Symmetry blocking

S02. Figure 9 shows how the SACs of 02 can be coupled to the MOs of H2S to give the MOs of H2S02, for each symmetry block, but the reader may work out the equivalent case, in which the MOs of S02 are combined with the SACs of H2. The final result must be qualitatively the same the MO shapes are given in Figure 10. Within the AS block, it is easy to recognize that the number of nodal surfaces increases from 1 to 4 as the energy rises. 

FIGURE 9. SACs of the OXO group (Figure 8a) interacting with the MOs of H2S to give the MOs of H2S02, for each symmetry block (labels with respect to OXO and HSH planes, respectively). 

Figure 19. Relation between and for a bound-state system. The functions in the single space (a) can be mapped onto the double space (b) where they have opposite symmetries under R2n, and belong to different symmetry blocks of the double-space Hamiltonian matrix (c). Unhke reactive wave functions, bound-state functions cannot be unwound from around the Cl.

The largest matrix blocks obtained with this basis set are the E symmetry blocks at 7—18 they have the dimension N(E)—9150. 

To be precise, we should point out that we have symgenn treat N2 as a D4/, system, rather than the completely correct Daoh In projecting symmetry blocks out of Hamiltonian matrices, it is never wrong to use a subgroup of the full symmetry, merely inefficient. It would be a serious error, of course, to use too high a symmetry. It happens for the ST03G basis that there is no difference between Aig D41, and + Dooh projections. 

Each of these four 2x2 symmetry blocks generate identical Hamiltonian matrices. This will be demonstrated for the B3g symmetry, the others proceed analogously ... 

This 4x4 symmetry block factors to two 2x2 blocks where one 2x2 block includes the SALC-AOs... 

Direct evaluation of this four-fold sum would require 4didi dgdpjdgdftd c multiplications to form a symmetry block of R integrals. By constrast, sequential one-index transformations... 

The MCSCF optimization process is only the last step in the computational procedure that leads to the MCSCF wave function. Normally the calculation starts with the selection of an atomic orbital (AO) basis set, in which the molecular orbitals are expanded. The first computational step is then to calculate and save the one- and two-electron integrals. These integrals are commonly processed in different ways. Most MCSCF programs use a supermatrix (as defined in the closed shell HF operator) in order to simplify the evaluation of the energy and different matrix elements. The second step is then the construction of this super-matrix from the list of two-electron integrals. The MCSCF optimization procedure includes a step, where these AO integrals are transformed to MO basis. This transformation is most effectively performed with a symmetry blocked and ordered list of AO integrals. Step... 

The CASSCF method has been mentioned at several places in these lecture notes. Here we shall give a more detailed account of this method, which is probably the most widely used MCSCF method today. It is based on a partitioning of the occupied molecular orbitals into subsets, corresponding to how they are used to build the wave function. We define for each symmetry block of MO s the following subsets ... 

In the ground K L state of the above fluorides F 2p orbitals mix with the outer orbitals of the metals to form the valence band. An example is 18a, and 15a, for the NiF, A, symmetry block shown in Fig. 7(a). Here the... 

Besides the restrictions imposed on the orbital transformations to preserve spin symmetries, it is also useful to preserve spatial symmetry. This is done by allowing transformations only within sets of orbitals having the same symmetry properties and by not allowing these different sets of orbitals to mix. This restriction is accomplished by forcing the off-diagonal symmetry blocks of the K matrix, those labeled by spatial orbitals belonging to different symmetry types, to be zero. The notation required to label the symmetry species of the orbitals is somewhat cumbersome and will not be used except when explicitly required. 

The symmetry projection of the wavefunction is equivalent to a particular orbital transformation among the occupied orbitals of the wavefunction. If the CSF expansion is full within these sets of symmetry-related orbitals, no new CSFs will be generated by this orbital transformation. This type of wavefunction could have been computed directly in terms of symmetry orbitals with no loss of generality. (In fact, the CSF expansion expressed in terms of symmetry orbitals will usually result in fewer expansion terms because the symmetry blocking of the individual CSFs allows those of the incorrect symmetries to be deleted from the expansion.) However, if the CSF expansion is not full within these orbital sets, it is possible that the symmetry transformation of the orbitals will generate new CSF expansion terms. The coefficients of these new CSF expansion terms are determined by the old expansion coefficients and the symmetry transformation coefficients. For example, consider the case of two H2 molecules, described in terms of localized orbitals, separated by a reflection plane. Assume that the localized description of the two H2 molecules is of the form... 

The use of atomic symmetry block-diagonalizes the matrix representation of the DHFB Hamiltonian of (95) giving the generalized matrix eigenvalue equations... 

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