Operators energy shift

Equation (7) shows the significance of the effective Hamiltonian which is directly related to the spectroscopic and dynamical observables, as line-shapes (see the end of this section) and transition probabilities. The effective Hamiltonian can be written as the sum of the projection of the exact Hamiltonian into the inner space and of the energy-shift operator [3, 7] ... 

Using (23), the energy-shift operator (16) can be transformed into... 

Note that Rzz z) = (3 i (z) 3) is the unique non-zero matrix element representing the energy-shift operator (14). Let us now introduce a notation which will display the physical meaning of the matrix elements of the effective Hamiltonian. We write (53) in the form... 

The correlation described by second-order perturbation theory is now included variationally by means of the energy shift operator. 

This term describes a shift in energy by Acim rn, for an orbital with quantum numbers I — 2, mi and that is proportional to the average orbital angular momentum (/z) for the TOj-spin subsystem and the so-called Racah parameters Bm, that in turn can be represented by the Coulomb integrals and The operator that corresponds to this energy shift is given by... 

In addition to energy eigenvalues it is of interest to calculate intensities of infrared and Raman transitions. Although a complete treatment of these quantities requires the solution of the full rotation-vibration problem in three dimensions (to be described), it is of interest to discuss transitions between the quantum states characterized by N, m >. As mentioned, the transition operator must be a function of the operators of the algebra (here Fx, Fy, F7). Since we want to go from one state to another, it is convenient to introduce the shift operators F+, F [Eq. (2.26)]. The action of these operators on the basis IN, m > is determined, using the commutation relations (2.27), to be... 

Note that SPODS is nearly always operative in resonant strong-held excitation using modulated ultrashort laser pulses, the only exception being so-called real laser pulses [72, 77] (i.e., electric helds with only one quadrature in the complex plane) that are usually hard to achieve in ultrafast laser technology. This is why many different pulse shapes can lead to comparable dressed state energy shifts and... 

In this connection it is hence convenient to introduce the reaction operator t by the relation t = VW, which gives the energy shift... 

Following this formalism, three different QM operators appear, namely V , yne (it may be shown that t/ne and t/en are formally identical), and ycc. These have a correspondence, respectively, to zero-, one-, and two-electron terms of H. We note that the zero-order term gives rise to an energetic contribution Unn which is analogous to the nuclei-nuclei repulsion energy ym and thus it is generally added as a constant energy shift term in H. The conclusion of this analysis is that we may define four operators (reduced in practice to two, plus a constant term) which constitute the operator y,nt of Equation (1.105). 

The spherical parts of the operators in Eq. (4-9) define the mean one-electron potential Vc(r)usedto generate the orbitals V>i (Eq. (3-11)), and hence when taken together, lead to a constant energy term in all of the diagonal matrix elements u = u this constant energy shift is of no interest and may be discarded. 

The operator between the brackets is the harmonic oscillator Hamiltonian, which has the basis functions (112) as its eigenfunctions the remaining term is taken into account via Eq. (114). The rotational kinetic energy operator L(cuP) [Eq. (26)] can be written in terms of the shift operators J = J, + iJi, and the operator J, which act on the basis as... 

Details of the derivation of general expressions for energy shifts at a given order can be found in Mohr et al. (1998). Contractions between pairs of fermion or boson field operators AM lead to electron and photon propagator functions. The exact electron propagator in a static external field is homogeneous in time and appears as... 

With the Coulomb and exchange parts of the MP discussed so far, the core-like solutions of the valence Fock equation would still fall below the energy of the desired valence-like solutions. In order to prevent the valence-orbitals collapsing into the core during a variational treatment and to retain an Aufbau principle for the valence electron system, the core-orbitals are moved to higher energies by means of a shift operator... 

The energy shift of orbital ij/j calculated to second order is given by Equation 4.15. Second-order perturbations are not additive for operators ti that contain several elements p 0. 

A U(1,2)A with A the product of free positive -energy projection operators, as in (2.9). The operator U(l,2) = u + U must be chosen so that if the external potential is turned off, the one-photon and two-photon exchange scattering amplitudes are reproduced. (I note in passing that if one uses U = Uj-r rather that one must already include U to get the a Ry level shift correctly ) On comparing the eigenvalues of (4.3) with those obtained from,... 

Gell-Mann and Low [23] derived a formula which yields the energy shift due to the interaction (149) in terms of the matrix elements of the operator 5. (0, —00) where S. is the electron operator (108) obtained from I5q(109) with replaced by operator (149). Later Sucher [24] derived a symmetrized version of the energy shift formula, containing the matrix elements of the operator 5y(oo, —00) and which is more suitable for the renormalization procedure. 

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