Effective operators commutation relations

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

Previous work has not investigated if commutation relations are conserved upon transformation to effective operators. Many important consequences emerge from particular commutation relations, for example, the equivalence between the dipole length and dipole velocity forms for transition moments follows from the commutation relation between the position and Hamiltonian operators. Hence, it is of interest to determine if these consequences also apply to effective operators. In particular, commutation relations involving constants of the motion are of central importance since these operators are associated with fundamental symmetries of the system. Effective operator definitions are especially useful... 

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. 

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... 

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... 

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. 

Section IV proves that the conservation of the commutation relation (4.12) between H and the position operator f leads to the equivalence of the dipole length and dipole velocity transition moments computed with certain effective operator definitions. Contrary to the similar equivalence for transition moments computed with true operators, however, this does not yield a sum rule. Many other sum rules follow from commutation relations between true operators. In view of the many useful applications of sum rules [141, 142] the existence of sum rules for quantities computed using effective operators is of interest and will be studied elsewhere [79]. A potential application lies in determining the amount, or proportion, of transition strengths carried by a particular state or group of states [142, 143]. 

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... 

Using any other effective operator definition (mentioned in the first paragraph) and proceeding similarly yields a relation analogous to (D.9) whose right hand side contains a matrix element of the commutator between the effective operators corresponding to z and p. This commutator cannot be transformed any further for the same reason that applies to the commutator in (D.9). 

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

The derivations of Theorems V and VI for the preservation of [ A, B] consist of (1) the replacement by P of the product KK that is present between A and B in [A, B], (2) the commutation of A with P, and (3) the incorporation of P into K and K. Step (1) uses Eq. (4.6), which is valid only for norm-preserving mappings. With other definitions, however, the products analogous to KK may be replaced by P if (criterion 1) they satisfy the fundamental relation (2.13). Step (2) clearly applies to any effective operator definitions. So does step (3) since P can be combined with any mapping operators using Eqs. (2.3) and (2.6). Hence, all effective operator definitions that fulfill criterion 1 conserve [A, B] if [A, B] =0. When these definitions, like A, produce the associated effec-... 

When relaxation occurs among r.f. pulses, the t dependence in the above sequence provides a means to study spin-lattice relaxation. It is known [2.20] that the Zeeman Tiz) and quadrupole (Tig) spin-lattice relaxation times can be measured for deuterons in liquid crystals from the sum and difference of the doublet signal strengths after the third pulse, respectively. This can be shown using the formalism of rotation operators. The rotation operator R(0, ) provides a general method to describe the effect of a r.f. pulse with rotation angle 0 and phase angle when there are no cyclic commutation relations, such that the simple relation in Eq. (2.60) exists. Now... 

Comparing with the commutation relations above, we see that for r and p at least, K has the effect of an antiunitary operator. Expressing orbital angular momentum as f = r X p, we see that = —1. For spin we can draw on the analogies between the transformation of commutation relations for spin and orbital angular momentum. From these we see that the transformed commutation relations are consistent with Cs0 = -s, and the spin has the same transformation properties as orbital angular momentum. 

Matrix blocking will already be effected by the point-group symmetry of the molecule. We do not expect time reversal to reduce the symmetry because the time-reversal operator K commutes with all operations of the point group. If time reversal introduces new symmetries, we will gain further blocking. We must therefore establish the relation between time-reversal symmetry and point-group symmetry. 

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