# Dictionary Definition

intermolecular adj : existing or acting between
molecules; "intermolecular forces"; "intermolecular
condensation"

# User Contributed Dictionary

# Extensive Definition

In physics, chemistry, and biology, intermolecular forces
are forces that act between stable molecules or between functional
groups of macromolecules.
Intermolecular forces (the weakest of which are van
der Waal's forces) include momentary attractions between
molecules, diatomic free elements, and individual atoms. They
differ from covalent and ionic bonding in that they are not stable,
but are caused by momentary polarization of particles. Because
electrons have no fixed position in the structure of an atom or
molecule, but rather are distributed in a probabilistic fashion
based on quantum probability, there is a positive chance that the
electrons are not evenly distributed and thus their electrical
charges are not evenly distributed. See Schrödinger
equation for the theories on wave functions and descriptions of
position and velocity of quantum particles.

In general one distinguishes short and long range
van der Waal's forces. The former are due to intermolecular
exchange and charge penetration. They fall off exponentially as a
function of intermolecular distance R and are repulsive for
interacting closed-shell
systems. In chemistry they are well known, because they give rise
to steric
hindrance, also known as Born or Pauli repulsion. Long range
forces fall off with inverse powers of the distance, R-n, typically
3 ≤ n ≤ 10, and are mostly attractive.

The sum of long and short range forces gives rise
to a minimum, referred to as Van der Waal minimum. The position and
depth of the Van der Waal's minimum depends on distance and mutual
orientation of the molecules. "General theory" This is because
before the advent of quantum
mechanics the origin of intermolecular forces was not well
understood. Especially the causes of hard sphere repulsion,
postulated by
Van der Waals, and the possibility of the liquefaction
of noble
gases were difficult to understand. Soon after the formulation
of quantum
mechanics, however, all open questions regarding intermolecular
forces were answered, first by S.C. Wang and then more completely
and thoroughly by Fritz
London.

The quantum mechanical basis for the majority of
intermolecular effects is contained in a nonrelativistic energy
operator, the molecular
Hamiltonian. This operator consists only of kinetic energies
and Coulomb interactions. Usually one applies the
Born-Oppenheimer approximation and considers the electronic
(clamped nuclei) Hamilton operator only. For very long
intermolecular distances the retardation of the Coulomb force
(first considered in 1948 for intermolecular forces by Hendrik
Casimir and Dirk Polder)
may have to be included. Sometimes, e.g., for interacting paramagnetic or
electronically excited
molecules, electronic spin and
other magnetic effects may play a role. In this article, however,
retardation and magnetic effects will not be considered.

We will distinguish four fundamental
interactions:

- exchange
- electrostatic
- induction
- dispersion.

## Perturbation theory

The last three of the fundamental interactions are most naturally accounted for by Rayleigh-Schrödinger perturbation theory (RS-PT). In this theory—applied to two monomers A and B—one uses as unperturbed Hamiltonian the sum of two monomer Hamiltonians, H^ \equiv H^+ H^.\,In the present case the unperturbed states are
products \Phi_n^A \Phi_m^B\quad \hbox\quad H^A \Phi_n^A =
E_n^A\Phi_n^A\quad\hbox\quad H^B \Phi_m^B = E_m^B\Phi_m^B

## Supermolecular approach

The early theoretical work on intermolecular forces was invariably based on RS-PT and its antisymmetrized variants. However, since the beginning of the 1990s it has become possible to apply standard quantum chemical methods to pairs of molecules. This approach is referred to as the supermolecule method. In order to obtain reliable results one must include electronic correlation in the supermolecule method (without it dispersion is not accounted for at all), and take care of the basis set superposition error. This is the effect that the atomic orbital basis of one molecule improves the basis of the other. Since this improvement is distance dependent, it gives easily rise to artifacts.## Exchange

The monomer functions ΦnA and ΦmB are antisymmetric under permutation of electron coordinates (i.e., they satisfy the Pauli principle), but the product states are not antisymmetric under intermolecular exchange of the electrons. An obvious way to proceed would be to introduce the intermolecular antisymmetrizer \tilde^. But, as already noticed in 1930 by Eisenschitz and London, this causes two major problems. In the first place the antisymmetrized unperturbed states are no longer eigenfunctions of H(0), which follows from the non-commutation \big[ \tilde^, H^\big] \ne 0 .In the second place the projected excited states
\tilde^ \Phi^A_n \Phi^B_m

become linearly
dependent and the choice of a linearly independent subset is
not apparent. In the late 1960s the Eisenschitz-London approach was
revived and different rigorous variants of symmetry adapted
perturbation theory were developed. (The word symmetry refers here
to permutational symmetry of electrons). The different approaches
shared a major drawback: they were very difficult to apply in
practice. Hence a somewhat less rigorous approach (weak symmetry
forcing) was introduced: apply ordinary RS-PT and introduce the
intermolecular antisymmetrizer at appropriate places in the RS-PT
equations. This approach leads to feasible equations, and, when
electronically correlated monomer functions are used, weak symmetry
forcing is known to give reliable results.

The first-order (most important) energy including
exchange is in almost all symmetry-adapted perturbation theories
given by the following expression E^_\mathrm = \frac .

The main difference between covalent and
non-covalent forces is the sign of this expression. In the case of
chemical bonding this interaction is attractive (for certain
electron-spin state, usually spin-singlet) and responsible for
large bonding energies—on the order of a hundred
kcal/mol. In the case of intermolecular forces between closed shell
systems, the interaction is strongly repulsive and responsible for
the "volume" of the molecule (see Van
der Waals radius). Roughly speaking, the exchange interaction
is proportional to the differential overlap between Φ0A and Φ0B.
Since the wavefunctions decay exponentially as a function of
distance, the exchange interaction does too. Hence the range of
action is relatively short, which is why exchange interactions are
referred to as short range interactions.

## Electrostatic interactions

By definition the electrostatic interaction is given by the first-order Rayleigh-Schrödinger perturbation (RS-PT) energy (without exchange):E^_\mathrm = \langle \Phi_0^A \Phi_0^B| V^|
\Phi_0^A \Phi_0^B \rangle .

Let the clamped nucleus α on A have position
vector Rα, then its charge times the Dirac
delta function, Zα δ(r-Rα), is the charge density of this
nucleus. The total charge density of monomer A is given by

\rho^A_\mathrm(\mathbf) = \sum_ Z_\alpha
\delta(\mathbf-\mathbf_\alpha) - \rho^A_\mathrm(\mathbf)

with the electronic charge density given by an
integral over nA - 1 primed electron coordinates:
\rho^A_\mathrm(\mathbf) = n_A \int |\Phi^A_0(\mathbf, \mathbf'_2,
\ldots, \mathbf'_)|^2 d\mathbf'_2 \cdots d\mathbf'_.

An analogous definition holds for the charge
density of monomer B. It can be shown that the first-order quantum
mechanical expression can be written as E^_\mathrm = \int\int
\rho^A_\mathrm(\mathbf_1)\frac \rho^B_\mathrm(\mathbf_2) d\mathbf_1
d\mathbf_2,

which is nothing but the classical expression for
the electrostatic interaction between two charge distributions.
This shows that the first-order RS-PT energy is indeed equal to the
electrostatic interaction between A and B.

#### Multipole expansion

At present it is feasible to compute the electrostatic energy without any further approximations other than those applied in the computation of the monomer wavefunctions. In the past this was different and a further approximation was commonly introduced: VAB was expanded in a (truncated) series in inverse powers of the intermolecular distance R. This yields the multipole EXPANSION of the electrostatic energy. Since its concepts still pervade the theory of intermolecular forces, we will present it here. In this article the following expansion is provedV^ = \sum_^\infty \sum_^\infty (-1)^ \binom^
\sum_^ (-1)^ I_(\mathbf_)\; \left[\mathbf^ \otimes \mathbf^
\right]^_M

with the Clebsch-Gordan
series defined by \left[\mathbf^ \otimes \mathbf^ \right]^_M \equiv
\sum_^ \sum_^\; Q_^ Q_^\;\langle \ell_A, m_A; \ell_B, m_B|
\ell_A+\ell_B, M \rangle.

and the irregular solid
harmonic is defined by I_(\mathbf_) \equiv
\left[\frac\right]^\; \frac.

The function YL,M is a normalized spherical
harmonic, while

Q^_ and Q^_ are
spherical multipole moment operators. This expansion is
manifestly in powers of 1/RAB.

Insertion of this expansion into the first-order
(without exchange) expression gives a very similar expansion for
the electrostatic energy, because the matrix element factorizes,
\begin E^_\mathrm = & \sum_^\infty \sum_^\infty (-1)^ \binom^
\\ &\sum_^ (-1)^ I_(\mathbf_)\; \left[\mathbf^ \otimes \mathbf^
\right]^_M, \end

with the permanent multipole moments defined by
M^_ \equiv \langle \Phi_0^A | Q^_| \Phi_0^A\rangle \quad\hbox\quad
M^_ \equiv \langle \Phi_0^B | Q^_| \Phi_0^B\rangle .

We see that the series is of infinite length,
and, indeed, most molecules have an infinite number of
non-vanishing multipoles. In the past, when computer calculations
for the permanent moments were not yet feasible, it was common to
truncate this series after the first non-vanishing term.

Which term is non-vanishing, depends very much on
the symmetry of the molecules constituting the dimer. For instance,
molecules with an inversion center such as a homonuclear diatomic
(e.g.,
molecular nitrogen N2), or an organic molecule like ethene (C2H4) do not possess a
permanent dipole moment (l=1), but do carry a quadrupole moment
(l=2). Molecules such a hydrogen
chloride (HCl) and water (H2O) lack an inversion
center and hence do have a permanent dipole. So, the first
non-vanishing electrostatic term in, e.g., the N2—H2O
dimer, is the lA=2, lB=1 term. From the formula above follows that
this term contains the irregular solid harmonic of order L = lA +
lB = 3, which has an R-4 dependence. But in this dimer the
quadrupole-quadrupole interaction (R-5) is not unimportant either,
because the water molecule carries a non-vanishing quadrupole as
well.

When computer calculations of permanent multipole
moments of any order became possible, the matter of the convergence
of the multipole series became urgent. It can be shown that, if the
charge distributions of the two monomers overlap, the multipole
expansion is formally divergent.

#### Ionic interactions

It is debatable whether ionic interactions are to be seen as intermolecular forces, some workers consider them rather as special kind of chemical bonding. The forces occur between charged atoms or molecules (ions). Ionic bonds are formed when the difference between the electron affinity of one monomer and the ionization potential of the other is so large that electron transfer from the one monomer to the other is energetically favorable. Since a transfer of an electron is never complete there is always a degree of covalent bonding.Once the ions (of opposite sign) are formed, the
interaction between them can be seen as a special case of
multipolar attraction, with a 1/RAB distance dependence. Indeed,
the ionic interaction is the electrostatic term with lA = 0 and lB
= 0. Using that the irregular harmonics for L = 0 is simply
I_(\mathbf_) = \frac,

and that the monopole moments and their
Clebsch-Gordan coupling are M^_0 = q_A,\quad M^_0 = q_B
\quad\hbox\quad [\mathbf^\otimes \mathbf^]^0_0 = q_A q_B ,

(where qA and qB are the charges of the molecular
ions) we recover—as to be expected—Coulomb's
law E^_\mathrm = \frac + \hbox.

For shorter distances, where the charge
distributions of the monomers overlap, the ions will repel each
other because of inter-monomer exchange of the electrons.

Ionic compounds have high melting and boiling
points due to the large amount of energy required to break the
forces between the charged ions. When molten they are also good
conductors of heat and electricity, due to the free or delocalized
ions.

#### Dipole-dipole interactions

Dipole-dipole interactions, also called Keesom interactions or Keesom forces after Willem Hendrik Keesom, who produced the first mathematical description in 1921, are the forces that occur between two molecules with permanent dipoles. They result from the dipole-dipole interaction between two molecules. An example of this can be seen in hydrochloric acid:The molecules are depicted here as two point
dipoles. A point dipole is an idealization similar to a point
charge (a finite charge in an infinitely small volume). A point
dipole consists of two equal charges of opposite sign δ+ and δ-,
which are a distance d apart. This distance d is so small that at
any distance R from the point dipole it can be assumed that d/R
>> (d/R)2. In this idealization the electrostatic field
outside the charge distribution consists of one (R-3) term only,
see
this article. The electrostatic interaction between two point
dipoles is given by the single term lA = 1 and lB = 1 in the
expansion above.

Obviously, no molecule is an ideal point dipole,
and in the case of the HCl dimer, for instance, dipole-quadrupole,
quadrupole-quadrupole, etc. interactions are by no means negligible
(and neither are induction or dispersion interactions). Note that
almost always the dipole-dipole interaction between two atoms is
zero, because atoms rarely carry a permanent dipole, see atomic
dipoles.

Writing \boldsymbol^A = (\mu_x^A, \mu_y^A,
\mu_z^A) \quad\hbox\quad \mu_z^A = \boldsymbol^A\cdot \hat_ \equiv
\boldsymbol^A\cdot \frac

and similarly for B, we get the well-known
expression E_ = \frac\left[ \boldsymbol^A\cdot\boldsymbol^B - 3
(\boldsymbol^A\cdot \hat_) (\hat_\cdot \boldsymbol^B)
\right].

As a numerical example we consider the HCl dimer
depicted above. We assume that the left molecule is A and the right
B, so that the z-axis is along the molecules and points to the
right. Our (physical) convention of the dipole moment is such that
it points from negative to positive charge. Note parenthetically
that in organic chemistry the opposite convention is used. Since
organic chemists hardly ever perform vector computations with
dipoles, confusion hardly ever arises. In organic chemistry dipoles
are mainly used as a measure of charge separation in a molecule.
So, \boldsymbol^A = \boldsymbol^B = \mu_\mathrm \begin 0 \\ 0 \\ -1
\end \quad\hbox\quad E_ = \frac.

The value of μHCl is 0.43 (atomic
units), so that at a distance of 10 bohr the
dipole-dipole attraction is -3.698 10-4 hartree
(-0.97 kJ/mol).

If one of the molecules is neutral and freely
rotating, the total electrostatic interaction energy becomes zero.
(For the dipole-dipole interaction this is most easily proved by
integrating over the spherical polar angles of the dipole vector,
while using the volume element sinθ dθdφ). In gases and liquids
molecules are not rotating completely freely—the rotation
is weighted by the Boltzmann
factor exp(-Edip-dip/kT), where k is the Boltzmann
constant and T the absolute temperature. It was first shown by
Lennard-Jones
that the temperature-averaged dipole-dipole interaction is
\overline_\mathrm = -\frac.

Since the averaged energy has an R-6 dependence,
it is evidently much weaker than the unaveraged one, but it is not
completely zero. It is attractive, because the Boltzmann weighting
favors somewhat the attractive regions of space. In HCl-HCl we find
for T = 300 K and RAB = 10 bohr the averaged attraction -62 J/mol,
which shows a weakening of the interaction by a factor of about 16
due to thermal rotational motion.

#### Hydrogen bonding

Hydrogen bonding is an intermolecular interaction with a hydrogen atom being present in the intermolecular bond. This hydrogen is covalently (chemically) bound in one molecule, which acts as the proton donor. The other molecule acts as the proton acceptor. In the following important example of the water dimer, the water molecule on the right is the proton donor, while the one on the left is the proton acceptor:The hydrogen atom participating in the hydrogen
bond is often covalently bound in the donor to an electronegative
atom. Examples of such atoms are nitrogen, oxygen, or fluorine. The electronegative
atom is negatively charged (carries a charge δ-) and the hydrogen
atom bound to it is positively charged. Consequently the proton
donor is a polar molecule with a relatively large dipole moment.
Often the positively charged hydrogen atom points towards an
electron rich region in the acceptor molecule. The fact that an
electron rich region exists in the acceptor molecule, implies
already that the acceptor has a relatively large dipole moment as
well. The result is a dimer that to a large extent is bound by the
dipole-dipole force.

For quite some time it was believed that hydrogen
bonding required an explanation that was different from the other
intermolecular interactions. However, reliable computer
calculations that became possible during the 1980s have shown that
only the four effects listed above play a role, with the
dipole-dipole interaction being particularly important. Since the
four effects account completely for the bonding in small dimers
like the water dimer, for which highly accurate calculations are
feasible, it is now generally believed that no other bonding
effects are operative.

Hydrogen bonds are found throughout nature. In
water the dynamics of these bonds produce unique properties
essential to all known lifeforms. Hydrogen bonds, between hydrogen
atoms and nitrogen atoms, of adjacent DNA base pairs generate
intermolecular forces that improve binding between the strands of
the molecule. Hydrophobic
effects between the double-stranded DNA and the solute
nucleoplasm prevail in sustaining the double-helix structure of
DNA.

## London dispersion forces

Also called London forces, instantaneous dipole (or multipole) effects (spatially variable δ+) or Van der Waals forces, these involve the attraction between temporarily induced dipoles in nonpolar molecules (often disappear within an instant). This polarization can be induced either by a polar molecule or by the repulsion of negatively charged electron clouds in nonpolar molecules. An example of the former is chlorine dissolving in water:(+)(-)(+) (-) (+) [Permanent Dipole]
H-O-H-----Cl-Cl [Induced Dipole]

Note added by other author: Sketched is an
interaction between the permanent dipole on water and an induced
dipole on chlorine. The latter dipole is induced by the electric
field offered by the permanent dipole of water (see
field from an electric dipole).

This permanent dipole-induced dipole interaction
is referred to as induction (or polarization) interaction and is to
be distinguished from the London dispersion interaction. The latter
is sometimes described as an interaction between two instantaneous
dipoles, see molecular
dipole. The Cl2—Cl2 interaction that now follows is an example
of a proper London dispersion interaction.

(+) (-) (+) (-) [instantaneous dipole]
Cl-Cl------Cl-Cl [instantaneous dipole]

Note added by other author: It must be pointed
out that the London interaction is not the only interaction between
two chlorine molecules in the region where the overlap between the
respective charge distributions may be neglected. Each chlorine
molecule carries permanent multipole
moments of even order, the first one being a permanent quadrupole moment (order 2).
The interaction between two permanent multipole moments also
contributes to the intermolecular force and the first term
(quadrupole-quadrupole) is as important as the London dispersion
force.

London dispersion forces exist between all atoms.
London forces are the only reason for rare-gas atoms to condense at
low temperature.

## Quantum mechanical theory of dispersion forces

The first explanation of the attraction between noble gas atoms was given by Fritz London in 1930. He used a quantum mechanical theory based on second-order perturbation theory. The perturbation is the Coulomb interaction V between the electrons and nuclei of the two monomers (atoms or molecules) that constitute the dimer. The second-order perturbation expression of the interaction energy contains a sum over states. The states appearing in this sum are simple products of the excited electronic states of the monomers. Thus, no intermolecular antisymmetrization of the electronic states is included and the Pauli exclusion principle is only partially satisfied.London developed the perturbation V in a Taylor
series in \frac, where R is the distance between the nuclear
centers of mass of the monomers.

This Taylor expansion is known as the multipole
expansion of V because the terms in this series can be regarded
as energies of two interacting multipoles, one on each monomer.
Substitution of the multipole-expanded form of V into the
second-order energy yields an expression that resembles somewhat an
expression describing the interaction between instantaneous
multipoles (see the qualitative description above). Additionally an
approximation, named after Albrecht
Unsöld, must be introduced in order to obtain a description of
London dispersion in terms of dipole polarizabilities
and ionization
potentials.

In this manner the following approximation is
obtained for the dispersion interaction E_^ between two atoms A and
B. Here \alpha^A and \alpha^B are the dipole polarizabilities of
the respective atoms. The quantities I_A and I_B are the first
ionization potentials of the atoms and R is the intermolecular
distance. E_^ \approx - R^

Note that this final London equation does not
contain instantaneous dipoles (see molecular
dipoles). The "explanation" of the dispersion force as the
interaction between two such dipoles was invented after London gave
the proper quantum mechanical theory. See the authoritative work
for a criticism of the instantaneous dipole model and for a modern
and thorough exposition of the theory of intermolecular
forces.

The London theory has much similarity to the
quantum mechanical theory of light
dispersion, which is why London coined the phrase "dispersion
effect" for the interaction that we described in this lemma.

## Anisotropy and non-additivity of intermolecular forces

Consider the interaction between two electric point charges at position \vec_1 and \vec_2. By Coulomb's law the interaction potential depends only on the distance |\vec_1-\vec_2| between the particles. For molecules this is different. If we see a molecule as a rigid 3-D body, it has 6 degrees of freedom (3 degrees for its orientation and 3 degrees for its position in R3). The interaction energy of two molecules (a dimer) in isotropic and homogeneous space is in general a function of 2×6−6=6 degrees of freedom (by the homogeneity of space the interaction does not depend on the position of the center of mass of the dimer, and by the isotropy of space the interaction does not depend on the orientation of the dimer). The analytic description of the interaction of two arbitrarily shaped rigid molecules requires therefore 6 parameters. (One often uses two Euler angles per molecule, plus a dihedral angle, plus the distance.) The fact that the intermolecular interaction depends on the orientation of the molecules is expressed by stating that the potential is anisotropic. Since point charges are by definition spherical symmetric, their interaction is isotropic. Especially in the older literature, intermolecular interactions are regularly assumed to be isotropic (e.g., the potential is described in Lennard-Jones form, which depends only on distance).Consider three arbitrary point charges at
distances r12, r13, and r23 apart. The total interaction U is
additive; i.e., it is the sum

- U = u(r_)+ u(r_)+u(r_).

- U = u(r_)+ u(r_)+u(r_) +u(r_,r_,r_),

## References

## External links

### Software for calculation of intermolecular forces

- Quantum 3.2
- SAPT: An ab initio quantumchemical package.

intermolecular in Arabic: تفاعل غير
ارتباطي

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