5.3 Electronic and Nuclear Spin States
227
at this point, we do not necessarily know what interchange symmetries, if any,
are possessed by the electron spin product functions. We do know, however, that
the Pauli Principle for a many-electron system requires that the total wavefunction
be antisymmetric to the interchange of any pair (here specifically, the 3p pair)
of electrons. Hence, for total (1, 2) to be antisymmetric when the orbital angular
momentum pair-state corresponds to L = 2, it is necessary that the spin angular
momentum pair-state be antisymmetric to the interchange of the two 3p electrons.
Similarly, for L = 1, the spin angular momentum pair-state must be symmetric to
this interchange, and for L = 0, it must again be antisymmetric. Let us now examine
how this can happen.
Two electron spins (each of magnitude
1
2 ) can be coupled vectorially to give
total electron spins S = s 1 + s 2 of magnitudes 1 and 0 (corresponding to s 1 and
s 2 combined collinearly or anticollinearly, respectively). These total electron spins
have corresponding M S values 1, 0, −1 for S = 1, and M S = 0 for S = 0. The
same arguments that we used above for the M L = +2 pair-state for L = 2 hold
for the M S = +1 pair-state for S = 1, so that the three total spin S = 1 pairstates with M S = +1, 0,−1 are all symmetric to the interchange of the two 3p
electrons, and the single pair-state for S = 0 (with M S = 0) is antisymmetric to
this interchange. Thus, the Pauli Principle requires in the case of the two equivalent
(indistinguishable) 3p valence electrons of Si that the (symmetric) L = 2 total
orbital angular momentum states must combine with the antisymmetric S = 0 total
spin angular momentum state, and similarly, the (antisymmetric) L = 1 orbital
states combine with the symmetric S = 1 spin states and the (symmetric) L = 0
orbital state with the antisymmetric S = 0 spin state. These collections of states
are designated by term symbols 1 D, 3 P , and 1 S, respectively, and are called the
atomic terms of the ground electronic configuration of the Si atom. 3 At this stage of
development, all states associated with a particular atomic term will be degenerate
in energy.
It happens that the terms themselves are split further by the interaction between
the spin and orbital angular momenta (a process referred to as spin–orbit coupling).
For the Si atom, this coupling only affects the 3 P term (because S = 0 for the D
and S terms, so that it vanishes for them). The 3 P term is, however, affected and
splits into three components (referred to as levels), corresponding to magnitudes of
the total electronic angular momentum, traditionally written as J = L + S, having
values J = 2, 1, 0. A level with total electronic angular momentum of magnitude
J has 2J + 1 degenerate states associated with it; note that in the present case, the
three levels obtained from the 3 P term correspond to a total of 1 + 3 + 5 = 9 states.
A determination of the relative energies for the three terms ( 1 S, 1 D, 3 P ) would
require us to delve too deeply into quantum mechanics. Those who would like to
pursue this topic in greater detail should consult a modern quantum chemistry text,
such as those by Levine [3] and Levin [4]. However, perhaps the most readable yet
fairly complete explanations of the term energies remain those given by Pauling and
3 For additional information on these atomic terms and the term symbols, see Appendix D.
227
at this point, we do not necessarily know what interchange symmetries, if any,
are possessed by the electron spin product functions. We do know, however, that
the Pauli Principle for a many-electron system requires that the total wavefunction
be antisymmetric to the interchange of any pair (here specifically, the 3p pair)
of electrons. Hence, for total (1, 2) to be antisymmetric when the orbital angular
momentum pair-state corresponds to L = 2, it is necessary that the spin angular
momentum pair-state be antisymmetric to the interchange of the two 3p electrons.
Similarly, for L = 1, the spin angular momentum pair-state must be symmetric to
this interchange, and for L = 0, it must again be antisymmetric. Let us now examine
how this can happen.
Two electron spins (each of magnitude
1
2 ) can be coupled vectorially to give
total electron spins S = s 1 + s 2 of magnitudes 1 and 0 (corresponding to s 1 and
s 2 combined collinearly or anticollinearly, respectively). These total electron spins
have corresponding M S values 1, 0, −1 for S = 1, and M S = 0 for S = 0. The
same arguments that we used above for the M L = +2 pair-state for L = 2 hold
for the M S = +1 pair-state for S = 1, so that the three total spin S = 1 pairstates with M S = +1, 0,−1 are all symmetric to the interchange of the two 3p
electrons, and the single pair-state for S = 0 (with M S = 0) is antisymmetric to
this interchange. Thus, the Pauli Principle requires in the case of the two equivalent
(indistinguishable) 3p valence electrons of Si that the (symmetric) L = 2 total
orbital angular momentum states must combine with the antisymmetric S = 0 total
spin angular momentum state, and similarly, the (antisymmetric) L = 1 orbital
states combine with the symmetric S = 1 spin states and the (symmetric) L = 0
orbital state with the antisymmetric S = 0 spin state. These collections of states
are designated by term symbols 1 D, 3 P , and 1 S, respectively, and are called the
atomic terms of the ground electronic configuration of the Si atom. 3 At this stage of
development, all states associated with a particular atomic term will be degenerate
in energy.
It happens that the terms themselves are split further by the interaction between
the spin and orbital angular momenta (a process referred to as spin–orbit coupling).
For the Si atom, this coupling only affects the 3 P term (because S = 0 for the D
and S terms, so that it vanishes for them). The 3 P term is, however, affected and
splits into three components (referred to as levels), corresponding to magnitudes of
the total electronic angular momentum, traditionally written as J = L + S, having
values J = 2, 1, 0. A level with total electronic angular momentum of magnitude
J has 2J + 1 degenerate states associated with it; note that in the present case, the
three levels obtained from the 3 P term correspond to a total of 1 + 3 + 5 = 9 states.
A determination of the relative energies for the three terms ( 1 S, 1 D, 3 P ) would
require us to delve too deeply into quantum mechanics. Those who would like to
pursue this topic in greater detail should consult a modern quantum chemistry text,
such as those by Levine [3] and Levin [4]. However, perhaps the most readable yet
fairly complete explanations of the term energies remain those given by Pauling and
3 For additional information on these atomic terms and the term symbols, see Appendix D.