Download density becomes larger between the two nuclei. This re

342
9. Diatomic Molecules
Fig. 9.28. Valence
bond as increased
electron charge between the two nuclei
E
R > Rc
Separated atoms
Re
Rc = rA + rB
R
Chemical
binding
Multipole
interaction
Fig. 9.27. Chemical binding with overlap of atomic orbitals
is important for R < Rc . For R > Rc multipole interaction
dominates
density becomes larger between the two nuclei. This results in an electrostatic attraction between the positive
cores of the two atoms (for H2 these are the two protons)
and the negative electron charge distribution between
them. This effect is emphasized in the valence bond
model used in chemistry. In the chemically bound molecule both atoms share one or more valence electrons in
a common molecular orbital. This is also described in
the LCAO approximation where the molecular orbital is
represented by a linear combination of atomic orbitals.
The second reason is of quantum mechanically nature and cannot be explained by a classical model.
The molecular orbital has a larger spatial extension
then the atomic orbitals. This increases the spatial
uncertainty for the electrons and therefore decreases
their average momentum !| p|" and their kinetic energy
!E kin " = ! p2 "/2m, according to Heisenberg’s uncertainty relation. The combination of both effects leads
to a minimum in the potential curve E(R), since the potential energy E(R) contains the average kinetic energy
of the electrons (see Sect. 9.1). This second contribution
to the molecular binding is called the exchange interaction, because the two electrons in the atomic orbitals
of the LCAO can be exchanged since they cannot be
distinguished in their common molecular orbital.
Both effects are important at internuclear distances
R # !rA " + !rB " that are smaller than the sum of the
Valence bond
mean atomic radii !rA " and !rB ", which give the extension of the electron clouds in the separated atoms.
For distances smaller than this sum, the orbitals of
the two atoms can overlap forming molecular orbitals
and sharing electrons (Fig. 9.28). Molecular bonds that
are formed due to this effect are called covalent or
homopolar.
One can also describe the chemical binding by
energy conservation. If the energy increase necessary to deform the atomic orbitals when the
two atoms approach each other is smaller than
the decrease of the total energy (potential energy
and mean kinetic energy of the electrons) in the
rearranged molecular charge distribution, a stable
molecule is formed. The nuclear distance Re and
the electron charge distribution always arrange
themselves in such a way that the total energy
becomes a minimum.
9.4.2 Multipole Interaction
For larger internuclear distances R > !rA " + !rB ", where
the electron clouds of the two atoms no longer overlap,
the chemical binding based on the two effects discussed above looses its importance. Nevertheless stable
molecules are possible with such large internuclear distances R, although their binding energy is smaller. The
correct treatment of the effects responsible for these
interactions at large distances is based on quantum
mechanics. However, good physical insight is already
provided by the classical model, which starts from the
multipole expansion of an arbitrary charge distribution ρ(r) for an observer at a point P at a distance R
9.4. The Physical Reasons for Molecular Binding
qi
A
r
ri
→
A
B
→
→
R
P
Fig. 9.29. Illustration of multipole expansion in (9.46) and
(9.47)
from the center of the charge distribution, which is large
compared to the extension of ρ(r) (Fig. 9.29). We will
discuss this model shortly.
The potential φ(R) at the point P(R) generated by
a distribution of charges qi (ri ) at the locations ri is
1 ! qi (ri )
φ(R) =
.
(9.46)
4πε0
|R− ri |
If we choose the origin of our coordinate system to
coincide with the nucleus of atom A, the positive charge
q(ri = 0) = +Z A · e is the nuclear charge of atom A and
q(ri ) = −e gives the charge of the ith electron in the
electron shell. For R % r we can expand (9.46) into
the Taylor series and obtain for the potential at a point
P(X, Y, Z)
R = {X, Y, Z} and r = {x, y, z}
"!
#
1 X!
1
qi +
φ(P) =
qi xi
4πε0 R i
R R
$
Y !
Z!
+
qi yi +
qi z i
R
R
&!
#% 2
1
3X
+ 2
−1
qi xi2
2R
R2
% 2
&!
3Y
+
−1
qi yi2
R2
% 2
&!
$
3Z
2
+
−
1
q
z
i i
R2
'
1
+ 3 [. . . ] + . . .
R
(! )
(! )
= φM
qi + φD
pi
(!
)
*i + . . .
QM
+ φQM
(9.47)
Fig. 9.30. Deformation and shift of
electron charge distribution of atom A
by the interaction
with atom B
The first term φM represents the monopole+contribution,
which is zero for neutral atoms where
qi = 0. For
ions it gives the main contribution.
The second term φD describes the potential
+ of an
electric dipole with a dipole moment p = qi · ri ,
which is the vector sum of the dipole moments pi = ei ri
formed by the different electrons and the nucleus at
r = 0.
The third term φQM gives the contribution of the
quadrupole moment to the potential, the next terms the
higher moments, such as the octopole or hexadecapole
which are here neglected.
If another atom B with total charge qB , electric
dipole moment pB and quadrupole moment Q MB is
placed at the position P(R), the potential energy of the
interaction between A and B can be written as the sum
E pot (A, B) = E pot (qB ) + E pot ( pB )
* B) + . . .
+ E pot (QM
(9.48)
where
E pot (qB ) = qB · φ( p) ,
E pot ( pB ) = + pB · grad φ( p) ,
* B · grad EA .
E pot (QMB ) = QM
(9.49)
The vector gradient grad EA of the electric field E A ,
produced by atom A is the tensor
'
"
∂E ∂E ∂E
,
,
.
(9.50)
grad E =
∂x ∂y ∂z
From the expression (9.49) we obtain the following
contributions to the interaction energy between A and B.
Two ions with charge qA and qB have long-range
interactions
E pot (qA , qB , R) =
1 qA · qB
4πε0 R
(9.51)
which decrease only with 1/R.
An ion with charge qA and a neutral atom or molecule with a permanent dipole moment pB , pointing in
a direction with an angle ϑ against the z-axis through
343
344
9. Diatomic Molecules
ϑ
R
→
pB
qA
Fig. 9.31. Interaction between a charge qA and an electric
dipole moment pB
Fig. 9.33. Interaction between two dipoles
A and B (Fig. 9.31) experience the interaction energy
where
1 qA pB cos ϑ
.
E pot ( pA , pB , R) =
4πε0
R2
(9.52)
The interaction potential between an ion and
a neutral atom with permanent dipole moment p
is proportional to 1/R2 and is zero in the direction
perpendicular to the dipole axis.
Two permanent dipoles pA and pB with angles ϑA
and ϑB against the z-axis and angles ϕA and ϕB against
the x-axis (Fig. 9.33) have the interaction energy
E pot ( pA , pB , R) = − pA · E( pB ) = − pB · E( pA )
(9.53)
1
(3 pB · R̂ · cos ϑ p − pB )
(9.54)
4πε0 R3
is the electric field generated by the dipole pB
(Fig. 9.32). The interaction energy then becomes
E( pB ) =
E pot ( pA , pB , R)
(9.55)
1
=−
[3 pA pB cos ϑA cos ϑB − pA · pB ]
4πε0 R3
pA · pB
[2 cos ϑA cos ϑB
=−
4πε0 R3
− sin ϑA sin ϑB · cos(ϕA − ϕB )] .
The interaction energy between two permanent
electric dipoles is proportional to the product of
the two dipole moments and depends on their
relative orientation. It decreases as 1/R3 with
increasing distance much faster than the 1/R
Coulomb interaction between two charges.
ER
→
p
Eϑ
→
+q
R
→
E
ϑ
S
−q
ER =
ϕ
2p ⋅ cos ϑ
4 πε 0 ⋅ R3
Eϑ =
p ⋅ sin ϑ
4 πε 0 ⋅ R3
Eϕ = 0
Fig. 9.32. Electric field of a dipole. It has axial symmetry
around the dipole axis
9.4.3 Induced Dipole Moments
and van der Waals Potential
If a neutral atom without permanent dipole moment
is placed in an electric field, the opposite forces on
the negative electrons and the positive nucleus shift
the electron charge distribution into the opposite direction than the nucleus. The centers of the positive and
the negative charges no longer coincide as in a neutral
atom without permanent dipole moment, and a dipole
moment
pind
A = αA E
(9.56a)
is induced by the electric field , which is proportional
to the field (Fig. 9.34). The constant αA is the electric
9.4. The Physical Reasons for Molecular Binding
e−
R
→
r(t)
B
+ qA
Fig. 9.35. Momentary dipole
moment of an atom with
spherically symmetric charge
distribution
→
PB
+
Fig. 9.34. The charge qA produces an induced dipole
moment pB
→
pA ( t) = e ⋅ r( t)
polarizability of atom A and is a measure of the restoring forces in the atom against the displacement and
deformation of the electron shell. If the electric field is
produced by an ion with charge qB , the induced dipole
moment of A becomes
αA qB
pind
R̂
(9.56b)
A =
4πε0 R2
where R̂ is the unit vector pointing into the direction
from B to A.
The potential energy of a neutral atom without
permanent dipole moment in an electric field E is
E pot = − pind
A · E = −(αA E) · E .
(9.57)
If the electric field is produced by an ion with charge qB ,
the potential becomes
E pot = −
αA qB2
(4πε0 )2 R4
(9.58)
If the field is generated by an atom B with permanent
dipole moment pB we obtain from (9.54) and (9.57) the
potential energy
E pot = −
αA p2B
(3 cos2 ϑB + 1) .
(4πε0 R3 )2
(9.59)
In molecular physics the interaction between two neutral atoms is of particular importance. For a charge
distribution on the electron shell that is spherically symmetric on the time average (e. g., for 1s electrons in the
H atom) the time averaged dipole moment has to be
zero. However, there still exists a momentary dipole
moment (Fig. 9.35) that produces according to (9.54)
a momentary electric field
EA =
1
(3 pA R̂ cos ϑA − pA ) ,
4πε0 R2
(9.60)
which is statistically pointing in all directions and has
a time average of zero. However, if we place another
→
pA
=0
atom B in the vicinity of A, this field induces a dipole
moment in atom B
pind
B = +αB EA ,
(9.61)
which in turn generates an electric field at atom A inducing a dipole moment in A. Now the time average of
p or E is no longer zero, because the interaction energy
between the two induced dipoles depend on their relative orientation and the positions with minimum energy
have a larger probability than those with higher energies. This leads to an attraction between A and B which
is called a van der Waals interaction and is an interaction between two induced dipoles. We will now treat
this more quantitatively.
According to (9.55) the negative interaction energy
between the two dipoles is a maximum when the two dipoles are either parallel (ϑA = ϑB = 180◦ ) (Fig. 9.36a)
with their dipole moments pointing into the −zdirection, or antiparallel (ϑ = 90◦ , ϑ = 270◦ ), pointing
in a direction perpendicular to the z-axis (Fig. 9.33).
In the case of induced dipole-dipole interactions both
dipole moments are directed along the axis through
the two nuclei, which we choose as z-axis. This means
that cos ϑA = cos ϑB = 180◦ and p ' R. From (9.60) we
obtain
2 pA
2 pB
EA =
R̂ , EB =
R̂ .
(9.62)
3
4πε0 R
4πε0 R3
The potential energy of the interaction between the two
ind
induced dipole moments pind
A and pB is then
ind
E pot (R) = − pind
B · EA = − pA · EB .
(9.63)
With pA = αA · E B and pB = αB · E A we get from (9.62)
ind
2
E pot (R) ∝ − pind
A · pB = −αA · αB · |E| ,
(9.64)
345
346
9. Diatomic Molecules
−
−
−
−
+
+
+
+
A
−
+
+
+
→
B
+
−
−
E
R
a)
→
pAind
S
S
b)
→
pBind
+
−
S
→
pAind
→
pBind
A
S
If higher order terms in the multipole expansion are
taken into account, the interaction energy between two
atoms includes terms with 1/R8 , 1/R10 , 1/R12 , . . . for
the induced quadrupole or octupole interactions. For
homonuclear molecules only even powers n of 1/Rn
can appear for symmetry reasons.
−
+
B
Fig. 9.36a,b. Possible orientations of two attraction-induced
dipoles with (a) parallel (b) antiparallel orientation
which can be written as
E pot (R) = −C1
αA αB
C6
=− 6
R6
R
,
interaction energy is also negative if the two induced dipoles are orientated antiparallel but both perpendicular
to the z-axis through their centers (Fig. 9.36b).
The quantum mechanical treatment of the van der
Waals interaction is based on the calculations of the
atomic charge distributions, perturbed by the mutual
interaction between A and B. Since only this perturbation leads to an attraction between the two atoms one
needs a perturbation calculation of second order [9.8, 9],
which is beyond the scope of this book.
(9.65)
αA ·αB
with C1 = (4πε1 )2 and C0 = (4πε
2.
0
0)
This is the van der Waals interaction potential between two neutral atoms with the polarizabilities αA and
αB . The constant C6 , which is proportional to the product αA · αB of the atomic polarizabilities, is called the
van der Waals constant.
The multipole interaction between two neutral
atoms is only important for internuclear distances R > !rA " + !rB ". For smaller values of R
the overlap of the electron shells of A and B
has to be taken into account, which results in
the above-mentioned exchange interaction and
the electrostatic interaction due to the increased
electron density between the two nuclei.
The total range of R-values can, however, be covered
by the empirical Lenard–Jones potential
LJ
E pot
(R) =
a
b
−
R12 R6
(9.66)
Epot
Epot (R) =
The interaction potential between two neutral
atoms scales for large separations R as R−6 . The
attraction is much weaker than between charged
particles.
a
R12
−
b
R6
∝ R −12
R
Note that the interaction is attractive (because of the
negative sign) and decreases as 1/R6 with increasing distance R. It is therefore a short range interaction compared with the Coulomb-interaction that is ∝ 1/R, but has
a longer range than the interaction of the chemical bond,
which falls of exponentially with increasing R. The
R0
EB =
b2
2a
∝ R− 6
Fig. 9.37. Lenard–
Jones potential
9.4. The Physical Reasons for Molecular Binding
where a and b are two parameters that depend on the
two atoms A and B and which are adapted to fit best
the potential curve obtained either experimentally or by
accurate and extensive calculations (Fig. 9.37).
From (9.66) it follows that E pot (R) = 0 for
R = (a/b)1/6 . The minimum of E pot (R) is obtained for
dE/ dR = 0. This gives the distance
Re = 2(a/b)1/6 = 21/6 R0
(9.67)
for the minimum. The binding energy at Re is then
E B = −E pot (Re ) = b2 /2a .
(9.68)
9.4.4 General Expansion
of the Interaction Potential
The potential energy E pot (R) of a diatomic molecule
can be expanded for |R − Re |/Re < 1 into a Taylor series around the equilibrium distance Re of the potential
minimum:
%
&
∞
!
1 ∂ n E pot
E pot (R) =
(R − Re )n . (9.69a)
n
n!
∂R
Re
n=0
Because (∂ 0 E/∂R0 ) Re = E pot (Re ) and (∂E/∂R) Re = 0,
this gives
%
&
1 ∂ 2 E pot
E pot (R) = E pot (Re ) +
(9.69b)
2
∂R2 Re
× (R − Re )2 + . . . .
In molecular physics the minimum of the ground
state potential is generally chosen as E pot (Re ) = 0.
Instead of the negative binding energy E B (which
is used if the zero point is chosen as the energy
of the separated ground state atoms) the positive
energy E D = −E B is now used, which gives the
energy necessary to dissociate the molecule from its
energy minimum at R = Re to the separated atoms at
R = ∞.
For |R − Re |/Re # 1 the higher order terms with
n > 2 can be neglected and we obtain a parabolic
potential in the vicinity of the potential minimum.
The potential energy of a diatomic molecule can
be approximated in the vicinity of the potential
minimum by a parabolic potential (Fig. 9.38).
Fig. 9.38. Comparison of parabolic and Morse potentials with
the real (experimental) potential
9.4.5 The Morse Potential
In 1929 P.M. Morse had already proposed an empirical
potential form
,
-2
E pot (R) = E D 1 − e−a(R−Re ) ,
(9.70)
which represents the attractive part of the potential
with a much better approximation to the experimental values than the parabolic potential. This potential
converges for R → ∞ correctly towards the dissociation energy E D , while the parabolic potential
goes to infinity for R → ∞ (Fig. 9.38). The repulsive part of the potential for R < Re deviates more
severely from the experimental data. We obtain from
(9.70)
,
-2
lim E pot (R) = E D 1 − e+aRe
R→0
(9.71)
while the experimental potential must converge towards the energy levels of the united atom (see
Fig. 9.25).
The Morse potential has the great advantage that
the Schrödinger equation of two atoms vibrating in this
potential can be solved exactly (see Sect. 9.6).
347