Canonical shift transformation

The most general approach to the formation of the effective operator of the intersite virtual phonon exchange interaction is based on the canonical shift transformation of the Hamiltonian... 

Let us begin by studying the relative dynamics in the Hamiltonian case, that is, for deterministic driving. The shift (42) to the moving origin is a time-dependent canonical transformation [98]. It transforms the linearized Hamiltonian (33) into... 

The linear terms can be removed from the canonical model by shifting the origin of the zx. .. zk system to the stationary point by the transformation... 

The Hamiltonian of the crystal after the shift canonical transformation is [10]... 

An out-of-space string Kp can only contribute if it is produced by K m)h and if E j transforms it back into an allowed string (Jp). But if the orbitals are numbered consecutively within each RAS space, and if equation (139) is used to define the canonical index (ij), such contributions are impossible. For example, consider the case where a maximum of two electrons are allowed in RAS III. If string Ip already contains two electrons in RAS III, then a single replacement E (Ip)) which promotes a third electron to RAS III will yield an out-of-space string. The shift operator E could produce an allowed string again if it moves any of the three electrons in RAS III back down into RAS I or RAS II. However, if the summation over orbitals is restricted to (ij) > (kl), this is impossible, since k > i. 

An efficient treatment of the time evolution operator defined in Eq. (2.33) can be achieved by performing a canonical transformation of coordinates acting on the field X. We define the shifted vector X— X -p.,Uj — p.2 2 a new set of field coordinates. The potential is now decoupled... 

We introduce the shifted coordinates X2 = x - fi. This canonical transformation enables us to obtain a more suitable form for the operator, in which V is decoupled. Neglecting the tilde symbol in the following, we identify with (aJj, p,) and with xj, P2)... 

So the FC integral is added to the very few physical systems [18] which are realizations of this particular algebra. Using the Taylor theorem for shift operators due to Sack [19], and the Cauchy relation mentioned above, we can apply this very general idea to the specific case of the harmonic oscillator to obtain the closed formula (5). Recurrence relations can also be obtained by noticing that O is in reality a superoperator which maps normal ladder operators by the canonical transformation ... 

As the second example, let in addition the variable be measured. The Jacobi matrix evaluated again at the standard operation conditions and transformed to canonical format can be found immediately. In (10.6.9), we only shift the y -column and place it after the column y Then for instance in (10.6.9a) we have... 

As a consequence of these two problems, a naive calculation will yield chemical shifts which are small differences of large numbers, and which are consequently subject to large numerical uncertainties. These problems can be remedied by the IGLO approach. In this method, the canonical molecular orbitals Xi transformed into localized molecular orbitals (pj by means of a unitary matrix U... 

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