For many-electron atoms

You are probably used to this idea from descriptive chemistry, where we build up the configurations for many-electron atoms in terms of atomic wavefunctions, and where we would write an electronic configuration for Ne as... 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

Boys, S. F., Proc. Roy. Soc. London) A207, 181, Electronic wave functions. IV. Some general theorems for the calculation of Schrodinger integrals between complicated vector-coupled functions for many-electron atoms."... 

A similar model for many-electron atoms has been developed,6 by considering each electron to be hydrogen-like, but under the influence of an effective nuclear charge (Z — Ss)e, in which Ss is called the size-screening constant. It is found that atoms and ions containing only 5 electrons (with the quantum number l equal to zero) and completed sub-groups of... 

Screening Constants for Many-electron Atoms. The Calculation and Interpretation of X-ray Term Values, and the Calculation of Atomic Scattering... 

Pauling L. and Sherman J. (1932). Screening constants for many-electrons atoms The calculation and interpretation of X-ray term values and the calculation of atomic scattering factors. Zeit. Krist., 81 1-29. 

D. A. Mazziotti, Variational two-electron reduced-density-matrix theory for many-electron atoms and molecules implementation of the spin- and symmetry-adapted Z2 condition through first-order semidefinite programming. Phys. Rev. A 72, 032510 (2005). 

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... 

More generally, for many-electron atoms we define U as the energy required to transfer an electron from one atom to another, so that... 

Having decided to use AOs (or combinations of them) for yrA and pB> we will now look at the form these take. They are approximate solutions to the Schrodinger equation for the atom in question. The Schrodinger equation for many-electron atoms is usually solved approximately by writing the total electronic wavefunction as the product of one-electron functions (these are the AOs). Each AO 4>i is a function of the polar coordinates r, 0, and single electron and can be written as... 

Because of interelectronic repulsions, the Schrodinger equation for many-electron atoms and molecules cannot be solved exactly. The two main approximation methods used are the variation method and perturbation theory. The variation method is based on the following theorem. Given a system with time-independent Hamiltonian //, then if

well-behaved function that satisfies the boundary conditions of the problem, one can show (by expanding

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While using (4.14) and exact wave functions. This supports the conclusion of Drake [83] that for electric dipole transitions, by considering the commutator of with the atomic Hamiltonian in the Pauli approximation, we obtain Qwith relativistic corrections of order v2/c2 (see (4.18)-(4.20)). However, for many-electron atoms and ions, one has to use approximate (e.g., Hartree-Fock) wave functions, and then this term gives non-zero contribution, conditioned by the inaccuracy of the model adopted. 

The elements of the theory of angular momentum and irreducible tensors presented in this chapter make a minimal set of formulas necessary when calculating the matrix elements of the operators of physical quantities for many-electron atoms and ions. They are equally suitable for both non-relativistic and relativistic approximations. More details on this issue may be found in the monographs [3, 4, 9, 11, 12, 14, 17]. 

The condition S(n, n ) for many-electron atoms is normally ensured only approximately unlike the other d(a,a), which are rigorous. The principle of the orthogonality of the wave functions reflects the fact that at one time only one state described by a given set of exact quantum numbers is realized, an electron cannot occupy simultaneously several physical states. 

Better approximations can be made, and numerical calculations leave no doubt that Schrodinger s equation works very accurately for many-electron atoms, as it does for hydrogen. However, the orbital approximation is good enough for most purposes, and it leads to the very appealing picture of a many-electron atom in which each electron occupies an orbital which is similar to, although not identical with, the orbitals which form the exact solutions of the hydrogen atom. 

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