Author: Fred J' Grieman

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Localized Bonding Model Hybridization Bond
Angle Extent of Hybridization
Author Fred J. Grieman
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Hybridized Orbitals Similar to BeH2 CH4
Localized Bonding Treatment 4 bonding orbitals
bi fi Hi(1s) fi sp3
hybrid 4 anti-bonding orbitals ai fi -
Hi(1s) i 1 4 R. M. Pitzer, J.C.P
46 (1967) 4871 Variational Calculation b1
0.02 C(1s) 0.292 C(2s) 0.277 C(2px) C(2py)
C(2pz) 0.57 HA(1s) 0.07
HB(1s) HC(1s) HD(1s) Approximation - sp3
hybrids with equal coefficients
f1 ½ (s px py pz)
4sp3 f2 ½ (s px - py
- pz) orthogonal set
f3 ½ (s - px py - pz)
equal coeff.s f4
½ (s - px - py pz) Bond Angle? sp3
hybrids give direction What have you been
told? 109.5o Prove it Take f1 ½ ( s
px py
pz ) ½
(4p)-½ (3/4p)½sin?cos? sin?sin?
cos? Find maximum wrt ? ?
3
(?f1/??)? -sin?sin? sin?cos? 0 ?
sin? cos? so ? p/4 45o (?f1/??)?
cos?cos? cos?sin? - sin? 0 ? cos?
(cos? sin?) sin? so
(sin?/cos?) tan? (cos? sin?) (cosp/4
sinp/4) (2)1/2 arctan (2)1/2
54.7356o Similar treatment for f4 ½ (s -
px - py pz) ? ? 225o ? -54.7356o
f4 -54.7o 54.7o f1
Bond Angle 54.7356o (-54.7356o)
109.4712o ?

s-orbital in hybrid gives
bond angle gt 90o 45o
Conversely, bond angle gives s-orbital
contribution to hybrid
hybrid ?i A(?s pi) ? leads to s
contribution pi cixpx ciypy
cizpz p orbital combination gives direction
Normalize 1 ltfifigt A2 ?2ltssgt 2 ?
ltspigt ltpipigt A2?2(1) 0 1
A2(?2 1) 1 A (1 ?2)-½
so, fi (1 ?2)-½ (?s pi) Square to
get s p contribution cs ?2/(1?2) cp
1/(1?2)
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Now Bond Angle ? ? Consider fi fj at angle ?
Use orthogonality ltfifjgt 0 (1
?2)-1 lt(?s pi)(?s pi)gt (1
?2)-1 ?2ltssgt ?ltspigt ?ltspjgt ltpipjgt
?2 ltpipjgt ? ?2 -
ltpipjgt Values of ltpipjgt, Two cases we
know ? 90o ? 0o

ltpipjgt 0 ltpipjgt 1 At some other
?? Consider pi pj in x,y plane
put pj on x axis then pj
px then pi
(cos?) px (sin?) py
cos ? pi
sin ? y ?
pj px So, ltpipjgt lt (cos?) px
(sin?) py px gt cos ? ltpx px gt sin ? ltpy
px gt cos ? -?2
Then, s contribution cs ?2/(1?2) -cos?
/ (1-cos?) ? cs cos? / (cos? - 1) or
cos ? cs / (cs-1) also cp 1 /
(1-cos?) ? ? cs, cp
0
0
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Applications 1) ? 109.47o cos ? -? ? cs
-? / (-?-1) ? / ? ?
sp3 has ?
s character ? 2) sp2 cs ? cos ? ? /
(?-1) ? /-? -? arccos(-?) 120o ? 3) sp
cs ? cos ? ? / (? -1) -1 arccos(-1)
180o ? 4) H2O ? 104.5o cos (104.5o) -
? cs - ? / (- ?-1) - ?/(-5/4)
? (20 s character) hybrid ?1 (?)? s
(?)? pi (?)? s (4/15)? (px py pz)
?2 found via orthogonality
Non-bonding hybrid angle!! cs for ?1,
?2 20 20 40 ?3, ?4 60 so cs 0.30
cos? (.3)/(.3-1) .3/-.7 -0.4286
arccos(-0.4286) 115.38o lone pair
?3 l1 (.3)? s (.7)? pi
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5) NH3 ? 107o cos (107o) -0.2924 ? cs
0.226 (22.6) f1 (0.226)? s (0.774)? pi
f2, f3 via orthogonality


112o non-bonding f4 l1 (0.322)? s
(0.678)? pi
(found via



ltf1l1gt) 6) Ethene Model
121.3o
117.4o Approximate
as Cs sp2 hybridized (120o bond angles) sCC
CA (sp2) CB (sp2) sCH CA(sp2)
HA(1s) p CA(pz) CB(pz)


4sCH hybrid-orbitals for s bonding
E p-orbitals for p-bonding

sCC

pCC Orbital
Energy Diagram (Local Bonding Model)

pCC

sCC


4sCH