A program for the helium ground-state

We shall take a similar variational approach as the one used for the hydrogen atom. Let us take a wave-function of the form

$$\phi({\bf r}) = \sum_{p=1}^N C_p \chi_p({\bf r}).$$ (71)

Replacing into (70) we obtain

$$\left[-\frac{1}{2}\nabla^2_{r_1} - \frac{Z}{r_1} + \sum_{r,s=1}^N... ...ac{1}{\vert{\bf r}_2-{\bf r}_2\vert} \right] \sum_{q=1}^N C_q \chi_q({\bf r}_1)$$ (72)

$$= E' \sum_{q=1}^N C_q \chi_q({\bf r}_1).$$ (73)

Note that the $C_p$ are real as the functions $\chi_p$ are also real. Multiplying from the left by $\chi_p({\bf r}_1)$ and integrating over ${\bf r}_1$ leads to

$$\sum_{pq} \left( h_{pq} + \sum_{rs} C_r C_s Q_{pqrs} \right) C_q = E' \sum_{pq} S_{pq}C_q$$ (74)

with

$$h_{pq} = \left\langle \chi_p \left\vert -\frac{1}{2}\nabla^2 - \frac{Z}{r}\right\vert \chi_q \right\rangle$$ (75)

$$Q_{pqrs} = \int d^3r_2d^3r_1 \chi_p({\bf r}_2)\chi_r({\bf r}_2)\frac{1}{\vert{\bf r}_2-{\bf r}_2\vert} \chi_q({\bf r}_1)\chi_s({\bf r}_2)$$ (76)

$$S_{pq} = \langle \chi_p\vert\chi_q\rangle$$ (77)

Unfortunately, thsi is not a generalized eigenvalue equation because of the presence of the variables $C_r$ and $C_s$ inside the brackets on the left hand side. However, we can carry out a self-consistency iteration process as described earlier. By keeping $C_r$ and $C_s$ fixed, we solve the equationo for the $C_q$'s. We then replace the coefficients by the new solutiono, and iterate until convergence is achieved.

In order to calculate the matrix elements, we shall use Gaussian $l=0$ basis functions, just as in the case of the hydrogen atom:

$$\chi_p({\bf r}) = e^{- \alpha_p r^2}.$$ (78)

We shall take the optimal values of $\alpha_p$ found by solving the non-linear problem:

$$\alpha_1 = 0.298073$$ (79)

$$\alpha_2 = 1.242567$$ (80)

$$\alpha_3 = 5.782948$$ (81)

$$\alpha_4 = 38.474970.$$ (82)

The matrix elements for the kinetic and Coulomb terms are similar to those calculated for the hydrogen atom, except for an extra factor of $Z$ in the nuclear attraction. The matrix elements ofr $Q_{pqrs}$ are

$$Q_{pqrs} = \frac {2\pi^{5/2} }{(\alpha_p+\alpha_q)(\alpha_r+\alpha_s)\sqrt{\alpha_p+\alpha_q+\alpha_r+\alpha_s} }.$$ (83)

The program is consructed as follows:

$\bullet$ First, the $N\times N$ matrices $h_{pq}$, $S_{pq}$ and the $N \times N \times N \times N$ tensor $Q_{pqrs}$ are calculated.

$\bullet$ Initial values for the $C_p$ coefficients are chosen (all equal, for instance).

$\bullet$ These values are used to contruct the $F_{pq}$ matrix given by

$$F_{pq} = h_{pq} + \sum_{rs} C_r C_s Q_{pqrs}.$$ (84)

$\bullet$ We solve the generalized eigenvalue problem. For the ground-state, we keep only the eigenvector with the lowest eigenvalue.

$\bullet$ We calculate the ground-state energy as:

$$E = 2 \sum_{pq} C_p C_q h_{pq} + \sum_{pqrs} Q_{pqrs} C_p C_q C_r C_s.$$ (85)

$\bullet$ The new solution of the geenralized eigenvalue problem is then used to contruct the new matrix $F_{pq}$ and we repeat until the energy converges.