Semiclassical Research Articles

Explanation of Raman Scattering Spectrum on the Basis of Semi Classical Model and Klein Gordon Equation 2

Explanation of Raman Scattering Spectrum on the Basis of Semi Classical Model and Klein Gordon Equation 2

Electronically Nonadiabatic Dynamics in Complex Molecular Systems: An Efficient and Accurate Semiclassical Solution

Chemical reaction dynamics is always a central theme in chemistry research. In many important chemical processes, reaction dynamics is electronically nonadiabatic, i.e., dynamics involves coupled multiple electronic states. We demonstrate in this paper that a semiclassical (SC) treatment based on an initial value representation methodology and a classical mapping formalism for the electronic degrees of freedom is now able to provide a rigorous and practical solution to electronically nonadiabatic dynamics in complex molecular systems. The key component of this treatment is to incorporate a correlated importance sampling protocol in nonadiabatic SC calculations, which results in a speedup factor of 100 or more in comparison with that using the standard sampling approach. This is illustrated by application to a two-state model coupled with up to 10 nuclear bath modes for a benchmark nonadiabatic excitation energy transfer problem. This work provides great opportunities for the effectively theoretical investigations on reaction mechanisms in complex molecular systems, in which electronically nonadiabatic dynamics plays an importance role.

Time-dependent importance sampling in semi-classical initial value representation calculations for time correlation functions. III. A state-resolved implementation to electronically non-adiabatic dynamics

The recently developed time-dependent (TD) Monte Carlo (MC) importance sampling method [Tao and Miller; JCP 135, 024104 (2011)] provides an efficient implementation of the semi-classical (SC) initial value representation (IVR) methodology for the evaluation of time correlation functions. The key idea in this TD-SC-IVR method is to perform importance sampling of trajectories for the MC averages in the SC calculations with a sampling function that includes information about the final (time-evolved) values of the coordinates and momenta of trajectories in addition to (the usual) information of their initial values. This paper shows how this approach deals with electronically non-adiabatic dynamics, i.e. dynamics that involves coupled multiple electronic states. We suggest that a state-resolved sampling function may facilitate the efficient implementation of TD-SC-IVR for such processes. This is illustrated by application to the calculation of the nuclear momentum distribution function (i.e. the nuclear energy-loss spectrum) for a benchmark non-adiabatic scattering problem.

Communication: Importance sampling including path correlation in semiclassical initial value representation calculations for time correlation functions

Full semiclassical (SC) initial value representation (IVR) for time correlation functions involves a double phase space average over a set of two phase points, each of which evolves along a classical path. Conventionally, the two initial phase points are sampled independently for all degrees of freedom (DOF) in the Monte Carlo procedure. Here, we present an efficient importance sampling scheme by including the path correlation between the two initial phase points for the bath DOF, which greatly improves the performance of the SC-IVR calculations for large molecular systems. Satisfactory convergence in the study of quantum coherence in vibrational relaxation has been achieved for a benchmark system-bath model with up to 21 DOF.

A Cartesian classical second-quantized many-electron Hamiltonian, for use with the semiclassical initial value representation

A new classical model for the general second-quantized many-electron Hamiltonian in Cartesian coordinates and momenta is presented; this makes semiclassical (SC) calculations using an initial value representation (IVR) more useful than the classical Hamiltonian in action-angle variables given earlier by Miller and White [J. Chem. Phys. 84, 5059-5066 (1986)]. If only 1-electron terms are included in this Hamiltonian, the classical equations of motion for the Cartesian variables are linear, and the SC-IVR gives exact results for the propagator (and thus for transition probabilities, the energy spectrum, etc.), as confirmed by analytic proof and numerical calculations. Though this new Hamiltonian is not exact when 2-electron interactions are included, we observe good results for the SC-IVR transition probabilities for times that are not too long. Test calculations, for example, show that the SC-IVR is accurate for times long enough to obtain good result for the eigenvalue spectrum (i.e., the energy levels of the electronic system).

Time-dependent importance sampling in semiclassical initial value representation calculations for time correlation functions. II. A simplified implementation

An efficient time-dependent (TD) Monte Carlo (MC) importance sampling method has recently been developed [G. Tao and W. H. Miller, J. Chem. Phys. 135, 024104 (2011)] for the evaluation of time correlation functions using the semiclassical (SC) initial value representation (IVR) methodology. In this TD-SC-IVR method, the MC sampling uses information from both time-evolved phase points as well as their initial values, and only the "important" trajectories are sampled frequently. Even though the TD-SC-IVR was shown in some benchmark examples to be much more efficient than the traditional time-independent sampling method (which uses only initial conditions), the calculation of the SC prefactor-which is computationally expensive, especially for large systems-is still required for accepted trajectories. In the present work, we present an approximate implementation of the TD-SC-IVR method that is completely prefactor-free; it gives the time correlation function as a classical-like magnitude function multiplied by a phase function. Application of this approach to flux-flux correlation functions (which yield reaction rate constants) for the benchmark H + H(2) system shows very good agreement with exact quantum results. Limitations of the approximate approach are also discussed.

Semiclassical glory analyses in the time domain for the H + D2(vi = 0, ji = 0) → HD(vf = 3, jf = 0) + D reaction

We make the first application of semiclassical (SC) techniques to the plane-wavepacket formulation of time-domain (T-domain) scattering. The angular scattering of the state-to-state reaction, H + D(2)(v(i) = 0, j(i) = 0) → HD(v(f) = 3, j(f) = 0) + D, is analysed, where v and j are vibrational and rotational quantum numbers, respectively. It is proved that the forward-angle scattering in the T-domain, which arises from a delayed mechanism, is an example of a glory. The SC techniques used in the T-domain are: An integral transitional approximation, a semiclassical transitional approximation, a uniform semiclassical approximation (USA), a primitive semiclassical approximation and a classical semiclassical approximation. Nearside-farside (NF) scattering theory is also employed, both partial wave and SC, since a NF analysis provides valuable insights into oscillatory structures present in the full scattering pattern. In addition, we incorporate techniques into the SC theory called "one linear fit" and "two linear fits", which allow the derivative of the quantum deflection function, Θ̃(')(J), to be estimated when Θ̃J exhibits undulations as a function of J, the total angular momentum variable. The input to our SC analyses is numerical scattering (S) matrix data, calculated from accurate quantum collisional calculations for the Boothroyd-Keogh-Martin-Peterson potential energy surface No. 2, in the energy domain (E-domain), from which accurate S matrix elements in the T-domain are generated. In the E-domain, we introduce a new technique, called "T-to-E domain SC analysis." It half-Fourier transforms the E-domain accurate quantum scattering amplitude to the T-domain, where we carry out a SC analysis; this is followed by an inverse half-Fourier transform of the T-domain SC scattering amplitude back to the E-domain. We demonstrate that T-to-E USA differential cross sections (DCSs) agree well with exact quantum DCSs at forward angles, for energies where a direct USA analysis in the E-domain fails.

Time-dependent importance sampling in semiclassical initial value representation calculations for time correlation functions

An efficient time-dependent importance sampling method is developed for the Monte Carlo calculation of time correlation functions via the initial value representation (IVR) of semiclassical (SC) theory. A prefactor-free time-dependent sampling function weights the importance of a trajectory based on the magnitude of its contribution to the time correlation function, and global trial moves are used to facilitate the efficient sampling the phase space of initial conditions. The method can be generally applied to sampling rare events efficiently while avoiding being trapped in a local region of the phase space. Results presented in the paper for two system-bath models demonstrate the efficiency of this new importance sampling method for full SC-IVR calculations.

Application of a semiclassical model for the second-quantized many-electron Hamiltonian to nonequilibrium quantum transport: The resonant level model

A semiclassical approach is developed for nonequilibrium quantum transport in molecular junctions. Following the early work of Miller and White [J. Chem. Phys. 84, 5059 (1986)], the many-electron Hamiltonian in second quantization is mapped onto a classical model that preserves the fermionic character of electrons. The resulting classical electronic Hamiltonian allows for real-time molecular dynamics simulations of the many-body problem from an uncorrelated initial state to the steady state. Comparisons with exact results generated for the resonant level model reveal that a semiclassical treatment of transport provides a quantitative description of the dynamics at all relevant timescales for a wide range of bias and gate potentials, and for different temperatures. The approach opens a door to treating nontrivial quantum transport problems that remain far from the reach of fully quantum methodologies.

Comparison of quantum, semiclassical and classical methods in hydrogen broadening of nitrogen lines

We use an accurate N 2–H 2 ab initio potential energy surface (PES) in order to inter-compare various methods commonly employed to calculate pressure broadening coefficients. Close-coupling (CC) calculations of the collisional linewidths of the isotropic Raman lines of N 2 perturbed by H 2 are performed for temperatures between 77 and 2000 K. The CC results compare well with available experimental values. Three less exact methods of calculation are also used: the full classical (FC) model of Gordon, the semiclassical (SC) formalism of Robert and Bonamy and the quantum dynamical coupled states (CS) method. The CS method provides good agreement with CC calculations for all studied temperatures, FC calculations can be considered as accurate above room temperature while the SC method gives overestimated values by about 20–30% in all cases. The temperature dependences of pressure broadening coefficients provided by each method are very similar at elevated (above room) temperatures.

Detailing Coherent, Minimum Uncertainty States of Gravitons, as Semi Classical Components of Gravity Waves, and How Squeezed States Affect Upper Limits To Graviton Mass

We present what is relevant to squeezed states of initial space time and how that affects both the composition of relic GW, and also gravitons. A side issue to consider is if gravitons can be configured as semi classical particles, which is akin to the Pilot model of Quantum Mechanics as embedded in a larger non linear deterministic background.

Semiclassical Description of Electronic Excitation Population Transfer in a Model Photosynthetic System

Recently, Fleming’s group observed direct evidence that the electronic excitation energy (and population) transfers coherently rather than through incoherent hopping motions in the Fenna−Mathews−Olson pigment−protein complex. Ishizaki and Fleming further developed a hierarchy equation approach to describe this excitation population transfer dynamics in a model photosynthetic system. Here, we treat this same model system via the linearized approximation to the semiclassical (SC) initial value representation (IVR) for time correlation functions, in combination with the Meyer−Miller−Stock−Thoss model for the electronic degrees of freedom. Our approach is able to describe the long-lived quantum coherent dynamics, in excellent agreement with Ishizaki−Fleming’s results. Moreover, the advantage of the linearlized SC-IVR approach is that it can be applied to any molecular model for which classical MD simulations are feasible, all the way up to a full all-atom MD simulation, while at the same time treating the electronic and nuclear dynamics in a consistent fashion.

Gaussian approximation for the structure function in semiclassical forward-backward initial value representations of time correlation functions

Initial value representations (IVRs) of semiclassical (SC) theory provide a general approach for adding quantum mechanical effects to classical molecular dynamics simulations of large molecular systems. Of the various versions of SC-IVR methodology for evaluating time correlation functions, the Fourier transform forward-backward (FB) approach is the simplest one that is able to describe true quantum coherence effects, so it is of considerable importance to find efficient and systematic ways for implementing it. It is shown in this paper that a Gaussian approximation for the "structure function"-the dependence of the correlation function on the (typically) momentum jump parameter-provides an efficient and accurate way for doing so. The approach is illustrated by an application to the time-dependent radial distribution function of I(2) (after photoexcitation) in a cluster of (up to 16) argon atoms.

Semiclassical description of vibrational quantum coherence in a three dimensional I2Arn (n≤6) cluster: A forward-backward initial value representation implementation

The semiclassical (SC) initial value representation (IVR) has been applied to describe true quantum coherence effects in a complex molecular system in full three dimensional space. The specific quantity considered is the time-dependent probability distribution of the I(2) vibrational coordinate following photoexcitation of I(2) in a rare gas cluster. The "forward-backward" version of the IVR method is shown to be capable of capturing detailed quantum coherence in this quantity, coherence that cannot be described by a classical Wigner model (which is equivalent to a linearized approximation to the more general SC-IVR). Solvent effects on this vibrational quantum coherence have also been investigated for a I(2)Ar(n) (n=1,6) cluster. A solvent cage consisting of six argon atoms reduces the fraction of iodine molecules that dissociate (an example of the "cage effect") and also diminishes, but does not entirely eliminate, quantum coherence in the vibrational motion of the molecules that remain undissociated.

Electronically Nonadiabatic Dynamics via Semiclassical Initial Value Methods

In the late 1970s Meyer and Miller (MM) [J. Chem. Phys. 1979, 70, 3214.] presented a classical Hamiltonian corresponding to a finite set of electronic states of a molecular system (i.e., the various potential energy surfaces and their couplings), so that classical trajectory simulations could be carried out by treating the nuclear and electronic degrees of freedom (DOF) in an equivalent dynamical framework (i.e., by classical mechanics), thereby describing nonadiabatic dynamics in a more unified manner. Much later Stock and Thoss (ST) [Phys. Rev. Lett. 1997, 78, 578.] showed that the MM model is actually not a "model", but rather a "representation" of the nuclear-electronic system; i.e., were the MMST nuclear-electronic Hamiltonian taken as a Hamiltonian operator and used in the Schrodinger equation, the exact (quantum) nuclear-electronic dynamics would be obtained. In recent years various initial value representations (IVRs) of semiclassical (SC) theory have been used with the MMST Hamiltonian to describe electronically nonadiabatic processes. Of special interest is the fact that, though the classical trajectories generated by the MMST Hamiltonian (and which are the "input" for an SC-IVR treatment) are "Ehrenfest trajectories", when they are used within the SC-IVR framework, the nuclear motion emerges from regions of nonadiabaticity on one potential energy surface (PES) or another, and not on an average PES as in the traditional Ehrenfest model. Examples are presented to illustrate and (hopefully) illuminate this behavior.

Simulations of the emission spectra of fac-tris(2-phenylpyridine) iridium and Duschinsky rotation effects using the Herman–Kluk semiclassical initial value representation method

The phosphorescent emission spectra of fac-tris(2-phenylpyridine) iridium [fac-Ir(ppy)(3)] due to the lowest triplet T(1) and T(2) states are simulated using the harmonic oscillator approximation for the S(0), T(1), and T(2) potential energy surfaces (PESs) and taking the Duschinsky rotation into account. The simulations involve the propagation of 177-dimensional wave packets on the coupled PES according to the Herman-Kluk (HK) semiclassical (SC) initial value representation (IVR) method. The HK SC-IVR method is employed because of its accuracy for the PES with mode mixing and its efficiency in dealing with coupled degrees of freedom for large systems. The simulated emission spectrum due to T(1) reproduces the structures of the emission spectra observed experimentally, while T(2) is found very unlikely to participate in the phosphorescent emission. Although the effect of the Duschinsky mode mixing is small for the T(1) state, neglecting it blueshifts the spectrum due to the T(2) state by 800 cm(-1) and changes the relative intensities, indicating that the importance of the Duschinsky rotation is rather unpredictable and should not be overlooked. The present simulations demonstrate that the simple harmonic oscillator approximation combined with the Duschinsky rotation can adequately describe the photophysics of fac-Ir(ppy)(3) and that the HK SC-IVR method is a powerful tool in studies of this kind.

Ro-vibrating energy states of a diatomic molecule in an empirical potential

An approximate analytical solution of the Schrodinger equation is obtained to represent the rotational–vibrational (ro-vibrating) motion of a diatomic molecule. The ro-vibrating energy states arise from a systematical solution of the Schrodinger equation for an empirical potential (EP) V±(r) = De{1 − (ɛ/δ)[coth (ηr)]±1/1 − (ɛ/δ)}2 are determined by means of a mathematical method so-called the Nikiforov–Uvarov (NU). The effect of the potential parameters on the ro-vibrating energy states is discussed in several values of the vibrational and rotational quantum numbers. Moreover, the validity of the method is tested with previous models called the semiclassical (SC) procedure and the quantum mechanical (QM) method. The obtained results are applied to the molecules H2 and Ar2.

Semiclassical description of electronically nonadiabatic dynamics via the initial value representation

The initial value representation (IVR) of semiclassical (SC) theory is used in conjunction with the Meyer-Miller/Stock-Thoss description of electronic degrees of freedom in order to treat electronically nonadiabatic processes. It is emphasized that the classical equations of motion for the nuclear and electronic degrees of freedom that emerge in this description are precisely the Ehrenfest equations of motion (the force on the nuclei is the force averaged over the electronic wave function) but that the trajectories given by these equations of motion do not have the usual shortcomings of the traditional Ehrenfest model when they are used within the SC-IVR framework. For example, in the traditional Ehrenfest model (a mixed quantum-classical approach) the nuclear motion emerges from a nonadiabatic encounter on an average potential energy surface (a weighted average according to the population in the various electronic states), while the SC-IVR describes the correct correlation between electronic and nuclear dynamics, i.e., the nuclear motion is on one potential energy surface or the other depending on the electronic state. Calculations using forward-backward versions of SC-IVR theory are presented to illustrate this behavior. An even more approximate version of the SC-IVR, the linearized approximation (LSC-IVR), is slightly better than the traditional Ehrenfest model, but since it cannot describe quantum coherence effects, the LSC-IVR is also not able to describe the correct correlation between nuclear and electronic dynamics.

Quantum-mechanical vs. semi-classical spectral-line widths and shifts from the line core in the non-impact region for the Ar-perturbed/ K-radiator system

New quantum-mechanical (QM) and semi-classical (SC) shifts (d's) and widths (HWHM's, w's) were measured from the line core of computed full spectral-line shapes for the Ar-perturbed/K-radiator system (K/Ar). The initial state of our model was based on a 4 p 2 P 3/2,1/2 pseudo-potential for the K/Ar system, and the final state on a zero potential. The Fourier transform of the line shape formed the basis for the computations. Excellent agreement was found between the QM and SC values of d and of w in a high-pressure ( P) non-impact region, which was characterized by a √ P dependence of w and a P dependence of d. These agreements were shown to be another example of a correspondence between classical (SC) quantities and QM quantities in the limit of large quantum numbers. Typically at P=1×10 6 Torr and T=400 K, w QM= 448 cm −1 and w SC =479 cm −1, where the deviation from the mean is ±3.3%. Also, d QM =−3815 cm −1 and d SC =−3716 cm −1, where the deviation from the mean is ±1.3%. A new general method was formulated which yielded a definite pressure P 0, which was defined as an upper limit to the low-pressure impact approximation and a lower limit to the non-impact region.

Using the thermal Gaussian approximation for the Boltzmann operator in semiclassical initial value time correlation functions

The thermal Gaussian approximation (TGA) recently developed by Frantsuzov et al. [Chem. Phys. Lett. 381, 117 (2003)] has been demonstrated to be a practical way for approximating the Boltzmann operator exp(-betaH) for multidimensional systems. In this paper the TGA is combined with semiclassical (SC) initial value representations (IVRs) for thermal time correlation functions. Specifically, it is used with the linearized SC-IVR (LSC-IVR, equivalent to the classical Wigner model), and the "forward-backward semiclassical dynamics" approximation developed by Shao and Makri [J. Phys. Chem. A 103, 7753 (1999); 103, 9749 (1999)]. Use of the TGA with both of these approximate SC-IVRs allows the oscillatory part of the IVR to be integrated out explicitly, providing an extremely simple result that is readily applicable to large molecular systems. Calculation of the force-force autocorrelation for a strongly anharmonic oscillator demonstrates its accuracy, and calculation of the velocity autocorrelation function (and thus the diffusion coefficient) of liquid neon demonstrates its applicability.