# 10.10.1 Vibration Configuration Interaction Theory

(June 30, 2021)

To solve the nuclear vibrational Schrödinger equation, one can only use direct integration procedures for diatomic molecules. 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

 $\psi^{(n)}_{i}=\phi^{(n)}_{i}\prod^{m}_{j\neq i}\phi^{(0)}_{j}$ (10.39)

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

 ${\phi}^{(n)}_{i}=\left(\frac{\omega_{i}^{\frac{1}{2}}}{{\pi}^{\frac{1}{2}}2^{n% }n!}\right)^{\frac{1}{2}}{e^{-\frac{\omega_{i}q_{i}^{2}}{2}}}H_{n}(q_{i}{\sqrt% {\omega_{i}}})$ (10.40)

With the basis function defined in Eq. (10.39), the $n$th wave function can be described as a linear combination of the Hermite polynomials:

 $\Psi^{(n)}=\sum_{i=0}^{n_{1}}\sum_{j=0}^{n_{2}}\sum_{k=0}^{n_{3}}\cdots\sum_{m% =0}^{n_{m}}c^{(n)}_{ijk\cdots m}\psi_{ijk\cdots m}^{(n)}$ (10.41)

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.:

 $n=n_{1}+n_{2}+n_{3}+\cdots+n_{m}$ (10.42)

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

 $c^{(n)}=E^{(n)}_{\mathrm{VCI}(n)}=\langle\Psi^{(n)}|\hat{H}|\Psi^{(n)}\rangle$ (10.43)

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. (10.43) 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.