2.2 The Eigenstates of Many-Electron Atoms
37
2.2 Give the Rusell–Saunders term symbol for the ground state of an atom
with three electrons in the 2p orbitals.
In cases where the spin-orbit coupling is strong compared to the electron-electron
interactions it is more reasonable to account for the spin-orbit coupling by using
the so-called j-j coupling scheme. Here the orbital moment l and the spin s of each
electron are coupled to give an individual angular moment for each electron. The
individual j for each electron are then coupled to give a total angular moment J.
Modern relativistic many-electron quantum mechanical computational treatments
are able to treat the entire range of angular moment couplings, from negligible to
dominant spin-orbit coupling. The results can be expressed in either terms of RussellSaunders states or in j-j coupled states, whatever representation gives better insight.
For core-excited states where the core spin-orbit coupling is much larger than the
valence spin-orbit coupling, a mixed notation is sometimes used, in which the open
core shell is j-j coupled and the open valence shell is Russell–Saunders coupled.
The many-electron states of an atom in a crystal field or a molecule can obviously
not be labelled by the IRs of SO(3), since the Hamilton operator, the angular moment
operator and therefore also the many-electron wave functions transform according
to the IRs of a less symmetric point group. The lower symmetry may also remove
the degeneracies of the LS terms. For example, the 2 P ground term of a boron atom
becomes 2 T 1u in an octahedral crystal field so that the three fold degeneracy is
retained, but splits into two LS terms of 2 E and 2 A 1 symmetry when the crystal field
symmetry is lowered to C 3v .
Orbital moment quenching: The eigenfunctions of the angular moment operator
ˆ l 2 are the spherical harmonics, characterized by the quantum numbers l and m.
ˆ l
2 Y l,m = l(l + 1)Y l,m
(2.5)
where is put to 1. On the other hand, these functions cannot be eigenfunctions simultaneously of the three components ˆ l x,y,z , because these operators do not commute.
Choosing z to be the quantization axis gives
ˆ l z Y l,m = mY l,m
ˆ l x Y l,m =
1
2
(l − m)(l + m + 1)Y l,m+1 +
1
2
(l − m + 1)(l + m)Y l,m−1
ˆ l y Y l,m =
1
2i
(l − m)(l + m + 1)Y l,m+1 −
1
2i
(l − m + 1)(l + m)Y l,m−1
(2.6)
In general the spherical harmonics Y l,m are complex functions, but linear combination
can be made such that the eigenfunctions of ˆ l 2 become real. For example for the
spherical harmonics with l = 1ofap 1 electronic configuration:
37
2.2 Give the Rusell–Saunders term symbol for the ground state of an atom
with three electrons in the 2p orbitals.
In cases where the spin-orbit coupling is strong compared to the electron-electron
interactions it is more reasonable to account for the spin-orbit coupling by using
the so-called j-j coupling scheme. Here the orbital moment l and the spin s of each
electron are coupled to give an individual angular moment for each electron. The
individual j for each electron are then coupled to give a total angular moment J.
Modern relativistic many-electron quantum mechanical computational treatments
are able to treat the entire range of angular moment couplings, from negligible to
dominant spin-orbit coupling. The results can be expressed in either terms of RussellSaunders states or in j-j coupled states, whatever representation gives better insight.
For core-excited states where the core spin-orbit coupling is much larger than the
valence spin-orbit coupling, a mixed notation is sometimes used, in which the open
core shell is j-j coupled and the open valence shell is Russell–Saunders coupled.
The many-electron states of an atom in a crystal field or a molecule can obviously
not be labelled by the IRs of SO(3), since the Hamilton operator, the angular moment
operator and therefore also the many-electron wave functions transform according
to the IRs of a less symmetric point group. The lower symmetry may also remove
the degeneracies of the LS terms. For example, the 2 P ground term of a boron atom
becomes 2 T 1u in an octahedral crystal field so that the three fold degeneracy is
retained, but splits into two LS terms of 2 E and 2 A 1 symmetry when the crystal field
symmetry is lowered to C 3v .
Orbital moment quenching: The eigenfunctions of the angular moment operator
ˆ l 2 are the spherical harmonics, characterized by the quantum numbers l and m.
ˆ l
2 Y l,m = l(l + 1)Y l,m
(2.5)
where is put to 1. On the other hand, these functions cannot be eigenfunctions simultaneously of the three components ˆ l x,y,z , because these operators do not commute.
Choosing z to be the quantization axis gives
ˆ l z Y l,m = mY l,m
ˆ l x Y l,m =
1
2
(l − m)(l + m + 1)Y l,m+1 +
1
2
(l − m + 1)(l + m)Y l,m−1
ˆ l y Y l,m =
1
2i
(l − m)(l + m + 1)Y l,m+1 −
1
2i
(l − m + 1)(l + m)Y l,m−1
(2.6)
In general the spherical harmonics Y l,m are complex functions, but linear combination
can be made such that the eigenfunctions of ˆ l 2 become real. For example for the
spherical harmonics with l = 1ofap 1 electronic configuration: