Showing papers in "Advances in Quantum Chemistry in 1964"
TL;DR: Perturbation theory is designed to deal systematically with the effects of small perturbations on physical systems when the effects are mathematically too difficult to calculate exactly, and the properties of the unperturbed system are known as mentioned in this paper.
Abstract: Publisher Summary The purpose of this chapter is to provide information on the recent developments in perturbation theory. In recent years, there is a great increase of interest in the application of perturbation theory to the fundamental problems of quantum chemistry. Perturbation theory is designed to deal systematically with the effects of small perturbations on physical systems when the effects of the perturbations are mathematically too difficult to calculate exactly, and the properties of the unperturbed system are known. The new applications have been mainly to atoms where the reciprocal of the atomic number, l/Z, provides a natural perturbation parameter. These may be divided into two groups. The first consists of calculations of energy levels, and is a natural outgrowth of Hylleraas's classic work on the 1/Z expansion for two-electron atoms. The applications in the second group are to the calculation of expectation values and other properties of atoms and molecules, and are of much more recent origin. There are two principal reasons for the success of these new applications: (1) sufficient accuracy is frequently obtained from knowledge of a first-order perturbed wave function, and (2) a great advantage of perturbation theory is that the functional form of the perturbed wave function is shaped by the perturbation itself.
551 citations
TL;DR: In this paper, a detailed quantum mechanical discussion of the problem in terms of second-order perturbation theory is provided, where the frequency of a molecular electronic absorption band is generally displaced when a molecule is immersed in a solvent medium such as a liquid or a foreign gas.
Abstract: Publisher Summary This chapter provides a detailed quantum mechanical discussion of the problem in terms of second-order perturbation theory. The frequency of a molecular electronic absorption band is generally displaced when a molecule is immersed in a solvent medium such as a liquid or a foreign gas. These shifts are usually towards longer wavelengths although the opposite is sometimes true. In the recent years, considerable progress is made in understanding the effect of solvent on various spectral characteristics of a solute using quantum mechanical methods. When a molecule absorbs or emits light undergoing an electronic transition from one state to another, it is important to realize that both the energy and the electronic charge distribution in the molecule change. The calculation of stabilization energy of an electronic state due to solvent-solute interaction was first made by Ooshika using quantum mechanical perturbation theory. The same quantum mechanical formulation was used by Longuet–Higgins and Pople and by McRae.
205 citations
TL;DR: A picture of reality drawn with a few sharp lines cannot be expected to be adequate to the variety of all its shades as mentioned in this paper. Yet even so the drafsman must have courage to draw the lines firm.
Abstract: A picture of reality drawn with a few sharp lines cannot be expected to be adequate to the variety of all its shades. Yet even so the drafsman must have courage lo draw the lines firm. H. Weyl in Philosophy of Mathematics and Natural Science
134 citations
TL;DR: In this paper, the authors discuss several particularly interesting problems that are more or less according to their historical development, such as the early state of the helium problem and the ground state problem.
Abstract: Publisher Summary This chapter discusses several particularly interesting problems that are more or less according to their historical development. This means, for instance, that the chapter concentrates mainly upon the ground state problem, because of its ease of comparison with experimental data. From the point of view of quantum chemistry, ground state is also a most important problem, since in the theory of chemical compounds one is mainly looking for ground states. It mentions the early state of the helium problem. The natural frequency unit in atomic spectroscopy is the Rydberg constant R. The practical unit however is a reciprocal length R/c, which is termed as R. Because of the slight motion of the nucleus with respect to the electron–nucleus center of mass, the R (or R/c) varies a little from atom to atom. At the early stage of the Bohr theory, even in its primitive shape of circular electron orbit, a full set of energy levels was found. When the theory was supplemented with the conception of elliptical orbit, as quantized even with respect to orientation in space, the energy formula remained unchanged.
120 citations
TL;DR: In this paper, the authors describe some of the energy band calculations that have been made at M.I.T. to carry out the calculation in a practical fashion, one derives a secular equation connecting the various plane waves, each supplemented by its spherical part inside the spheres.
Abstract: Publisher Summary This chapter describes some of the energy band calculations that have been made at M.I.T. To carry out the calculation in a practical fashion, one derives a secular equation connecting the various plane waves, each supplemented by its spherical part inside the spheres; it is now customary to refer to such a function as an augmented plane wave (APW). A few of these have been published, but a number of them are still in a preliminary stage, and the description given in the chapter is not intended to do more than indicate in a general way the type of result found. The chapter provides the calculation on iron carried out by Wood and shows the energy bands. The characteristic is of course the bands of 3d character interposed in the middle of the 4s-like band. Wood's calculations are made for a nonmagnetic potential; that is, the same potential is used for electrons of both spin. However, Wood has also estimated the effect of using different potentials for the two spins, to give an energy band description of ferromagnetism.
56 citations
TL;DR: The wave function method as discussed by the authors is based on a more fundamental approach to the problem and leads to calculations of properties in which the errors are of order η 2, which is unsatisfactory as a general solution because it may never be available for elaborate systems and because it evades the real problem.
Abstract: Publisher Summary This chapter discusses the wave function method that is based on a more fundamental approach to the problem and leads to calculations of properties in which the errors are of order η2. The usual method recommended for improving the calculations is to improve the wave function by further minimization of the energy. This is unsatisfactory as a general solution because it may never be available for elaborate systems and because it evades the real problem. The essence of the method is that molecular properties can be defined in a number of ways which are entirely equivalent when an exact eigenfunction is used but have different dependence on the error when an approximate wave function is used. The problem becomes one of finding the appropriate formula for a property which minimizes its dependence on the error. There are two other types of functional transformation that are also of importance. The hypervirial operators investigated by Hirschfelder and his associates—Epstein and Hirschfelder. Apart from the small number of atoms and the small number of approximate wave functions for which an analytical treatment is possible, the problem of calculating accurate values of atomic and molecular properties has no easy solution.
25 citations
TL;DR: The Huckel method is a successful method for calculating the electronic structure and electronic spectra of certain types of molecules, specifically of complex unsaturated molecules However its usefulness is to correlate or estimate observed phenomena, not to predict quantitatively the unknown properties as mentioned in this paper.
Abstract: Publisher Summary This chapter examines some of the approximations and assumptions that are incorporated into the generalized Huckel method and also describes several devices to improve some weaknesses of this method The Huckel method is a successful method for calculating the electronic structure and electronic spectra of certain types of molecules, specifically of complex unsaturated molecules However its usefulness is to correlate or estimate observed phenomena, not to predict quantitatively the unknown properties of complex molecules In particular, various assumptions are shown to have sound bases and to be convenient method for making complicated problems manageable These are—(1) π electrons are treated apart from the rest, (2) Configuration interactions by one-electron excitations, (3) determination of each Z value for the corresponding molecular state, (4) evaluation of one-center core integral α , (5) compution of two-center core integral β, (6) overlapping of Zero differential, (7) One-center repulsion integrals, and (8) estimation of two-center repulsion integrals
24 citations
TL;DR: In this article, a discussion of the pi-electron approximation in terms of quantum mechanics is presented. But the appropriate basis for such a discussion is not quantum mechanics, but quantum mechanics itself.
Abstract: Publisher Summary This chapter focuses on the pi-electron approximation. The appropriate basis for a discussion of the pi-electron approximation is in terms of quantum mechanics. Pi-electron systems represent the most fruitful class of molecular systems to which quantum mechanics can be applied using a semi empirical approach. There are countless numbers of them, and pi-electron can be classified by forming homologous series where many variables can be suppressed, thus these electrons lend themselves well to theoretical approximation methods. The quantum mechanical approach to them can be related to solid state physics as well as to atomic and molecular physics. The study of pi-electron systems constitutes an important part of the history of quantum chemistry. Any analysis which does not relate to that underlying structure cannot provide firm footing for any a priori predictions about chemical or physical behavior. Some attention is given to the pi-electron approximation on that basis and its essential features seem to be fairly well understood.
16 citations
TL;DR: In this paper, the authors focus on the notion of orbital that represents the wave function associated with an isolated electron in the field of the nuclei of the molecule and show that there are an infinite number of sets of orbitals corresponding to exactly the same total wave function.
Abstract: Publisher Summary This chapter focuses on the notion of orbital that represents the wave function associated with an isolated electron in the field of the nuclei of the molecule. When the self-consistent field approach is used it can see that there are an infinite number of sets of orbitals corresponding to exactly the same total wave function. The notion of orbital is not invariant. Some sets are particularly interesting. The set that diagonalizes the self-consistent field operator corresponds, in the case of atoms, to orbitals localized in the various loges associated with the various shells. The energies of these orbitals are approximations of the ionization energies. In the case of molecules the corresponding set is called the set of molecular orbitals. Group theory permits us to separate the molecular orbitals between various classes that can occupy the same loge. Furthermore, the molecular orbitals are usually largely delocalized and do not correspond to the localized bonds. A particular unitary transformation permits to pass from these orbitals to a set of orbitals which corresponds to the loges of the localized bonds when such bonds exist.
12 citations