This article discusses the evaluation of molecular overlap integrals for Gaussian-type functions with arbitrary angular dependence. As an example, we calculate the overlap matrix for the water molecule in the STO-3G basis set.
where is an unnormalized Cartesian Gaussian function centered at :
where is normally taken at the nucleus, is the orbital exponent, and the polynomial represents the angular part, in that the sum of the Cartesian angular momenta corresponds to functions of type , , , , …. When the operator is 1, one simply has the overlap/density integral; otherwise it can be the energy operator for kinetic energy , electron-nuclear attraction , or electron-electron repulsion (which would involve double integrals). Other molecular properties involving external fields (response functions) or transition moments can also be computed from integrals of this form.
Gaussian-type functions are not the most natural choice for expanding the wavefunction. Slater-type functions, where the exponent is instead, can describe atomic systems more realistically. However, complications quickly arise in molecular calculations, which has led to the use of Gaussian functions in the overwhelming majority of current computational programs. Gaussian functions possess several desirable computational properties [4, 5, 6]. (Much credit is due to S. F. Boys for the introduction of the Cartesian Gaussian function into computational chemistry and its early development in a series of 12 papers under the general title “Electronic Wavefunctions,” the first being .) A given Slater function can be approximated by a linear combination of several Gaussians.
The first useful property is that the product of two Gaussian functions located at and is another Gaussian located at a point somewhere between and . (The proof of this can be found in .) The product of two Gaussians and is:
with and .
A second desirable property is that a derivative of a Gaussian can be expressed as a sum of Gaussians of lower and higher Cartesian angular values.
The simplest molecular integral is the overlap integral . We first separate the integral into its orthogonal components:
where the notation expresses its functional dependence on the Cartesian angular components. The component, for instance, is:
Using a binomial expansion in the polynomial part,
Odd values of result in odd functions whose integrals vanish. For even values of , a solution  for the integrals is given by , and, in those cases:
We keep in mind that for the summations only terms of even values of survive. We have thus obtained an expression to evaluate the overlap of two Gaussians with arbitrary Cartesian angular factors. Equation (6) should be sufficient for programming or even for manual evaluation with small Cartesian angular values. One can, however, further reduce the number of operations by using a recurrence relation, one of the most useful techniques of computational chemistry. Recurrence relations let us efficiently calculate molecular integrations of higher angular values using previously obtained results with lower angular values. Recurrence relations are used in most computational chemistry programs and their application to other molecular integrations will be shown here.
The derivation of the following relations involves straightforward algebraic manipulations, but is rather lengthy. Its omission does not impede our understanding of the recurrence relation, but helps maintain a reasonable continuity in our discussion. Interested readers can find a detailed derivation in the Appendix.
We start by defining the function
and combining the result with the same derivative of , but using the definition of in equation (6) instead,
For , we have the index recursion relation
Starting with the initial values and , the recurrence relation and the transfer equation, we can build up the overlap of functions of higher Cartesian angular values from lower ones. This is particularly useful for contracted Gaussian basis primitives with different functions sharing the same exponent.
The function Ov evaluates the overlap integral of two Gaussian functions; here alpha, beta, RA, RB, LA, and LB are , , , , , and as defined earlier.
The published contracted Gaussian basis sets (see, for example, ) are usually not normalized; in our first example, we will calculate the normalization factor of the and Cartesian Gaussian functions that we will need later on. The normalization factor is simply the inverse square root of the overlap integral. Here we calculate the overlap between two functions with the Cartesian angular vector ,
and observe that . The normalization factor for functions is . Similarly, the overlap between two functions () is
where we have analogously used the recurrence relations to obtain and afterward, . Similar results are obtained for and . The normalization factor for the function is then . These results are special cases of the more general formula of the normalization factor
and we note that this value depends only on the orbital exponent and the Cartesian angular values.
In the second example, we will calculate the overlap matrix of the water molecule (, , geometry optimized at the HF/STO-3G level). The molecule lies in the - plane with Cartesian coordinates in atomic units.
In the STO-3G basis set, each atomic orbital is described by a sum of three Gaussians; here are their primitive contraction coefficients and orbital exponents (taken from ).
Here are the centers and Cartesian angular values of the orbitals, in the following order: , , , , , , and .
For instance, the atomic orbital of the hydrogen atom 1 is described as
Similarly, the orbital of the oxygen atom is
The overlap integral between the two orbitals expands to nine integrals involving the primitives. Using the function Ov, for example, the overlap integral between the first two primitives of equations (13) and (14) is
And the overlap between and is
The overlap matrix for the entire water molecule in the STO-3G basis set can be calculated in a similar manner.
Since the overlap matrix is symmetrical, we need to calculate only the elements above the main diagonal. The basis functions are normalized, as indicated by the unit diagonal elements. We note that equals zero. This is the overlap between the orbital of the first hydrogen atom and the of the oxygen atom. The molecule lies in the - plane, so this overlap vanishes due to symmetry. Similar cases occur in the overlaps between the and orbitals of () and among the orbitals (), etc. The strongest overlaps are those between the hydrogen orbitals and the oxygen orbital.
We next plot the atomic-orbital overlap between of and of ,
in the - plane, superimposing the molecule structure.
We observe a strong distortion of the positive (lower) lobe of the function toward the hydrogen atom. The negative lobe shows less deformation, whereas the node remains precisely at the atomic position. Note that this is simply the orbital overlap between noninteracting atoms, such as in the case of the “promolecule.”
We have provided an introduction to the evaluation of molecular integrals involving Gaussian-type basis functions both analytically and by use of recurrence relations. The results are general and relatively straightforward; the simple algorithm makes it suitable for implementation in a number of programming languages. Together with the kinetic, nuclear-electron attraction, and electron-electron repulsion energies, this is the first step toward the calculation of molecular energies and electronic properties.
Consider the derivative term
the first term is simply
and, using the chain rule, the derivative of the second term is
recalling that . Substitute the results into the expression for ,
Writing in the second term inside the bracket, after expanding we have
Comparing the three integrals with the definitions of and we have the desired equation (7).
We make use of the relationship
to reduce the derivative above to
This is equation (8), which is what was needed to prove.
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About the Authors
Minhhuy Hô received his Ph.D. in theoretical chemistry at Queen’s University, Kingston, Ontario, Canada in 1998. He is currently a professor at the Centro de Investigaciones Químicas at the Universidad Autónoma del Estado de Morelos in Cuernavaca, Morelos, México.
Julio-Manuel Hernández-Pérez obtained his Ph.D. at the Universidad Autónoma del Estado de Morelos in 2008. He has been a professor of chemistry at the Facultad de Ciencias Químicas at the Benemérita Universidad Autónoma de Puebla since 2010.
Universidad Autónoma del Estado de Morelos
Centro de Investigaciones Químicas
Ave. Universidad, No. 1001, Col. Chamilpa
Cuernavaca, Morelos, Mexico CP 92010
Benemérita Universidad Autónoma de Puebla
Facultad de Ciencias Químicas
Ciudad Universitaria, Col. San Manuel
Puebla, Puebla, Mexico CP 72570