Brian C. Hoffman: SCF Theory

The derivation of the Hartree-Fock equations involved the minimization of the expectation value of the electronic Hamiltonian and required the introduction of a set of undetermined multipliers, $\epsilon_{ab}$ . The notation for the undetermined multipliers was chosen with the foreknowledge that they would correspond to orbital energies, but no proof was given and no definition of an orbital energy and its relation to the electronic energy was provided. The energy of a canonical SCF orbital is defined by equation (1.48) and may written in terms of the one-electron and two-electron integrals as follows:

	$\textstyle \hat{f} \mid \chi_i >$	$\displaystyle = \epsilon_i \vert \chi_i \rangle$	(74)
$\displaystyle \Rightarrow$	$\textstyle \langle \chi_i \vert \hat{f} \vert \chi_i \rangle$	$\displaystyle = \langle \chi_i \vert \epsilon_i \vert \chi_i \rangle$	(75)
$\displaystyle \Rightarrow$	$\textstyle \langle \chi_i \vert \hat{f} \vert \chi_i \rangle$	$\displaystyle = \epsilon_i \langle \chi_i \vert \chi_i \rangle$	(76)
$\displaystyle \Rightarrow$	$\textstyle \epsilon_i$	$\displaystyle = \langle \chi_i \vert \hat{f} \vert \chi_i \rangle$	(77)
		$\displaystyle = \langle \chi_i \vert \hat{h} + \sum_b (\hat{J}_b -\hat{K}_b) \vert \chi_i \rangle$	(78)
		$\displaystyle = \langle \chi_i \vert \hat{h} \vert \chi_i \rangle + \sum_b \langle \chi_i \vert \hat{J}_b \vert \chi_i \rangle$	(79)
		$\displaystyle - \langle \chi_i \vert \hat{K}_b \vert \chi_i \rangle$	(80)
		$\displaystyle = \langle i \vert \hat{h} \vert i \rangle + \sum_b \langle ib \vert ib \rangle - \langle ib \vert bi \rangle$	(81)
		$\displaystyle = \langle i \vert \hat{h} \vert i \rangle + \sum_b \langle ib \vert \vert ib \rangle.$	(82)

Therefore, the energy of an occupied spin orbital, $\chi_a$ , is a sum of the kinetic and nuclear attraction energies for an electron in $\chi_a$ plus a Coulomb $\langle ib \vert ib \rangle$ and exchange $\langle ib \vert bi \rangle$ interaction with each of the remaining N-1 electrons. Note that while the summation runs over the spin orbitals of all the electrons including the electron in $\chi_a$ , the Coulomb and exchange interaction, $\langle aa \vert \vert aa \rangle$ , is zero. The energy of an unoccupied orbital, $\chi_r$ , differs from that of an occupied orbital because while it has the same kinetic and nuclear energy component, it has a Coulomb and exchange interaction with all N electrons in the system; thus, it is as if an extra electron had been added to $\chi_r$ to form an (N+1) electron system. Given these interpretations, it is not surprising that the electron affinities (EAs) and ionization potentials (IPs) for an N-electron system described by a single Slater determinant are equal to the negative of the virtual orbital energy, $\epsilon_r$ , and occupied orbital energy, $\epsilon_a$ , respectively, assuming that the orbitals are not allowed to relax. This relationship between orbital energies, IPs, and EAs is known as Koopman's theorem.[20] In general, IPs determined via Koopman's Theorem are fairly reliable whereas Koopman's Theorem EAs are quite poor due to the much larger impact of orbital relaxation on the EAs.

Orbital Energies and Koopman's Theorem