D.4 Term Symbols for Nonlinear Molecules
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multi-electron molecule. This symmetry label is then identified by the appropriate
upper case Roman letter. One-dimensional irreducible representations are commonly labelled by symmetry types A or B (sometimes, though rarely, C or D), while
two-dimensional irreducible representations are typically labelled by E, and threedimensional ones by T . As there may be, and often is, more than one independent
irreducible representation of a particular symmetry type, the various occurrences
may be distinguished by subscripts 1, 2, . . . , as well as by primed and unprimed
letters.
The total electronic spin for the ground term in most nonlinear polyatomic
molecules is most commonly zero (all spins paired). Occasionally, doublets or even
triplets (relatively uncommon) also occur, corresponding to unpaired spins in the
molecules. Electronic term symbols like 2 A 1 , 3 B 1g , 3 T 1u , 2 E , and so on can be
found: everything but the spin multiplicity (in the upper left-hand corner of the
term symbol) is related to point-group symmetry information on the irreducible
representation.
Two Examples The H 2 O and NH 2 molecules both have the same symmetry
point group, C 2v , with four one-dimensional irreducible representations,
{A 1 , A 2 , B 1 , B 2 }. They also have very similar ground state electronic configurations, namely,
H 2 O : (1a 1 )
2 (a 1 )
2 (1b 2 )
2 (3a 1 )
2 (1b 1 )
2 ,
NH 2 : (1a 1 )
2 (a 1 )
2 (1b 2 )
2 (3a 1 )
2 (1b 1 )
1 .
Note that, as for atoms and linear molecules, the orbitals are designated by lower
case Roman letters; the numbers 1, 2, 3, preceding the letters identify the number
of independent orbitals having the same symmetry type (basically the same role as
played by the principal quantum number for atoms and linear molecules). A configuration in general represents a reducible representation, and we wish to know which
irreducible representations of the point symmetry group it contains. One method
which we can use is the method of outer products within a group. As we are working
with a group that has only one-dimensional irreducible representations, we do not
need to worry too much about the formalities: all that we really need to know for the
moment is that the product of two one-dimensional irreducible representations in the
same group is again a one-dimensional irreducible representation of that group, that
the product of any one-dimensional irreducible representation with itself is always
the totally symmetric a 1 or equivalent irreducible representation in that symmetry
point group, and that the product of the a 1 representation with any other irreducible
representation gives that representation (which is another way of saying that the
a 1 representation is the identity for the outer product process). This means, for
example, that in every case in which there are two electrons in a given orbital (i.e.,
a filled orbital), it will behave as the a 1 representation. For the H 2 O molecule, this
clearly leads to an overall a 1 irreducible representation within the C 2v symmetry
point group: we label this as an A 1 term according to the spectroscopic convention
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multi-electron molecule. This symmetry label is then identified by the appropriate
upper case Roman letter. One-dimensional irreducible representations are commonly labelled by symmetry types A or B (sometimes, though rarely, C or D), while
two-dimensional irreducible representations are typically labelled by E, and threedimensional ones by T . As there may be, and often is, more than one independent
irreducible representation of a particular symmetry type, the various occurrences
may be distinguished by subscripts 1, 2, . . . , as well as by primed and unprimed
letters.
The total electronic spin for the ground term in most nonlinear polyatomic
molecules is most commonly zero (all spins paired). Occasionally, doublets or even
triplets (relatively uncommon) also occur, corresponding to unpaired spins in the
molecules. Electronic term symbols like 2 A 1 , 3 B 1g , 3 T 1u , 2 E , and so on can be
found: everything but the spin multiplicity (in the upper left-hand corner of the
term symbol) is related to point-group symmetry information on the irreducible
representation.
Two Examples The H 2 O and NH 2 molecules both have the same symmetry
point group, C 2v , with four one-dimensional irreducible representations,
{A 1 , A 2 , B 1 , B 2 }. They also have very similar ground state electronic configurations, namely,
H 2 O : (1a 1 )
2 (a 1 )
2 (1b 2 )
2 (3a 1 )
2 (1b 1 )
2 ,
NH 2 : (1a 1 )
2 (a 1 )
2 (1b 2 )
2 (3a 1 )
2 (1b 1 )
1 .
Note that, as for atoms and linear molecules, the orbitals are designated by lower
case Roman letters; the numbers 1, 2, 3, preceding the letters identify the number
of independent orbitals having the same symmetry type (basically the same role as
played by the principal quantum number for atoms and linear molecules). A configuration in general represents a reducible representation, and we wish to know which
irreducible representations of the point symmetry group it contains. One method
which we can use is the method of outer products within a group. As we are working
with a group that has only one-dimensional irreducible representations, we do not
need to worry too much about the formalities: all that we really need to know for the
moment is that the product of two one-dimensional irreducible representations in the
same group is again a one-dimensional irreducible representation of that group, that
the product of any one-dimensional irreducible representation with itself is always
the totally symmetric a 1 or equivalent irreducible representation in that symmetry
point group, and that the product of the a 1 representation with any other irreducible
representation gives that representation (which is another way of saying that the
a 1 representation is the identity for the outer product process). This means, for
example, that in every case in which there are two electrons in a given orbital (i.e.,
a filled orbital), it will behave as the a 1 representation. For the H 2 O molecule, this
clearly leads to an overall a 1 irreducible representation within the C 2v symmetry
point group: we label this as an A 1 term according to the spectroscopic convention