Symmetry-adapted basis functions

There are 60 symmetry adapted basis functions of A symmetry. 

The phrase symmetry adapted basis functions refers to those linear combinations of basis functions (on several atoms) that transform like the particular irreducible representation of the appropriate point group. Molecular symmetry is used at various points in these calculations twenty years ago I would have had to write several chapters on molecular symmetry, point groups, constructing symmetry-adapted combinations of basis functions, factoring a Hamiltonian matrix using symmetry and related topics. The point is that twenty... 

They will both produce identical energy levels and identical values for the coefficients of the symmetry-adapted basis functions in the final MOs. Again, it is necessary to solve eqn (12-2.1) for only one of the blocks. Overall we will have three doubly-degenerate energy levels (6,-type). 

We can go beyond the classification of atomic basis functions of the last subsection to obtain explicit symmetry-adapted basis functions by projection. We can use either the full matrix projection and shift operators (Eqs. 1.28 and 1.31, which sure repeated here for convenience) ... 

In either case it is helpful to have a table of transformation properties of spherical harmonics like that given in the Supplementary Notes. We shall illustrate the procedure by finding symmetry-adapted basis functions arising from an s function on each F atom in BF3, using full matrix projection operators. We already know that the three atomic basis functions transform as a 0 s, and since the behaviour with respect to the horizontal plane is already known, we can, without loss of generality, work with the subgroup C3u only. We denote the s functions on Fi, F2, and F3 as si, S2, and s3, respectively, and apply our C3u projection operators... 

We can also consider using the character projector of Eq. 3.9 to obtain our symmetry-adapted basis functions. There is no need to consider the Ai case again since the irrep is one-dimensional. For VE we obtain... 

There are 44 symmetry- adapted basis functions of Al symmetry... 

The Hamiltonian (58) commutes with the operator that displaces all the spins by one unit cell cyclically. Therefore, the eigenfunctions of (58) must be characterized by hole quasi-impulse k = 2nmlN (m=l,2...N). The symmetry adapted basis functions corresponding to a fixed value of k can be constructed by the usual group theory technique. 

Symmetry considerations play a role on several levels in the analysis of Hartmann-Hahn experiments. In the presence of rotational symmetry and permutation symmetry, the effective Hamiltonian often can be simplified by using symmetry-adapted basis functions (Banwell and Primas, 1963 Corio, 1966). For example, any zero-quantum mixing Hamiltonian can be block-diagonalized in a set of basis functions that have well defined magnetic quantum numbers. Block-diagonalization of the effective Hamiltonian simplifies the analysis of Hartmann-Hahn experiments (Muller and... 

We circumvent this problem in 2fa-like methods by lumping the non-linear terms into the local V and determining its symmetry. Now we can use an expression like Eq. (20) to simplify matrix elements over atom-centered basis functions when there are many symmetry-related atoms. For a given symmetry-adapted basis function centered on atom type C we expand the basis function into terms centered on the N symmetry-equivalent atoms,... 

Avery JS, Rettrup S, Avery JE (2011) Symmetry-adapted basis functions automatic generation for problems in physics and chemistry. World Scientific (in press)... 

If these are symmetry-adapted basis functions, then the local potential must transform according to the invariant representation. Thus for use with localized basis sets it must be expressible as a superposition of identical single-center terms,... 

---