Download Intermolecular Interactions and Potentials

Intermolecular Interactions and
Potentials
Origin of intermolecular force
• Is it Gravitational?
No, is of the order of 10-52 J and so negligible
• Strong force?
No, significant in the range of 10-4 nm, but molecular
dimensions are typically 5×10-1 nm
• Weak force?
No, although electromagnetic in origin but
exits only between nuclear particles
• Electromagnetic?
Yes, since they are charged particles, attractive at
long range and repulsive at short range
Types of intermolecular interaction
• Electrostatic
• Induction
• Dispersion
In terms of the coordinates r, θ of the point P and the axial
separations of the charge from O this may be written as


Q1
Q2
Φ=
+
 2 2

1
/
2
2
2
1
/
2
4 πε 0 

( r + z 2 − 2 z 2 r cos θ )
 ( r + z1 + 2 z1 r cos θ )
1
r1
r
P
r2
θ
-z1
Q1
O
z2
z
Q2
expanded in powers of z1/r and z2/r
1
Φ =
4 πε
o
{
Q
1
+ Q
r
2
+
(Q z − Q z ) cos θ
2 2
1 1
2
r
+
2
2
2
(Q z − Q z )( 3 cos θ − 1)
1 1
2 2
3
2r
+
}
...
previous eq. may be written as
1
Φ =
4πε 0
{
+
r
2
µ cos θ
Q
r
2
θ (3 cos θ − 1)
+
2
r
3
}
+ ...
Q = Q1+ Q2 , total charge of the molecule, zeroth
moment of the charge distribution
µ = Q2Z2-Q1Z1, is the dipole moment of the charge
distribution
Θ = Q1Z21+Q2Z22 , is the quadrupole moment of the
charge distribution
the above eq. is valid only for r >> z1, z2 . Since we are interested in the longrange interactions this approximation is valid.
Interaction of two linear charge distribution
Now we consider the energy of interaction of two charge
distribution
-Z1
-Z’1
Z2
O
Q1
-Z1
Q2
-Z1
Q’1
Q’2
Q2
(a)
Q’2
Z’2
-Z’1
P
(b)
Q’1
Q’2
Z2
O
Q1
P
Z2
O
Q1
Z’2
P
(c)
Q2
Q’1
θ1
Q1
Q2
θ2
Ф
(d)
For the configuration (a) the electrostatic energy of the two distribution is Uael(r) =
Q’2Ф(Q’2) + Q’1Ф(Q’1)
Ф(Q’2) is the electrostatic potential at Q’2 due to the first (unprimed) molecule,
Ф(Q’1) is electrostatic potential at Q’1 arising from the same source
Since these electrostatic potentials can be expressed in terms of z’2 and z’1
U
el
{
1
a
(r ) =
4 πε
0
( µQ '− Qµ '
QQ '
+
2 µµ '
−
+
2
r
r
+
3
r
(3 Θ ' µ − 3 µ ' Θ )
( Q Θ ' + ΘQ '
+
3
r
6 ΘΘ '
r
} → ( A)
+ ...
4
5
r
various term in the electronic energy is the dipole-dipole interaction which varies
as r-3 and is given by
Q’ = Q’1+ Q’2 , total charge of the second molecule
µ’ = Q’2z’2-Q’1z’1, is the dipole moment
Θ’ = Q2’z2’2+Q1’z1’2 , is the quadrupole moment
various term in the electronic energy is the dipole-dipole interaction which
varies as r-3 and is given by
−2 µµ '
U µµ ' (r ) =
4πε 0 r 3
a
Thus in the configuration the dipole-dipole contribution to the interaction
energy is negative and there is an attractive force between the neutral
molecules
Similar development can be carried out for configurations (b) and (c) that
shown earlier for the dipole-dipole interaction of two neutral molecules and
can obtain
+2µµ '
U µµ ' (r ) =
4πε 0 r 3
b
Which corresponds to repulsive force and
Ucμμ’(r) =0.
i.e) dipole-dipole interaction energy is zero
These illustrations show that the dipole-dipole interaction energy is proportional
to r-3 for a fixed configuration of the molecules, and that it is strongly dependent
on orientation varying from attractive to repulsive as one molecule is rotated
An analysis of the interaction of two linear charge distributions in the general
configuration in fig (d) wherein Ф denotes rotation of the second dipole about
the line joining them leads to the dipole-dipole interaction energy
U
Where
µµ
'
(r , θ1 , θ 2 , φ ) =
− µµ
'
4 πε 0 r
3
ζ (θ 1 , θ 2 , φ )
ζ (θ , θ , φ ) = (2 cos θ cos θ − sin θ sin θ cos φ )
1
2
1
2
1
describes the dependence of the energy on orientation
2
If it is gas phase the two molecules are free to rotate it is often the average of
the dipole-dipole energy over all possible orientations <Uel>μμ‘ is required.
The probability of observing a configuration of energy U is proportional to the
Boltzmann factor exp(-U/KT)
U el
µµ
'
=−
2
µ µ
2
6
'2
3 r kT (4πε 0 )
2
+ ...
The leading term of the orientation-averaged dipole-dipole contribution to
the electrostatic energy of interaction to two neutral molecules is therefore
attractive and inversely proportional to the sixth power of their separation
The temperature dependence of <Uel>μμ‘ is a result solely of the orientational
averaging with the Boltzmann weighting
Eq. A shows the variation of energy as
r-4 for dipole-quadrupole interactions
r-5 for quadrupole-quadrupole interactions for fixed orientation
but for rotational case the corresponding interaction energy is given by
1
U el
µΘ
U el
8
r kT (4πε 0 )
ΘΘ
'
=−
{µ Θ
2
= −
2
2
ΘΘ
14
10
'2
'2
+µ Θ
2
} + ...
Dipole-quadrupole
'2
5 r kT (4πε 0 )
2
+ ...
Quadrupole-Quadrupole
Above expressions show that these two contributions to the orientation-averaged
electrostatic interaction energy are also attractive
Induction energy
Any molecule is placed in a uniform, static electric field E there is a
polarization of its charge distribution
Electric field E
----
++
++
++
Process of induction
Induced dipole moment in the molecule is given by
μind=αE (a) α is static polarizability of the molecule
Energy of a neutral dipolar molecule in an electric field E is
π
U = − ∫ µ ⋅ dE
Using eq (a) in (b)
(b)
0
Since μ and E are parallel
U ind
π
1
0
2
= − ∫ α EdE = −
αE
2
(c)
Interaction of dipolar molecule with non-polar
molecule
Neutral molecule possessing permanent dipole moment can induce a dipole
moment in a second molecule which is nearby whether the second molecule is
itself polar or not
Non-polar molecule
r
O
P
θ
Interaction of a dipolar molecule with
non-polar molecule
µ
µ cos θ
1
Φ =
The electrostatic potential due to the dipole at P is

E =| E | =


( ) ( )
∂φ
2
1 ∂φ
+
∂r
r ∂θ
2



1
2
4πε
0
1
µ
=
4πε 0 r
3
[ 4 cos
2
2
θ + sin θ
]
2
r
2
Magnitude of the dipole induced in the molecule at P is thus α'E and the energy is
-(1/2) α'E so that the interaction energy of induction is
U ind = −
1
2
α
'
µ
r
2
6
( 3 cos
2
θ +1
(4πε 0 )
2
)
(i)
which is attractive for all configurations and inversely proportional to the sixth
power of the intermolecular separation for a fixed orientation
If the above potential energy is averaged over all possible orientations, giving
each orientation the Boltzmann weight exp(-U/KT), the average induction energy
between a dipolar molecule and a non-polar molecule is, at sufficiently high
temperatures
2
U ind =
−µ α
2
'
(4πε 0 ) r
6
This term is not temperature dependent unlike the corresponding term for the electrostatic
energy
Interaction of two polar molecule
•
•
If interaction is between two polar molecules each molecule induces a dipole
moment in the other so that there are two contributions to the total
interaction energy of the eq. (i).
The orientationally-averaged induction energy for two dipolar molecules is
U ind
•
µµ
=−
2
{
(4πε 0 ) r
6
2
'
'
2
}
µ α + µ α + ...
For two identical polar molecules this reduces to
U ind
•
'
1
µµ
'
=
−2αµ
2
2
6
+ ...
(4πε 0 ) r
The general treatment of induction forces is very much more complicated than
that given here. This because for many molecules the polarizability is not
isotropic but is tensorial character. In addition higher-order multipoles in polar
molecules can induce higher-order multipoles in other molecules and there can
be other contributions to the induced dipole moment.
Dispersion energy
• So far the we analysed contributions to the long range energy
by means of classical electrostatics and classical mechanics
• For the interaction of two molecules possessing no
permanent electric dipole or higher-order moments,
electrostatic and induction contributions are absent and the
interaction arises solely from the dispersion energy
• The dispersion energy cannot be analysed by classical
mechanics since as we shall see its origins are purely quantum
mechanical
• Consequently dispersion energy is well explained by the
Drude Model
Drude model
Considering each molecule is composed of two charges +Q and –Q.
We imagine charge +Q to be stationary and that the negative charge oscillates
about the positive charge with an angular frequency ω0 in the
Z-direction which is along the line joining the positive charges of the two
molecules as show in fig.
r
1-d Drude model of the dispersion interaction
za
+Q
zb
-Q
-Q
+Q
Animation
If we denote the displacement of the negative charge of molecule a from its
positive charge by za, we note that at any time t the moelcules possess
instantaneous dipole moments μa=Qza(t) and μb=Qza(t)
Denoting F as force constant of the harmonic oscillator by k and the
mass of the oscillating charge by M, the frequency of the oscillation is
ω = k/M
thus
When the two molecules are infinitely separated, the Schrödinger
wave eq. for molecule a is
0
2
1 ∂ Ψa
M ∂z a
2
+
2
2
(Ε a −
1
2
which is the equation of SHO, where ½ kZ2a
is the potential energy of the oscillator
2
kz a ) Ψ a = 0
The eigen values of the energy for molecules a and b are given by
Ε
a
=
( n
a
+
Ε
b
=
( n
b
+
1
2
1
2
) ω
0
) ω
0
When the two molecules are infinitely separated and both are in their
ground states the total energy of the two-molecule system is
Ε ( ∞ ) = Ε a + Ε b = ω 0
When the molecules are separated by a finite distance r, which is still
considered to be large by comparison with the dimensions of the
molecules, there is an energy of interaction between the two dipoles at
any instant and this interaction energy is included in the Schrödinger
wave eq.
2
1 ∂ Ψ
M ∂z a
2
2
+
1 ∂ Ψ
M ∂z b
2
+
2
2
(Ε −
1
2
2
kz a −
1
2
2
kz b −
2 za zb Q
4 πε 0 r
2
3
)Ψ = 0
Where ψ is the wave function for the two-molecule system
If we make the transformations
Z
1
=
za + zb
2
Z2 =
za − zb
2
After transformation Schrödinger eq. becomes
2
1 ∂ Ψ
M ∂Z 1
2
2
+
1 ∂ Ψ
M ∂Z 2
2
+
2
2
[Ε −
1
2
2
k1 Z 1 −
1
2
2
k 2 Z 2 ]Ψ = 0
where
k1 = k −
2Q
2
4 πε 0 r
k2 = k +
3
2Q
2
4 πε 0 r
3
Above Schrödinger wave equation is for two independent SHO in the
coordinates Z1 and Z2. Thus the eigenvalues for the total energy of the system
are
1
1
) ω1 + ( n 2 +
Ε ( r ) = ( n1 +
2
) ω 2
2
If we consider the two molecules to be in their ground states then
1
( ω1 + ω 2 )
Ε( r ) =
2
or
ω 1 = ω 0 {1 −
1/ 2
ω1 = ( k1 / M )
where
2Q
2
}
1 / 2
ω 2 = ω 0 {1 +
3
4πε 0 r k
1/ 2
ω2 = ( k2 / M )
2Q
2
}
1/ 2
3
4 πε 0 r k
We are interested in the long-range interaction for which the perturbation
potential is small so we can expand ω1,ω2 by the binomial theorem which
allows to write E(r) as
4
Ε ( r ) = ω 0 −
Q ω 0
2
6
2 ( 4 πε 0 ) r k
+ ...
2
The energy of interaction of the two molecules for our model is
4
U disp = Ε ( r ) − Ε (∞ ) = −
Q ω0
2
3
4(4πε 0 ) r k
2
+ ...
The motion of the oscillating charge can be resolved into three oscillations
of identical frequency along three Cartesian coordinates centered on the
positive charge. Thus
4
U disp = Ε( r ) − Ε (∞ ) = −
Q ω0
2
3
4(4πε 0 ) r k
2
+ ...
The force constant, k is related to the polarizability of the molecules
If we place the single Drude molecule to an external electric field E a force
of magnitude QE acts on each charge to produce a displacement z’a which
attains a static value when the restoring force kz’a is equal to the imposed
'
electrical force. Then
Q = kz n / Ε
so that static dipole moment induced in the molecule by the field is μind=
Qz’n=Q2E/ k
2
α = Q /k
Now, Polarizability
So that we can write the dispersion energy for two identical molecules in
their ground states for this model as Udisp =C6/r6
where
2
C
6
= −
3 α
ω
0
4 (4πε 0 )
2
The dipole-dipole dispersion energy for two molecules in their ground states is
therefore attractive and inversely proportional to the sixth power of the
intermolecular separation
If we treat the same problem classically the interaction energy is zero. The
existence of the dispersion energy is therefore a consequence of the zero-point
energy of the oscillators, a purely quantum mechanical concept.
Considering additional contributions to the dispersion energy arising from
instantaneous dipole-quadrupole, quadrupole-quadrupole interactions, etc. we
get
U disp =
C6
r
6
+
C8
r
8
+
C10
r
10
+ ...
where, for interactions between ground-state molecules, each coefficient is
negative so that each contributions is attractive
London first estimated the magnitude of the coefficient C6 by means of the
oscillator model and he obtained for argon atoms as -5 × 10 -78 J m6 which is only
30 percent smaller than the value obtained by the most refined calculations
Long-range energy: Summary
Order of magnitude
For the most general case of the interaction of two
polar molecules the total long-range intermolecular
energy is
U=Uel+Uind+Udisp
For two identical, neutral molecules, free to rotate
with a dipole moment μ and static polarizability α,
the leading contributions are
−1
U =
(4πε 0 )
2
{
2 µ
4
2
3
+ 2µ α +
3 kT
2
α ω 0
4
}
r
−6
Table 1.1 contains a list of the magnitudes of the three coefficients of
the r-6 term for the different contributions to the interaction of like pairs
of simple molecules for a temperature of 300K
In order to compare the magnitude of the various
attractive contributions to the intermolecular energy,
calculation were made for 1 mole of a gas on the
assumption that the molecules interact in pairs at a
separation, σ, where the total potential energy
U(σ) = 0 and the values are tabulated in table 1.2
Limitations
The description of the electrostatic energy has been
confined to axially-symmetric charge distributions
The discussion of induction energy has been
restricted to molecules with an isotropic
polarizability
The treatment of the dispersion energy limited to
that for the interaction of spherically symmetric
atoms in their ground states
Finally, the retardation effect, which means by the
time field acts at the second molecule the dipole in
the first molecule will have changed
Short-range energy
Short-range repulsive forces between molecules is much more
complicated the the long-range forces Here we discuss briefly the HeitlerLondon or valence-bond method, a method for the evaluation of shortrange forces
The interaction of
two hydrogen atoms
Fig show the two hydrogen atoms at a separation r and defines the
coordinates of the system
The Hamiltonian, H for the system can be written H = Ha+Hb+Ve
where Ve is the electrostatic energy which arises from the interaction of
the two atoms and is given by
Ve = −
e
2
4πε 0
1

r
a2
+
1
rb 1
−
1
r 12
−
1

r
The integral
S, the overlap integral, measures the degree of
overlap of the wave functions of the two atoms
J describes the columbic interaction between the
electron 1 in the orbital A(1) with nucleus b
J’ describes the columbic interaction between the
two electrons
K and K’ are exchange integrals
Contours of equal electron density, in a plane containing the two nuclei
for the interaction of two hydrogen atoms
In fig a the two positively charged protons are attracted towards negatively
charged region and hence there is a net attractive force on them according to
classical electrostatics. This wave function ψ+ therefore corresponds to that
of a bonding orbital for a hydrogen molecule.
In fig b there is decreased electron density between the two nuclei and thus
two nuclei are incompletely shielded from each other and an electrostatic
repulsion results. The wave function ψ- is therefore anti-bonding.
Potential energy for the interaction of two hydrogen atoms
U(r)
e
U+ =
Uo
r
U+
'
'
+
4πε 0 r
e
U− =
J + K − 2 ( J + SK )
2
1+ S
2
J − K − 2 ( J − SK )
2
'
'
−
4πε 0 r
1+ S
2
Evaluating the energies U+ and U- and plotting for U(r) confirms the qualitative
discussion that done before
The energy U+ possesses an attractive minimum region and corresponds to a
stable hydrogen molecule
The energy U- is repulsive and because of the spin symmetry of the wave
function is similar to the interaction energy for closed shell atoms.
At short range the energy U- varies as 1/r owing to the internuclear repulsion
At larger separations the energy decays as e-2r/a, where a is the radius of the
Bohr orbit of the hydrogen atom. This final exponential form has been used as
an analytic representation of the behavior of short –range forces
Representation of the intermolecular pair
potential energy function
• The earliest and simplest representation was molecule viewed as a hard
sphere of diameter σ so that the intermolecular potential energy is written
U (r ) = ∞
r ≤ σ
U (r ) = 0
r > σ
•
This form of potential has the single disposable parameter σ. In view of
our description of the nature of intermolecular forces this model is
evidently unrealistic
• The most frequently used model potential is that due to Lennard-Jones(LJ)
 6
U (r ) = ε 
n − 6
()
rm
r
n
−
n
n−6
()
rm
r
6



• This function possesses the general features of the true intermolecular
potential energy in tat it has a repulsive short-range region joined to a
long-range attractive region by a single minimum which occurs at rm where
the energy is –ε
• The attractive component of the function is theoretically based on the
dispersion energy contribution. But the form of the repulsive term has no
theoretical justification. In the above form the LJ potential has one
disposable parameter n in addition to ε and rm
• Most often the repulsive exponent has been given the value n=12 and the
potential is then written

U ( r ) =∈


( ) ( )
12
rm
−2
r
rm
r
6



• Or, equivalently,

U (r ) = 4 ∈ 

() ()
σ
r
12
−
σ
r
6



• Where σ (=2-1/2rm) is the intermolecular separation for which the energy is
zero
• L-J(12-6) potential has no adjustable parameters other than σ and ε,
whose values can be determined by forcing agreement between
experimental data for a physical property and calculated values for the
potential model
LJ potential
  σ 12  σ  6 
U (r ) = 4 ∈   −   
 r  
  r 
Interatomic pair potential for argon
molecules
Interatomic pair potential for Cesium
molecules
Other pair potential functions are
Buckingham potential (buck)
(6-exp)
Born-Huggins-Meyer potential (bhm)
Hydrogen-bond (12 - 10) potential (hbnd)
Other topics required for MD
• Intermolecular interactions
• Classical Mechanics
• Statistical Mechanics
Thank you