Gel’fand states

In this section the simplicity of freeon dynamics is illustrated by its application to polyenes [6]. Here I relate molecular-orbital Gel fand states to atomic-orbital Gel fand states and then relate the atomic-orbital Gel fand states to valence bond states. Note that this construction provides a theoretical basis for the Rumer rule i.e., for the number of linearly independent valence bond states. We then use this freeon dynamics to explicate the spin paradigm. 

The procedure for the construction of the freeon Gel fand states of polyenes consists of the following simple steps ... 

We have outlined above the procedure for the construction of orthonormal molecular-orbital and atomic-orbital Gel fand states and for the conversion of the latter to the non-orthogonal valence bond states. We require, in addition, a freeon Hamiltonian to compute the spectra of the several polyene systems. For this we employ the freeon, reduced Hiickel-Hubbard Hamiltonian which has the following form ... 

Freeon Molecular-Orbital Freeon Gel fand States... 

A covalent bond between two atoms is defined as one which is invariant under the exchange of the two orbitals. We can see this most simply by expressing the singlet and triplet Gel fand states in terms of atomic orbital products ... 

Gel fand construction a triplet Gel fand state, 5. cannot be constructed from two identical freeon atomic orbitals. 

The valence bond structures of the doublet allyl radical are more complicated. The transformation properties under orbital permutation of the doublet, covalent Gel fand states are shown below ... 

Like the allyl radical only the first Gel fand state, II , is a pure valence state... 

The spin-free Gel fand states span the several irreducible representation spaces. They are uniquely labeled by Gel fand diagrams constructed by adding spin-free orbitals in nondescending order along rows and ascending order down... 

One of the early great triumphs of atomic theory was the aufbau principle which explained the periodic table. In it the atomic orbital energies were assigned the following order Is < 2s < 2p < 3s <. ..The occupation number of freeon orbitals was limited to no more than two as was discussed in Section 1. Further Hund s rule was imposed i. e. states of highest spin lie the lowest. The Gel fand state reconstruction of the aufbau for the second row of the periodic table (with mi = +1 or zero) is shown in Fig. 10.1 where 2p+ = + 1,2po = 0 and 2p.i = -1 ... 

The superfreeon Is orbital can carry up to 4 nucleons. The lsA isofreeon Gel fand state are listed Fig. 11.1 together with the isospin, the ordspin and the nuclear species which are associated with each Gel fand state. 

We use three-particle Young diagrams for the construction of flavor Gel fand states ... 

The flavor Gel fand states for the octet and the decuplet spaces are shown as follows. 

Use of unitary group algebra to choose the model space CSFs as Gel fand states and compute the various matrix elements of the generators between them. 

The matrix element of the Hamiltonian, H, taken between two Gel fand states, and 0 can be written as... 

It is clear that the matrix elements of the one- and two-body generators between Gel fand states and generate one- and two-particle transition density matrix elements respectively yf " and F with appropriate indices ... 

The Ansatz we have chosen for our unitary group-based MRCC methods [45-48] is designed to closely mimic the Jeziorski-Monkhorst (JM) Ansatz [51, 83] in order to follow quite closely the developments in the analogous non-spin-adapted theories. As mentioned, we choose a set of Gel fand states, 0, to denote the model functions. We next introduce our spin-free JM-inspired Ansatz in Eq. (10) for the wave operator acting on 0 s. Our choice differs in two aspects from the traditional spinorbital-based JM... 

Unlike in the spinorbital-based formulation, the UGA-based MRCC theories use spin-free Gel fand states as the model functions and generate projection equations leading from each model function 4> to virtual functions... 

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