The LCAO-MO-SCF Equation

Having a hamiltonian and a trial wavefunction, we are now in a position to use the linear variation method. The detailed derivation of the resulting equations is complicated and notationally clumsy, and it has been relegated to Appendix 7. Here we discuss the results of the derivation. 

For our restricted case of a closed-shell single-determinantal wavefunction, the variation method leads to 

These equations are sometimes called the Hartree-Fock equations, and F is often called the Fock operator. The detailed formula for F is (from Appendix 7) 

The symbols Jj and Kj stand for operators related to the l/n-y operators in H. Jj is called a coulomb operator because it leads to energy terms corresponding to charge cloud repulsions. It is possible to write Jj explicitly  

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Semirigorous LCAO-MO-SCF methods start with the complete many-electron Hamiltonian and make certain approximations for the integrals and for the form of the matrices to be solved. Several years ago, such a method was derived starting with the correct many electron Hamiltonian (in which interelectronic interactions are included explicitly) and the LCAO-MO-SCF equations of Roothaan and then making a consistent series of systematic... 

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