11.11.1 Vibration Configuration Interaction Theory

To solve the nuclear vibrational Schrödinger equation, one can only use direct integration procedures for diatomic molecules.^{729, 141} For larger systems, a truncated version of full configuration interaction is considered to be the most accurate approach. When one applies the variational principle to the vibrational problem, a basis function for the nuclear wave function of the $n$th excited state of mode $i$ is

$${\psi }_i^{(n)}={\varphi }_i^{(n)}\prod _{j\ne i}^m{\varphi }_j^{(0)}$$

(11.49)

where the ${\varphi }_i^{(n)}$ represents the harmonic oscillator eigenfunctions for normal mode $q_i$. This can be expressed in terms of Hermite polynomials:

$${\varphi }_i^{(n)}={\left(\frac{{\omega }_i^{\frac{1}{2}}}{{\pi }^{\frac{1}{2}}{2}^{n}n!}\right)}^{\frac{1}{2}}{e}^{-\frac{{\omega }_i{q}_i^2}{2}}{H}_n(q_i\sqrt{{\omega }_i})$$

(11.50)

With the basis function defined in Eq. (11.49), the $n$th wave function can be described as a linear combination of the Hermite polynomials:

$${\mathrm{\Psi }}^{(n)}=\sum _{i=0}^{n_1}\sum _{j=0}^{n_2}\sum _{k=0}^{n_3}\mathrm{\cdots }\sum _{m=0}^{n_m}{c}_{ijk\mathrm{\cdots }m}^{(n)}{\psi }_{ijk\mathrm{\cdots }m}^{(n)}$$

(11.51)

where $n_i$ is the number of quanta in the $i$th mode. We propose the notation VCI($n$) where $n$ is the total number of quanta, i.e.:

$$n=n_1+n_2+n_3+\mathrm{\cdots }+n_m$$

(11.52)

To determine this expansion coefficient $c^{(n)}$, we integrate the $\widehat{H}$, as in Eq. (4.1), with ${\mathrm{\Psi }}^{(n)}$ to get the eigenvalues

$$c^{(n)}=E_{\mathrm{VCI}(n)}^{(n)}=⟨{\mathrm{\Psi }}^{(n)}|\widehat{H}|{\mathrm{\Psi }}^{(n)}⟩$$

(11.53)

This gives us frequencies that are corrected for anharmonicity to $n$ quanta accuracy for a $m$-mode molecule. The size of the secular matrix on the right hand of Eq. (11.53) is ${((n+m)!/n!m!)}^2$, and the storage of this matrix can easily surpass the memory limit of a computer. Although this method is highly accurate, we need to seek for other approximations for computing large molecules.