Diatomic molecules analytic solution

In order to obtain an analytic solution they subsequently took the Wvfl belonging to a quantized harmonic oscillator, which is not unreasonable for the vibrations of a diatomic molecule ). Thus their M-equation has an artificial boundary ... 

Another very important analytically solvable case is the harmonic oscillator. This term is used for a mechanical system in which potential energy depends quadratically on displacement from the equilibrium position. The harmonic oscillator is very important, as it is an interacting system (i.e., a system with nonzero potential energy), which admits an analytical solution. A diatomic molecule, linked by a chemical bond with potential energy described by Eq. (2), is a typical example that is reasonably well described by the harmonic oscillator model. A chain with harmonic potentials along its bonds (bead-spring model), often invoked in polymer theories such as the Rouse theory of viscoelasticity, can be described as a set of coupled harmonic oscillators. 

The standard approaches usually use the quasiclassical approximation, that is, the initial conditions are selected in accordance with quantum mechanics and then the evolution of the phase space points is treated purely classically. The quasiclassical approximation thus requires that the trajectories are initiated in the semiclassical eigenstates of molecules. This is easily accomplished for diatomic molecules. For example, approximate analytical solutions for the rotating Morse oscillator have been derived which allow for straightforward selections of initial conditions. If the molecular vibrations are such that the harmonic approximation can be made, then the required analytical relationship is even simpler. ... 

I discuss how other equations of state can be derived theoretically using information about the interactions at the atomic level. I do this analytically for non-ideal gases, liquids, and sohds of single components of monoatomic and of diatomic molecules. I then introduce computer simulation techniques that enable us numerically to coimect the microcosm with the macrocosm for more complex systems, for which analytical solutions are intractable. 

Light J.C. and Walker. E.B. (1976) An B matrix approach to the solution of coupled equations for atom-molecule reactive scattering, J. Chem. Phys. 65. 4272-4282. Halavee, U. and Shapiro, M. (1976) A collinear analytic model for atom-diatom chemical reactions, J. Chem. Phys. 64, 2826-2839. 

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