XIII–63
Polyatomic bonding -2014, Notes (13) – Atkins Ch.10(p.395-407),
Engel 16-17, House (Ch. 9.6, 11)
Balance:
nuclear repulsion, positive R0, V∞
with e-n attract, neg.  united atom AO i
applies to all bonding, just more nuclei in polyatomic
repulsion and attract maximum at low separation, repulsion dominate -get balance – minimum total (e & n) energy
Extend LCAO-MO for Simple Polyatomics,
1st - Hetero atom diatomic bonding – Atkins, p.384-86,
requires adjustment to homonuclear diatomic picture
Consider H-F - F more electronegative, higher IP than H
(If ionize, electron goes to vacuum, free to fly away)
bigger IP means more negative energy
Ex. HF bonding
Result: bonding MO is closer in energy to F 2p than H 1s,
b will look more like F-2pz, and with 2 electrons, will have
more electron density closer to F than to H
 create a dipole moment by displace e2pFx,y are non-bonding in HF, ┴ to -bond, no overlap 1sH
these are then effectively -lone pairs, but if used hybrids one
sp3 would from bond overlap, other three sp3 would be lone pairs
XIII–64
Same idea for polyatomics, keep track of identical atom
e.g.NH3 –VB--consider valence electrons: F(2s,2p)+3H(1s)
split of bonding from non bond is large, since -bonds are strong
Valence bond (VB) model, local (like diatomic) bonds all , in NH3
easiest way for chemist – hybrid idea
– bonding interaction bigger energy than s-p split
– “promote” s equivalent to p
treat s, p as virtually degenerate—can mix
sp3  4 orbitals point at apex tetrahedral
sp2  3 orbitals – 120 apart(leaves pz  ) double bond
sp  2 orbitals – 180 apart (px,y orbital 2) triple bond
linear combinations again, like px = p1-p-1, but not degenerate
sp31 = s + pz + px + py
sp21 = s + px + py
sp32 = s + pz - px - py
sp22 = 2s - px - py
sp33 = s - pz + px - py
sp23 = px - py
sp34 = s - pz - px + py
sp1 = s + pz sp1 = s - pz
(these coefficients not normalized)
NH3 – pyramidal, use sp3 hybrids, one orbital non bonding
BH3 – planar, sp2: 2pFx, 2pFy with 2sF, empty  - 2pFz no overlap 1sH
XIII–65
These take chemical intuition (scheme assume sp3 degenerate)
balance electronegativity and how molecules form
C2H2 –linear, C2H4 –planar, {NH3 non planar, BH3 planar}
A little more depth
We took simplistic separation approach
How do we evaluate energies / realistic wave/fct
imagine AO’s as a basis set of orbitals – “guesses”
H =  iHi /  ii = Hii/Sii expectation value (i,i)-diagonal
H =  iHj /  ij = Hij/Sij interaction value (i,j)–off-diagonal
These are called Coulomb integrals (Hij) – energy operator
which are normalized to overlap integrals (Sij)
So if  =  ci i i.e. if MO made of AOs, then want to
optimize ci -- use variation principle: H/ci = 0
e.g. let:  = cAA + cBB (as in b) then
-- demonstrate
E = [cA2HAA + 2cAcBHAB + cB2HBB]/[cA2 + 2cAcBSAB + cB2]
set E/cA = 0, E/cB = 0  2 equations, 2 unknowns
easiest differentiate both sides  secular equations,
{E[cA2+2cAcBSAB+cB2]}/cA={cA2HAA+2cAcBHAB+cB2HBB}/cA
E/cA[cA2+2cAcBSAB+cB2] + E[2cA+2cBSAB] = [2cAHAA+2cBHAB]
First term = 0 (min), rest: 0 = 2cA(HAA - E)+ 2cB(HAB - ESAB)
same for HBB
XIII–66
Solution:
energies obtained from secular determinant:
(polynomial, quadratic
2 solution – E1,E2)
0 0=
general wave function (LCAO-MO):
HAA – E
HAB – ESAB
HAB – ESAB
HBB – E
 =  ci i
“Secular Determinant” n-AOs give n-MOs  n - EK values
0=
H11 – E
H12 – ES12
H13 – ES13
:
:
H1n – ES1n
H21 – ES21  
H22 – E
Hn1 – ESn1

Hnn – E
:
:
:
Then insert EK’s into secular equations (n - equations)
ij (Hij - EkSij)ckj = 0
n-Ek makes n separate sol’n
solve set of simultaneous equations for coefficients
 get a set of n - ciK – for each (n-values) EK
Delocalization—major aspect of MO model, see above,
all MOs are solutions to secular determinant (energies) and
associated set of secular equations (wave functions),
so each has contributions from all the AOs - delocalized
Valence Bond method, localized view, chemical appeal
Idea, AOs on one atom overlap those on another
and share and electron pair  bond
H2VB (1,2) = A(1)B(2) + A(2)B(1)
electrons identical, need both forms,
bonding - increase e- density between
XIII–67
Contrast this with the MO picture
MO(1,2) = b(1)b (2) = [A+B](1) [A+B](2)
= A(1)B(2)+A(2)B(1) + A(1)A(2)+B(2)B(1)
____________________________________
-
VB 2 e covalent state
_____________________________________
ionic states (excited)
Difference, MO too much emphasis on ionic states,
VB too much bond strength (e- all between nuclei)
Bring them together
VB add excited state (ionic) contribution (e- promoted)
VB’(1,2) = A(1)B(2)+A(2)B(1) + [A(1)A(2)+B(2)B(1)]
MO add excited configuration, *(1)*(2)
b(1)b (2) '*(1)*(2)] =
A(1)B(2)+A(2)B(1) + (1-'[A(1)A(2)+B(2)B(1)]
Adjusting ' will bring these to the same functional form
Can use variation method to optimize '
Poly atomic VB  select AOs with valence e- to mix,
share e-, form 2e- bond, promote e- to get hybrids
C (2s)2(2p)2  C (2s)1(2p)3  C (sp3)4
Then can form local bonds: C-H:  = C (sp3)1+H(1s)1
And C-C:  = Ca(sp3)1 + Cb(sp3)1 etc.
Problem: VB completely local, so misses out on delocalized
systems like  and interactions between bond networks
XIII–68
VSEPR model — describe molecular geometry
based on e- pair repulsion, in VB – bonds, lone pairs
Minimize energy by distance, spread, open angles
Diatomic – linear, trivial, one bond
Triatomic – linear or bent, depend on lone pairs
O=C=O - two bonds around C, both double, linear
H-O-H - two bonds and two lone pairs around O
Approx. tetrahedral, pairs more space
Rules: 1. bonds to ligands and lone pairs repel
2. lone pairs require more angular space than bonds
3. More electronegative ligand—increase space,
more electronegative central atom, decrease space
4. multiple bond needs more space than single bond
Ex. CH4 –tetrahedral, angle between C-H bonds, 109º
NH3 – pyramidal, lone pair 4th position, angle ~107º
H2O – bent, two lone pairs other positions, angle ~105º
CO2 – linear, SO2 – bent (lone pair)
C2H2 – linear, C2H4 – planar, C2H6 – each ~ tetrahedral
XIII–69
Solving MO problem for real systems:
Clearly all of this gets complex for large molecules –
most quantum chemistry problems solved using computers
 ~ i ciiAO
iAO ~ iakfk
fk – functions, e.g. STO  e-r,
2
Gaussian  e-’r
sometimes fix ak’s by some optimization (solve for atomic Emin)
always optimize ci by minimizing energy
can also vary geometry to optimize energy
 structure determination
Hückel Model of -systems -- a little different
-bonds – weaker – so usually highest E filled MO - HOMO
symmetry – out of the plane – not mix with -orbital
Hückel Theory – empirical approximation
— solve -system separately, idea is  not mix with 
 = i cii
i = 2pzi just sum over pz orbital on atoms i
this ignores the b orbitals - lower in energy (* higher)
Ex. Butadiene
k = ck11 + ck22 + ck33 + ck44
normal secular determinant of just C 2pz or  electrons:
H11 – 
H12 – S12
H13 – S13
H14 – S14
H21 – S21 H22 – 
H23 – S23
H24 – S24
0 = H31 – S31 H32 – S32
H33 – 
H34 – S34
H41 – S41 H42 – S42
H43 – S43
H44 – 
Rules: 1. Let Sij = 0 i  j -- no overlap between atoms
2. let Hii = (i = j) -- All the same, diagonal, Sii = 1
3. let Hij = 
i = j  1, near neighbor interact
4. let Hij = 0
non-near neighbor do not interact
XIII–70
Hückel secular determinant:
0
–


–

0=
0

–
0
0


0=
x
1
0
0
1
x
1
0
0
1
x
1
0
0
1
x
0
0

–

solution – divide entire
matrix (determinant) by 
then let – x = ( – )/
det - expand by minors (show):
 x4 – 3x2 + 1 = 0
solution (just quadratic eq. let y=x2):
x = 0.618, 1.618
Plug into: x = ( – )/to get
expressions for 
1 =   1.618 
2 =   0.618 
Butadiene has 4 -electrons,
could have 2 in each orbital:
Eg = 21+ + 22+
= 4 + 4.472  - ground state: (1)2 (2)2
Eex = 21+ + 2+ + 2= 4 + 3.236 excited state: (1)2 (2)1 (3)1
E = 1.236 
– transition in uv:
XIII–71
4
Plug  into:  (Hij – kij)cjk = 0 for j = 1 – 4
k
i 1
(ij - no Sij)
(k)c1 + c2 = 0
c1 + (k)c2 + c3 = 0
c2 + (k)c3 + c4 = 0
c3 + (k)c4 = 0
4 equations (j) solve simultaneously for cj , repeat each k
Ex:
1 = 0.37 p1 + 0.60 p2 + 0.60 p3 + 0.37 p4
2 = 0.60 p1 + 0.37 p2 – 0.37 p3 – 0.60 p4
3 = 0.60 p1 – 0.37 p2 – 0.37 p3 + 0.60 p4
4 = 0.37 p1 – 0.60 p2 + 0.60 p3 – 0.37 p4
(0 nodes)
(1 node)
(2 nodes)
(3 nodes)
Hückel MO’s (phasing, shapes - side view) for butadiene:
# nodes  E
Hückel MO’s for benzene very similar, but since cyclic get an
extra term in determinant, i.e. H16 and H61≠0  try it!
What if system not -system?
QM calculational Complexity builds-up
– semi empirical—extend the idea above
Extended Hückel – let atoms differ, calculate Sij
but valence electrons interaction – empirical parameter
(i.e. include  +  but parameterize Hij)
CNDO – MNDO – INDO
– basically variations of neglecting overlap (NDO)
and including parameters for Hij
XIII–72
ab initio  Compute solution to Schrödinger Equation –
no parameters but many approximations –
SCF {typically Hartree-Fock – set up w/f and
use to calculate potential (V(ri)) and
re-solve -- keep cycling until no change in V(ri)}
Being a little more precise—add spin:
Multielectron wave function - obey Pauli (anti-symmetric w/r/t
exchange of electrons) -- use determinant form:
(r1,r2,r3,…,rN) =
A(r1)
A(r2)
:
A(r4)
B(r1)
B(r2)
C(r1)

N
N(rn)
This will be anti-symmetric w/r/t exchange of electron
i.e. (r1, r2, r3, …) = -(r1, r3, r2, …) etc.
since exchange electron  exchange rows in determinant
Still product of one-electron functions (MO) – average V(rij)
Solving ab-initio problem involves 1-2-3-4 center integrals
example of what this means: ∫ A(1)B(2) H C(1) D(2) d
big problem with 3 + 4 center  sol’n: often neglect some
2
Gaussian orbitals gk(r) = ( rk) e-r
make integrals simpler, rep. AOs as: STO =  cngn(r)
Calculations get very big – tend to scale like n4 – n5
n = number basis functions (the gn(r) above)
so small molecules very quick (do on PC)
big molecules (e.g. proteins) become almost impossible
Methods work well for molecules up to ~100 atoms
(or ~1000 basis functions  represent AO’s)
multi center integrals  need large memory / disk
 takes time (now dedicated PC’s or clusters)
XIII–73
Alternate approach: Density Functional Theory (DFT)
Hohenberg-Kohn  (r) electron density for grd state
in principle can determine w/f:  (r1, …, rn)
Advantage – calculations become easier / bigger system
Disadvantage – involves parameters / approximations
(variation works differently)
[now part of standard Quantum Chemistry programs]
Solution – density orbital density etc. (self-consist)
QM Goals in Chemistry:
1. molecular properties  structure, bonding,
charge distribution, dipole moments
spectral transitions –IR(vibration) -grd st property
UV (electronic) - more difficult, excited state
2. Structure – optimize geometry – minimum E
3. Kinetics – reaction surface – E as fct geometry
Very large molecules  e.g. proteins / nucleic
Use results of Quantum Mechanics  create Uel (R)
an energy function that varies with nuclear position
 minimize geometries: molecular mechanic model
 dynamics / trajectories: molecular dynamics