Collisional energy transfer with statistical energy exchange: an analytical solution in the statistical limit

Collisional energy transfer between highly vibrationally excited molecules and bath gas is considered with a statistical kernel, describing energy exchange in complex-forming collisions. Knowledge of the bilinear formula for the Laguerre polynomials offers a means for determining eigenvalues and eigenfunctions of the kernel. An exact solution to master equation for the conditional probability is given as an expansion in terms of these eigenfunctions. The bulk averages of internal energy moments and energy transfer moments are calculated analytically.     

The relationship between physical observables and the potential energy surface in the He-HF system

This paper examines the effect of infinitesimal functional variations in a rigid rotor He-HF potential surface on several different types of observables: inelastic cross sections, rate constants, and rotational energy level populations. The dynamics and kinetic observables studied were found to be sensitive to a large number of Legendre components of the potential with the region of highest sensitivity dependent upon the energy or temperature as well as the states related by the individual observable. Sensitivity to the entire surface tends to show a large degree of structure due to competition among sensitivities to the individual potential components. Significant information loss has been observed in the transition from microscopic to macroscopic observables.     

Density matrix methods in orbital optimization for MCSCF calculations

The problem considered is that of selecting the finite orbital basis which will minimize the energy in a given size CI calculation. (1) A one-body operator is defined which has as eigenfunctions the desired optimal basis. The operator is defined in terms of the basis which leads to a self-consistency problem of Hartree-Fock type. (2) A method of successive orbital rotations is defined which is shown to have desirable convergence properties.     

Usage de l'Algègbre de Lie su (n) Dans l'Etude des Systèmes Quantiques à n Etats. I. Quelques Conséquences Générales

In an n level quantum system there is a relation between each density operator and an element of su (n) Lie algebra. This relation is also established between Cartan's subalgebras and the complete sets of compatible observables. A scalar product is then defined in this algebra in order to introduce orthonormal bases and to simplify many calculations about expectation values of observables. Therefore simple general rules were established which show how to determine (completely or partly), from linearly independent observables, the density operator of the system.     

The extended Koopmans' theorem Fock operator and the generalized overlap amplitude one-electron operator

The wave function of a system may be expanded in terms of eigenfunctions of the N −1 electron Hamiltonian times one-particle functions known as generalized overlap amplitudes (GOAS). The one-electron operator whose eigenfunctions are the GOAS is presented, without using an energy-dependent term as in the one-particle Green function or propagator approach. It is shown that this operator and the extended Koopmans' theorem (EKT) one-electron operator are of similar form, but perform complementary roles. The GOA operator begins with one-electron densities and total energies of N −1 electron states to generate the two-matrix and total energy of an N-electron state. The EKT operator begins with the two-matrix of an N-electron state to generate one-electron densities and ionization potentials (or approximations thereto) for N −1 electron states. However, whereas the EKT orbitals must be linearly independent, no such restriction applies to the GOAS. © 1996 John Wiley & Sons, Inc.     

Molecular states of atoms. I. Orbital energy parameters

Atomic valence state energies are analyzed to obtain values of orbital energy parameters that may be used in semiempirical molecular orbital calculations. Difficulty in defining the interaction between orbitals with non-integer electron populations is systematically avoided by distinguishing between a valence state and a molecular state of an atom, only the latter state having non-integer spin paired orbital occupancy. Application of the virial theorem to the molecular state enables a value for the orbital kinetic energy to be obtained from the valence state orbital energy parameters once an arbitrary configuration is defined as reference. The orbitals then are eigenfunctions of the atomic Fock operator for that reference molecular state and, with their energy parameters, may be employed as a fixed basis set for molecular orbital calculations.     

Improving the convergence of closed and open path integral molecular dynamics via higher order Trotter factorization schemes

Higher order factorization schemes are developed for path integral molecular dynamics in order to improve the convergence of estimators for physical observables as a function of the Trotter number. The methods are based on the Takahashi-Imada and Susuki decompositions of the Boltzmann operator. The methods introduced improve the averages of the estimators by using the classical forces needed to carry out the dynamics to construct a posteriori weighting factors for standard path integral molecular dynamics. The new approaches are straightforward to implement in existing path integral codes and carry no significant overhead. The Suzuki higher order factorization was also used to improve the end-to-end distance estimator in open path integral molecular dynamics. The new schemes are tested in various model systems, including an ab initio path integral molecular dynamics calculation on the hydrogen molecule and a quantum water model. The proposed algorithms have potential utility for reducing the cost of path integral molecular dynamics calculations of bulk systems.     

The extended Koopmans' theorem Fock operator

By expanding the wave function of a system of N particles in terms of products of functions of one and (N-1) particles, the one-particle, nonlocal operator F?_EKT (extended Koopmans' theorem) is determined. It is shown that although this operator is nonhermitian, its eigenvalues and eigenfunctions represent the ionization energies and occupied orbitals, respectively. The eigenfunctions of F?_EKT are the one-particle functions that enter into the expansion of the wave function of the system as partners of the (N-1)-particle wave functions. The eingenvalues are also one-particle energies that, multipled by the orbital occupancy probalities, enter the expression for the total N-particle energy of the system.     

Construction of orbital angular momentum eigenfunctions for many-particle systems by harmonic projection

Formulas are derived which allow the direct construction of total orbital angular momentum eigenfunctions for many-particle systems without the use of Clebsch–Gordan coefficients. One of the equations is closely analogous to Dirac' identity for the total spin operator. This equation describes the action of L² on a function of the particle coordinates in terms of a class operator of the symmetric group and a "contraction operator." A general projection operator for constructing symmetric eigenfunctions of L² is presented.     

A canonical averaging in the second-order quantized Hamilton dynamics

Quantized Hamilton dynamics (QHD) is a simple and elegant extension of classical Hamilton dynamics that accurately includes zero-point energy, tunneling, dephasing, and other quantum effects. Formulated as a hierarchy of approximations to exact quantum dynamics in the Heisenberg formulation, QHD has been used to study evolution of observables subject to a single initial condition. In present, we develop a practical solution for generating canonical ensembles in the second-order QHD for position and momentum operators, which can be mapped onto classical phase space in doubled dimensionality and which in certain limits is equivalent to thawed Gaussian. We define a thermal distribution in the space of the QHD-2 variables and show that the standard beta=1/kT relationship becomes beta'=2/kT in the high temperature limit due to an overcounting of states in the extended phase space, and a more complicated function at low temperatures. The QHD thermal distribution is used to compute total energy, kinetic energy, heat capacity, and other canonical averages for a series of quartic potentials, showing good agreement with the quantum results.     

A density functional representation of quantum chemistry. II. Local quantum field theories of molecular matter in terms of the charge density operator do not work

A comprehensive review of the attempts to rephrase molecular quantum mechanics in terms of the particle density operator and the current density or phase density operator is given. All pertinent investigations which have come to our attention suffer from severe mathematical inconsistencies and are not adequate to the few-body problem of quantum chemistry. The origin of the failure of these attempts is investigated and it is shown that a realization of a local quantum field theory of molecular matter in terms of observables would presuppose the solution of many highly nontrivial mathematical problems.     

Vibrational anharmonicity in methyl bromide

The vibrational spectra of the methyl halides have been extensively studied and for many of these studies some estimate of the vibrational anharmonicity has been desirable. Accordingly, we have used a simple model for the anharmonic force field to calculate the anharmonic corrections to the observed fundamental wavenumbers and the cubic normal-coordinate force constants. The latter are important in the interpretation of vibrational resonances, in the interpretation of the intensities of overtone, combination and difference bands and in the calculation of vibrational averages of molecular physical observables.     

Reduced hamiltonian orbitals. III. Unitarily invariant decomposition of hermitian operators

A decomposition of an N-particle operator as a sum of N + 1 components is defined such that, in the case of a model system employing a finite one-particle basis set, the decomposition is invariant under unitary transformations of the basis set. Applied to a two-particle Hamiltonian, this decomposition gives rise to the distinction between the independent-particle energy and the coupling energy defined in previous papers. Applied to the reduced density operator for a quantum state, the decomposition corresponds to partitioning the density into irreducible components. This partitioning is illustrated by graphs of electron density for the water molecule.     

A direct relaxation method for calculating eigenfunctions and eigenvalues of the schrödinger equation on a grid

Eigenfunctions and eigenvalues of the Schrodinger equation are determined by propagating the Schrodinger equation in imaginary time. The method is based on representing the Hamiltonian operation on a grid. The kinetic energy is calculated by the Fourier method. The propagation operator is expanded in a Chebychev series. Excited states are obtained by filtering out the lower states. Comparative examples include: eigenfunctions and eigenvalues of the Morse oscillator, the Hénon-Heiles system and weakly bound states of He on a Pt surface.     

Studies of electronic configurations in the emission spectra of lanthanides and actinides: application to the interpretation of Es I and Es II, predictions for Fm I

The interpretation of the spectra of free atoms and gaseous ions in the 4f^N and 5f^N periods became less active after critical compilations of energy levels appeared. However, several spectra are still under study and the application of the Racah-Slater and HFR methods to extended sets of configurations leads to revisions and additions. In doubly charged ions of lanthanides, the treatment of configuration interaction by means of effective parameters and by extension of the basis of states are both important. Concerning actinides, calculations of several observables (Landé factors and isotope shifts in Pu I, hyperfine constants, transition probabilities) prove the quality of eigenfunctions. The classification of Es I and Es II has been extended and radial parameters for fine and hyperfine structures have been derived. Level predictions for the next element fermium are supported by parameter extrapolations.     

An exactly soluble base problem for atomic systems

An exactly soluble base problem for atomic systems is presented which approximates the atomic Hamiltonian as a sum of identical one‐electron operators. The eigenfunctions of the one‐electron operator consist of a radial function multiplied by a spherical harmonic. Energies for many‐electron atoms are found by summing the one‐electron energy eigenvalues according to the Pauli principle. These energies are rigorous lower bounds to the exact energies. This revised version was published online in July 2006 with corrections to the Cover Date.     

Lower bound to the ground-state expectation value of a positive unbounded operator using related bounded operators

Error bars around observables of quantum mechanical systems are extremely lacking; in most cases only an upper bound to the energy is practical. We present a new lower bound to the expectation value of an operator that is most similar to the lower bound of Weinhold. While Weinhold’s bound has flexibility by incorporating expectation values (some of which may not exist) of different moments of the operator to be bounded, the flexibility of our lower bound relies on the form of a similar, but bounded, operator. Like Weinhold’s bound, ours is limited to non-negative operators and the ground-state of the system. Our lower bound is shown to have properties which allow it to converge to the true expectation value of the ground state, but a practical application to the Helium atom shows that Weinhold’s bound is superior in this case.     

Eigenvalues of the Boltzmann collision operator for binary gases: Mass dependence

Eigenvalues of the Boltzmann collision operator are calculated versus mass ratio with two different methods. One method involves the expansion of the eigenfunctions in speed polynomials, whereas with the second method the eigenfunctions are evaluated at discrete points based on a particular gaussian quadrature rule. The discrete ordinate method proved to be superior provided the mass ratio was neither too large nor too small. The approach of the eigenvalues to the continuum boundary was also studied for several mass ratios.