Harmonic oscillator vibrational eigenfunctions

The vibrational part of the molecular wave function may be expanded in the basis consisting of products of the eigenfunctions of two 2D harmonic oscillators with the Hamiltonians ffj = 7 -I- 1 /2/coiPa atid 7/p = 7p - - 1 /2fcppp,... 

Any linear combination of the three functions (6.85) is an eigenfunction of the vibrational Hamiltonian (within the harmonic-oscillator approximation) and represents a possible state. There are three linearly independent linear combinations of special importance these are... 

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. 

The unavailability of an RKR-like inversion (hence the impossiblity of obtaining the potential energy surface, V(Q), and exact vibrational eigenfunctions directly from experimental data) makes it convenient to use products of simple harmonic or Morse-oscillator basis functions as vibrational basis states... 

Evaluate fm H f ) if (a) H is the harmonic-oscillator Hamiltonian operator and f and f are harmonic-oscillator stationary-state wave functions with vibrational quantum numbers m and n (b) H is the particle-in-a-box H and / and f are particle-in-a-box energy eigenfunctions with quantum numbers m and n. 

The leading term in Eq. (6.9), li 0) Xm Xt is zero for because of the orthogonality of the eigenfunctions. The term (dnldx)o xm x x, thus dominates the series, provided that (3/i/3x)o 0. Examination of the eigenfunctions of a harmonic oscillator shows that the integral Xm x Xn) in this term is non-zero only if m = n 1 (Box 6.2). The formal selection mles for excitation of a harmonic vibration are, therefore ... 

---