7. Complex many-electron atoms

Scientist that have contributed to the understanding of the structure of complex many electron-atoms are:

W. Pauli, J.C. Slater, D.R. Hartree, and V.A. Fock

The structure of complex atoms is in principle treated in the same way as that of the hydrogen atom or the hydrogenlike atoms. It boils down to solving the Schrödinger equation. There are however two problems that complicate the picture:

All constituting particles have their three degrees of freedom complicating immensely the solution of the Schrödinger equation: exact solutions cannot be found.
A fundamental theorem of quantum mechanics related to the indistinguishability of particles (here the electrons) comes into play: this gives restrictions on the wave functions.

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Approximate treatments of the Schrödinger equation

The underlying physical problem of the complex atom is that the electrons, distibuted in the atom, undergo electrostatic repulsion. All electrons are attracted to the nucleus by a central Coulomb force by a charge +Ze, similar as in the hydrogen atom. The repulsive force between the electrons, represented by a potential V_ee involves the spatial coordinates of all electrons and is not centrally directed.

In a simplified picture the binding energy can be calculated for the helium atom, involving two electrons. The binding of two electrons in the central field of a +2e nucleus adds to eight equivalents of Rydberg energies. If the two electrons are assumed to stay at an average distance of 2 Bohr radii the repulsive energy is 2 Rydberg equivalents. This simple model (read) gives almost the correct binding energy of -79 eV for the helium atom. Of course no two-electron wave functions can be derived in this way. For the helium atom variational calculations ( see the variational priniple) can be performed that predict the binding energies of the quantum levels up to 10 digits or more.

The effects of the non-central potential V_ee is treated in the Central Field Approximation model. Also a formal treatment can be given.
Here the Z(Z-1)/2 repulsive electrostatic contributions in V_ee are replaced by a single effective central potential V(r). In this model the attractive potential (with +Ze) due to the nucleus and the repulsive potential (due to the sum of the electrons) is taken as one Coulomb potential that scales as Z_eff.
So each electron sees a screened charge Z_eff, which is:
- at large distance Z_eff = 1
- at short distance, near the nucleus Z_eff = Z

Note that as a consequence electrons in orbitals with differing l quantum numbers undergo different screening:
- they come closer to the nucleus (low l numbers)
- or they stay away (high l numbers)
The screened charge distribution Z_eff can be written as a radially dependent function Z_eff(r). So the potential is no longer purely Coulombic ! This is the reason why electron energies depend on the orbital angular momentum.

Slater developed a method to derive screening constants and calculate the binding energies of electrons in various quantum states within the atom.
Slater's rules
These rules can also be applied to calculate Atomic radii.

Hartree and Fock developed the self-consistent field method to numerically calculate wave functions and energies in multi-electron systems. Particulerly sice the availability of powerful computers ab initio calculations of atomic (and molecular) structure has become feasible.
Self-consistent field method

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The aufbau principle for the Periodic system

The electronic structure of many-electron systems is based on the following ingredients:

A calculation of the electronic orbitals in the central field model; note that the level energies do depend on the quantum number l as discussed.

Filling of these one-electron orbitals with the number of electrons in the atom, taking into account Pauli's exclusion principle that no two electrons can occupy the same state. Shells are defined for each principal quantum number n with a degeneracy factor 2n².

For filled shells L = S = 0. (read proof) Hence only the electrons outside filled shells will contribute to the angular momentum of the atom. This shell-filling procedure is called the Aufbau principle. Note that this procedure holds for the electronic ground states of the atoms. Note that 2, 10, 18, 36 and 54 are the magic numbers associated with filled shells. It is not always clear what the lowest orbital is, particlarly when filling 3d and 4s orbitals that are almost energetic, depending on the specific Z number. There is some irregularity, because of this, for chromium.

In this way the elements can be ordered according to their shells in the Periodic Table of the Elements. See also the Web-elements version. The elements in column have the same number of electrons in the outer shell and also have the same angular momentum in the ground state configuration. The elements in the rightmost column, the noble gases, have filled shells.

Based on this model the ionization energies of the various atoms can be understood as well. (read).

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Another consequence of the Pauli principle for electrons

The Pauli exclusion principle was used already as the ordering principle for the construction of the Peirodic table. The concept of indistinguishability has however some deeper consequences for atomic structure. Study Electron Asymmetry (Brehm and Mullin 9-5) and the consequences for the helium atom (Brehm and Mullin 9-6).

Ortho- and parahelium
Helium spectrum

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The alkali atoms

Spin-orbit interaction in many-electron systems. (read)
Lithium energy levels
Na energy levels and spectrum and an interesting experiment to show the yellow Na-emission from an electrocuted pickle.

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Various couplings, Term values and Hund's rules

LS-Coupling
Spin-orbit coupling
JJ-coupling
Term values and Hund's rules

Last change: 21 February 2001