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10.8 Anharmonic Vibrational Frequencies

10.8.2 Vibration Configuration Interaction Theory

(July 10, 2023)

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 nth excited state of mode i is

ψi(n)=ϕi(n)jimϕj(0) (10.49)

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) (10.50)

With the basis function defined in Eq. (10.49), 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) (10.51)

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 (10.52)

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) (10.53)

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. (10.53) 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.