Harmonic oscillator eigenfunctions

From equations (4.34) and the orthonormality of the harmonic oscillator eigenfunctions n), we find that the matrix elements of a and are... 

The resultant optimization equations cannot be solved analytically. Numerically, it is convenient to expand each field in an orthonormal basis set un(a>,x) (e.g., harmonic oscillator eigenfunctions) ... 

The four lowest harmonic-oscillator eigenfunctions are plotted in Fig. 5.3. Note the topological resemblance to the corresponding particle-in-a-box eigenfunctions. The eigenvalues are given by the simple formula... 

Equation (10.38) is recognized as the Schrodinger equation (4.13) for the one-dimensional harmonic oscillator. In order for equation (10.38) to have the same eigenfunctions and eigenvalues as equation (4.13), the function Slq) must have the same asymptotic behavior as in (4.13). As the intemuclear distance R approaches infinity, the relative distance variable q also approaches infinity and the functions F(R) and S(q) = RF(R) must approach zero in order for the nuclear wave functions to be well-behaved. As 7 —> 0, which is equivalent to q —Re, the potential U(q becomes infinitely large, so that F(R) and S(q rapidly approach zero. Thus, the function S(q) approaches zero as q -Re and as Roo. The harmonic-oscillator eigenfunctions V W decrease rapidly in value as x increases from x = 0 and approach zero as X —> oo. They have essentially vanished at the value of x corresponding to q = —Re. Consequently, the functions S(iq in equation (10.38) and V ( ) in... 

The harmonic potential is a model of last resort for diatomic molecules. Its behavior at R = 0 and R = oo is unphysical, as is the sign of ae. Exact diatomic molecule vibrational wavefunctions for levels above v = 0, except for their number of nodes, differ from harmonic oscillator eigenfunctions (Hermite polynomials with an exponential factor) in that they are not symmetric about Re and, increasingly so at high v, are skewed toward the outer turning point. 

This idea occurred to me during the Dimensional Scaling Workshop itself. I should like to state here that Martin Dunn independently and essentially simultaneously realised the benefits of computing the l/Z -expansion coefficients perturba-tively in a basis of harmonic oscillator eigenfunctions. 

Tb see how this displacement depends on the GTO exponent, let us recall its relation to the harmonic oscillator force constant k (cf. Chapter 4). The harmonic oscillator eigenfunction corresponds to a Gaussian orbital with an exponent equal to a/2, where (in a.u.). Therefore, if we have a GTO with exponent equal to A,... 

Transition amplitudes between harmonic osdllator states, and the averages present in the TCFs can be conveniently evaluated in the coordinate representation using the algebra of the operators and some properties of harmonic oscillator eigenfunctions [126]. Compact expressions have been derived in this way for thermally averaged TCFs in terms of m = 1,2. ... 

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