An analysis is made of the way in which the behavior of various wave functions approximates that of exact vibrational eigenfunctions of an anharmonic crystal. The criterion used in determining the closeness of approximation is the order of magnitude of the difference between the second moment and the square of the first moment of the Hamiltonian operator. The functions chosen are products of harmonic oscillator wave functions, three different linear combinations of products of harmonic oscillator functions differing slightly in form from the exact eigenfunction, and a function which is the result of operating on an exact eigenfunction with a dipole moment operator.

The first of these functions gives a value of the criterion which is proportional to the number of unit cells in the crystal. For the rest the criterion is independent of the size of the crystal but is still not zero.

It is shown that, despite the fact that product functions are very poor approximations to eigenfunctions, quantities such as vibrational absorption intensities and frequencies calculated with them should give approximate agreement with experiment and that perturbation theory calculations using them can also give approximately correct results.

It is also shown that the results obtained by using the product wave functions have intrinsic limitations on their accuracy that arise from the collective interaction of all the phonons or modes of vibration.

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