Carrier space

The linear operators on a vector space forms themselves a vector space, called operator space. In this context, the original vector space is called the carrier space for the operators. The operator space is sometimes normed, but usually not. Since operator products are defined, we have here a vector space where a product of vectors to give a vector is defined. Such a vector space is also called a linear algebra. Operations and functions can be defined in the operator space thus we can define superoperators for which the operator space is the carrier space. The hierarchy is not usually driven any further. Functions are usually named in analogy to their analytical counterparts. To be specific, assume that A has a spectral resolution... 

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. 

It leaves intact the fermion operators related to the /1-th group itself. By virtue of this the two-electron operators WBA result in a renormalization of one-electron terms in the Hamiltonians for each group. <4 = 1,..., M. The expectation values ((b+b ))B are the one-electron densities. The Schrodinger equation eq. (1.193) can be driven close to the standard HFR form. This can be done if one defines generalized Coulomb and exchange operators for group A by their matrix elements in the carrier space of group A ... 

The geminals defined in the carrier space spanned by HOs constructed by the above formulae are termed to be strictly local geminals (SLG). 

As mentioned previously, the specifics of the central atoms in CCs are determined by the structure of the supermatrix II, which is in its turn predefined by the structure of the carrier space of the CLS group and by the number of electrons in it. Indeed, the supermatrix II of the polarization propagator is particularly simple in the basis of the eigenstates of the Fock operator Fq. Its matrix elements then are ... 

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. 

A remarkable feature is that the derivative of the one-electron part of the Fock operator with respect to the symmetry adapted nuclear shift Sqr < (an operator acting on the one-electron states in the CLS carrier space) itself transforms according to the irreducible representation T and its row 7. That means that applying the deformation T7) to a complex results in a perturbation of the Fock operator having the same symmetry 1 7. This allows us to write the vibronic operator in a symmetry-adapted form ... 

To clarify the physics relevant to the operation of a-Si H solar cells, we shall restrict our discussion to the stnicture shown in Fig. 6a. An energy band diagram is shown in Fig. 11 for a p-i-n cell in the short-circuit mode. In many high-performance a-Si H solar cells, the electric field in the i layer is almost uniform since both the trapped charge and the free-carrier space charge are negligible. In this case, Crandall (1982) has shown that the photocurrent can be written as... 

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. 

This subduction shows that the two equisymmetric hg cluster deformations may be distinguished by a different parentage. The construction of the modes with (5,1) and (4,2) parentage proceeds as follows. One first defines a carrier space of Hg symmetry, based on the six pentagonal sites. The components of this space are labeled 9, e, r],... 

The operators T in ordinary quantum mechanics are defined as mappings of the elements in the wave function space P which serves as a carrier space for the operators. Similarly, one may consider mappings M of operators on operators, and they have obviously the operator space 7 as their carrier space. One of the most fundamental superoperators is the Liouvillian L, which is defined through the relation... 

The eigenoperators C and Cf are apparently excitation and de-excita-tion operators and, if one introduces a Fock space as a carrier space for the operators, one may also include ionization phenomena—removing or adding particles to the system under consideration. 

The classical operator space, which has been thoroughly investigated in mathematics almost since the beginning of this century, is the Hilbert-Schmidt (HS) space A consisting of all operators A for which the product A fA is a trace-class operator. A review of the use of the HS space as a carrier space for the superoperators in quantum theory has recently been given.25... 

In the previous discussion, it was assumed that we could start from any truncated basis B = B, B, . . . , Bm consisting of m linearly independent elements in the operator space. In this subsection, we will instead use a very special operator basis constructed by means of ket-bra operators ft)(a in general, such operators have turned out to be very useful tools in going from a carrier space with a binary product (a b) to the associated operator space. 

It is easily shown4 that, if

[Pg.311]

If

truncated basis of order n in the carrier space, then the corresponding basis P = Pk in the operator space is of order m = n2. It may further be shown4 that, if n —> °° and the basis tp = < becomes complete in the carrier space, then the basis P = Pki becomes complete in the HS operator space. It should perhaps be observed that this completeness theorem is somewhat different in nature from the completeness theorems for products of particle-hole operators which are proven to be... 

In the computational technique of today, one may be able to handle secular problems of order n = 106 in the carrier space, and, since this implies m = 1012, it seems at first sight as if the Liouvillian eigenvalue... 

Starting out from the truncated ON set

approximate eigenfunction P to the Hamiltonian H in the form ... 

---