Partition function of the hindered rotor: Analytical solutions

A simple numerical method that has considerably simplified the calculations of torsional energy levels is proposed. Approximate expressions of the hindered-rotor partition function have been derived, analytically interpreted, and compared with numerically exact values. The accuracy of the previously developed approximations is estimated.     

The semiclassical and the quasiclassical partition function of a quartic anharmonic oscillator

After a substitution a known Laplace-type integral is used to derive quantum corrections to the classical partition function of a quartic anharmonic oscillator in the framework of the Wigner—Kirkwood perturbation expansion. By straightforward calculations results are given in a closed form allowing the analytical formulation of the thermodynamic functions H, E, S, C_υ. The numerical results agree for arbitrary anharmonicity and for high and intermediate temperatures with the numerical partition function calculated from the Hioe—Montroll eigenvalues. Furthermore, the same integral type is used for the analytical calculation of a “quasiclassical” partition function and of “quasiclassical” moments. In the trace formulation of the partition function all commutators are neglected. The harmonic oscillator density matrix is applied to the evaluation of the truncated trace expressions. The “quasiclassical” partition function is an exact upper bound and lies always below the classical partition function.     

Molecular chains with fixed bond lengths and angles: a study based on quantum statistical mechanics

This work studies large three-dimensional open molecular chains at thermal equilibrium in which bond lengths and angles are fixed (hard variables), based upon quantum statistics. A model for a chain formed by N particles interacting through harmonic-like vibrational potentials is treated in the high-frequency limit in which all bond lengths and angles become constrained, while other N angles (soft variables) remain unconstrained. The associated quantum partition function is bounded rigorously, using a variational inequality (related to the Born-Oppenheimer approximation), by another quantum partition function, Z. The total vibrational zero-point energy is shown to be independent of the soft variables thereby solving for this model a generic difficulty in the elimination of hard variables. Z depends only on soft variables and, under certain conditions, it can be approximated by a classical partition function Z_c. The latter satisfies the equipartition principle and it differs from other classical partition functions for related molecular chains. The extension of the model when only part of the bond angles become fixed in the high-frequency limit is outlined. As another generalization, a systematic study of macromolecules, as composed of electrons and heavy particles with Coulomb interactions, is also presented. Its exact quantum partition function is bounded, supposing that the effective molecular potential also tends to constrain all bond lengths and angles, and under suitable assumptions, by another quantum partition function. The latter depends only on the remaining soft variables and it generalizes the one obtained for the first model.     

Fluctuation-induced interactions between dielectrics in general geometries

We study thermal Casimir and quantum nonretarded Lifshitz interactions between dielectrics in general geometries. We map the calculation of the classical partition function onto a determinant, which we discretize and evaluate with the help of Cholesky factorization. The quantum partition function is treated by path integral quantization of a set of interacting dipoles and reduces to a product of determinants. We compare the approximations of pairwise additivity and proximity force with our numerical methods. We propose a "factorization approximation" that gives rather good numerical results in the geometries that we study.     

Quantized RRK theory of unimolecular reaction rates

The main difference between the simple RRK theory and the better based but more complex RRKM theory is explained. Starting from the premise that the classical versus quantum mechanical estimation of the density of states is the major source of the difference, earlier attempts to incorporate the quantum effects in an effective value for the number of oscillators s are noted. By examining the expression for the RRKM rate coefficient it is found that a single effective s value will generally not suffice, but a much better representation of the quantum effects can be obtained if it is recognized that the problem inherently contains two different effective s values. A theory based on this analysis is constructed. It reproduces RRKM results to much improved accuracy, removing difficulties found earlier with single-s-value theories.     

Path integral approach to correlation energy

The Feynman path integral method is applied to the many-electron problem of quantum chemistry. We begin with investigating the partition function of the system in question; then, “a classical path of electron” that corresponds to the Hartree–Fock approximation is obtained by minimizing the thermodynamic potential of the system with respect to the electron coordinate. The next-order approximation is obtained by evaluating the deviation from this classical path, which is approximately written by an easily integrable Gaussian integral. The result is expected to be the random-phase approximation. As numerical examples, the hydrogen molecule and butadiene are treated. © 1994 John Wiley & Sons, Inc.     

Quantum state reconstruction for rigid rotors

We describe a quantum state reconstruction scheme for dipolar rigid rotors based on determination of the expectation value of the molecular orientation. A key feature is the use of half-cycle pulses to excite the rotor prior to the orientation measurement. The set of expectation values obtained by varying the intensity and polarization of the laser and the time interval between excitation and measurement can be inverted directly to yield the rotor density operator. When the density operator corresponds to the admixture of relatively few rotor states, our procedure successfully reconstructs both pure and mixed states.     

Effects of olefin group and its position on the kinetics for intramolecular H-shift and HO2 elimination of alkenyl peroxy radicals

Two classes of unimolecular reactions of peroxy radicals are key to autoignition, namely, intramolecular H-atom shift (which promotes autoignition) and concerted HO(2) elimination (which inhibits autoignition). Olefin groups are prominent functional groups in biodiesel fuels. This paper explores the effects of the presence of an olefin group and its position on the kinetics of unimolecular reactions of peroxy radicals. CBS-QB3 calculations were carried out for 10 selected alkyl- and alkenylperoxy radicals. Transition-state theory was used to determine the temperature dependence of the high-pressure limiting rate constants, and Rice-Ramsperger-Kassel-Marcus/master equation simulations were performed to determine the pressure dependence of selected rate constants. Tunneling effects were computed using the asymmetric Eckart potential. The contribution of internal rotors to partition functions were included by using the hindered-rotor treatment.     

What happens when translational energy levels are quantized?

The conventional translational partition function (CTPF) that is widely used in textbooks is essentially semiclassical, whose form is found by using the particle‐in‐a‐box model to represent the particles motion of an ideal gas. This form assumes continuum translational energy levels, thereby replacing the sum over the energy levels with an integral at high temperature. This is only valid if de Broglie wavelength is much shorter than the container dimension in which the particle is placed. Additionally, de Broglie wavelength must also be much smaller than the mean separation of the constituent particles of the gas. To ensure this, one will have to assume large mass and container size and high temperature (T). This assumption is a restriction in itself. Should de Broglie wavelength be larger than the microscopic size of interest, the CTPF will be invalid to use which in turn will lead to erroneous conclusions, and a pure quantum mechanical treatment must then be employed for accurate results. A perfect example of the failure of using the CTPF would be the conduction electrons in a metal or in conjugated dienes, whereby the translational energy levels are significantly quantized. This quantization manifests itself in the quantum translational partition function (TPF) as its curve starts at zero and remains zero at low T till further increase in T stimulates accessible translational thermal states to be populated, at which point a rise in quantum TPF is observed, whereas the CTPF is only zero at T = 0, and then starts rising as T increases since energy levels are continuum and their discretization is insensitive to the CTPF. Therefore this article explores quantum mechanical treatment of the TPF and its effects on thermodynamic functions using our closed‐form expression of quantum TPF developed in [M. Toutounji, Int J Quantum Chem Early View (unpublished)] (IJQC) which is valid at all Ts, sizes, and masses. Although this TPF appeared in IJQC in an abstract form before, there was not any discussion of its applicability and physical aspects therein. Also, some model calculations are presented to show the failure of the CTPF in the current literature in case of a purely quantum conditioned environment. Thermodynamic functions such as Helmholtz free energy, entropy, and heat capacity are explored using the herein closed form quantum partition function. A closer look at the quantum translational heat capacity shows a curve starts at zero at low T, and then starts sharply rising as T increases going through a maximum after which it levels off at its classical value k/2, where k is Boltzmann constant. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Classical and quantum mechanics of diatomic molecules in tilted fields

We investigate the classical and quantum mechanics of diatomic molecules in noncollinear (tilted) static electric and nonresonant linearly polarized laser fields. The classical diatomic in tilted fields is a nonintegrable system, and we study the phase space structure for physically relevant parameter regimes for the molecule KCl. While exhibiting low-energy (pendular) and high-energy (free-rotor) integrable limits, the rotor in tilted fields shows chaotic dynamics at intermediate energies, and the degree of classical chaos can be tuned by changing the tilt angle. We examine the quantum mechanics of rotors in tilted fields. Energy-level correlation diagrams are computed, and the presence of avoided crossings quantified by the study of nearest-neighbor spacing distributions as a function of energy and tilting angle. Finally, we examine the influence of classical periodic orbits on rotor wave functions. Many wave functions in the tilted field case are found to be highly nonseparable in spherical polar coordinates. Localization of wave functions in the vicinity of classical periodic orbits, both stable and unstable, is observed for many states.     

A quantum mechanical Hamiltonian for unhindered macromolecular chains: consistency with Gaussian models

A model for an open unhindered three-dimensional macromolecular chain, based upon quantum mechanics and proposed in previous works, is studied in order to investigate its physical properties and consistency. The chain is formed by N particles interacting through harmonic-like vibrational potentials in the high-frequency limit (in which all successive bond lengths become fixed). This formulation leads to a specific Hamiltonian for the chain: it constitutes an improvement in comparison with standard Gaussian models, which do not. The classical partition function Z_c resulting from the quantum formulation is represented through an integral, which exhibits explicitly rotational invariance in the integrand and provides the basis for further approximations for large N. Approximate formulae are obtained for correlations between pairs of bond vectors, the distribution function for the end-to-end vector, distribution functions for individual bond vectors, “rubber eleasticity” (when stretching forces act) and the structure factor for small wave vector. In all cases, the results which have arisen from the quantum mechanical formulation coincide with those obtained for the standard Gaussian chain. This agreement appears to confirm the physical consistency of the quantum Hamiltonian characterizing the model.     

Nonlinear response properties of atomic hydrogen under quantum plasma environment: A time-dependent variation perturbation study on hyperpolarizability and two-photon excitations

Pilot calculations on the frequency-dependent nonlinear response property, viz. the electric dipole hyperpolarizability of atomic hydrogen under quantum plasma environment, have been performed using an external oscillatory electric field. Fourth-order perturbation theory within a variational scheme is adopted to obtain the hyperpolarizability within and beyond normal dispersion region. Two-photon absorption from the ground state is explicitly obtained from the pole positions of nonlinear response of the system and studied up to principal quantum number n = 4 . Ground and perturbed wave functions of appropriate symmetries are represented by linear combination of Slater-type orbitals. Exponential cosine-screened Coulomb potential is used to simulate the quantum plasma environment. With respect to plasma strength, the nonlinear response properties are considerably enhanced. Results are compared with those under classical plasma environment represented by screened Coulomb potential. Departure from Coulomb potential results in lifting of the accidental degeneracy in the respective two-photon excited states beyond n = 2 . For free hydrogen atom, the transition energies and the radial density profiles of the respective two-photon excited states match exactly with those obtained from analytical wave functions.     

An overview of coupled cluster theory and its applications in physics

Summary What has since become known as the normal coupled cluster method (NCCM) was invented about thirty years ago to calculate ground-state energies of closed-shell atomic nuclei. Coupled cluster (CC) techniques have since been developed to calculate excited states, energies of open-shell systems, density matrices and hence other properties, sum rules, and the sub-sum-rules that follow from imbedding linear response theory within the NCCM. Further extensions deal both with systems at nonzero temperature and with general dynamical behaviour. More recently, a new version of CC theory, the so-called extended coupled cluster method (ECCM) has been introduced. It has the potential to describe such global phenomena as phase transitions, spontaneous symmetry breaking, states of topological excitation, and nonequilibrium behaviour. CC techniques are now widely recognized as providing one of the most universally applicable, most powerful, and most accurate of all microscopicab initio methods in quantum many-body theory. The number of successful applications within physics is now impressively large. In most such cases the numerical results are either the best or among the best available. A typical case is the electron gas, where the CC results for the correlation energy agree over the entire metallic density range to within less than 1 millihartree (or <1%) with the essentially exact Green's function Monte Carlo results. The role of CC theory within modern quantum many-body theory is first surveyed, by a comparison with other techniques. Its full range of applications in physics is then reviewed. These include problems in nuclear physics, both for finite nuclei and infinite nuclear matter; the electron gas; various integrable and nonintegrable models; various relativistic quantum field theories; and quantum spin chain and lattice models. Particular applications of the ECCM include the quantum hydrodynamics of a zero-temperature, strongly-interacting condensed Bose fluid; a charged impurity in a polarizable medium (e.g., positron annihilation in metals); and various anharmonic oscillator and spin systems.     

Theoretical Shaping of Femtosecond Laser Pulses for Molecular Photodissociation with Control Techniques Based on Ehrenfest′s Dynamics and Time‐Dependent Density Functional Theory

The combination of nonadiabatic Ehrenfest‐path molecular dynamics (EMD) based on time‐dependent density functional theory (TDDFT) and quantum optimal control formalism (QOCT) was used to optimize the shape of ultra‐short laser pulses to achieve photodissociation of a hydrogen molecule and the trihydrogen cation H₃⁺. This work completes a previous one [A. Castro, ChemPhysChem, 2013 , 14, 1488–1495], in which the same objective was achieved by demonstrating the combination of QOCT and TDDFT for many‐electron systems on static nuclear potentials. The optimization model, therefore, did not include the nuclear movement and the obtained dissociation mechanism could only be sequential: fast laser‐assisted electronic excitation to nonbonding states (during which the nuclei are considered to be static), followed by field‐free dissociation. Here, in contrast, the optimization was performed with the QOCT constructed on top of the full dynamic model comprised of both electrons and nuclei, as described within EMD based on TDDFT. This is the first numerical demonstration of an optimal control formalism for a hybrid quantum–classical model, that is, a molecular dynamics method.     

Mutual capture of dipolar molecules at low and very low energies. II. Numerical study

The low-energy rate coefficients of capture of two identical dipolar polarizable rigid rotors in their lowest nonresonant (j(1) = 0 and j(2) = 0) and resonant (j(1) = 0, 1 and j(2) = 1, 0) states are calculated accurately within the close-coupling (CC) approach. The convergence of the quantum rate coefficients to their quantum-classical counterparts is studied. A comparison of the present accurate numerical with approximate analytical results (Nikitin, E. E.; Troe, J. J. Phys. Chem. A 2010, 114, 9762) indicates a good performance of the previous approach which was based on the interpolation between s-wave fly wheel quantal and all-wave classical adiabatic channel limits. The results obtained apply as well to the formation of transient molecular species in the encounter of two atoms at very low collision energy interacting via resonance dipole-dipole interaction.     

Introducing the γ function in quantum theory

Through a series of postulates, we define a function γ(x) whose square acts as Dirac's δ(x) and exhibits several unusual properties. Though the square root of δ cannot be defined among distributions, it appears in quantum theory if one wants to associate a wave function to a (quasi)classical particle having charge distribution δ(x) . The newly defined function γ(x) serves to describe quasi-classical particles using part of the quantum formalism (eg, wave functions, operators, expectation values) but exhibiting classical properties. The function γ(x) appears to be useful to define model wave functions for simple (quasi)quantum systems. In a spherical coordinate system, γ(r − r₀) leads to a quasi-classical “bubble” model of the hydrogen atom, where the electron is distributed on the surface of a sphere with radius r₀, and it provides exact quantum mechanical energies of its total symmetric levels. For other simple quantum systems, it provides approximate but meaningful energies. In particular, exact energy differences for harmonic oscillator levels are obtained, with the zero-point energy missing.     

Spectral density analysis of nuclear spin—spin coupling: II. Hartree-Fock LCAO studies for homonuclear coupling constants

A spectral density function has been calculated for the indirect nuclear spin—spin coupling constant for homonuclear coupling. The ground state wavefunction is obtained with a normal ab-initio calculation. The sum over states approach for calculating the reduced coupling constant K is replaced by an integration over a spectral density function where the integration variable is the orbital exponent of a “scanning molecular orbital”. This results in a stable method for calculating K with reasonable accuracy. The spectral density function also gives information about which excited states give important contributions to K.Furthermore a residual spectral density function is defined that can be used as a test for the completeness of a set of virtual orbitals in a sum over states calculation.     

Path integral based calculations of symmetrized time correlation functions. I

In this paper, we examine how and when quantum evolution can be approximated in terms of (generalized) classical dynamics in calculations of correlation functions, with a focus on the symmetrized time correlation function introduced by Schofield. To that end, this function is expressed as a path integral in complex time and written in terms of sum and difference path variables. Taylor series expansion of the path integral's exponent to first and second order in the difference variables leads to two original developments. The first order expansion is used to obtain a simple, path integral based, derivation of the so-called Schofield's quantum correction factor. The second order result is employed to show how quantum mechanical delocalization manifests itself in the approximation of the correlation function and hinders, even in the semiclassical limit, the interpretation of the propagators in terms of sets of guiding classical trajectories dressed with appropriate weights.