Effective operators state-independence

Actions and operations describe interactions between objects an effect describes state transitions. You use effects to factor a specification or to describe important transitions before the actual units of interaction are known. Just as attributes are introduced when convenient to simplify the specification of an operation, independent of data storage, so can effects be introduced to simplify or defer the specification of operations. 

The preceding analysis is just a transformation of one representation of the n-state problem to another representation. To be useful, the new representation must admit simplifying approximations not suggested by the original representation. One such approximation is to replace the frequency variable 03 in M( ) in (7.10) by a typical P space eigenfrequency, say We thereby obtain the frequency-independent effective operator... 

The second-quantization counterpart of this approach is the replacement (for the lN configuration) of operator (14.65) by some effective operator, whose two-particle submatrix elements are independent of characteristics L12, S12 of the pairing state of electrons. To this end, we introduce the submatrix element averaged over the number of various antisymmetric pairing states in shell, equal to (4/ + 2)(4/ + l)/2 ... 

This is the Markovian memory-less approximation to the Master Equation. In this approximation, the effective time evolution operator becomes independent of t and the integral may be extended to infinity. It is also consistent to assume that the system lost memory of the initial state of the reservoir, whatever this was. In the limit when Uq is calculated in perturbation theory and pq(0) = 0, we obtain the conventional Born-Markov time evolution which has a long and successful history. 

Appendix E Preservation of Commutation Relations by State-Independent Effective Operator Definitions Other than A Acknowledgments References... 

A fundamental aspect of semi-empirical chemical bonding theories is their requirement that the model operators be state independent [56]. This property is, of course, not required of effective operators if only the numerical values are desired for the matrix elements of operators. Indeed, some semi-empirical theories, used in other areas of physics, do not impose the requirement of state independence. For instance, LS-dependent parameters are employed in describing the hyperfine coupling of two-electron atoms [31]. However, whenever effective operators themselves are the quantities of interest, as when studying semi-empirical theories of chemical bonding, state independence of effective operators becomes a necessity. This paper thus examines conditions leading to the generation of state-independent effective operators. 

The operator K AK = A in (2.24) does not depend on the indices a and /8. Thus, this case produces the state-independent effective operator definition A of Table I. Equation (2.3) implies that A may be rewritten as k PAPK, a form which emphasizes that A gives matrix elements of A only for the states in Cl (a property of all effective operator definitions). 

Mappings Norms Conserved" Effective Hamiltonian Effective operator definition Effective Hamiltonian is the A = H Case State- Independent... 

We now turn to the effective operator definitions produced by (2.14) with model eigenfunctions that incorporate the normalization factors of (2.16) so their true counterparts are unity normed. Equations (2.27) and (2.38) show these model eigenfunctions to be the a)o and ( that are defined in (2.33) and (2.34). Substituting Eqs. (2.27) and (2.38) into (2.14) and proceeding as in the derivation of the forms / = I-I1I, yields the state-independent definitions A, A" and A" of Table I. Notice that the effective Hamiltonian H is identically produced upon taking A = // in the effective operator A". Table I indicates that this convenient property is not shared by all the effective operator definitions. 

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. 

This section studies the previously unaddressed problem of commutation relation conservation upon transformation to effective operators [77]. State-independent effective operators are treated first. 

Section II and Table I show that state-independent effective operators can be obtained with norm-preserving mappings K, or with any of the three kinds of non-norm-preserving mappings K, L), K, L), and K, L). This section first proves that the commutation relations between two arbitrary operators cannot generally be conserved upon transformation to any of these state-independent effective operators. A determination is then made of operators whose commutation relations are preserved by at least some state-independent effective operator definitions, and a few applications are then presented. Particular interest is focused on operators which commute with H, including constants of the motion. 

Theorem IV pertains to commutation relations for all possible state-independent effective operators Let A and B be two arbitrary operators and let F be their commutator F = [A,B. The commutation relation between A and B is, in general, not conserved upon transformation to state-independent effective operators. This theorem is first demonstrated... 

The proof is then modified to apply to the other state-independent effective operator definitions. To prove (4.1) it is first shown that... 

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. 

The key step in deriving (4.7) is the commutation of P with H. Clearly, a similar reasoning applies when replacing H with any operator that commutes with H because such an operator also commutes with P. Therefore, this leads to Theorem V as follows state-independent effective operators produced by norm-preserving mappings conserve the commutation relations between H and an arbitrary operator B and between B and any operator that commutes with H. Given particular choices of P,... 

Hence, the commutation relation between A and B is conserved iff the right hand sides of Eqs. (4.8) and (4.9) are equal to each other, thereby leading to Theorem VII as follows the commutation relation between two operators A and B is preserved upon transformation to state-independent effective operators obtained with norm-preserving mappings iff A and B satisfy... 

Another important application of Theorem V is that (Corollary V.2) the dipole length and dipole velocity transition moments are equivalent when computed with state-independent effective operators obtained with norm-preserving mappings. According to definition A (see Table I), these computations evaluate o( p /8)o, and (a r )3)oWith... 

Table III summarizes Theorems V-VII and their corollaries along with similar results for the other state-independent effective operator definitions. Appendix E demonstrates the analogs of Theorems V-VII, except the conservation by definitions A" and A " of [H, C] for C a constant of the motion which commutes separately with and V. This last point is proven in paper II. The analogs of Corollaries V.l and V.2 are obtained similarly to, respectively. Corollaries V.l and V.2. Just as with Corollary V.2, none of the equivalences between the dipole length and dipole velocity transition moments for definitions A", A , or A , / = I-IV, produces a sum rule for transition moments (see Appendix D).

Complications arise here that are absent with state-independent effective operators. The effective operators in (4.16) cannot merely be replaced by their definitions from Table I since the latter may not be applied directly to any vector of the space Oq because of their normalization factors. These factors are associated with the model eigenvectors on which the operators act to produce the matrix elements A. Consequently, arbitrary bras and kets of flp must first be expanded in the basis of these eigenvectors before a state-dependent definition can be used with them. This represents a serious drawback with the use of state-dependent effective operators. 

The commutation relation between two arbitrary operators is not conserved upon transformation to effective operators by any of the definitions. Many state-independent effective operator definitions preserve the commutation relations involving // or a constant of the motion, as well as those involving operators which are related to P in a special way, for example, A with [P, 4] = 0. Many state-dependent definitions also conserve these special commutation relations. However, state-dependent definitions are not as convenient for formal and possibly computational reasons. The most important preserved commutation relations are those involving observables, since, as discussed in Section VII, they ensure that the basic symmetries of the system are conserved in effective Hamiltonian calculations. 

State-independent effective operators can be applied directly to arbitrary bras and kets of the model space fig- However, when used with state-dependent effective operators these vectors must be expanded in the basis of the model eigenbras and eigenkets, respectively, that are utilized to produce the with the latter operators. Consequently, as shown in Section IV.B, general expressions for state-dependent effective operators are more complicated than for state-independent ones. State dependence of effective operators is thus a eonceptual drawback. 

Semi-empirical Hamiltonians and operators are taken to be state independent [56] and have the same Hermiticity as their true counterparts. Consequently, the valence shell effective Hamiltonians and operators they mimic must also have these two properties. Table I shows that the effective Hani iltonian and operator definitions H and A, as well as H and either A or a fulfill these criteria. Thus, these definition pairs may be used to derive the valence shell effective Hamiltonians and operators mimicked by the semi-empirical methods. Table III indicates that the commutation relation (4.12) is preserved by all three definition pairs. Hence, the validity of the relations derived from the semi-empirical version of (4.12) depends on the extent to which the semi-empirical Hamiltonians and operators actually mimic, respectively, exact valence shell effective Hamiltonians and operators. In particular, the latter Hamiltonians and operators contain higher-body terms which are neglected, or ignored, in semi-empirical theories. These nonclassical higher body interactions have been shown to be nonnegligible for the valence shell Hamiltonians of many atoms and molecules [27, 145-149] and for the dipole moment operators of some small molecules [56-58]. There is no a... 

Equation (4.14) provides the equivalence between the dipole length and dipole velocity transition moments for a system of n identical particles of mass m with state-independent effective operator definition A. To see that this equivalence does not produce a sum rule, consider first the usual derivation of the Thomas-Reiche-Kuhn sum rule for the true operators. Left- and right-multiplying equation (4.12) by ( l and ), respectively, the z component yields... 

APPENDIX E PRESERVATION OF COMMUTATION RELATIONS BY STATE-INDEPENDENT EFFECTIVE OPERATOR DEFINITIONS OTHER THAN A... 

Theorems V and VI of Section IV show that definition A conserves [H, B and [A, B], with B an arbitrary operator and A an operator commuting with H or, more generally, with P. Theorem VII provides the necessary and sufficient condition for the conservation of a general commutator by definition A. Analogous results for the other state-independent effective operator definitions are presented in columns 3-7 of Table III and are derived in this appendix by modifying the proofs of Theorems V-VII. 

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