V(r)
Repulsive region
r=σ
rm
V(rm)=-ε, Fattr=Frep
V(σ)=0, Vattr=Vrep
r
Attractive region
εm
Short range interactions are dominated by interelectronic
repulsion, The quantitative treatment of such interactions
necessarily requires QM description (Pauli effects).
These interactions substantiate the concept of sterical hindrance.
V(r)
r
σ
Vrep(r≤σ)=∞
Vrep(r>σ)=0
r
r
σ=2r
Vatt=0
An analytical function reproducing the results of quantum
calculations is:
V (R ) 
C
R
n
n  10,12
C = constant characteristic of the atom-atom pairwise interaction
Van der Waals radius = it represents the minimum contact
distance between two atoms. It can be approximately considered as
the interatomic distance beyond the repulsive energy rapidly rises
rVdW ≈ σ
The Van der Waals radius also defines a Van der Waals area
(SVdW) and volume (VVdW). They represent areas and volumes
that cannot be penetrated (excluded areas and volumes).
The Van der Waals volume of a molecule is approximately
given by the sum of of the Van der Waals volumes of the atoms
or groups of atoms forming the molecule.
S VdW   
2
4
VVdW    3
3
Contact interatomic distances
Atom pair
C--C
Average distance
(Å)
3.2
Minimum distance
(Å)
3.0
C--O
2.8
2.7
C--N
2.9
2.8
C--H
2.4
2.2
O--O
2.8
2.7
O--N
2.7
2.6
O--H
2.4
2.2
N--N
2.7
2.6
N--H
2.4
2.2
H--H
2.0
1.9
At long range intermolecular distances we can approximately
neglect quantum effects and describe nuclei and electrons as
point charges following the laws of classical electrostatics
(Coulomb law).
The interaction energy between a charge qi and a charge
distribution qj is simply given by the sum of the pair coulomb
contributions:
Vi = q i 
j
qj
R ij
Fcoul
1
q1q2

 2
40 R 1 2
1
q1q2
V

40 R 12
The energy of coulomb interaction between ions is of the order of
250 kJ·mol-1.
q = electric charge = 1.60·10-19 C = 4.8·10-10 ues
1 ues = 1 electrostatic unit. It is the charge that at 1 cm of distance
from an other unit charge exerts 1 dyne force (CGS units).
 ( H 2O )  78
 ( esano )  2
The dipole moment is a vector measuring the distance between
the center of the positive and negative charge distributions:
l
-
+


  qr
μ
Unit positive and negative charges separated by a 1 Å distance:
 = q · l = 4.8·10-10·1·10-8 = 4.8·10-18 ues·cm = 4.8 D (Debye)
1 Debye = 10-18 ues·cm=3.336·10-30C·m
Dipole moment of some molecular bond
bond
(D)
bond
(D)
H-F
1.9
C-F
1.4
H-Cl
1.1
C-Cl
1.5
H-N
1.3
C-N
0.2
H-O
1.5
C-O
0.1
Which is the polarity of CO?
δ-C—Oδ+
H2O:
exp=1.85 D
(OH) = 1.52D  = 2 (OH) cos(52.5°)
O
H
H
Isomers of dichlorobenzene:
=0
=2.25D
=1.48D
R12
q1
μ2
q1  2
V=
2
40 R 12
1
The order of magnitude of this interaction is 15 kJ·mol-1. The
interaction can be attractive or repulsive depending on the
nature of the charge q1 and the charge–dipole relative
orientation.
l
q1
-q1

1  q1 q2
q1 q2

V=


l
40  r  l
r

2
2

l
x=
2r
r
q2

 q q 
1
1 
  1 2 


 40 r  1  x 1  x 


Per
l
 1
2r
1
 1  x  x 2  ......
1 x
1
 1  x  x 2  ......
1 x

 

q1 q2 
1
1  q1 q2
2
2
V=




1

x

x

1

x

x



40 r  1  x 1  x  40 r
q1 q2
q1 q2l
1 q2
 2 x  
=

2
40 r
40 r
40 r 2
Q = monopole. Ex. Na+
l
-
+
μ
dipole
Quadrupole
CO2
Octupole
The interaction energy decreases with the distance as faster
as the order of the multipolar interaction increases.
For an n-pole interacting with an m-pole:
V
1
r
n  m 1
The interaction energy can be attractive or repulsive
depending on the relative orientation between the two
molecular dipoles:
+ +
+-
-+
- For two non-rotating dipoles (fixed orientation, like in solids):
1  2
V=3
40 R 12
 int  2kJ  mol -1
For rotating dipoles (solution, gas) the interaction energy
should be averaged among all possible orientations following
the Boltzmann distribution.
Because attractive energy orientations are slightly favoured
with respect to orientations giving rise to repulsive
interactions, dipole-dipole interactions in solution or in gas
phases are attractive, depending on the temperature:
V =-C
12  22
kT R 612
At 25°C for two HCl molecules (μ=1D) at 5Å:
 int  0.07 kJ  mol -1
An apolar molecule under the effect of an external electric field
could be polarized.
Electric field
------
+++
-----+++
-----+++
Polarized molecule
 1
2
indotto  α  E  β : E  ......
2

polarizability
hyperpolarizability
Polarizability is a molecular property that increases with the
number of electrons belonging to the molecule and decreases
with the increase of the ionization potential :
Z (atomic number)
 
I(ionizati on potential)

' 
40
Volume polarizability
[m3]
polarizability [C2m2J-1]
Vacuum permittivity
[C2m-1J-1]
q=0
z
p=0
Q = -7.510-40 Cm2
x
CO2
xx= //
p// = //E//
// = 4.0510-24 cm3
 = 2.0210-24 cm3
yy=zz= 
p= E
The electric field generated by a permanent dipole moment
gives rise to a dipole moment (induced dipole moment) on a
nearby apolar molecule.
Cl
H
(HCl)= 1 Debye
At 3Å ε ≈ -0.8 kJ·mol-1
Benzene  = 110-29 m3
V=-
12  2
4 0 R 6
 int  0.8 kJ  mol -1
12
As the orientation of the induced dipole depends on the
orientation of the inducing dipole, the dipole-induced dipole
interaction does not depend on the thermal energy (kT).
A pure quantistic effect arising from the correlation between
the electron motions of the interacting atoms at large distances.
These interactions, named dispersion or London interactions,
occur in all systems, even between apolar molecules.
They are always attractive. Semiclassically can be described as
the interaction between istantaneous dipoles arising from the
fluctuations of electronic charge distributions.
Molecular polarizability
V =-C
 12  22
6
R 12
 C
Ionization energy
1  2
R 612
I1I 2

I1  I 2
For two CH4 molecules (=2.6·10-30 m3, I7 eV) separated
by 3 Å: εint  -2 kJ·mol-1.
Interaction
Ion-ion
Distance
dependence
R-1
Energy
(kJ mol-1)
250
Ion-dipole
R-2
15
Dipole-dipole
R-3
R-6
2
0.3
Dipoleinduced
dipole
Dispersion
(London)
R-6
0.3
R-6
2
Type
ion
Fixed dipoles
Rotating
dipoles
All the
molecules
In basence of ions and for rotating systems in solution, dipolar
interactions are attractive nad depend on the sixth inverse
power of the distance.
These contributions to the potential energy can be described
by an analytical finction as :
V=-
C
R6
The electrostatic origin of H-bond interaction is emphasized by
the involvement of strong electronegative atoms in competition
with the same H-atom:
D-(donor)  H+ - - - -A-(acceptor)
H-bond interaction can de described as a dipole-dipole
interaction between fixed dipoles:
VHB = -
C
3
R D- H - A
The order of magnitude of H-bond interaction is 20 kJ·mol-1
(R  2Å).
Length and strenght
of H-bond depend on the
electronegativity of the donor-acceptor pair and on the
geometry of the atomic groups:
1.03 Å
N
O
H
1.9-2.0 Å
C
 r0 12  r0 6 
V ( r )  4      
 r  
 r 
13
7

V ( r ) 24  r0   r0  
F 

2     
r
r0   r   r  
The attractive force is maximum at :
26
r
r0  1.244r0
7
6
At this distance:
Tipically around 10 pN.
Fmax  -2.397

r0