Symmetry Adapted Linear Combinations SALCs of the Determinants

Symmetry adapted linear combinations (SALCs) of the determinants 

The VB Hamiltonian matrix can split into N+l blocks characterized by 

After the factorization, one can derive SALCs within each invariant subspace via the action of the projection operator. The projection operator for irreducible representation j is defined as 

For medium-sized molecules such as hexacene of 26 r-electrons, n(0) is equal to 10,400,600. By means of symmetry, the ground state is solved from the block of Ag under point group D2hwith a dimension of 2,600,612. Thus the Hamiltonian matrix can be considerably reduced for those molecules with higher symmetry. For hexacene and two other benzenoids, the dimensions of onedimensional irreducible representations in subspaces of M=0, 1, and 2 are listed in Table 2. 

In addition, the spin-inversion symmetry is available for M=0. The operator 6 is defined when it acts on a Slater determinant, transforming all spin-up sites to spin-down sites and vice versa. Thus the space of M=0 is invariant under the group [7,0] which consists of identity and spin-inversion operations. Let us discuss the 20 determinants of M=0 for trimethylene-cyclopropane, the spin-inversion operator transforms A to B, C to D, and E to F, or vice versa, respectively. Under the compounded groups [/,d], a pair of Slater determinants that are transformable via the spin-inversion operation should combine, as represented by A B, C D and E F respectively. Thus blocks A and A2 are further reduced to two 3x3 and two lxl blocks 

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