MADNESS
0.10.1
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The Hartree-Fock wave function is computed for the helium atom in three dimensions without using spherical symmetry.
The atomic orbital is an eigenfunction of the Fock operator
that depends upon the orbital via the Coulomb potential ( ) arising from the electronic density ( ).
Per the usual MADNESS practice, the equation is rearranged into integral form
where and is the Green's function for the Helmholtz equation
The initial guess is , which is normalized before use. Each iteration proceeds by
The kinetic energy operator is denoted by . Thus, one would expect to compute the kinetic energy with respect to a wave function by . In this particular example, the wave functions goes to zero smoothly before the boundary so we apply the product rule for differentiation and the kinetic energy is equal to .
The source is here.