Abelian point group

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... 

In the case of a symmetry-broken solution, these weights can not only diagnose the problem, but also quantify its extent. As for the projection operator considered in Section 3.2, this analysis is used primarily with Abelian point groups. 

M. Kollwitz, M. Haser, and J. Gauss,/. Chem. Phys., 108, 8295 (1998). Non-Abelian Point Group Symmetry in Direct Second-Order Many-Body Perturbation Theory Calculations of NMR Chemical Shifts. 

Efficient use of symmetry can greatly speed up localized-orbital density-functional-exchange-and-correlation calculations. The local potential of density functional theory makes this process simpler than it is in Hartree-Fock-based methods. The greatest efficiency can be achieved by using non-Abelian point-group symmetry. Such groups have multidimensional irreducible representations. Only one member of each such representation need be used in the calculation. However efficient localized-orbital evaluation of the chosen matrix element requires the sum of the magnitude squared of the components of all the members on one of the symmetry inequivalent atoms, based on Eq. 13. 

Basic rule Standard basis functions for irreducible representations that occur in antisymmetrized (skew) direct products are chosen out of the spherical harmonics with I odd and the remaining ones out of those with I even. This rule is immediately applicable to 2 3 and without difficulty to all non-abelian point groups except the icosahedral group which, however, is not a simply reducible group" ). 

If we restrict attention to Abelian point groups for simplicity, then the operations of the point groups commute and, therefore, so do their matrix representations. Two (or more) matrices which commute... 

I. Shavitt, The Utilization of Abelian Point Group Symmetry in the Graphical Unitary Group Approach to the Calculation of Correlated Electronic Wave-functions, Chem. Phys. Lett. 63, 421-427 (1979). 

For Abelian point groups, the use of symmetry is particularly simple. Under each operation R in the group, each basis function transforms entirely into another, apart from a possible change of sign, i.e. 

If A and A differ, the coupling mode g is non-totally symmetric (coined g ). It can thus not coincide with (first-order) Franck-Condon active modes which are characterized by the decomposition of A <8> A or A <8 A and are totally symmetric (in Abelian point groups). The latter are also called tuning modes in the LVC scheme which forms the body of our early work in the field. The conical intersection is usually termed symmetry-allowed in such a case since it normally occurs (for a single coupling mode) in the subspace g = 0 where A 7 A and a free crossing is possible. 

The CCSD approach has been in recent years efficiently implemented in several program systems (ACES II, CADPAC, DALTON, GAUSSIAN, MOLCAS, MOLPRO, PSI, TITAN). In particular, vectorization, exploitation of Abelian point-group symmetry, and (partial) AO-based algorithms have been used to accomplish this task. Calculations for molecules with up to ten nonhydrogen atoms... 

All these methods benefit from the pseudospectral treatment of the two-electron integrals and exhibit significant time savings over conventional implementations. In addition, Jaguar also uses full point group symmetry to speed up calculations, including both Abelian and non-Abelian point groups. 

The order in which the representations are presented in this table is not significant, except that it is customary to list the totally symmetric representation first. The totally symmetric representation is the one for which all the eigenvalues are 1. Every point group has a totally symmetric representation, including non-Abelian point groups. 

Key information about molecular symmetry is often given in tables of symmetry representations. Before considering these, we need to recognize differences in non-Abelian point groups. The mathematical analysis for non-Abelian point groups is somewhat more... 

Some operators may commute, but there is at least one pair of operators fhat do not commute in a non-Abelian point group. 

Non-Abelian point groups give rise to the possibility of degeneracy in quantum mechanical systems. 

With Abelian point groups, we find the representation of the product of two things by taking the products of their characters and determining the representation that corresponds. For example, in the 2 symmetry of the water molecule, the product of something that transforms as with something that transforms as B2 is of the A2 representation. 

We used the DIRAC program suite. Time-reversal symmetry [30] and Abelian point groups, including C2v, are fully exploited in the DIRAC program with the help of quaternion algebra. [31] We briefly summarize quaternion algebra for the case of C2v... 

While this labelling is a technical issue, it appears sufficiently important to be briefly outlined. The necessity for this lies in the fact that especially for non-Abelian point groups, existing Cl implementations are not always able to assign symmetry labels to the states obtained in a determinant-based calculation, such as CASSCF or MRCI. In particular, the non-Abelian GUGA approach does not appear to be widely used. Consider a wavefunction... 

For more general orbital-rotation operators, see Section 3.3.3. If we also wish to conserve the spatial symmetry of the CSF, we must retain in the orbital-rotation operator (10.1.9) only those excitation operators that transform as the totally symmetric irreducible representation of the molecular point group. For Abelian point groups, this is accomplished by summing over only those pairs pq where p and q transform as the same irreducible representation. 

In Abelian point groups, the requirement that is totally symmetric is easily satisiied by summing only pairs of MOs p and q that belong to the same irreducible representation of the point group. 

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