then shown that the quantum rod, the rod constructed from the classical potentials, and the composite classical/quantum rod all have the same equilibrium structure and response to elastic strain. In more detail, the charge density and forces in the quantum subdomain are accurately reproduced by the proposed modeling of the environmental effects even for strains beyond the linear domain. The accuracy of the modeling is shown to apply for two quite different choices for the underlying quantum chemical method: transfer Hamiltonian and density functional methods.     

Assignment and extracting dynamics from experimentally and theoretically obtained spectroscopic hamiltonians in the complex spectral and classically chaotic regions

An analysis of existing algebraic multiresonance spectroscopic Hamiltonians, derived by fitting to experimental data or from classical canonical or quantum Van Vleck perturbation theory, allows without any significant further classical or quantum calculation the assignment of quantum numbers and motions to states observed in spectra that were previously thought to be irregular or just unassignable. In such cases, inspection of the amplitude and phase of eigenfunctions previously calculated in the Hamiltonians derivation process but now transformed to a reduced dimension semiclassical action-angle representation reveals extremely simple albeit unfamiliar topologies that give quantum numbers by simply counting nodes and phase advances. The topology allows these simple wave functions to be sorted into dynamically different excitation ladders or classes of states which are associated with different regions of phase space. The rungs of these ladders or the states in the classes intersperse in energy causing the spectral complexity. No experimental procedure allows such sorting. The success of the work stems from (1) the qualitative insights of nonlinear dynamics, (2) the conversion of the quantum problem in full dimension to a semiclassical one in reduced dimension by use of a canonical transform that takes advantage of the polyad and other constants of the motion, and (3) the judicious choice of the reduced angle variables to reflect rational ratio resonance frequency conditions. This leads to localization of those semiclassical wave functions, which are affected by the particular resonance. In reverse, the localized appearance of the reduced dimension wave function reveals which resonances govern it and makes sorting visually simple. The success of the work also stems from (4) the revealing use of plots of phase advances as well as the usual densities of the eigenstates for sorting and assignment purposes. Even in classically chaotic regions, organizing trajectories, which correspond to averages over regional phase space structures that need not be computed, can easily be drawn as the structure about which eigenfunction localization takes place. The organizing trajectories when transformed back to the full dimensional configuration space reveal the internal molecular motions. The complexity of the usual quantum stationary and propagated wave functions and associated classical trajectories forbids most often such assignments and sorting. This procedure brings the ability to interpret complex vibrational spectra to a degree previously thought possible only for lower excitations. The new methodology replaces and extends the computationally more difficult parts of a procedure used by the authors that was applied successfully to several molecules during the past few years. The new methodology is applied to DCO, CHBrClF, and the bending of acetylene.     

A semiclassical study of the thermal conductivity of low temperature liquids

The conventional classical energy current auto-correlation function has been extended into a quantum mechanical version and then approximated by the linearized semiclassical initial value representation approach. Comparison of the thermal conductivity to simulation results shows that about 15% quantum correction to the classical molecular dynamics results for liquid neon are quantitatively predicted. For liquid para-hydrogen the quantum effects are sufficiently large that the linearized semiclassical approach is only 20% accurate, while for both liquid He(4) and He(3) the thermal conductivity disagrees by a factor of 2, although exchange effects appear to play a minor role.     

Semiclassical quantum unimolecular reaction rate theory revisited

We describe a semiclassical quantum unimolecular reaction rate theory derived from the corresponding classical theory developed by Davis, Gray, Rice and Zhao (DGRZ). The analysis retains the intuitively useful mechanistic distinctions between intramolecular energy transfer and reaction, with the consequence that the semiclassical quantum theory version neglects some interference effects in the reaction dynamics. In the limiting case that intramolecular energy transfer is very fast compared to the rate of reaction we show that the DGRZ representation of the rate constant can be transformed, using the Weyl correspondence between quantum operators and classical variables, to the quantum flux–flux correlation function representation of the rate constant. In the more general case that the rate of intramolecular energy transfer influences the reaction dynamics, the semiclassical representation of the Wigner function for a classical system with both quasiperiodic and chaotic motion is used to obtain the reaction rate constant. Our analysis identifies the quantum analogue of the classical bottleneck to intramolecular energy transfer with the scars of unstable periodic orbits; it leads to a flux–flux correlation function representation of the rate constant for intramolecular energy transfer.     

Generalization of classical mechanics for nuclear motions on nonadiabatically coupled potential energy surfaces in chemical reactions

Classical trajectory study of nuclear motion on the Born-Oppenheimer potential energy surfaces is now one of the standard methods of chemical dynamics. In particular, this approach is inevitable in the studies of large molecular systems. However, as soon as more than a single potential energy surface is involved due to nonadiabatic coupling, such a naive application of classical mechanics loses its theoretical foundation. This is a classic and fundamental issue in the foundation of chemistry. To cope with this problem, we propose a generalization of classical mechanics that provides a path even in cases where multiple potential energy surfaces are involved in a single event and the Born-Oppenheimer approximation breaks down. This generalization is made by diagonalization of the matrix representation of nuclear forces in nonadiabatic dynamics, which is derived from a mixed quantum-classical representation of the electron-nucleus entangled Hamiltonian [Takatsuka, K. J. Chem. Phys. 2006, 124, 064111]. A manifestation of quantum fluctuation on a classical subsystem that directly contacts with a quantum subsystem is discussed. We also show that the Hamiltonian thus represented gives a theoretical foundation to examine the validity of the so-called semiclassical Ehrenfest theory (or mean-field theory) for electron quantum wavepacket dynamics, and indeed, it is pointed out that the electronic Hamiltonian to be used in this theory should be slightly modified.