Oscillator Basis Functions Research Articles

6He as a Three-Cluster System: Investigation of the Ground State and Continuum 0+ States

With the use of microscopic approaches, i.e., the algebraic version of the resonating group method employing the hyperspherical harmonic oscillator basis functions, the general properties of weakly bound light nuclei with the lowest three-body decay threshold are studied. As a way of truncating the basis space, the minimal approximation based on the Pauli-allowed harmonic oscillator basis states is proposed. For the case of the 6He nucleus, the basis states are constructed, and the wave function of the ground state as well as the scattering S-matrix for the continuum spectrum is found. Although the exchange mixture parameter of the Minnesota potential is chosen rather large to fit the experimental ground state energy, all basic features of the system are well reproduced. The resonance behavior of the latter indicates that a 0+ resonance state exists at ∼ 8 MeV over the threshold. Calculations of the probability of the monopole transition from the ground state to the continuum states confirm this result.

Recursive computation of Hamiltonian matrix elements using harmonic oscillator eigenfunctions: Application to the inversion of ammonia and to the methyl torsion + aldehydic hydrogen wagging of acetaldehyde

A program able to use hybrid free rotor plus harmonic oscillator basis functions for the variational study of large and small amplitude vibrations is developed. The Hamiltonian matrix elements between harmonic oscillator eigenfunctions and polynomial terms are calculated using a recursive algorithm. This technique permits use of only one basic algorithm to compute the kinetic and potential parts of the Hamiltonian. In addition, the program can handle potential functions perturbed with Gaussian barriers and obtain the quantum mechanical average of a magnitude. The program is used to test the efficiency of Taylor series vs polynomial + Gaussian potential functions for the description of the ammonia inversion mode. The data for the construction of the potential functions are obtained by ab initio methodology at the QCISD/6-311G+ +(3 df, 3 dp) level. The quantum mechanical average values for the structural parameters of ammonia are computed and compared to the fully optimized ab initio results. The simultaneous methyl torsion + aldehydic hydrogen wagging motions in the S 0 state of acetaldehyde are used to illustrate the efficiency of mixed free rotor + harmonic oscillator basis functions.

Potential energy surface and vibrational analysis along the stretching vibrations of the ArHAr + ion

An ab initio calculated potential energy surface along the stretching coordinates of the linear ArHAr + ion presented. The potential surface has been obtained using the MP2/15s11p3d1f/7s2p1d level of the ab initio electronic structure calculation theory. Qualitatively the potential is characterised by a development of double minimum structure at elongated Ar---Ar distances at higher vibrational energies and by a large amplitude for the motion of the hydrogen atom. The stretching vibrational energy levels and wavefunctions have been calculated variationally using harmonic oscillator basis functions. The vibrational analysis indicated extensive mixing of the zeroth-order harmonic oscillator vibrational states already at the bottom of the well giving rise to strong absorption “progression” starting at 1208 cm −1. The Einstein A coefficients for spontaneous emission transitions have also been calculated and the IR emission spectrum is predicted to be particularly rich in the 1100–1900 cm −1 spectral region.

A weak-mode representation of floppy molecules. II. Kinetic energy and total Hamiltonian matrix

The total Hamiltonian matrix of a floppy molecule (i.e. in a bound state for which large-amplitude isomerization-like deformations are possible), is derived in an entirely analytical manner which rests on the basic idea that the weakest mode of internal deformation is separated from the other stronger modes. The latter are supposed to undergo throughout only small-amplitude vibrations with respect to equilibrium positions which vary adiabatically with the former. In using (i) for the potential energy function, a Taylor expansion in the strong-mode displacements around the weak-mode isomerization reaction pathway, and (ii) for the kinetic energy operator matrix representation, a basis that includes for the strong modes adiabatically displaced harmonic oscillator basis functions, the total Hamiltonian operator can be represented by a sparse matrix whose diagonal elements are very like analytical spectroscopic terms, and the off-diagonal elements are also analytical expressions in terms of the quantum numbers of the basis functions. This adiabatic representation, which is known to be efficient for numerical diagonalization, is also particularly interesting from the physical point of view insofar as it allows to bring very clearly to the fore the coupling scheme between the internal deformations. The J = 0 case (no angular momentum) is developed first. Then the case where J ≢ 0 is treated (also analytically), in using for the first time in dynamical calculations the principal-axis system as the body-fixed (BF) frame of reference. This last feature is important, since it allows to distinguish (at least formally, if not physically) the Coriolis contribution to the total energy from the rotational one.

Harmonic oscillator tensors. I. The nondegenerate case

AbstractIt is shown that the Heisenberg Lie algebra of the nondegenerate harmonic oscillator leads to a basis {J+, J0, J−} of LASU(2). The Hamiltonian of the system is proportional to J0, and the basis elements give rise to irreducible tensors in the associative enveloping algebra of the Heisenberg Lie algebra. The construction of these irreducible tensors is studied with special attention being paid to the case in which they act upon a single vector space spanned by the harmonic oscillator basis functions. A tensor coupling rule is developed, and useful application is made of it in the calculation of general expressions for vibrational operators and their matrix elements. Throughout, the value of the additional algebraic quantum numbers (l, m) is emphasized.

Potential energy surfaces for the ground electronic state of water, hydrogen sulfide, and hydrogen selenide

A curvilinear internal coordinate Hamiltonian is used to calculate vibrational term values of water, hydrogen sulfide, and hydrogen selenide variationally with Morse oscillator basis functions for the stretches and harmonic oscillator basis functions for the bend. Rotational parameters, a constants, are calculated with perturbation theory expressions. Potential energy parameters are optimized with the nonlinear least-squares method using vibrational term values, a constants, and Coriolis coupling coefficients as data. The model is applied to five isotopic species of water, five isotopic species of hydrogen sulfide, and to two isotopic species of hydrogen selenide. The observed vibrational and rotational data of the studied molecules are reproduced well

Fermi resonances and local modes in water, hydrogen sulfide, and hydrogen selenide

A simple vibrational curvilinear internal coordinate Hamiltonian for bent H2X molecules is constracted by expanding the g matrix elements and the potential energy function in terms of the Morse variable y=1−exp(−ar) and retaining important local mode and Fermi resonance terms. The eigenvalues of this Hamiltonian are calculated variationally using Morse oscillator basis functions for the stretches and harmonic oscillator basis functions for the bend. The nonlinear least-squares method is used to optimize the potential energy parameters. The model is applied to water, hydrogen sulfide, and hydrogen selenide. Experimental vibrational levels up to 18 500 cm−1 for five symmetrical isotopic species of water are reproduced with a standard deviation of about 4 cm−1. For both hydrogen sulfide and hydrogen selenide two symmetrical isotopic species were included in the optimization procedure and standard deviations of 1.0 and 0.66 cm−1 were obtained. The potential energy parameters obtained agree well with previous anharmonic force field calculations.

Sextic force field of nitrous oxide

The sextic force field in the curvilinear internal coordinates has been studied for the nitrous oxide molecule from the spectroscopic data of 14N 2 16O, 14N 15N 16O, and 15N 14N 16O. The bands below 6600 cm −1 have been used. The force constants in the internal coordinates are converted to those in dimensionless normal coordinates by two successive transformations. The vibration Hamiltonian matrix for each symmetry species of a given isotopic species has been constructed from the harmonic oscillator basis functions, and it is then diagonalized numerically to give the vibrational energy levels and the wavefunctions. The latter have been used for the evaluation of ratational constants. The least-squares refinement has been very successful in the present study, and it is shown that the general quartic force field supplemented by the quintic and sextic stretching diagonal force constants estimated from the Morse function, provided that the terms up to sextic are kept in the dimensionless normal coordinate space, well reproduces the spectroscopic constants such as the vibrational levels, rotational constants, l-type doubling constants, and centrifugal distortion constants. The spectroscopic constants of the isotopic molecules which are excluded from the refinement process are also in good agreement with the computed ones. The bond dissociation energies of the NN and NO bonds estimated from the present results have been critically examined.

Kinetic anharmonic coupling in the trihalomethanes: A mechanism for rapid intramolecular redistribution of CH stretch vibrational energy

The coupling of CH stretching and CH bending vibrations in trisubstituted methanes is analyzed. Improved spectroscopic constants, especially the cubic anharmonic stretch–bend coupling constants, are extracted from Fermi resonances in the overtone spectra of HCF3 and HCCl3. Both harmonic oscillator and Morse oscillator basis functions are used in the analysis and the results compared. That part of the coupling which arises from the kinetic energy as expressed in curvilinear coordinates is calculated and compared to the coupling calculated using the more conventional rectilinear treatment. Use of curvilinear coordinates is found to provide significant advantages. The formalism for curvilinear normal coordinates is clarified and generalized. From these calculations and the spectral analysis, one of the cubic anharmonic constants of the potential energy surface is extracted for comparison with ab initio calculations. The curvilinear model of the CH stretch–bend interaction tested for these isolated CH chromophores is expected to be useful in understanding CH bonds and vibrational energy flow in larger hydrocarbons.

A comparison of computational and model approaches to the two-coupled anharmonic oscillator problem in five-membered ring compounds

The bend/twist eigenvalues and transitions of 2,5-dihydrofuran, cyclopentanone, cyclopentene, silacyclopentane, and germacyclopentane are calculated with a two-dimensional hamiltonian in several different ways, viz. using two-dimensional isotropic harmonic oscillator basis functions, a directproduct basis set consisting of two one-dimensional anharmonic oscillator functions, a similar set of basis functions improved in the fashion of partial SCF, evaluating explicitly the perturbation corrections up to third order, and seeking the series limit for these corrections by means of Padé approximants. These eigenvalues and transitions are compared for the four methods, and compared also with previous experimental measurements and with previous calculations. The merits or disadvantages of the methods are discussed in relation to the degree of coupling of the two modes. Substantially different values for the potential constants of sila- and germacyclopentane are suggested.

Vibrational angular momentum in the theory of vibration-rotation spectra

The theory of the vibrational angular momentum term\(\hat \pi _\alpha ^2 \) in the effective molecular rotational Hamilton operator is briefly reviewed. Novel vibrational matrix elements of\(\hat \pi _\alpha ^2 \), off-diagonal in the basis of the coupled linear oscillator basis functions, are derived for asymmetric top molecules. These matrix elements contain two or three distinct vibrational quantum numbers, respectively. A group-theoretical scheme; the use of repeated Jahn’s rules, is given for testing the existence of the proposed matrix elements. Finally suggestions are made for the application of the latter.

Two-mode vibronic interaction between neighboring 1 2A2 and 2 2B2 excited electronic states of the benzyl radical

The 2 2B2–1 2B2 transition of the benzyl radical occurs in the same frequency region as the 1 2A2–1 2B2 transition. Vibronic coupling between the 2 2B2 and 1 2A2 excited electronic states via 6b and 18b vibrational modes is shown to be responsible for the unusual structure of the electronic spectrum of benzyl observed in the visible region. Calculations of vibronic interaction were carried out using a model constructed with crude adiabatic, harmonic oscillator basis functions. The model was expressed in terms of four dimensionless parameters: D, the energy separation between the deperturbed 2 2B2 and 1 2A2 states; Cα and Cβ, which are interaction constants for the α≡18b and β≡6b modes, respectively; the ratio ωα/ωβ of these zero-order mode frequencies. The possible range of these constants is discussed in detail. The model reproduces well the vibronic energies and relative transition intensities of bands observed within 700 cm−1 of the 1 2A2–1 2B2 transition origin for isotopic benzyl radicals. The deperturbed 2 2B2 state origin is found to lie between 430 and 485 cm−1 above the 1 2A2 state in gas phase benzyl-h7 and between 355 and 370 cm−1 in benzyl-d7.

Generalized N-body harmonic oscillator basis functions for the hyperspherical approach

The concept of harmonic oscillator functions is generalized to the multidimensional space of the N-body system. Explicit expressions are found for these functions. They are shown to constitute a complete orthonormal set of basis functions in which one can expand the radial functions of the hyperspherical approach. Because of the wrong asymptotic behavior of the harmonic oscillator basis a method for correcting the expansion is also given.

Theory of helium submonolayers: Formulation of the problem and solution in the zero coverage limit

We are concerned here with the physical adsorption of helium atoms on a crystal surface. A general formulation of the problem is first given, which takes into account both the static periodic potential of a model substrate and He-He interactions. This formulation leads in a natural way to expressions of surface phonon states obtained recently by Jackson. As the first step of carrying out our formalism, we extend an earlier single-particle calculation by Lennard-Jones and Devonshire. A set of basis functions consisting of products of Mathieu functions and eigenfunctions of a Morse Hamiltonian are defined, and a low-order perturbative calculation is performed. While the procedure provides a qualitative description of the energy bands, it appears to be quantitatively inadequate. To facilitate a basis for future work, which will include the effects of correlations between the helium atoms, we carry out variational calculations in this zero-coverage limit, using various combinations of Gaussian and Morse functions. A reasonably single combination results in a ground-state energy which is just 1% higher than that of Ricca, Pisani, and Garrone, who performed elaborate computations to solve a secular equation obtained with harmonic oscillator basis functions.

Nuclear Photodisintegration in the1sShell: A Perturbative Approach to the Dipole Sum Rules

The integrated and bremsstrahlung-weighted $E1$ photoabsorption cross sections, ${\ensuremath{\sigma}}_{\mathrm{int}}$ and ${\ensuremath{\sigma}}_{\mathrm{b}}$, have been calculated for the lightest nuclei, ${\mathrm{H}}^{2}$, ${\mathrm{H}}^{3}$, ${\mathrm{He}}^{3}$, and ${\mathrm{He}}^{4}$, within the framework of a second-order perturbation procedure. For the purpose of comparison, two Gaussian nucleon-nucleon potentials were employed: one containing a repulsive core and central attractive and tensor components, the other only central attractive and tensor components. The results for ${\ensuremath{\sigma}}_{\mathrm{int}}$ indicate that while reasonable over-all agreement with experiment may be achieved with either potential, the component contributions differ widely because of the admixture of the repulsive core. Furthermore, the results for ${\ensuremath{\sigma}}_{\mathrm{b}}$ for the deuteron seem to indicate a deficiency in the present choice of oscillator basis functions when applied to the loosely bound system. Considerable improvement for ${\ensuremath{\sigma}}_{\mathrm{b}}$ is noted for ${\mathrm{H}}^{3}$, ${\mathrm{He}}^{3}$, and ${\mathrm{He}}^{4}$, where the nuclei are more tightly bound and less extended structures. It is found by analyzing exact and approximate third-order contributions to both the deuteron binding energy and integrated cross section that the use of a wave function containing parameters determined by minimization of the perturbation expansion through second order is probably not acceptable, at least for this nucleus. This is further substantiated by comparison with the binding-energy results obtained from an exact numerical solution of the coupled $S$ and $D$ radial differential equations.