Generalized internal vibrational coordinates are optimized and used to describe highly excited vibrational motions in the N2O molecule. These coordinates are defined as the magnitudes of two vectors, which are expressed as linear combinations of the internal displacement vectors and the angle formed between them. They depend on two parameters and contain, as particular cases, valence and orthogonal (Jacobi, Radau, etc.) coordinate systems. The coordinates are optimized by minimizing unconverged variationally computed vibrational energies with respect to the external parameters. A comparison of the optimal internal coordinates derived for N2O with valence and hyperspherical normal coordinates is made. The optimal internal coordinates are also used to determine a new potential energy function for N2O from the observed vibrational frequencies up to 15 000 cm−1 using fully variational calculations. The quality of the adjusted potential energy function is checked by computing vibrational-rotation energy levels and rotational constants for Σ, Π, Δ, Φ, and Γ states and comparing them with the observed values.
In this article we develop the solutions of the time-independent Schrödinger equation analytically for a system of two identical harmonic oscillators linearly coupled in the kinetic and potential energies, using oblique coordinates. These coordinates are constructed by making a non-orthogonal rotation of the original coordinates that allows us to write the matrix representation of the Hamiltonian operator in a block-diagonal form characterized by the polyadic quantum number n = n1 + n2. The expression for the oblique rotation angle is first derived as a function of the kinetic and potential coupling parameters. Then the expression for the exact energy levels of the system is obtained by making an orthogonal rotation of the oblique coordinates, and the structure of the spectrum is discussed. Finally, the eigenvectors of the Hamiltonian polyadic blocks are identified with the discrete orthogonal polynomials of Krawtchouk, which provides an analytical expression for the coefficients of the linear combinations of the wave functions in oblique coordinates.
In this article we extend our previous quantum-mechanical treatment of the system of identical harmonic oscillators linearly coupled in the kinetic and potential energies using oblique coordinates (Zúñiga et al 2019 J. Phys. B: At. Mol. Opt. Phys. 52 055101) to the general system of coupled non-identical harmonic oscillators. Oblique coordinates are obtained by making non-orthogonal rotations of the original coordinates that convert the matrix representation of the quadratic Hamiltonian operator into a block diagonal matrix. Accordingly, we derive the analytical formula for the oblique rotation angles, and obtain the expressions, also analytical, for the energy levels and eigenfunctions of the system in terms of the oblique parameters. We also show that oblique coordinates are in fact dependent on one of the rotational angles whose value can be freely chosen, so they form, in fact, a continuous set of coordinates that are especially flexible in dealing with more complex vibrational systems. To illustrate this, we make a numerical application to a system of kinetically coupled Barbanis oscillators, which clearly shows the advantages of using oblique coordinates to determine the energy levels and wave functions of the system variationally.
m Analytical exact expressions are obtained for matrix elements of the modified Poschl-Teller oscillator over different operators including powers of the hyperbolic functions sinh( a x ) , cosh(a x ) , and tanh( a x) and the differential operators d / d x and d 2 / d x 2 . These expressions are derived using explicitly the Poschl-Teller eigenfunctions. In addition, several recursion relations connecting different Poschl-Teller matrix elements are obtained using the factorization and hypervirial techniques. It is shown that these relations can be used to make easier the computation of the matrix elements. 0 1996 fohn Wiley & Sons, Inc.Let us consider the evaluation of the matrix elements of powers of sirMax), cosh(ax), and 44 VOL. 57, NO. 1
The Morse–Pekeris oscillator model for the calculation of the vibration–rotation energy levels of diatomic molecules is revisited. This model is based on the realization of a second-order exponential expansion of the centrifugal term about the minimum of the vibrational Morse oscillator and the subsequent analytical resolution of the resulting approximate radial eigenvalue equation for the rotating Morse oscillator. It is, however, possible to develop the Morse–Pekeris model by transforming the second-order exponential expansion of the rotating Morse oscillator into an effective Morse oscillator and by then taking advantage of the well-known solutions for the Morse oscillator. Such a transformation can be carried out in a straightforward and instructive way merely by making a suitable change of variable of the internuclear coordinate. In addition, this alternative method allows us to correct the Morse–Pekeris model by simple use of first-order perturbation theory. The efficiency of this treatment is demonstrated by carrying out a numerical application to the H2 molecule.
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