Electronic wave functions: VI. Some theorems facilitating the evaluation of Schrödinger integrals of vector-coupled functions

1952 ◽  
Vol 245 (893) ◽  
pp. 95-115 ◽  
Keyword(s):  
General Theory ◽  
Wave Functions ◽  
Coupling Coefficients ◽  
Restricted Class ◽  
Vector Coupling ◽  
Electronic Wave ◽  
Atomic Wave ◽  
Variational Calculations ◽  
Electronic Wave Functions

The theorems reported here provide some powerful additional relations to the general theory of the reduction of Schrodinger integrals completed in part V. One result of these is to establish a number of relations between the various vector-coupling coefficients and thus to reduce considerably the labour of calculation of these. A second result is to provide a method of evaluating the two-electron electrostatic integrals which is a complete alternative to the c k method which has generally been used previously. This appears to be simpler and more powerful. A third result is only applicable to a restricted class of integrals and only to particular terms in the formulas for these, but where applicable it makes trivial the evaluation of the terms concerned and actually simplifies about three-quarters of the integrals normally occurring. These methods have been found to be extremely useful in the convergent variational calculations of atomic wave functions, and will also be applicable to all problems which require the evaluation of Schrödinger integrals between vector-coupled functions.

Electronic wave functions III. Some theorems on integrals of antisymmetric functions of equivalent orbital form

1951 ◽  
Vol 206 (1087) ◽  
pp. 489-509 ◽  
Keyword(s):  
Wave Functions ◽  
Present Series ◽  
Variational Procedure ◽  
Reduction Scheme ◽  
Electronic Wave ◽  
Atomic Wave ◽  
Practical Necessity ◽  
Electronic Wave Functions ◽  
Antisymmetric Functions ◽  
Systematic Reduction

A fresh formulation of the Slater methods of constructing antisymmetric functions is given. This formulation enables a systematic reduction scheme to be found for the integrals of all antisymmetric functions which occur in a completely convergent variational procedure. The theorems are purely mathematical, but the types of integral considered are those which would be obtained from functions representing both equal and different numbers of equivalent electrons. There have not previously been theorems which reduce the antisymmetry operators in the latter cases. These relations are a practical necessity for the calculations of atomic wave functions which are being reported in the present series. The theorems are equally applicable to functions which might be used for atoms, molecules, or nuclei.

Basis-Set Expansions of Relativistic Electronic Wave Functions

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Relativistic Quantum ◽  
Finite Basis ◽  
Wave Functions ◽  
Basis Set ◽  
Electronic Wave ◽  
Nonrelativistic Case ◽  
Hartree Fock ◽  
Electronic Wave Functions ◽  
Basis Set Expansion ◽  
Molecular Calculations

There have been several successful applications of the Dirac–Hartree–Fock (DHF) equations to the calculation of numerical electronic wave functions for diatomic molecules (Laaksonen and Grant 1984a, 1984b, Sundholm 1988, 1994, Kullie et al. 1999). However, the use of numerical techniques in relativistic molecular calculations encounters the same difficulties as in the nonrelativistic case, and to proceed to general applications beyond simple diatomic and linear molecules it is necessary to resort to an analytic approximation using a basis set expansion of the wave function. The techniques for such calculations may to a large extent be based on the methods developed for nonrelativistic calculations, but it turns out that the transfer of these methods to the relativistic case requires special considerations. These considerations, as well as the development of the finite basis versions of both the Dirac and DHF equations, form the subject of the present chapter. In particular, in the early days of relativistic quantum chemistry, attempts to solve the DHF equations in a basis set expansion sometimes led to unexpected results. One of the problems was that some calculations did not tend to the correct nonrelativistic limit. Subsequent investigations revealed that this was caused by inconsistencies in the choice of basis set for the small-component space, and some basic principles of basisset selection for relativistic calculations were established. The variational stability of the DHF equations in a finite basis has also been a subject of debate. As we show in this chapter, it is possible to establish lower variational bounds, thus ensuring that the iterative solution of the DHF equations does not collapse. There are two basically different strategies that may be followed when developing a finite basis formulation for relativistic molecular calculations. One possibility is to expand the large and small components of the 4-spinor in a basis of 2-spinors. The alternative is to expand each of the scalar components of the 4-spinor in a scalar basis. Both approaches have their advantages and disadvantages, though the latter approach is obviously the easier one for adapting nonrelativistic methods, which work in real scalar arithmetic.

Analytic Raman intensities from molecular electronic wave functions

1986 ◽  
Vol 84 (1) ◽  
pp. 531-532 ◽  
Author(s):  
Michael J. Frisch ◽  
Yukio Yamaguchi ◽  
Jeffrey F. Gaw ◽  
Henry F. Schaefer ◽  
J. Stephen Binkley
Keyword(s):  
Wave Functions ◽  
Molecular Electronic ◽  
Electronic Wave ◽  
Electronic Wave Functions ◽  
Raman Intensities
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