57
•
51
(a) (b) Hybrid orbitals are generated by mixing atomic orbitals which.are close m
energy. A set of spatially-directed orbitals, for application within VB theory, is
derived. The character of a hybrid orbital depends on the atomic orbitals involved
and their percentage contributions, e.g. an sp^ hybrid comprises V.s and V^ orbital
character Each hybrid orbital points along an intemuclear vector or towards a lone
pair within a molecule; e.g. in CH,, sp' hybridization is used to obtain 4 equiyalpnt
hybrid orbitals, each pointing along a different C-H intemuclear vector..
Hybridization provides a convenient way to develop a bonding picture lismg
localized ff-bonds; unused orbitals such asp atomic orbitals can be used to fonri ;rbonds, e.g.jn CO2 (see answer 5.9).
(c) Equations 5.1 and 5.2 in H&S are:
>/^sp hybrid = ~f~(y^2s
Normalization:
see Box 1.4 and discussion
with equation 2.2 in H&S
+ ¥2p, )
4i
y^sp hybrid
4i
{¥2s~¥2p)
The equations refer to two sp hybrid orbitals. For normalized wavefunctions, the
sum of the squares of the normalization factors must equal unity. Checkthis is true.
For the fii^t wavefunction:
1
1
¥sp hybrid = -J^¥2s^~-f^ ¥lp.
Vz
V/-
1 +1 = 1
.*. Normalized
2 2
Similarly for the second wavefunction.
Questions 5.2-5.4:
general notes
In hybridization schemes:
• mixing orbitals of the same phase corresponds to constructive interference of waves;
- mixing orbitals of opposite phases cotxesponds to destructive interference of waves;
• changing the sign of a wavefunction changes its phase.
^
5.2
(a) The information in Figure 5.4 in H&S only allows a qualitative answer.
It is easily seen that the first (additive) combination yields the hybrid orbital shown:
2s
2p^ •
For the second hybrid, take the combinations of atomic orbitals in 2 stages. Step 1:
X
Watch signs!
Think about phases!
2R
Now add in the 2p^ character:
r
2P.
>
58
Bonding in polyatomic molecules
For the last hybrid, again take the orbital combinations in 2 stages. Step I:
X
X X
2s
2/j.
Now add in the 2p character:
Watch signs!
Think about phases!
'2p.
(b) The method is as in answer 5. lb. Take each wavefunction.in turn: • ;.••.•
1
I//
•
-.-iiv If/
^sp^ hybrid ~ R^'^'
/T
A.:
I -.--If/
i-x ^2 P.
1 2
Sum of the squares of the normalization factors is: - 4--- = i
3 3
1
1
W.JI
sp^ hybrid "
/ r y^2s
VJ
Normalized
Vz
Sum of the squares ofthe normalization factors is: - + - + — = i .-. Normalized
3 6 2
1
¥^^21,,,5,^;^ -""Tr 5^2,s
rzWlp,
^¥l,
VI ^^"^
5.3
Sum of the squares of the normalization factors is: - -i- •- H- — = l .-. Noniialized
3 6 2
The method of working is as answer 5.2a, but now you must work in three
dimensions; use Figure 5.1 to help you. Equation 5.6 in H&S is:
^ sp^ hybrid
-1 W2s+V^2p,+W2p+W2p,
9
Take the combinations below in a stepwise manner as in answer 5.2a. Because of
the vector properties of the 2p orbitals, as each 2p contribution is added in, the
resultant hybiid orbital changes direction:
c=:>
Equation 5.7 in H&S is:
Vhybrid = 7 ^ 2 . +y^2p^ ~Vip^ ~¥2p.
Figure 5.1 The relationship
between a cube and a tetrahedron
(for answer 5.3), The edges of the
cube coincide with a set of
Cartesian axes.
Again, consider the contributions from the atomic orbitals in a stepwise manner
and watch the signs:
c=^
Equations 5.8 and 5.9 in H&S can be correlated to the last two diagrams in Figure
5.6a in H&S in a similar way to the worked answers above.
Bonding in polyatomic molecules
54
•
^5
59
(a)Taketheshadedlobesofthep,and,i^^,.orbitaltopoiBtalongthe+xms^^a^^^
the shaded lobe of the p^ orbital to point along the +y axis. In the xy plane, the
orbital combinations to give 4 spH hybrid orbitals are:
Watch signs!
Think about phases!
IT-;—>
1
(b) Available for hybridization are one ., two p, and one d
^^^^^'^^^^^
orbital must contain the same amount of.character;smce there are4hybndorbitals
each contains 25% . character. Each hybrid orbital must contam the same amount
of p character, i.e. 50% p character. Each hybrid contains 25% d character.
5.5
.Si I'll,
(5.1)
1-~
"
P . - "
a
ci'
ra)SiF -Si group 14.4valenceeIectrons,seestructure5.1.Molecularstructureis
tetrahedral! 4 substituents and no lone pairs, therefore sp^ hybridization. _
(b) fPdCl f- : Square planar Pd(II) complex 5.2, therefore spU is appropnate.
c NF - N group 15,5 valence electrons, see structure 5.3. Molecular structure is
daonal pyramidal; 3 substituents and 1 lone pair, therefore sp^ hybridization.
( T F 0 o j o u p 16,6 valence electrons, see stmcmre 5.4. Molecular structure .s
bent;'2 sub'stituents and 2 lone pairs, therefore sp" hybridization,
(e) rCoH.]'^ is a 5-coordinate Co(I) complex. Whether this xs tngonal bipyramidal
! ^ \ q u ^ e based pyramidal (5.5, see Figure 10.13 in H&S), spH hybridization xs
m F ^ H ^ r is an octahedral FedD complex (structure 5.6, see Figure 10.13 in
H&S)-5p3# hybridization is appropriate.
, , • ,•
(g) CS, C, group 14,4 valence electrons, see structure 5.7. tHe molecule is hnear
and the C Ltom is therefore sp hybridized.
fh) BF • B group 13,3 valence electrons, see structure 5.8. Molecular structure is
trigona'l planar; 3 substituents and no lone pairs, therefore sp^ hybridization.
(5.2)
H
4-
4-
H
l.o^H
S=
C=
S
F
,^^^^'F
B
'H
H
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
60
Bonding in polyatomic molecuies
5.6
(a) See 2.28 and 2.29 (p. 22) for structures of a V and trans-f^^F^. Both isomers
contain N in a non-Imear environment with I lone pair, so sp^ hybridization
(b) Figure 2.1 in H&S shows that each O atom in H^O^ is in a non-linear environment
with 2 lone pairs. Hence, sp^ hybridization is appropriate.
5.7
(a) Regular geometries for a 5-coordinate species are trigonal bipyramidal and
square-based pyramidal (5.9). C,^ symmetry for PF^ is consistent with a trigonal
bipyramidal structure, 5.10.
(b) VB theory describes the bonding in PF^ in tenns of a set of resonance structures
m which the P atom retains an octet of electrons:
F
F
F
I
F
*.o>'F
F
F-
(5.10)
F-
5.8
(a) Trigonal planar (isoelectronic with [NO3]-, see worked example 5 2 in H&S)
(b) The resonance structures which contribute the most are:
°
^v
O
C
o
C
(c) The bonding description is like that for [^03]- in worked example 5.2 in H&S
and can be summarized as follows:
Check that the number of
electrons used in the scheme
= number of valence
electrons available (24)
C - 0 cT-bond
SfP hybrid orbital
on oxygen,
occupied by a ione
pair of electrons
One pair of
electrons in
each of two
oxygen 2p
orbitais
One pair of electrons for C - 0 7C-hond
This scheme corresponds to one resonance structure (1 C=0 double and 2 C-O
single bonds). Each resonance structure may be similarly described.
5.9
(a) CO2 is linear (see answer 2.19b, p.21).
(b) For a Hnear triatomic molecule, the central atom can be considered to be sp
hybndized.
^
^tiW°W*'"
Bonding in polyatomic molecules
61
(c) An appropriate bonding scheme is summarized below:
Carbon sp
hybrid
orbitaJs
Oxygen
2p orbital
Oxygen sp^
hybrid orbital
;r~Bond
Oxygen
sp^ hybrid
• orbitals each
containing 2
electrons
;o ; c : o
a-Bond (T-Bond
-Q.
•Q
Oxygen'
2p orbital
7r-Bond
Q
Carbon 2p
orbitals
(5.11)
(d) The scheme shows the formation of two C^O double bonds.
(e) Lewis structures are shown in 5.11. These are consistent with the bonding
schemes developed using hybridized atomic orbitals in the VB model.
5.10
Read through the first part of Section 5.4 in H&S.
Your answer to this problem should be constructed from the information in Section
5.4 in H&S, and should include a simple example such as an MO diagram for
linear XH,.
5.11
In VB theory, localized bonds arise because a wavefunction is set up to describe
each X~H interaction. Using a hybrid orbital approach for linear XH^, the X atom
can be considered to be sp hybridized and each X-H interaction described as in
Figure 5.2. In MO theory, molecular orbitals are constructed using contributions
from atomic orbitals of all the atoms (where this is allowed by symmetiy). For
linear XH^, the interactions between the atomic orbitals of X and the ligand group
orbitals (LGOs) of the H—H fragment are considered (Figure 5.3). Each of the
bonding MOs possesses bonding character spread out (delocalized) over all 3 atoms.
Assuming X has at least 2 valence electrons (e.g. Be), then the bonding MOs are
filled. The TC^ M O S are non-bonding, and the 0^* and o;* MOs are antibonding.
•X
Figure 5.2 Foranswer5.11: in
linear XHg, each X™H bond can be
described by the overlap of an sp
hybrid orbltai and H Is orbital. Each
bond comprises a iocalized 2c-2e
interaction.
Out-of-phase 1.^ orbitals
Q = LG0(2), (7,
(^ In-phase Is orbitals
= LGO(l),a
Figure5.3 Foranswer5,11:an
MO diagram for the formation of
linear XH^, the atoms are defined to
iie along the z axis.
a^ (H-X~H bonding)
(7^ (H-X-H bonding)
X
XH.
H-X-H-
H--H
62
Bonding in polyatomic molecules
5.12
-* y
(a) First draw the HjO molecular framework with respect to the axis set specified
in the question: see 5.12. The left-hand column in Table 5.6 in H&S lists the atomic
or ligand group orbitals, and reading down each of the next 6 columns gives the %
composition of each MO along with the eigenvector (i.e. sign of each contributing
wavefunction). MOs can be constructed as follows, with the relative sizes of the
lobes reflecting the % contributions:
H'
(5.12)
V^i
%
¥,
¥.
•
%
'
%
'
(b) The number of valence electrons available = 6 (from O) + 2 (from 2H) = B.
Hence MOs )/,, xir^, y^ and y/^ are occupied, y/^ and t/^^ P^^o^ide the 0-H bonding
character, and y/^ and ^^ correspond to the lone pairs.
5.13
(a) The BH3 molecule is defined as lying in the xy plane. The H atoms lie in the
nodal plane of the B 2p^ orbital. There is no net overlap between the H Is and B 2p,
orbitals, so the B 2p^ orbital becomes a non-bonding MO in BHg.
(b) Schematic representations are:
=5 Bonding MOs
5.14
Antibonding
MOs
Schematic representations are:
The second e orbital is shown
in Figure 5.19 in H&S
5.15
[NH^]"^ is isoelectronic with CH^. Thus, the description of the bonding in [NH^]-^ is
essentially the same as in CH^. Refer to the discussion in Section 5.5 in H&S and to
Figures 5.20 and 5.21 in H&S. In each, replace C by N+ (isoelectronic species).
5.16
-j
r
I
-+
(a) The Lewis structures
give 2c-2e localized I-I
I^
bonds in each case. This
does not explain the variation in bond lengths.
(b) The bonding in I^ can be described as in Fj (Figure 2.7 in H&S, replacing 2s
and 2p orbitals by 55" and 5p). This gives a bond order of I. For bent [I3]*, an MO
diagram can be. constructed using that for HjO (Figure 5.15 in H&S) as a basis,
because I"" is isoelectronic (in terms of valence electrons) with O. Each I-I bond
order is therefore 1. For linear [1^]~, an MO diagram can be constructed using that
•
^nsn^
T - '
'
Bonding in polyatoinic molecules
63
for XeF, (Figure 5.30 in H&S) as a basis, because I^ is isoelectronic with Xe. This
™ o one occupied bonding MO which is deiocalized over all tee atoms, and
therefore an 1-1 bond order of % llie conclusion from MO theory is:
I-I bond orders: ^ ^ {l^V > [I3]
Expected trend in I-I bond lengths: I j « i h T < S^^' .
. ,, ,^
, ^Q^
This agrees with the experimental values of 267 pm in I,, 268 pm m [I3] , and 290
pm in [I3]".
5.17
(a)ForBa3tohaveD3,symmetryJtmustcontaina<r>rrorplane,aC3principal
S s , three \ axes andWee o^ mirror planes. T^e molecule is togonal planar.
ci
Each of the three Cl-B-Cl bond angles is 120°.
,, „
X!
cr
For NCl^ to possess C3. symmetry, it must contain a C, Pnncipal a>ds and tee a
mirror planes. Its structure is therefore trigonal pyramidal. Each Ci-N-Cl bond
ansle is <120^ but the exact value cannot be determined from the pomt group,
( b ^ e axes a e defmed in BCI3 so that the . axis coincides with the C axis, and
S e / a n d , axes are as shown in the margin structure.The boron 2 . orbual xs e
Unchanged by each of the symmetry operations of the J^,, pomt group and th.s
corresponds to the following row of characters:
OK
For character tables, see
Appendix 3 in H&S
This matches the row of characters for symmetry type A,' in the chmcter table
arTd tte 2 , orbital therefore has «/ symmetry. For the 2p, orbital, the results ofOperating on it are as follows:
F.
C-,
Cn
Dj,
53
o-v
ar=d this leads to an assignment of a'' symmetry for the 2p^ orbital. Similarly, the
2p and 2B orbitals can be shown to possess e'symmetry.
Th'e same procedure is applied to NCI,, C,, symmetry. Operating on the mtrogen
2s orbital with the relevant symmetry operators yields:
and from the Q , character table, the 2s orbital can be assigned a, symrnetry. The
"stltTo'operaJikg on the 2,, orbital can be summarized in fte row of characters.
and this orbital therefore also has . , symmetry. The 2p, and 2p,orbitals are
degenerate and possess e symmetry.