Per-Olov Löwdin

On the Non‐Orthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals

The treatment of molecules and crystals by the Heitler‐London method or by the collective electron model can be based on the atomic orbitals φμ of the system. These orbitals are in general overlapping, and the corresponding overlap integrals Sμν, given by (1), have almost universally been neglected in the literature as causing undesirable complications. Here we will take these overlap integrals into consideration and show that they, instead of being negligible, are of essential importance in molecules and in crystals. The problem is simply solved by considering the orthonormalized functions [open phi]μ, given by (21), as the real atomic orbitals. The solution is worked out in detail for (I) the molecular orbital method of treating molecules, (II) the Bloch orbital method of treating crystals, and (III) the Heitler‐London method of treating both these systems in some simple spin cases. Some numerical applications are given for ionic crystals, showing that the overlap effects are responsible for all the rep...

On the Nonorthogonality Problem

Publisher Summary This chapter discusses three orthonormalization procedures, such as successive orthonormalization, symmetric orthonormalization, and canonical orthonormalization. The simplest way of orthonormalizing a finite set of functions is by the classical Schmidt procedure, in which each member of the set in order is orthogonalized against all the previous members and subsequently normalized. In solid-state theory, one could probably construct orthonormal combinations of the atomic orbitals of the system, which would still preserve the natural symmetry. In such an approach, it would be necessary to treat the given functions ϕ = {ϕ 1 , ϕ 2 …., ϕ n } simultaneously, on an equivalent basis instead of successively as in the Schmidt procedure. In molecular and solid-state theory, there are cases when also the symmetric orthonormalization procedure will break down, depending on the fact that, even if the basis ϕ = {ϕ 1 , ϕ 2 …., ϕ n } is linearly independent from the mathematical point of view, it may be approximately linearly dependent from the computational point of view. This phenomenon causes a great many complications and may lead to very misleading results, since the associated secular equations may be almost identically vanishing. Unfortunately, it seems as if many of the conventionally used basic systems are strongly affected by approximate linear dependencies. In order to systematize this problem, it is convenient to study the metric matrix.

Studies in Perturbation Theory. IV. Solution of Eigenvalue Problem by Projection Operator Formalism

The partitioning technique for solving secular equations is briefly reviewed. It is then reformulated in terms of an operator language in order to permit a discussion of the various methods of solving the Schrodinger equation. The total space is divided into two parts by means of a self‐adjoint projection operator O. Introducing the symbolic inverse T = (1—O)/(E—H), one can show that there exists an operator Ω = O + THO, which is an indempotent eigenoperator to H and satisfies the relations HΩ = EΩ and Ω2 = Ω. This operator is not normal but has a form which directly corresponds to infinite‐order perturbation theory. Both the Brillouin‐ and Schrodinger‐type formulas may be derived by power series expansion of T, even if other forms are perhaps more natural. The concept of the reaction operator is discussed, and upper and lower bounds for the true eigenvalues are finally derived.

Scaling problem, virial theorem, and connected relations in quantum mechanics

Abstract A trial wave function is said to be subject to a scale transformation, if all its coordinate vectors from a given origin are uniformly stretched by a certain scale factor η in order to get a better fit of the trial function to the domain of space which is actually occupied by the system under consideration. The virial theorem may be derived from the variation principle by considering the scale factor as a variable parameter. It is shown that, for an equilibrium state as well as for fixed nuclei, one can by a proper choice of the scale factor always get the virial theorem satisfied for an arbitrary normalizable trial function. The fulfilment of the virial theorem is hence a necessary but not sufficient criterion that a wave function is an accurate solution of the Schrodinger equation. The first derivative of the energy with respect to an arbitrary parameter is considered, and a generalization of the Hellman-Feynman theorem is derived. It is shown that, in solving the Schrodinger equation by means of Ritzs variational method, one can for each state, in addition to the energy, obtain its derivative with respect to the scale factor η. For states of different symmetry types, the associated scale factors may be varied independently of each other, but for states of the same symmetry, the scaling problem becomes more difficult. Some applications to perturbation theory are given and, in conclusion, the total energy of an atom is separated into its three fundamental parts.

QUANTUM GENETICS AND THE APERIODIC SOLID. SOME ASPECTS ON THE BIOLOGICAL PROBLEMS OF HEREDITY, MUTATIONS, AGEING, AND TUMORS IN VIEW OF THE QUANTUM THEORY OF THE DNA MOLECULE

Publisher Summary This chapter discusses aspects on the biological problems of heredity, mutations, aging, and tumors in view of the quantum theory of the DNA molecule. Each hydrogen bond in DNA consists of a proton shared between two electron lone pairs, and the genetic code is essentially a proton code. The probability of proton transfer in the hydrogen bonds of DNA is further discussed. By using the available charge orders for the π electrons of the base pairs, it is shown that the double-well potentials acting on the protons are highly asymmetric. At normal temperature, there is practically no proton transfer above the barrier, explaining the enormous stability of the genetic code. According to quantum mechanics, however, a proton is not a classical particle but a “wave packet,” which may penetrate a potential barrier by means of the “tunnel effect.” Depending on this proton tunneling, there is hence a very small but with time increasing probability that the normal base pairs A-T and G-C may spontaneously go over into the tautomeric pairs A*-T* and G*-C* through a “proton exchange” along the hydrogen bonds. Since the tautomeric bases have another pairing pattern, the proton exchange leads inevitably to errors in the genetic base sequence in the next duplication. The various possibilities suggested for the transcription of the genetic code through the formation of messenger RNA are studied, and the present status of the coding problem in protein synthesis is briefly reviewed.

STUDIES IN PERTURBATION THEORY. PART I. AN ELEMENTARY ITERATION-VARIATION PROCEDURE FOR SOLVING THE SCHRODINGER EQUATION BY PARTITIONING TECHNIQUE

Abstract The fundamental Schrodinger equation in quantum mechanics may be transformed to a discrete representation by introducing a complete set of basis functions in which the eigenfunctions may be developed. The eigenvalues are then determined by solving a secular equation, and this problem is here attacked by a partitioning which leads to an implicit relation for the energy E of the form E = f ( E ), which corresponds to the Schrodinger-Brillouin perturbation formula but has a more condensed form. The first-order iteration process based on the relation E ( k +1) = f { E ( k ) } is studied, and it is shown that, independent of whether this process is convergent or not, one can go over to a second-order process which appears to be closely connected with the variational expression. This second-order iterative procedure turns out to be very convenient for numerical work, particularly since it treats degenerate eigenvalues just as easily as the single eigenvalues. The general behavior of the curve y = E − f ( E ) is discussed, and the method is illustrated by a few numerical examples. In an appendix, a brief survey of the classification of iteration processes is also given.

SUPERPOSITION OF CONFIGURATIONS AND NATURAL SPIN ORBITALS. APPLICATIONS TO THE He PROBLEM.

The method of superposition of configurations is examined in its application to the helium atom in two cases: a 21×21 matrix including all configurations up to 〈6s〉2, and a 20×20 matrix with all configurations up to the 4‐quantum level, including angular terms. A new radial limit is established at −2.87900±0.00003, and this is used to discuss the convergence of such expansions in Legendie functions. The variation with scale factor is discussed in detail. The wave functions are analyzed in terms of natural spin orbitals (NSOs), which seem to have many advantages. The first NSO bears a striking resemblance to the Hartree‐Fock function, and the first two together provide a close approximation to the solution of the extended Hartree‐Fock equations with different orbitals for different electrons. An energy of −2.877924 is obtained for the best (u, v) function found. An analysis of the results suggests that inner orbitals may be better represented by pure exponentials than by Hartree‐Fock orbitals whenever additional correlational degrees of freedom are permitted. Expressed in approximate NSO form, the wave function is almost invariant to choice of basis set, provided that the latter is reasonably chosen. In particular, the necessity of including continuum terms along with the discrete hydrogen‐like set is demonstrated.

Band Theory, Valence Bond, and Tight-Binding Calculations

In the theory of the electronic structure of crystals, the fundamental features of the band theory, the valence bond method, and the tight‐binding approximation are reviewed. The band theory is studied on the basis of the Hartree‐Fock scheme, and the Bloch functions are formed by a projection technique. The main methods for calculating Hartree‐Fock functions in a solid are briefly discussed. The advantages and disadvantages of the band theory and the valence bond method are emphasized, and special attention is paid to the correlation error.In connection with the tight‐binding approximation, the importance of the continuum part and of the approximate linear dependencies is stressed. It is shown that a complete orthonormal set of translationally connected atomic orbitals may be constructed as a convenient basis for this approach. The implication of the virial theorem in interpreting the cohesive properties of the ionic crystals is further emphasized.Some recent refinements of band theory are then discussed....

Studies in Perturbation Theory. V. Some Aspects on the Exact Self‐Consistent Field Theory

The independent‐particle model in the theory of many‐particle systems is studied by means of the self‐consistent‐field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree‐Fock schemes, the extension of the SCF method developed by Brueckner is further refined by introducing the exact reaction operator containing all correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formalism which is completely analogous to the Hartree scheme but which still renders the exact energy and the exact wave function. An elementary derivation of the linked‐cluster theorem is given, and finally the inclusion of various symmetry properties is discussed.

Studies on the Alternant Molecular Orbital Method. I. General Energy Expression for an Alternant System with Closed‐Shell Structure

The alternant molecular orbital method is used to derive a general energy expression for an arbitrary alternant system with closed‐shell structure. This expression exhibits a simple dependence on a single mixing parameter, minimization with respect to which yields the optimum energy. An analysis reveals the effectiveness of the method to depend on the relative magnitude of two quantities, one involving the one‐electron operator energies and the other related to electron interaction integrals which connect the bonding with the antibonding orbitals from the simple ASMO scheme. Some comments on the method, in the light of the results obtained, are added.