eq.1Uncorrelated Ab Initio Methods:
The exact Schrödinger Equation cannot be solved for polyelectronic systems due to various interactions among the electrons. Therefore approximations must be made instead. The first quantum mechanical treatment of molecules began with the development of Ab Initio methods. Ab Initio roughly means 'from the beginning'. The Hartee-Fock Equations, the first developed for this method, are based on the Schrödinger Equation, as applied to larger groupings of nuclei and electrons. Ab Initio methods are based on the assumption that any one electron in a polyelectronic system can be thought of as moving in a electrostatic field consisting of the nuclei and the remaining N-1 electrons.
The hamiltonian for the interaction of electron 1 in orbital y i with the field of nuclei is called the core hamiltonian. It is comprised of the kinetic energy operator and the sum of the electron-nucleus attraction operators.
eq.1Summation occurs over all nuclei A. The coulombic interaction, meaning the electrostatic repulsion, between electrons is absolutely necessary for determining the correct electronic configuration of the molecule. This is represented by the coulomb operator, denoted J.
eq.2Also, as seen in the case of the excited helium atom, electrons of parallel spins but in different spatial orbitals can exchange places. This exchange has no classical analouge and it helps to lower the total electronic energy.
eq.3The combination of the three terms above constitute the principle Hartee-Fock Equation for closed-shell systems.
eq.4Summation occurs over all occupied molecular orbitals m. The quantity containing the coulomb and exchange operators requires further explanation. An electron in one orbital will be repelled by and exchange with electrons in other orbitals. Since two electrons can occupy the same space orbital, pending they pair their spins, then any one electron will interact with two electrons in any other given orbital. While the one electron will experience repulsion due to both of the electrons in the other orbital, it will only exchange places with only one of the electrons, namely the one with the same spin. That is why a 2 is placed in front of the coulomb operator, so as to count repulsion by both electrons in the other orbital. A note of caution is needed here. The sum of the Ei's is not the correct total energy for the molecule. This is because in calculating the electron-electron interactions, each pair of electrons in different orbitals is counted twice, this has nothing to do with the previous statement about the coulomb and exchange operators by the way. Taking this into account results in an expression for the total electronic energy for the molecule.
eq.5This was obtained by doubling eq.4 to represent a filled orbital, summing over all possible orbitals, and then subtracting the unnecessary J and K interactions. Summation occurs over all occupied molecular orbitals m and n. If one cares enough about molecular modeling, which most people don't, one will notice that in order to solve these equations for the molecular orbitals, one must solve the coulomb and exchange integrals, which require orbitals as input. The obvious dilema is: how can one solve these equations when they seem to need the answer in order to begin solving them? The solution here is to give the equations a trial set of orbitals and solve the equations. A refined