The expansion of \(\frac{1}{r_{ij}}\) (the reciprocal of the inter-electron distance)

The expansion of enables the Hartree equations to be solved analytically. Consider two electrons and  as depicted in the diagram below:

According to the law of cosines , which is equal to or

where .

Due to inter-electronic repulsion,  if is acute and hence , which implies that we can express eq35 as a binomial series, where  . Substituting  in the series, rearranging and displaying the terms of up to , we have

Since the coefficients of the powers of are the Legendre polynomials and that

As mentioned in the previous article, is dependent on , , and (see diagram below). This implies that a function of like is a function of four angles. If we rotate the z-axis such that it lies on  , the vectors are expressed in a new coordinate system, where the new polar angle and azimuthal angle are   and  respectively. Consequently, we have , which is a simpler function to work with. We can also simplify the function in the original coordinate system by holding   and  as constants, resulting in a function of the variables and , or .

With reference to the new coordinate system, and its complex conjugate are eigenfunctions of with eigenvalue . For the original coordinate system, and its complex conjugate are eigenfunctions of with the same eigenvalue .



Show that the eigenvalue of   is in any three-dimensional coordinate system.


From eq108 and eq109, the quantum orbital angular momentum ladder operators for the original and new coordinate systems are and  respectively. The corresponding eigenfunctions of  and are and  respectively. Following the steps of determining the eigenvalues of and , we have and respectively.


Furthermore, the general solution of the eigenfunction of  is , where are the associated Legendre polynomials and are the coefficients of the basis polynomials in the linear combination. Therefore,


Why is ?


For a particular value of , the linear combination is a sum of basis functions of a particular quantum angular momentum value . For example, when , we have a linear combination of the hydrogenic -orbital wavefunctions. Therefore, the eigenvalues of each term of the sum is always degenerate, which ensures that is an eigenfunction of .


To solve for , we multiply both sides of eq37 by the eigenfunction  and integrate over all space:

Due to the orthogonal property of associated Legendre polynomials, only the  term in the RHS expansion of the above equation survives,

From eq98 of notes on Legendre functions, . Substitute this equation in the above equation and rearranging,

To evaluate the integral in eq38, we expand the eigenfunction as a linear combination of basis functions in the new coordinate system:



Why can we express in the original coordinate system as a linear combination of basis functions in the new coordinate system?


As mentioned earlier in the article,   and share the same set of eigenvalues. For a particular value of , is an eigenfunction of with a specific eigenvalue of , which is the same eigenvalue associated with for the same value of . In other words, , which implies that is invariant with respect to rotation of the coordinate system.


To solve for , multiply both sides of eq39 by , integrate over all space and repeat the steps in evaluating . We have

Next, eq39 must hold for the case of , with and . Expanding eq39,

From eq33b of notes on Legendre functions, , which is equal to zero if  and is equal to one if (because ), i.e.

So the only term that survives in eq41 is  and eq41 becomes

Substitute eq40, where and noting that , in eq43,

Substitute eq44 in eq38

Combining eq45, eq37 and eq36

Substitute in eq46, we have



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