Spatial symmetries

Spatial symmetry plays a role in a large number of the examples in Part II of this book. This can arise in a number ways, but the two most important are simplification of the calculations and labeling of the energy states. We have devoted considerable time and space in Chapter 5 to the effects of identical particle symmetry and spin. In this chapter we look at some of the ways spatial symmetry interacts with anti-symmetrization. 

We first note that spatial symmetry operators and permutations commute when applied to the functions we are interested in. Consider a multiparticle function 0( 1, r2, r ), where each of the particle coordinates is a 3-vector. Applying a permutation to / gives 

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If the atom or moleeule has additional symmetries (e.g., full rotation symmetry for atoms, axial rotation symmetry for linear moleeules and point group symmetry for nonlinear polyatomies), the trial wavefunetions should also eonform to these spatial symmetries. This Chapter addresses those operators that eommute with H, Pij, S2, and Sz and among one another for atoms, linear, and non-linear moleeules. 

As illustrated above, the spatial symmetries of these four S = 1 and S = 0 states are obtained by forming the direet produet of the "open-shell" orbitals that appear in this eonfiguration b2 x bi = A2. 

All four states have this spatial symmetry. In summary, the above eonfiguration yields A2... 

The teehniques used earlier for linear moleeules extend easily to non-linear moleeules. One begins with those states that ean be straightforwardly identified as unique entries within the box diagram. For polyatomie moleeules with no degenerate representations, the spatial symmetry of eaeh box entry is identieal and is given as the direet produet of the open-shell orbitals. For the formaldehyde example eonsidered earlier, the spatial symmetries of the nji and nn states were A2 and Ai, respeetively. 

For the ease given above, one finds n(ai) =1, n(a2) = 1, and n(e) =1 so within the eonfiguration e there is one Ai wavefunetion, one A2 wavefunetion and a pair of E wavefunetions (where the symmetry labels now refer to the symmetries of the determinental wavefunetions). This analysis tells one how many different wavefunetions of various spatial symmetries are eontained in a eonfiguration in whieh degenerate orbitals are fraetionally oeeupied. Considerations of spin symmetry and the eonstruetion of proper determinental wavefunetions, as developed earlier in this Seetion, still need to be applied to eaeh spatial symmetry ease. 

To generate the proper A, A2, and E wavefunetions of singlet and triplet spin symmetry (thus far, it is not elear whieh spin ean arise for eaeh of the three above spatial symmetries however, only singlet and triplet spin funetions ean arise for this two-eleetron example), one ean apply the following (un-normalized) symmetry projeetion operators (see Appendix E where these projeetors are introdueed) to all determinental wavefunetions arising from the e eonfiguration ... 

The resultant family of six eleetronie states ean be deseribed in terms of the six eonfiguration state funetions (CSFs) that arise when one oeeupies the pair of bonding a and antibonding a moleeular orbitals with two eleetrons. The CSFs are eombinations of Slater determinants formed to generate proper spin- and spatial symmetry- funetions. 

The spin- and spatial- symmetry adapted N-eleetron funetions referred to as CSFs ean be formed from one or more Slater determinants. For example, to deseribe the singlet CSF eorresponding to the elosed-shell orbital oeeupaney, a single Slater determinant... 

It is essential to realize that the energies (i H Oi> of the CSFs do n represent the energies of the true electronic states Ek the CSFs are simply spin- and spatial-symmetry adapted antisymmetric functions that form a basis in terms of which to expand the true electronic states. For R-values at which the CSF energies are separated widely, the true Ek are rather well approximated by individual (i H Oi> values such is the case near Rg. 

The UHF option allows only the lowest state of a given multiplicity to be requested. Thus, for example, you could explore the lowest Triplet excited state of benzene with the UHF option, but could not ask for calculations on an excited singlet state. This is because the UHF option in HyperChem does not allow arbitrary orbital occupations (possibly leading to an excited single determinant of different spatial symmetry than the lowest determinant of the same multiplicity), nor does it perform a Configuration Interaction (Cl) calculation that allows a multitude of states to be described. 

It is particularly interesting to consider the influence of the substituents R and Rj in diphenylol alkanes of the type shown in Figure 20.12. Such variations will influence properties because they affect the flexibility of the molecule about the central C-atom, the spatial symmetry of the molecule and also the interchain attraction, the three principal factors determining the physical nature of a high polymer. 

In order to specify the proper electronic state, ozone calculations should be performed as unrestricted calculations, and the keyword Gue s=Mix should always be included. This keyword tells the program to mix the HOMO and LUMO within the wavefunction in an effort to destroy a-P and spatial symmetries, and it is often useful in producing a UHF wavefunction for a singlet system. Running a UHF GuesssMix Stable calculation confirms that the resulting wavefunction is stable, and it predicts the same energy (-224.34143 hartrees) as the previous Stable=Opt calculations. 

This is shown schematically in Figure 6.1. For such molecules, the only degeneracies that occur are accidental ones and all the i/ s have the same spatial symmetry (their irreducible representation is a). 

I am assuming that this particular electronic state is the lowest-energy one of that given spatial symmetry, and that the i/f s are orthonormal. The first assumption is a vital one, the second just makes the algebra a little easier. The aim of HF theory is to find the best form of the one-electron functions i/ a,. .., and to do this we minimize the variational energy... 

Sokolov, A. V., and Shirokovski, V. P., Group Theoretical Methods in the Quantum Physics of Solids (Spatial Symmetry), Soviet Physics—Uspekhi, 3, 651 (1961). 

Instead of using repeated solution of a suitable eigenvalue equation to optimize the orbitals, as in conventional forms of SCF theory, we have found it more convenient to optimize by a gradient method based on direct evaluation of the ener functional (4), ortho normalization being restored after every parameter variation. Although many iterations are required, the energy evaluation is extremely rapid, the process is very stable, and any constraints on the parameters (e.g. due to spatial symmetry or choice of some type of localization) are very easily imposed. It is also a simple matter to optimize with respect to non-linear parameters such as orbital exponents. 

A non-abelian point-group contains irreducible representations of dimension larger than one. Since the degree of degeneracy caused by spatial symmetry equals the dimensionality of the corresponding irreducible... 

The basis of the expansion, ifra, are configuration state functions (CSF), which are linear combinations of Slater determinants that are eigenfunctions of the spin operator and have the correct spatial symmetry and total spin of the electronic state under investigation [42],... 

For odd electron systems in the absence of spatial symmetry H Eq. (12) becomes... 

As a particular example of materials with high spatial symmetry, we consider first an isotropic chiral bulk medium. Such a medium is, for example, an isotropic solution of enantiomerically pure molecules. Such material has arbitrary rotations in three dimensions as symmetry operations. Under rotations, the electric and magnetic quantities transform similarly. As a consequence, the nonvanishing components of y(2),eee, y 2)-een and y,2)jnee are the same. Due to the isotropy of the medium, each tensor has only one independent component of the xyz type ... 

This argument is not restricted to spatial symmetry and in fact the most familiar example of the phenomenon is the Different Orbitals for Different Spins (DODS) technique for open electronic shells where the total spin function S2 takes the role of our G (in the one-electron-group model). 

In our discussion so far we have used a group of spatial symmetries as an easily-visualised example. It is possible to put all the symmetries of the molecular Hamil-... 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

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