Generating functions and recurrence relations for harmonic oscillator matrix elements

A simple and straightforward way of obtaining generating functions and recurrence relations for harmonic oscillator integrals is presented. The method, which is based on the properties of unnormalized coherent states and canonical transformations, allows a unified treatment of several different physical problems. Matrix elements of Gaussian and exponential functions, Franck-Condon overlaps, and transition probabilities for time-dependent quadratic Hamiltonians are discussed as illustrative examples.     

Hypervirial theorem and ladder operators. Recurrence relations for harmonic oscillator integrals

A second-quantization formalism combined with a hypervirial theorem is used to derive new recurrence relations for one-dimensional harmonic oscillator matrix elements. The most general case of 〈m|f(â, â⁺)|n〉 is considered, and the recurrence relations forf(â, â) = X^k, exp(?βX), and exp(?X²) are given as examples. The relations obtained are considerably simpler than those derived by using only the hypervirial theorem; comparatively, the recurrence relations presented here have the advantage of avoiding the use of the quantum mechanical sum-rules when determining initial matrix elements. The proposed procedure can be used to determine the recurrence relations for other potentials as well as to evaluate the two-center integrals.     

Connections between perturbation theory and the Van Vleck transformation: Illustrative calculations on the perturbed harmonic oscillator

We continue to examine the connection between perturbation theory and the Van Vleck unitary transformation. Here we illustrate the formalism derived earlier by applying it to compute the stationary states of the perturbed harmonic oscillator. We find that each solution of the traditional Brillouin-Wigner perturbation theory equations gives rise to a different unitary transformation which, when operating on the unperturbed ground state, produces one or the other of the perturbed eigenstates. With any of the perturbed states able to be reached by a unitary transformation on the unperturbed ground state, we advise caution in using approximate solutions of the perturbation equations in general cases, lest an unexpected stationary state be obtained.     

The effects of static quartic anharmonicity on the quantum dynamics of a linear oscillator with time-dependent harmonic frequency: Perturbative analysis and numerical calculations

The effects of quartic anharmonicity on the quantum dynamics of a linear oscillator with time-dependent force constant (K) or harmonic frequency (ω) are studied both perturbatively and numerically by the time-dependent Fourier grid Hamiltonian method. In the absence of anharmonicity, the ground-state population decreases and the population of an accessible excited state (k = 2, 4, 6…) increases with time. However, when anharmonicity is introduced, both the ground- and excited-state populations show typical oscillations. For weak coupling, the population of an accessible excited state at a certain instant of time (short) turns out to be a parabolic function of the anharmonic coupling constant (λ), when all other parameters of the system are kept fixed. This parabolic nature of the excited-state population vs. the λ profile is independent of the specific form of the time dependence of the force constant, K_t. However, it depends upon the rate at which K_t relaxes. For small anharmonic coupling strength and short time scales, the numerical results corroborate expectations based on the first-order time-dependent perturbative analysis, using a suitably repartitioned Hamiltonian that makes H₀ time-independent. Some of the possible experimental implications of our observations are analyzed, especially in relation to intensity oscillations observed in some charge-transfer spectra in systems in which the dephasing rates are comparable with the time scale of the electron transfer. © 1995 John Wiley & Sons, Inc.     

Finite difference Poisson-Boltzmann electrostatic calculations: Increased accuracy achieved by harmonic dielectric smoothing and charge antialiasing

A common problem in the calculation of electrostatic potentials with the Poisson-Boltzmann equation using finite difference methods is the effect of molecular position relative to the grid. Previously a uniform charging method was shown to reduce the grid dependence substantially over the point charge model used in commercially available codes. In this article we demonstrate that smoothing the charge and dielectric values on the grid can improve the grid independence, as measured by the spread of calculated values, by another order of magnitude. Calculations of Born ion solvation energies, small molecule solvation energies, the electrostatic field of superoxide dismutase, and protein-protein binding energies are used to demonstrate that this method yields the same results as the point charge model while reducing the positional errors by several orders of magnitude. © 1997 by John Wiley & Sons, Inc.     

Note on open shell restricted SCF calculations for rotation barriers about C-C double bonds: Ethylene and allene

It is shown thatab initio open-shell SCF calculations yield acceptable results for rotation barriers about double bonds in contrast to more conventional closed-shell SCF calculations. Using basis sets of double zeta+polarization quality the SCF values for the rotation barriers of ethylene and allene are obtained to be 48 and 50 kcal/mole, respectively. An IEPA estimate of the influence of electron correlation leads to values of 64 and 52 kcal/mole, respectively, which are in reasonable agreement with the experimental values.     

An empirical potential function for metal centers: Application to molecular mechanics calculations on metalloproteins

At present, most molecular mechanics programs that model metalloproteins do not allow for specific geometric requirements at metal centers. An analysis of small-molecule crystal structures containing four-, five-, and six-coordinated zinc, retrieved from the Cambridge Structural Database (CSD), leads to a new metal-center potential function for use in molecular mechanisc programs. This potential function includes as variables the metal-ligand separations and the angles subtended at the metal and allows specifically for distortions from frequently occurring types of coordination geometries (e.g. tetrahedron square pyramid, trigonal bipyramid, and octahedron). The combination of such a metal-center potential function and one for hydrogen bonds allowing for Ione-pair directionality makes monopole electrostatic contributions to the force-field energy superfluous, thus circumventing many problems associated with the assignment to the force-field energy superfluous, thus circumventing many problems associated with the assignment of atomic apartial charges and a dielectric constant. The molecular mechanics program ‘YETI’, containing both types of potential functions, has been used to refine details of substrate binding of 16 complexes of human carbonic anhydrase II with small molecules. The stereochemistry of the refined complexes is in good agreement with data retrived from the CSD and hence allows realistic structure activity relationships.     

Fluorescence quenching by excimer formation: quenching constant approximations for excimer formation-dissociation by classical potential models

Fluorescence quenching by excimer formation is studied on the assumption that the excimer formation and dissociation can be modeled as overdamped motion in an attractive potential (classical potential models). An approach to the zeroth-order, concentration-independent quenching constants is proposed which starts from a mean reaction-time ansatz and reduces the calculation essentially to the solution of the eigenvalue problem for the Smoluchowski operator which describes the excimer equilibration. For a square-well potential model it is shown that a quenching constant expansion in terms of relaxation modes, truncated at the kinetic level, gives a satisfactory approximation of the recently obtained exact zeroth-order result under defined conditions. It is demonstrated how this two-mode approach can be applied for a quenching constant estimation if the excimer formation and dissociation are modeled by more realistic interaction potentials, as for instance, Morse- or Gaussian-type ones.     

Vibrational spectra and potential energy distributions for 3-cyclopropenecarboxaldehyde by density functional and normal coordinate calculations

The structural stability and internal rotation in 3-cylopropenecarboxaldehyde were investigated by ab initio calculations with 6-311++G^** basis set. The calculations were carried out at the restricted Hartree–Fock (HF) and the Density Functional B3LYP levels. The vibrational frequencies were computed at HF and DFT-B3LYP levels. Normal coordinate calculations were carried out and potential energy distributions were calculated for the cis and the trans conformers of the molecule.     

SCF-Xα-SW model calculations for metal clusters of nickel,copper, and silver and for the oxygen chemisorption on these metals

The oxygen chemisorption on Ni, Cu and Ag is studied by comparing PE spectra of these systems and SCF-Xα scattered-wave cluster models. Consideration of octahedral clusters M₆ (M = Ni, Cu, Ag) shows that they are large enough to reproduce trends in energy differences, such as the width of the d-bands and the distance from the top of the d-bands to the Fermi level, as found in experiment and in bulk energy band calculations. Substrate model clusters for the interaction of oxygen with different metal surfaces are derived from an octahedron by removing one ((100) face) or two adjacent metal atoms ((110) face). Comparing the UPS difference spectrum for O/Ag (110) with several Ag₄O cluster models makes it possible to interpret the peaks above the Ag d-band as O-Ag anti-bonding levels. These peaks are caused by O 2p-Ag 4d and O 2p-Ag 5s interaction. The corresponding bonding levels fall in the Ag d-bands and cannot therefore be identified with confidence in the spectra. The decreasing intensity of the oxygen derived peak below the metal d-band in the UPS spectra when going from Ni to Cu to Ag, and the simultaneously increasing O peaks above the d-band correlate with the changes of the localization of the corresponding bonding and anti-bonding levels in the oxygen sphere and the decreasing strength of the chemisorption bond.     

Anab initio potential energy surface and dynamics calculations for vibrational excitation of I2 by He

We present newab initio calculations of the interaction potential and the elastic and inelastic cross sections for He scattering by I₂. The electronic structure calculations of the interaction potential are based on an extensive one-electron basis set (triple zeta plus ad set on each I, ans function plus ap set at the I₂ bond center, and quadruple zeta plus twop sets on He), a two-configuration-SCF orbital set, and a configuration interaction calculation based on all single and double excitations out of the two-configuration reference space. The calculations are performed at 16He-I₂ distances for nine combinations of I₂ vibrational displacement and orientation. A new form of analytic representation is presented that is particularly well suited to efficient and accurate fitting ofab initio interaction potentials that include vibrational displacements. Scattering calculations are performed by the vibrational close-coupling, rotational-infinite-order-sudden approximation with a converged vibrational basis.     

Necessary and sufficient conditions for the appearance of extrema on concentration distribution curves in complex equilibrium systems

It is shown that species distribution curves with at least one minimum may be found in most equilibrium systems with three or more components. Whether such concentration minima are actually observed then depends on the values of the equilibrium constants and on the total (analytical) concentrations of the components. A general algorithm is given for the necessary and sufficient conditions for the appearance of extrema in multicomponent systems. Three-component systems are studied in more detail and special attention is given to the limiting case of a horizontal inflection, i.e., the point where the concentration minimum just disappears. Two well-studied chemical examples, the Cu(2+)-diethylenetriamine-OH(-) and Hg(2+)-Cl(-)-OH(-) systems are discussed, along with a simple model system showing as many as five extrema on a single distribution curve.     

Generalized Kubo relations and conditions for anomalous diffusion: physical insights from a mathematical theorem

The paper describes an approach to anomalous diffusion within the framework of the generalized Langevin equation. Using a Tauberian theorem for Laplace transforms due to Hardy, Littlewood, and Karamata, generalized Kubo relations for the relevant transport coefficients are derived from the asymptotic form of the mean square displacement. In a second step conditions for anomalous diffusion are derived for the asymptotic forms of the velocity autocorrelation function and the associated memory function. Both spatially unconfined and confined diffusion processes are considered. The results are illustrated with semi-analytical examples.