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The Journal of Chemical Physics

Equilibrium Spatial Distribution Function of Reaction Coordinates

Morton D. Kostin

Published Sep 15, 1969• 1 citation
Publisher

Abstract

A newly derived partial differential equation for the temperature-dependent spatial distribution function of quantum-statistical mechanics is applied to the case of calculating the spatial distribution function along reaction coordinates for a system at equilibrium. In the limit where Planck's constant tends to zero, it is shown that the partial differential equation yields the classical formula for the probability that the chemical complex is at a specified region of reaction coordinates. Explicit formulas are given for the quantum correction to the probability density as a function of temperature for stable chemical complexes, whose total energy is negative, and for unstable chemical complexes, whose total energy is positive.

DOI
10.1063/1.1672366

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A newly derived partial differential equation for the temperature-dependent spatial distribution function of quantum-statistical mechanics is applied to the case of calculating the spatial distribution function along reaction coordinates for a system at equilibrium. In the limit where Planck's constant tends to zero, it is shown that the partial differential equation yields the classical formula for the probability that the chemical complex is at a specified region of reaction coordinates. Explicit formulas are given for the quantum correction to the probability density as a function of temperature for stable chemical complexes, whose total energy is negative, and for unstable chemical complexes, whose total energy is positive.
An integral-differential equation was derived for the temperature-dependent spatial distribution function of quantum statistical thermodynamics and analytical solutions for several quantum systems were given. It was shown that the classical and quantum-mechanical equations for the temperature-dependent spatial distribution function for a separable potential differ by one term proportional to Planck's constant squared. In several limiting cases it was found that solutions obtained from the integral-differential equation agreed with those obtained by Uhlenbeck using other methods.
The Schrödinger equation for the wave functions is used to derive a linear partial differential equation for the time-dependent probability densities, ħ24m δ3P(x, t)δx3 = 2V(x) δPδx + V′P(x, t) − 2δε(x, t)δx − mδj(x, t)δt, where P(x, t), ε(x, t), j(x, t) are the time-dependent position, energy, and current densities, respectively. In the limiting case where the system is in the nth quantum state, it is shown that this partial differential equation reduces to the previously derived third-order ordinary differential equation for the probability position density.
A partial differential equation is derived for the Slater sum of the Dirac equation. Normalization of the Slater sum gives the temperature-dependent probability density. Another partial differential equation is derived for the Slater sum of the Klein–Gordon equation. This Slater sum yields the temperature-dependent charge density. The transition from quantum theory to classical theory for the two partial differential equations is carried out and the relation between the two resulting classical equations is investigated.
The Bloch equation is utilized to derive an integro-differential equation for the temperature dependence of the Wigner distribution function of a canonical ensemble. This equation is solved by two methods; one yields a power series in Planck's constant and the other a power series in the potential energy of the system. Transformation functions for the density matrix and the Wigner function are discussed and their possible application in the treatment of systems obeying Fermi-Dirac or Bose-Einstein statistics is investigated.
Quantum mechanical spatial probability densities may be obtained by solving the time independent Schrodinger equation. Statistical mechanical spatial densities may be obtained by maximizing Shannon´s entropy d(r)ln[d(r)] (where d(r) is the spatial density) subject to appropriate constraints. A classical spatial probability density may be obtained from 1/v(x) where v(x) is the velocity at position x obtained from kinetic energy + potential energy (x) =total energy. Recently, there have been attempts to link these various spatial probability densities. For example for the ground state wavefunction of a harmonic oscillator, the Schrodinger equation can be obtained from varying Shannon´s entropy with constraints because V(x) W*(x) W(x) (where W(x) is the wavefunction) can be written in the form of Shannon´s entropy times a constant [3]. Furthermore, Casal et al have maximized Tsalis´ entropy to obtain good approximations to the ground state wavefunctions of oscillators and Coulomb potentials. In [1], the classical density 1/v(x) has been calculated using energy levels from quantum mechanics, namely E=hbar w (n+.5). This density has in turn been used to calculate entropy and compared with results from wavefunctions of the Schrodinger equation with good results even for low n values even though for n=0, the wavefunction W(x)=1/C exp(-ax2) and 1/v(x) are not at all similar. The purpose of this note, is to attempt to examine the relationship between these different spatial densities.
Quantum statistical mechanics of spin systems is formulated by the use of the representation of atomic coherent state of spin, which is specified by a complex number. After introducing a newly defined distribution function of spin, the Bloch equation for the equilibrium density matrix, the von Neumann equation for the nonequilibrium density matrix and the Heisenberg equation of motion are converted into partial differential equations of the type of separating variables respectively. Formal solutions of them are presented. Any operator which is composed of spins can be converted uniquely into a differential operator. Thus we succeed in formulating equilibrium and nonequilibrium quantum statistical mechanics in the complex variable space, which differs from the previous one in the diagonal representation of density matrix. We further introduce the interaction representation for the perturbational expansion. The linear response formula is written down in the present representation.