Theory Notes

Helium Calculations

\( [-\frac{\hbar^2}{2m}(\nabla^2_1 +\nabla^2_2) - \frac{Ze^2}{r_1} - \frac{Ze^2}{r_2}+\frac{e^2}{r^2_{12}} ]\psi = E\psi\nonumber \)

Define \(\rho = \frac{Zr}{a_0}\) where \(a_0 = \frac{\hbar^2}{me^2}\) (Bohr radius). Then

\([-\frac{\hbar^2}{2m}Z^2(\frac{me^2}{\hbar^2})^2(\nabla^2_{\rho_1}+\nabla^2_{\rho_2}) - Z^2\frac{e^2}{a_0}\rho^{-1}_1 - Z^2\frac{e^2}{a_0}\rho^{-1}_2 + \frac{e^2}{a_0}Z\rho^{-1}_{12}]\psi= E\psi\nonumber\)

But \(\frac{\hbar^2}{m}(\frac{me^2}{\hbar^2})^2 = \frac{e^2}{a_0}\) is in atomic units (au) of energy. Therefore

\([-\frac{1}{2}(\nabla^2_{\rho_1}+\nabla^2_{\rho_2}) - \frac{1}{\rho_1} - \frac{1}{\rho_2} + \frac{Z^{-1}}{\rho_{12}}]\psi = \varepsilon\psi\nonumber\) where \(\varepsilon = \frac{Ea_0}{Z^2e^2}\)

The problem to be solved is thus \([\frac{1}{2}(\nabla^2_1+\nabla^2_2) - \frac{1}{r_1}-\frac{1}{r_2} + \frac{Z^{-1}}{r_{12}}]\psi = \varepsilon\psi\nonumber\)

[figure to be inserted]

The Hartree Fock Method

Assume that \(\psi({\bf r}_1,{\bf r}_2)\) can be written in the form

\(\psi({\bf r}_1,{\bf r}_2) = \frac{1}{\sqrt{2}}[u_1(r_1)u_2(r_2) \pm u_2(r_1)u_1(r_2)]\nonumber\)

for the \(1S^21S\) ground state

\([-\frac{1}{2}(\nabla^2_1+\nabla^2_2) - \frac{1}{r_1}- \frac{1}{r_2} + \frac{Z^{-1}}{r_{12}}]\psi(r_1,r_2) = E\psi(r_1,r_2)\nonumber\)

Substitute into \(<\psi|H-E|\psi>\) and require this expression to be stationary with respect to arbitrary infinitesimal variations \(\delta u_1\) and \(\delta u_2\) in \(u_1\) and \(u_2\). ie

\(\frac{1}{2}<\delta u_1(r_1)u_2(r_2) \pm u_2(r_1)\delta u_1(r_2)|H-E|u_1(r_1)u_2(r_2)\pm u_2(r_1)u_1(r_2)>\nonumber\)

\(=\int\delta u_1(r_1)d{\bf r}_1\{\int d{\bf r}_2u_2(r_2)(H-E)[u_1(r_1)u_2(r_2)\pm u_2(r_1)u_1(r_2)]\}\nonumber\)

\(= 0 \ \ \ for \ arbitrary \ \delta u_1(r_1).\nonumber\)

Therefore \(\{\int d{\bf r}_2 \ldots \} = 0\).

Similarly, the coefficient of \(\delta u_2\) would give

\(\int d{\bf r}_1 u_1(r_1)(H-E)[u_1(r_1)u_2(r_2) \pm u_2(r_1)u_1(r_2)] = 0\nonumber\)

Define

\(I_{12} = \int dru_1(r)u_2(r), \nonumber\)

\(I_{21} = \int dru_1(r)u_2(r), \nonumber\)

\(H_{ij} = \int d{\bf r}u_i(-\frac{1}{2}\nabla - \frac{1}{r})u_j(r), \nonumber\)

\(G_{ij}(r) = \int d{\bf r}^\prime u_i(r^\prime)\frac{1}{|{\bf r} - {\bf r}\prime|}u_j(r^\prime)\nonumber\)

Then the above equations become the pair of integro-differential equations

\([ H_0 - E + H_{22}+G_{22}(r)]u_1(r) = \mp [ I_{12}(H_0-E) + H_{12}+G_{12}(r)]u_2(r)\nonumber\)

\([H_0-E+H_{11}+G_{11}(r)]u_2(r) = \mp [I_{12}(H_0-E) + H_{12}+G_{12}(r)]u_1(r)\nonumber\)

These must be solved self-consistently for the "constants" \(I_{12}\) and \(H_{ij}\) and the function \(G_{ij}(r)\).

The H.F. energy is \(E \simeq -2.87\cdots a.u.\) while the exact energy is \(E = -2.903724\cdots a.u.\)

The difference is called the "correlation energy" because it arises from the way in which the motion of one electron is correlated to the other. The H.F. equations only describe how one electron moves in the average field provided by the other.

Configuration Interaction

Expand \( \psi({\bf r}_1,{\bf r}_2)= C_0u^{(s)}_1(r_1)u^{(s)}_1(r_2) + C_1u^{(P)}_1({\bf r}_1)u^{(P)}_1({\bf r}_2)\Upsilon^0_{1,1,0}(\hat{\bf r}_1, \hat{\bf r}_2)+C_2u^{(d)}_1({\bf r}_1)u^{(d)}_2({\bf r}_2)\Upsilon^0_{2,2,0}(\hat{\bf r}_1, \hat{\bf r}_2)+... \pm\) exchange where \( \Upsilon^M_{l_1,l_2,L}(\hat{\bf r}_1, \hat{\bf r}_2)=\Sigma_{m_1,m_2}\Upsilon^{m_1}_{l_1}({\bf r}_1)\Upsilon^{m_2}_{l_2}({\bf r}_2)\times <l_1l_2m_1m_2\mid LM> \).

This works, but is slowly convergent, and very laborious. The best CI calculations are accurate to \( ~10^{-7}\) a.u.

Hylleraas Coordinates

[E.A. Hylleraas, Z. Phys. \({\bf 48}, 469(1928)\) and \({\bf 54}, 347(1929)\)] suggested using the co-ordinates \(r_1\), \(r_2\) and \(r_{12}\) or equivalently

\(\begin{eqnarray} s &=& r_1 + r_2, \nonumber\\ t &=& r_1-r_2, \nonumber\\ u &=& r_{12}\nonumber \end{eqnarray}\)

and writing the trial functions in the form

\( \Psi({\bf r}_1,{\bf r}_2) = \sum^{1+j+k\leq N}_{i,j,k}c_{i,j,k}r_1^{i+l_1}r_2^{j+l_2}r_{12}^ke^{-\alpha r_1 - \beta r_2} \mathcal{Y}^M_{l_1,l_2,L}(\hat{r}_1,\hat{r}_2)\pm exchange\nonumber \)

Diagonalizing H in this non-orthogonal basis set is equivalent to solving \( \frac{\partial E}{\partial c_{i,j,k}} = 0\nonumber \) for fixed \(\alpha\) and \(\beta\).

The diagonalization must be repeated for different values of \(\alpha\) and \(\beta\) in order to optimize the non-linear parameters.

Completeness

The completeness of the above basis set can be shown by first writing \(r_{12}^2 = r_1^2 + r_2^2 - 2r_1r_2\cos(\Theta_{12})\) and \(\cos(\Theta_{12})=\frac{4\pi}{3}\sum^1_{m=-1}Y^{m*}_l(\theta_1,\varphi_1)Y^m_l(\theta_2,\varphi_2)\) consider first S-states. The \(r_{12}^0\) terms are like the ss terms in a CI calculation. The \(r_{12}^2\) terms bring in p-p type contributions, and the higher powers bring in d-d, f-f etc type terms. In general \( P_l(\cos(\theta_{12}) = \frac{4\pi}{2l+1}\sum^l_{m=-l}{Y^{m}_l}^*(\theta_1,\varphi_1)Y^m_l(\theta_2,\varphi_2)\nonumber \)

For P-states, one would have similarly

\(\begin{eqnarray} r_{12}^0\ \ \ \ \ \ \ \ (sp)P\nonumber\\ r_{12}^2\ \ \ \ \ \ \ \ (pd)P\nonumber\\ r_{12}^4\ \ \ \ \ \ \ \ (df)P\nonumber\\ \vdots \ \ \ \ \ \ \ \ \ \ \vdots\nonumber \end{eqnarray}\)

For D-states

\(\begin{eqnarray} r_{12}^0\ \ \ \ \ \ \ \ (sp)D\ \ \ \ \ \ \ \ (pp^\prime)D\nonumber\\ r_{12}^2\ \ \ \ \ \ \ \ (pd)D\ \ \ \ \ \ \ \ (dd^\prime)D\nonumber\\ r_{12}^4\ \ \ \ \ \ \ \ (df)D\ \ \ \ \ \ \ \ (ff^\prime)D\nonumber\\ \vdots \ \ \ \ \ \ \ \ \ \ \vdots\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots\nonumber\end{eqnarray}\)

In this case, since there are two ``lowest-order couplings to form a D-state, both must be present in the basis set. ie

\(\begin{eqnarray} \Psi(r_2,r_2) = \sum c_{ijk}r_1^ir_2^{j+2}r_{12}^ke^{-\alpha r_1-\beta r-2}\mathcal{Y}^M_{022}(\hat{r}_1,\hat{r}_2)\nonumber\\ +\sum d_{ijk}r_1^{i+1}r_2^{j+1}r_{12}e^{-\alpha^\prime r_1 - \beta^\prime r_2}\mathcal{Y}^M_{112}(\hat{r}_1,\hat{r}_2)\nonumber \end{eqnarray}\)

For F-states, one would need \((sf)F\) and \((pd)F\) terms.

For G-states, one would need \((sg)G\), \((pf)G\) and \((dd^\prime)G\) terms.

Completeness of the radial functions can be proven by considering the Stern-Liouville problem

\( \left(-\frac{1}{2}\nabla^2-\frac{\lambda}{r_s}-E\right)\psi({\bf r}) = 0\nonumber \) or \( \left(-\frac{1}{2}\frac{1}{r^2}\left({r^2}\frac{\partial}{\partial r}\right) - \frac{l\left(l+1\right)}{2r^2} - \frac{\lambda}{r} - E\right)u(r) = 0.\nonumber \) For fixed E and variable \(\lambda\) (nuclear charge).

The eigenvalues are \(\lambda_n = (E/E_n)^{1/2}\), where \(E_n =- \frac{1}{2n^2}\)

INSERT FIGURE HERE

\( u_{nl}(r) = \frac{1}{(2l+1)!}\left(\frac{(n+l)!}{(n-l-1)2!}\right)^{1/2}(2\alpha)^{3/2}e^{-\alpha r},\nonumber \)

with \(\alpha = (-2E)^{1/2}\) and \(n\geq l+1\).

Unlike the hydrogen spectrum, which has both a discrete part for \(E<0\) and a continuous part for \(E>0\), this forms an entirely discrete set of finite polynomials, called Sturmian functions. They are orthogonal with respect to the potential

-ie \(\int^\infty_0 r^2dru_{n^\prime l}(r)\frac{1}{r}u_{nl}(r) = \delta_{n,n^\prime}\nonumber \)

Since they become complete in the limit \(n\rightarrow\infty\), this assures the completeness of the variational basis set.

[see B Klahn and W.A. Bingel Theo. Chim. Acta (Berlin) \({\bf 44}, 9\) and \(27 (1977)\)].

Solutions of the Eigenvalue Problem

For convenience, write

\(\Psi({\bf r}_1,{\bf r}_2) = \sum^N_{m=1}c_m\varphi_m\nonumber \)

where \(m= m'th\) combinations of \(i,j,k\)

\(\varphi_{ijk}=r_1^ir_2^jr_{12}^ke^{-\alpha r_1 - \beta r_2}\mathcal{Y}^M_{l_1,l_2,L}(\hat{r}_1,\hat{r}_2) \pm exchange.\nonumber '"`UNIQ-MathJax1-QINU`"' '"`UNIQ-MathJax2-QINU`"' '"`UNIQ-MathJax3-QINU`"'\)

Therefore \( (\cos^2(\theta)-\sin^2(\theta))H_{12} = \cos(\theta)\sin(\theta)(H_{11}-H_{22})\nonumber \) and \( \tan(2\theta) = \frac{2H_{12}}{H_{11}-H{22}}\nonumber \)

ie

\(\begin{eqnarray} \cos(\theta)=\left(\frac{r+\omega}{2r}\right)^{1/2}\nonumber,\\ \sin(\theta)=-sgn(H_{12})\left(\frac{r-\omega}{2r}\right)^{1/2}\nonumber \end{eqnarray}\\\)

where

\(\begin{eqnarray} \omega = H_{22}-H_{11}\nonumber\\ r=\left(\omega^2+4H_{12}^2\right)^{1/2}\nonumber\\ E_1=\frac{1}{2}\left(H_{11}+H_{22}-r\right)\nonumber\\ E_2=\frac{1}{2}\left(H_{11}+H_{22}+r\right)\nonumber \end{eqnarray}\)

Brute Force Method

-Gives all the eigenvalues and eigenvectors, but it is slow.

-First orthonormalize the basis set - ie form linear combinations

\( \Phi_m = \sum_n\varphi_nR_{nm}\nonumber \) such that \(<\Phi_m|\Phi_n> = \delta_{m,n}\\\). This can be done by finding an orthogonal tranformation, T, such that

\( T^TOT=I=\left( \begin{array}{cccc} \ I_1\ 0 \ldots 0\\ 0\ \ I_2 \ldots 0 \\ \ \ \vdots \ \ \ \ \ \ I_3 \ldots \ 0 \\ 0\ \ 0\ \ 0\ \ddots \end{array}\right); \ \ O_{mn} = <\varphi_m|\varphi_n>\nonumber \) and then applying a scale change matrix \( S = \left(\begin{array}{cccc} \frac{1}{I_1^{1/2}} 0 \ldots 0\\ 0 \frac{1}{I_2^{1/2}} \ldots 0 \\ \vdots \ \ \frac{1}{I_3^{1/2}} 0 \\ 0\ 0\ 0\ \ddots \end{array}\right)= S^T\nonumber \)

Then \(S^TT^TOTS = 1\\\). ie \(R^TOR = 1\) with \(R=TS\).

If H is the matrix with elements \(H_{mn}=<\varphi_m|\varphi_n>\), then H expressed in the \(\Phi_m\) basis set is \( H^\prime = R^THR.\nonumber \)

We next diagonalize \(H^\prime\) by finding an orthogonal transformation W such that \( W^TH^\prime W = \lambda = \left( \begin{array}{cccc} \lambda_1 0 \ldots 0\\ 0\ \lambda_2 \ldots 0 \\ \ \ \vdots \ \ \ \ \ \ddots 0 \\ 0\ \ldots \lambda_N \end{array}\right)\nonumber \)

The q'th eigenvector is

\(\begin{eqnarray} \Psi^{(q)} = \sum_n\Phi_n W_{n,q}\nonumber \\ = \sum_{n,n^\prime}\varphi_{n^\prime}R_{n^\prime ,n}W_{n,q}.\nonumber \end{eqnarray}\) ie. \(c_{n^prime}^{(q)} = \sum_n R_{n^\prime n} W_{n,q}\).

The Power Method

-Based on the observation that if H has one eigenvalue, \(\lambda_M\), much bigger than all the rest, and \(\chi = \left(\nonumber\begin{array}{c}a_1\\a_2\\\vdots\nonumber\end{array}\right)\nonumber\) is an arbitrary starting vector, then \(\chi = \sum_q x_q\Psi^{(q)}\nonumber\).

\(\begin{eqnarray} (H)^n\chi = \sum_q x_q \lambda^n_q\Psi^{(q)}\nonumber\rightarrow x_M\lambda_M^n\Psi^{(M)}\nonumber \end{eqnarray}\\\) provided \(x_M\neq 0\\\).

To pick out the eigenvector correspondng to any eigenvalue, with the original problem in the form

\(\begin{eqnarray} H\Psi = \lambda O\Psi\\\nonumber\\\ (H-\lambda)qO)\Psi\nonumber = (\lambda - \lambda_q)O\Psi\nonumber\end{eqnarray}\)

Therefore, \( G\Psi = \frac{1}{\lambda-\lambda_q}\Psi\nonumber\\ \) where \(G=(H-\lambda_qO)^{-1}O\nonumber\) with eigenvalues \(\frac{1}{\lambda_n-\lambda_q}\nonumber\\\).

By picking \(\lambda_q\) close to any one of the \(\lambda_n\), say \(\lambda_{n^\prime}\), then \(\frac{1}{\lambda_n-\lambda_q}\) is much larger for \(n=n^\prime\nonumber\) than for any other value. The sequence is then

\(\begin{eqnarray} \chi_1=G\chi\nonumber\\ \chi_2=G\chi_1\nonumber\\ \chi_3=G\chi_2\nonumber\\ \vdots\nonumber \end{eqnarray}\\ \)

until the ratios of components in \(\chi_n\) stop changing.

- To avoid matrix inversion and multiplication, note that the sequence is equivalent to

\( F\chi_n = (\lambda-\lambda_q)O\chi_{n-1}\nonumber\\ \)

where \(F = H-\lambda_qO\). The factor of \((\lambda - \lambda_q)\) can be dropped because this only affects the normalization of \(\chi_n\). To find \(\chi_n\\\), solve \( F\chi_n = O\chi_{n-1}\nonumber\\ \) (N equations in N unknowns). Then

\( \lambda = \frac{<\chi_n|H|\chi_n>}{<\chi_n|\chi_n>}\nonumber \)

Matrix Elements of H

\( H=-\frac{1}{2}\nabla^2_1 -\frac{1}{2}\nabla^2_2 - \frac{1}{r_1} - \frac{1}{r_2} +\frac{Z^{-1}}{r_{12}}\nonumber\\ \)

Taking \(r_1, r_2\) and \(r_{12}\) as independent variables,

\(\begin{eqnarray} \nabla_1^2 = \frac{1}{r_1^2}\frac{\partial}{\partial r_1}\left(r_1^2\frac{\partial}{\partial r_1}\right)+ \frac{1}{r_{12}^2}\frac{\partial}{\partial r_{12}} \left(r_{12}^2\frac{\partial}{\partial r_{12}}\right)\nonumber -\frac{l_1(l_1+1)}{r_1^2}+2(r_1-r_2\cos(\theta))\frac{1}{r_{12}}\frac{\partial^2}{\partial r_1 \partial r} \nonumber - 2(\nabla_1 \cdot {\bf r}_2)\frac{1}{r}\frac{\partial}{\partial r}\nonumber \end{eqnarray}\)

where INSERT FIGURE HERE

The complete set of 6 independent variables is \(r_1, r_2, r_{12}, \theta_1,\varphi_1, \chi\).

If \(r_{12}\) were not an independent variable, then one could take the column element to be \( d\tau = r_1^2dr_1\sin(\theta_1)d\theta_1d\varphi_1r_2^2dr_2\sin(\theta_2)d\theta_2d\varphi_2.\nonumber \)

However, \(\theta_2\) and \(\varphi_2\) are no longer independent variables. To eliminate them, take the point \({\bf r}_1\) as the

origin of a new polar co-ordinate system, and write \( d\tau=-r_1^2dr_1\sin(\theta_1)d\theta_1d\varphi_1r_{12}^2dr_{12}\sin(\psi)d\psi d\chi\nonumber\\\)

and use \(r_2^2=r_1^2+r_{12}^2 +2r_1r_{12}\cos(\psi).\\ \)

Then for fixed \(r_1\) and \(r_{12}\\\), \( 2r_2dr_2 = -2r_1r_{12}\sin(\psi)d\psi\\ \)

Thus \( d\tau= r_1dr_1r_2dr_2r_{12}dr_{12}\sin(\theta_1)d\theta_1d\varphi_1d\chi\\ \)

The basic type of integral to be calculated is

\(\begin{eqnarray} I(l_1,m_1,l_2,m_2;R) =\int\sin(\theta_1)d\theta_1d\varphi_1d\chi Y^{m_1}_{l_1}(\theta_1,\varphi_1)^{*}Y^{m_2}_{l_2}(\theta_2,\varphi_2)\nonumber\\ \times\int r_1dr_1r_2dr_2r_{12}dr_{12}R(r_1,r_2,r_{12})\nonumber\\ \end{eqnarray}\\\)

Consider first the angular integral. \(Y^{m_2}_{l_2}(\theta_2,\varphi_2)\\\) can be expressed in terms of the independent variables \(\theta_1, \varphi_1,\chi\\\) by use of the rotation matrix relation

\( Y^{m_2}_{l_2}(\theta_2,\varphi_2) =\sum_m\mathcal{D}^{(l_2)}_{m_2,m}(\varphi_1,\theta_1,\chi)^*Y^m_{l_2}(\theta,\varphi)\nonumber\\\)

where \(\theta, \varphi\) are the polar angles of \({\bf r}_2\) relative to \({\bf r}_1\). The angular integral is then

\(\begin{eqnarray} I_{ang}=\int^{2\pi}_0d\chi\int^{2\pi}_0d\varphi_1\int^\pi_0\sin(\theta_1)d=theta_1Y^{m_1}_{l_1}(\theta_1,\varphi_1)^*\nonumber\\ \times\sum_m\mathcal{D}^{(l_2)}_{m_2,m}(\varphi_1,\theta_1,\chi)^*Y^m_{l_2}(\theta,\varphi)\nonumber\\ \end{eqnarray}\\\)

Use

\( Y^{m_1}_{l_1}(\theta_1,\varphi_1)^* = \sqrt{\frac{2l_1+1}{4\pi}}\mathcal{D}^{(l_1)}_{m_1,0}(\varphi_1,\theta_1,\chi)\nonumber\\ \)

together with the orthogonality property of the rotation matrices (Brink and Satchler, p 147)

\( \mathcal{D}^{(j)*}_{m,m^\prime}\mathcal{D}^{(J)}_{M,M^\prime}\sin(\theta_1)d\theta_1d\varphi_1d\chi = \frac{8\pi^2}{2j+1}\delta_{jJ}\delta_{mM}\delta_{m^\prime M^\prime}\nonumber\\ \)

to obtain

\(\begin{eqnarray} I_{ang}=\sqrt{\frac{2l_1+1}{4\pi}}\frac{8\pi^2}{2l_1+1}\delta_{l_1,l_2}\delta_{m_1,m_2}Y^0_{l_2}(\theta,\varphi)\nonumber\\ =2\pi\delta_{l_1,l_2}\delta_{m_1,m_2}P_{l_2}(\cos\theta)\nonumber\\ \end{eqnarray}\\\)

since

\(Y^0_{l_2}(\theta,\varphi)=\sqrt{\frac{2l_1+1}{4\pi}}P_{l_2}(\cos(\theta))\nonumber\\\).

Note that \(P_{l_2}(\cos\theta)\) is just a short hand expression for a radial function because \(\cos\theta = \frac{r_1^2+r_2^2-r_{12}^2} {2r_1r_2}\nonumber\\\).

The original integral is thus

\(\begin{eqnarray} I(l_1,m_1,l_2,m_2;R)=2\pi\delta_{l_1l_2}\delta_{m_1m_2}\nonumber\\ \times\int^\infty_0r_1dr_1\int^\infty_0r_2dr_2\int^{r_1+r_2}_{|r_1-r_2|} r_{12}dr_{12}R(r_1,r_2,r_{12})P_{l_2}(\cos\theta)\nonumber\\ \end{eqnarray}\\\)

where \(\cos\theta = (r_1^2 +r^2_2-r_{12}^2)/(2r_1r_2)\nonumber\) is a purely radial function.

The above would become quite complicated for large \(l_2\) because \(P_{l_2}(\cos\theta)\) contains terms up to \((\cos\theta)^{l_2}\\\). However, recursion relations exist which allow any integral containing \(P_l(\cos\theta)\) in terms of those containing just \(P_0(\cos\theta)=1\nonumber\\\) and \(P_1(\cos\theta)=\cos\theta\nonumber\\\).

Radial Integrals and Recursion Relations

- The basic radial integral is \( I_0(a,b,c) = \int^\infty_0r_1dr_1\int^\infty_{r_1}r_2dr_2\int^{r_1+r_2}_{r_2-r_1}r_{12}dr_{12}r_1^ar_2^br_{12}^ce^{-\alpha r_1-\beta r_2}+\int^\infty_0r_2dr_2\int^\infty_{r_2}r_1dr_1\int^{r_1+r_2}_{r_1-r_2}r_{12}dr_{12}r_1^ar_2^br_{12}^ce^{-\alpha r_1- \beta r_2}\\ =\frac{2}{c+2}\sum^{[(c+1)/2]}_{i=0}\left( c+2 2i+1 \right)\{\frac{q!}{\beta^{q+1}(\alpha+\beta)^{P+1}}\sum_{j=0}q\frac{(p+j)!}{j!}\left(\frac{\beta}{\alpha+\beta}\right)^j+\frac{q^\prime !}{\alpha^{q^\prime+1}(\alpha+\beta)^{p^\prime+1}}\sum^{q^\prime}_{j=0}\frac{(p^\prime+j)!}{j!}\left(\frac{\alpha}{\alpha+\beta}\right)^j\}\nonumber \)

where \( p=a+2i|2\\ p^\prime = b+2i+2\\ q=b+c-2i+2\\ q^\prime = a+c-2i+2 \)

The above applies for a,b \(\geq -2 c\geq -1\). [x] means ``greatest integer in x.

Then \( I_1(a,b,c)=\int d\tau_r r_1^ar_2^br_{12}^ce^{-\alpha r_1-\beta r_2}P_1(\cos\theta) =\frac{1}{2}\left(I_0(a+1,b-1,c)+I_0(a-1,b+1,c)-I_0(a-1,b-1,c+2)\right).\nonumber \)

The Radial Recursion Relation

Recall that the full integral is \(I(l_1m_1,l_2m_2;R)=2\pi\delta_{l_1l_2}\delta_{m_1m_2}I_{l_2}(R)\) where \(I_{l_2}(R) = \int d\tau_r R(r_1,r_2,r_{12})P_{l_2}(\cos\theta)\).

To obtain the recursion relation, use

\(\begin{eqnarray}P_l(x) = \frac{[P^\prime_{l+1}(x)-P^\prime_{l-1}(x)]}{2l+1}\end{eqnarray}\) with \(P_{l+1}^\prime (x) = \frac{d}{dx}P_{l+1}(x)\).

Here \(x=\cos\theta\) and \(\begin{eqnarray}\ \ \ \frac{d}{d\cos\theta}= \frac{dr_{12}}{d\cos\theta}\frac{d}{dr_{12}} = -\frac{r_1r_2}{r_{12}}\frac{d}{dr_{12}}\end{eqnarray}\).

Then, \(\begin{eqnarray}\ \ I_l(R)=-\int d\tau_r R \frac{r_1r_2}{r_{12}}\frac{d}{dr_{12}}\frac{[P_{l+1}(\cos\theta)-P_{l-1}(\cos\theta)]}{2l+1} \end{eqnarray}\).

The \(r_{12}\) part of the integral is \( \begin{eqnarray}\ \ \int^{r_1+r_2}_{|r_1-r_2|}r_{12}dr_{12}R\frac{r_1r_2}{r{12}}\frac{d}{dr_{12}}[P_{l+1}-P_{l-1}]=Rr_1r_2[P_{l+1}-P_{l-1}]^{r_1+r_2}_{|r_1-r_2|}-\int^{r_1+r_2}_{|r_1-r_2|}r_{12}dr_{12}\left(\frac{d}{dr_{12}}R\right)\frac{r_1r_2}{r_{12}}\frac{[P_{l+1}-P_{l-1}]}{2l+1}\nonumber \end{eqnarray}\)

The integrated term vanishes because \(\begin{eqnarray}\ \ \cos\theta = \frac{r_1^2+r_2^2-r_{12}^2}{2r_1r_2}\ \ \ \begin{array}{cc} = -1 & when \ \ r_{12}^2 = (r_1+r_2)^2\\ = \ \ 1 & when \ \ r_{12}^2 = (r_1-r_2)^2 \end{array} \end{eqnarray}\) and \(P_l(1)=1\), \(\ \ P_l(-1)=(-1)^l\).

Thus, \(\begin{eqnarray}\ \ I_{l+1}\left(\frac{r_1r_2}{r_{12}}R^\prime\right) = (2l+1)I_l(R)-I_{l-1}\left(\frac{r_1r_2}{r_{12}}R^\prime\right) \end{eqnarray}\)

If \(R=r_1^{a-1}r_2^{b-1}r_{12}^{c+2}e^{-\alpha r_1 - \beta r_2}\), then \(\begin{eqnarray}\ I_{l+1}(r_1^ar_2^br_{12}^c) = \frac{2l+1}{c+2}I_l(r_1^{a-1}r_2^{b-2}r_{12}^{c+2})+I_{l-1}(r_1^ar_2^br_{12}^c) \end{eqnarray}\)

For the case c=-2, take \(\begin{eqnarray}\ R=r_1^{a-1}r_2^{b-1}\ln r_{12} e^{-\alpha r_1 -\beta r_2}\end{eqnarray}\). Then \(\begin{eqnarray}\ I_{l+1}(r_1^ar_2^br_{12}^c) = I_l(r_1^{a-1}r_2^{b-1}\ln r_{12})(2l+1)-I_{l-1}(r_1^ar_2^br_{12}^{-2}). \end{eqnarray}\) This allows all \(I_l\) integrals to be calculated from tables of \(I_0\) and \(I_1\) integrals.

Graphical Representation

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Optimization of Non-linear Parameters

• Difficulties
• Cure

Small Corrections

• Mass Polarization