Determining quantum bound-state eigenvalues and eigenvectors as functions of parameters in the Hamiltonian: An efficient evolutionary approach

This paper addresses the problem of finding the quantum bound-state energy eigenvalues and eigenvectors as functions of a set of continuous parameters characterizing a Hamiltonian. A recent paper introduced a parametric equations of motion (PEM) method for this purpose, and the present work extends the method to allow for the analysis ofa single energy level and its wavefunction. After solving the Schrodinger equation for its nth eigenvalue and eigenvector, evaluated at a reference value of the Hamiltonian's parameters, the differential equations of the single-state PEM (ss-PEM) method are used to propagate the nth energy level and its eigenfunction through the entire parameter space of the Hamiltonian. The new ss-PEM method, which reduces the number of differential equations to be solved, appears more efficient than diagonalization when the energy is sought at a moderate number of values for the parameters in the Hamiltonian. The PEM methods are extended to treat non-orthogonal basis sets that facilitate more rapid convergence of the solutions. The energy of the ss-PEM, which is always an upper bound to the true energy, is exact in the limit of a complete basis set. Connections of the method are made to linear variational calculations, Dalgarno-Lewis perturbation theory and the original PEM methods. Sets of non-orthogonal Chebyshev polynomials are employed in illustrations of the ss-PEM method to determine (a) the ground-state energy as a function of internuclear separation in the hydrogen molecular ion, and (b) the ground-state energy of two electron ions as a function of nuclear charge. The calculation withthe two-electron ions involves two parameters, the nuclear charge and a basis set parameter that influences the distribution of the nodes of the Chebyshev basis functions. Evolution of the basis set parameter to improve the energies of the ions suggests an additional application of the ss-PEM method in which quantum energies are minimized with respect to nonlinear basis set parameters. The ss-PEM method offers an effective tool for mapping the solutions of the Schro dinger equation as a function of model parameters in the Hamiltonian.