Orbital carrier spaces

The given formulae contain all the necessary results, but cannot be easily qualitatively interpreted. The necessary interpretation has been done by Levin and Dyachkov and is based on clarifying the interplay of the effects produced by substitution and vibronic operators upon the solution of the Hiickel-like problem in the 10-dimensional orbital carrier space using symmetry considerations. This will be done in the next section. 

The notation concerns are easily overcome by the following simple construct bearing the name of second quantization formalism.21 Let us consider the space of wave functions of all possible numbers of electrons and complement it by a wave function of no electrons and call the latter the vacuum state vac). This is obviously the direct sum of subspaces each corresponding to a specific number of electrons. It is called the Fock space. The Slater determinants eq. (1.137) entering the expansion eq. (1.138) of the exact wave function are uniquely characterized by subsets of spin-orbitals K = k,, k2,..., fc/v which are occupied (filled) in each given Slater determinant. The states in the list are the vectors in the carrier space of spin-orbitals (linear combinations of the functions of the (pk (x) = ma (r, s) basis. We can think about the linear combinations of all Slater determinants, may be of different numbers of electrons, as elements of the Fock space spanned by the basis states including the vacuum one. 

The set <5) = (pi) i=i,n forms a carrier space which is in one to one correspondence with the elements of the orbit Q, H c G). An orthogonal basis set for ) may then always be defined by forming the h — 1 traceless combinations of these n components. As an example in the case of a tetrahedron an arbitrary function space, transforming as T2, will have exactly one component which is totally symmetric under a Csv subgroup, and which we will label as a)- Four such components can be formed, one for each trigonal site. The T2 basis may then be expressed (up to... 

At present we have found that for the degenerate point group irreps which are listed in the table the basis functions can be expressed by means of a carrier space which exactly matches the orbit of a maximal subgroup of the point group, and counts G / H = n elements. The one-particle Hamiltonian operating in this carrier space can easily be constructed as follows ... 

So far the analysis has lead to the concept of a carrier space which links the degeneracy to a doubly transitive orbit of cosets of maximal subgroups. Interactions in this space are expressed as transition operators between the cosets. The final part of the treatment should bring in the vibrational degrees of freedom which are responsible for the Jahn-Teller activity. 

In this subsection, we will briefly discuss how one may construct a basis

carrier space which is adapted not only to the treatment of the ground state of the Hamiltonian H but also to the study of the lowest excited states. In molecular and solid-state theory, it is often natural and convenient to start out from a set of n linearly independent wave functions = < > which are built up from atomic functions (spin orbitals, geminals, etc.) involved and which are hence usually of a nonorthogonal nature due to the overlap of the atomic elements. From this set O, one may then construct an orthonormal set tp = d>A by means of successive, symmetric, or canonical orthonormalization.27 For instance, using the symmetric procedure, one obtains... 

Besides the apparent similarities. Table 8.1 illustrates also the obvious formal differences between bras and kets and their second quantized counterparts. Namely, the corresponding symbols are mathematically very different. The bra and ket vectors are elements of a linear vector space over which quantum-mechanical operators are defined, while the creation and annihilation operators are defined over the abstract space of particle number represented wave functions serving as their carrier space. This carrier space leads to the concept of the vacuum state, which has no analog in the bra-ket formalism. Moreover, an essential difference is that the effect of second quantized operators depends on the occupancies of the one-electron levels in the wave function, since no annihilation is possible from an empty level and no electron can be created on an occupied spinorbital. At the same time, the occupancies of orbitals play no role in evaluating bra and ket expressions. Of course, both formalisms yield identical results after calculating the values of matrix elements. 

As another example, the Hilbert space L (R3) is the carrier space for the direct product group 0(3, R) X SU(2) where the unitary operators U(C0,i°) = U co )U i°) with (T= 0,1 that are defined by Equations [7] and [8] represent the orbital part together with the spin part a unitary operator representation of 0(3, R) X SU(2) on the Hilbert space in question. The spin part reads ... 

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In other words, the entire space originally spanned by the orbitals ipm is represented as the direct sum of orthogonal subspaces serving as the carrier subspaces to different groups, but the numbers of residing electrons are not fixed for each carrier subspace and all their possible distributions enter the expansion. In the expansion eq. (1.214) each distribution a of electrons among the groups satisfies the condition ... 

The charge-carrier mobility is based on the overlap of the n orbitals of the DCNQI molecules in the stacks. The high conductivity is due as in every conductor to the partial filling of the conduction band. Conduction along the Cu chains is not possible, because the Cu ions have closed electronic shells and because their spacing is too large. In the Cu salts of DCNQI, the Cu-Cu distance is about 50% greater than in Cu metal. 

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