10.10.3 Transition-Optimized Shifted Hermite Theory

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

$$\widehat{H}=-\frac{1}{2}\frac{{\partial }^2}{\partial q^2}+\frac{1}{2}{\omega }^2{q}^2+{\eta }_{iii}{q}^3+{\eta }_{iiii}{q}^4$$

(10.37)

and the wave function is expressed as in Eq. (10.28). Now, if we shift the center of the wave function by $\sigma$, which is equivalent to a translation of the normal coordinate $q$, 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

$${\varphi }^{(0)}={\left(\frac{\omega }{\pi }\right)}^{\frac{1}{4}}{e}^{-\frac{\omega }{2}{\left(q-\sigma \right)}^2}$$

(10.38)

Similarly, for the first excited vibrational state, we have

$${\varphi }^{(1)}={\left(\frac{4{\omega }^3}{\pi }\right)}^{\frac{1}{4}}\left(q-\sigma \right)e^{\frac{\omega }{2}{\left(q-\sigma \right)}^2}$$

(10.39)

Therefore, the energy difference between the first vibrational excited state and the ground state is

$$\mathrm{\Delta }{E}_{\mathrm{TOSH}}=\omega +\frac{{\eta }_{iiii}}{8{\omega }^2}+\frac{{\eta }_{iii}\sigma }{2\omega }+\frac{{\eta }_{iiii}{\sigma }^2}{4\omega }$$

(10.40)

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

$$\mathrm{\Delta }{E}_{\mathrm{VPT1}}=\omega +\frac{{\eta }_{iiii}}{8{\omega }^2}$$

(10.41)

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:

$$\mathrm{\Delta }{E}_{\mathrm{VPT2}}=\omega +\frac{{\eta }_{iiii}}{8{\omega }^2}-\frac{5\eta _{iii}{}^2}{24{\omega }^4}$$

(10.42)

Since $\sigma$ is a very small quantity compared with the other variables, we ignore the contribution of ${\sigma }^2$ and compare $\mathrm{\Delta }{E}_{\mathrm{TOSH}}$ with $\mathrm{\Delta }{E}_{\mathrm{VPT2}}$, which yields an initial guess for $\sigma$:

$$\sigma =-\frac{5}{12}\frac{{\eta }_{iii}}{{\omega }^3}$$

(10.43)

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 $i$th mode be expressed as:

$$\mathrm{\Delta }{E}_i^{\mathrm{TOSH}}={\omega }_i+\frac{1}{8{\omega }_i}\sum _j\frac{{\eta }_{iijj}}{{\omega }_j}+\frac{1}{2{\omega }_i}\sum _j{\eta }_{iij}{\sigma }_{ij}+\frac{1}{4{\omega }_i}\sum _{j,k}{\eta }_{iijk}{\sigma }_{ij}{\sigma }_{ik}$$

(10.44)

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

$${\sigma }_{ij}=\frac{({\delta }_{ij}-2)({\omega }_i+{\omega }_j){\eta }_{iij}}{4{\omega }_i{\omega }_j^2(2{\omega }_i+{\omega }_j)}-\sum _k\frac{{\eta }_{kkj}}{4{\omega }_k{\omega }_j^2}$$

(10.45)