Antisymmetry elements

Not only a symmetry plane but also other symmetry elements may serve as antisymmetry elements. We have already seen the contour of the oriental symbol Yin/Yang representing twofold rotational... 

Figure 4-12 illustrates different combinations of symmetry elements, for example, twofold, fourfold, and sixfold antirotation axes together with other symmetry elements after Shubnikov [15], The fourfold antirotation axis includes a twofold rotation axis, and the sixfold antirotation axis includes a threefold rotation axis. The antisymmetry elements have the same notation as the ordinary ones except that they are underlined. Antimirror rotation axes characterize the rosettes in the second row of Figure 4-12. The antirotation axes appear in combination with one or more symmetry planes perpendicular to the plane of the drawing in the third row of Figure 4-12. Finally, the ordinary rotation axes are combined with one or more antisymmetry planes in the two bottom rows of Figure 4-12. In fact, symmetry 1 m here is the symmetry illustrated in Figure 4-11. The black-and-white variation is the simplest case of color symmetry.

More general situations have also been considered. For example. Mead [21] considers cases involving degeneracy between two Kramers doublets involving four electronic components a), a ), P), and P ). Equations (4) and (5), coupled with antisymmetry under lead to the following identities between the various matrix elements... 

In the case of conrotatory mode, the symmetry is preserved with respeo to C2 axis of rotation. On 180° rotation along this axis, F goes to H. and H2 to H, and the new configuration is indistinguishable from the original. An orbital symmetric with respect to rotation is called a and antisymmetric as b. On the other hand, in the case of disrotatory moot-the elements of symmetry are described with respect to a mirror plane. Tilt symmetry and antisymmetry of an orbital with respect to a mirror plant of reflection is denoted by a and a" respectively (Section 2.9). The natun of each MO of cyclobutene with respect to these two operations is shov. n in the Table 8.4 for cyclobutene and butadiene. 

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

Another important physical interpretation of the molecular-orbital determinant follows from an application of a similar argument to the columns. The elements of two columns become identical if two electrons have the same spin (a or [>) and are at the same point (, y, z). The determinant then vanishes and consequently the probability of such a configuration is zero. Such an argument does not apply to electrons of different spin, however. The antisymmetry principle operates, therefore, in such a way that electrons of the same spin are kept apart. We shall see in later sections that this is an important factor in determining stereochemical valence properties. 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

The first factor under the summation sign takes care of the antisymmetry of the two-electron states of the shell. The submatrix elements are summed in accordance with the statistical weights of these states. Using (14.66), (6.25), (6.26) and (6.18), we sum the right side of (14.67) in the explicit form... 

Further, one must pay attention to Eq. (5.10) for /mm - In isotropic collisions, when we have Tmm> — ram=m-M i and in the absence of an external field, when we have jkjJmm1 — 0) linearly polarized excitation yields /mm i = -f-M i-M- And since /mm i and f-M i-M enter into the sum (5.41) with equal coefficients, the aforesaid implies the absence of transversal orientation f v More precisely, the contention concerning the antisymmetry of the respective density matrix elements /mm i = —f-M i-M follows from the explicit form of the cyclic components of the polarization vector (see Appendix A), and from the symmetry properties of the Clebsch-Gordan coefficients (see Appendix C). 

Qise A. Consider the HT coupling between two doubly-degenerate excited states, le> and s>, as for example between two distinct states in a molecule of D4h symmetry. Transitions to each state are allowed, from an. <4 xg ground state, with both X mdy polarisation. In addition to the zero-order transition moments x , y , the first-order borrowed moments x, y, must be considered. In order to demonstrate the origin of the antisymmetry of the tensor in this case ( xy — yx) the non-zero off-diagonal elements are written as follows ... 

Returning now to Eqs.(28), it can be seen that, because of the antisymmetry relations, the resonant and non-resonant parts of each tensor element tend to cancel in the off-resonance limit Eq Ej > El > hco. In addition, the contributions from le> and s> states to each term in Eqs.(28) are of comparable magnitude off resonance. These contributions tend to cancel because of the difference in sign of the HT perturbation energy denominators, viz. (Ee - E )" = - (Es - Ee) . The net result is the disappearance of antisymmetric scattering in the off-resonance region. 

A consequence of the antisymmetry property of the determinantal wave functions is that the matrix elements can be evaluated according to the Slater rules. Before the application of the Slater rules the determinantal functions should be ordered in such a way that maximum coincidence among the positions of the spinorbitals is secured. Notice that each transposition in the position of spinorbitals alters the sign of the determinantal function. 

Slater pointed out in 1929 that a determinant of the form (10.40) satisfies the antisymmetry requirement for a many-electron atom. A determinant like (10.40) is called a Slater determinant. All the elements in a given column of a Slater determinant involve the same spin-orbital, whereas elements in the same row all involve the same electron. (Since interchanging rows and columns does not affect the value of a determinant, we could write the Slater determinant in another, equivalent form.)... 

In some linear series (with segments of high intrinsic symmetry) the antisymmetry of the numeric code relative to some digits gives an indication that additional symmetry elements exist in the structure. 

The forerunner of Cl is the self-consistent field (SCF) method [1, 2]. A version that properly accounts for the antisymmetry of the electronic wave function was developed independently by Fock [3] and Slater [4] shortly after Schrodinger s papers. It is characterized by an approximate wave function that is a single determinant whose elements are one-electron functions (spin orbitals). The latter orbitals are optimized under two conditions minimization of the energy expectation value and mutual orthonormality. The method produces both the occupied orbitals appearing in the determinant but also a potentially infinite number of unoccupied functions that prove to be the basis for the Cl method. One can look upon a Slater determinant formed by substituting unoccupied for occupied one-electron functions as a representation of an excited state of the molecular system. The possible applications to spectroscopy were obvious. 

The antisymmetry of d (X) is a consequence of the orthonormality of the molecular orbitals, Eq. (43a). Here the adjective square has been emphasized in reference to the one-particle transition density matrix. The one particle transition density matrix is in general not symmetric, that is, the full or square matrix must be retained. However, in most electronic structure applications the associated one electron integrals, for example are symmetric, permitting the off-diagonal density matrix element to be stored in folded or triangular form. Since d is not symmetric, it is necessary to construct and store the transition density matrix in its unfolded or square form. 

Each of the MOs in Fig. 10-3 is symmetric or antisymmetric for some of the operations that apply to a tetrahedron. 02 is symmetric for rotations about the z axis by 27t/3, and also for reflection through the xz plane. This same reflection plane is a symmetry plane for 03 and 04, but neither of these MOs shows symmetry or antisymmetry for rotation about the z axis. Each MO contains one p AO and, perforce, has the symmetry of that AO. We shall refer to these as p-type MOs. Note that hydrogen Is AOs lying in the nodal plane of a p AO do not mix with that p AO in formation of MOs. This results from zero interaction elements in H, which, in turn, results from zero overlap elements in S. Note also that the MO 02 is the MO that we anticipated earlier on the basis of inspection of the matrix H. Because of phase agreements between the hydrogen Is AOs and the adjacent lobes of the p AOs, 0j, 02, and 03 are C-H bonding MOs. 

---