Three-electron atoms

Now we turn to lithium and three-electron lithium-like ions. Again we start with the normally-ordered no-pair Hamiltonian given in Eq. (132), and choose the starting potential to be the Hartree-Fock potential of the (Is) helium-like core. We expand the energy of an atomic state in powers of the interaction potential 

The terms give the fe-th order contributions to the energy of the helium-like core, and are independent of the valence state. These are precisely the terms evaluated in the previous section. The terms are the fe-th order contributions to the energy of the atom relative to the ionic core in other words is the fc-th order contribution to the 

The lowest approximation to the removal energy is seen to be — e , where e is the eigenvalue of the frozen-core Hartree-Fock equation. It should be emphasized that the valence orbital is not treated self consistently. The orbitals of the closed-shell core are determined self-consistently, then the valence electron HF equation is solved in the frozen potential of the core. From Eq. (145) it follows that there is no first-order correction to the removal energy in the frozen-core HF potential. 

It is interesting to note that the second-order energy E can be written as the diagonal matrix element of the second-order self-energy operator defined by 

With this definition, Ei = [E )(e )]. It is an elementary exercise to show that, in configuration-space, the self-energy operator has the asymptotic form 

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Others (e.g., Fukashi Sasaki s upper bound on eigenvalues of 2-RDM [2]). Claude Garrod and Jerome Percus [3] formally wrote the necessary and sufficient A -representability conditions. Hans Kummer [4] provided a generalization to infinite spaces and a nice review. Independently, there were some clever practical attempts to reduce the three-body and four-body problems to a reduced two-body problem without realizing that they were actually touching the variational 2-RDM method Fritz Bopp [5] was very successful for three-electron atoms and Richard Hall and H. Post [6] for three-nucleon nuclei (if assuming a fully attractive nucleon-nucleon potential). 

T. Morishita, C.D. Lin, Comprehensive analysis of electron correlations in three-electron atoms, Phys. Rev. A 59 (1999) 1835. 

Fig. 2. Three-electron Feynman diagrams representing electron interaction in three-electron atoms in the second order. The designations are the same as in Fig.l...

Coulomb-Coulomb interaction for three-electron atom is represented by two-electron Feynman diagrams Fig. la, b and three-electron diagrams Fig.2a. 

Besides taking into account the two-electron diagrams Fig.lc, d, e, f, Coulomb-Breit interaction for three-electron atom represented by the three-electron diagrams Fig.2b, c (as it is mentioned above the contribution of the diagrams Fig.2b, c is considered as doubled contribution of the diagram Fig.2b). The formulas for irreducible part of the diagram Fig.2b is (see the Appendix)... 

Figure 10. Ionization energy, 1, for three-electron atoms as a function of X for N = 1, 2,..., 8. (N = 1 means that 5 basis functions were used in the calculations, and N = 8 means that 1589 basis functions were used.)...

Now, let us use the data collapse method to test the hypothesis of finite-size scaling used to obtain the critical parameter for this system and estimate the critical exponent v for the lithium-like atoms. Using data collapse to the ionization energy of the three-electron atom in its ground state, /3(A) = Eq1(A)—... 

Figure 11. The correlation length, tfN as a function of % for the three-electron atoms for E = 1,2,..., 8.

Figure 12. Data collapse for the ionization energy of the three-electron atom with a — 1.64 and V3 = 0.8.

Lithium Three-electron atom, Beryllium Four-electron atom, Boron Five-electron atom. ... 

However, application of rij-correlated basis sets is confined to two- and three-electron atoms. In general, the most often used are configuration expan-... 

E. Holoien, S. Geltman, Variational calculations for quartet states of three-electron atomic systems, Phys. Rev 153 (1967) 81. 

On the other hand, as early as 1968, Somorjai [4] had introduced an integral transform method, closely related to GCM, for atomic and molecular systems. An extensive review of this method may be found in Ref. [5]. In this context, accurate correlated functions for two- and three-electron atomic systems were obtained by Thakkar and Smith [6]. Nonetheless, Somorjai used the integration limits of the integral transform as variational parameters. While this was a very innovative option, to some extent it masked the full potential of the HW equation. 

The same variational techniques can be applied to lithium and other three-electron atomic systems. In this case, the terms in the Hylleraas correlated basis set have the form... 

Figure 3.9 A three-electron atom. Distance of the electrons from the nucleus/3 < q < r. The nucleus exerts a strong attractive force on electron 1. The effect of the nucleus on...

Write the Hamiltonian for a three-electron atom, including the kinetic energy and Coulomb potential terms. Briefly deflne aU the symbols you use. 

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