To solve the nuclear vibrational Schrödinger equation, one can only use direct integration procedures for diatomic molecules.755, 126 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 th excited state of mode is
where the represents the harmonic oscillator eigenfunctions for normal mode . This can be expressed in terms of Hermite polynomials:
With the basis function defined in Eq. (10.27), the th wave function can be described as a linear combination of the Hermite polynomials:
where is the number of quanta in the th mode. We propose the notation VCI() where is the total number of quanta, i.e.:
To determine this expansion coefficient , we integrate the , as in Eq. (4.1), with to get the eigenvalues
This gives us frequencies that are corrected for anharmonicity to quanta accuracy for a -mode molecule. The size of the secular matrix on the right hand of Eq. (10.31) is , 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.