Symmetry-adapted basis

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

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... 

In addition one can always find a transformation leading to a symmetry adapted basis [4] e, so that T is brought to the block diagonal form T via the associated similarity transformation. The T matrix can be written as a direct sum... 

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 UHF formalism becomes inconvenient for open-shell configurations of atoms or molecules with point-group symmetry. Unless specific restrictions are imposed, the self-consistent occupied orbitals fall into sets that are nearly but not quite transformable into each other by operations of the symmetry group. By imposing equivalence and symmetry restrictions, these sets become symmetry-adapted basis states for irreducible representations of the symmetry group. This makes it possible to construct symmetry-adapted /V-clcctron functions, as described in Section 4.4. The constraints in general invalidate the theorems of Brillouin and Koopmans. This restricted theory (RHF) is described in detail for atoms by Hartree [163] and by Froese Fischer [130],... 

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. 

The coefficients determine the number of times a particular molecular multiplet 2s+ y1 occurs in this decomposition. This is just the dimension of the secular equation (23) which has to be solved in the symmetry-adapted basis. The cSA can be determined in the usual manner from character tables for the groups SAn and St. ... 

The projection operator is one of the most useful concepts in the application of group theory to chemical problems [25, 26], It is an operator which takes the non-symmetry-adapted basis of a representation and projects it along new directions in such a way that it belongs to a specific irreducible representation of the group. The projection operator is represented by P in the following form ... 

Because the Hamiltonian can always be diagonalized numerically, transfer functions can be conveniently calculated with the help of a computer. Analytical solutions can often be derived if the effective Hamiltonian is expressed in a symmetry-adapted basis (Banwell and Primas, 1963), where it often has a simple block structure that facilitates its diagonalization (Corio, 1966). 

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... 

---