Harmonic oscillator tensors. III: Efficient algorithms for evaluating matrix elements of vibrational operators

Succinct expressions for the matrix elements of various vibrational operators have been derived in the basis of the nondegenerate harmonic oscillator. Among these are the matrix elements of\([q^k e^{\lambda q} ,q^k \sin ^l (\mu q),q^k \cos ^l (\mu q),q^k \sinh ^l (\mu q)\) and\(q^k \cosh ^l (\mu q)\), which are found to be dependent upon two quantities and their derivatives. Furthermore, the derivative property of the commutator is used to obtain an explicit expression for the derivatives of an operator in terms of its nested commutator with the conjugate momentum. It may be applied to any of the above cases to obtain the matrix representatives of expressions such as the mixed products\(\sin ^l (\mu q)\cos ^{l - m} (\mu q)\), for example. In addition, a simple expression for 1/q is given and its derivatives may be evaluated by this commutator technique. Also the matrix elements of a Gaussian-type operator \(q^k e^{\lambda q^2 } \) has been evaluated.