ANALYTICAL SOLUTIONS OF ARBITRARY ORDERS TO THE CLASSICAL AND QUANTUM OSCILLATORS WITH VELOCITY-DEPENDENT QUARTIC ANHARMONICITIES

The classical oscillator with velocity-dependent anharmonicity (COVDA) arises when the velocity of the oscillator is reasonably high. By using an intuitive approach, we obtain an approximate analytical solution of arbitrary order to the problem of a COVDA. In addition to the third harmonic generation manifested by the nonlinear interaction, it is found that the solution contains the secular terms since the intuitive approach basically depends on the perturbation method. By assuming the small anharmonic constant, the secular terms are summed up for all orders and we obtain the renormalization of the frequency. The frequency of the oscillator decreases with the increase of the anharmonic constant. Interestingly, the magnitude of the shifts of the frequency of the oscillator with velocity-dependent quartic anharmonicity is identical with those of the oscillator with q-dependent quartic anharmonicity. However, the sign of the shifts for those two types of anharmonic oscillator is opposite in nature. These results indicate that the frequency shifts of the oscillator are actually the resultant effects (shifts) due to the q-dependent and the velocity-dependent anharmonicities. Finally, with the help of the correspondence principle, the solution of a quantum oscillator with velocity-dependent quartic anharmonicity (QOVDA) is obtained from the knowledge of the solution of its classical counterpart.

Empirical interionic potentials for alkali halide molecules

A systematic study of families of empirical internuclear potential energy functions based on the Rittner potential is presented for alkali halide molecules, and comparison is made with recent binding energy and anharmonic constant data. It is concluded that the Rittner formalism, however parameterized, is incapable of consistent prediction of several molecular properties simultaneously, and that the potential seems to be failing to model some aspect of the detailed bonding of the molecule. Inclusion of a Gaussian form of repulsion potential in the model gives excellent agreement for the dissociation energies, but in general extreme care must be taken when applying empirical potentials of the Rittner form to more complex systems such as ionized molecules or clusters.

The 1650-1350 Å absorption spectrum of nitrogen dioxide

An analysis of the 1650-1350 Å band system of nitrogen dioxide has been carried out. A pattern of band spacings and intensities is found that is complex but regular. It is shown that this pattern is qualitatively, and to a large extent quantitatively, just what would be expected for a transition in which the shape of the molecule changes from bent to linear. The transition is a parallel one and the upper state has 2 Σ + u symmetry. The symmetrical stretching frequency is increased from its ground-state value to ca. 1420 cm -1 in the upper state. The upper-state bending frequency is ca. 600 cm -1 . The N — O length is decreased from its groundstate value, probably to 1·1(3) Å. The upper state resembles closely the ground state of NO + 2 . The transition is to be classed as one of the Rydberg transitions leading to the first ionization potential of NO 2 ; and the orbital to which the odd electron is transferred in the transition is (pσ) in type. The anharmonic constant g 22 for the linear upper state is found to be 2·(3) cm -1 . Other Rydberg transitions may well be present in the region, but have not been definitely identified.

Solvent effects in infra-red spectroscopy

The effects of solute-solvent interactions on the vibrational spectrum of a dissolved molecule are evaluated by supposing that the interaction energy U can be expanded as a power series in the normal co-ordinates of the active molecule. By treating U and the anharmonic terms in the potential energy function of the free molecule as small perturbations to the harmonic oscillator Hamiltonian, the solvent shifts, ∆ ω , in the vibrational frequencies are found to be proportional to ( U" — 3 U' A / ω e ), where U' and U" are the first and second derivatives of U with respect to the normal co-ordinates and A / ω e is an anharmonic constant obtainable from the spectrum of the gas. The theory indicates that ∆ ω / ω is independent of isotopic sub­stitution as well as of the order of the transition; experimental data for HCl and DCl support these conclusions. The intensities of vibrational bands of dissolved molecules are shown to be proportional to a factor involving the refractive index of the solvent and to be dependent upon the derivatives with respect to the normal co-ordinates of the dipole moment of the solute molecule and its near neighbours. It is predicted that for diatomic molecules the intensity of the ( n — 1)th overtone, ( A s ) 0, n' is related to the frequency ω so that ( A s ) 0, n / ω n +1 is independent of isotopic substitution, as in the gas phase.