So far, every aspect of solving the nuclear wave equation has been considered, except the wave function. Since Schrödinger proposed his equation, the nuclear wave function has traditionally be expressed in terms of Hermite functions, which are designed for the harmonic oscillator case. Recently a modified representation has been presented.584 To demonstrate how this approximation works, we start with a simple example. For a diatomic molecule, the Hamiltonian with up to quartic derivatives can be written as
and the wave function is expressed as in Eq. (10.28). Now, if we shift the center of the wave function by , which is equivalent to a translation of the normal coordinate , the shape will still remain the same, but the anharmonic correction can now be incorporated into the wave function. For a ground vibrational state, the wave function is written as
Similarly, for the first excited vibrational state, we have
Therefore, the energy difference between the first vibrational excited state and the ground state is
This is the fundamental vibrational frequency from first-order perturbation theory.
Meanwhile, We know from the first-order perturbation theory with an ordinary wave function within a QFF PES, the energy is
The differences between these two wave functions are the two extra terms arising from the shift in Eq. (10.40). To determine the shift, we compare the energy with that from second-order perturbation theory:
Since is a very small quantity compared with the other variables, we ignore the contribution of and compare with , which yields an initial guess for :
Because the only difference between this approach and the ordinary wave function is the shift in the normal coordinate, we call it “transition-optimized shifted Hermite” (TOSH) functions.584 This approximation gives second-order accuracy at only first-order cost.
For polyatomic molecules, we consider Eq. (10.40), and propose that the energy of the th mode be expressed as:
Following the same approach as for the diatomic case, by comparing this with the energy from second-order perturbation theory, we obtain the shift as