A Many-Electron Atom

As we already know, it is impossible to solve analytically the Schrodinger equation for atoms with two and more electrons. This prompted the development of new quantum mechanical methods of approximate solutions or modified solutions, equitable for the hydrogen atom, by introducing empirical adjustments. In this chapter we will consider intra-atomic interactions between electrons and the complication that this interaction causes. (Here we must emphasize that any complication in the theory forces the development of more sophisticated experimental methods of investigation. 

The wavefunctions of many-electron atoms will not be studied in this book these are the subject of quantum chemistry. 

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The true value of tk for a many-electron atom or a molecule is unknown. If we could set it equal ( expand it) to a linear combination of an infinite number of basis functions, each defined in a space of infinite dimensions, we could carry out an exact calculation of (k. Such a set of basis functions would be a complete set. 

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... 

For the following pairs of orbitals, indicate which is lower in energy in a many-electron atom. 

Figure 15-11 shows a schematic energy level diagram of a many-electron atom. Blue patterns... 

It is possible to remove two or more electrons from a many-electron atom. Of course it is always harder to remove the second electron than the first because the second electron to come off leaves an ion with a double positive charge instead of a single positive charge. This gives an additional electrical attraction. Even so, the values of successive ionization energies have great interest to the chemist. 

Turn back to Figure 15-11, the energy level diagram of a many-electron atom, and consider the occupied orbitals of the element potassium. With 19 electrons placed, two at a time, in the orbitals of lowest energy, the electron configuration is... 

The notion of electrons in orbitals consists essentially of ascribing four distinct quantum numbers to each electron in a many-electron atom. It can be shown that this notion is strictly inconsistent with quantum mechanics (7). Definite quantum numbers for individual electrons do not have any meaning in the framework of quantum mechanics. The erroneous view stems from the original formulation of the Pauli principle in 1925, which stated that no two electrons could share the same four quantum numbers (8), This version of the principle was superseded by a new formulation that avoids any reference to individual quantum numbers for separate electrons. The new version due to the independent work of Heisenberg and Dirac in 1926 states that the wave function of a many-electron atom must be antisymmetrical with respect to the interchange of any two particles (9,10). 

A many-electron atom is also called a polyelectron atom. 

The next step in our journey takes us from hydrogen and its single electron to the atoms of all the other elements in the periodic table. A neutral atom other than a hydrogen atom has more than one electron and is known as a many-electron atom. In the next three sections, we build on what we have learned about the hydrogen atom to see how the presence of more than one electron affects the energies of... 

FIGURE 1.41 The relative energies of the shells, subshells, and orbitals in a many-electron atom. Each of the boxes can hold at most two electrons. Note the change in the order of energies of the 3d- and 4s-orbitals after Z = 20. 

As well as being attracted to the nucleus, each electron in a many-electron atom is repelled by the other electrons present. As a result, it is less tightly bound to the nucleus than it would be if those other electrons were absent. We say that each electron is shielded from the full attraction of the nucleus by the other electrons in the atom. The shielding effectively reduces the pull of the nucleus on an electron. The effective nuclear charge, Z lle, experienced by the electron is always less than the actual nuclear charge, Ze, because the electron-electron repulsions work against the pull of the nucleus. A very approximate form of the energy of an electron in a many-electron atom is a version of Eq. 14b in which the true atomic number is replaced by the effective atomic number ... 

In a many-electron atom, because of the effects of penetration and shielding, the order of energies of orbitals in a given shell is s < p < d < f. 

Describe the factors affecting the energy of an electron in a many-electron atom (Section l.f2). 

As a first approximation, each electron in a many-electron atom can be considered to have the distribution in space of a hydrogen-like electron under the action of the effective nuclear charge (Z—Ss)e, in which 5s represents the screening effect of inner electrons. In the course of a previous investigation,6 values of S5 for a large number of ions were derived. 

The labelling of terms as S,L,J,Mj) is preferable when one takes into account the effect of spin-orbit coupling, since / and Mj remain good quantum numbers even after this perturbation is accounted for. In detail, the effect of spin-orbit coupling over a many-electron atomic term is evaluated by writing the spin-orbit operator in terms of the total angular and spin momentum, L and 5 ... 

Let us assume a many-electron atom enclosed within an infinitely hard spherical cavity of radius R and develop the method and calculations so that we may recover the free-atom case when . This procedure also allows to look at the evolution of its ground state energy as the cage volume shrinks as has been done elsewhere [25,53]. The TFDW energy-density functional for an atom enclosed within a spherical cavity of radius R is written as... 

Including resonance effects, the atomic scattering factor for a many-electron atom is written as... 

For an atom with many electrons, the first electron fills the lowest energy orbital, and the second electron fills the next lowest energy orbital, and so forth. For a one-electron atom or ion, the energy depends only on n, the principal quantum number but for a many-electron atom or ion, the value of I also plays a role in the energy. The order of atomic orbital energy is given by... 

The wave functions describing the elechonic states of a many-electron atom or molecule are funchons of all the coordinates of all the elechons ... 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

Their energies vary as -Z2/n2, so atoms with higher Z values will have more tightly bound electrons (for the same n). In a many-electron atom, one often introduces the concept of an effective nuclear charge Zeff, and takes this to be the full nuclear charge Z minus the number of electrons that occupy orbitals that reside radially "inside" the orbital in question. For example, Zeff = 6-2=4 for the n=2 orbitals of Carbon in the ls22s22p4... 

The parity of atomic states is important in spectroscopy. A radial function is an even function [see (1.113)] the spherical harmonic Y(m is found to be an even or odd function of the Cartesian coordinates according to whether / is an even or odd number. For a many-electron atom, it follows that states arising from a configuration for which the sum of the / values of all the electrons is an even number are even functions when 2,/, is odd, the state has odd parity. 

With internal nuclear motion neglected, the Hamiltonian for a many-electron atom with atomic number Z is... 

Figure 1-5 The single-electron energies for a many-electron atom. The drawing is only very approximate. The single s, p, and d levels are separated for the sake of clarity.

Thus, the state of each electron in a many-electron atom is conditioned by the Coulomb field of the nucleus and the screening field of the charges of the other electrons. The latter field depends essentially on the states of these electrons, therefore the problem of finding the form of this central field must be coordinated with the determination of the wave functions of these electrons. The most efficient way to achieve this goal is to make use of one of the modifications of the Hartree-Fock self-consistent field method. This problem is discussed in more detail in Chapter 28. 

Thus, we have expressed the non-relativistic Hamiltonian of a many-electron atom with relativistic corrections of order a2 in the framework of the Breit operator (formulas (1.15), (1.18)—(1.22)) in terms of the irreducible tensorial operators (second term in (1.15), formulas (19.5)—(19.8), (19.10)— (19.14), (19.20), respectively). 

Dependence of the main terms of the Hamiltonian of a many-electron atom on nuclear charge Z may be easily revealed in the following fashion. Let us consider the Hamiltonian of the kind (1.15)... 

There exist a number of attempts to generalize hydrogenic radial orbitals to cover the case of a many-electron atom. In [180] the so-called generalized analytical radial orbitals (Kupliauskis orbitals) were proposed ... 

The main ideas of the book are described in seven Parts divided into 33 Chapters, which are subdivided into Sections. In Part 1 we present the initial formulas to calculate the energy spectrum of a many-electron atom in non-relativistic and relativistic approximations, accounting for the relativistic effects as corrections and use perturbation theory in order to describe the energy spectra of an atom. Radiative and autoionizing... 

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