Characters and operations
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص417-418
2025-12-02
61
Characters and operations
The characters in the rows labelled A and B and, in the columns, headed by symmetry operations other than the identity E indicate the behaviour of an orbital under the corresponding operations: a +1 indicates that an orbital is unchanged, and a −1 indicates that it changes sign. It follows that we can identify the symmetry label of the orbital by comparing the changes that occur to an orbital under each operation, and then comparing the resulting +1 or −1 with the entries in a row of the character table for the point group concerned. For the rows labelled E or T (which refer to the behaviour of sets of doubly and triply degenerate orbitals, respectively), the characters in a row of the table are the sums of the characters summarizing the behaviour of the individual orbitals in the basis. Thus, if one member of a doubly degenerate pair remains unchanged under a symmetry operation but the other changes sign (Fig. 12.20), then the entry is reported as χ=1−1=0. Care must be exercised with these characters because the transformations of orbitals can be quite complicated; nevertheless, the sums of the individual characters are integers. As an example, consider the O2px orbital in H2O. Because H2O belongs to the point group C2v, we know by referring to the C2v character table (Table 12.2) that the labels available for the orbitals are a1, a2, b1, and b2. We can decide the appropriate label for O2px by noting that under a 180° rotation (C2) the orbital changes sign (Fig. 12.21), so it must be either B1 or B2, as only these two symmetry types have character −1 under C2. The O2px orbital also changes sign under the reflection σv ′, which identifies it as B1. As we shall see, any molecular orbital built from this atomic orbital will also be a b1 orbital. Similarly, O2py changes sign under C2 but not under σv ′; therefore, it can contribute to b2 orbitals. The behaviour of s, p, and d orbitals on a central atom under the symmetry operations of the molecule is so important that the symmetry species of these orbitals are generally indicated in a character table. To make these allocations, we look at the symmetry species of x, y, and z, which appear on the right-hand side of the character table. Thus, the position of z in Table 12.3 shows that pz (which is proportional to zf (r)), has symmetry species A1 in C3v, whereas px and py (which are proportional to xf (r) and yf(r), respectively) are jointly of E symmetry. In technical terms, we say that px and py jointly span an irreducible representation of symmetry species E. An sorbital on the central atom always spans the fully symmetrical irreducible representation (typically labelled A1 but sometimes A1 ′) of a group as it is unchanged under all symmetry operations.
The five d orbitals of a shell are represented by xy for dxy, etc, and are also listed on the right of the character table. We can see at a glance that in C3v, dxy and dx2 −y2 on a central atom jointly belong to E and hence form a doubly degenerate pair.
Fig. 12.20 The two orbitals shown here have different properties under reflection through the mirror plane: one changes sign (character −1), the other does not (character +1).
Fig. 12.21 A px orbital on the central atom of a C2v molecule and the symmetry elements of the group.
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