(c) trigonal bipyramid

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Chapter 4Answers to Problems

4.1 The reducible representation, symmetry-allowed AOs, and possible combinations are listedfor each part.

(a) trigonal planar

3h 3 2 h 3 vD E 2C 3C F 2S 3F

' 3 0 1 3 0 1

1' = A ' + E'

1 z x y xy x -yA ': s, d 2 E': (p , p ), (d , d 2 2)

1A ' E' Notation

x ys (p , p ) sp2

xy x -ys (d , d 2 2) sd2

z x yd 2 (p , p ) p d / dp2 2

z xy x -yd 2 (d , d 2 2) d3

(b) square planar

4h 4 2 2 2 4 h v dD E 2C C 2C ' 2C " i 2S F 2F 2F

' 4 0 0 2 0 0 0 4 2 0

1g 1g u' = A + B + E

1g z 1g x -y u x yA : s, d 2 B : d 2 2 E : (p , p )

1g 1g uA B E Notation

x -y x ys d 2 2 (p , p ) dsp2

z x -y x yd 2 d 2 2 (p , p ) d p2 2

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(c) trigonal bipyramid

3h 3 2 h 3 vD E 2C 3C F 2S 3F

' 5 2 1 3 0 3

1 2' = 2A ' + A " + E'

1 z 2 z x y xy x -yA ': s, d 2 A ": p E': (p , p ), (d , d 2 2)

1 22A ' A " E' Notation

z z x ys, d 2 p (p , p ), dsp3

z z xy x -ys, d 2 p (d , d 2 2) d sp3

(d) octahedral

h 3 2 4 2 4 6 h dO E 8C 6C 6C 3C i 6S 8S 3F 6F

' 6 0 0 2 2 0 0 0 4 2

1g g 1u' = A + E + T

1g g z x -y 1u x y zA : s E : (d 2, d 2 2) T : (p , p , p )

Only possibility is d sp .2 3

2d4.2 The indicated distortion takes the symmetry down to D .

2d 4 2 2 dD E 2S C 2C ' 2F

' 4 0 0 0 2

1 2' = A + B + E

1 z 2 z xy x y xz yzA : s, d 2 B : p , d E: (p , p ), (d , d )

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1 2A B E Notation

z x ys p (p , p ) sp3

z xz yzs p (d , d ) spd / d sp2 2

xy x ys d (p , p ) sp d / dsp2 2

xy xz yzs d (d , d ) sd3

z z x yd 2 p (p , p ) dp / p d3 3

z z xz yzd 2 p (d , d ) pd / d p3 3

z xy x yd 2 d (p , p ) p d / d p2 2 2 2

z xy xz yzd 2 d (d , d ) d4

dAs with the perfect tetrahedron (T ), the sp and sd are still possible hybrids, although the3 3

actual wave functions would differ, because the contributions of individual p or d orbitalswould depend upon the angles in the distorted tetrahedron. The lowered symmetry admitsthe possibility of the other six combinations, but in most cases these would be less likelycontributors to bonding than the modified sp and sd hybrids.3 3

4.3 (a) Taking the three hydrogen 1s orbitals as a basis for a representation of SALCs:

3h 3 2 h 3 vD E 2C 3C F 2S 3F

' 3 0 1 3 0 1

1' = A ' + E'

The equations for the corresponding SALCs are:

1 1 a b cA ': M = 1//3(R + R + R )

2 a b cE': M = 1//6(2R – R – R )

3 b cM = 1//2(R – R )

1 x y z 2 zSymmetries of the AOs of B are s = A ', (p , p ) = E', and p = A ". The p orbital has nomatch in SALCs and must remain nonbonding. The following MO scheme results:

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1(b) Disregarding ionizations from the F (a ') core level, which would probably fall outsiden

the range of u.v.-P.E.S., the spectrum should consist of two bands, both with vibrationalfine structure. The lower-energy band, from the doubly degenerate F (e') level, should be

1roughly twice as big as the higher-energy band, from the nondegenerate F (a ') 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 MOdescription would place all three bonding pairs in equivalent localized sigma-bondingMOs. The suggestion of energetic equivalence of all three pairs is at odds with thesymmetry restrictions of the system.

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4.4 (a) Taking the two hydrogens as a basis for the SALCs

we obtain the following reducible representation:

2v 2 v vC E C F F N

H' 2 0 0 2

H 1 2for which ' = A + B . The equations for the SALCs are

Oxygen AO symmetries are

1 x 1 y 2 z 1s = A p = B p = B p = A

x 1 yThe p AO (B ) has no matching SALC and must be nonbonding. The p orbital can form

2 zbonding and antibonding combinations with the B SALC. Both s and p orbitals on

1oxygen match with the A SALC, so s-p mixing can be expected. If we formed bondingand antibonding combinations for both of these, we would end up with more MOs in thefinal scheme (seven) than there are available AOs on the component atoms (six). To avoid

1this, we can only make three MOs from the A AOs and SALC. For simplicity, we will

zassume that the s and p orbitals on oxygen both form bonding MOs, and together they formone s-p mixed antibonding orbital. Representations of the LCAOs are shown below.

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The qualitative MO scheme is shown below, in which hashed lines indicate lessercontributions arising from s-p mixing.

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(b) The P.E.S. results are consistent with the MO scheme. Only the B (x) MO is strictlyn

nonbonding, while the F(z) MO is weakly bonding, as indicated by the vibrational fine structureon its P.E.S. band.

(c) Rather than two lone pairs in approximately sp hybrids, the MO scheme suggests a3

single region of electron density protruding from the back side of the molecule. This iscreated by the combination of the nonbonding B (x) MO and the weakly bonding F(z) MO,n

whose LCAOs are shown above.

2(d) In a linear molecule like BeH , the degenerate MOs that are essentially the two p

x yorbitals perpendicular to the molecular axis must be nonbonding (the p and p orbitals,

u 4hdesignated A in D ). If the molecule acquires an additional pair of electrons, there couldbe 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.

1The result is the F(z) a 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 energylowering accompanying this additional bonding can be seen as the driving force for the

2molecule to assume a bent geometry, as in H O. The energetics of the changing MO

4h 2v 2schemes on descent from D to C for MX molecules can be represented by a Walshdiagram [cf. R. L. DeKock and W. B. Bosna, J. Chem. Educ. 1988, 65, 194-197.]

3 24.5 The approach to NH is similar to that for H O (see 4.4).

3v 3 vC E 2C 3F

SALC' 3 0 1

SALC 1' = A + E

Nitrogen AO symmetries are

1 z 1 x ys = A p = A (p , p ) = E

zAll AOs match with SALCs, so there are no nonbonding levels. Both s and p AOs match

1the A SALC, so s-p mixing is likely. If bonding and antibonding combinations were

zformed for both s and p AOs, we would end up with eight MOs, but only seven AOs on Nand H are available (disregarding the 1s AO on N). We must make only three MOs from

z 1 zthe s and p AOs and the A SALC. For simplicity, we will assume that the s and p AOs

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each form essentially separate bonding MOs, but that together they form a single mixedantibonding MO. The resulting MO scheme is shown below, where hashed lines indicatelesser contributions from s-p mixing.

(b) The P.E.S. has three bands with vibrational fine structure, indicative of ionizationsfrom bonding MOs. This is consistent with the MO scheme above.

(c) The VB model assumes one nonbonding lone pair in an sp hybrid. As the scheme3

above indicates, this pair is weakly bonding. This is not inconsistent with the well-known

3Lewis base character of NH , because the F(z) MO has considerable electron density above

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the nitrogen, not unlike the customary picture of the VB model's lone-pair sp hybrid. A3

rough sketch of the MO is shown below:

3 3 z(d) In a planar MH molecule, such as BH , the p AO on the central atom is a nonbonding

2 3hB (a ") MO in D . If the three hydrogen atoms move downward to form a pyramidaln

3vstructure (C ), they can form the F(z) bonding MO. There is no advantage in doing thisunless there exists an electron pair to form the bonding and thereby lower the overallenergy of the molecule. Consequently, the addition of another electron pair to an otherwise

3planar MH 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 shownbelow.

The reducible representation and its decomposition are

2v 2 v vC E C F F N

B' 3 -1 1 -3

B 2 1' = A + 2B

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On the basis of the symmetry implied by the irreducible representations (as indicated in theMulliken symbols), we could conceive of the following SALCs.

2 1 3Q(A ) = 1//2(R – R )

1 1 3Q(B ) = 1//2(R + R )

1 2Q(B ) = R

Although these functions are normalized and orthogonal to each other (cf. Chapter 5), it isdifficult to understand the B bonding in terms of them. A more intuitively satisfying set, whichis also normalized and mutually orthogonal, is the following:

2 1 3Q(A ) = 1//2(R – R ) = Bn

1 1 2 3Q(B ) = 1//3(R + R + R ) = B

1 1 2 3Q(B ) = 1//6(R – 2 R + R ) = B*

These combinations are shown below:

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The first of these, which has no nodes perpendicular to the molecular plane, is the bondingcombination; the second, which has a nodal plane through the middle carbon atom, isnonbonding; and the third, which has perpendicular nodes between each C-C bond, is theantibonding 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

zconsider the 2p orbitals. Of these we will assume that on each F atom the 2p is

3 zperpendicular to the BF plane and capable of forming out-of-plane pi interactions (B ) ; the

x y2p points toward the B atom and forms sigma interactions (F); and the 2p orbital is

3 2parallel to the BF plane and has the potential to form in-plane pi interactions (B ). Thesymmetries of the central boron AOs are as follows:

1 x y z 2s = A ' (p , p ) = E' p = A "

z(a) The effective pB interactions are the out-of-plane type formed from combinations of 2pAOs on both B and F atoms. The formation of the SALCs is based on the following vectorset.

The reducible representation and its decomposition are

3h 3 2 h 3 vD E 2C 3C F 2S 3F

B' 3 0 -1 -3 0 1

B 2' = A " + E"

2 zThe A " SALC matches with the "empty" 2p AO on B. The E" SALCs have no match onB and remain nonbonding. The pi-only MO scheme that results is shown below.

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(b) The development of the three fluorine 2s SALCs, assumed to be nonbonding, is shownbelow.

3h 3 2 h 3 vD E 2C 3C F 2S 3F

2s' 3 0 1 3 0 1

2s 1' = A ' + E'

x(c) The vector set of 2p sigma bonding is identical to the set shown in part (b). Therefore,

F 1 1' = A ' + E'. The A ' SALC matches with the boron 2s AO, and the E' SALCs match with

x ythe degenerate boron 2p and 2p AOs.

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(d) The development of the in-plane B-SALCs is shown below.

3h 3 2 h 3 vD E 2C 3C F 2S 3F

2' 3 0 -1 3 0 -1

2 2' = A ' + E'

2The A ' SALC has no match in B AOs and must be strictly nonbonding. The E' SALCs do

x ymatch with born 2p and 2p , which are involved in sigma bonding. Assuming that thesigma interactions are more effective, the in-plane pi interactions of the E' SALCs can betaken to be essentially nonbonding.

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(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.

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x(f) See answer to part (d). If the in-plane e' SALCs had effective interactions with the 2p

yand 2p AOs on the boron the B 3e' level would be slightly bonding and would probablyfall below the energy of the B level. The reported P.E.S. assignment does not support this. n

Evidently the much more effective sigma interactions preclude effective in-plane piinteractions.

3 3(g) In BH the formal bond order is 1 for each B-H bond. In BF the delocalized pair in the

2 3B 1a " MO adds 1/3 to each bond order (1-1/3 total), giving additional stability to BF .

3 2(h) In terms of frontier orbital theory, the LUMO in BF is the B 2a " MO, which can*

receive an electron pair from a Lewis base. This MO has the form

1 2 a b cM = c N(B) + c [N(F ) + N(F ) + N(F )]

The LCAO for this is

3 1In the case of the base NH , the HOMO is the weakly bonding F a MO (cf. answer to

2problem 4.5), which can provide electrons to the boron B 2a " MO in the following*

manner:

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4.8 (a) The B-MO scheme is developed as follows:


B' 4 0 0 -2 0 0 0 -4 2 0

B 2u g 2u' = A + E + B

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 hasonly one pair in a bonding pi-MO (the other two pairs are nonbonding). The C-C bondorder (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

2 4h 2losing the 2C ' axes (which define x and y in D ) while retaining 2C " to become x and y in

2h g 2g 3g 2hD . The descent in symmetry lifts the E degeneracy to become B and B in D . The

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vector model, reducible representation, its decomposition, and the resulting LCAOs areshown below.

2h 2 2 2D E C (z) C (y) C (x) i F(xy) F(xz) F(yz)

B' 4 0 0 0 0 -4 0 0

B 2g 3g u 1u' = B + B + A + B

The resulting MO scheme, filled for the neutral species, is

2g 3g zThe B (b ) MO lies lower than the B (b ) MO, because it matches p wave function signs*

3galong the two short C-C distances. The B (b ) MO makes the match along the two long*

C-C distances. (The Mulliken designations of these MOs would be reversed if the

4 4assignment of x and y were reversed.) The square model of C H would be a diradical (two

g 4 4unpaired electrons in the nonbonding e MOs), and like C H would have an average C-C2-

bond order of 1¼ -- insufficient to hold the ring together, given the ring strain. Therectangular model is diamagnetic and allows formation of two pi-bonds, as shown in theMO 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.

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4.9 (a)

3h 3 2 h 3 vD E 2C 3C F 2S 3F

B' 3 0 -1 -3 0 1

B 2' = A " + E"

N one ofthese is particularly stable, given the ring strain. The pi contributions to the C-C bond order

3 3are 1, ½, and 0, respectively. Thus, C H is marginally more stable.+

(b)

5h 5 5 2 h 5 5 vD E 2C 2C 5C F 2S 2S 5F2 3

B' 5 0 0 -1 -5 0 0 1

B 2 1 2' = A " + E " + E "

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The pi contributions to the C-C bond order are 0.4, 0.5, and 0.6, respectively. The anion istherefore most stable.

B 1 2 3(c) ' = A + E + E + E

7 7The pi-contributions to the C-C bond order are 3/7, 5/14, 2/7, respectively. C H is+

therefore most stable. Moreover, it is the only one that is not a free radical.

4.10 (a) C-ring SALCs (2s)

3h 3 2 h 3 vD E 2C 3C F 2S 3F

2s' 3 0 1 3 0 1

2s 1' = A ' + E'

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6H SALCs

3h 3 2 h 3 vD E 2C 3C F 2S 3F

6H' 6 0 0 0 0 2

6H 1 2' = A ' + E' + A " + E"

1The A ' + E' carbon and hydrogen SALCs will form bonding and antibonding F MOs.

C-H 1 C-HThe bonding MOs are the first two levels in the configuration: [(F ) = (2a ') ][(F )2 2 4

1 2= (2e') ]. The notations 2a ' and 2e' are those of Turner et al. The A " + E" hydrogen4

SALCs are nonbonding with respect to sigma bonding, although they do match C-ringB-SALCs [cf. part (d) below]. Ring B-bonding is assumed to be preferential.

(b) C-ring in-plane 2p SALCs

3h 3 2 h 3 vD E 2C 3C F 2S 3F

ex' 3 0 -1 3 0 -1

ex 2' = A ' + E'

2The A ' SALC has no match and forms an unoccupied nonbonding level. The E' pair of

C-HSALCs form the HOMO level (last listed) in the given configuration: [(F ) = (3e') ]. 4 4

This is described by Turner et al. as "external F", meaning ring bonding MOs formed byin-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 ringSALCs would result in S = 0 with those SALCs. Thus, the in-plane ring SALCs arenonbonding with respect to hydrogen, and really should be regarded as ring bonding.

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(c) C-ring F-SALCs

3h 3 2 h 3 vD E 2C 3C F 2S 3F

F' 3 0 1 3 0 1

F 1' = A ' + E'

1 C-C 1The A ' level is the fourth level in the given configuration: [(F ) = (3a ') ]. It is the2 2

energetically more favorable of the F-ring MOs [cf. part (b) above]. The E' C-ring F-SALCs evidently form a higher-energy unoccupied level. It may be noted that they dohave appropriate form for S � 0 overlap with the 6H SALCs of the same symmetry.

(d) C-ring B SALCs

3h 3 2 h 3 vD E 2C 3C F 2S 3F

B' 3 0 -1 -3 0 1

B 2' = A " + E"

C-C 2These SALCs form the third and fifth levels in the given configuration: [(B ) = (a ") ]2 2

C-C... [(B ) = (1e") ]. As noted in part (a), these match 6H SALCs of the same4 4

symmetries, but ring bonding is presumed to be preferential.

(e) Summarizing the identities given in the preceding parts:

C-H 1 C-H C-C 2 C-C 1[(F ) = (2a ') ][(F ) = (2e') ][(B ) = (a ") ][(F ) = (3a ') ]2 2 4 4 2 2 2 2

C-C C-H[(B ) = (1e") ][(F ) = (3e') ]4 4 4 4

C-CAs noted in part (b), the last level would be better designated F , because it is actuallyan external ring bonding MO.

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(f) The LCAOs for all occupied levels are shown below.

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4.11 Boron sp SALCs.3


B' 4 0 0 0 0 0 4 0

B g 2g 1u 3u' = A + B + B + B

H SALCs


H' 2 0 0 2 0 2 2 0

H g 3u' = A + B

The H SALCs have matching B SALCs and will form bonding and antibonding

1u 2gcombinations. The B and B B SALCs remain nonbonding. The LCAOs are shownbelow for the bonding and nonbonding MOs.

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These give the following MO scheme for the two bridges:

4.12 (a) Sigma SALCs for 2p orbitals on fluorines.

h 3 2 4 2 4 6 h dO E 8C 6C 6C 3C i 6S 8S 3F 6F

' 6 0 0 2 2 0 0 0 4 2

1g g 1u' = A + E + T

Available AOs on sulfur.

1g 1u x y z g z x -y 2g xz yz xyA : s T : (p , p , p ) E : (d 2, d 2 2) T : (d , d , d )

bonding/antibonding

bonding/antibonding

virtuallynonbonding

strictlynonbonding

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The LCAOs for the bonding MOs are shown below.

6(b) The approximate bond order is 4 pairs/6 bonds = 2/3. However, the S-F bonds in SFare short (156 pm), implying that they are stronger than "normal" S-F single bonds. Thebonds would be expected to have a great deal of ionic character, which would giveadditional strength. Note that the energies of the fluorine AOs are considerably lower thanthose of the S AOs. Thus, the pairs assigned to bonding and nonbonding levels are closerin energy and distribution to pure fluorine AOs. In effect, electrons in these MOs aresubstantially localized on the fluorine atoms.

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(c) A linear F-S-F 3c-4e bond would have the following MO scheme:

1uThe LCAO for the bonding MO is the same as any one of the three degenerate t MOs,whose LCAOs are shown above. To incorporate this model into the full MO scheme would

1u 1grequire that only the t levels be bonding, making the a level nonbonding, along with the

gpreviously identified nonbonding e 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:

3h 3 2 h 3 vD E 2C 3C F 2S 3F

' 5 2 1 3 0 3

1 2' = 2A ' + E' + A "

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Symmetry of phosphorus AOs:

1 x y z 2 z 1s = A ' (p , p ) = E' p = A " [d 2 = A ']

z 1If we assume virtually no d 2 participation, then one of the two A ' fluorine SALCs will be

1nonbonding. As the LCAOs shown below indicate, this A ' fluorine SALC has appropriate

zform for bonding and antibonding combinations with d 2 on phosphorus.

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5The complete sigma-only MO scheme for PF is shown below.

z 1With phosphorus d 2 participation, the F a ' HOMO would be stabilized (lowered in energy)to become a bonding MO.

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4.14 Sigma bonding.


F' 4 0 0 2 0 0 0 4 2 0

F 1g 1g u' = A + B + E

Matching AOs on Xe:

1g x y us = A (p , p ) = E

1g x -yThe B SALC is nonbonding, unless d 2 2 participation is considered. The forms of theSALCs are shown below.

Pi (z) bonding.


z' 4 0 0 -2 0 0 0 -4 2 0

z g 2u 2u' = E + A + B

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Matching AO on Xe:

z 2up = A

g 2uThe E and B SALCs are nonbonding. The SALCs are shown below.

Pi (2) bonding.


2' 4 0 0 2 0 0 0 4 2 0

2 2g 2g u' = A + B + E

2g 2g uThe A and B SALCs have no match with Xe AOs and must be nonbonding. The E SALC is

x ypotentially bonding with p and p on Xe, but sigma bonding is more favorable. The SALCs areshown below.

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MO scheme.

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4.15 Sigma bonding.

5h 5 5 2 h 5 5 vD E 2C 2C 5C F 2S 2S 5F2 3

F' 5 0 0 1 5 0 0 1

F 1 1 2' = A ' + E ' + E '

Matching AOs on Xe:

1 1 x yA ' = s E ' = (p , p )

2The E ' SALC has no match on Xe and must be nonbonding.

Pi (z) bonding.

5h 5 5 2 h 5 5 vD E 2C 2C 5C F 2S 2S 5F2 3

z' 5 0 0 -1 -5 0 0 1

z 2 1 2' = A " + E " + E "

Matching AO on Xe:

2 zA " = p

1 2The E " and E " SALCs have no match on Xe and must be nonbonding.

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Pi (2) bonding.

5h 5 5 2 h 5 5 vD E 2C 2C 5C F 2S 2S 5F2 3

2' 5 0 0 -1 -5 0 0 1

2 2 1 2' = A ' + E ' + E '

2 2 1The A ' and E ' SALCs have no matching AOs on Xe. The E ' SALC matches the

x ydegenerate pair (p , p ) on Xe, but sigma bonding is expected to be more effective (seeabove), so these can be considered to be virtually nonbonding with respect to Xe.

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MO scheme.

(c) trigonal bipyramid

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