Model eigenfunction

Hartree-Fock M.O. previous models, eigenfunction of the with Watson s correlation effects1 -25 already core-field (Ux,y,z). found in gaseous monatomic ions. ... 

Define k as the (linear) operator, called the wave operator, transforming the model eigenfunctions into the corresponding true ones,... 

Consider a time-independent operator A whose matrix elements, yf a, /3 d (both expectation values and transition moments), in the space fl we wish to compute. This goal is to be achieved by transforming the calculation from 0 into one in O, resulting in an effective operator a whose matrix elements, taken between appropriate model eigenfunctions of an effective Hamiltonian h, are the desired As we now discuss, numerous possible definitions of a arise depending on the type of mapping operators that are used to produce h and on the choice of model eigenfunctions. 

Without loss of generality, the right model eigenfunctions of h are taken to be unity normalized. If k is norm preserving, then the corresponding true eigenfunctions are also unity normalized. Thus, matrix elements of A are then given by... 

Norm-preserving mappings are denoted by K, fC) and, as discussed in Section II.B, generate a Hermitian effective Hamiltonian K HK = i. The orthonormalized model eigenfunctions of H are written as a)o and the corresponding true eigenfunctions are designated by I Pa). Thus, Eq. (2.2) specializes to... 

As Section II.B explains, expression (2.14) or (2.16) for matrix elements Aai3 applies here depending on which model eigenfunctions are selected... 

We now turn to the effective operator definitions generated by the non-norm-preserving mappings K, L). As mentioned in Section II.B, expression (2.16), in which is replaced by ), applies if the a )o are used, while expression (2.14), with (/) ) replaced by ), applies if the are incorporated into new model eigenfunctions. We first discuss the effective operator expressions that may be obtained from (2.16). 

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. 

Following the notation of Section II.D.l, the right model eigenfunctions b)o of Hg correspond to orthonormalized true eigenfunctions (TThe notation lag), just appends the subscript B to that used by Bloch.) The form of Lg implies upon use of (2.28) that... 

The relation (5.2) applies for any normalization of ag)o, provided is its corresponding true eigenfunction, since right model and true eigenfunctions scale together. In particular, introducing the definition of Section II.D.l for unity normed model eigenfunctions converts Eq. (5.2) into... 

The simple projection relation between the right model eigenfunctions of Hg and their true counterparts is an appealing aspect of Bloch s formalism. However, the non-Hermiticity of the resulting effective Hamiltonian represents a strong drawback, as discussed in Section VII. This has led many, beginning with des Cloizeaux [7], to derive Hermitian effective Hamiltonians, des Cloizeaux s method transforms the lag)ol not the... 

Because h is non-Hermitian, the mappings (s, T) do not conserve angles. As described in Section II.D, degenerate model eigenfunctions may be taken such that their true counterparts are mutually orthogonal. Thus, the true eigenvectors form an orthonormal set,... 

Now introduce a set of new model eigenfunctions o )(, through the scaling transformation... 

Consider the set of orthonormalized model eigenfunctions a)o of the effective Hamiltonian H defined in Table I. As explained in section II.C, the corresponding true eigenfunctions ) also form an orthonormalized set. Multiplying by an arbitrary complex number c , with c l l for at least one a, generates another set of orthogonal true eigenfunctions... 

The Hermiticity of an effective Hamiltonian is first shown to be equivalent to the orthogonality of its nondegenerate eigenfunctions. This does not impose any conditions on the conservation of true eigenvector norms or of angles between degenerate model eigenfunctions, which are next demonstrated to be independent of one another. Theorem II.b then follows. 

Let ,)o,.. . , 0 )o be arbitrarily chosen degenerate eigenfunctions of h with true counterparts ,) =/c (/),)o. Applying an arbitrary linear transformation to the produces other arbitrary model eigenfunctions i ) 0 = S I ) 0 with overlaps... 

Hence, mappings satisfying conservation B preserve norms of arbitrarily chosen degenerate eigenvectors [154], but not necessarily those of nondegenerate eigenfunctions. Thus, conservation B does not imply conservation C. The converse is also true since the condition ( <, ) = implied by conservation C for a set of model eigenfunctions that includes the <, )o, differs from Eq. (B.12). 

Consider the orthonormalized model eigenfunctions a )o and their true counterparts in terms of which the mappings K, L) are formally expressed in Eqs. (B.7) and (B.8). A set of orthonormalized true eigenfunctions may be obtained by normalizing to unity. 

Because the effective Hamiltonian H is Hermitian, there is only one set of model eigenfunctions. More than one fl(, expression for (B.14) can nonetheless be obtained because Eqs. (B.7) and (B.8) yield, respectively. 

Ellis and Osnes [32], Jorgensen and co-workers [35, 36], Lindgren [41], and Ratcliff [59] have shown that if a constant of the motion C commutes with the zeroth order Hamiltonian then it commutes with particular effective Hamiltonians, truncated at any order, whose model eigenfunctions, therefore, have the symmetry due to C. These authors do not consider this symmetry preservation in terms of commutation relation conservation, and paper II shows that the former can be obtained as a special case of the latter. 

Let us define the projector on to the ground-state model eigenfunction as... 

It can be easily shown, via equation (73), that contributions (178) and (179) reduce to the conventional paramagnetic terms whenever the hypervirial theorem is fulfilled (e.g., in the case of exact model eigenfunctions). By inserting identity (169), rewritten in the form... 

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