The interelectronic Coulomb cusp condition

The electronic Coulomb interaction u(r 12) = greatly complicates the task of formulating and carrying out accurate computations of iV-electron wave functions and their physical properties. Variational methods using fixed basis functions can only with great difficulty include functions expressed in relative coordinates. Unless such functions are present in a variational basis, there is an irreconcilable conflict with Coulomb cusp conditions at the singular points ri2 - 0 [23, 196], No finite sum of product functions or Slater determinants can satisfy these conditions. Thus no practical restricted Hilbert space of variational trial functions has the correct structure of the true V-electron Hilbert space. The consequence is that the full effect of electronic interaction cannot be represented in simplified calculations. 

The interelectronic Coulomb cusp can be analyzed by transforming a two-electron Hamiltonian to relative coordinates. The one-electron potential function is regular at the singularity ri2 - 0 and does not affect the cusp behavior. Given coordinates ri and r2, mean and relative coordinates are defined, respectively, by 

Given relative angular momentum , the singular part of the Schrodinger equation is 

When substituted into the differential equation, this gives 

To avoid a singularity that cannot be cancelled by any one-electron potential, the coefficient ofV/-1 must vanish when t = 0. This implies the Coulomb cusp condition fo(q) = food + q H------). A similar expansion is valid for any i 0. Because the 

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