Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

Although all of the nuclear coordinates participate in this kinetic energy operator, and in our previous discussions, all of the nuclear coordinates are expanded, with respect to an equivalent position, in power series of the parameter k, here in the specific case of a diatomic molecule, we found that only the R coordinate seems to have an equilibrium position in the molecular fixed coordinates. This means that actually we only have to, or we can only, expand the R coordinate, but not the other coordinates, in the way that

In other words, for calculating the second-order energy (the vibrational energy), we only have to keep the term to do with the interatomic distance. The other terms, then, will enter the total Schro¨dinger equation in higher orders.

Now let us use a more specific coordinate system shown in Figure 2 for H2 molecule. The z axis is taken to be along the internuclei vector R. Adapting this coordinate-system, the consequence is that only the z coordinate of the nucleus are to be expanded in powers of k. That is, we define

Figure 2. Molecular-fixed coordinates.

Due to our choice of the coordinates, Z is actually identical with R. Therefore, we rewrite the following definitions:

Thus, the equation we are going to solve for the zeroth-order nuclear motion is

Equation (60) is a standard equation of motion of a harmonic oscillator. It can be easily solved as long as the term corresponding to the force constant can be evaluated. To do that with Eq. (66), we need to know how to calculate the

In modern quantum chemistry packages, one can obtain molecular basis set at the optimized geometry, in which the wave functions of the molecular basis are expanded in terms of a set of orthogonal Gaussian basis set. Therefore, we need to derive efficient formulas for calculating the above-mentioned matrix elements, between Gaussian functions of the first and second derivatives of the Coulomb potential terms, especially the second derivative term that is not available in quantum chemistry packages. Section IV is devoted to the evaluation of these matrix elements.

In the work of King, Dupuis, and Rys [15,16], the matrix elements of the Coulomb interaction term in Gaussian basis set were evaluated by solving the differential equations satisfied by these matrix elements. Thus, the Coulomb matrix elements are expressed in the form of the Rys’ polynomials. The potential problem of this method is that to obtain the matrix elements of the higher derivatives of Coulomb interactions, we need to solve more complicated differential equations numerically. Great effort has to be taken to ensure that the differential equation solver can solve such differential equations stably, and to

make sure that the result is accurate. In this work, we have carried out the integrals explicitly up to the second-order derivatives of the Coulomb interaction. After the lengthy derivations, it should be clear that the results we obtain, in the form of a finite series (with number of terms <12 for the extreme case in which h orbitals are taken into the basis set) of error functions and exponential functions, are simple to calculate, and the numerical accuracy is high, thanks to the existing numerical libraries providing accurate computation of error function.

The starting point of our technique is to make use of the equality

By using the geometry defined in Figure 2, the coordinates of the electrons with respect to the nucleus will be written as

The matrix elements of these derivatives are to be evaluated with R equal to its equilibrium value R0. However, to keep the notation simple, we shall still write R in place of R0 in later text unless ambiguity may occur.

With the following discussions, it can also be seen that higher order derivatives can be evaluated with similar technique.

to represent a normalized Gaussian wave function. Here, a is the label to specify the nuclei, which will be either A or B in our cases, and i is the label to specify the electron 1 or 2. In the context in which the label of electron is immaterial, we shall drop the i index.

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical harmonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form

This type of basis functions is frequently used in popular quantum chemistry packages. We shall discuss the way to evaluate different kinds of matrix elements in this basis set that are often used in quantum chemistry calculation.

A.Normalization Factor