Adaptation spatial symmetry

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. 

Strength of positivity conditions Spin and spatial symmetry adaptation 1. Spin adaptation and S-representabiUty Open-shell molecules... 

Similar to spin adaptation each 2-RDM spin block may further be divided upon considering the spatial symmetry of the basis functions. Here we assume that the 2-RDM has already been spin-adapted and consider only the spatial symmetry of the basis function for the 2-RDM. Denoting the irreducible representation of orbital i as T, the 2-RDM matrix elements are given by... 

To illustrate the advantage of spin- and spatial-symmetry adaptation, consider the BH molecule in a minimal basis set. If only is considered, the largest block of the two-electron RDM (i.e., is of dimension 36. Spin adaptation divides into two blocks, with sizes 15 and 21,... 

We may disregard the closed-shell cores of the atoms since these play no role in the construction of symmetry-adapted wavefunctions, and concentrate attention upon the valence electrons. In the simplest case, with one valence electron per atom, we have a configuration 0102 n of N singly-occupied, non-degenerate valence orbitals which is then said to form a covalent structure for the molecule. Then under any spatial symmetry operation (%, a VB function Vsu-.k transforms as... 

The usually well-localised nature of the orbitals appearing in VB wavefunction makes spatial symmetry more difficult to use than in the MO case. In MO theory, symmetry can be introduced and utilised at the orbital level Each delocalised MO can be constructed as a symmetry-adapted linear combination (SALQ of basis functions, which is straightforward to implement in program code and can be exploited to achieve substantial computational savings. As a rule, the individual localised orbitals from VB wavefunctions are not S5mimetry-adapted, but transform into one another under the symmetry operations of the molecular point group. The use of symmetry of this type normally requires prior knowledge of the orbital shapes and positions and is very difficult to handle without human intervention. 

Consequently two normalised, symmetry-adapted spatial functions result... 

One can take advantage of any spin or spatial symmetry in the Hamiltonian by symmetry adapting the metric matrices and thereby reducing the size of the 2-RDM to be optimized [16]. For the ladder model, we transform the RDMs to bonding and antibonding spaces and then Fourier transform to take advantage of the translational symmetry. We consider linear combination of creation and annihilation operators to form two disjoint one-electron subspaces... 

From this picture it is obvious that the nuclear framework has to relax in order to adopt the same spatial symmetry as the density matrix of the CDW state. The subsequent optimization with different bond lengths and R2 leads to the bond-alternating structure (Peierls distortion), which is singlet-stable and properly symmetry-adapted. We also observe that the major part of the energetic stabilization of the bond-alternating state (AE 7 kcal/mol) originates from the relaxation of the nuclear framework, since the CDW state lowers the energy only by approximately 2 kcal/mol. 

The second approach is EOM-CC theory, described above. The dimension of the problem is the same as the analogous Cl problem, but the use of the similarity transformed Hamiltonian rather than the bare Hamiltonian yields better numerical results. The penalty is that the matrix elements are more complicated and the matrix is not symmetric, but generalizations of the Davidson scheme for matrix diagonalization are available. If the ground state is a closed-shell singlet, EOM-CC energies may be obtained for singlet and triplet excited slates, and these are spin-adapted and have proper spatial symmetry. 

Within the Bom-Oppenheimer approximation, the exact stationary states form a basis for an irreducible representation of the molecular point group. We may enforce the same spatial symmetry on the approximate state by expanding the wave function in determinants constmcted from a set of symmetry-adapted orbitals. For atoms, in particular, the use of point-group... 

Z (-1)) CSFs are, by no means, the true eleetronie eigenstates of the system they are simply spin and spatial angular momentum adapted antisymmetrie spin-orbital produets. In prineiple, the set of CSFs i of the same symmetry must be eombined to form the proper eleetronie eigenstates Fk of the system ... 

The GUGA-Cl wavefunctions are spatial and spin symmetry-adapted, thus the projections of total orbital angular momentum and total spin of a hydrogen molecule in a particular electronic state are conserved for all the values of R. Therefore, the term remains constant for an electronic state, and it causes a... 

The use of the conventional spin formulation in conjunction with a spin-free Hamiltonian HSF merely assures symmetry adaptation to a given spin-free permutational symmetry [Asp] without recourse to group theory. In fact, one may symmetry adapt to a given spin-free permutational symmetry without recourse to spin. This is the motivation behind the Spin-Free Quantum Chemistry series.107-116 In this spin-free formulation one uses a spatial electronic ket which is symmetry adapted to a given spin-free permutational symmetry by the application of an appropriate projector. The Pauli-allowed partitions are given by eq. (2-12) and the correspondence with spin by eqs. (2-14) and (2-15). Finally, since in this formulation [Asp] is the only type of permutational symmetry involved, we suppress the superscript SF on [Asp],... 

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