Redundant parameters Hartree-Fock theory

The fundamental variational parameters of our theory are the elements of the rotation matrix k (0- ii Hartree-Fock theory, the non-redundant rotations are those between occupied and unoccupied orbitals. Equation (40) implies that the individual Kohn-Sham spin orbitals obey the transformation law... 

In Hartree-Fock theory, complications arise only for open-shell systems, where the active-active rotations are in some cases redundant, in other cases nonredundant For instance, for open-shell states constructed by distributing two electrons between two orbitals, we found in Section 10.1.2 that the active-active rotations are redundant for the triplet state but nonredundant for the singlet state. We can easily imagine that the situation becomes even more complicated in the MCSCF case, where the wave function is generated by optimizing simultaneously the orbital-rotation parameters and a (potentially) laige number of Cl coefficients. Fortunately, for the more common MCSCF models such as those based on the CAS and RAS concepts, the question of redundancies is simple and unexpected redundancies will only rarely arise. 

Although we may keep the redundant parameters fixed (equal to zero) during the optimization of the Hartree-Fock state, we are also free to vary them so as to satisfy additional requirements on the solution - that is, requirements that do not follow from the variational conditions. In canonical Hartree-Fock theory (discussed in Section 10.3), the redundant rotations are used to generate a set of orbitals (the canonical orbitals) that diagonalize an effective one-electron Hamiltonian (the Fock operator). This use of the redundant parameters does not in any way affect the final electronic state but leads to a set of MOs with special properties. 

For variational methods, such as Hartree-Fock (HF), multi-configurational self-consistent field (MCSCF), and Kohn-Sham density functional theory (KS-DFT), the initial values of the parameters are equal to zero and 0) thus corresponds to the reference state in the absence of the perturbation. The A operators are the non-redundant state-transfer or orbital-transfer operators, and carries no time-dependence (the sole time-dependence lies in the complex A parameters). Furthermore, the operator A (t)A is anti-Hermitian, and tlie exponential operator is thus explicitly unitary so that the norm of the reference state is preserved. Perturbation theory is invoked in order to solve for the time-dependence of the parameters, and we expand the parameters in orders of the perturbation... 

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