The two-dimensional anharmonic oscillator II: the microwave spectrum of silyl isocyanate, SiH3NCO

The J + 1 ← J transitions (J = 2, 3, 4, 5, and 6) in the microwave spectrum of SiH3NCO have been assigned for the vibrational ground state and for the vibrational states v10 = 1, 2, and 3. The results for v10 = 0 confirm earlier work. The vibration-rotation constants show a remarkable variation with v10 and l10. To a large extent the anomalous behavior of these constants has been explained in terms of a strongly anharmonic potential function for the ν10 vibrational mode.

The two-dimensional anharmonic oscillator III: the CCC bending mode of C3O2

Newly observed data on the rotational constants of carbon suboxide in excited vibrational states of the low-wavenumber bending vibration ν7 have been successfully interpreted in terms of the two-dimensional anharmonic oscillator wavefunctions associated with this vibration. By combining these results with published infrared and Raman spectra the vibrational assignment has been extended and a refined bending potential for ν7 has been derived: this has a minimum at a bending angle of about 24° at the central C atom, with an energy maximum at the linear configuration some 23 cm−1 above the minimum. From similar data on the combination and hot bands of ν7 with ν4 (1587 cm−1) and ν2 (786 cm−1) the effective ν7 bending potential has also been determined in the one-quantum excited states of ν4 and ν2. The effective ν7 potential shows significant changes from the ground vibrational state; the central hump in the ν7 potential surface is increased to about 50 cm−1 in the v4 = 1 state, and decreased to about 1 cm−1 in the v2 = 1 state. In the light of these results vibrational assignments are suggested for most of the observed bands in the infrared and Raman spectra of C3O2.

The two-dimensional anharmonic oscillator I: vibration-rotation constants of fulminic acid, HCNO

The lowest-wavenumber vibration of HCNO and DCNO, ν5, is known to involve a largeamplitude low-frequency anharmonic bending of the CH bond against the CNO frame. In this paper the anomalous vibrational dependence of the observed rotational constants B(v5, l5), and of the observed l-doubling interactions, is interpreted according to a simple effective vibration-rotation Hamiltonian in which the appropriate vibrational operators are averaged in an anharmonic potential surface over the normal coordinates (Q5x, Q5y). All of the data on both isotopes are interpreted according to a single potential surface having a minimum energy at a slightly bent configuration of the HCN angle ( 170°) with a maximum at the linear configuration about 2 cm−1 higher. The other coefficients in the Hamiltonian are also interpreted in terms of the structure and the harmonic and anharmonic force fields; the substitution structure at the “hypothetical linear configuration” determined in this way gives a CH bond length of 1.060 Å, in contrast to the value 1.027 Å determined from the ground-state rotational constants. We also discuss the difficulties in rationalizing our effective Hamiltonian in terms of more fundamental theory, as well as the success and limitations of its use in practice.

Two-dimensional anharmonic oscillator: application to 2,5-dihydrofuran

A program has been developed to calculate the energy levels and corresponding wavefunctions for a two‐dimensional anharmonic potential surface of at least C2v symmetry. This program has been employed to explain the high resolution splittings observed in the far infrared spectrum of 2,5‐dihydrofuran. The magnitude of the cross term connecting the ring‐twisting and ring‐puckering modes of 2,5‐dihydrofuran is sufficiently large to be significant. The potential surface determined also suggests that the ring‐twisting mode may be slightly anharmonic.

Approximate expression for the energy of a D-dimensional anharmonic oscillator

An approximate expression is constructed for the energy of an anharmonic potential with centrifugal barrier. In order to obtain such an analytical expression, the quasi-exact solvability is used and then a fitting of these exact solutions is done.

Approximate analytic expression for the eigenenergies of the anharmonic oscillator V(x)=Ax6+Bx2

An approximate analytic expression for the eigenenergies of the anharmonic oscillator V(x)=Ax6+Bx2 is introduced, starting from particular analytic solutions which are valid when certain relations between the parameters A and B are held. © 1995 The American Physical Society.

A Nonlinear System Under Combined Periodic and Random Excitation

The anharmonic oscillator under combined sinusoidal and white noise excitation is studied using the Gaussian closure approximation. The mean response and the steady-state variance of the system is obtained by the WKBJ approximation and also by the Fokker Planck equation. The multiple steadystate solutions are obtained and their stability analysis is presented. Numerical results are obtained for a particular set of system parameters. The theoretical results are compared with a digital simulation study to bring out the usefulness of the present approximate theory.

On the determination of energy eigenvalues and coupling strengths using duality

The Shifman-Vainshtein-Zakharov method of determining the eigenvalues and coupling strengths, from the operator product expansion, for the current correlation functions is studied in the nonrelativistic context, using the semiclassical expansion. The relationship between the low-lying eigenvalues, and the leading corrections to the imaginary-time Green function is elucidated by comparing systems which have almost identical spectra. In the case of an anharmonic oscillator it is found that with the procedure stated in the paper, that inclusion of more terms to the asymptotic expansion does not show any simple trend towards convergence to the exact values. Generalization to higher partial waves is given. In particular for the P-level of the oscillator, the procedure gives poorer results than for the S-level, although the ratio of the two comes out much better.

Rotational cut-off and anharmonicity effects on thermodynamic properties of gases at high temperatures

The exact expressions for the partition function (Q) and the coefficient of specific heat at constant volume (Cv) for a rotating-anharmonic oscillator molecule, including coupling and rotational cut-off, have been formulated and values of Q and Cv have been computed in the temperature range of 100 to 100,000 K for O2, N2 and H2 gases. The exact Q and Cv values are also compared with the corresponding rigid-rotator harmonic-oscillator (infinite rotational and vibrational levels) and rigid-rotator anharmonic-oscillator (infinite rotational levels) values. The rigid-rotator harmonic-oscillator approximation can be accepted for temperatures up to about 5000 K for O2 and N2. Beyond these temperatures the error in Cv will be significant, because of anharmonicity and rotational cut-off effects. For H2, the rigid-rotator harmonic-oscillator approximation becomes unacceptable even for temperatures as low as 2000 K.

Application of dynamical Lie algebraic method to atom-diatomic molecule scattering

The dynamical Lie algebraic approach developed by Alhassid and Levine combined with intermediate picture is applied to the study of translational-vibrational energy transfer in the collinear collision between an atom and an anharmonic oscillator. We find that the presence of the anharmonic terms indeed has an effect on the vibrational probabilities of the oscillator. The computed probabilities are in good agreement with those obtained using exact quantum method. It is shown that the approach of dynamical Lie algebra combining with intermediate picture is reasonable in the treating of atom-anharmonic oscillator scattering.

Microwave spectrum of deuterated silyl isocyanate, SiD3NCO

The microwave spectrum of SiD3NCO has been observed and analyzed for 18 different vibrational states in the ν10 manifold. Some accidental resonances have been observed and analyzed. The vibrational dependence of the rotational and l-doubling constant and centrifugal distortion constant DJK has been successfully interpreted in terms of the two-dimensional anharmonic oscillator model.

Borel-Leroy summability of a nonpolynomial potential

We investigate the perturbation series for the spectrum of a class of Schrodinger operators with potential V = 1/2 x(2) + g(m-1)x(2m)/(1 + alpha gx(2)) which generalize particular cases investigated in the literature in connection with models in laser theory and quantum field theory of particles and fields. It is proved that the series obey a modified strong asymptotic condition of order (m - 1) and have an order (m - 1) strong asymptotic series in g which are shown to be summable in the sense of Borel-Leroy method.

Quantization rules for bound states in quantum wells

An algebraic reformulation of the Bohr-Sommerfeld (BS) quantization rule is suggested and applied to the study of bound states in one-dimensional quantum wells. The energies obtained with the present quantization rule are compared to those obtained with the usual BS and WKB quantization rules and with the exact solution of the Schrodinger equation. We find that, in diverse cases of physical interest in molecular physics, the present quantization rule not only yields a good approximation to the exact solution of the Schrodinger equation, but yields more precise energies than those obtained with the usual BS and/or WKB quantization rules. Among the examples considered numerically are the Poeschl-Teller potential and several anharmonic oscillator potentials. which simulate molecular vibrational spectra and the problem of an isolated quantum well structure subject to an external electric field.