Chapter 4 Answers to Problems

Chapter 4 Answers to Problems 4.1 The reducible representation, symmetry-allowed AOs, and possible combinations are listed for each part. (a) trigonal planar D3h E 2C3 3C2 Fh 2S3 3Fv ' 3 0 1 3 0 1 ' = A1' + E' A1': s, dz2 E': (px, py), (dxy, dx2-y2) A1' E' Notation s (px, py) sp2 s (dxy, dx2-y2) sd2 dz2 (px, py) p2d / dp2 dz2 (dxy, dx2-y2) d3 (b) square planar D4h E 2C4 C2 2C2' 2C2" i 2S4 Fh 2Fv 2Fd ' 4 0 0 2 0 0 0 4 2 0 ' = A1g + B1g + Eu A1g: s, dz2 B1g: dx2-y2 Eu: (px, py) A1g B1g Eu Notation s dx2-y2 (px, py) dsp2 dz2 dx2-y2 (px, py) d2p2 (c) trigonal bipyramid D3h E 2C3 3C2 Fh 2S3 3Fv ' 5 2 1 3 0 3 ' = 2A1' + A2" + E' A1': s, dz2 A2": pz E': (px, py), (dxy, dx2-y2) 2A1' A2" E' Notation s, dz2 pz (px, py), dsp3 s, dz2 pz (dxy, dx2-y2) d3sp (d) octahedral Oh E 8C3 6C2 6C4 3C2 i 6S4 8S6 3Fh 6Fd ' 6 0 0 2 2 0 0 0 4 2 ' = A1g + Eg + T1u A1g: s Eg: (dz2, dx2-y2) T1u: (px, py, pz) Only possibility is d2sp3. 4.2 The indicated distortion takes the symmetry down to D2d. D2d E 2S4 C2 2C2' 2Fd ' 4 0 0 0 2 ' = A1 + B2 + E A1: s, dz2 B2: pz, dxy E: (px, py), (dxz, dyz) A1 B2 E Notation s pz (px, py) sp3 s pz (dxz, dyz) spd2 / d2sp s dxy (px, py) sp2d / dsp2 s dxy (dxz, dyz) sd3 dz2 pz (px, py) dp3 / p3d dz2 pz (dxz, dyz) pd3 / d3p dz2 dxy (px, py) p2d2 / d2p2 dz2 dxy (dxz, dyz) d4 As with the perfect tetrahedron (Td), the sp3 and sd3 are still possible hybrids, although the actual wave functions would differ, because the contributions of individual p or d orbitals would depend upon the angles in the distorted tetrahedron. The lowered symmetry admits the possibility of the other six combinations, but in most cases these would be less likely contributors to bonding than the modified sp3 and sd3 hybrids. 4.3 (a) Taking the three hydrogen 1s orbitals as a basis for a representation of SALCs: D3h E 2C3 3C2 Fh 2S3 3Fv ' 3 0 1 3 0 1 ' = A1' + E' The equations for the corresponding SALCs are: A1': M1 = 1//3(Ra + Rb + Rc) E': M2 = 1//6(2Ra – Rb – Rc) M3 = 1//2(Rb – Rc) Symmetries of the AOs of B are s = A1', (px, py) = E', and pz = A2". The pz orbital has no match in SALCs and must remain nonbonding. The following MO scheme results: (b) Disregarding ionizations from the Fn (a1') core level, which would probably fall outside the range of u.v.-P.E.S., the spectrum should consist of two bands, both with vibrational fine structure. The lower-energy band, from the doubly degenerate F (e') level, should be roughly twice as big as the higher-energy band, from the nondegenerate F (a1') level. (c) All models lead to the same overall electron distribution. The VB model is based on sp2 hybridized boron AOs forming 2c-2e bonds with hydrogen 1s orbitals. The localized MO description would place all three bonding pairs in equivalent localized sigma-bonding MOs. The suggestion of energetic equivalence of all three pairs is at odds with the symmetry restrictions of the system. 4.4 (a) Taking the two hydrogens as a basis for the SALCs we obtain the following reducible representation: C2v E C2 Fv Fv N 'H 2 0 0 2 for which 'H = A1 + B2. The equations for the SALCs are Oxygen AO symmetries are s = A1 px = B1 py = B2 pz = A1 The px AO (B1) has no matching SALC and must be nonbonding. The py orbital can form bonding and antibonding combinations with the B2 SALC. Both s and pz orbitals on oxygen match with the A1 SALC, so s-p mixing can be expected. If we formed bonding and antibonding combinations for both of these, we would end up with more MOs in the final scheme (seven) than there are available AOs on the component atoms (six). To avoid this, we can only make three MOs from the A1 AOs and SALC. For simplicity, we will assume that the s and pz orbitals on oxygen both form bonding MOs, and together they form one s-p mixed antibonding orbital. Representations of the LCAOs are shown below. The qualitative MO scheme is shown below, in which hashed lines indicate lesser contributions arising from s-p mixing. (b) The P.E.S. results are consistent with the MO scheme. Only the Bn(x) MO is strictly nonbonding, while the F(z) MO is weakly bonding, as indicated by the vibrational fine structure on its P.E.S. band. (c) Rather than two lone pairs in approximately sp3 hybrids, the MO scheme suggests a single region of electron density protruding from the back side of the molecule. This is created by the combination of the nonbonding Bn(x) MO and the weakly bonding F(z) MO, whose LCAOs are shown above. (d) In a linear molecule like BeH2, the degenerate MOs that are essentially the two p orbitals perpendicular to the molecular axis must be nonbonding (the px and py orbitals, designated Au in D4h). If the molecule acquires an additional pair of electrons, there could be a bonding combination between one of these p orbitals and the totally symmetric SALC, if the molecule bends to allow overlap between one lobe of the p orbital with this SALC. The result is the F(z) a1 bonding MO. (Note: By the conventions of assigning axis labels, the z axis of the linear case becomes the y axis in the bent case, and vice versa.) The energy lowering accompanying this additional bonding can be seen as the driving force for the molecule to assume a bent geometry, as in H2O. The energetics of the changing MO schemes on descent from D4h to C2v for MX2 molecules can be represented by a Walsh diagram [cf. R. L. DeKock and W. B. Bosna, J. Chem. Educ. 1988, 65, 194-197.] 4.5 The approach to NH3 is similar to that for H2O (see 4.4). C3v E 2C3 3Fv 'SALC 3 0 1 'SALC = A1 + E Nitrogen AO symmetries are s = A1 pz = A1 (px, py) = E All AOs match with SALCs, so there are no nonbonding levels. Both s and pz AOs match the A1 SALC, so s-p mixing is likely. If bonding and antibonding combinations were formed for both s and pz AOs, we would end up with eight MOs, but only seven AOs on N and H are available (disregarding the 1s AO on N). We must make only three MOs from the s and pz AOs and the A1 SALC. For simplicity, we will assume that the s and pz AOs each form essentially separate bonding MOs, but that together they form a single mixed antibonding MO. The resulting MO scheme is shown below, where hashed lines indicate lesser contributions from s-p mixing. (b) The P.E.S. has three bands with vibrational fine structure, indicative of ionizations from bonding MOs. This is consistent with the MO scheme above. (c) The VB model assumes one nonbonding lone pair in an sp3 hybrid. As the scheme above indicates, this pair is weakly bonding. This is not inconsistent with the well-known Lewis base character of NH3, because the F(z) MO has considerable electron density above the nitrogen, not unlike the customary picture of the VB model's lone-pair sp3 hybrid. A rough sketch of the MO is shown below: (d) In a planar MH3 molecule, such as BH3, the pz AO on the central atom is a nonbonding Bn (a2") MO in D3h. If the three hydrogen atoms move downward to form a pyramidal structure (C3v), they can form the F(z) bonding MO. There is no advantage in doing this unless there exists an electron pair to form the bonding and thereby lower the overall energy of the molecule. Consequently, the addition of another electron pair to an otherwise planar MH3 molecule becomes the driving force to assume a pyramidal geometry. 4.6 A model for the reducible representation of B-SALCs, which form the MOs, is shown below. The reducible representation and its decomposition are C2v E C2 Fv Fv N 'B 3 -1 1 -3 'B = A2 + 2B1 On the basis of the symmetry implied by the irreducible representations (as indicated in the Mulliken symbols), we could conceive of the following SALCs. Q(A2) = 1//2(R1 – R3) Q(B1) = 1//2(R1 + R3) Q(B1) = R2 Although these functions are normalized and orthogonal to each other (cf. Chapter 5), it is difficult to understand the B bonding in terms of them. A more intuitively satisfying set, which is also normalized and mutually orthogonal, is the following: Q(A2) = 1//2(R1 – R3) = Bn Q(B1) = 1//3(R1 + R2 + R3) = B Q(B1) = 1//6(R1 – 2 R2 + R3) = B* These combinations are shown below: The first of these, which has no nodes perpendicular to the molecular plane, is the bonding combination; the second, which has a nodal plane through the middle carbon atom, is nonbonding; and the third, which has perpendicular nodes between each C-C bond, is the antibonding combination. The resulting MO scheme is shown below: 4.7 We will assume that fluorine 2s orbitals are not involved in the bonding and will only consider the 2p orbitals. Of these we will assume that on each F atom the 2pz is perpendicular to the BF3 plane and capable of forming out-of-plane pi interactions (Bz) ; the 2px points toward the B atom and forms sigma interactions (F); and the 2py orbital is parallel to the BF3 plane and has the potential to form in-plane pi interactions (B2). The symmetries of the central boron AOs are as follows: s = A1' (px, py) = E' pz = A2" (a) The effective pB interactions are the out-of-plane type formed from combinations of 2pz AOs on both B and F atoms. The formation of the SALCs is based on the following vector set. The reducible representation and its decomposition are D3h E 2C3 3C2 Fh 2S3 3Fv 'B 3 0 -1 -3 0 1 'B = A2" + E" The A2" SALC matches with the "empty" 2pz AO on B. The E" SALCs have no match on B and remain nonbonding. The pi-only MO scheme that results is shown below. (b) The development of the three fluorine 2s SALCs, assumed to be nonbonding, is shown below. D3h E 2C3 3C2 Fh 2S3 3Fv '2s 3 0 1 3 0 1 '2s = A1' + E' (c) The vector set of 2px sigma bonding is identical to the set shown in part (b). Therefore, 'F = A1' + E'. The A1' SALC matches with the boron 2s AO, and the E' SALCs match with the degenerate boron 2px and 2py AOs. (d) The development of the in-plane B-SALCs is shown below. D3h E 2C3 3C2 Fh 2S3 3Fv '2 3 0 -1 3 0 -1 '2 = A2' + E' The A2' SALC has no match in B AOs and must be strictly nonbonding. The E' SALCs do match with born 2px and 2py, which are involved in sigma bonding. Assuming that the sigma interactions are more effective, the in-plane pi interactions of the E' SALCs can be taken to be essentially nonbonding. (e) The ordering of levels in the following complete MO scheme is based on the P.E.S. data of G. H. King, et al., as cited in the problem. (f) See answer to part (d). If the in-plane e' SALCs had effective interactions with the 2px and 2py AOs on the boron the B 3e' level would be slightly bonding and would probably fall below the energy of the Bn level. The reported P.E.S. assignment does not support this. Evidently the much more effective sigma interactions preclude effective in-plane pi interactions. (g) In BH3 the formal bond order is 1 for each B-H bond. In BF3 the delocalized pair in the B 1a2" MO adds 1/3 to each bond order (1-1/3 total), giving additional stability to BF3. (h) In terms of frontier orbital theory, the LUMO in BF3 is the B* 2a2" MO, which can receive an electron pair from a Lewis base. This MO has the form M = c1N(B) + c2[N(Fa) + N(Fb) + N(Fc)] The LCAO for this is In the case of the base NH3, the HOMO is the weakly bonding F a1 MO (cf. answer to problem 4.5), which can provide electrons to the boron B* 2a2" MO in the following manner: 4.8 (a) The B-MO scheme is developed as follows: D4h E 2C4 C2 2C2' 2C2" i 2S4 Fh 2Fv 2Fd 'B 4 0 0 -2 0 0 0 -4 2 0 'B = A2u + Eg + B2u The resulting MO scheme, using the "polygon rule" for ordering, is (b) The LCAOs are (c) Benzene has three pairs in bonding pi-MOs, whereas the cyclobutadiene dianion has only one pair in a bonding pi-MO (the other two pairs are nonbonding). The C-C bond order (F + B) is 1½ for benzene, but only 1¼ for the cyclobutadiene dianion. (d) Relative to the square configuration shown in part (a), forming a rectangle involves losing the 2C2' axes (which define x and y in D4h) while retaining 2C2" to become x and y in D2h. The descent in symmetry lifts the Eg degeneracy to become B2g and B3g in D2h. The vector model, reducible representation, its decomposition, and the resulting LCAOs are shown below. D2h E C2(z) C2(y) C2(x) i F(xy) F(xz) F(yz) 'B 4 0 0 0 0 -4 0 0 'B = B2g + B3g + Au + B1u The resulting MO scheme, filled for the neutral species, is The B (b2g) MO lies lower than the B* (b3g) MO, because it matches pz wave function signs along the two short C-C distances. The B* (b3g) MO makes the match along the two long C-C distances. (The Mulliken designations of these MOs would be reversed if the assignment of x and y were reversed.) The square model of C4H4 would be a diradical (two unpaired electrons in the nonbonding eg MOs), and like C4H42- would have an average C-C bond order of 1¼ -- insufficient to hold the ring together, given the ring strain. The rectangular model is diamagnetic and allows formation of two pi-bonds, as shown in the MO scheme above. This gives an average C-C bond order of 1½, the same as in benzene. However, the ring is still strained, and cyclobutadiene is not a stable molecule. 4.9 (a) D3h E 2C3 3C2 Fh 2S3 3Fv 'B 3 0 -1 -3 0 1 'B = A2" + E" N one of these is particularly stable, given the ring strain. The pi contributions to the C-C bond order are 1, ½, and 0, respectively. Thus, C3H3+ is marginally more stable. (b) D5h E 2C5 2C52 5C2 Fh 2S5 2S53 5Fv 'B 5 0 0 -1 -5 0 0 1 'B = A2" + E1" + E2" The pi contributions to the C-C bond order are 0.4, 0.5, and 0.6, respectively. The anion is therefore most stable. 'B = A + E1 + E2 + E3 (c) The pi-contributions to the C-C bond order are 3/7, 5/14, 2/7, respectively. C7H7+ is therefore most stable. Moreover, it is the only one that is not a free radical. 4.10 (a) C-ring SALCs (2s) D3h E 2C3 3C2 Fh 2S3 3Fv '2s 3 0 1 3 0 1 '2s = A1' + E' 6H SALCs D3h E 2C3 3C2 Fh 2S3 3Fv '6H 6 0 0 0 0 2 '6H = A1' + E' + A2" + E" The A1' + E' carbon and hydrogen SALCs will form bonding and antibonding F MOs. The bonding MOs are the first two levels in the configuration: [(FC-H)2 = (2a1')2][(FC-H)4 = (2e')4]. The notations 2a1' and 2e' are those of Turner et al. The A2" + E" hydrogen SALCs are nonbonding with respect to sigma bonding, although they do match C-ring B-SALCs [cf. part (d) below]. Ring B-bonding is assumed to be preferential. (b) C-ring in-plane 2p SALCs D3h E 2C3 3C2 Fh 2S3 3Fv 'ex 3 0 -1 3 0 -1 'ex = A2' + E' The A2' SALC has no match and forms an unoccupied nonbonding level. The E' pair of SALCs form the HOMO level (last listed) in the given configuration: [(FC-H)4 = (3e')4]. This is described by Turner et al. as "external F", meaning ring bonding MOs formed by in-plane 2p orbitals. The LCAOs are shown below: Note that these have the same symmetry as the E' 6H-SALCs, but the form of these ring SALCs would result in S = 0 with those SALCs. Thus, the in-plane ring SALCs are nonbonding with respect to hydrogen, and really should be regarded as ring bonding. (c) C-ring F-SALCs D3h E 2C3 3C2 Fh 2S3 3Fv 'F 3 0 1 3 0 1 'F = A1' + E' The A1' level is the fourth level in the given configuration: [(FC-C)2 = (3a1')2]. It is the energetically more favorable of the F-ring MOs [cf. part (b) above]. The E' C-ring FSALCs evidently form a higher-energy unoccupied level. It may be noted that they do have appropriate form for S 0 overlap with the 6H SALCs of the same symmetry. (d) C-ring B SALCs D3h E 2C3 3C2 Fh 2S3 3Fv 'B 3 0 -1 -3 0 1 'B = A2" + E" These SALCs form the third and fifth levels in the given configuration: [(BC-C)2 = (a2")2] ... [(BC-C)4 = (1e")4]. As noted in part (a), these match 6H SALCs of the same symmetries, but ring bonding is presumed to be preferential. (e) Summarizing the identities given in the preceding parts: [(FC-H)2 = (2a1')2][(FC-H)4 = (2e')4][(BC-C)2 = (a2")2][(FC-C)2 = (3a1')2] [(BC-C)4 = (1e")4][(FC-H)4 = (3e')4] As noted in part (b), the last level would be better designated FC-C, because it is actually an external ring bonding MO. (f) The LCAOs for all occupied levels are shown below. 4.11 Boron sp3 SALCs. D2h E C2(z) C2(y) C2(x) i F(xy) F(xz) F(yz) 'B 4 0 0 0 0 0 4 0 'B = Ag + B2g + B1u + B3u H SALCs D2h E C2(z) C2(y) C2(x) i F(xy) F(xz) F(yz) 'H 2 0 0 2 0 2 2 0 'H = Ag + B3u The H SALCs have matching B SALCs and will form bonding and antibonding combinations. The B1u and B2g B SALCs remain nonbonding. The LCAOs are shown below for the bonding and nonbonding MOs. These give the following MO scheme for the two bridges: 4.12 (a) Sigma SALCs for 2p orbitals on fluorines. Oh E 8C3 6C2 6C4 3C2 i 6S4 8S6 3Fh 6Fd ' 6 0 0 2 2 0 0 0 4 2 ' = A1g + Eg + T1u Available AOs on sulfur. A1g: s T1u: (px, py, pz) Eg: (dz2, dx2-y2) T2g: (dxz, dyz, dxy) bonding/ antibonding bonding/ antibonding virtually nonbonding strictly nonbonding The LCAOs for the bonding MOs are shown below. (b) The approximate bond order is 4 pairs/6 bonds = 2/3. However, the S-F bonds in SF6 are short (156 pm), implying that they are stronger than "normal" S-F single bonds. The bonds would be expected to have a great deal of ionic character, which would give additional strength. Note that the energies of the fluorine AOs are considerably lower than those of the S AOs. Thus, the pairs assigned to bonding and nonbonding levels are closer in energy and distribution to pure fluorine AOs. In effect, electrons in these MOs are substantially localized on the fluorine atoms. (c) A linear F-S-F 3c-4e bond would have the following MO scheme: The LCAO for the bonding MO is the same as any one of the three degenerate t1u MOs, whose LCAOs are shown above. To incorporate this model into the full MO scheme would require that only the t1u levels be bonding, making the a1g level nonbonding, along with the previously identified nonbonding eg MOs. This model gives an even lower S-F bond order (½), requiring an even greater ionic character to account for the short bond distances. 4.13 Fluorine F-SALCs: D3h E 2C3 3C2 Fh 2S3 3Fv ' 5 2 1 3 0 3 ' = 2A1' + E' + A2" Symmetry of phosphorus AOs: s = A1' (px, py) = E' pz = A2" [dz2 = A1'] If we assume virtually no dz2 participation, then one of the two A1' fluorine SALCs will be nonbonding. As the LCAOs shown below indicate, this A1' fluorine SALC has appropriate form for bonding and antibonding combinations with dz2 on phosphorus. The complete sigma-only MO scheme for PF5 is shown below. With phosphorus dz2 participation, the F a1' HOMO would be stabilized (lowered in energy) to become a bonding MO. 4.14 Sigma bonding. D4h E 2C4 C2 2C2' 2C2" i 2S4 Fh 2Fv 2Fd 'F 4 0 0 2 0 0 0 4 2 0 'F = A1g + B1g + Eu Matching AOs on Xe: s = A1g (px, py) = Eu The B1g SALC is nonbonding, unless dx2-y2 participation is considered. The forms of the SALCs are shown below. Pi (z) bonding. D4h E 2C4 C2 2C2' 2C2" i 2S4 Fh 2Fv 2Fd 'z 4 0 0 -2 0 0 0 -4 2 0 'z = Eg + A2u + B2u Matching AO on Xe: pz = A2u The Eg and B2u SALCs are nonbonding. The SALCs are shown below. Pi (2) bonding. D4h E 2C4 C2 2C2' 2C2" i 2S4 Fh 2Fv 2Fd '2 4 0 0 2 0 0 0 4 2 0 '2 = A2g + B2g + Eu The A2g and B2g SALCs have no match with Xe AOs and must be nonbonding. The Eu SALC is potentially bonding with px and py on Xe, but sigma bonding is more favorable. The SALCs are shown below. MO scheme. 4.15 Sigma bonding. D5h E 2C5 2C52 5C2 Fh 2S5 2S53 5Fv 'F 5 0 0 1 5 0 0 1 'F = A1' + E1' + E2' Matching AOs on Xe: A1' = s E1' = (px, py) The E2' SALC has no match on Xe and must be nonbonding. Pi (z) bonding. D5h E 2C5 2C52 5C2 Fh 2S5 2S53 5Fv 'z 5 0 0 -1 -5 0 0 1 'z = A2" + E1" + E2" Matching AO on Xe: A2" = pz The E1" and E2" SALCs have no match on Xe and must be nonbonding. Pi (2) bonding. D5h E 2C5 2C52 5C2 Fh 2S5 2S53 5Fv '2 5 0 0 -1 -5 0 0 1 '2 = A2' + E1' + E2' The A2' and E2' SALCs have no matching AOs on Xe. The E1' SALC matches the degenerate pair (px, py) on Xe, but sigma bonding is expected to be more effective (see above), so these can be considered to be virtually nonbonding with respect to Xe. MO scheme.