Form factors, symmetric/antisymmetric

Here, the space-spin function is a one-term product of a spatial part and a spin factor. The overall function is antisymmetric, the spin factor is antisymmetric with respect to exchange of spin coordinates thus the spatial part must be symmetric with respect to exchange of spatial coordinates. Having used this formalism to ensure that the total wavefunction is antisymmetric, we are finished with spin completely. This result is clearly independent of the specific form of the HL trial function and ... 

We know from Section 6.7 that the true wave function i/q,. . . , t/r4 are linear combinations of the basis functions. If we begin with symmetrized functions, such as and then each of the ip s can be formed exclusively from symmetric functions or exclusively from antisymmetric functions. Stated another way, functions of different symmetry do not mix. The result is that, like the situation with Fz, many off-diagonal elements of the secular equation must be zero, and the equation factors into several equations of lower order. We shall study an example of this factoring in Section 6.13, when we consider the A2B system. 

For E terms, the superscript denotes the sign change of the wavefimction under a reflection in a plane containing the internuclear axis. This is equivalent to a sign change in the variable

needed when we deal with spectroscopic selection rules. In a spin-paired Tr -subshell, the triplet spin function is symmetric so that the orbital factor must be antisymmetric, of the form... 

We have adopted the convention that the position vector be written in component form as r = XiCj. Now we recognize that upon factoring out Xj, three of these terms are a restatement of linear momentum balance, and are thus zero. In addition, the term involving vj also clearly yields zero since it is itself symmetric and is contracted with the antisymmetric Levi-Cevita symbol, The net result is that we are left with = 0 which immediately implies that the stress tensor itself... 

When fhe lasf fwo indices are expanded from lower triangles to squares, symmetric and antisymmetric matrices are formed. The superscripts S and A denote these cases. Diagonal elements for the symmetric case have a different normalization factor. The rank of the spin-adapted H matrix for the closed-shell case is o- -v- -vo - -ov. ... 

The covariance groups underlying the tensor forms of the respective Einstein and the Maxwell held equations are reducible. This is because they entail reflection symmetry, not required by relativity theory, as well as the required continuous symmetry of the Einstein group E. When the Einstein held equations are factorized, they yield the irreducible form, which are then in terms of the quaternion and spinor variables, rather than the tensor variables. Such a generalization must then extend the physical predictions of the usual tensor forms of general relativity of gravitation and the standard vector representation of the Maxwell theory (both in terms of second-rank tensor helds, one symmetric and the other antisymmetric) because the new factorized variables have more degrees of freedom than did the earlier version variables. 

The factor (1/V ) ensures that the wavefunction is normalised. Of the three acceptable spatial forms that we have described so far, two are symmetric (i.e. do not change sign when the electron labels are exchanged) and one is antisymmetric (the sign changes when the electrons are exchanged) ... 

Consider the forms of the wave functions for the terms. We shall call the two subshells tt and tt and shall use a subscript to indicate the m value. For the A terms, both electrons have m = +1 or both have m = -1. For Ml = +2, we might write as the spatial factor in the wave function - r+i(l)- ri,(2) or ir+i(2)irii(l). However, these functions are neither symmetric nor antisymmetric with respect to exchange of the indistinguishable electrons and are unacceptable. Instead, we must take the linear combinations (we shall not bother with normalization constants)... 

The minus sign for the exchange term comes from the factor of (-1X in the antisymmetrizing operator, eq. (3.21). The energy may also be written in a more symmetrical form as in eq. (3.32). 

The shifts in the C—H stretching frequency noted above at the SCF level are not affected appreciably when correlation is included. Dimerization increases the intensity of the C—H stretch of the proton-donating molecule by a factor of 4 or 5. In the trimer, the C—H stretches of both proton-donating molecules combine together. The intensity of the lower frequency mode (the symmetric combination) is enhanced by a factor of 10 with respect to the monomer while that of the antisymmetric combination vanishes almost completely. The C—H stretch of the terminal molecule which does not act as proton donor has the highest frequency its intensity is not much affected by complexation. The intensity of the H N stretch in the dimer is calculated to be quite low, 1.3 km/mol at the SCF level and 2.8 km/mol when correlation is included. The two H N stretches in the trimer form symmetric and antisymmetric combinations. The former has zero intensity and the other is quite weak as well with an intensity of 2 km/mol at either level. 

In the development of the Slater method (Section 3.1) it was noted that the Pauli principle in the form (1.2.27) could always be satisfied by constructing the electronic wavefunction from determinants (i.e. antisymmetrized products) of spin-orbitals. In an earlier section, however, it was shown that for a two-electron system the antisymmetry principle could also be satisfied by writing the wavefunction as a product of individually symmetric or antisymmetric factors—one for spatial variables and the other for spin variables. Since, in the usual first approximation the Hamiltonian does not contain spin variables, it is natural to enquire whether a corresponding exact N-electron wavefunction might be written as a space-spin product in which the spatial factor is an exact eigenfunction of the spinless Hamiltonian (1.2.1). To investigate this possibility, we need a few basic ideas from group theory (Appendix 3). 

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