Intemuclear distances, calculation

Fig. 3. This figure shows the total energy of the H2 molecule as a function of intemuclear distance, calculated from the electronic energies shown in Figure 2. For small values of R, the calculation using 15 atomic orbitals per atom agrees well with the values of Kolos and Wolniewicz but another configuration would be needed for agreement at large R.

Cade and Huo (13) have calculated near Hartree-Fock limit wavefunctions for LiH, BH, HF and HCl. The molecular orbital coefficients c j are available in Hie literature again at only the equilibrium Intemuclear distance. Thus again Cl values cannot be completely calculated. In Table I, the C values at the experimental equilibrium intemuclear distances calculated for LiH, BH, HF, and HCl with (1) minimum basis set wavefunctions, with (2) our extended basis set wavefunctions and with (3) the near Hartree-Fock limit wavefunctions are compared. In order to assess the quality of the various wavefunctions, the respective electronic energies are compared with those of the corresponding near Hartree-Fock limit wavefunctions. For the minimum basis set and the near Hartree-Fock limit calculations, the correct CL values of the extended basis set calculations were employed to calculate C. It is seen that both the C values and the A C values for the tended basis set calculations approach closely those of the near Hartree-Fock limit calculations. For H2, we carried out calculations not only for an extended basis set but also for a large extended basis set which... 

If we were to calculate the potential energy V of the diatomic molecule AB as a function of the distance tab between the centers of the atoms, the result would be a curve having a shape like that seen in Fig. 5-1. This is a bond dissociation curve, the path from the minimum (the equilibrium intemuclear distance in the diatomic molecule) to increasing values of tab describing the dissociation of the molecule. It is conventional to take as the zero of energy the infinitely separated species. 

With the above in mind, it is sensible to modify the atomic orbital by treating the orbital exponent as a variational parameter. What we could do is vary for each value of the intemuclear separation 7 ab, and for each value of 7 ab calculate the energy with that particular orbital exponent. Just for illustration, I have calculated the energies for a range of orbital exponent and intemuclear distance pairs, and my results are shown as energy contours in Figure 3.3. 

All three states were described by a single set of SCF molecular orbitals based on the occupied canonical orbitals of the X Z- state and a transformation of the canonical virtual space known as "K-orbitals" [10] which, among other properties, approximate the set of natural orbitals. Transition moments within orthogonal basis functions are easier to derive. For the X state the composition of the reference space was obtained by performing two Hartree-Fock single and double excitations (HFSD-CI) calculations at two typical intemuclear distances, i.e. R. (equilibrium geometry) and about 3Re,and adding to the HF... 

The last is known as the overlap integral as it is determined by the volume common to the atomic orbitals a and b at a given intemuclear distance. In general, 5 < 1, an integral that is often set equal to zero in approximate calculations. 

The results of these calculations are summarised in Figs. 1,2 and 3. Fig. 1 has plots of the total energy for the three wave functions (together with the results of Kolos and Roothaan (8) for reference). The optimised orbital exponents are plotted in Fig. 2. In Fig. 3 the orbital exponents for case (iii) for the short intemuclear distance region R = 0 to R = 0.5 a.u. 

For a more precise calculation of intensities of infrared bands it is necessary to take into account the variation of the dipole function with intemuclear distance,... 

Functions (46) have been succesfully used in numerous quantum-mechanical variational calculations of atomic and exotic systems where there is, at most, one particle (nuclei), which is substantially heavier than other constituents. However, as is well known, simple correlated Gaussian functions centered at the origin cannot provide a satisfactory convergence rate for nearly adiabatic systems, such as molecules, containing two or more heavy particles. In the diatomic case, which we we will mainly be concerned with in this section, one may introduce in basis functions (46) additional factors of powers of the intemuclear distance. Such factors shift the peaks of Gaussians to some distance from the origin. This allows us to adequately describe the localization of nuclei around their equilibrium position. 

In the present variation-perturbation calculations the first order corrections were expanded in 600-term ECG basis defined in equations (15) and (16). The components of the polarizability were computed from equation (11) using the optimized The optimization was performed separately for each component and intemuclear distance. The values of aj, (co) are arithmetic sums of the plus and minus components (equation (12)) computed from two separate first-order corrections. For a given component v (either or ), and are expanded in the same basis but, because they are solutions to two different equations (equation (9)) they differ in the linear expansion coefficients. The computed components of the static polarizability an(/ ) and a R) are drawn in Fig. 2 and their numerical values at selected intemuclear distances are listed in Table 1. 

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