2.41: Third Trial Wavefunction
- Page ID
- 157323
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\[ \Psi = \text{exp} \left( - \alpha r_1 \right) \text{exp} \left( - \beta r_2 \right) + \text{exp} \left( - \beta r_1 \right) \text{exp} \left( - \alpha r_2 \right) \nonumber \]
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: \( \begin{matrix} \alpha = Z & \beta = Z + 1 \end{matrix}\)
Variational energy expression:
\[ E ( \alpha,~ \beta ) = \frac{64 \alpha^3 \beta^3 [ \alpha \beta - Z ( \alpha + \beta)] \frac{ \alpha \beta}{ \alpha + \beta} + \frac{ \alpha^2 beta^2}{( \alpha + \beta)^3} + \frac{20 \alpha^3 \beta^3}{ ( \alpha + \beta)^5}}{1 + \frac{64 \alpha^3 \beta^3}{( \alpha + \beta)^6}} \nonumber \]
\[ \begin{matrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \text{Minimize} (E,~ \alpha,~ \beta ) & \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} 1.1885 \\ 2.1832 \end{pmatrix} & E ( \alpha,~ \beta ) = -2.8757 \end{matrix} \nonumber \]
Experimental ground state energy:
\[ E_{exp} = -2.9037 \nonumber \]
Calculate error in calculation:
\[ \begin{matrix} \text{Error} = \left| \frac{E_{exp} - E ( \alpha,~ \beta)}{E_{exp}} \right| & \text{Error} = 0.9656 \% \end{matrix} \nonumber \]
Summarize the calculations in the following table.
\[ \begin{pmatrix} \Psi & \text{H} & \text{He} & \text{Li} & \text{Be} \\ \alpha & 0.28322 & 1.18853 & 2.07898 & 2.98472 \\ \beta & 1.023923 & 2.18317 & 3.29491 & 4.38972 \\ E_{atom} & -0.5133 & -2.8757 & -7.2488 & -13.6230 \\ E_{atom} ( \text{exp} ) & -5.277 & -2.9037 & -7.2838 & -13.6640 \\ \% \text{Error} & 2.73 & 0.964 & 0.481 & 0.300 \end{pmatrix} \nonumber \]
Fill in the table below and explain why this trial wave function gives better results than the previous trial wave function.
\[ \begin{matrix} T ( \alpha,~ \beta ) = \frac{ \frac{ \alpha^2 + \beta^2}{2} + \frac{64 \alpha^3 \beta^3 ( \alpha \beta )}{( \alpha + \beta)^6}}{1 + \frac{64 \alpha^3 \beta^3}{( \alpha + \beta)^6}} & V_{ne} ( \alpha,~ \beta ) = \frac{ -Z ( \alpha + \beta ) + \frac{64 \alpha^3 \beta^3 ( \alpha \beta )}{( \alpha + \beta)^6}}{1 + \frac{64 \alpha^3 \beta^3 [- Z ( \alpha + \beta)]}{( \alpha + \beta)^6}} \\ V_{ee} ( \alpha,~ \beta ) = \frac{ \frac{ \alpha \beta}{ \alpha + \beta} + \frac{ \alpha^2 \beta^2}{( \alpha + \beta)^3} + \frac{20 \alpha^3 \beta^3}{( \alpha + \beta)^5}}{1 + \frac{64 \alpha^3 \beta^3}{( \alpha + \beta)^6}} & \begin{matrix} T ( \alpha,~ \beta ) = 2.8757 \\ V_{ne} ( \alpha,~ \beta ) = -6.7434 \\ V_{ee} ( \alpha,~ \beta ) = 0.9921 \end{matrix} \end{matrix} \nonumber \]
\[ \begin{pmatrix} \text{WF3} & \text{E} & \text{T} & \text{V}_{ne} & \text{V}_{ne} \\ \text{H} & -0.5133 & 0.5133 & -1.3225 & 0.2958 \\ \text{He} & -2.8757 & 2.8757 & -6.7434 & 0.9921 \\ \text{Li} & -7.2487 & 7.2487 & -16.1217 & 1.6242 \\ \text{Be} & -13.6230 & 13.620 & -29.4978 & 2.2519 \end{pmatrix} \nonumber \]
Demonstrate that the virial theorem is satisfied.
\[ \begin{matrix} E ( \alpha,~ \beta ) = -2.8757 & -T ( \alpha,~ \beta ) = -2.8757 & \frac{V_{ne} ( \alpha,~ \beta) + V_{ee} ( \alpha,~ \beta)}{2} = -2.8757 \end{matrix} \nonumber \]
Add the results for this wave function to your summary table for all wave functions.
\[ \begin{matrix} \begin{pmatrix} \text{H} & \text{E} & \text{T} & \text{V}_{ne} & \text{V}_{ee} \\ \text{WF1} & -0.4727 & 0.4727 & -1.375 & 0.4297 \\ \text{WF2} & -0.4870 & 0.4870 & -1.3705 & 0.3965 \\ \text{WF3} & -0.5133 & 0.5133 & -1.3225 & 0.2958 \end{pmatrix} & \begin{pmatrix} \text{He} & \text{E} & \text{T} & \text{V}_{ne} & \text{V}_{ee} \\ \text{WF1} & -2.8477 & 2.8477 & -6.7500 & 1.0547 \\ \text{WF2} & -2.8603 & 2.8603 & -6.7488 & 1.0281 \\ \text{WF3} & -2.8757 & 2.8757 & -6.7434 & 0.9921 \end{pmatrix} \\ \begin{pmatrix} \text{Li} & \text{E} & \text{T} & \text{V}_{ne} & \text{V}_{ee} \\ \text{WF1} & -7.2227 & 7.2227 & -16.1250 & 1.6797 \\ \text{WF2} & -7.2350 & 7.2350 & -16.1243 & 1.6544 \\ \text{WF3} & -7.2487 & 7.2487 & -16.1217 & 1.6242 \end{pmatrix} & \begin{pmatrix} \text{Be} & \text{E} & \text{T} & \text{V}_{ne} & \text{V}_{ee} \\ \text{WF1} & -13.5977 & 13.5977 & -29.5000 & 2.3047 \\ \text{WF2} & -13.6098 & 13.6098 & -29.4995 & 2.2799 \\ \text{WF3} & -13.6230 & 13.6230 & -29.4978 & 2.2519 \end{pmatrix} \end{matrix} \nonumber \]
These tables show that the improved agreement with experimental results (the lower total energy), is due to a reduction in electron-electron repulsion.