Electronic computation of first-order wave functions using green's functions

Green's functions and the symbol manipulative computer language LISP have been used to obtain exact, closed form, first-order functions and second-order energies for the first fourteen states of the hydrogen atom in electric and magnetic fields.     

Numerical instabilities in the computation of pseudopotential matrix elements

Steep high angular momentum Gaussian basis functions in the vicinity of a nucleus whose inner electrons are replaced by an effective core potential may lead to numerical instabilities when calculating matrix elements of the core potential. Numerical roundoff errors may be amplified to an extent that spoils any result obtained in such a calculation. Effective core potential matrix elements for a model problem are computed with high numerical accuracy using the standard algorithm used in quantum chemical codes and compared to results of the MOLPRO program. Thus, it is demonstrated how the relative and absolute errors depend an basis function angular momenta, basis function exponents and the distance between the off-center basis function and the center carrying the effective core potential. Then, the problem is analyzed and closed expressions are derived for the expected numerical error in the limit of large basis function exponents. It is briefly discussed how other algorithms would behave in the critical case, and they are found to have problems as well. The numerical stability could be increased a little bit if the type 1 matrix elements were computed without making use of a partial wave expansion.     

The calculation of matrix elements for valence bond functions

Pauling's formulas for the calculation of matrix elements for valence bond functions are derived using a simple substitutional process. The results generalize and simplify the formulas. In particular, the formulas do not depend upon orthogonality of atomic orbitals nor upon the nature of the choice of bond structures (canonical or not). The results are particularly adaptable to automatic computation.     

Valence-Bond calculations and matrix elements between two tableau functions of non-orthogonal orbitals

An integral formula is derived involving a new type of determinantal function which reproduces the effect of the Young operator θNPN. This result is used to calculate matrix elements between tableau functions of non-orthogonal orbitals. Matrices which transform this representation into the traditional valence-bond scheme are also given.     

Evaluation of recurrence relationships and numerical solutions for finite rotation matrix elements by using auxiliary functions

In this article, we derived several new recurrence relations of the rotation matrix elements by using Gauss’ recurrence formulas for hypergeometric functions and auxiliary functions , and . The aim of this contribution is to obtain general algorithm to compute the rotation matrix elements, paying attention to the use recurrence relationships of the auxiliary functions that allow the treatment of the functions with high angular momentum quantum numbers.     

On the “level-shifting” method for converging hartree-fock wave functions

A new formulation of level-shifting method is presented, which provides a more economical procedure for computations.     

On a variational method for determining excited state wave functions

A variational method is proposed which results in an approximate wave function for an excited state which has the maximum overlap with the true excited state eigenfunction. The method involves the calculation of the quantities E=¦H¦ and =¦(H-E)²¦, but is free of the constraint that the trial function must remain orthogonal to all states of the same symmetry which lie beneath it. One must know, however, an approximation to the true eigenvalue. A discussion is given on how one might gain the latter information, lacking knowledge of the spectrum, from the repeated application of the method.

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Zusammenfassung Es wird eine Variationsmethode vorgeschlagen, die auf eins genäherte Wellenfunktion für einen angeregten Zustand führt, welche sich maximal mit der wahren Eigenfunktion des angeregten Zustandes überlappt. Die Methode ermöglicht die Berechnung der Größen: E=¦H¦ und =¦(H-E)²¦, ist aber frei von der Nebenbedingung, daß die Versuchsfunktion orthogonal zu allen benachbarten Zuständen mit derselben Symmetrie bleiben muß. — Man will jedoch eine Näherung des wahren Eigenwertes wissen. — Eine Diskussion darüber, wie man durch wiederholte Anwendung der Methode letztere Information gewinnen kann, ohne Kenntnisse über das Spektrum zu besitzen, wird durchgeführt.

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Résumé Méthode variationnelle pour la détermination d'une fonction d'onde d'un état excité présentant le recouvrement maximum avec la fonction d'onde exacte. La méthode implique le calcul de E=¦H¦ et =¦(H-E)²¦ mais ne comporte pas la contrainte d'orthogonalité de la fonction d'essai à tous les états inférieurs de même symétrie. On doit cependant connaître une valeur approchée de la valeur propre exacte. Discussion sur la manière dont on peut obtenir cette valeur approchée sans connaître le spectre, par application répétée de la méthode.

     

A method of calculation of the matrix elements between the spin-projected nonorthogonal Slater determinants

In the resonating Hartree–Fock calculation, we have to calculate the matrix element between spin-projected nonorthogonal Slater determinants (S dets). The matrix element between coherently spin-rotated S dets are given by a linear transformation of the spin-projected ones. Using the inverse transformation, we get the projected matrix elements from the coherently spin-rotated ones. By appropriately choosing the angles of the spin rotation, the required computational time is considerably reduced. © 1995 John Wiley & Sons, Inc.     

On the transferability of Fock matrix elements

The transferability of Fock matrix elements in the linear combination of atomic orbitals molecular orbital scheme is analysed using localized orbitals. It is shown that this transferability is dependent on the transferability of these localized orbitals and the neglect of long-range contributions from partially cancelling Coulomb nuclear attraction and electron repulsion terms. A theoretical basis is thus provided for the simulated ab initio molecular orbital and related methods. Various corrections previously introduced in an ad hoc manner are shown to be justified. Transferability in both the closed shell and open shell schemes is analysed.     

Gaussian matrix elements of the operator X: Reduction to confluent hypergeometric functions

A new method has been proposed for calculating the Gaussian matrix elements of the interaction operator represented by an arbitrary degree of interelectron distance X. The method is based on the expansion of two-electron integrals as the sum of one-electron integrals which in turn admit compact operator representation in terms of confluent hypergeometric functions. The generating differential operator has been shown to be related to the modified Hermitian polynomials. The standard structure of the special functions encountered in this approach is useful in studying the analytical behavior of the integrals and makes it possible to obtain for these integrals recurrence relations, direct algebraic expressions in the forms of finite sum of confluent hypergeometric functions, integral representations, and asymptotic properties. Unlike the usual methods based on integral transformation of the interaction operator, the proposed approach has a wider field of application, and in addition, leads to compact and convenient analytical expressions. The idea of using differential properties of integrals to simplify the integrand structure gives the proposed approach a certain resemblance to that suggested by Boys but not developed in detail in his pioneer work.     

On the theory of electron correlation in atoms and molecules: Relation between cluster expansion theory and the correlated wave functions method

A relation between the cluster expansion theory of many electron wave functions and the correlated wave functions method is established. In this way, the theoretical basis of the method is elucidated and the approximations involved in its application become apparent. General forms of the correlated wave function, differing in certain important respects from that form usually assumed, are derived.     

On the calculation of expectation values and transition matrix elements by coupled cluster method

Summary Finite order expressions are derived for expectation values and transition matrix elements within the framework of the coupled cluster method.     

On the calculation of vibronic coupling matrix elements

The non-diagonal matrix elements in the adiabatic Born-Oppenheimer approximation are considered. The effect of the Q-dependence of the electronic energy denominator is calculated explicitly for an arbitrary initial and final state. It is shown that the inclusion of this effect does not change the relative values of the coupling matrix elements for different initial vibronic states.     

On the summations involving Wigner rotation matrix elements

Two new methods to evaluate the sums over magnetic quantum numbers, together with Wigner rotation matrix elements, are formulated. The first is the coupling method which makes use of the coupling of Wigner rotation matrix elements. This method gives rise to a closed form for any kind of summation that involves a product of two Wigner rotation matrix elements. The second method is the equivalent operator method, for which a closed form is also obtained and easily implemented on the computer. A few examples are presented, and possible extensions are indicated. The formulae obtained are useful for the study of the angular distribution of the photofragments of diatomic and symmetric-top molecules caused by electric-dipole, electric-quadrupole and two-photon radiative transitions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Evaluation of the second-order reduced density matrix for correlated electronic wave functions

Formulas are presented for the evaluation of the second-order reduced density matrix for correlated wave functions, including wave functions constructed from pairwise nonorthogonal orbitals. Numerical results are provided for the correlation holes and conditional nuclear spin densities for the water molecule.     

On the closed form of Wigner rotation matrix elements

The closed forms of some rotation matrix elementsd _{m m} ^j (/2) are presented. The closed forms of summation involved two binomials and some special hypergeometric functions are also obtained. The MAPLE V program which calculates d _{m m} ^j (), dm,. (/2) and the help file are given in appendix.Supported in part by the Chiu Feng-Chia research Fund. Thanks are due to the Board of Trustees Chairman Dr. Ying-Ming Liao as well as the Trustee Dr. Charles Chiu-Hsiong Huang and President Dr. Ted C. Yang of Feng-Chia University, Taichung, Taiwan, ROC.