Fock expansion

The second and third terms handle two-body collisions while the fourth term is related to the three-body collision. The term second-order in R in the Fock expansion is also known, and Myers, et a/. [12] have verified that this term eliminates the discontinuity in the local energy at the origin. This article also contains an analysis of the behavior of the wave function in the vicinity of these singular points. 

A Cl basis cannot describe the third term in the Fock expansion very easily while it can be included explicitly in a correlated basis, explaining the superior convergence of the latter over any one-electron basis. Studies of the partial wave expansion,... 

We can treat FD quantum-optical states as those of a real single-mode electromagnetic field, which fulfill the condition of truncated Fock expansion. These states can directly be generated by the truncation schemes (the quantum scissors) proposed by Pegg et al. [44] and then generalized by other authors [45-47]. Alternatively, one can analyze states obtained by a direct truncation of operators rather then of their Fock expansion. Such an operator truncation scheme, proposed by Leonski et al. [48-50], will be discussed in detail in the next chapter [51]. 

Kuang et al. [17] defined the normalized FD coherent states by truncating the Fock expansion of the conventional ID coherent states or equivalently by the action of the operator exp( v a ) (with proper normalization) on vacuum state. The Kuang et al. approach is similar to the Vaccaro-Pegg treatment [57] of the... 

Analogously to the generalized, CS in a FD Hilbert space, analyzed in Section IV. A, other states of the electromagnetic field can be defined by the action of the FD displacement or squeeze operators. In particular, FD displaced phase states and coherent phase states were discussed by Gangopadhyay [28]. Generalized displaced number states and Schrodinger cats were analyzed in Ref. 21 and generalized squeezed vacuum was studied in Ref. 34. A different approach to construction of FD states can be based on truncation of the Fock expansion of the well-known ID harmonic oscillator states. The same construction, as for the... 

One can propose another definition of a FD squeezed vacuum, such as by truncation of the Fock expansion of the conventional squeezed vacuum at the state. v). Thus, we define the truncated squeezed vacuum as follows [34]... 

A very compact and highly accurate wave function for the ground state of the He atom has already been constructed by Hylleraas long ago [21]. He expressed this in terms of the coordinates ri,r2 and ru with ri and T2 the distances of the first and second electron from the nucleus, and ri2 the distance between the electrons. Thus the cusp conditions [20] could be satisfied. Essentially in the same philosophy Pekeris performed a calculation on the He-ground state [22], that remained an undisputed landmark for quite some time. A progress beyond this was possible when analytic properties of the exact wave function of a three-particle system (one nucleus and two electrons) were talren into account, which were ignored in earlier formulations. The keyword to this is Fock expansion and it requires terms that are logarithmic in the coordinates [23, 24, 25]. 

One can include the leading terms of three-particle nonanalytic points with logarithmic terms from the Fock expansion, but their impact on the variational energy is small [20]. Several forms for the correlation terms have been proposed and tested. For example, Umrigar et al. [16] used a Fade form of polynomials in a linear combination of electron-electron (r y) and electron-ion (r,) distances. 

Union [9] and the West [10-13]. The Fock expansion features an expansion... 

It helps even more to build into the basis terms which describe the singularity in the wavefunction at the 3-particle coalescence, as was first demonstrated by Frankowski and Pekeris [16], who with only 246 functions, including some suggested by the Fock expansion, obtained a variational upper bound of... 

By including some higher-order terms from the Fock expansion which Frankowski and Pekeris should not have omitted, and including many more powers of (r2 — ri)/(ri - -r2) and ri2/(ri - -r2) to reflect electron-electron correlation on short to moderate length scales, Freimd, Hux-table, and Morgan [14] obtained with 230 functions an upper bound of... 

Abbott PC, Maslen EN (1987) Coordinate systems and analytic expansions for three-body atomic wavefunctions I. Partial summation for the Fock expansion in hyperspherical coordinates. J Phys A Math Gen 20 2043-2075... 

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