Commutation Model operation

Some semi-empirical theories of chemical bonding assume without proof that the commutation relations between particular model operators are the same as the ones between the corresponding original operators [60-62]. Criteria for the validity of this assumption are investigated here for the first time based on our results for conservation of commutation relations by effective operators. 

In particular, having the model operator (34) and the wave operator (40) in second quantization, we can evaluate the commutator on the Ihs of the Bloch Eq. (16) and bring it into its normal-order form by analyzing term by term,... 

Eq. (22) have been derived from the variation principle alone (given the structure of H) they contain only the single model approximation of Eq. (9) the typically chemical idea that the electronic structure of a complex many-electron system can be (quantitatively as well as qualitatively) understood in terms of the interactions among conceptually identifiable separate electron groups. In the discussion of the exact solutions of the Schrodinger equation for simple systems the operators which commute with the relevant H ( symmetries ) play a central role. We therefore devote the next section to an examination of the effect of symmetry constraints on the solutions of (22). 

It is found empirically and of course is predictable theoretically that, when using a model for molecular electronic structure, the set of eigenfunction equations associated with the operators commuting with H are constraints on the action of the variation principle if Et is computed from R subject to symmetry constraints and E2 is computed in the same model with no such constraints then (2)... 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

Some of the SRPA values read as averaged commutators between T-odd and T-even operators. This allows to establish useful relations with other models. For example, (18), (19) and (44) give... 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

Exercise. Since in this model the operators B ( ) at different t commute, the time ordering in (4.3) may be omitted. It is therefore also possible to supply the higher terms and to construct the corresponding equation (4.7a) to all orders. In fact, it can even be solved. Show in this way that if p(0) = 8(n, n0) the k-th moment correlation functions gm with m> k. 

It has been shown that the time-volume and volume-time averaging operators are mathematically commutative [43, 47]. Nevertheless, Sha and Slattery [189] argued that in experimental analysis the instrumentation often records space average followed by a time average data. For this reason it might be convenient to formulate a consistent model formulation for the theoretical... 

The initial conditions at Z = 0 are the corresponding Schrodinger operators. This model is seen to be particularly simple All operators in Eq. (9.46) commute with each other, therefore this set of equations can be solved as if these operators are scalars. 

The effective operator A is the state-independent part of the definition AL/3, i = I-III. The operator A can thus be obtained by combining the perturbation expansions of its normalization factors and of A into a single expression [73] or by computing these normalization factors and A separately. These combined and noncombined forms of A[, may differ when computed approximately (see Section VI and paper II). The calculation of with the noncombined form is the same as with A since the model eigenvectors used with A are obtained by multiplying those utilized with A[,p by the above normalization factors. The operators and A are nevertheless different and, thus, do not have necessarily the same properties, for example, the conservation of commutation relations studied in Section IV. 

We now determine particular classes of commutation relations that are, indeed, conserved upon transformation to state-independent effective operators. The proof of (4.1) demonstrates that the preservation of [A, B] by definition A requires the existence of a relation between K, K, or both and one or both of the true operators A or B. Likewise, there must be a relation between the appropriate wave operator, the inverse mapping operator, or both, and A, B, or both for other state-independent effective operator definitions to conserve [A, B]. All mapping operators depend on the spaces and fl. Although the model space is often specified by selecting eigenfunctions of a zeroth order Hamiltonian, it may, in principle, be arbitrarily defined. On the other hand, the space fl necessarily depends on H. Therefore, the existence of a relation between mapping operators and A, B, or both, implies a relation between H and A, B, or both. 

Some semi-empirical theories of chemical bonding [60-62] use the commutation relation (4.12) with the true operators replaced by model ones to obtain additional relations between the parameters of the theories. This yields a reduction in the number of these parameters and, often a simpler determination of their values. It is also hoped [62] that this improves the theory by building in the right physics. This belief that useful theorems retain their validity upon replacement of nonem-pirical quantities by semi-empirical ones is commonly held by semiempiricists [144] but remains, in fact, an assumption. Our analysis of the conservation of this commutation relation in Section IV is useful in this context. 

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