Nonredundant rotations

For a closed-shell wave limction, therefore, the only nonredundant rotations are those that mix inactive and virtual orbitals. 

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. 

For the nonredundant inactive-active, inactive-virtual and active-virtual rotations of a Hartree-Fock state, the occupations are different and (10.9.19) can be used to fix the Fock-matrix elements. For closed-shell systems, for example, the only nonredundant rotations are those that mix occupied and virtual orbitals. The corresponding elements of the Foek matrix may be identified uniquely as... 

For open shells, the identification of the Fock-matrix elements is more difficult. For high-spin states, it is straightforward to identify the elements that correspond to the nonredundant rotations since these rotations mix orbitals of different occupations. The identification of the elements that correspond to redundant rotations is less straightforward, hi some cases, such as for the two-electron singlet state, an unambiguous identification cannot be made since rotations between singly occupied orbitals are not redundant. 

For H2 in the STO-3G basis, the only nonredundant rotation is the one that mixes the gerade and ungerade orbitals. The singlet and triplet Hessians thus contain only one element ... 

One may also wish to impose an additional requirement on the connection, namely that it is translationally and rotationally invariant. This may seem to be a trivial requirement. However, a connection is conveniently defined in terms of atomic Cartesian displacements rather than in terms of a set of nonredundant internal coordinates. This implies that each molecular geometry may be described in an infinite number of translationally and rotationally equivalent ways. The corresponding connections may be different and therefore not translationally and rotationally invariant. In other words, the orbital basis is not necessarily uniquely determined by the internal coordinates when the connections are defined in terms of Cartesian coordinates. Conversely, a rotationally invariant connection picks up the same basis set regardless of how the rotation is carried out and so the basis is uniquely defined by the internal coordinates. [For a discussion of translationally and rotationally invariant connections, see Carlacci and Mclver (1986).]... 

SOLUTION A three-fold improper axis coincident with the regular three-fold rotational axis appears to have the correct symmetry characteristics It rotates each hydrogen to the next position and then interchanges the spaces above and below the plane. Do we need it in the group Only if there is no other single operation that accomplishes the same thing. There is no such operation, so this improper axis is indeed a nonredundant symmetry element. ... 

The energy at the new point depends on the Hamiltonian matrix elements and the parameters of the rotation. Not all of these parameters will necessarily cause a change in the energy, and we need to determine which parameters are the nonredundant ones. Substituting k into the first term in the energy expansion, (8.10),... 

The summation in (12.2.6) should be interpreted as involving only the nonredundant orbital rotations. A discussion of Hartree-Fock orbital redundancies was given in Section 10.1.2. In MCSCF theory, the question of orbital redundancies is more difficult because of the coupling to the configuration space. The MCSCF redundancies are therefore considered in more detail in Section 12.2.6, where we develop a set of simple conditions that allow us to identify the redundant operators in (12.2.6). 

Oibital redundancies were discussed in connection with Hartree-Fock theory in Sections 10.1.2 and 10.2.2. We recall that, in Haitree-Fock theory, all rotations that mix the inactive orbitals among one another (i.e. the inactive-inactive rotations) were identified as redundant, as were the virtual-virtual rotations. On the otho hand, the inactive-virtual, inactive-active and active-virtual rotations were all classified as nonredundant. 

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. 

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 ... 

The number of nonredundant orbital rotations for n > 2 is V and there is one nonredundant Cl parameter. The total number of nonredundant parameters is therefore V -I- 1. 

The purpose of the present chapter is to discuss the structure and construction of restricted Hartree-Fock wave functions. We cover not only the traditional methods of optimization, based on the diagonalization of the Fock matrix, but also second-order methods of optimization, based on an expansion of the Hartree-Fock eneigy in nonredundant orbital rotations, as well as density-based methods, required for the efficient application of Hartree-Fock theory to large molecular systems. In addition, some important aspects of the Hartree-Fock model are analysed, such as the size-extensivity of the energy, symmetry constraints and symmetry-broken solutions, and the interpretation of orbital energies in the canonical representation. 

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. 

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. 

As for closed shells, it follows that the inactive-inactive and virtual-virtual rotations are redundant. On the other hand, all rotations that mix different classes of orbitals (i.e. the inactive-active, inactive-virtual and active-virtual rotations) are nonredundant. It remains only to investigate the active-active rotations. 

We conclude that the active-active rotations are nonredundant for the singlet state (10.1.7) but redundant for the triplet state (10.1.4)-(10.1.6). 

More generally, we conclude that the inactive-inactive and secondary-secondary rotations are always redundant, whereas the inactive-active, inactive-secondary and active-secondary rotations are always nonredundant. The active-active rotations, by contrast, may or may not be redundant depending on the structure of the CSF, and their redundancy must be established for each CSF separately. In Section 12.2.6, we shall see how redundant parameters can be identified for multiconfigurational wave functions and present a simple general prescription for identifying the nonredundant active-active rotations for single-configuration CSF wave functions. 

To characterize the stationary point and to distinguish local minima from saddle points, we must consider the second-order variation of the energy E k). For ground states, in particular, we must require the electronic Hessian in (10.1.30) to be positive definite (with respect to the nonredundant orbital rotations). At a stationary point, the expression for this Hessian simphfies somewhat. Invoking the Jacobi identity (1.8.17), we obtain... 

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 ... 

The electronic gradient for the nonredundant occupied-virtual orbital rotations is therefore... 

Equating (10.9.17) and (10.9.18), we arrive at the following expression for those elements of the Fock matrix that correspond to the nonredundant orbital rotations ... 

The conditions on the zero-order approximate GET vector determine those elements that correspond to nonredundant oibital rotations. To fix the remaining elements of the Fock matrix, we consider the Jacobian. By matching as closely as possible the elements of the exact and approximate Jacobians (10.9.5) and (10.9.10), we hope to improve the local description of the GET vector and thus the convergence of the optimization. In general, however, we cannot reproduce the Jacobian exactly by this procedure. This is not too critical since the form of the Jacobian does not affect the identification of the stationary point, only the path that is taken towards this point Let us first consider closed-shell states. For such states, the elements of the ap xoximate Jacobian (10.9.10)... 

In the optimization of a closed-shell electronic system, the nonredundant orbital-rotation parameters... 

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