Intermolecular Forces

--> In Physics , Chemistry , and Biology , intermolecular forces are forces that act between stable Molecule s or between functional groups of Macromolecule s. These Non-covalent forces, which give rise to bonding energies of less than a few kcal/mol, are generally much weaker than the Chemical Bonding forces. Nevertheless, intermolecular forces are responsible for a wide range of physical, chemical, and biological phenomena. For instance, they play a role in the deviation from the Ideal Gas Law for real gases, the Tertiary Structure of Macromolecules and signal induction in Neurotransmitters . In general one distinguishes short and long range intermolecular 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 Waals minimum''. The position and depth of the Van der Waals minimum depends on distance and mutual orientation of the molecules. GENERAL THEORY 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 Gas es 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 thorougly 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, : $H^{(0)} \equiv H^{A}+ H^{B},\,$ while the perturbation is : $V^{AB} \equiv H^{AB} - H^{A}- H^{B} = \sum_{i\in A} \sum_{j \in B} rac{q_i q_j}{r_{ij}},$ where ''q''_''i'' indicates the charge (in units ''e'' of Elementary Charge ) of a particle of monomer ''A''; ''q''_''j'' belongs to monomer ''B''. For electrons we take ''q'' = -1, for a nucleus we take ''q'' equal to its Atomic Number ''Z''. The quantity ''r''_''ij'' is the distance between particle ''i'' and particle ''j''. In this equation and further in this article Atomic Units are used. Perturbation theory is based on expansions of perturbed states in terms of unperturbed states (eigenstates of ''H''⁽⁰⁾). In the present case the unperturbed states are products : $\Phi_n^A \Phi_m^B\quad \hbox{with}\quad H^A \Phi_n^A = E_n^A\Phi_n^A\quad\hbox{and}\quad H^B \Phi_m^B = E_m^B\Phi_m^B$ 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 Method s 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 artefacts. Supermolecule calculations must be performed with very high precision, because the problem, known as ''weighing the captain'', arises here. First we weigh the ship with the captain aboard (total energy of molecules in interaction) and then we weigh the ship with the captain ashore (total energy of molecules at an infinite distance apart); the difference gives the captain's weight. This parable is due to the late Charles Coulson . To understand it we must remember that the total energy of molecules is six to seven orders of magnitude larger than a typical intermolecular interaction. That is, the significant digits in the results of supermolecule calculations start to appear beyond the sixth or seventh decimal place. A disadvantage of the supermolecule method is that it yields the interaction as a lump sum. It does not give an interaction energy separated in the four fundamental contributions mentioned above. Therefore, we will not discuss the supermolecule method any further in this article. EXCHANGE The monomer functions Φ_''n''^''A'' and Φ_''m''^''B'' 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 $ilde{\mathcal{A}}^{AB}$ . But, as already noticed in 1930 by Eisenschitz and London,R. Eisenschitz and F. London, Zeitschrift für Physik, vol. 60, p. 491 (1930). English translations in H. Hettema, ''Quantum Chemistry, Classic Scientific Papers,'' World Scientific, Singapore (2000), p. 336. 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 : $\big[ ilde{\mathcal{A}}^{AB}, H^{(0)}\big] e 0 .$ In the second place the projected excited states : $ilde{\mathcal{A}}^{AB} \Phi^A_n \Phi^B_m$ 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.B. Jeziorski, R. Moszynski, and K. Szalewicz, Chemical Reviews, vol. 94, pp. 1887-1930 (1994).K. Szalewicz and B. Jeziorski, in: ''Molecular Interactions'', editor S. Scheiner, Wiley, Chichester (1995). ISBN 0471 959219. The first-order (most important) energy including exchange is in almost all symmmetry-adapted perturbation theories given by the following expression : ${border 0 cellpadding="2" cellspacing="0" style="width:300px" E^{(1)} \mathrm{electrostatic} \langle \Phi_0^A \Phi_0^B V^{AB} \Phi_0^A \Phi_0^B angle Ho^A \mathrm{el}(\mathbf{r}) n_A \int \Phi^A_0(\mathbf{r}, \mathbf{r}'_2, \ldots, \mathbf{r}'_{n_A})^2 d\mathbf{r}'_2 \cdots d\mathbf{r}'_{n_A}$
	{border	0 cellpadding="2" cellspacing="0" style="width:300px"
	E^{(1)} \mathrm{electrostatic}	\langle \Phi_0^A \Phi_0^B V^{AB} \Phi_0^A \Phi_0^B angle
	Ho^A \mathrm{el}(\mathbf{r})	n_A \int \Phi^A_0(\mathbf{r}, \mathbf{r}'_2, \ldots, \mathbf{r}'_{n_A})^2 d\mathbf{r}'_2 \cdots d\mathbf{r}'_{n_A}

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: ''V''^''AB'' 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 proved

:

^{\ell_A+\ell_B}_M

with the Clebsch-Gordan series defined by
:

^{\ell_A+\ell_B}_M \equiv \sum_{m_A=-\ell_A}^{\ell_A} \sum_{m_B=-\ell_B}^{\ell_B}\;

The function Y_''L,M'' is a normalized Spherical Harmonic , while

Q^{\ell_A}_{m_A}

and

Q^{\ell_B}_{m_B}

are Spherical Multipole Moment operators. This expansion is manifestly in powers of 1/''R_AB''.

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,
:

^{\ell_A+\ell_B}_M, \end{align}

with the ''permanent multipole moments'' defined by
:

and that the monopole moments and their Clebsch-Gordan coupling are
:

M^{0_A}_0 = q_A,\quad M^{0_B}_0 = q_B \quad\hbox{and}\quad \mathbf{M}^{0_A}^0_0 = q_A q_B ,

(where ''q''_''A'' and ''q''_''B'' are the charges of the molecular ions)
we recover—as to be expected— Coulomb's Law
:

E^{(1)}_\mathrm{electrostatic} = rac{q_A q_B}{R_{AB}} + \hbox{higher terms}.

For shorter distances, where the charge distributions of the monomers overlap, the ions will repel each other because of intermonomer 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 delocalised 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 Molecule s. 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 ''l''_''A'' = 1 and ''l''_''B'' = 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 .

To get the mathematical equation for the dipole-dipole interaction we must consider the term with ''l''_''A'' = 1 and ''l''_''B'' = 1 in the expansion of the electrostatic energy. Because this expansion is termwise rotational invariant, we can choose a convenient system of axes to evaluate the term. We choose a coordinate system centered on ''A'' with its ''z''-axis coinciding with the intermolecular vector R_''AB''.
Under this circumstance it holds for the irregular solid harmonic that
:

I^m_{\ell}(\mathbf{R}_{AB}) \equiv \sqrt{rac{4\pi}{2\ell+1}} \; rac{ Y^m_{\ell}(0,0)}{R^{\ell+1}_{AB}} = rac{\delta_{0m}}{R^{\ell+1}_{AB}} .

Hence, the dipole-dipole term becomes after substitution of two Clebsch-Gordan Coefficients
:

^{2}_0 = - rac{\sqrt{6}}{R^3_{AB}} \left[ rac{1}{\sqrt{6}} (\mu^A_{1} \mu^B_{-1} + \mu^A_{-1} \mu^B_{1}) +\sqrt{ rac{2}{3}} \mu^A_0 \mu^B_0 ight],

where
:

\mu^A_{\pm 1} \equiv M^{1_A}_{\pm 1} = \mp rac{1}{\sqrt{2}} (\mu^A_x \pm i \mu^A_y)\quad
\hbox{and}\quad \mu^A_0 \equiv  M^{1_A}_{0} = \mu^A_z.

Analogous relations hold for the permanent dipole moments on ''B''. Then
:

E_{\mathrm{dip-dip}} = rac{1}{R^{3}_{AB}}\left \mu_x^A\mu_x^B + \mu_y^A\mu_y^B - 2\mu_z^A\mu_z^B 
ight .

Writing
:

\boldsymbol{\mu}^A = (\mu_x^A, \mu_y^A, \mu_z^A) \quad\hbox{and}\quad
\mu_z^A = \boldsymbol{\mu}^A\cdot \hat{\mathbf{R}}_{AB} \equiv \boldsymbol{\mu}^A\cdot rac{\mathbf{R}_{AB}}{R_{AB}}

and similarly for ''B'', we get finally the well-known expression
:

E_{\mathrm{dip-dip}} = rac{1}{R^{3}_{AB}}\left \boldsymbol{\mu}^A\cdot\boldsymbol{\mu}^B  - 3 (\boldsymbol{\mu}^A\cdot \hat{\mathbf{R}}_{AB}) (\hat{\mathbf{R}}_{AB}\cdot \boldsymbol{\mu}^B) 
ight .

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{\mu}^A = \boldsymbol{\mu}^B = \mu_\mathrm{HCl} 
\begin{pmatrix} 0 \ 0 \ -1 \end{pmatrix}
\quad\hbox{and}\quad E_{\mathrm{dip-dip}} = rac{-2\mu^2_\mathrm{HCl}}{R^{3}_{AB}}.

The value of μ_HCl 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(-''E''_dip-dip/''kT''), where ''k'' is the Boltzmann Constant and ''T'' the absolute temperature.
It was first shown by Lennard-Jones J. E. Lennard-Jones, Proc. Royal Society (London), vol. 43, p. 461 (1931). that the temperature-averaged dipole-dipole interaction is
:

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME

	MORE SHOPPING...

	CATEGORIES ABOUT INTERMOLECULAR FORCE

	chemical bonding

	wikipedia articles with ascii art

Information About

Intermolecular Forces