Large and Small Distances

The range of application of the method of comparison-matrices for the calculation of brackets to eigenvalues of the Schrodinger equation is extended to infinite bandmatrices, H. It is shown by construction that the elements αi and βi+1 of an infinite tridiagonal matrix may be calculated up to some finite maximum index, i, in a finite calculation. The tridiagonal matrix is a unitary transform of H and consequently has the same eigenvalues.

Brackets to the eigenvalues of the schrödinger equation, Part 2. Partial tridiagonalization of bandmatrices

The parameter optimization method of Part I is applied to the exponents of real STOS of first row atoms. In addition to minimum basis ground states, some independently optimized excited states are discussed in the case of Be. Local minima on the energy versus parameter surface are found in 4-configuration functions for the ground state of N. They are not present in either the simpler minimum basis function or in a more complete 8-configuration function.

Direct minimization of the energy by simultaneous variation of parameters in nonorthogonal basis functions. II. Real STOS for first row atoms

We present a state-interaction approach for matrix product state (MPS) wave functions in a nonorthogonal molecular orbital basis. Our approach allows us to calculate, for example, transition and spin–orbit coupling matrix elements between arbitrary electronic states, provided that they share the same one-electron basis functions and size of the active orbital space, respectively. The key element is the transformation of the MPS wave functions of different states from a nonorthogonal to a biorthonormal molecular orbital basis representation, by exploiting a sequence of nonunitary transformations, following a proposal by Malmqvist [Int. J. Quantum Chem. 1986, 30, 479]. This is well-known for traditional wave function parametrizations but has not yet been exploited for MPS wave functions.

A Nonorthogonal State-Interaction Approach for Matrix Product State Wave Functions

We illustrate here an efficient approach to the implementation of configuration interaction (CI) methodologies in the case of general purpose semiempirical Hamiltonians such as MINDO/3, MNDO, PM3, AM1, and SAM1. The strategy described here has been implemented in the commercial program AMPAC and successfully blends well-known and tested algorithms. The primary purpose is to derive a reliable method for determining a CI solution with minimal user input. The success and intrinsic accuracy of the approach is discussed.

An Implementation of Configuration Interaction in a General Purpose Semiempirical Context

The problem of upper and lower bounds to the first few eigenvalues of a very large or infinite tridiagonal matrix H is studied. Those eigenvalues of a comparison-matrix Mn which are lower than a characteristic limit, together with the corresponding eigenvalues of the variational matrix Hn are shown to bracket exact eigenvalues of H. Mn differs from Hn only in the last off-diagonal element and is easily obtained from H. Sufficient conditions for lower bounds are based on a low estimate of the characteristic limit. For increasing dimensions n, the lower bounds approach the exact eigenvalues from below. As a numerical illustration, brackets to the known eigenvalues of the harmonic oscillator with a linear perturbation are calculated.

Brackets to the eigenvalues of the Schrödinger equation, part 1. Tridiagonal matrices

A direct method to optimize parameters in nonorthogonal basis orbitals is discussed. The partial derivatives of the energy of the state of interest, not necessarily the ground state, with respect to the orbital parameters are calculated analytically. The required cofactors up to third order of the overlap matrices over spin-space orbitals in pairs of Slater determinants are calculated by biorthogonalization. All parameters are varied simultaneiously in the direction of the negative gradient (steepest descent). A search logic with dynamically adjusted step-size is shown, in the example of the minimum basis Ne ground state, to find the energy minimum in an efficient manner.

Direct minimization of the energy by simultaneous variation of parameters in nonorthogonal basis functions. I. Method

Non-linear parameters in orbitals are optimized by a search for a minimum on the total energy versus parameter surface. In a nonorthogonal CI these parameters are independent. The partial derivatives of the energy are then evaluated analytically and used in a simultaneous variation of all parameters in the direction of steepest descent. Some results for atomic wave functions with real s and p STOS are presented.

Direct optimization of nonlinear parameters

Iterative methods for the calculation of the lowest eigenvalue and the corresponding eigenvector are derived from the partitioned Schrodinger equation in matrix form. While they have a formal similarity to the Brillouin-Wigner perturbation expansion, they converge for a much wider class of matrices. The double iteration methods (DIM and DIMA) lead to a series of upper and lower bounds to the desired lowest eigenvalue in accordance with Lowdin's bracketing theorem and can be used in a strategy which insures convergence to the solution for arbitrarily large perturbations. A much more efficient single iteration method (SIMA) produces the proper solution even for numerically very ill-conditioned matrices (large offdiagonal elements and small separation of the diagonal elements from the lowest). An important feature of the iterative methods is their insensitivity toward roundoff errors which makes them especially suited for very large matrices, where direct diagonalisation, for example by the Jacobi method, is impracticable.

Convergent iterative solutions for the lowest state of a system with large perturbations

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