Joseph O. Hirschfelder

Recent Developments in Perturbation Theory

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 Hylleraass 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.

Transport Properties of Multicomponent Gas Mixtures

Equations are derived for the viscosity, ordinary (pressure) diffusion, and thermal diffusion of multicomponent mixtures of gases. The ordinary diffusion velocities are expressed in terms of the usual diffusion coefficients for binary mixtures. The analysis is an extension of the work of Chapman and Cowling. It is shown that the Chapman‐Cowling and Enskog procedure can be justified on the basis of a variational principle. This variational principle should be generally applicable to a large class of problems.

Potential Energy Functions for Diatomic Molecules

The usual Morse functions are determined from the energy of dissociation, the equilibrium separation of the nuclei, and the fundamental vibration frequency. Two additional spectroscopic constants, ωexe and αe, are available for most of the common diatomic molecules and permit us to add a two‐parameter correction term to the Morse curve. Both the potential V/D=(1−e−x)2+cx3(1+bx)e−2x and the extended Morse curve of the Coolidge, James and Vernon type, V/D=C2(1−e−x)2+C3(1−e−x)3+C4(1−e−x)4 agree with accurate potentials in those cases where they are known. Here x = 2β(r—re)/re. The constants for the first of these potentials are easy to evaluate and are given for 25 common diatomic molecules. With only a few exceptions, the improved potentials lie above the Morse curves and the corrections for moderately large internuclear separations may amount to ten percent of the energy of dissociation. Our treatment is based on the work of Dunham and the analysis of Coolidge, James and Vernon.

Contribution of Bound, Metastable, and Free Molecules to the Second Virial Coefficient and Some Properties of Double Molecules

The second virial coefficient for molecules interacting with a spherically symmetric potential is divided into three parts: (1) a contribution Bb, related to the equilibrium constant for the formation of bound double molecules; (2) a contribution Bm, related to the equilibrium constant for the formation of metastably‐bound double molecules; and (3) a contribution Bf, due to molecules which interact but are free to separate after the interaction. Equations are given for determining each of the three parts of the second virial coefficient. A detailed treatment of these three contributions together with numerical tables on a reduced temperature basis is given for the square‐well, Sutherland, and Lennard‐Jones (6–12) potentials.The mean lifetimes of metastably bound double molecules are discussed, and numerical values are given for the special case of argon. Tables for computing mean lifetimes in other Lennard‐Jones gases are given. It is found that most metastably bound double molecules have mean lifetimes c...

Classical and Quantum Mechanical Hypervirial Theorems

If W is a function of the coordinates and momenta, then in classical mechanics the time average of the Poisson bracket (H, W) is zero. In quantum mechanics it follows from the Heisenberg equation of motion that the expectation value of (WH — HW) for any wave function, corresponding to a stationary energy state of the system, is zero. For each selection of W there is a dynamical relationship, in a time‐average or space‐average sense, which the system must obey. Two classes of Ws are considered in detail: W as a function only of the coordinates, and W as a function of the coordinates times the first power of a momentum. For the first class, neither the classical mechanical nor the quantum mechanical results provide useful information. The usual virial theorem of Clausius is a special case of the second class of relations. The classical treatment should be useful for the determination of the equation of state of liquids. The use of the quantum‐mechanical relations for determining the constants in an approxi...

A Theory of Liquid Structure

A simple model of the liquid is used to extend the equation of state previously obtained and to treat the process of fusion, viscous flow, and binary liquid systems. Our equation of state which applies to dense liquids has been fitted on to Happels modification of van der Waals equation to give a single equation applicable over the entire range from gas to liquid. A liquid differs from a solid in that the surplus volume in one part of the liquid becomes available in another part without an activation energy appreciable as compared to kT. This communal sharing of volume gives rise to an entropy of fusion R modified, of course, if there are other structural changes. Other entropy changes arise from expansion, changes of librations into free rotations and from polymerization. Double molecules held together by van der Waals forces are considered quantitatively and used in the explanation of viscous flow and deviations of the equation of state at the critical point. An explicit expression is given for the osm...

The Theoretical Treatment of Chemical Reactions Produced by Ionization Processes Part I. The Ortho‐Para Hydrogen Conversion by Alpha‐Particles

The possible individual processes that may occur in a gaseous system under irradiation from alpha‐particles have been examined from the theoretical standpoint. The ionization, clustering and the fate of the ions have been studied. The ortho‐para hydrogen conversion under the influence of alpha‐particles has been chosen as an example with which to illustrate the method of treatment. Recent experimental data by Capron have been analyzed to confirm his conclusion that atomic hydrogen is responsible for the large ratio of molecules converted to ions produced (M/N = (700 to 1000)/1). It is also shown that clustering is unimportant in this case and that the paramagnetism of the ions is of negligible importance in producing the spin‐isomerization. The role of mercury atoms in removing atomic hydrogen from the system is found to be negligible. Removal of atoms by three body collision of two atoms with molecular hydrogen is slow compared with removal at the walls of the reaction vessel. Analysis of this latter pro...

Some Quantum-Mechanical Considerations in the Theory of Reactions Involving an Activation Energy

The activated complex or transition state method for calculating the absolute rate of a chemical reaction with an activation energy would be rigorously valid if classical mechanics applied to all degrees of freedom. In quantum mechanics, two kinds of limitations must be considered. First, because of Heisenbergs uncertainty principle, the transition state itself can be defined only if the potential surface is sufficiently flat around the highest point of the reaction path. Second, even if this condition is fulfilled, the transmission coefficient can differ from the value expected on the basis of classical mechanics, because a wave packet can be reflected both on its way up, and also on its way down the potential barrier separating the initial and final states. In fact, the transmission coefficient is, in many cases, a rapidly fluctuating function of the energy of the system. If the temperature distribution of the energy is sufficiently broad to cover several periods of this fluctuation, an average transmi...

Lennard‐Jones and Devonshire Equation of State of Compressed Gases and Liquids

The reduced equation of state for compressed gases and liquids is computed according to the theory of Lennard‐Jones and Devonshire for a large number of temperatures and densities. The corrections due to gas imperfection of internal energy, specific heat, and entropy as well as the compressibility factor (pv/RT) are expressed in terms of reduced variables. The calculations were made by punched‐card methods. A comparison is made between experiment and theory. It is shown that the theory of Lennard‐Jones and Devonshire is unsatisfactory at densities near the critical point and lower but improves at higher densities becoming better for normal liquids. Such results were to be expected from the nature of the cell or free‐volume method.

Heat Conductivity in Polyatomic or Electronically Excited Gases. II

The usual Eucken equation for the heat conductivity of a molecule with internal degrees of freedom is derived and improved. In the improved form the correction to the coefficient of thermal conductivity for the internal degrees of freedom of the molecule is given by the factor 0.115+0.354 Cp/R. With the help of Part I, we prove that this approximation is valid only if the electronic states are not metastable and if the coefficients of diffusion of all the molecular quantum states are equal. The metastable electronic species should lead to anomalously large coefficients of thermal conductivity, whereas the usual excited electronic states, because of their gigantic size, should give a much smaller contribution to the coefficient of thermal conductivity than would be expected on the basis of the Eucken assumption.