In this paper we present explicit and simple analytical formulae for the energy eigenvaluesE_n (λ) of one-dimensional anharmonic oscillators characterized by the potentials 1/2mω²x²+λx^2α withα=2, 3 and 4. A simple intuitive criterion supplemented by the requirement of correct asymptotic behaviour, has been employed in arriving at the formulae. Our energy values over a wide range ofn andλ are in good agreement with the numerical values computed by earlier workers through very elaborate techniques. To our knowledge this is the first time that formulae of such wide validity have been given. The results for pure power oscillators are trivially obtained by going over to theω→0 limit. Approximate analytic expressions for the low order even moments ofx are also given.

The nature of inverse velocity surfaces as well as energy surfaces for elastic wave propagation in the (111) plane have been studied for a number of cubic crystals. The sections of inverse velocity surfaces by the (111) plane exhibit six-fold symmetry in all cases. Cuspidal edges are exhibited with a six-fold symmetry by both the slow transverse and fast transverse shear modes in the (111) plane, unlike the case of the (100) and (110) planes for which only the slow transverse shear mode exhibits cuspidal edges. The slow transverse mode energy surface exhibits cuspidal edges along$$(\bar 1\bar 12)$$ direction or an equivalent symmetry direction. The inverse velocity surfaces of the A-15 compounds exhibit unusually large inflexions for the slow transverse mode, whereas their energy surfaces have large cuspidal edges which intersect each other resulting in common regions of cusps.

The radial distribution function for a fluid in which the molecules interactvia a triangular well potential is considered. Expanding the radial distribution function in pwoers of βɛ, where ɛ is the depth of the potential andβ=1/k_BT the first-order terms are calculated analytically using the Percus-Yevick theory in the Baxter’s formulation. The first-order terms in the direct correlation functionc(r) are also calculated. The first- and second-order terms in the free energy obtained from the energy equation of state are calculated and compared with other calculations.

The effective potential is used to calculate the viscosity of complex polar vapours like water and ammonia. The agreement between theory and experiment confirms the validity of the effective potential.

The high resolutionnπ* electronic emission spectra ofo- andm-fluoro-benzaldehyde vapours in the region 365–560 nm excited in a discharge are reported for the first time. The spectra of both the compounds consist of theA¹A″ -X¹A′ fluorescence as well as thea³A″ -X¹A′ phosphorescence bands. In the case ofo-isomer, all the eleven out-of-plane vibrations have been observed in the fluorescence and the phosphorescence, though weakly in the latter, whereas in the case ofm-isomer, only ten have been observed in the fluorescence and nine in the phosphorescence. It is found that the most intense bands in both the fluorescence and the phosphorescence spectra of these molecules belong to the trans-O rotamer.

A possible explanation is given of the chemical shifts of x-ray K-absorption edges of metals when they undergo a chemical combination and form compounds. It is proposed that when a metal forms a compound its Fermi edge changes. It explains the numerical order as well as the nature of the chemical shifts. A fairly good agreement between the calculated and observed values has been obtained.

An analytical expression is given for the mixing parameter that characterizes the vibrational eigen vector matrix in the second-order case based on the criterion of minimization of average bending energy of molecular vibration. The expression, which is derived on the assumption that the parameter is small, contains only molecular geometry and the two normal frequencies of the vibrational species. Bent symmetric XY₂ molecular system is treated as an example and the calculated values of the parameter are in good agreement with the standard results.

Total charge transfer cross-sections for protons impinging on hydrogen has been calculated in Coulomb-projected Börn approximation. It is seen that the present calculation compares well with the available experimental data.