Distributed Gaussian orbitals for the description of electrons in an external potential

Abstract

In this work, we demonstrate the viability of using distributed Gaussian orbitals as a basis set for the calculation of the properties of electrons subjected to an external potential. We validate our method by studying one-electron systems for which we can compare to exact analytical results. We highlight numerical aspects that require particular care when using a distributedGaussian basis set. In particular, we discuss the optimal choice for the distance between two neighboring Gaussian orbitals. Finally, we show how our approach can be applied to many-electron problems.

Appendix: Energy contribution from the transverse components of the Gaussians

In this appendix, we derive the contribution to the energy due to the transverse components of the 3D Gaussian functions.

The electron wavefunction ψ^α(r) is expanded as a linear combination of the $\phi _{i}^{\alpha }$ orbitals:

$$ \psi^{\alpha}(\mathbf{r}) = \sum\limits_{i} C_{i} \phi_{i}^{\alpha} (\mathbf{r}). $$

(23)

By inserting Eq. 15 into Eq. 23, we obtain

$$\psi^{\alpha}(\mathbf{r}) = \sum\limits_{i} C_{i} \exp (-\alpha\|\mathbf{r}-\mathbf{R}_{i})\|^{2}) = $$

$$ = \exp (-\alpha (x^{2}+y^{2})) \sum\limits_{i} C_{i} \exp (-\alpha(z-Z_{i})^{2}). $$

(24)

By acting on ψ^α(r) with the Hamiltonian, given in Eq. 8, and scalar multiplying the resulting equation by 〈ϕ^α(r)| one finally gets, after a separation of variables, the three independent equations

$$\begin{array}{@{}rcl@{}} \langle\exp (-\alpha x^{2}) |T_{x}| \exp (-\alpha x^{2})\rangle &=& E_{x} \langle\exp (-\alpha x^{2})|\exp (-\alpha x^{2})\rangle, \end{array} $$

(25)

$$\begin{array}{@{}rcl@{}} \langle\exp (-\alpha y^{2}) |T_{y}| \exp (-\alpha y^{2})\rangle &=& E_{y} \langle\exp (-\alpha y^{2})|\exp (-\alpha y^{2})\rangle, \end{array} $$

(26)

$$\begin{array}{@{}rcl@{}} \langle\exp (-\alpha(z-Z_{j})^{2}) |T_{z} + V_{z}| \sum\limits_{i} C_{i} \exp (-\alpha(z-Z_{i})^{2})\rangle &=& E_{z} \langle \exp (-\alpha(z-Z_{i})^{2}) | \sum\limits_{i} C_{i} \exp (-\alpha(z-Z_{i})^{2})\rangle. \end{array} $$

(27)

The first two equations have the solution E_x = E_y = α/2. This can be easily verified by explicitly performing the second derivative contained in the kinetic energy, and integrating the resulting expression. Alternatively, we note that a one-dimensional Gaussian with exponent α is the ground-state eigenfunction of a harmonic oscillator with eigenvalue α. Because of the virial theorem, the mean value of the kinetic energy is equal to the mean value of the potential energy, and hence equal to one half of the total energy, i.e., E_x = E_y = α/2.

The third equation corresponds to the projection of a true 1D eigen-equation for H_z onto the set of non-orthogonal basis functions $|\exp (-\alpha (z-Z_{i})^{2})\rangle$. This means that the wavefunction is expressed as the product of three independent functions, depending on x, y, and z, respectively. Since the Hamiltonian is separable, the time-independent Schrödinger equation will have an energy given by the sum of three independent terms. The functions ψ^α(x) and ψ^α(y) do not contain any parameter, and their energy contributions will be those of a single Gaussian. The ψ^α(z) function can be computed, via the variational principle, by minimizing its energy. In the limit $M \rightarrow \infty$, ψ^α(z) will be an arbitrary function on the [0,L] interval, vanishing in all points z∉[0,L].