Tetsuo Morikawa

Method of the orthonormality-constrained variation

Abstract A method for carrying out the variation subject to the orthonormality conditions is described, which is an alternative to the method of Lagrange multipliers used for the derivation of the Hartee-Fock equations. An application to some types of atomic and molecular systems is also described.

Unitary variational method for atoms and molecules

Abstract A new scheme for treatment of unitary variation in orthonormal orbitals is presented. Applications of the method to various atomic and molecular systems are discussed.

Method of the orthonormality‐constrained variation: A pair of basis sets and the pair‐orthogonality

The orthogonality between sets of basis vectors, referred below to as pair‐orthogonality, is examined from a variational point of view. Both how to constitute pair‐orthogonal vectors and how to retain the pair‐orthogonality conditions in the course of variation are presented. The relation of the vectors to canonically orthonormalized Lowdin vectors is noted.

The symmetric deorthogonalization and the constrained variation method for atoms and molecules

Abstract A method is presented which constructs symmetrically a set of deorthogonalized ket-vectors from a set of given orthonormalized ket-vectors in atomic and molecular theory by means of variational minimization of the sum of squared distances between the sets. A new procedure of constraining the metric of deorthogonalization throughout the variational process is also described.

Erratum: Characterization of eigenvalues by use of orthonormality‐constrained variation methods

Various kinds of methods are presented for performing variational characterization of eigenvalues of a self‐adjoint operator F bounded from below. The methods are derived from construction of ket vectors under the orthogonality and normality constraints through the process of variation. Application of the methods to the expectation value of the operator leads to variational equations, in each of these equations an extremum condition yielding an eigenvalue equation.

Variational Approach to Orthogonality and Normality of Basis Ket-Vectors

Publisher Summary This chapter provides a general description of a new variational approach for ketvectors with the orthonormality constraint named orthonormality-constrained variation, (ONCV), a method of characterization of eigenvalues by use of ONCV, comments on the stability of solutions, and derivation of SCF equations with comparison to the usual Lagrange multipliers method, clarifying significance of some unrevealed aspects of Lowdins orthonormalization method. Lowdin first explained the cohesive and elastic properties of alkali halide crystals by using the orthogonalized wave functions. The nonorthogonality problem in molecular orbital theory of atoms and molecules was studied by Pauling, Slater and others, who developed the so-called maximum overlapping principle from the consideration that the strength of a chemical bond should be measured by the degree of the overlap between the orbitals involved. Paulings hybrid orbitals are constructed from an atomic orbital set so as to satisfy, first, the normalization, second, the orthogonality, and third, an implicit condition that hybrid orbitals can be spanned completely by a set of atomic orbitals. In addition, the chapter also describes a relation of the ONCV method to the Edmiston–Ruedenberg localized orbital method and also expands the ONCV method to a pair of ket-vector sets and proposes a method for constraining pairorthogonality- orthogonality between two basis sets-in the course of variation.

Parameter representation of first-and second-order variations preserving orthonormality

Abstract A parameter expression that preserves orthonormality under variation up to second order, is derived by means of the Lowdin orthonormalization. This representation removes the restrictions on the fluctuation parameters, brought within the usual variation methods. Application to various variation problems in atomic and molecular theory proves the superiority of the present method.

Construction of orthonormal vector sets for atoms and molecules by means of recursive variation

It is shown using a recursive variation method that the Schmidt orthonormalized vectors are optimal in the sense that the least squared distances are minimized, and that the use of maximal squared overlaps yields a Schmidt canonical orthonormalized vector set which corresponds to the Lowdin canonical vector set. By introducing the Krylov sequence, the method is applied to the derivation of a variational expression for the Lanczos vectors which tridiagonalize any self‐adjoint operator. The repeated use of two‐step variations is shown to derive a pseudopotential which is useful for the calculations of correlated pair functions.

Reinvestigation of Brillouin’s theorem in open‐shell SCF theory by means of constrained orbital variation

The orthonormality‐constrained variation (ONCV) method is compared with the usual variation methods with respect to the so‐called Brillouin’s theorem. Within the restricted open‐shell Hartree–Fock approximation, the ONCV method yields a modified form of Brillouin’s theorem which is not in a form such that one bra–ket vanishes, together with the well‐known forms of Brillouin’s theorem. Brillouin’s theorem for the excited singlet state constructed from nonorthogonal spatial orbitals is examined using a parameter representation of the ONCV variation and is found to be not in a typical form. What has caused these results is discussed in some detail.

Variation approach to the Löwdin canonical orthonormalization

Abstract The canonically orthonormalized vector set is examined from a vector-variational point of view. A new variation equation, eliminating the freedom of the choice of unitary transformation in various variation equations for construction of the set, is proposed. A method that holds canonical orthonormality unchanged during the process of variation, is also presented.