In physics, chemistry, and biology, intermolecular forces are forces that act between stable molecules or between functional groups of macromolecules. 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. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses, see Chemistry (disambiguation). ... Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle Biology (from Greek: βίος, bio, life; and λόγος, logos, knowledge), also referred to as the biological sciences, is the study of living organisms utilizing the scientific method. ... 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... Illustration of a polypeptide macromolecule Structure of a polyphenylene dendrimer macromolecule reported by Müllen and coworkers in Chem. ... Noncovalent bonding refers to a variety of interactions, that are not covalent in nature, between molecules or parts of molecules that provide force to hold the molecules or parts of molecules together usually in a specific orientation or conformation. ... In chemistry, a chemical bond is the force which holds together atoms in molecules or crystals. ... Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ... In biochemistry, the tertiary structure of a protein is its overall shape. ... A macromolecule is a molecule composed of a very large number of atoms. ... Neurotransmitters are chemicals that are used to relay, amplify and modulate electrical signals between a presynaptic and a postsynaptic neuron. ...

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. Steric effects are the interaction of molecules dictated by their shape and/or spatial relationships. ...

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 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 thorougly by Fritz London. For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Johannes Diderik van der Waals, a 1910 Nobel Prize winner, was responsible for a number of advances in physical chemistry which are named after him. ... Liquefaction of gases includes a number of processes used to convert a gas into a liquid state at a temperature above the normal boiling point of the substance. ... Neon, like all noble gases, has a full valence (outermost) electron shell. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Fritz Wolfgang London (March 7, 1900–March 30, 1954) was a German-born American physicist for whom the London force is named. ...

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. The molecular Hamiltonian is an operator in quantum chemistry and atomic, molecular, and optical physics which describes the motions of electrons and nuclei in a polyatomic molecule. ... The Born-Oppenheimer approximation, also known as the adiabatic approximation, is a technique used in quantum chemistry and condensed matter physics in order to de-couple the motion of nuclei and electrons (i. ... Hendrik Brugt Gerhard Casimir (July 15, 1909 – May 4, 2000) was a Dutch physicist. ... Dirk Polder (August 23, 1919, The Hague — March 18, 2001, Iran) was a Dutch physicist who, together with Hendrik Casimir, first predicted the existence of what today is known as the Casimir-Polder force,[1] sometimes also referred to as the Casimir effect or Casimir force. ... Simple Illustration of a paramagnetic probe made up from miniature magnets. ... After absorbing energy, an electron may jump from the ground state to a higher energy excited state. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ...

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, In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ...

while the perturbation is

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 The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ... See also: List of elements by atomic number In chemistry and physics, the atomic number (also known as the proton number) is the number of protons found in the nucleus of an atom. ... Atomic units (au) form a system of units convenient for electromagnetism, atomic physics, and quantum electrodynamics, especially when the focus is on the properties of electrons. ...

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 artefacts. Computational chemistry is a branch of chemistry that uses the results of theoretical chemistry incorporated into efficient computer programs to calculate the structures and properties of molecules and solids, applying these programs to complement the information obtained by actual chemical experiments, predict hitherto unobserved chemical phenomena, and solve related problems. ... Electronic correlation refers to the interaction between electrons in a quantum system whose electronic structure is being considered. ...

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. Charles Alfred Coulson (1910-1974) was a prominent researcher in the field of theoretical chemistry. ...

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 . But, as already noticed in 1930 by Eisenschitz and London,^[1] 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 The Pauli exclusion principle, commonly referred to simply as the exclusion principle, is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state. ... In quantum mechanics, an antisymmetrizer (also known as antisymmetrizing operator[1] ) is a linear operator that makes a wave function of N identical fermions antisymmetric under the exchange of the coordinates of any pair of fermions. ...

In the second place the projected excited states

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.^[2][3] In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

The first-order (most important) energy including exchange is in almost all symmmetry-adapted perturbation theories given by the following expression

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 Φ₀^A and Φ₀^B. 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. The van der Waals radius of an atom is the radius of an imaginary hard sphere which can be used to model the atom for many purposes. ...

Electrostatic interactions

By definition the electrostatic interaction is given by the first-order Rayleigh-Schrödinger perturbation (RS-PT) energy (without exchange):

Let the clamped nucleus α on A have position vector R_α, then its charge times the Dirac delta function, Z_α δ(r-R_α), is the charge density of this nucleus. The total charge density of monomer A is given by The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...

with the electronic charge density given by an integral over n_A - 1 primed electron coordinates:

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

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 This article is in need of attention from an expert on the subject. ...

with the Clebsch-Gordan series defined by This article may be too technical for most readers to understand. ...

and the irregular solid harmonic is defined by In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates. ...

The function Y_L,M is a normalized spherical harmonic, while and are spherical multipole moment operators. This expansion is manifestly in powers of 1/R_AB. Spherical Harmonic is a fantasy novel by Catherine Asaro which tells the story of Pharaoh Dyhianna (Dehya) Selei, ruler of the Skolian Imperialate, after the Radiance War fought by the Imperialate and their enemy Eubian Concord. ... Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, i. ...

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,

with the permanent multipole moments defined by

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 posses 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 l_A=2, l_B=1 term. From the formula above follows that this term contains the irregular solid harmonic of order L = l_A + l_B = 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. General Name, symbol, number nitrogen, N, 7 Chemical series nonmetals Group, period, block 15, 2, p Appearance colorless gas Standard atomic weight 14. ... Ethylene or ethene is the simplest alkene hydrocarbon, consisting of two carbon atoms and four hydrogens. ... R-phrases , S-phrases , , , , Flash point non-flammable Supplementary data page Structure and properties n, εr, etc. ... Impact from a water drop causes an upward rebound jet surrounded by circular capillary waves. ...

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. This article is about the electrically charged particle. ... The electron affinity, Eea, of an atom or molecule is the energy required to detach an electron from a singly charged negative ion, i. ... The ionization potential, ionization energy or EI of an atom or molecule is the energy required to remove one mole of electrons from one mole of isolated gaseous atoms or ions. ...

Once the ions (of opposite sign) are formed, the interaction between them can seen as a special case of multipolar attraction, with a 1/R_AB distance dependence. Indeed, the ionic interaction is the electrostatic term with l_A = 0 and l_B = 0. Using that the irregular harmonics for L = 0 is simply

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

(where q_A and q_B are the charges of the molecular ions) we recover—as to be expected—Coulomb's law Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ...

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 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 molecules. An example of this can be seen in hydrochloric acid: Willem Hendrik Keesom (1876-1956) was a Dutch scientist who, in 1926, invented a method to solidify helium. ... The Earths magnetic field, which is approximately a dipole. ... 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... The chemical compound hydrochloric acid is the aqueous (water-based) solution of hydrogen chloride gas (HCl). ...

Image File history File links Download high-resolution version (1100x263, 24 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Intermolecular force ...

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. The Earths magnetic field, which is approximately a dipole. ...

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. The Earths magnetic field, which is approximately a dipole. ...

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

Hence, the dipole-dipole term becomes after substitution of two Clebsch-Gordan coefficients This article may be too technical for most readers to understand. ...

where

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

Writing

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

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,

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). Atomic units (au) form a system of units convenient for electromagnetism, atomic physics, and quantum electrodynamics, especially when the focus is on the properties of electrons. ... In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. ... A Hartree (symbol Eh) is the atomic unit of energy and is named after physicist Douglas Hartree. ...

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^[4] that the temperature-averaged dipole-dipole interaction is In physics, the Boltzmann factor is a weighting factor determining the relative probability of a system in thermodynamic equilibrium at a temperature T being in a state with energy E: (kB is Boltzmanns constant. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Sir John Edward Lennard-Jones Sir John Edward Lennard-Jones KBE, FRS (October 27, 1894 - November 1, 1954) was a mathematician who held a chair of theoretical physics at Bristol University, and then a chair of theoretical science at Cambridge University. ...

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 R_AB = 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

Main article: Hydrogen bond

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: An example of a quadruple hydrogen bond between a self-assembled dimer complex reported by Meijer and coworkers. ... In chemistry, a hydrogen bond is a type of attractive intermolecular force that exists between two partial electric charges of opposite polarity. ... Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. ... H2O and HOH redirect here. ...

Image File history File links Download high-resolution version (1100x671, 41 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Intermolecular force ...

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. Electronegativity is a measure of the ability of an atom or molecule to attract electrons in the context of a chemical bond. ... General Name, symbol, number nitrogen, N, 7 Chemical series nonmetals Group, period, block 15, 2, p Appearance colorless gas Standard atomic weight 14. ... General Name, symbol, number oxygen, O, 8 Chemical series nonmetals, chalcogens Group, period, block 16, 2, p Appearance colorless (gas) very pale blue (liquid) Standard atomic weight 15. ... Distinguished from fluorene and fluorone. ...

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 since 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. They give water its unique properties that are so important to life on earth. Hydrogen bonds between hydrogen atoms and nitrogen atoms of adjacent base pairs provide the intermolecular force that help more precisely bind together the two strands in a molecule of DNA. Hydrophobic effects between the double-stranded DNA and the surrounding aqueous environment, however, are more important in maintaining the DNA in its double stranded form. The structure of part of a DNA double helix Deoxyribonucleic acid, or DNA, is a nucleic acid molecule that contains the genetic instructions used in the development and functioning of all known living organisms. ...

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: In chemistry, the term van der Waals force originally referred to all forms of intermolecular forces; however, in modern usage it tends to refer to intermolecular forces that deal with forces due to the polarization of molecules. ...

 (+)(-)(+) (-) (+) [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). The Earths magnetic field, which is approximately a 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. The Earths magnetic field, which is approximately a dipole. ...

 (+) (-) (+) (-) [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. Multipole moments in mathematics and mathematical physics are an orthogonal basis for the decomposition of a function, based on the response of a field to point sources that are brought infinitely close to each other. ... Schematic quadrupole magnet(four-pole) used to focus particle beams in a particle accelerator. ...

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.^[5] 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. Fritz Wolfgang London (March 7, 1900–March 30, 1954) was a German-born American physicist for whom the London force is named. ... In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ...

London developed the perturbation V in a Taylor series in , where R is the distance between the nuclear centers of mass of the monomers. As the degree of the Taylor series rises, it approaches the correct function. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

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. This article is in need of attention from an expert on the subject. ... Albrecht Otto Johannes Unsöld (April 20, 1905 – September 23, 1995) was a German astronomer. ... Polarizability is the relative tendency of the electron cloud of an atom to be distorted from its normal shape by the presence of a nearby ion or dipole--that is, by an external electric field. ... The ionization potential, ionization energy or EI of an atom or molecule is the energy required to remove one mole of electrons from one mole of isolated gaseous atoms or ions. ...

In this manner the following approximation is obtained for the dispersion interaction between two atoms A and B. Here α^A and α^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.

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^[6] for a criticism of the instantaneous dipole model and^[7] for a modern and thorough exposition of the theory of intermolecular forces. The Earths magnetic field, which is approximately a dipole. ...

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. Dispersion of a light beam in a prism. ...

Anisotropy and non-additivity of intermolecular forces

Consider the interaction between two electric point charges at position and . By Coulomb's law the interaction potential depends only on the distance 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). The Coulomb barrier, named after physicist Charles-Augustin de Coulomb (1736–1806), is the energy barrier due to electrostatic interaction that two nuclei need to overcome so they can get close enough to undergo nuclear fusion. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... Euler angles are a means of representing the spatial orientation of an object. ... Neutral atoms and molecules are subject to two distinct forces in the limit of large distance and short distance: an attractive force at long ranges (van der Waals force, or dispersion force) and a repulsive force at short ranges (the result of overlapping electron orbitals, referred to as Pauli repulsion...

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(r₁₂) + u(r₁₃) + u(r₂₃).

Again for molecules this can be different. Pretending that the interaction depends on distances only—but see above—the interaction of three molecules takes in general the form

U = u(r₁₂) + u(r₁₃) + u(r₂₃) + u(r₁₂,r₁₃,r₂₃),

where u(r₁₂,r₁₃,r₂₃) is a non-additive three-body interaction. Such an interaction can be caused by exchange interactions, by induction, and by dispersion (the Axilrod-Teller triple dipole effect). In physics, the exchange interaction is a quantum mechanical effect which increases or decreases the energy of two or more electrons when their wave functions overlap. ... The Axilrod-Teller potential is a three-body potential that results from a third-order perturbation correction to the attractive London dispersion interactions where is the distance between atoms and , and is the angle between the vectors and . ...

See also

• Hydrophobic effect
• Intramolecular force
• Polymer

The hydrophobic effect is the property that nonpolar molecules like to self-associate in the presence of aqueous solution. ... An intramolecular force is any force that holds together the atoms making up a molecule. ... A polymer (from Greek: πολυ, polu, many; and μέρος, meros, part) is a substance composed of molecules with large molecular mass composed of repeating structural units, or monomers, connected by covalent chemical bonds. ...

References

1. ^ 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.
2. ^ B. Jeziorski, R. Moszynski, and K. Szalewicz, Chemical Reviews, vol. 94, pp. 1887-1930 (1994).
3. ^ K. Szalewicz and B. Jeziorski, in: Molecular Interactions, editor S. Scheiner, Wiley, Chichester (1995). ISBN 0471 959219.
4. ^ J. E. Lennard-Jones, Proc. Royal Society (London), vol. 43, p. 461 (1931).
5. ^ Cite error 8; No text given.
6. ^ J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954
7. ^ A. J. Stone, The Theory of Intermolecular Forces, 1996, (Clarendon Press, Oxford)

External links

Software for calculation of intermolecular forces

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