Orbital rotations redundant parameters

Thus the Hessian will become singular if we include rotations between the active orbitals. Redundant parameters must not be included in the Newton-Raphson procedure.They are trivial to exclude for the examples given above, but in more general cases a redundant variable may occur as a linear combination of S and T and it might be difficult to exclude them. One of the advantages of the CASSCF method is that all parameters except those given above are non-redundant. 

Further redundancies are represented by rotations amongst the inactive (occupied) orbitals or amongst the secondary (virtual) orbitals as they do not change the quasienergy they are straightforwardly eliminated within the chosen parameterization. We will reserve indices i, j, k and I for inactive orbitals, indices a, b, c and d for secondary orbitals and indices p, q, r and, v for general orbitals. In terms of non-redundant parameters the k operator can then be written... 

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. 

Clearly, our MCSCF parametrization is redundant in the sense that (12.2.52) is satisfied for c = C and ic = 0. However, as noted above, we are not interested in this redundancy and shall instead concentrate on identifying redundancies among the orbital-rotation parameters. 

Let us first demonstrate explicitly the automatic fulfilment of the stationary conditions for the redundant rotations. Inserting the expansion (12.2.53) in the expression for the orbital gradient (12.2.59), we find that the gradient with respect to a redundant orbital rotation may be expressed as a linear combination of the gradients with respect to the nonredundant orbital and configuration parameters ... 

In the present section, we discuss the param zation of the restricted Hartrcc—Fock wave function and its electronic energy. We begin by reviewing CSFs in Section 10.1.1 and then go on to consider orbital rotations of such CSFs in Section 10.1.2, paying special attention to the concept of redundant and nonredundant orbital rotations. Finally, in Section 10.1.3, we discuss the parametrization of the electronic energy, expanding it to second order in the orbital-rotation parameters. 

As it stands, the orbital-rotation operator in (10.1.9) contains parameters that mix all classes of MOs among one another - the inactive orbitals, the active orbitals and the virtual orbitals. These parameters are all necessary in order to carry out a general rotation of the individual MOs. For a general transformation of the wave function, however, not all of these parameters are needed those parameters that are not needed for a general transformation of the wave function are referred to as redundant. We shall here use the term redundant in a more specialized sense, applying it to those parameters Kpq that are not needed for a general first-order transformation of the wave function — CSF). A parameter Kp is thus said to be redundant if the corresponding operator in (10.1.9) satisfies the condition... 

By group-theoretical arguments, it may be shown that, if the set of redundant orbital-rotation operators (10.1.10) constitute a group, then their elimination from the orbital-rotation operator (10.1.9) will not affect our description of the wave function to any order in - that is. any state that can be reached with the full set of redundant and nonredundant parameters in (10.1.8) can then also be reached with the set of nonredundant parameters. 

To illustrate the identification of redundant parameters, we consider the three CSFs of Section 10.1.1. Since, according to Section 3.3, no diagonal elements Kpp appear in the special orthogonal orbital transformation, we need to examine only the nondiagonal elements. For the closed-shell CSF (10.1.2), we have for rotations among inactive orbitals... 

In Section 10.1.2, the redundant orbital-rotation parameters Kpg were defined by the following condition on the associated excitation operators ... 

Hence, there is no need to include the redundant orbital-rotation parameters in the optimization - any state that we may arrive at in the space of the nonredundant rotations will remain stationary in the full space of redundant and nonredundant rotations. 

In any Newton-based optimization - which, as discussed in Section 10.8, implicitly or explicitly requires the inversion of the Hessian matrix - the inclusion of redundant parameters is not only unnecessary but also undesirable since, at stationary points, these parameters make the electronic Hessian singular. The singularity of the Hessian follows from (10.2.8), which shows that the rows and columns corresponding to redundant rotations vanish at stationary points. Away from the stationary points, however, the Hessian (10.1.30) is nonsingular since the gradient elements that couple the redundant and nonredundant operators in (10.2.5) do not vanish. Still, as the optimization approaches a stationary point, the smallest eigenvalues of the Hessian will tend to zero and may create convergence problems as the stationary point is approached. Therefore, for the optimization of a closed-shell state by a Newton-based method, we should consider only those rotations that mix occupied and virtual orbitals ... 

Having established the need to avoid the redundant rotations in Newton-based optimizations of the wave function, let us briefly consider rotations that are purely redundant. According to the discussion in Section 10.1.2, an antisymmetric matrix containing the redundant orbital-rotation parameters has the following block-diagonal stmeture... 

The unitary matrix exp(—ic ) transforms the inactive orbitals among themselves, the active orbitals among themselves, and the virtual orbitals among themselves. For the optimization of the wave function, it is convenient to set the redundant parameters equal to zero in each iteration since this choice of k simplifies the orbital-rotation operator ic. In Exercise 10.2, it is shown that, for a closed-shell state, the choice of k = 0 leads to transformed MOs that (in the least-squares sense) are as similar as possible to the original MOs - that is, to the transformed MOs that have the largest possible overlap with the original ones. 

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. 

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... 

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