Generating Symmetry Orbitals

It might seem as though three orbitals having E symmetry were generated, but only two... 

Symmetry enters the approximate solution of the electronic Schrd-dinger equation in two ways. In the first place, the exact MOs are eigenfunctions of an operator which commutes with all Om of the point group concerned, they therefore generate irreducible representations of that point group (see Chapter 8) and can be classified accordingly. The same is true for the approximate MOs and consequently one constructs them from combinations of atomic orbitals (symmetry orbitals) which generate irreducible representations. 

The symmetry group to which an A(B, C,. . . ) molecule belongs is determined by the arrangement of the pendent atoms. The A atom, being unique, must lie on all planes and axes of symmetry. The orbitals that atom A uses in forming the A—(B, C,. . . ) bonds must therefore be discussed and classified in terms of the set of symmetry operations generated by these axes and planes—that is, in terms of the overall symmetry of the molecule. Thus, our first order of business is to examine the wave functions for AOs and consider their transformation (symmetry) properties under the various operations which constitute the point group of the A(B, C,. . . ) molecule. 

Summary. We consider the Josephson effect in a ballistic Superconductor/ Quantum Wire/ Superconductor junction. It is shown that the interplay of chiral symmetry breaking generated by Rashba spin-orbit interaction and Zeeman splitting results in the appearance of a Josephson current even in the absence of any phase difference between the superconductors. 

Figure 6-22. Generation of the A symmetry orbital of the 3H group orbitals of...

Since there is only one B2u symmetry orbital among the SALCs, the one in [Pg.283]

On the lowest level, using symmetry and nodal properties of the MOs of H2C=0 as a representative of the inherently symmetric carbonyl chromophore, an octant rule can be derived for the influence of a (static) perturber of the (n, n ) excitation The geometrical symmetry (C2v) of the H2C=0 unit with the two symmetry planes, yz and xz, leads to a quadrant rule. If the orbitals involved in the (n, n ) excitation of ketones are described as " = 2p and n = N 2p — 2pJ) (N being a normalization constant), the nodal plane (xy) of the virtual orbital generates the octant diagram for the contributions of substituents of chirally perturbed compounds (Figure 10). 

The complete set of local d-orbitals on a set of vertices generates a, tt and (5 characters under actions of the symmetry operations of the point group. The set of local f-orbitals generates a, n and 8 and characters, and so on. 

When the basis set was enlarged, almost linearly dependent basis functions appeared. They were akin to some orbitals generated on different atoms in the moleeule beeause the linear eombination of many basis functions at multi-centers spanned the space repeatedly. The linear dependence among the symmetry-adapted orbitals was removed before the orthogonalization. 

Fig. 6. Illustrative use of generator orbitals (GO s) in obtaining symmetry orbitals (SO s) and molecular orbitals for N2.

Fig. 11. Hydrogen symmetry orbitals for BH3 generated by two different axis systems ((a)-(c) and (d) (e)).

Fig. 20. Generation of the ten symmetry orbitals for ferrocene using generator atomic orbitals. Full and dotted lines in the C5H5 1 MO s and in the GO S represent opposite signs in the wave functions.

Fig. 25.. Sigma-type symmetry orbitals for the CoF6 3 ion obtained by means of generator orbitals.

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 symmetry problem is trivially solved in the local-scaling version of density functional theory because we can include the symmetry conditions in our choice of orbit-generating or initial wavefunction W [33]. Since it is from this initial wave-function that we obtain and xc,gy namely, the non-local quantities appearing in the energy functional of Eq. (50), it follows that the symmetry properties of the parent wavefunction are transferred to the variational functional. Notice, therefore, that symmetry is not as important for the density as it is for the Fermi and Coulomb holes, which are related, to and jtyc,gy respectively. 

The generalization is obvious. We partition our orbital space into three sub-spaces inactive, active, and external. The inactive orbitals are doubly occupied in all CSFs (like the ls-3p orbitals in Cr2). The choice of the active orbitals must be based on our knowledge about the system and the chemical process we are studying. This is not a trivial task and a CASSCF program can never be treated as a black box. Once we have chosen the active orbitals, we distribute the active electrons among the active orbitals in all ways possible for a given total spin and space symmetry. This generates a set of CSFs in which we expand the wave function ... 

In order to find out how large the effect is, we need to perform a calculation. We shall make it a little bit more extensive just for the fun of it. We choose as active orbitals all nine orbitals generated from the oxygen 2p orbitals with 12 active electrons. The calculations are run in 2 symmetry where the orbital labels are oj and / 2 for the CT-orbitals. The active space is then 3fli, 3/ 2, 2bi, and lfl2 orbital, which we write as (3321). We use a basis set of the Atomic Natural Orbital (ANO) type with the contraction... 

The parity (see Box 1.9) of a rr-orbital is m, and that of a it -orbital is g. These labels are the reverse of those for a and a -orbitals, respectively (Figure 1.20). The overlap between two Py atomic orbitals generates an MO which has the same symmetry properties as that derived from the combination of the two p,. atomic orbitals, but the ttuipy) MO lies in a plane perpendicular to that of the TtuiPx) MO. The tr ipx) and TTuipy) MOs lie at the same energy they are degenerate. The Ttg py) and Ttg (j>x) MOs are similarly related. 

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