We have pointed out that the hydrogen-like wave functions have trigonometric factors which correspond to various projections of a vector and that such projec
tions if carried through the symmetry operations of a group, generate a represen
tation. We may also take a set of vectors through the symmetry operations of a group, thus generating matrices which will again be representations, usually reducible ones. The detailing of such matrices is lengthy and, fortunately, it turns out that a simple rule allows the determination of the trace of the matrix for each symmetry operation.
The rule is that any vector of the set which remains unchanged by the symmetry operation contributes unity to the trace. A vector which is unchanged in position but reversed in sign contributes — 1 . These rules will always work for the cases considered here, but it should be mentioned that they have been simplified. If, for example, a set of vectors is rotated, each into the other's position, by a sym
metry operation, then it may be necessary to assign appropriate fractional numbers in adding up the trace of the matrix.
A. Sigma Bonds
The term sigma bond is conventionally used to describe a type of bond between two atoms which results from the overlap of an atomic orbital of each atom. The It should be mentioned that several complexities have been side-stepped in the presentation of symmetry groups and their character tables. As a consequence a number of types have been omitted from Table 17-7 as involving entries which there is insufficient space to explain. The omissions are not of vital importance here, however; the omitted aspects have more the nature of complications than of new principles.
Returning to the first column, the naming scheme is one developed by R. Mulliken. Irreducible representations of a group may be one-, two- or three-dimensional (we will not encounter any of higher dimension), and may also possess various symmetry properties themselves.
One-dimensional representations are named A if they are symmetric to rotation by 2π/η about the principal or Cn axis, and Β if antisymmetric to such rotation.
They have subscripts 1 or 2 to designate whether they are symmetric or anti
symmetric to rotation about a secondary C2 axis, or if this is missing, to a vertical plane of symmetry.
For example, the Ax representation in C2V [Table 17-5] is generated by a vector on the ζ axis, and such a vector is unchanged by C2 and σν . The A2 representation is generated by a circular vector in the xy plane, centered at the origin; it is unchanged by rotation but inverted on reflection by σν . The Βλ representation is like a vector on the JC axis; it is inverted by C2 but unchanged by σν (taken to be the xz plane), and B2 is like a vector on the y axis and is inverted by both C2 and σν .
In more complicated symmetry groups primes and double primes are attached to all letters to indicate symmetry or antisymmetry with respect to ah . Also the subscripts g and u (from the German gerade and ungerade) may be present to show whether the representation is symmetric or antisymmetric with respect to inversion.
17-7 BONDS AS BASES FOR REDUCIBLE REPRESENTATIONS 745
further requirement is that the orbitals and hence their region of overlap be a figure of revolution about the line between the two nuclei. As illustrated in Fig. 1 7 - 1 1 , two s orbitals can form a sigma bond, and if the atoms A and Β lie on the χ axis, then the sigma-bonding combination SA P ^ B is possible, as well as Ρ^,ΑΡΖ,Β >
The orbital pictures shown in the figure are just the polar plots of the angular portions—these must be multiplied by the radial function R(r) to give the corre
sponding electron density. As two atoms are brought together, the product of the electron amplitudes determines what is called the overlap; the mathematical operation is the evaluation of the i n t e g r a ld r for the atoms at some given distance of separation, where dr is the element of volume (note Section 16-10).
This integral normally provides a guide to the degree of bonding that is expected.
For two atoms to bond, it is first necessary that their respective orbitals overlap, that is, have appreciable values in some common region of space. There must also be a reinforcement of the two orbital wave functions; their phase relation must be suitable. First, it is conventional to indicate the phase relationships between the lobes of the angular portion of a given orbital by plus and minus signs. These signs are usually chosen so that if the orbitals of two atoms overlap in regions for which their sign is the same, then a bonding interaction is indicated
—the potential energy of the atoms close together should be less than when sepa
rated. Conversely, if the overlapping orbitals are of opposite sign, the potential energy of the system is higher than for the separated atoms. The two situations are referred to as bonding and antibonding, respectively.
It may happen alternatively that the overlap integral is small or zero either because the overlap is small or because positive and negative domains of the integral cancel. The usual consequence is that neither bonding nor repulsion is expected and the situation is called one of nonbonding. The three cases are illus
trated in Fig. 17-11.
H i N - N H ,
H » N
-Co3+ /
\ /
- N H3
t
H '
F I G . 1 7 - 1 2 . Sigma bond vectors.
It should be emphasized that the overlap integral does not in itself give the degree of bonding; as discussed in Section 18-2, the actual integrals involve the Hamiltonian of the system. However, the preceding qualitative use of the overlap idea will usually work—it is a convenient quantum mechanical rule of thumb.
By using group theory, it is possible to determine which pairs of wave functions can give a nonzero overlap integral. The approach is the following. The symmetry properties of the set of sigma bonds form a representation of the symmetry group of the molecule and one which will contain various irreducible representations of the group, as determined by the use of Eq. (17-16). The wave functions which are bases for these irreducible representations will have the necessary symmetry properties for the overlap integral to be nonzero and either positive or negative, depending on the signs given to the orbital lobes.
As illustrated in Fig. 17-12, we draw a set of vectors corresponding to the sigma bonds of the molecule. The trace of the matrix generated by each symmetry operation is then determined by applying the rule described at the beginning of this section. Equation (17-16) gives the irreducible representations contained by, or "spanned" by, the reducible one, and consultation of the character table for the group determines which hydrogen-like orbitals have the right symmetry for bonding.
The following examples should help to clarify matters. The planar B F3 molecule has Z)3h symmetry. Consultation of Table 17-7 gives the symmetry classes as E, 2 C3, 3 C2, oh , 2S3, and 3 σν. The operation Ε leaves all three bond vectors unchanged, C3 changes all of them, C2 leaves one unchanged, oh leaves all unchanged, S3 changes all, and σν leaves one unchanged. The character for the reducible representation Γ3σ generated by the set of bonds is therefore
Ε 2 C3 3C2 oh 2*S3 3 σν
Γ3σ 3 0 1 3 0 1
17-7 BONDS AS BASES FOR REDUCIBLE REPRESENTATIONS 747
Γ4σ 4 1 0 0 2
Application of Eq. (17-16) gives Γ4σ = Ax + T2. The IR A1 corresponds to x2 + y2 + z2, or to a sphere, and hence to an s orbital. The IR T2 is generated by (x, y, z) or (xy, xz, yz). The bonding can thus be spxp^pz or sàxyàxzayz ; it can be some combination of both types if the ρ and d orbitals are similar in energy.
The first choice is obvious in the case of C H4 and is the set given in Section 16-10 in the discussion of hybrid orbitals. The tetrahedral molecule CoCl^- might, however, use the s d ^ d ^ d ^ combination, in view of the availability of cobalt d orbitals. Decisions of this type rest on quantitative calculations.
For a molecule having Oh symmetry, such as C o ( N H3) e+ (considering only the Co and Ν atoms) the result is
E 8 C3 6 C2 6 C 4 3 C2 / 6.S4 8 f S6 3 σ ΐ ι 6σα
Γ6σ 6 0 0 2 2 0 0 0 4 2
We now find Γβσ = Alg + Tlu + Eg , which corresponds to the orbital set sd^dtf-.yzPxPypz (often reported as just d2sp3). A linear combination of these orbitals will indeed generate a set of new orbitals having lobes pointing to the corners of an octahedron.
The Αι irreducible representation is thus contained once. On repeating the calcula
tion for each irreducible representation of the Z>3h group in turn, it is found that the coefficient is zero for all except E', for which it is again unity. Thus JH3 cT spans the Αλ' + E' irreducible representations. Consultation of the last two columns of the character table shows that the algebraic functions of the correct symmetry are (x2 + y2) or z2 for AÎ and (x, y) or (x2 — y2, xy) for E'. The E' irreducible representation is a two-dimensional one and the functions must be taken together.
The function (x2 + y2) corresponds to a circle, and since the molecule is planar, this is the projection of an s orbital. Thus a combination s, p x, p y would do for bonding—this is just the hybrid bond combination discussed in Section 16-10.
Another possibility is s, dX2_y2, dxy . The choice between these two combinations is made on the basis of whether it is the ρ or d orbitals that are the more stable.
This is easy in the case of B F3 since the outer electrons of boron occupy 2p orbitals, and the lowest d ones, 3d, would be very high in energy. In a less extreme case it may be appropriate to use a linear combination of the two sets of orbitals.
Notice that in this approach the fluorine atoms are really irrelevant. We are determining what orbital combinations for the central atom will have the proper symmetry to form sigma bonds in the observed bonding directions.
A second example is C H4 (or any tetrahedral molecule), for which the point group is Ta . Turning to the character table for this group, we find
Ε 8 C3 3C2 6.S4 6σα
with the conclusion that Γ2σ = Ax + B1. From the character table the bonding might involve some combination of s, px, and pz. This answer does not seem very helpful, that is, it does not suggest why the experimental bond angle in water is 104°27'. It should be recognized at this point that the oxygen atom has two additional pairs of electrons; these are not used in bonding in HaO but are used, for example, in the hydrogen-bonded structures present in liquid water and in ice (see Section 8-CN-2). Since the treatment centers on the oxygen atom it is more realistic to view it as having four more or less tetrahedrally disposed pairs of electrons, of which two happen to be shared in a bond. The point is that the treatment described here works best for coordinatively saturated atoms; a useful alternative approach for an A B2 type of molecule is given in Section 18-6.
β . Pi Bonds
The term pi bond applies to a bonding overlap where there is a node along the bonding axis, that is, the electron density is zero along the line of the sigma bond.
Figure 17-13 illustrates some typical pi bonding orbital combinations, assuming the bond to lie on the χ axis. The figures suggest, and calculation confirms, that the value of a pi bonding overlap integral is in general smaller than that of a sigma bonding one (see Section 18-2). As a consequence, pi bonding is considered not so much as providing the primary bonding holding a molecule together as supplementing an already present sigma bond.
The symmetry approach may again be used. As shown in Fig. 17-14, pi bonds must be represented by vectors since there is a plus-to-minus direction. Usually, two such vectors at right angles to each other must be considered. Various sym
metry operations of the molecular point group may now leave pi bond vectors unchanged in position but reversed in sign; as noted at the beginning of this
&yz dyz
F I G . 1 7 - 1 3 . Arrangements of orbitals showing pi bonding for the case of an A —Β molecule.
17-7 BONDS AS BASES FOR REDUCIBLE REPRESENTATIONS 749
F 1 G . 1 7 - 1 4 . Pi bonding vectors.
section, the rule is then that —1 is contributed to the trace of the representation generated by the pi bonding set. Also, one may consider each set of mutually perpendicular vectors separately, provided that the two sets are not mixed by any symmetry operation.
For example, a D3h molecule such as B F3 or N 03~ generates the traces shown in Table 17-8, where Γ3πη and Γ37Τ± denote the sets of pi bonding vectors parallel to and perpendicular to the molecular plane, respectively. Application of Eq. (17-16) leads to the result
Γ3π± = A2 + E\ Γ3 π || = Λ2' + E\
which gives p ^ as suitable bonding orbitals. We exclude dxz, dyz, listed for E\
as energetically unavailable, and the other irreducible representations do not correspond to any hydrogen-like orbitals. However, p^p^ is also suitable for sigma bonding, and we assume that this type of overlap will take precedence over the pi bonding type. It appears that molecules such as B F3 have mainly sigma bonding in the molecular plane. However, pz belongs to A2 so π± bonding is possible.
T A B L E 1 7 - 8 .
Ε 2C3 3C2 2S3 3σν
•^STTJL 3 0 - 1 - 3 0 1
-^WII 3 0 - 1 3 0 - 1
T A B L E 1 7 - 9 .
Turning next to theZ>4h molecule PtCl^
Application of Eq. (17-16) yields
we find the situation shown in Table 17-9.
= ^lg + ^lg + or dz2 , dx*_yt, (pxpy), 4 77 II — ^2g + ^2g + or ^xy 5 (PœPl/) ?
- Γ| 7 Γ ± = v42u + i ?2u + or
The sigma bonds are thus dsp2 in type, leaving the di5 set and pz available for some pi bonding. We assume in the latter case that the ligands are suitably disposed, that is, have available electrons in pi bonding orbitals to enter empty metal orbitals, or vice versa. Since Pt(II) has eight d electrons of its own and chloride ion has a full octet of electrons, neither the ligands nor the central metal ion has empty pi-type orbitals to accommodate the other's electrons. The conclusion, in the case of PtCl^", is that pi bonding is unimportant. Cyanide ion, however, does have some empty pi bonding type orbitals, and the P t ( C N )2" complex is considered to have both sigma and pi bonding.
Finally, and in abbreviated fashion, the situation for the combined sets of pi bonds of an Oh complex is
Ε C3 6 C2' 6 C4 3C2 / 6*S4 SSG 3ah 6 ad
Γ12π 12 0 0 0 - 4 0 0 0 0 0
This corresponds to Tlg + T2g + Tlu+ T2u or ( d ^ d ^ d ^ ) , ( p^ P z ) . There are only six appropriate orbitals, of which the set (pxPypz) is preempted by sigma bonding. We conclude that only the set d{j is available for pi bonding in an Oh complex. The actual degree of such bonding will depend on the particular central metal ion and the ligands. The ion Cr(III) has only three d electrons and could accept or donate pi bonding electron density from or to ligand pi-type orbitals. If the ligand is N H3, as in C r ( N H3) e+, the nitrogen is surrounded by hydrogen atoms, so no pi bonding at all is possible. Ligands such as CI" might donate some pi bonding electrons to Cr(III) and ligands such as C N_ might accept some.
These examples illustrate how symmetry plus ancillary considerations allow the physical chemist to draw qualitative conclusions as to the bonding in coordina
tion compounds. The treatment of pi bonding in organic molecules generally involves multicenter situations, as in C H2= C H2 or benzene, and these are usually treated by a molecular orbital approach (see Chapter 18).
COMMENTARY AND NOTES, SECTION 1 751