10.10 Anharmonic Vibrational Frequencies

10.10.1 Vibration Configuration Interaction Theory

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 nth excited state of mode i is

ψi(n)=ϕi(n)jimϕj(0) (10.27)

where the ϕi(n) represents the harmonic oscillator eigenfunctions for normal mode qi. This can be expressed in terms of Hermite polynomials:

ϕi(n)=(ωi12π122nn!)12e-ωiqi22Hn(qiωi) (10.28)

With the basis function defined in Eq. (10.27), the nth wave function can be described as a linear combination of the Hermite polynomials:

Ψ(n)=i=0n1j=0n2k=0n3m=0nmcijkm(n)ψijkm(n) (10.29)

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

n=n1+n2+n3++nm (10.30)

To determine this expansion coefficient c(n), we integrate the H^, as in Eq. (4.1), with Ψ(n) to get the eigenvalues

c(n)=EVCI(n)(n)=Ψ(n)|H^|Ψ(n) (10.31)

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. (10.31) 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.