36
2 One Magnetic Center
The relativistic Hamilton operator for an electron can be derived, using the correspondence principle, from its relativistic classical Hamiltonian and this leads to
the one-electron Dirac equation, which does contain spin operators. From the oneelectron Dirac equation it seems trivial to define a many-electron relativistic equation, but the generalization to more electrons is less straightforward than in the
non-relativistic case, because the electron-electron interaction is not unambiguously
defined. The non-relativistic Coulomb interaction is often used as a reasonable first
approximation. The relativistic treatment of atoms and molecules based on the manyelectron Dirac equation leads to so-called four-component methods. The name stems
from the fact that the electronic wave functions consist of four instead of two components. When the couplings between spin and orbital angular moment are comparable
to the electron-electron interactions this is the preferred way to explain the electronic
structure of the lowest states.
In most cases, however, the relativistic effects are rather weak and may be separated into spin-orbit coupling effects and scalar effects. The latter lead to compression
and/or expansion of electron shells and can rather accurately be treated by modifying the one-electron part of the non-relativistic many-electron Hamiltonian. With this
scalar-relativistic Hamiltonian the (modified) energies and wave functions are computed and subsequently an effective spin-orbit part ˆ
H SO is added to the Hamiltonian.
The effects of the spin-orbit term on the energies and wave functions are commonly
estimated using second-order perturbation theory. More information for the interested reader can be found in excellent textbooks on relativistic quantum chemistry
[2, 3].
The standard way to include relativistic angular moment couplings in the notation
of eigenvalues and eigenfunctions of the thus obtained energies and wave functions is
the so-called Russell–Saunders coupling scheme. It is adequate if the spin-orbit coupling is considered to be weak compared to the electron-electron interactions. For a
free atom or ion the Russell–Saunders scheme implies that the one-electron moments
l and s are first coupled to a many-electron angular moment L and spin moment S,
which are subsequently coupled to a total angular moment J. Due to the spin-orbit
coupling the wave functions are no longer eigenfunctions of the ˆ
L and ˆ
S operators
(L and S are no longer “good quantum numbers”) but only of the ˆ
J operator and the
degeneracy of the states belonging to one LS term is partly removed. Only the states
corresponding to a particular J eigenvalue are degenerate, but nevertheless the states
of one LS term are close in energy. Such a set of nearly-degenerate states originating
from one LS term is called a Russell–Saunders (RS) term and commonly denoted
by a Russell-Saunders term symbol 2S+1 L J . For example, the lowest energy RS term
of an atom with a single valence p-electron is 2 P 1/2 , with the RS term 2 P 3/2 having
a slightly higher energy. The 2 P 1/2 term is two-fold degenerate (M J = 1/2, −1/2)
while the 2 P 3/2 term is four-fold degenerate (M J = 3/2, 1/2, −1/2, −3/2).
2 One Magnetic Center
The relativistic Hamilton operator for an electron can be derived, using the correspondence principle, from its relativistic classical Hamiltonian and this leads to
the one-electron Dirac equation, which does contain spin operators. From the oneelectron Dirac equation it seems trivial to define a many-electron relativistic equation, but the generalization to more electrons is less straightforward than in the
non-relativistic case, because the electron-electron interaction is not unambiguously
defined. The non-relativistic Coulomb interaction is often used as a reasonable first
approximation. The relativistic treatment of atoms and molecules based on the manyelectron Dirac equation leads to so-called four-component methods. The name stems
from the fact that the electronic wave functions consist of four instead of two components. When the couplings between spin and orbital angular moment are comparable
to the electron-electron interactions this is the preferred way to explain the electronic
structure of the lowest states.
In most cases, however, the relativistic effects are rather weak and may be separated into spin-orbit coupling effects and scalar effects. The latter lead to compression
and/or expansion of electron shells and can rather accurately be treated by modifying the one-electron part of the non-relativistic many-electron Hamiltonian. With this
scalar-relativistic Hamiltonian the (modified) energies and wave functions are computed and subsequently an effective spin-orbit part ˆ
H SO is added to the Hamiltonian.
The effects of the spin-orbit term on the energies and wave functions are commonly
estimated using second-order perturbation theory. More information for the interested reader can be found in excellent textbooks on relativistic quantum chemistry
[2, 3].
The standard way to include relativistic angular moment couplings in the notation
of eigenvalues and eigenfunctions of the thus obtained energies and wave functions is
the so-called Russell–Saunders coupling scheme. It is adequate if the spin-orbit coupling is considered to be weak compared to the electron-electron interactions. For a
free atom or ion the Russell–Saunders scheme implies that the one-electron moments
l and s are first coupled to a many-electron angular moment L and spin moment S,
which are subsequently coupled to a total angular moment J. Due to the spin-orbit
coupling the wave functions are no longer eigenfunctions of the ˆ
L and ˆ
S operators
(L and S are no longer “good quantum numbers”) but only of the ˆ
J operator and the
degeneracy of the states belonging to one LS term is partly removed. Only the states
corresponding to a particular J eigenvalue are degenerate, but nevertheless the states
of one LS term are close in energy. Such a set of nearly-degenerate states originating
from one LS term is called a Russell–Saunders (RS) term and commonly denoted
by a Russell-Saunders term symbol 2S+1 L J . For example, the lowest energy RS term
of an atom with a single valence p-electron is 2 P 1/2 , with the RS term 2 P 3/2 having
a slightly higher energy. The 2 P 1/2 term is two-fold degenerate (M J = 1/2, −1/2)
while the 2 P 3/2 term is four-fold degenerate (M J = 3/2, 1/2, −1/2, −3/2).