Minhhuy Hô, Julio Manuel Hernández-Pérez

II. Kinetic-Energy Integrals

This article carries out the evaluation of kinetic energy integrals using Gaussian-type functions with arbitrary Cartesian angular values. As an example, we calculate the kinetic matrix for the water molecule in the STO-3G basis set.


In this article, the second of a series describing algorithms for evaluating molecular integrals, we detail the evaluation of the kinetic energy integrals. Detailed accounts of molecular integrals can be found in the references of [1]. The electronic kinetic energy in atomic units () involves integrals of the type


in which is an unnormalized Cartesian Gaussian primitive centered at the nucleus :


Here 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 , , , , …. One notable property of the Gaussian function, which will be used here, is that the derivative of a Gaussian function can be expressed as a sum containing Gaussians of lower and higher Cartesian angular values. In particular,


Similarly, . This property is useful in integrals involving differential operators, such as the kinetic energy; in calculations involving the gradient of the energy or Hamiltonian; and in deriving one of the most important algorithms of Gaussian function computation, the recurrence relation.

The kinetic operator, expressed in Cartesian coordinates as , enables separation of the kinetic integrals into three components . For example,


Furthermore, if we also separate into its Cartesian components, becomes a product of three integrals


Derivative of Gaussian Functions

The last two integrals in are simply overlap integrals and can be evaluated as outlined in [1]. Integrating by parts, the first integral is


where, according to the property mentioned above, the first term can be expanded into two Gaussian products, which vanish at the integral limits.

Next, we consider the first term of the integrand. Expressing equation (3) in terms of (1) and rearranging, we get


Substituting the result into equation (6), we have


and expanding the polynomial part yields


We can now express the first integral of equation (5) in terms of one-dimensional Gaussian primitives:


Kinetic-Energy Integral

We have now shown that the kinetic-energy integral can be written as a product of overlap integrals. The component, for example, is given by


Recurrence Relations

Using the notation of [1], where the overlap integral of two Gaussians is expressed in terms of its orthogonal components,


the component of the kinetic-energy integral, equation (11), can be written as


where the factor has been absorbed into the definition of the kinetic integral . We will derive the recurrence relation for this term . For , the first three terms of equation (13) are zero, and we are left with . Analogously, for or , we have


and the recurrence relation for the kinetic integral function with any two Cartesian angular momenta generalizes to



The function Kin evaluates the kinetic integral of two Gaussian primitives; here alpha, beta, RA, RB, LA, and LB are , , , , , and as defined earlier. The first step is the evaluation of the overlap integral as described in [1].

We describe in detail the evaluation of the kinetic-energy matrix for the water molecule (, , the 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 approximated by a sum of three Gaussians; their unnormalized primitive contraction coefficients and orbital exponents (taken from [2]) are as follows.

Here are basis function origins and Cartesian angular values of the orbitals, listed in the order , , , , , , and .

Specifically, for the kinetic-energy integral of the first primitive of the orbital of hydrogen atom 1, , and the first primitive of the orbital of the oxygen atom, ,


where the indices indicate primitive of basis function . The kinetic integral in terms of the kinetic and overlap function is then


We start with and , and from the first four equations of the module, we use the recurrence scheme to build up the overlap values needed for . We will need values up to ,


leading to .

Similarly, for , we will need which, in turn, comes from and . We have




leading to .

For the component,




leading to .

With , the kinetic integral of equation (16) is , which we can also obtain from the module Kin.

Three Gaussian primitives for each atomic orbital result in nine integrals of the type that we have just evaluated. For example, the element of the kinetic-energy matrix,


is , derived via a contraction scheme that requires the following normalization factor.

This calculates .

For larger basis sets, one needs only to replace the summation upper limit 3 in equation (19) with the appropriate number of primitives belonging to a particular basis function and an additional summation for the basis functions.

Finally, here is the resulting kinetic energy matrix.

Since the kinetic-energy matrix is symmetrical, we need only to calculate the upper elements. The value of the elements hartree is the electronic kinetic energy of the hydrogen atom described by the STO-3G basis set (compared to the exact value of 0.5 hartree). Analogously, the element is the corresponding kinetic energy of an electron in the orbital of the oxygen atom, and so forth.


We have provided an introduction to the evaluation of kinetic-energy integrals involving Gaussian-type basis functions both analytically and by use of recurrence relations. The results are sufficiently general so that no modification of the algorithm is needed when larger basis sets with more Gaussian primitives or primitives with larger angular momenta are employed.


[1] M. Hô and J. M. Hernández-Pérez, “Evaluation of Gaussian Molecular Integrals I,” The Mathematica Journal, 2012. doi:10.3888/tmj.14-3.
[2] “Basis Set Exchange.” (April 12, 2012) bse.pnl.gov/bse/portal.

M. Hô and J. M. Hernández-Pérez, “Evaluation of Gaussian Molecular Integrals II,” The Mathematica Journal, 2013. dx.doi.org/doi:10.3888/tmj.15-1.

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.

Minhhuy Hô
Universidad Autónoma del Estado de Morelos
Centro de Investigaciones Químicas
. Universidad, No. 1001, Col. Chamilpa
, Morelos, Mexico CP 92010

Julio-Manuel Hernández-Pérez
Benemérita Universidad Autónoma de Puebla
Facultad de Ciencias Químicas
Ciudad Universitaria
, Col. San Manuel
, Puebla, Mexico CP 72570