2.40: Second Trial Wavefunction

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\[ \Psi = \text{exp} \left( - \alpha r_1 \right) \text{exp} \left( - \alpha r_2 \right) + \text{exp} \left( - \alpha r_1 \right) \text{exp} \left( - \beta r_2 \right) + \text{exp} \left( - \beta r_1 \right) \text{exp} \left( - \alpha r_2 \right) + \text{exp} \left( - \beta r_1 \right) \text{exp} \left( - \beta r_2 \right) \nonumber \]

When the wavefunction shown above is used in a variational method calculation for the ground state energy for two-electron atoms or ions the two-parameter equation shown below for the energy is obtained. This equation is then minimized simultaneously with respect to the adjustable parameters, α and β.

Nuclear charge: Z = 2

Seed values for scale factors: \( \begin{matrix} \alpha = 2 & \beta = Z + 1 \end{matrix}\)

Variational energy expression:

\[ E ( \alpha,~ \beta ) = \frac{ \begin{matrix} \left[ \frac{ \frac{ \alpha^2 + \beta^2}{2} - Z ( \alpha + \beta ) - \frac{8 \alpha^{1.5} \beta^{1.5}}{( \alpha + \beta)^2} \left( Z - \frac{ \alpha \beta}{ \alpha + \beta} \right)}{1 + \frac{8 \alpha^{1.5} \beta^{1.5}}{( \alpha + \beta)^3}} \right] ... \\ \frac{5}{8} ( \alpha + \beta ) + \frac{2 \alpha \beta \left( \alpha^2 + 3 \alpha \beta + \beta^2 \right)}{( \alpha + \beta)^3} + 4 \begin{bmatrix} \frac{8 \alpha^{2.5} \beta^{1.5} \left( 11 \alpha^2 + 8 \alpha \beta + \beta^2 \right)}{( \alpha + \beta)^2 (3 \alpha + \beta)^3} ... \\ + \frac{8 \alpha^{1.5} \beta^{2.5} \left( 11 \beta^2 + 8 \alpha \beta + \alpha^2 \right)}{( \alpha + \beta)^2 (3 \beta + \alpha)^3} ... \\ \frac{20 \alpha^3 \beta^3}{( \alpha + \beta)^5} \end{bmatrix} \end{matrix}}{4 \left[ 1 + \frac{8 \alpha^{1.5} \beta^{1.5}}{( \alpha + \beta)^3} \right]^2} \nonumber \]

\[ \begin{matrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \text{Minimize} (E,~ \alpha,~ \beta) & \begin{pmatrix} \alpha \\ beta \end{pmatrix} = \begin{pmatrix} 1.2141 \\ 2.1603 \end{pmatrix} & E ( \alpha,~ \beta ) = -2.8603 \end{matrix} \nonumber \]

Experimental ground state energy:

\[ E_{exp} = -2.9037 \nonumber \]

Calculate error in calculation:

\[ \begin{matrix} \text{Error} = \begin{vmatrix} \frac{E_{exp} - E( \alpha,~ \beta)}{E_{exp}} \end{vmatrix} & \text{Error} = 1.4931 \% \end{matrix} \nonumber \]

Fill in the table and answer the questions below:

\[ \begin{pmatrix} \Psi & \text{H} & \text{He} & \text{Li} & \text{Be} \\ \alpha & 0.3703 & 1.2141 & 2.0969 & 2.9993 \\ \beta & 1.0001 & 2.1603 & 3.2778 & 4.3756 \\ E_{atom} & -0.487 & -2.8603 & -7.235 & -13.6098 \\ E_{atom} ( \text{exp} ) & -0.5277 & -2.9037 & -7.2838 & -13.6640 \\ \% \text{Error} & 7.72 & 1.49 & 0.670 & 0.397 \end{pmatrix} \nonumber \]

Fill in the table below and explain why this trial wave function gives better results than the first trial wave function.

\[ \begin{matrix} T( \alpha,~ \beta ) = \left[ \frac{ \frac{ \alpha^2 + \beta^2}{2} + \frac{8 \alpha^{1.5} \beta^{1.5}}{( \alpha + \beta)^2} \left( \frac{ \alpha \beta}{ \alpha + \beta} \right)}{1 + \frac{8 \alpha^{1.5} \beta^{1.5}}{( \alpha + \beta)^2}} \right] & V_{ne} ( \alpha,~ \beta ) = \left[ \frac{ -Z ( \alpha + \beta) - \frac{8 \alpha^{1.5} \beta^{1.5}}{( \alpha + \beta)^2} Z}{1 + \frac{8 \alpha^{1.5} \beta^{1.5}}{( \alpha + \beta)^3}} \right] \end{matrix} \nonumber \]

\[ \begin{matrix} T( \alpha,~ \beta) = 2.8603 & V_{ne} = -6.7488 \\ V_{ee} ( \alpha,~ \beta ) = E( \alpha,~ \beta ) - T( \alpha,~ \beta ) - V_{ne} ( \alpha,~ \beta ) & V_{ee} ( \alpha,~ \beta ) = 1.0281 \end{matrix} \nonumber \]

\[ \begin{pmatrix} \text{WF2} & \text{E} & \text{T} & \text{V}_{ne} & \text{V}_{ee} \\ \text{H} & -0.4870 & 0.4780 & -1.3705 & 0.3965 \\ \text{He} & -2.8603 & 2.8603 & -6.7488 & 1.0281 \\ \text{Li} & -7.2350 & 7.2350 & -16.1243 & 1.6544 \\ \text{Be} & -13.6098 & 13.6098 & -29.4995 & 2.2799 \end{pmatrix} \nonumber \]

Demonstrate that the virial theorem is satisfied.

\[ \begin{matrix} E ( \alpha,~ \beta ) = -2.8603 & - T ( \alpha,~ \beta ) = -2.8603 & \frac{V_{ne} ( \alpha,~ \beta ) + V_{ee} ( \alpha,~ \beta)}{2} = -2.8603 \end{matrix} \nonumber \]

Add the results for this wave function to your summary table for all wave functions.

\[ \begin{matrix} \begin{pmatrix} \text{H} & \text{E} & \text{T} & \text{V}_{ne} & \text{V}_{ee} \\ \text{WF1} & -0.4727 & 0.4727 & -1.375 & 0.4297 \\ \text{WF2} & -0.4870 & 0.4870 & -1.3705 & 0.3965 \end{pmatrix} & \begin{pmatrix} \text{He} & \text{E} & \text{T} & \text{V}_{ne} & \text{V}_{ee} \\ \text{WF1} & -2.8477 & 2.8477 & -6.7500 & 1.0547 \\ \text{WF2} & -2.8603 & 2.8603 & -6.7488 & 1.0281 \end{pmatrix} \\ \begin{pmatrix} \text{Li} & \text{E} & \text{T} & \text{V}_{ne} & \text{V}_{ee} \\ \text{WF1} & -7.2227 & 7.2227 & -16.1250 & 1.6797 \\ \text{WF2} & -7.2350 & 7.2350 & -16.1243 & 1.6544 \end{pmatrix} & \begin{pmatrix} \text{Be} & \text{E} & \text{T} & \text{V}_{ne} & \text{V}_{ee} \\ \text{WF1} & -13.5977 & 13.5977 & -29.5000 & 2.3047 \\ \text{WF2} & -13.6098 & 13.6098 & -29.4995 & 2.2799 \end{pmatrix}\end{matrix} \nonumber \]

These tables show that the improved agreement with experimental results (the lower total energy), is due to a reduction in electron-electron repulsion.