15.7: Appendix
- Page ID
- 372795
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Triplets. A triplet wavefunction is defined by the Slater determinate: \({\mathit{\Psi}}^3\left(1,2\right)=\)
\[det\frac{1}{\sqrt{2}}\left|{\mathit{\Psi}}_1\left(1\right)\alpha \left(1\right)\ {\mathit{\Psi}}_1\left(2\right)\alpha \left(2\right)\ {\mathit{\Psi}}_2\left(1\right)\alpha \left(1\right)\ {\mathit{\Psi}}_2\left(2\right)\alpha \left(2\right)right|=\frac{1}{\sqrt{2}}{\mathit{\Psi}}_1\left(1\right)\alpha \left(1\right){\mathit{\Psi}}_2\left(2\right)\alpha \left(2\right)-\frac{1}{\sqrt{2}}{\mathit{\Psi}}_2\left(1\right)\alpha \left(1\right){\mathit{\Psi}}_1\left(2\right)\alpha \left(2\right) \nonumber \]
We now apply this to the electron-electron repulsion operator \(\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}\) as follows:
\[\int{}
Callstack:
at (Under_Construction/Free_Energy_(Snee)/15:_The_Hydrogen_Atom/15.07:_Appendix), /content/body/p[4]/span, line 1, column 1
\[\int{}\left\{{\psi }^*_1\left(1\right){\alpha }^*\left(1\right){\psi }^*_2\left(2\right){\alpha }^*\left(2\right)-{\psi }^*_2\left(1\right){\alpha }^*\left(1\right){\psi }^*_1\left(2\right){\alpha }^*\left(2\right)\right\}\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}\left\{{\mathit{\Psi}}_1\left(1\right)\alpha \left(1\right){\mathit{\Psi}}_2\left(2\right)\alpha \left(2\right)-{\mathit{\Psi}}_2\left(1\right)\alpha \left(1\right){\mathit{\Psi}}_1\left(2\right)\alpha \left(2\right)\right\}\cdot \partial \tau \nonumber \]
Next the expression is FOIL’ed out and the spin wavefunctions are factored out:
\[\frac{1}{2}\int{}{\psi }^*_1\left(1\right){\mathit{\Psi}}_1\left(1\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}
Callstack:
at (Under_Construction/Free_Energy_(Snee)/15:_The_Hydrogen_Atom/15.07:_Appendix), /content/body/p[8]/span, line 1, column 1
Since \(\int{}{\alpha }^*\alpha =1\) and \({\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}_1\left(1\right)={\left|{\mathit{\Psi}}_1\left(1\right)\right|}^2\) etc., the above can be factored into:
\[\frac{1}{2}\left(\int{}{\left|{\mathit{\Psi}}_1\left(1\right)\right|}^2\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\left|{\mathit{\Psi}}_2\left(2\right)\right|}^2\cdot \partial \tau +\int{}{\left|{\mathit{\Psi}}_1\left(2\right)\right|}^2\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\left|{\mathit{\Psi}}_2\left(1\right)\right|}^2\cdot \partial \tau \right) \nonumber \]
\[-\frac{1}{2}\left(\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}^*_2\left(2\right){\mathit{\Psi}}_2\left(1\right)\cdot \partial \tau +\int{}{\mathit{\Psi}}^*_1\left(2\right){\mathit{\Psi}}_1\left(1\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}^*_2\left(1\right){\mathit{\Psi}}_2\left(2\right)\cdot \partial \tau \right) \nonumber \]
The terms in parentheses are equal because the labels “1” and “2” are arbitrary, and the integral results are the same. The result is the Coulomb integral minus the exchange integral:
\[\int{}
Callstack:
at (Under_Construction/Free_Energy_(Snee)/15:_The_Hydrogen_Atom/15.07:_Appendix), /content/body/p[14]/span, line 1, column 1
\[\int{}{\left|{\mathit{\Psi}}_1\left(1\right)\right|}^2\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\left|{\mathit{\Psi}}_2\left(2\right)\right|}^2\cdot \partial \tau -\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}^*_2\left(2\right){\mathit{\Psi}}_2\left(1\right)\cdot \partial \tau \nonumber \]
Singlets: A singlet wavefunction is defined by two Slater determinates:
\[det\frac{1}{2}\left(\left|{\mathit{\Psi}}_1\left(1\right)\alpha \left(1\right)\ {\mathit{\Psi}}_1\left(2\right)\alpha \left(2\right)\ {\mathit{\Psi}}_2\left(1\right)\beta \left(1\right)\ {\mathit{\Psi}}_2\left(2\right)\beta \left(2\right)right|-\left|{\mathit{\Psi}}_1\left(1\right)\beta \left(1\right)\ {\mathit{\Psi}}_1\left(2\right)\beta \left(2\right)\ {\mathit{\Psi}}_2\left(1\right)\alpha \left(1\right)\ {\mathit{\Psi}}_2\left(2\right)\alpha \left(2\right)right|\right) \nonumber \]
\[={\frac{1}{2}\mathit{\Psi}}_1\left(1\right)\alpha \left(1\right){\mathit{\Psi}}_2\left(2\right)\beta \left(2\right)-\frac{1}{2}{\mathit{\Psi}}_2\left(1\right)\beta \left(1\right){\mathit{\Psi}}_1\left(2\right)\alpha \left(2\right)-\frac{1}{2}{\mathit{\Psi}}_1\left(1\right)\beta \left(1\right){\mathit{\Psi}}_2\left(2\right)\alpha \left(2\right)+\frac{1}{2}{\mathit{\Psi}}_2\left(1\right)\alpha \left(1\right){\mathit{\Psi}}_1\left(2\right)\beta \left(2\right) \nonumber \]
We now apply this to the electron-electron repulsion operator \(\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}\) as:
\[\int{}
Callstack:
at (Under_Construction/Free_Energy_(Snee)/15:_The_Hydrogen_Atom/15.07:_Appendix), /content/body/p[22]/span, line 1, column 1
The expression is FOIL’ed and the spin wavefunctions are factored out on the following page. Since \(\int{}{\alpha }^*\alpha =1\), \(\int{}{\alpha }^*\beta =\int{}{\beta }^*\alpha =0\) and \({\psi }^*_1\left(1\right){\mathit{\Psi}}_1\left(1\right)={\left|{\mathit{\Psi}}_1\left(1\right)\right|}^2\) etc., half the terms can be removed, and the remainder factored into:
\[\frac{1}{2}\left(\int{}{\left|{\mathit{\Psi}}_1\left(1\right)\right|}^2\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\left|{\mathit{\Psi}}_2\left(2\right)\right|}^2\cdot \partial \tau +\int{}{\left|{\mathit{\Psi}}_1\left(2\right)\right|}^2\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\left|{\mathit{\Psi}}_2\left(1\right)\right|}^2\cdot \partial \tau \right) \nonumber \]
\[+\frac{1}{2}\left(\int{}{\psi }^*_1\left(1\right){\mathit{\Psi}}_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\psi }^*_2\left(2\right){\mathit{\Psi}}_2\left(1\right)\cdot \partial \tau +\int{}{\psi }^*_1\left(2\right){\mathit{\Psi}}_1\left(1\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\psi }^*_2\left(1\right){\mathit{\Psi}}_2\left(2\right)\cdot \partial \tau \right) \nonumber \]
The terms in parentheses are equal because the labels “1” and “2” are arbitrary. Thus, we have the Coulomb integral plus the exchange integral:
\[\int{}
Callstack:
at (Under_Construction/Free_Energy_(Snee)/15:_The_Hydrogen_Atom/15.07:_Appendix), /content/body/p[28]/span, line 1, column 1
\[\int{}{\left|{\mathit{\Psi}}_1\left(1\right)\right|}^2\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\left|{\mathit{\Psi}}_2\left(2\right)\right|}^2\cdot \partial \tau +\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}^*_2\left(2\right){\mathit{\Psi}}_2\left(1\right)\cdot \partial \tau \nonumber \]
which proves that the paramagnetic triplet state is lower in energy than the singlet.
\[\frac{1}{4}\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}^*_2\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_1\left(1\right){\mathit{\Psi}}_2\left(2\right)\cdot \partial \tau \int{}{\alpha }^*\left(1\right)\alpha \left(1\right)\int{}{\beta }^*\left(2\right)\beta \left(2\right)-\frac{1}{4}\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}^*_2\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_2\left(1\right){\mathit{\Psi}}_1\left(2\right)\cdot \partial \tau \int{}{\alpha }^*\left(1\right)\beta \left(1\right)\int{}{\beta }^*\left(2\right)\alpha \left(2\right)-\frac{1}{4}\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}^*_2\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_1\left(1\right){\mathit{\Psi}}_2\left(2\right)\cdot \partial \tau \int{}{\alpha }^*\left(1\right)\beta \left(1\right)\int{}{\beta }^*\left(2\right)\alpha \left(2\right)+\frac{1}{4}\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}^*_2\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_2\left(1\right){\mathit{\Psi}}_1\left(2\right)\cdot \partial \tau \int{}{\alpha }^*\left(1\right)\alpha \left(1\right)\int{}{\beta }^*\left(2\right)\beta \left(2\right)-\frac{1}{4}\int{}{\mathit{\Psi}}^*_2\left(1\right){\mathit{\Psi}}^*_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_1\left(1\right){\mathit{\Psi}}_2\left(2\right)\cdot \partial \tau \int{}{\beta }^*\left(1\right)\alpha \left(1\right)\int{}{\alpha }^*\left(2\right)\beta \left(2\right)+\frac{1}{4}\int{}{\mathit{\Psi}}^*_2\left(1\right){\mathit{\Psi}}^*_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_2\left(1\right){\mathit{\Psi}}_1\left(2\right)\cdot \partial \tau \int{}{\beta }^*\left(1\right)\beta \left(1\right)\int{}{\alpha }^*\left(2\right)\alpha \left(2\right)+\frac{1}{4}\int{}{\mathit{\Psi}}^*_2\left(1\right){\mathit{\Psi}}^*_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_1\left(1\right){\mathit{\Psi}}_2\left(2\right)\cdot \partial \tau \int{}{\beta }^*\left(1\right)\beta \left(1\right)\int{}{\alpha }^*\left(2\right)\alpha \left(2\right)-\frac{1}{4}\int{}{\mathit{\Psi}}^*_2\left(1\right){\mathit{\Psi}}^*_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_2\left(1\right){\mathit{\Psi}}_1\left(2\right)\cdot \partial \tau \int{}{\beta }^*\left(1\right)\alpha \left(1\right)\int{}{\alpha }^*\left(2\right)\beta \left(2\right)-\frac{1}{4}\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}^*_2\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_1\left(1\right){\mathit{\Psi}}_2\left(2\right)\cdot \partial \tau \int{}{\beta }^*\left(1\right)\alpha \left(1\right)\int{}{\alpha }^*\left(2\right)\beta \left(2\right)+\frac{1}{4}\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}^*_2\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_2\left(1\right){\mathit{\Psi}}_1\left(2\right)\cdot \partial \tau \int{}{\beta }^*\left(1\right)\beta \left(1\right)\int{}{\alpha }^*\left(2\right)\alpha \left(2\right)+\frac{1}{4}\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}^*_2\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_1\left(1\right){\mathit{\Psi}}_2\left(2\right)\cdot \partial \tau \int{}{\beta }^*\left(1\right)\beta \left(1\right)\int{}{\alpha }^*\left(2\right)\alpha \left(2\right)-\frac{1}{4}\int{}{\mathit{\Psi}}^*_1\left(1\right){\mathit{\Psi}}^*_2\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_2\left(1\right){\mathit{\Psi}}_1\left(2\right)\cdot \partial \tau \int{}{\beta }^*\left(1\right)\alpha \left(1\right)\int{}{\alpha }^*\left(2\right)\beta \left(2\right)+\frac{1}{4}\int{}{\mathit{\Psi}}^*_2\left(1\right){\mathit{\Psi}}^*_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_1\left(1\right){\mathit{\Psi}}_2\left(2\right)\cdot \partial \tau \int{}{\alpha }^*\left(1\right)\alpha \left(1\right)\int{}{\beta }^*\left(2\right)\beta \left(2\right)-\frac{1}{4}\int{}{\mathit{\Psi}}^*_2\left(1\right){\mathit{\Psi}}^*_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_2\left(1\right){\mathit{\Psi}}_1\left(2\right)\cdot \partial \tau \int{}{\alpha }^*\left(1\right)\beta \left(1\right)\int{}{\beta }^*\left(2\right)\alpha \left(2\right)-\frac{1}{4}\int{}{\mathit{\Psi}}^*_2\left(1\right){\mathit{\Psi}}^*_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_1\left(1\right){\mathit{\Psi}}_2\left(2\right)\cdot \partial \tau \int{}{\alpha }^*\left(1\right)\beta \left(1\right)\int{}{\beta }^*\left(2\right)\alpha \left(2\right)+\frac{1}{4}\int{}{\mathit{\Psi}}^*_2\left(1\right){\mathit{\Psi}}^*_1\left(2\right)\frac{e^2}{4\pi {\varepsilon }_0\left|r_1-r_2\right|}{\mathit{\Psi}}_2\left(1\right){\mathit{\Psi}}_1\left(2\right)\cdot \partial \tau \int{}{\alpha }^*\left(1\right)\alpha \left(1\right)\int{}{\beta }^*\left(2\right)\beta \left(2\right) \nonumber \]