True eigenfunction

The commutator relations (21) show that the true eigenfunctions can be chosen such that they are simultaneously eigenfunctions of the operators H, S, eind Sz. ... 

PROBLEM 3.13.1. Prove Eq. (3.13.20) by expanding the trial wavefunction Ptrial in terms of a complete orthonormal basis set of the true eigenfunctions T rini. nm where H 1 lb En T. . 

Since ip, involves R as well as r, the functions XtnWi are not true eigenfunctions of the exact Hamiltonian, but only of an approximate H0. The perturbation operator for this problem is defined by the equation... 

Finally, we then use well-known mathematical procedures to determine the energy levels of the system and to describe the true wave functions in terms of linear combinations of the basis functions. As we shall see, this means that we set up the Hamiltonian as a matrix in terms of the spin product basis functions, then manipulate it into a form in which it is diagonal. The functions used to represent the Hamiltonian in diagonal form are, then, the true eigenfunctions, which are linear combinations of the basis functions. 

Equation 11.16 is a general result that can be applied to any spin system. In addition, it is helpful to have an expression for the time dependence of each element of p. Equation 11.12 is applicable to a set of basis functions that are otherwise not limited. However, we know from Chapter 6 that we can (in principle, at least) solve the Schrodinger equation to obtain the true eigenfunctions of 96. If these eigenfunctions are used as the basis functions, then Eq. 11.12 simplifies to give... 

Designate the space defined by the d true eigenfunctions as Cl, and let P and Q = l- P be the projection operators onto Cl and its orthogonal complement Cl, respectively. Note that k operates only on functions in fig and transforms them to functions only in O. Thus, we have the conditions on k that... 

The space Hq is usually chosen by selecting d eigenfunctions of a zeroth order Hamiltonian and k, I and h are obtained using perturbation theory or one of its formally exact reformulations, for example, an iterative scheme. In general, only these operators are perturbatively expanded and not the model or the true eigenfunctions. By multiplying (2.2) on the left by / and (2.5) on the left by k, it follows that / and k are related to one another by [37]... 

In accord with previous works, the overlaps, between degenerate true eigenfunctions are assumed to be null. This is, of course, useful for computing the but it often requires additional calculations when mappings do not conserve overlaps, as discussed in Sections II.D and II.E. Henceforth, all greek indices lie in the set (d), unless otherwise indicated. 

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... 

Notice that the condition [A, P] = 0 in Theorems V and VI is indeed sufficient for (4.10). There clearly exist many other sufficient conditions on A and B causing (4.10) to be fulfilled, for example, PAQ = 0 = PBQ, PA = 0 = PB, PA = 0 = BQ, etcetera. All such conditions necessarily imply a relation between A, B, or both, and P, Q, or both. Consequently, although more general. Theorems VI and VII have fewer applications than Theorem V because all the sufficient conditions obtainable from (4.10), except that of Theorem V, cannot be verified exactly. This is because the d true eigenfunctions that define P are not known a priori and can be determined a posteriori only approximately. Except for H and operators commuting with H, it is thus in general impossible to determine if a particular operator commutes with P. Also, conditions that involve Q explicitly, for example, PAQ = 0 = PBQ are not readily established. 

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 bi-orthonormal complement of the right eigenvectors ag)o and b)o re denoted, respectively, by o( bI and o(aBl - Bloch [6] shows that the unity normed true eigenfunction P ) is also the projection onto fl of lag)(),... 

Klein [65] notes that the des Cloizeaux transformation is a different formulation of the symmetric transformation of Lowdin [99] and uses this fact to prove that orthonormal functions in Hq which differs minimally (in a least squares sense) from the d true eigenfunctions of H. This is often referred to as a maximum... 

The left hand side of (6.13) is the desired diagonal matrix element of A only if ) is the true eigenfunction of interest with eigenvalue... 

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... 

Let a )o be the set of orthonormalized eigenfunctions of H. The corresponding true eigenfunctions are orthogonal Jo one another but are not necessarily normalized to unity. Denote by the norm of the ath corresponding true eigenfunction ),... 

The angles between the corresponding true eigenfunctions (/) ) = /c < ) are given by... 

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. 

The only unknowns in (2.12), (2,13) are k and I because even though P is defined in terms of unknown true eigenfunctions, its perturbation expansion is known (see [6, 107]) and is independent of k and /. Therefore, if either fe or / is chosen, then the other is uniquely determined by (2.12) or (2.13). Although no previous work has taken advantage of this simplification, we will show elsewhere [66] that Eq. (2.12) or (2.13) can indeed be used to simplify the derivation of the operators k and I for many of the known different forms of effective Hamiltonians. 

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