Mass-polarization operator

In second-quantization the mass-polarization operator takes the form... 

Owing to the similarity of the mass-polarization operator and the Breit operator, the mass-polarization corrections in MBPT can be classified using the scheme described for the Breit interaction in the previous subsection. Correspondingly, we write the valence contribution to the expectation value of P as... 

Figure 11. Scaled values of the mass-polarization operator P 2s)/ Z — 2) for the 2s state and P 2p)/ Z — 2) for 2pi/2 and 2p3/2 states are plotted agetinst Z for n = 2 states of lithium-like ions. Units a.u..

The expectation value of the mass-polarization operator, calculated as described above, is presented for 2s and 2p states of lithium-like ions in Fig. 11. It should be mentioned that for the special case of lithium-like ions, with a (Is) core, the RPA corrections to P identically vanish. Therefore, for lithium-like ions, Prpa = P - Moreover, for ns states of lithium-like ions, both P > and P q vanish. Thus, P(2s) = P (2s) for lithium-like ions. 

Other examples of two-electron operators are the two-electron part of the spin-orbit operator and the mass-polarization operator. A two-electron operator gives nonvanishing matrix elements between Slater determinants if the determinants contain at least two electrons and if they differ in the occupations of at most two pairs of electrons. The second-quantization representation of a two-electron operator therefore has the structure... 

Here Hg is the electronic Hamilton operator and H p is called the mass-polarization (Mtot is the total mass of all the nuclei and the sum is over all electrons). We note that He depends only on the nuclear positions (via Vne and Vnn, see eq. (3.23)) and not on their momenta. 

Neglecting the mass polarization and reintroducing the kinetic energy operator gives... 

The mass spectrometer is also incompatible with the HPLC system, but for a different reason. The ordinary mass spectrometer operates under very low pressure (a high vacuum see Chapter 10), and thus the liquid detection path must rapidly convert from a very high pressure and large liquid volume to a very low pressure and a gaseous state. Several approaches to this problem have been used, but probably the most popular is the thermospray (TS) technique. In this technique, the column effluent is converted to a fine mist (spray) as it passes through a small-diameter heated nozzle. The analyte molecules, which must be thermally stable, are preionized with the presence of a dissolved salt. A portion of the spray is introduced into the mass spectrometer. The analyte and mobile phase must be polar if the TS technique is used because the mobile phase must dissolve the required salt and the components must interact with the analyte molecule. See Workplace Scene 13.5. 

The leading polarization operator contribution to the Lamb shift in Fig. 2.2 was already calculated above in (2.6). Restoring the reduced mass factors which were omitted in that qualitative discussion, we easily obtain... 

In the case of the polarization insertions the calculations may be simplified by simultaneous consideration of the insertions of both the electron and muon polarization loops [18, 19]. In such an approach one explicitly takes into account internal symmetry of the problem at hand with respect to both particles. So, let us preserve the factor 1/(1 - - m/M) in (9.9), even in calculation of the nonrecoil polarization operator contribution. Then we will obtain an extra factor m /m on the right hand side in (9.12). To facilitate further recoil calculations we could simply declare that the polarization operator contribution with this extra factor m /m is the result of the nonrecoil calculation but there exists a better choice. Insertion in the external photon lines of the polarization loop of a heavy particle with mass M generates correction to HFS suppressed by an extra recoil factor m/M in comparison with the electron loop contribution. Corrections induced by such heavy particles polarization loop insertions clearly should be discussed together with other radiative-recoil... 

The subtracted radiatively corrected electron factor may be obtained from the subtracted one-loop electron factor in (9.10). To this end, one should restore the radiative photon mass in the one-loop electron factor, and then the polarization operator insertion in the photon line is taken into account with the help of the dispersion integral like one in (3.44) for the spin-independent... 

From the present calculations, the expectation value of the operator r 2 may provide a direct physical picture about the thermodynamic stability and dissociation of Hj-like molecules. As shown in Fig. 16, there is a vertical jump of the mean value ru at Xc. We note that there are similarities and differences between helium-like atoms and Hj-like molecules. In Section V.A of heliumlike systems, based on an infinite mass assumption, we show that the electron at the critical point leaves the atom with zero kinetic energy in a first-order phase transition. This limit corresponds to the ionization of an electron as the nuclear charge varies. For the Hj-like molecules, the two protons move in an electronic potential with a mass-polarization term. They move apart as X approaches its critical point and the system approaches its dissociation limit through a first-order phase transition. 

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