Potential energy many-electron atom

In order to retain the orbital model for a many-electron atom, Hartree assumed that each electron came under the influence of the nuclear charge and an average potential due to the remaining electrons. He therefore retained the form of the radial equation for a one-electron atom, equation 12.2, but assumed that the mutual potential energy U was the sum of... 

Flambaum, V.V. and Ginges, J.S.M. (2005) Radiative potential and calculations of QED radiative corrections to energy levels and electromagnetic amplitudes in many-electron atoms. Physical Review A, 72, 052115-1-052115-13. 

We conclude this section by providing a comparison in Figures 1—5of the kinetic-energy potentials of the CGE and several better GGA OF-KEDF s, using accurate densities for H, He, Be, Ne, and Ar atoms. For many-electron atoms, highly accurate densities (from atomic configuration interaction calcu-lations) are fed into the OF-KEDF s. Accurate potentials are obtained via a two-step procedure the exact [p]) is obtained for a given accurate... 

Although equation (2) appears concise in the form above, it cannot be solved exactly for a molecular system. Electron-electron repulsions present a practical problem in the same way that they do in many-electron atoms the repulsive interaction between two electrons is a function of the position coordinates of each electron and cannot be separated into two functions dependent on each set of coordinates individually. However, even if a comphcated wavefunction that was expressed in the coordinates of aU the electrons in a molecule could be developed, it would have to be reported as an extensive grid of points in 3-D space, each associated with a different potential energy with unique solutions for... 

We begin by considering the forces that act within and the potential energy functions for many-electron atoms. Consider Li, for which Z = 3. 

These Hartree orbitals resemble the atomic orbitals of hydrogen in many ways. Their angular dependence is identical to that of the hydrogen orbitals, so quantum numbers and m are associated with each atomic orbital. The radial dependence of the orbitals in many-electron atoms differs from that of one-electron orbitals because the effective field differs from the Coulomb potential, but a principal quantum number n can still be defined. The lowest energy orbital is a Is orbital and has no radial nodes, the next lowest s orbital is a 2s orbital and has one radial node, and so forth. Each electron in an atom has associated with it a set of four quantum numbers (n, , m, mfj. The first three quantum numbers describe its spatial distribution and the fourth specifies its spin state. The allowed quantum numbers follow the same pattern as those for the hydrogen atom. However, the number of states associated with each combination of (n, , m) is twice as large because of the two values for m. ... 

The physical structure of the atom, as determined by experiments, is summarized in the planetary model. In an atom with atomic number Z, there are Z electrons moving around a dense nucleus that has positive charge +Ze. Coulomb s law describes the forces and potential energy of interaction between each electron and the nucleus and between the electrons in many-electron atoms. 

Whereas, for non-Coulombic potentials, one can define nr and , n is then no longer related simply to the binding energy. Indeed, for a complex, many-electron atom, it is not at all obvious how one should set about quantising the system, since there is no guarantee that the orbits of individual electrons will close.6 In fact, conservation of the angular momentum for individual electrons is, at best, only an approximation. It would hold exactly for central fields. Even then, the same simple, precise relationship between n and as for H is not to be expected for many-electron atoms. As we shall see, the very meaning of n (the principal or most important quantum number) becomes less clear-cut for many-electron systems. In a nutshell the n and quantum numbers of... 

The total relativistic and QED energy shift for many-electron atoms consists of two parts. The first part contains the Bethe logarithm and the other is the average value of some effective potential. Throughout the exact nonrelativistic (Schrddinger) wave functions for the many-electron atom are used. The energy shift is [50] ... 

We thus anticipate that because of the rather different form Eq. (18) takes, its capability to reproduce accurate exchange energy results for many-electron atoms and molecules may be limited. Also, the exchange potential stemming from Eq. (18) is divergent at a... 

One approximate method for obtaining Eq. 7.182 in a central field form wa.s introduced by Hartree and named the self-consistent field approach. This method regards each electron in a many-electron atom as moving in the temporarily fixed field of the remaining electrons. The system can now be described in terms of one-electron wavefunctions (or orbitals) j(rj). The non-Coulomb potential energy for the jth electron is then V y(ry) atid this contains the other electronic coordinates only as parameters. Vjirj) can be chosen to be spherically symmetric. The computational procedure is to solve the Schrodinger equation for every electron in its own central field and then to make the wavefunctions, so found, self-consistent with their potential fields. The complete wavefunction for the system is a product of the one-electron functions. 

Abstract In this chapter I discuss some aspects of relativistic theory, the accuracy of the infinite order two-component relativistic lOTC method and its advantage over the infinite order Douglas-Kroll-Hess (DKHn) theory, in the proper description of the molecular spectroscopic parameters and the potential energy curves. Spin-free and spin dependent atomic mean filed (AMFI) two-component theories are presented. The importance of the quanmm electrodynamics (QED) corrections and their role in the correct description of the spectroscopic properties of many-electron atoms for the X-ray spectra is discussed as well. Some examples of the molecular QED calculations will be discussed here as well. 

In Chapter 2, the development of atomic orbitals for many-electron atoms was built upon an understanding of the orbitals obtained from the exact solution for the one-electron hydrogen atom. Similarly, it is useful to examine the electronic structure for the simplest molecular species—the hydrogen molecule ion (Hj)—to begin our discussion of molecular orbitals. Like the H atom, the ion contains only one electron. The difference is the presence of two nuclei instead of one. The potential energy of the system is a sum of the Coulomb attraction of the electron to each of the two nuclei and Coulomb repulsion between the positively charged nuclei. 

In section 6.9 we already introduced finite-size models of the atomic nucleus and analyzed their effect on the eigenstates of the Dirac hydrogen atom. This analysis has been extended in the previous sections to the many-electron case. It turned out that neither the electron-electron interaction potential functions nor the inhomogeneities affect the short-range behavior of the shell functions already obtained for the one-electron case. Table 9.5 now provides the total electronic energies calculated for the hydrogen atom and some neutral many-electron atoms obtained for different nuclear potentials provided by Visscher and Dyall [439], who also provided a list of recommended finite-nucleus model parameters recommended for use in calculations in order to make computed results comparable. 

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