Lagrange multiplier orbital energy

The canonical MOs are convenient for the physical interpretation of the Lagrange multipliers. Consider the energy of a system with one electron removed from orbital number Ic, and assume that the MOs are identical for the two systems (eq. (3.32)). 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

The reference state T is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions orbital energy functional E = To + Ec is to be made stationary, subject to the orbital orthonormality constraint (i j) = StJ, imposed by introducing a matrix of Lagrange multipliers. The variational condition is... 

Nonzero Lagrange multipliers are required for orthogonalization of orbitals x° to Xc. The energy functional (4> // 4>) is... 

In Eq. 4.7, ij/j is the jth one-electron orbital accommodating the /rth electron with spatial coordinate and spin coordinate Likewise, the rth electron resides in the ith orbital denoted by (/r,. The Lagrange multiplier, Sj, guarantees that the solutions to this equation forms an orthonormal set and is the expectation value for the equation. Hence, it is the quantized one-electron orbital energy. The second term within the... 

In this last expression, /j, is an additional, non-physical parameter, which represents the fictitious mass assigned to the additional degrees of freedom, Cj(G) s, of the system. The potential energy of the system as a whole is (c, R) = ( j(c),R), the electron+nuclei total energy functional in the Khon-Sham framework. Finally, Aij are Lagrange multipliers introduced to satisfy at all times the orthonormality constraints of the Kohn-Sham orbitals. 

In the Hartree-Fock equations the Lagrange multipliers are actually written —TSy to reflect the fact that they are related to the molecular orbital energies. The equation to be solved is thus ... 

To find a minimum energy, one can vary the orbitals under the condition that they remain orthogonal by using the method of Lagrange multipliers. This leads to the definition of the Fock operator, Fi, which describes the kinetic energy, the nuclear attraction energy, and the electron repulsion energy of one electron in the field of the other electrons ... 

---