IX–59
Bonding – Diatomics - 2012 (Ch.10-Atkins, also
Ch. 16 Engel, House Ch. 9, Tinoco p.479-90)
For multi electron atoms we built up from H-atom
For molecules take same approach
Can solve problem for H2+
(1 – e- in non-central potential – 2 nuclei)
get something like s and pz -- g and *u
bonding -- 
anti-bonding—*
As go out with contours
Alternate – has nodal plane
gets more spherical
recall—nodes energycurvature
Both -orbitals have cylindrical symmetry about z-axis –
this is equivalent to mℓ = 0  call this: -state
e- attracted to both nuclei, far away looks like atom
 - bonding,g  * (e- density) increase between nuclei
* - anti-bonding, u  * decrease between nuclei
Energetics:
Reference state – 2
u
atoms
(if H2+ – atom and proton)
As atoms approach,
their electrons
– attract other nucleus
H + H+
reference
energy
bg
IX–60
– repel each other (not H2+)
-- nuclei always repel
Full Hamiltonian -need to know! – for units SI add (40)-1:
Balance
n-n repuls, e-n
2
2
2
2
2
2
2 attr.
H = -   /2M -   /2mei -  Ze /ri+  e /rij +  ZZe /R
N
n
 1
i 1
n-KE
i 
e-KE
Bonding balance:
ij
n-e attr.
, 
e-e repul.
n-n repul
e – n attract with n – n, e – e repel
Equation for N nuclei and n electrons
MO : For one electron, can drop sums on i and 4th term,
solution is a 1 e- wave fct. – i.e. orbital with 2 nuclei
Plot as fct of R (n-n separation) go from united to
separate atoms, or He2H, with H2 between
1-elctron plot:
He+
H + H+
(n=2 balso b
Note: only
electron terms,
need to add n-n
repulsion for
bonding energy
R0: equilibrium
bond distance
(minimum energy)
IX–61
H2+ orbitals as function of R
LCAO-MO Linear Combination of Atomic Orbitals
MO theory-- Simpler more transparent method
AO’s – form a basis, centered on one nucleus—
i.e. describe all the functions possible for 1 eFor molecule need MO –
orbital (1e-) delocalize over several nuclei
Form linear combination AO’s  LCAO–MO
Multi-electron MOs:
So if lowest E AO’s for H2+ are 1s
Then b ~ (1sA) + (1sB)
* ~ (1sA) – (1sB)
Build up e-density
between nuclei
Decrease
density
Continue on this track:
more electrons – just put 2 in
each MO (opposite spin)
H2+  (b)1 — weak bond
(1/2)
H2  (b)2 — strongly bound (1)
( )=
bond order
Bomd prder (B.O.) = (#e-bond - #e-antibond)/2
IX–62
Move on past H2 – multi-electron molecules:
He2+
 (b)2 (*)1
— weaker (1/2)
He2
 (b)2 (*)2 — no bond (0)
B.O = [(number)b – (number)*]/2
Can continue to 2nd row in same manner
(2sA) + (2sB)  b
discriminate by 2b, 2*
(2sA) – (2sB)  *
or b(2s), *(2s), or 2g, 2u
(2pz)A + (2pz)B  b molec. axis  z
(2pz)A – (2pz)B  * symm. about z  
(2px,y)A + (2px,y)B  b,u 2 possibilities x or y
(2px,y)A – (2px,y)B  *,g degenerate, weaker overlap
Pictures of orbitals:
Li2 bs*s
Li2 bs*s
Combination of 1s orbitals – H2
1s based *
Compare to 2s -- Li2 *
Inner contours opp. sign, node
IX–63
Representations
of bonding and
anti-bonding,
 and *,
molecular orbitals
Learn: recognize,
interpret shapes
LCAO-MO Scheme diagram --
Correlation Diagram
IX–64
Multi-electron wavefunction/state—follow atom model
method again – approx. H as sum of 1-elect hi
N
Hel =  (-2/2m i2 + V(ri)) V(ri) – elect. Potential
i 1
part depend on ri (all nuclei) only [H–F typical method]
Summed H  product total w/f:  =
N

i 1
i (ri)
i  MO = 1e- w/f
N
Energy is sum of occupied orbital energies: E =  i
i 1
Overlap of orbitals makes bonding (E dec.) antibonding (E inc.) interaction (Engel)
Typical E-levels homo nuclear diatomic
(rt. side, ex. O2, F2)
Aufbau—fill in order of increase
energy, 2 elect. each MO
1s interaction strong H2,
weak for 2nd row
Ordering changes as cross
periodic table interaction of
2s  and 2p  causes a shift
 see next page
IX–65
early in series – AO--less s-p separation, MO--b < b
later – more shielding– AO - s-p separate, MO - b < b
Homonuclear Diatomic MO diagrams (LCAO approach)
Express as configuration  2e- each , 4e each :
(1b)2 (1*)2 (2b)2 (2*)2 (3b(2p))2 (1b)4 (1*)4 (3*)2
3rd row series – same idea – these will be shrunk
IX–66
F2 MO diagram
N2 MO diagram (Engel)
Second row filling diagram (Engel) – see energies change
IX–67
Orbital overlap
in 1- and 2-D
(Engel)
Bond energy,
length (note
He2, Be2 no
bond!?) and
force constant
(Engel)
IX–68
Homonuclear Diatomics – bond energies, distances,
orders, configurations, terms
De
Re bond
Configuration
Molc (eV) (Å) order
(lowest energy)
H2+ 2.78 1.06 1/2 (g1s)1
H2 4.76 0.74
1 (g1s)2
Grd
term
2
g+
1
g+
He2
---
---
0
(g1s)2(u1s)2
1
g
Li2
1.1
2.67
1
(g1s)2 (u1s)2 (g2s)2
1
g
Be2
---
---
0
[KK]4(g2s)2(u2s)2
1
g
3
g
1
g
g
B2
2.9
1.6
1
KK(g2s)2(u2s)2(u2s)2
alt: KK(g2s)2(u2s)2
(u2s)1(g2p)1
C2
N2
6.4
10
1.24
1.12
2
3
KK (g)2 (u*)2 (u)4
KK g2 u*2 u4 (g2p)2
2
KK g2 u*2 u4 g2
(g*)2
O2
5.2
1.21
F2
1.6
1.42
1
Ne2
---
---
0
KK g2 u*2 g2 u4
(g*)4 (u*)0
KK g2 u*2 g2 u4
u*4 u*2
1
3
g
1
g
1
g
Comment
ground state
unbound
weak
ground state
unbound
2 possible
configurations
paramagnetic
ground state
strongest bond
paramagnetic
relative stability
of g2p and u2p
unbound
Note: a) “bond order” reflected in De, Re trends
b) He2,Be2,Ne2 all possible as excited states
– e.g. He2* – (g)2(u*)1(g2s)1 – partial bond
c) More than half-fill less well bound yet shorter
bond than less than half-fill for same bond order
d) Open shells have more than one possible term
(2S+1, where  = mL) (u4)  1g , 3g , 1g