MODELING MATTER AT
NANOSCALES
2. Energy of intermolecular processes
1
Understanding molecular
interactions
Rough classification
Strong interactions:
• r  2 Å (r  0.2 nm)
• geometry and electronic states of interacting molecules are noticeably
perturbed
• chemical bonds could be formed or destroyed
Rough classification
Strong interactions:
• r  2 Å (r  0.2 nm)
• geometry and electronic states of interacting molecules are noticeably
perturbed
• chemical bonds could be formed or destroyed
Weak interactions:
• 2 Å < r < 6 Å (0.2 nm < r < 0.6 nm)
• geometry and electronic states of interacting molecules are very slightly
perturbed
• appearance or destruction of chemical bonds is quite impossible
Modeling the approach between
two molecules
Hypersurfaces are multidimensional functions where the
potential energy of a system is depending on coordinates and
mass of each and every particle.
Modeling the approach between
two molecules
Hypersurfaces are multidimensional functions where the
potential energy of a system is depending on coordinates and
mass of each and every particle.
They must contain information to describe the behavior of
interactions that can occur between the involved molecular
bodies:
E  E (Z , R)
Modeling the approach between
two molecules
Hypersurfaces are multidimensional functions where the
potential energy of a system is depending on coordinates and
mass of each and every particle.
They must contain information to describe the behavior of
interactions that can occur between the involved molecular
bodies:
E  E (Z , R)
Finding wells and valleys in hypersurfaces means owning the
most probable pathways and structures involved in molecular
interactions.
Modeling the approach between
two molecules
A process of approaching a body A to B,
interacting, and giving an associated AB in the
nanospace can be represented as:
A + B  AB
Modeling the approach between
two molecules
A process of approaching a body A to B,
interacting, and giving an associated AB in the
nanospace can be represented as:
A + B  AB
The association energy hypersurface can then
be expressed, generally, by:
Eassoc  Eassoc ( Z A , Z B , R)
Quantum variational modeling
of molecular associations
Quasi exact solution of molecular
associations
A “quasi exact” solution consists in the direct
quantum calculation of the many body system
by means of an adequate wave function Y0.
Hˆ Y0  E0 Y0
11
Quasi exact solution of molecular
associations
A “quasi exact” solution consists in the direct
quantum calculation of the many body system
by means of an adequate wave function Y0.
Hˆ Y0  E0 Y0
This kind of calculation uses to be limited to
consider other than very small systems because
the huge computational requirements.
12
The quantum expression
The wave equation of the associated species
could be written as:
AB
AB
AB
ˆ
HY  E Y
and the corresponding to the isolated bodies:
A
A
A
ˆ
HY  E Y
B
B
B
ˆ
HY  E Y
13
Variational solution
The calculation of energy differences between the tied
species and those corresponding to the original
isolated components provides a reliable and “a priori”
value of the association energy:
Eassoc ( Z A , Z B , R )  E
E
AB
E E
A
B
14
Variational solution
It must be observed that searching variables for a
variational energy optimization will strongly depend on
the character and size of AB.
15
Variational solution
It must be observed that searching variables for a
variational energy optimization will strongly depend on
the character and size of AB.
The calculation of EAB must be determined with “size
consistency” respect to the reference of the isolated
species A and B.
16
Variational solution
A general overview, still valid, can be found in:
17
Variational solution
A general overview with more recent tools:
18
Variational solution
A variational optimization of EAB has been proposed
several times in literature, ranging from first principle
ab initio calculations of the associated species to the
selection of relevant geometrical parameters during the
interaction process to monitor the evolution of relevant
quantum properties.
•
•
Ayers, P. W.; Parr, R. G., Variational Principles for Describing Chemical Reactions. Reactivity Indices Based
on the External Potential. J. Am. Chem. Soc. 2001, 123 (9), 2007-2017.
Specchio, R.; Famulari, A.; Sironi, M.; Raimondi, M., A new variational coupled-electron pair approach to
the intermolecular interaction calculation in the framework of the valence bond theory: The case of the
water dimer system. J. Chem. Phys. 1999, 111 (14), 6204-6210.
19
Quantum perturbational
understanding of molecular
associations
The quantum expression in
terms of a perturbation
Departing from the wave equation:
Hˆ Y AB  E AB Y AB
the Hamiltonian can be considered as:
A
B
ˆ
ˆ
ˆ
H  H 0  H 0  Hˆ assoc
 Hˆ 0  Hˆ assoc
And the energy eigenvalue can be decomposed as:
AB
0
assoc
E
E E
23
The quantum expression in
terms of a perturbation
Departing from the wave equation:
Hˆ Y AB  E AB Y AB
the Hamiltonian can be considered as:
A
B
ˆ
ˆ
ˆ
H  H 0  H 0  Hˆ assoc
 Hˆ 0  Hˆ assoc
And the energy eigenvalue can be decomposed as:
AB
0
assoc
We must take into account that this approach allows a direct
identification of the Hamiltonian and the energy originated from
molecular associations and, consequently, facilitates the understanding
of the process.
E
E E
24
The quantum expression in
terms of a perturbation
The Hamiltonian of association according the theory of
perturbations applied to non reactive molecular interactions can
be developed following the “electrostatic approach” as:
Hˆ assoc   A  B rij1
i
j
  A  B Z k rik1   A  B Z l rlj1
i
k
l
j
  A  B Z l Z k Rlk1
l
k
where i,j are sub-indexes for electronic wave functions and k,l
for nuclei in the A and B systems, respectively.
25
Case A. Wave functions of non
overlapping bodies (long distances):
According to the Rayleigh - Schrödinger perturbation theory
(RSPT), the wave function of the system is:
Y0AB  Y0A Y0B
A
B
A
B
ˆ

Y
Y
|
H
|
Y
Y
A
B
A
B
a
b
assoc
0
0 
 
Y
Y
a
b
A
B
A
B
E0  E0  Ea  Eb
a
b
 ...
being the sums on ground and excited vibronic states a,b of
systems A and B, respectively.
26
Case A. Wave functions of non
overlapping bodies (long distances):
In this case the energy of the system can be expressed in terms
of the previous general expression:
Hˆ Y AB  E AB Y AB
as the eigenvalue developed in series from the non perturbed
energy of the system E0:
E AB
 Y0AB | Hˆ | Y0AB 

 E0  E1  E2  ...
AB
AB
 Y0 | Y0 
27
Case A. Wave functions of non
overlapping bodies (long distances):
In this case the energy of the system can be expressed in terms
of the previous general expression:
Hˆ Y AB  E AB Y AB
as the eigenvalue developed in series from the non perturbed
energy of the system E0:
E AB
 Y0AB | Hˆ | Y0AB 

 E0  E1  E2  ...
AB
AB
 Y0 | Y0 
Then, it can be considered that:
Eassoc  E AB  E0  E1  E2  ...
28
Case A. Wave functions of non
overlapping bodies (long distances):
The zeroth order energy is then:
E0  E0A  E0B
29
Case A. Wave functions of non
overlapping bodies (long distances):
The zeroth order energy is then:
E0  E0A  E0B
and the association energy is then:
Eassoc  E1  E2  ...
  Y0A Y0B | Hˆ assoc | Y0A Y0B 
A
B
2
ˆ
A
B |  Y Y | H assoc | Y0 Y0  |
 
 ...
A
B
A
B
Ea  Eb  E0  E0
a
b
A
a
B
b
where E1 expresses the first order perturbation, E2 the second
and so on.
30
The first order: electrostatic energy
First order perturbation energy is given by purely
electrostatic interactions (Coulombic) only depending on
A B
charge distributions ( Y0 Y0 ) of the ground state:
E1   Y0A Y0B | Hˆ assoc | Y0A Y0B   ECoul
The first order: electrostatic energy
First order perturbation energy is given by purely
electrostatic interactions (Coulombic) only depending on
A B
charge distributions ( Y0 Y0 ) of the ground state:
E1   Y0A Y0B | Hˆ assoc | Y0A Y0B   ECoul
It can be expressed after some algebra as:
A 
B 
r (rn ) r (rm ) 3  3 
E1   
d rn d rm
 
| rn  rm |
nm
X
Y
where is the charge distribution of the ground
  state 0 for
each pair of particles placed according the rn , rm position
rX
vectors .
The first order: electrostatic energy
It can be shown that the E1 energy
corresponding to interactions
between permanent dipole
moments of two interacting bodies
with a neglected net charge is:
B
B
R
A
A
A B
A 
B 
E1  R  a  b  3 a  RAB b  RAB 
3

A

where a is the A molecule dipole moment in the a state,

R is the distance between their charge centers, and RAB is
a non-dimensional unitary vector in the direction of R.
33
The first order: electrostatic energy
• Accurate expressions for electrostatic energy are only
useful in cases and distances where other perturbing
effects are absent.
34
The first order: electrostatic energy
• Accurate expressions for electrostatic energy are only
useful in cases and distances where other perturbing
effects are absent.
• The simplest representation is to find or assign a
constant fractional charge to each atom in the
system, in the framework of the position of nuclei:
q A qB
ECoul  
A B rAB
35
The first order: electrostatic energy
• Accurate expressions for electrostatic energy are only
useful in cases and distances where other perturbing
effects are absent.
• The simplest representation is to find or assign a
constant fractional charge to each atom in the
system, in the framework of the position of nuclei:
q A qB
ECoul  
A B rAB
• More accurate representations must take into
account anisotropy of orbitals in molecules and
therefore, include a multipole analysis.
36
The second order: induction and
dispersion
Second order perturbation energy is intrinsic negative
(stabilizing) and it is given by:
A
B
A
B
2
ˆ
|

Y
Y
|
H
|
Y
Y

|
A
B
a
b
assoc
0
0
E2    
A
B
A
B
Ea  Eb  E0  E0
a
b
37
The second order: induction and
dispersion
Second order perturbation energy is intrinsic negative
(stabilizing) and it is given by:
A
B
A
B
2
ˆ
|

Y
Y
|
H
|
Y
Y

|
A
B
a
b
assoc
0
0
E2    
A
B
A
B
Ea  Eb  E0  E0
a
b
This term can be decomposed as:
where:
E2  E
A
ind
E
B
ind
 Edisp
A
Eind
is the polarization or induction energy because a
redistribution of charge in A owing to the interaction with the
charge distribution in system B.
Edisp is the “dispersive energy”.
38
Inductive effect
The expressions for inductive terms are:
A
B
A
B
2
ˆ
A
A |  Ya Y0 | H assoc | Y0 Y0  |
Eind   
A
A
E

E
a 0
a
0
A
B
A
B
2
ˆ
|

Y
Y
|
H
|
Y
Y

|
B
0
b
assoc
0
0
Eind
  B
EbB  E0B
b0
39
Inductive effect
The expressions for inductive terms are:
A
B
A
B
2
ˆ
A
A |  Ya Y0 | H assoc | Y0 Y0  |
Eind   
A
A
E

E
a 0
a
0
A
B
A
B
2
ˆ
|

Y
Y
|
H
|
Y
Y

|
B
0
b
assoc
0
0
Eind
  B
EbB  E0B
b0
They are related with the effects of non – perturbed charge
X
distributions in a molecule with a Y0 associated wave
Y
function on the Ya other, as developed in terms of their
excited vibronic states a, b,… to model distortions in their
charge distribution.
40
Inductive effect
This term becomes very important if ions are present.
41
Inductive effect
This term becomes very important if ions are present.
For an ion of net charge qB and a molecule with aA
2
polarizability:
qa
A
ind
E

B
2R
A
4
AB
42
Inductive effect
This term becomes very important if ions are present.
For an ion of net charge qB and a molecule with aA
2
polarizability:
qa
A
ind
E

B
2R
A
4
AB
In the case of neutral molecules this term is not important.
43
Inductive effect
A molecule with B dipole moment
acting together with a non polar
molecule A develops and inductive
energy given by:
A
ind
E

 a A (3 cos q B  1)
2
B
2
2R
B
B
6
AB
where qB is the angle between B and
the internuclear axis.
qB
RAB
A
44
Dispersive effect
Dispersive effects can be modeled in the framework of
perturbation theory as originated in the overlapping space
of excited (a, b,…) and fundamental (0) vibronic states
unperturbed molecules.
45
Dispersive effect
Dispersive effects can be modeled in the framework of
perturbation theory as originated in the overlapping space
of excited (a, b,…) and fundamental (0) vibronic states
unperturbed molecules.
This is a very theoretical expression of dynamic correlation
between electrons in both interacting systems on the
grounds of their local electronic configurations:
Edisp
A
B
2
ˆ
A
B |  Y Y | H assoc | Y0 Y0  |
  
A
B
B
( E  E0 )  ( Eb  E0 )
a 0 b0
A
B
a
b
A
a
46
Dispersive effect
The energy of dispersive term
can be expanded as:

Edisp   Cn R
n
n 6
47
Dispersive effect
The energy of dispersive term
can be expanded as:

Edisp   Cn R
n
n 6
The leading term of this series results to be:
C6    
a 0 b0
| 2 A  B   A  B   A  B |
z
z
x
x
y
2
y
(E  E )  (E  E )
A
a
A
0
B
b
B
0
where  Az is the modular component in z coordinate of the
electric moment of transition a  0.
48
Dispersive effect
The energy of dispersive term
can be expanded as:

Edisp   Cn R
n
n 6
The leading term of this series results to be:
C6    
a 0 b0
| 2 A  B   A  B   A  B |
z
z
x
x
y
2
y
(E  E )  (E  E )
A
a
A
0
B
b
B
0
where  Az is the modular component in z coordinate of the
electric moment of transition a  0.
It arises from the correlation of the instantaneous dipole
fluctuations in the charge density of the two nanoscopic
49
bodies.
Dispersive effect
This term brings up the London dispersion energy as:
E London  C6 R
6
50
Dispersive effect
This term brings up the London dispersion energy as:
E London  C6 R
6
For most effects, the London dispersive term is
approached as the total dispersive energy
Edisp  ELondon
51
Dispersive effect
Higher terms in the expansion:

Edisp   Cn R  n
n6
are related dipole - quadrupole interactions (R-8), both
quadrupole - quadrupole and dipole - octopole (R-10), etc.
52
Dispersive effect
Higher terms in the expansion:

Edisp   Cn R  n
n6
are related dipole - quadrupole interactions (R-8), both
quadrupole - quadrupole and dipole - octopole (R-10), etc.
They are not frequently taken into account.
53
Case B. Overlapping bodies
(intermediate distances):
• It must be observed that exchange interaction Eex is
absent as either a first or second order term because
bonding is a negligible contribution to energy at large
separations.
54
Case B. Overlapping bodies
(intermediate distances):
• It must be observed that exchange interaction Eex is
absent as either a first or second order term because
bonding is a negligible contribution to energy at large
separations.
• Overlapping of electronic clouds during the approach of
two nanoscopic bodies means the increase of electronic
repulsion (because both electrostatic effects and the
Pauli principle) and attraction (because electron
exchange effects arise).
55
Case B. Overlapping bodies
(intermediate distances):
• It must be observed that exchange interaction Eex is
absent as either a first or second order term because
bonding is a negligible contribution to energy at large
separations.
• Overlapping of electronic clouds during the approach of
two nanoscopic bodies means the increase of electronic
repulsion (because both electrostatic effects and the
Pauli principle) and attraction (because electron
exchange effects arise).
• At intermediate distances, repulsion is usually
predominant, although the true behavior is strongly
dependent on the character of the system.
56
Case B. Overlapping bodies
(intermediate distances):
• It must be observed that exchange interaction Eex is
absent as either a first or second order term because
bonding is a negligible contribution to energy at large
separations.
• Overlapping of electronic clouds during the approach of
two nanoscopic bodies means the increase of electronic
repulsion (because both electrostatic effects and the
Pauli principle) and attraction (because electron
exchange effects arise).
• At intermediate distances, repulsion is usually
predominant, although the true behavior is strongly
dependent on the character of the system.
• The use of perturbative procedures for accounting such
effects must necessarily take into account isotropy
features of the interacting electron clouds.
57
Case B. Overlapping bodies
(intermediate distances):
These cases are treated nowadays by variational
procedures as implemented in current program
packages.
58
Case C. Prereactive interactions:
• At very small separations between different nanoscopic
bodies the unperturbed Hamiltonian is very similar to the
one corresponding to a united molecule. Therefore,
effects are mainly originated in Coulomb – exchange
interactions among electrons.
59
Case C. Prereactive interactions:
• At very small separations between different nanoscopic
bodies the unperturbed Hamiltonian is very similar to the
one corresponding to a united molecule. Therefore,
effects are mainly originated in Coulomb – exchange
interactions among electrons.
• A comprehensive treatment is more appropriate than the
perturbative solution because pre-reactive effects are
usually present and they are hardly considered as a
factor in any kind of pairwise development.
60
Case C. Prereactive interactions:
• A detailed study of the method and the quality of the
hypesurface is usually very important.
61
Approximate calculations of
molecular associations
Resumé on the physical origin of
the main interactions
Eassoc ( Z A , Z B , R )  E
 E AB  E A  E B
Eassoc = ECoul + Eex + Eind + Edisp
63
Resumé on the physical origin of
the main interactions
Eassoc ( Z A , Z B , R )  E
 E AB  E A  E B
Eassoc = ECoul + Eex + Eind + Edisp
The important terms for strong interactions at short
distances are:
• Ecoul is the Coulombic or purely electrostatic
interaction energy between polarized zones of
molecules
64
Resumé on the physical origin of
the main interactions
Eassoc ( Z A , Z B , R )  E
 E AB  E A  E B
Eassoc = ECoul + Eex + Eind + Edisp
The important terms for strong interactions at short
distances are:
• Ecoul is the Coulombic or purely electrostatic
interaction energy between polarized zones of
molecules
• Eex is the orbital exchange energy if electronic
overlap is present
65
Resumé on the physical origin of
the main interactions
Eassoc ( Z A , Z B , R )  E
 E AB  E A  E B
Eassoc = ECoul + Eex + Eind + Edisp
The active terms in the case of weak interactions at
intermediate distances are:
• Eind means inductive effect energy of charges in one
molecule on the electronic cloud of the other
66
Resumé on the physical origin of
the main interactions
Eassoc ( Z A , Z B , R )  E
 E AB  E A  E B
Eassoc = ECoul + Eex + Eind + Edisp
The active terms in the case of weak interactions at
intermediate distances are:
• Eind means inductive effect energy of charges in one
molecule on the electronic cloud of the other
• Edisp means dispersive or London energy,
characterizing the so called van der Waals forces
67
Aproximation by a power series
Finding Eassoc must be solved in a practical way and is
usually developed in a power series of distances among
involved nanoscopic bodies: 
Eassoc   Vn R  n
n 1
where Vn values will depend on the kind of interactions to
consider in each case.
68
Practical solutions for neutral
spherical or quasi-spherical
systems:
Lennard - Jones potential:
 s 12  s 6 
  
 
Eassoc ( RAB )  4e 
 RAB 
 RAB  
where e is the deepness of the energy well between
the two bodies, and s is the distance where Eassoc is
zero.
69
Practical solutions for neutral
spherical or quasi-spherical
systems:
Buckingham potential:
Eassoc ( RAB )  ae
bRAB
c6
c8
 6  8 ...
RAB RAB
where a, b, c6 y c8 are constants.
70
Practical solutions for neutral
isotropic systems:
Stockmayer potential:
c6
c12
Eassoc ( RAB )  12  6
RAB RAB

B
B
qB
 A  B (2 cos q A cos q B  sin q A sin q B cos f )
RAB
qA
A
A
3
RAB
where qA is the angle between RAB joining the centers of
mass and the direction of the dipole moment
A, and f




is the dihedral angle between planes  A , R AB  and  B , RAB 
Practical solutions for general
cases:
If the energy of association is expressed as a series in terms of
interactions among atoms i, j,… :
Eassoc   V2 (i, j )
i j
  V3 (i, j , k )
i  j k
 ...
 V0 (i, j , k , l ,..., N )
it will converge over N atoms, being among them aggregate
entities that could qualify as molecules or clusters.
72
Practical solutions for general
cases:
The “pairwise additive approximation” also allows to write:
Eassoc   V (i, j )
i j
where potentials are different to those of the series with
interactions of more than two bodies.
73
Practical solutions for general
cases:
The “pairwise additive approximation” also allows to write:
Eassoc   V (i, j )
i j
where potentials are different to those of the series with
interactions of more than two bodies.
Nevertheless, this pairwise additive approximation is expected
to prevent convergence.
74
Practical solutions for general
cases:
Clementi potentials:
AijAB BijAB CijABqi q j
V (i , j )   6  12 
rij
rij
rij
where AijAB, BijABand CijAB are adjusted constants for each
pair of atoms of types A and B with a net electrostatic
charge q.
75
Practical solutions for general
cases:
Clementi potentials:
AijAB BijAB CijABqi q j
V (i , j )   6  12 
rij
rij
rij
where AijAB, BijABand CijAB are adjusted constants for each
pair of atoms of types A and B with a net electrostatic
charge q.
Constants are adjusted against accurate quantum
mechanical ab initio calculations.
76
Practical solutions for general
cases:
Fraga potentials:
V (i, j ) 
Aqi q j
rij
B ( f ia i q 2j  f ja j qi2 )
Cf i f ja ia j

rij4
 f a   f a 
 i i    j j 
 ni   n j 
Dci c j
 12
rij
1
2
1
2

 rij6

where A, B, C, and D are adjusted to give the appropriate
units, ai is the atomic polarizability, qi the net charge and ni
= Zi - qi the number of electrons on atom i.
77
Practical solutions for general
cases:
Fraga potentials:
V (i, j ) 
Aqi q j
rij
B ( f ia i q 2j  f ja j qi2 )
Cf i f ja ia j

rij4
 f a   f a 
 i i    j j 
 ni   n j 
Dci c j
 12
rij
1
2
1
2

 rij6

where A, B, C, and D are adjusted to give the appropriate
units, ai is the atomic polarizability, qi the net charge and ni
= Zi - qi the number of electrons on atom i.
Parameters ci and fi are adjusted to fit previous accurate ab
78
initio values.
Atom types
Either the Clementi’s and Fraga’s potentials, as well as
many other approaches for approximate non bonding
and bonding energies, are not parameterized by
elements but by atom types that mean elements
according their particular bonding conditions in the
considered nanoscopic system.
79
Hydrogen bond treatments
“Hydrogen bond” is an ubiquitous case of molecular interactions
behaving sometimes as electron exchange driven (quasi covalent
bonding) as well as other times being described as electrostatic
associations.
80
Hydrogen bond treatments
There are some potential formulas fitting hydrogen
bond properties in the case of different classes of
atoms involved. Solutions are:
1. A separate hydrogen bond potential, i.e. 10 - 12
Lennard - Jones, or a Buckingham potential with
different parameters.
81
Hydrogen bond treatments
There are some potential formulas fitting hydrogen
bond properties in the case of different classes of
atoms involved. Solutions are:
1. A separate hydrogen bond potential, i.e. 10 - 12
Lennard - Jones, or a Buckingham potential with
different parameters.
2. The use of normal dispersive potentials with
parameters dedicated to hydrogen bonding classes
of atoms.
82
Main conclusions
Covalent vs. non – covalent
associations
The border line for the separation between
molecules in covalent and non-covalent
interactions could be considered as 2 Å for most
organic systems.
84
Associative factors
The main interacting forces at larger distances
could be classified as those between:
• permanent multipoles (electrostatic)
85
Associative factors
The main interacting forces at larger distances
could be classified as those between:
• permanent multipoles (electrostatic)
• permanent multipoles and induced multipoles
(inductive)
86
Associative factors
The main interacting forces at larger distances
could be classified as those between:
• permanent multipoles (electrostatic)
• permanent multipoles and induced multipoles
(inductive)
• instantaneous variable multipoles (dispersive)
87
Energy scales for molecular
associations
• Eassoc ranges between 1 and 20 kcal mol-1 (4 and 85 kJ mol-1 or
350 and 7000 cm-1) in non covalent molecular aggregates
88
Energy scales for molecular
associations
• Eassoc ranges between 1 and 20 kcal mol-1 (4 and 85 kJ mol-1 or
350 and 7000 cm-1) in non covalent molecular aggregates
• van der Waals interactions reach values between 0.1 and 0.2
kcal mol-1 (0.4 and 0.8 kJ mol-1 or 35 and 70 cm-1)
89
Energy scales for molecular
associations
• Eassoc ranges between 1 and 20 kcal mol-1 (4 and 85 kJ mol-1 or
350 and 7000 cm-1) in non covalent molecular aggregates
• van der Waals interactions reach values between 0.1 and 0.2
kcal mol-1 (0.4 and 0.8 kJ mol-1 or 35 and 70 cm-1)
• Hydrogen bonding behaves between 2 and 5 kcal mol-1 (8 and 21
kJ mol-1 or 700 and 1750 cm-1)
90
Energy scales for molecular
associations
• Eassoc ranges between 1 and 20 kcal mol-1 (4 and 85 kJ mol-1 or
350 and 7000 cm-1) in non covalent molecular aggregates
• van der Waals interactions reach values between 0.1 and 0.2
kcal mol-1 (0.4 and 0.8 kJ mol-1 or 35 and 70 cm-1)
• Hydrogen bonding behaves between 2 and 5 kcal mol-1 (8 and 21
kJ mol-1 or 700 and 1750 cm-1)
• Covalent bonds range around 100 kcal mol-1 (420 kJ mol-1 or
35000 cm-1).
91