Bloch wave-operator

As one can see, the operator has a property of the wave operator (it transforms the projection of the exact wave function into the exact wave function), however, it should be stressed that the operator converts just one projected wave function into the corresponding exact wave function so we will denote it as a state-specific wave operator in contrast to the so-called Bloch wave operator [46] that transforms all d projections into corresponding exact states. From definition (11) it is iimnediately seen that the state-specific wave operators obey the following system of equations for a = 1,..., d... 

Offermann ejt al /70/ formulated an open—shell CC formalism in Nuclear Physics in which a Bloch wave-operator Cl n for an n-valence problem is obtained recursively from those of the lower valence problems ... 

Furthermore, the field operator is expanded in the Bloch waves with wave vector k in the band denoted by b as... 

Mukherjee/69/, use of the sufficiency conditions (7.3.9) amounts in effect to assuming that ft is a valence-universal wave-operator. In fact Haque has explicitly demonstrated/123/ that the use of a valence-universal ft in the Fock-space Bloch equation leads automatically to eqn (7.3.9) with the ad-hoc sufficiency requirement. We give the sketch of a general proof here, since it shows that the extra information content of a Fock-space ft, as opposed to a Hilbert space, can be used to advantage for ensuring the connectivity of the cluster amplitudes of S/93/. For a valence-universal ft, the Fock-space Bloch equation (6.1.15) leads to... 

For an example, see [8] where definition A is used with K as Bloch s wave operator [6]. 

The Bom-von Karman contour condition demonstrates that the Bloch wave vector of free electrons in a cubic lattice is, according to Sommerfeld, constituted only by real components. The number of k values (k = p/h) admitted in a primitive cell of a reciprocal lattice is equal to the number of sites in the crystal. The linear momenta operator, p, is... 

The Hilbert space multireference CC (see e.g. Refs. [36-40]), based on the Jeziorski-Monkhorst ansatz for the wave operator [36]. This ansatz can be either combined with the standard (Rayleigh-Schrodinger) Bloch equation, or with the Brillouin-Wigner Bloch equation (cf. Section 18.4), or with a linear combination of both... 

The main question now is How can we approximate this powerful operator which allows us to generate the exact eigenvectors The aim of the Bloch formalism presented below is to develop a perturbative expansion for the wave operator. 

We are now ready to develop approximate solutions for the wave operator using this later equation, appropriately called the Bloch equation. 

Perturbative solution of the Bloch equation. We will first introduce the reduced wave operator X by assuming intermediate normalization for the exact wave function and writing fl as... 

This original approach, first proposed by Bloch (22) in 1958, follows a pedagogical approach to obtain both the wave operator and the effective Hamiltonian. However, from a computational point of view, the perturbative expansion [Eq. (41)] frequently diverges and the first few terms give only an approximation to the exact solution. In vibrational... 

The generalized Bloch equation (12) is the basis of the RS perturbation theory. This equation determines the wave operator and, together with Eq. (11), the energy corrections for all states of interest especially, it leads to perturbation expansions which are independent of the energy of the individual states, just referring the unperturbed basis states. Another form, better suitable for computations, is to cast this equation into a recursive form which connects the wave operators of two consecutive orders in the perturbation V. To obtain this form, let us start from the standard representation of the Bloch equation (16) in intermediate normalization and define... 

For a simple treatment of the (RS) perturbation expansions, it is often desirable to have just one wave operator 2, which is defined on the whole model space. This desire requests however that the distinction of internal and external excitations must apply for all determinants < ) e A4 in the same way [43]. This can be seen, for instance, from the second term (on the rhs) of the Bloch Eq. (16) which contains the operator product PVS2P. The projector P, standing left of this product, eliminates all contributions of the operator which leads out of the model space. For a single wave operator, acting on the whole model space, this projection must be the same for all determinants < ) e M. We therefore find that an unique dispartment of the excitations into internal and external ones is a necessary and sufficient condition. The importance of the proper choice of the model space and their classification has been discussed in detail by Lindgren [43]. 

The intermediate projection of operators or states upon the vacuum can be used also in order to classify the wave operator (and other perturbation expansions) in a different way. In the Bloch Eq. (16) and the effective Hamiltonian Heg in Eq. (10), namely, the order of the individual terms in the perturbation expansion is determined just by the number of interactions V, which occur in each term. A different classification of the diagrams is obtained in the (so-called) Q-box formalism as applied, for instance, in nuclear physics. In this formalism, the order of a perturbation expansion results from the number of foldings, i.e. the number of projections o)(o upon the vacuum. In each of these new orders, the number of interaction lines is no longer fixed but may vary from application to application. 

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,... 

An analogue representation applies independently also for the wave operator in every order n of fhe order-by-order expansion. Thaf is, if we are able fo bring the operator products on the rhs of fhe corresponding Bloch equafions (20)-(23) into normal form, we can identify the terms on the left- and right-hand side in order to express the amplitudes of the wave operator. For each order n, this finally... 

Solves the energy diagrams ("+l) of the order-byorder Bloch Eq. (23) by starting from the nth order solution of the wave operator. ... 

With these definitions of the perturbation and the model space, we can now solve successively the Bloch equation (16) for the energy corrections (e ) and wave operator (w ). As discussed above, we shall need the wave operator up to order (n — 1) if we wish to determine the energy correction to order n ... 

We emphasize that the wave operator (A) is applied to the /xth state only. It is state specific. Equation 2.29 is termed the Bloch equation [7] in Brillouin-Wigner form. Having introduced the Brillouin-Wigner wave operator, we turn our attention now to the corresponding reaction operator (A). 

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