5.3 Electronic and Nuclear Spin States
225
the Ne atom. We also know from introductory quantum mechanics that electronic
configurations can be associated with what is called the independent-electron model
and described in terms of a central-field Hamiltonian in which an individual electron
moves under the influence of a central potential energy function that is due to the
net motion of all other electrons about the positively charged nucleus. At this stage,
all inter-electron repulsions are ignored.
We shall consider the (3s) 2 component of the [Ne](3s) 2 (3p) 2 electronic configuration of the Si atom first. According to the Pauli (Exclusion) Principle for electrons,
no two electrons in a given atom may have the same set of values for all four of the
quantum numbers, n, l, m l , and m s . Both (3s) 2 electrons have principal quantum
numbers n = 3, azimuthal quantum numbers l = 0, and magnetic quantum numbers
m l = 0, with the consequence that the Pauli Principle therefore now requires that
they have different values for the spin quantum number m s . We also know that
electrons have ‘spin’ one-half, so that the quantum number m s may have only one
of the two values ±
1
2 . The two 3s electrons are thus said to be ‘paired’. This pair of
electrons can be represented as a pair-state (or outer product state) for which the total
spin S is zero: the associated magnetic spin quantum number is designated as M S
and also necessarily has the value zero. Let us now consider the two 3p electrons:
we have n 1 = n 2 = 3 and l 1 = l 2 = 1, but in this case, we have m l = 0, ±1 and m s
= ±
1
2 available to us as choices to be made. Thus, each of the two 3p-electrons in
principle has six possible pairs of values (m l , m s ) available to it; however, the Pauli
Principle in this case reduces this choice from six to five for the second 3p-electron
in order that not all four of the quantum numbers associated with an individual
electron be the same. This means that instead of 36 possible quantum pair-states
for the two 3p electrons, there will only be 30; however, we are not finished yet,
because the two p-electrons are indistinguishable so that we must now halve the
number of possible pair-states from 30 to 15. All 15 of these pair-states will have
the same (configurational) energy at this level of description. In other words, the
ground electronic configuration of the Si atom is 15-fold degenerate in the centralfield approximation.
At the next stage of description, we shall consider the effect of including the
inter-electron repulsion terms in the Hamiltonian. The most important consequence
arising from taking these repulsions into account is that the orbital angular momenta
of the valence electrons are coupled, so that if l i represents the orbital angular
momentum (vector) for electron i, then the only orbital angular momentum that is
valid for a multi-electron atom at this level of description is the total orbital angular
momentum given by the vector sum L =
i l i for all electrons in the atom. We
have saved some effort, however, in that the contribution from the electrons of the
Ne ‘core’ for the configuration gives a net orbital angular momentum of zero, as do
the two electrons in the 3s sub-shell, with the consequence that we have to consider
in detail only those contributions arising from the 3p valence electrons of the Si
atom. Two vectors l 1 and l 2 , each of ‘length’ 1, can be added vectorially to give
resultant vectors L of ‘length’ 0, 1, and 2, corresponding to addition of the vectors
anticollinearly, at an angle of 60 ◦ , and collinearly.
225
the Ne atom. We also know from introductory quantum mechanics that electronic
configurations can be associated with what is called the independent-electron model
and described in terms of a central-field Hamiltonian in which an individual electron
moves under the influence of a central potential energy function that is due to the
net motion of all other electrons about the positively charged nucleus. At this stage,
all inter-electron repulsions are ignored.
We shall consider the (3s) 2 component of the [Ne](3s) 2 (3p) 2 electronic configuration of the Si atom first. According to the Pauli (Exclusion) Principle for electrons,
no two electrons in a given atom may have the same set of values for all four of the
quantum numbers, n, l, m l , and m s . Both (3s) 2 electrons have principal quantum
numbers n = 3, azimuthal quantum numbers l = 0, and magnetic quantum numbers
m l = 0, with the consequence that the Pauli Principle therefore now requires that
they have different values for the spin quantum number m s . We also know that
electrons have ‘spin’ one-half, so that the quantum number m s may have only one
of the two values ±
1
2 . The two 3s electrons are thus said to be ‘paired’. This pair of
electrons can be represented as a pair-state (or outer product state) for which the total
spin S is zero: the associated magnetic spin quantum number is designated as M S
and also necessarily has the value zero. Let us now consider the two 3p electrons:
we have n 1 = n 2 = 3 and l 1 = l 2 = 1, but in this case, we have m l = 0, ±1 and m s
= ±
1
2 available to us as choices to be made. Thus, each of the two 3p-electrons in
principle has six possible pairs of values (m l , m s ) available to it; however, the Pauli
Principle in this case reduces this choice from six to five for the second 3p-electron
in order that not all four of the quantum numbers associated with an individual
electron be the same. This means that instead of 36 possible quantum pair-states
for the two 3p electrons, there will only be 30; however, we are not finished yet,
because the two p-electrons are indistinguishable so that we must now halve the
number of possible pair-states from 30 to 15. All 15 of these pair-states will have
the same (configurational) energy at this level of description. In other words, the
ground electronic configuration of the Si atom is 15-fold degenerate in the centralfield approximation.
At the next stage of description, we shall consider the effect of including the
inter-electron repulsion terms in the Hamiltonian. The most important consequence
arising from taking these repulsions into account is that the orbital angular momenta
of the valence electrons are coupled, so that if l i represents the orbital angular
momentum (vector) for electron i, then the only orbital angular momentum that is
valid for a multi-electron atom at this level of description is the total orbital angular
momentum given by the vector sum L =
i l i for all electrons in the atom. We
have saved some effort, however, in that the contribution from the electrons of the
Ne ‘core’ for the configuration gives a net orbital angular momentum of zero, as do
the two electrons in the 3s sub-shell, with the consequence that we have to consider
in detail only those contributions arising from the 3p valence electrons of the Si
atom. Two vectors l 1 and l 2 , each of ‘length’ 1, can be added vectorially to give
resultant vectors L of ‘length’ 0, 1, and 2, corresponding to addition of the vectors
anticollinearly, at an angle of 60 ◦ , and collinearly.