A proposal for the calculation of the vibrational states of small molecules where L=0, P=+

The adiabatic Gaussians used for molecular systems, where L=0, P=+, are generalized to nonadiabatic Gaussians in essentially the following way. Let gsub(s)sup(ad) = PIsub(i=Nsub(a)+1)sup(N) exp [-Asub(i)sup(2)(s)/anti rsub(i)-anti Rsub(gi)(s)/2] be a typical member of the adiabatic Gaussians. Here Nsub(a) is the number of the nuclei and N-Nsub(a) the number of the electrons. Now we replace anti Rsub(gi)(s) by (1/Asub(i)(s))SIGMAsub(j=1)sup(Nsub(a))asub(ij)(s)anti rsub(j), where Asub(i)(s) = SIGMAsub(j=1)sup(Nsub(a))asub(ij)(s), and the obtained functions gsub(s) is then multiplied by the function fsub(s)(anti n) = PIsub(1<=p<q<=Nsub(a)) (rsub(pq)sup(2)-Rsub(pq)sup(2))sup(n)sub(pq) (exp [-Asub(pq)sup(2) (s).(rsub(pq)+Rsub(pq))2] + exp [-Asub(pq)sup(2)(s)(rsub(pq)-Rsub(pq))2]). The function fsub(s) will determine the molecular structure (mean distances and mean angles) and we observe that in the limit Asub(pq)(s)→infinity, when anti n=0, the nonadiabatic Gaussian approaches the adiabatic Gaussian. The matrix elements [gsub(s)fsub(s)(n12)Qfsub(s')(n'12)gsub(s')], Q = 1 and H, for a biatomic molecule with any number of electrons, are given in closed form. Finally, the formulae are applied to the case H2+, where the lowest vibrational states are calculated. (author)