Redundant rotations

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... 

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 ... 

It should be understood that there are many solutions to equations (10.2.1) - for two reasons. First, as for any nonlinear set of equations, there are many independent solutions to the variational conditions, each of which represents a different stationary point (local minimum or saddle point) of the Hartree-Fock energy function E k). Second, because of the presence of the redundant rotations, each stationary point of E(k) may be represented in infinitely many ways, related to one another by means of unitary transformations, as will be discussed in Section 10.2.2. In the following, we shall (unless otherwise stated) assume that the solution to (10.2.1) corresponds to the global minimum of E(k), as appropriate for an approximation to the electronic ground state. 

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... 

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 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. 

This table gives the displacements for the normal mode corresponding to the imaginary frequency in terms of redundant internal coordinates (several zero-valued coordinates have been eliminated). The most significant values in this list are for the dihedral angles D1 through D6. When we examine the standard orientation, we realize that such motion corresponds to a rotation of the methyl group. 

The parameter redundancy is also the reason that care should be exercised when trying to decompose energy differences into individual terms. Although it may be possible to rationalize the preference of one conformation over another by for example increased steric repulsion between certain atom pairs, this is intimately related to the chosen functional form for the non-bonded energy, and the balance between this and the angle bend/torsional terms. The rotational banier in ethane, for example, may be reproduced solely by an HCCH torsional energy term, solely by an H-H van der Waals repulsion or solely by H-H electrostatic repulsion. Different force fields will have (slightly) different balances of these terms, and while one force field may contribute a conformational difference primarily to steric interactions, another may have the... 

It should be obvious that if A vanishes, the phase degree of freedom has to become redundant, as seen later. It would be worth mentioning that similar configuration has been studied in other contexts [21-23], Note that the configuration in (44) breaks rotational invariance as well as translational invariance, but the latter invariance is recovered by an isospin rotation [26]. 

Five isotopomers of Sia were studied in Ref (20), and are labeled as follows Si- Si- Si (I) Si- Si- Si (II) Si- Si- Si (III) Si- "Si- Si (IV) Si- Si- °Si (V). Rotational constants for each (both corrected and uncorrected for vibration-rotation interaction) can be found towards the bottom of Table I. Structures obtained by various refinement procedures are collected in Table II. Two distinct fitting procedures were used. In the first, the structures were refined against all three rotational constants A, B and C while only A and C were used in the second procedure. Since truly planar nuclear configurations have only two independent moments of inertia (A = / - 4 - 7. = 0), use of B (or C) involves a redundancy if the other is included. In practice, however, vibration-rotation effects spoil the exact proportionality between rotational constants and reciprocal moments of inertia and values of A calculated from effective moments of inertia determined from the Aq, Bq and Co constants do not vanish. Hence refining effective (ro) structures against all three is not without merit. Ao is called the inertial defect and amounts to ca. 0.4 amu for all five isotopomers. After correcting by the calculated vibration-rotation interactions, the inertial defect is reduced by an order of magnitude in all cases. 

Point of inversion. The action of a point of inversion is described above in the context of improper rotation axes. Note that planes of symmetry and points of inversion are somewhat redundant symmetry elements, since they are already implicit in improper rotation axes. However, they are somewhat more intuitive as separate phenomena than are S axes, and thus most texts treat them separately. 

The unitary diagonal matrix exp [i d ] induces phase shifts of the orbitals. These phase shifts are redundant in our future derivations where only special unitary matrices (Eq. (4.15)) need to be considered. In time-independent theory it is practise to use real orbitals. All rotations of real orbitals into real orbitals can be written... 

The state rotation parameters S are obtained from the variations of the individual Cl coefficients as S = C SCq, but the number of parameters in 5Cq is M, while the number of linearly independent parameters in S is only M-l. This redundancy in 5C0 can be removed by adding one row to equation... 

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. 

Oi contains the identity operation k2 contains three 180° rotations about x, y, z axes, respectively k3 contains six 90° rotations (+ and —) about the x, yt z axes k4 contains six 180° rotations about the six (110) axes k6 contains eight 120° rotations (4- and —) about the four (111) axes. The d wave functions are even and therefore operations involving inversion provide a redundant set. The degeneracy within a representation is given by ci. The Bethe (66) and Mulliken (457a) notations are compared.]... 

We require only three nuclear coordinates to define the nuclear motion and we choose these to be R, the internuclear distance, and 0 the third Euler angle x is a redundant coordinate. In fact, because there are no nuclei lying off-axis in a diatomic molecule, X is undefineable it is, however, expedient to retain it because of simplification in the final form of the rotational Hamiltonian. We shall examine this point in more detail in... 

In Section VIII we described a method for finding the most probable rotationally symmetric shape given measurements of point location. The solution for mirror symmetry is similar. In this case, given m measurements (where m - 2q), the unknown parameters are fyjpj, (0 and 0 where 0 is the angle of the reflection axis. However these parameters are redundant and we reduce the dimensionality of the problem by replacing two-dimensional (0 with the one dimensional x0 representing the x-coordinate at which the reflection axis intersects the x-axis. Additionally we replace Rt, the rotation matrix with ... 

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