Application of Finite Orthogonal Polynomials to the Thermal Functions of Harmonic Oscillators. I. Reduced Partition Function of Isotopic Molecules

Abstract

The utility as well as limitations of the Taylor series expansion of the quantum-mechanical partition function of a harmonic oscillator is discussed. Finite orthogonal polynomials are suggested as a basis for the approximation of the thermodynamic functions of an assumbly of harmonic oscillators. Shifted Chebyshev polynomials of the first kind are used to obtain finite expansions of arbitrary order of the reduced partition functions of isotopic molecules. The resulting series is similar, except for modulating coefficients, term by term to the Bernoulli series obtained from the Taylor expansion. The modulating coefficients of each term in the Chebyshev expansion approach unity as the order of the expansion increases. Thus, the Chebyshev series developed here approaches the Bernoulli series asymptotically. The Chebyshev expansion converges much faster than the Bernoulli expansion. The radius of convergence can be made arbitrarily large in contrast to the limit of 2π which exists for the Bernoulli expansion. These features of the Chebyshev expansion provide a basis for the extension of the principles of isotope effects, derived from quantum corrections of order (ℏ / kT)2n, to the complete spectrum of energy states. The mathematical behavior of the modulating Chebyshev coefficients is examined. Detailed application of the Chebyshev expansion to typical polyatomic molecules is given through numerical calculations.