632
D Atomic and Molecular Term Symbols
associated with the total electronic spin angular momentum quantum number S, and
a subscript at the lower right of the letter designating the value of the total electronic
angular momentum quantum number J associated with the total electronic angular
momentum (vector) operator J = L + S (in the simplest angular momentum
coupling scheme, referred to as Russell–Saunders coupling). The unique atomic
term symbol associated with the ground electronic configuration (1s) 2 is thus 1 S 0 .
The simplest excited electronic configurations of the He atom are those in
which one electron remains in the 1s (hydrogenic) atomic orbital and the second
He electron is in an (excited) higher-energy orbital. The lowest such excited
configuration is the (1s) 1 (2s) 1 configuration, for which n 1 = 1 and n 2 = 2, so
that the Pauli Principle is already satisfied for this configuration. The orbital angular
momentum quantum numbers are 1 = 2 = 0 (because both electrons lie in sorbitals), so that L = 0 only is allowed, as for the ground electronic configuration.
Because 1 and 2 are both 0, the quantum numbers m 1 and m 2 are both 0 also.
However, as the Pauli Principle has already been satisfied, the m s -values for the
two electrons are no longer required to have opposite signs, and there will be four
possible pair combinations
m s 1 , m s 2
, namely
+
1
2 , +
1
2
,
+
1
2 , −
1
2
,
−
1
2 , +
1
2
,
−
1
2 , −
1
2
. Two of these four pairs of m s 1 , m s 2 -values clearly correspond to a total
spin angular momentum quantum number S = 1 (namely, the pairs for which
M S = m s 1 + m s 2 = 1, −1): the other two pairs correspond to M S = 0, and must
somehow be associated with the S = 1, M S = 0 and S = 0, M S = 0 states.
A more general version of the Pauli Principle for electrons states that the total
wavefunction for a system of electrons must be antisymmetric to the interchange
of any pair of electrons. This version of the Pauli Principle allows us to determine
that the S = 1, M S = 0 wavefunction must then be the sum of the wavefunctions
corresponding to the primitive pairs (m s 1 , m s 2 ) =
+
1
2 , −
1
2
and
−
1
2 , +
1
2
, while
the S = 0, M S = 0 wavefunction must be the difference between the wavefunctions
for these two primitive pairs, and the four linearly independent spin states for
the (1s) 1 (2s) 1 He configuration then correspond to a symmetric triplet of spin
pair-states and an antisymmetric singlet spin pair-state. L = 0 and S = 1 give
J = L + S = 1, while L = 0, S = 0 gives J = L + S = 0, so that we have two
electronic terms, 3 S 1 and 1 S 0 , associated with the (1s) 1 (2s) 1 He configuration.
Similar arguments applied to the more highly excited configurations give rise by
the same arguments that we have utilized for the (1s) 1 (2s) 1 configuration to pairs of
terms ( 3 S 1 , 1 S 0 ) for other (1s) 1 (ns) 1 configurations, ( 3 P 2,1,0 , 1 P 1 ) for (1s) 1 (np) 1
configurations, ( 3 D 3,2,1 , 1 D 2 ) for (1s) 1 (nd) 1 configurations, and so on. The set of
energies thereby obtained can be displayed in an energy level diagram referred to as
a Grotrian diagram: a typical diagram of this type for the He atom can be found, for
example, on p. 65 of G. Herzberg, Atomic Spectra and Atomic Structure (Dover,
New York, 1944). The Herzberg figure also includes the set of electric dipoleallowed spectroscopic transitions, with selection rules S = 0, L = ±1: the
electronic spin selection rule S = 0 means that triplet and singlet terms cannot
interconvert by absorption or emission of photons, and hence the singlet and triplet
D Atomic and Molecular Term Symbols
associated with the total electronic spin angular momentum quantum number S, and
a subscript at the lower right of the letter designating the value of the total electronic
angular momentum quantum number J associated with the total electronic angular
momentum (vector) operator J = L + S (in the simplest angular momentum
coupling scheme, referred to as Russell–Saunders coupling). The unique atomic
term symbol associated with the ground electronic configuration (1s) 2 is thus 1 S 0 .
The simplest excited electronic configurations of the He atom are those in
which one electron remains in the 1s (hydrogenic) atomic orbital and the second
He electron is in an (excited) higher-energy orbital. The lowest such excited
configuration is the (1s) 1 (2s) 1 configuration, for which n 1 = 1 and n 2 = 2, so
that the Pauli Principle is already satisfied for this configuration. The orbital angular
momentum quantum numbers are 1 = 2 = 0 (because both electrons lie in sorbitals), so that L = 0 only is allowed, as for the ground electronic configuration.
Because 1 and 2 are both 0, the quantum numbers m 1 and m 2 are both 0 also.
However, as the Pauli Principle has already been satisfied, the m s -values for the
two electrons are no longer required to have opposite signs, and there will be four
possible pair combinations
m s 1 , m s 2
, namely
+
1
2 , +
1
2
,
+
1
2 , −
1
2
,
−
1
2 , +
1
2
,
−
1
2 , −
1
2
. Two of these four pairs of m s 1 , m s 2 -values clearly correspond to a total
spin angular momentum quantum number S = 1 (namely, the pairs for which
M S = m s 1 + m s 2 = 1, −1): the other two pairs correspond to M S = 0, and must
somehow be associated with the S = 1, M S = 0 and S = 0, M S = 0 states.
A more general version of the Pauli Principle for electrons states that the total
wavefunction for a system of electrons must be antisymmetric to the interchange
of any pair of electrons. This version of the Pauli Principle allows us to determine
that the S = 1, M S = 0 wavefunction must then be the sum of the wavefunctions
corresponding to the primitive pairs (m s 1 , m s 2 ) =
+
1
2 , −
1
2
and
−
1
2 , +
1
2
, while
the S = 0, M S = 0 wavefunction must be the difference between the wavefunctions
for these two primitive pairs, and the four linearly independent spin states for
the (1s) 1 (2s) 1 He configuration then correspond to a symmetric triplet of spin
pair-states and an antisymmetric singlet spin pair-state. L = 0 and S = 1 give
J = L + S = 1, while L = 0, S = 0 gives J = L + S = 0, so that we have two
electronic terms, 3 S 1 and 1 S 0 , associated with the (1s) 1 (2s) 1 He configuration.
Similar arguments applied to the more highly excited configurations give rise by
the same arguments that we have utilized for the (1s) 1 (2s) 1 configuration to pairs of
terms ( 3 S 1 , 1 S 0 ) for other (1s) 1 (ns) 1 configurations, ( 3 P 2,1,0 , 1 P 1 ) for (1s) 1 (np) 1
configurations, ( 3 D 3,2,1 , 1 D 2 ) for (1s) 1 (nd) 1 configurations, and so on. The set of
energies thereby obtained can be displayed in an energy level diagram referred to as
a Grotrian diagram: a typical diagram of this type for the He atom can be found, for
example, on p. 65 of G. Herzberg, Atomic Spectra and Atomic Structure (Dover,
New York, 1944). The Herzberg figure also includes the set of electric dipoleallowed spectroscopic transitions, with selection rules S = 0, L = ±1: the
electronic spin selection rule S = 0 means that triplet and singlet terms cannot
interconvert by absorption or emission of photons, and hence the singlet and triplet