Franck–Condon principle

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Figure 1. Franck–Condon principle energy diagram. Since electronic transitions are very fast compared with nuclear motions, the vibrational states to and from which absorption and emission occur are those that correspond to a minimal change in the nuclear coordinates. As a result, both absorption and emission produce molecules in vibrationally excited states. The potential wells are shown favoring transitions with changes in ν.

The Franck–Condon principle describes the intensities of vibronic transitions, or the absorption or emission of a photon. It states that when a molecule is undergoing an electronic transition, such as ionization, the nuclear configuration of the molecule experiences no significant change.

Overview

Figure 2. Schematic representation of the absorption and fluorescence spectra corresponding to the energy diagram in Figure 1. The symmetry is due to the equal shape of the ground and excited state potential wells. The narrow lines can usually only be observed in the spectra of dilute gases. The darker curves represent the inhomogeneous broadening of the same transitions as occurs in liquids and solids. Electronic transitions between the lowest vibrational levels of the electronic states (the 0–0 transition) have the same energy in both absorption and fluorescence.

Figure 3. Semiclassical pendulum analogy of the Franck–Condon principle. Vibronic transitions are allowed at the classical turning points because both the momentum and the nuclear coordinates correspond in the two represented energy levels. In this illustration, the 0–2 vibrational transitions are favored.

The Franck–Condon principle has a well-established semiclassical interpretation based on the original contributions of James Franck. Electronic transitions are relatively instantaneous compared with the time scale of nuclear motions, therefore if the molecule is to move to a new vibrational level during the electronic transition, this new vibrational level must be instantaneously compatible with the nuclear positions and momenta of the vibrational level of the molecule in the originating electronic state. In the semiclassical picture of vibrations (oscillations) of a simple harmonic oscillator, the necessary conditions can occur at the turning points, where the momentum is zero.

Classically, the Franck–Condon principle is the approximation that an electronic transition is most likely to occur without changes in the positions of the nuclei in the molecular entity and its environment. The resulting state is called a Franck–Condon state, and the transition involved, a vertical transition. The quantum mechanical formulation of this principle is that the intensity of a vibronic transition is proportional to the square of the overlap integral between the vibrational wavefunctions of the two states that are involved in the transition.

— IUPAC Compendium of Chemical Terminology, 2nd Edition (1997)

In the quantum mechanical picture, the vibrational levels and vibrational wavefunctions are those of quantum harmonic oscillators, or of more complex approximations to the potential energy of molecules, such as the Morse potential. Figure 1 illustrates the Franck–Condon principle for vibronic transitions in a molecule with Morse-like potential energy functions in both the ground and excited electronic states. In the low temperature approximation, the molecule starts out in the v = 0 vibrational level of the ground electronic state and upon absorbing a photon of the necessary energy, makes a transition to the excited electronic state. The electron configuration of the new state may result in a shift of the equilibrium position of the nuclei constituting the molecule. In Figure 3 this shift in nuclear coordinates between the ground and the first excited state is labeled as q₀₁. In the simplest case of a diatomic molecule the nuclear coordinates axis refers to the internuclear separation. The vibronic transition is indicated by a vertical arrow due to the assumption of constant nuclear coordinates during the transition. The probability that the molecule can end up in any particular vibrational level is proportional to the square of the (vertical) overlap of the vibrational wavefunctions of the original and final state (see Quantum mechanical formulation section below). In the electronic excited state molecules quickly relax to the lowest vibrational level of the lowest electronic excitation state (Kasha's rule), and from there can decay to the electronic ground state via photon emission. The Franck–Condon principle is applied equally to absorption and to fluorescence.

The applicability of the Franck–Condon principle in both absorption and fluorescence, along with Kasha's rule leads to an approximate mirror symmetry shown in Figure 2. The vibrational structure of molecules in a cold, sparse gas is most clearly visible due to the absence of inhomogeneous broadening of the individual transitions. Vibronic transitions are drawn in Figure 2 as narrow, equally spaced Lorentzian line shapes. Equal spacing between vibrational levels is only the case for the parabolic potential of simple harmonic oscillators, in more realistic potentials, such as those shown in Figure 1, energy spacing decreases with increasing vibrational energy. Electronic transitions to and from the lowest vibrational states are often referred to as 0–0 (zero zero) transitions and have the same energy in both absorption and fluorescence.

Development of the principle

In a report published in 1926 in Transactions of the Faraday Society, James Franck was concerned with the mechanisms of photon-induced chemical reactions. The presumed mechanism was the excitation of a molecule by a photon, followed by a collision with another molecule during the short period of excitation. The question was whether it was possible for a molecule to break into photoproducts in a single step, the absorption of a photon, and without a collision. In order for a molecule to break apart, it must acquire from the photon a vibrational energy exceeding the dissociation energy, that is, the energy to break a chemical bond. However, as was known at the time, molecules will only absorb energy corresponding to allowed quantum transitions, and there are no vibrational levels above the dissociation energy level of the potential well. High-energy photon absorption leads to a transition to a higher electronic state instead of dissociation. In examining how much vibrational energy a molecule could acquire when it is excited to a higher electronic level, and whether this vibrational energy could be enough to immediately break apart the molecule, he drew three diagrams representing the possible changes in binding energy between the lowest electronic state and higher electronic states.

Diagram I. shows a great weakening of the binding on a transition from the normal state n to the excited states a and a'. Here we have D > D' and D' > D". At the same time the equilibrium position of the nuclei moves with the excitation to greater values of r. If we go from the equilibrium position (the minimum of potential energy) of the n curve vertically [emphasis added] upwards to the a curves in Diagram I. the particles will have a potential energy greater than D' and will fly apart. In this case we have a very great change in the oscillation energy on excitation by light...

— James Franck, 1926

James Franck recognized that changes in vibrational levels could be a consequence of the instantaneous nature of excitation to higher electronic energy levels and a new equilibrium position for the nuclear interaction potential. Edward Condon extended this insight beyond photoreactions in a 1926 Physical Review article titled "A Theory of Intensity Distribution in Band Systems".^[3] Here he formulates the semiclassical formulation in a manner quite similar to its modern form. The first joint reference to both Franck and Condon in regard to the new principle appears in the same 1926 issue of Physical Review in an article on the band structure of carbon monoxide by Raymond Birge.

Figure 5. Figure 1 in Edward Condon's first publication on what is now the Franck–Condon principle [Condon 1926]. Condon chose to superimpose the potential curves to illustrate the method of estimating vibrational transitions.

Quantum mechanical formulation

Consider an electrical dipole transition from the initial vibrational state (υ) of the ground electronic level (ε), ${\displaystyle |ϵv⟩}$, to some vibrational state (υ′) of an excited electronic state (ε′), ${\displaystyle |{ϵ}^{\prime }{v}^{\prime }⟩}$ (see bra–ket notation). The molecular dipole operator μ is determined by the charge (−e) and locations (r_i) of the electrons as well as the charges (+Z_je) and locations (R_j) of the nuclei:

${\displaystyle \mathit{\mu }={\mathit{\mu }}_e+{\mathit{\mu }}_N=-e\sum _i{\mathit{r}}_i+e\sum _j{Z}_j{\mathit{R}}_j.}$

The probability amplitude P for the transition between these two states is given by

${\displaystyle P=⟨{\psi }^{\prime }|\mathit{\mu }|\psi ⟩=\int {\psi }^{\prime \ast }\mathit{\mu }\psi d\tau ,}$

where ${\displaystyle \psi }$ and ${\displaystyle {\psi }^{\prime }}$ are, respectively, the overall wavefunctions of the initial and final state. The overall wavefunctions are the product of the individual vibrational (depending on spatial coordinates of the nuclei) and electronic space and spin wavefunctions:

${\displaystyle \psi ={\psi }_e{\psi }_v{\psi }_s.}$

This separation of the electronic and vibrational wavefunctions is an expression of the Born–Oppenheimer approximation and is the fundamental assumption of the Franck–Condon principle. Combining these equations leads to an expression for the probability amplitude in terms of separate electronic space, spin and vibrational contributions:

${\displaystyle P=⟨{\psi }_e^{\prime }{\psi }_v^{\prime }{\psi }_s^{\prime }|\mathit{\mu }|{\psi }_e{\psi }_v{\psi }_s⟩=\int {\psi }_e^{\prime \ast }{\psi }_v^{\prime \ast }{\psi }_s^{\prime \ast }({\mathit{\mu }}_e+{\mathit{\mu }}_N){\psi }_e{\psi }_v{\psi }_{s}d\tau }$

${\displaystyle =\int {\psi }_e^{\prime \ast }{\psi }_v^{\prime \ast }{\psi }_s^{\prime \ast }{\mathit{\mu }}_e{\psi }_e{\psi }_v{\psi }_{s}d\tau +\int {\psi }_e^{\prime \ast }{\psi }_v^{\prime \ast }{\psi }_s^{\prime \ast }{\mathit{\mu }}_N{\psi }_e{\psi }_v{\psi }_{s}d\tau }$

${\displaystyle =\underset{{\displaystyle \genfrac{}{}{0ex}{}{\text{Franck–Condon}}{\text{factor}}}}{\underbrace{\int {\psi }_v^{\prime \ast }{\psi }_{v}d{\tau }_n}}\underset{{\displaystyle \genfrac{}{}{0ex}{}{\text{orbital}}{\text{selection rule}}}}{\underbrace{\int {\psi }_e^{\prime \ast }{\mathit{\mu }}_e{\psi }_{e}d{\tau }_e}}\underset{{\displaystyle \genfrac{}{}{0ex}{}{\text{spin}}{\text{selection rule}}}}{\underbrace{\int {\psi }_s^{\prime \ast }{\psi }_{s}d{\tau }_s}}+\underset{{\displaystyle 0}}{\underbrace{\int {\psi }_e^{\prime \ast }{\psi }_{e}d{\tau }_e}}\int {\psi }_v^{\prime \ast }{\mathit{\mu }}_N{\psi }_{v}d{\tau }_v\int {\psi }_s^{\prime \ast }{\psi }_{s}d{\tau }_s.}$

The spin-independent part of the initial integral is here approximated as a product of two integrals:

${\displaystyle \iint {\psi }_v^{\prime \ast }{\psi }_e^{\prime \ast }{\mathit{\mu }}_e{\psi }_e{\psi }_{v}d{\tau }_{e}d{\tau }_n\approx \int {\psi }_v^{\prime \ast }{\psi }_{v}d{\tau }_n\int {\psi }_e^{\prime \ast }{\mathit{\mu }}_e{\psi }_{e}d{\tau }_e.}$

This factorization would be exact if the integral ${\displaystyle \int {\psi }_e^{\prime \ast }{\mathit{\mu }}_e{\psi }_{e}d{\tau }_e}$ over the spatial coordinates of the electrons would not depend on the nuclear coordinates. However, in the Born–Oppenheimer approximation ${\displaystyle {\psi }_e}$ and ${\displaystyle {\psi }_e^{\prime }}$ do depend (parametrically) on the nuclear coordinates, so that the integral (a so-called transition dipole surface) is a function of nuclear coordinates. Since the dependence is usually rather smooth it is neglected (i.e., the assumption that the transition dipole surface is independent of nuclear coordinates, called the Condon approximation is often allowed).

The first integral after the plus sign is equal to zero because electronic wavefunctions of different states are orthogonal. Remaining is the product of three integrals. The first integral is the vibrational overlap integral, also called the Franck–Condon factor. The remaining two integrals contributing to the probability amplitude determine the electronic spatial and spin selection rules.

The Franck–Condon principle is a statement on allowed vibrational transitions between two different electronic states; other quantum mechanical selection rules may lower the probability of a transition or prohibit it altogether. Rotational selection rules have been neglected in the above derivation. Rotational contributions can be observed in the spectra of gases but are strongly suppressed in liquids and solids.

It should be clear that the quantum mechanical formulation of the Franck–Condon principle is the result of a series of approximations, principally the electrical dipole transition assumption and the Born–Oppenheimer approximation. Weaker magnetic dipole and electric quadrupole electronic transitions along with the incomplete validity of the factorization of the total wavefunction into nuclear, electronic spatial and spin wavefunctions means that the selection rules, including the Franck–Condon factor, are not strictly observed. For any given transition, the value of P is determined by all of the selection rules, however spin selection is the largest contributor, followed by electronic selection rules. The Franck–Condon factor only weakly modulates the intensity of transitions, i.e., it contributes with a factor on the order of 1 to the intensity of bands whose order of magnitude is determined by the other selection rules. The table below gives the range of extinction coefficients for the possible combinations of allowed and forbidden spin and orbital selection rules.

Intensities of electronic transitions

	Range of extinction coefficient (ε) values (mol−1 cm−1)
Spin and orbitally allowed	103 to 105
Spin allowed but orbitally forbidden	100 to 103
Spin forbidden but orbitally allowed	10−5 to 100

Franck–Condon metaphors in spectroscopy

The Franck–Condon principle, in its canonical form, applies only to changes in the vibrational levels of a molecule in the course of a change in electronic levels by either absorption or emission of a photon. The physical intuition of this principle is anchored by the idea that the nuclear coordinates of the atoms constituting the molecule do not have time to change during the very brief amount of time involved in an electronic transition. However, this physical intuition can be, and is indeed, routinely extended to interactions between light-absorbing or emitting molecules (chromophores) and their environment. Franck–Condon metaphors are appropriate because molecules often interact strongly with surrounding molecules, particularly in liquids and solids, and these interactions modify the nuclear coordinates of the chromophore in ways closely analogous to the molecular vibrations considered by the Franck–Condon principle.

Figure 6. Energy diagram of an electronic transition with phonon coupling along the configurational coordinate q _i, a normal mode of the lattice. The upwards arrows represent absorption without phonons and with three phonons. The downwards arrows represent the symmetric process in emission.

Franck–Condon principle for phonons

The closest Franck–Condon analogy is due to the interaction of phonons (quanta of lattice vibrations) with the electronic transitions of chromophores embedded as impurities in the lattice. In this situation, transitions to higher electronic levels can take place when the energy of the photon corresponds to the purely electronic transition energy or to the purely electronic transition energy plus the energy of one or more lattice phonons. In the low-temperature approximation, emission is from the zero-phonon level of the excited state to the zero-phonon level of the ground state or to higher phonon levels of the ground state. Just like in the Franck–Condon principle, the probability of transitions involving phonons is determined by the overlap of the phonon wavefunctions at the initial and final energy levels. For the Franck–Condon principle applied to phonon transitions, the label of the horizontal axis of Figure 1 is replaced in Figure 6 with the configurational coordinate for a normal mode. The lattice mode ${\displaystyle q_i}$ potential energy in Figure 6 is represented as that of a harmonic oscillator, and the spacing between phonon levels (${\displaystyle \hslash {\mathrm{\Omega }}_i}$) is determined by lattice parameters. Because the energy of single phonons is generally quite small, zero- or few-phonon transitions can only be observed at temperatures below about 40 kelvins.

See Zero-phonon line and phonon sideband for further details and references.

Franck–Condon principle in solvation

Figure 7. Energy diagram illustrating the Franck–Condon principle applied to the solvation of chromophores. The parabolic potential curves symbolize the interaction energy between the chromophores and the solvent. The Gaussian curves represent the distribution of this interaction energy.

Franck–Condon considerations can also be applied to the electronic transitions of chromophores dissolved in liquids. In this use of the Franck–Condon metaphor, the vibrational levels of the chromophores, as well as interactions of the chromophores with phonons in the liquid, continue to contribute to the structure of the absorption and emission spectra, but these effects are considered separately and independently.

Consider chromophores surrounded by solvent molecules. These surrounding molecules may interact with the chromophores, particularly if the solvent molecules are polar. This association between solvent and solute is referred to as solvation and is a stabilizing interaction, that is, the solvent molecules can move and rotate until the energy of the interaction is minimized. The interaction itself involves electrostatic and van der Waals forces and can also include hydrogen bonds. Franck–Condon principles can be applied when the interactions between the chromophore and the surrounding solvent molecules are different in the ground and in the excited electronic state. This change in interaction can originate, for example, due to different dipole moments in these two states. If the chromophore starts in its ground state and is close to equilibrium with the surrounding solvent molecules and then absorbs a photon that takes it to the excited state, its interaction with the solvent will be far from equilibrium in the excited state. This effect is analogous to the original Franck–Condon principle: the electronic transition is very fast compared with the motion of nuclei—the rearrangement of solvent molecules in the case of solvation. We now speak of a vertical transition, but now the horizontal coordinate is solvent-solute interaction space. This coordinate axis is often labeled as "Solvation Coordinate" and represents, somewhat abstractly, all of the relevant dimensions of motion of all of the interacting solvent molecules.

In the original Franck–Condon principle, after the electronic transition, the molecules which end up in higher vibrational states immediately begin to relax to the lowest vibrational state. In the case of solvation, the solvent molecules will immediately try to rearrange themselves in order to minimize the interaction energy. The rate of solvent relaxation depends on the viscosity of the solvent. Assuming the solvent relaxation time is short compared with the lifetime of the electronic excited state, emission will be from the lowest solvent energy state of the excited electronic state. For small-molecule solvents such as water or methanol at ambient temperature, solvent relaxation time is on the order of some tens of picoseconds whereas chromophore excited state lifetimes range from a few picoseconds to a few nanoseconds. Immediately after the transition to the ground electronic state, the solvent molecules must also rearrange themselves to accommodate the new electronic configuration of the chromophore. Figure 7 illustrates the Franck–Condon principle applied to solvation. When the solution is illuminated by light corresponding to the electronic transition energy, some of the chromophores will move to the excited state. Within this group of chromophores there will be a statistical distribution of solvent-chromophore interaction energies, represented in the figure by a Gaussian distribution function. The solvent-chromophore interaction is drawn as a parabolic potential in both electronic states. Since the electronic transition is essentially instantaneous on the time scale of solvent motion (vertical arrow), the collection of excited state chromophores is immediately far from equilibrium. The rearrangement of the solvent molecules according to the new potential energy curve is represented by the curved arrows in Figure 7. Note that while the electronic transitions are quantized, the chromophore-solvent interaction energy is treated as a classical continuum due to the large number of molecules involved. Although emission is depicted as taking place from the minimum of the excited state chromophore-solvent interaction potential, significant emission can take place before equilibrium is reached when the viscosity of the solvent is high, or the lifetime of the excited state is short. The energy difference between absorbed and emitted photons depicted in Figure 7 is the solvation contribution to the Stokes shift.

Molecule

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Molecule

Atomic force microscopy (AFM) image of a PTCDA molecule, in which the five six-carbon rings are visible

A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings

AFM image of 1,5,9-trioxo-13-azatriangulene and its chemical structure

A molecule is a group of two or more atoms that are held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemistry, and biochemistry, the distinction from ions is dropped and molecule is often used when referring to polyatomic ions.

A molecule may be homonuclear, that is, it consists of atoms of one chemical element, e.g. two atoms in the oxygen molecule (O₂); or it may be heteronuclear, a chemical compound composed of more than one element, e.g. water (two hydrogen atoms and one oxygen atom; H₂O). In the kinetic theory of gases, the term molecule is often used for any gaseous particle regardless of its composition. This relaxes the requirement that a molecule contains two or more atoms, since the noble gases are individual atoms. Atoms and complexes connected by non-covalent interactions, such as hydrogen bonds or ionic bonds, are typically not considered single molecules.

Concepts similar to molecules have been discussed since ancient times, but modern investigation into the nature of molecules and their bonds began in the 17th century. Refined over time by scientists such as Robert Boyle, Amedeo Avogadro, Jean Perrin, and Linus Pauling, the study of molecules is today known as molecular physics or molecular chemistry.

Etymology

According to Merriam-Webster and the Online Etymology Dictionary, the word "molecule" derives from the Latin "moles" or small unit of mass. The word is derived from French molécule (1678), from Neo-Latin molecula, diminutive of Latin moles "mass, barrier". The word, which until the late 18th century was used only in Latin form, became popular after being used in discussions of the natural philosophy of René Descartes.

History

Main article: History of molecular theory

The definition of the molecule has evolved as knowledge of the structure of molecules has increased. Earlier definitions were less precise, defining molecules as the smallest particles of pure chemical substances that still retain their composition and chemical properties. This definition often breaks down since many substances in ordinary experience, such as rocks, salts, and metals, are composed of large crystalline networks of chemically bonded atoms or ions, but are not made of discrete molecules.

The modern concept of molecules can be traced back towards pre-scientific and Greek philosophers such as Leucippus and Democritus who argued that all the universe is composed of atoms and voids. Circa 450 BC Empedocles imagined fundamental elements (fire (), earth (), air (), and water ()) and "forces" of attraction and repulsion allowing the elements to interact.

A fifth element, the incorruptible quintessence aether, was considered to be the fundamental building block of the heavenly bodies. The viewpoint of Leucippus and Empedocles, along with the aether, was accepted by Aristotle and passed to medieval and renaissance Europe.

In a more concrete manner, however, the concept of aggregates or units of bonded atoms, i.e. "molecules", traces its origins to Robert Boyle's 1661 hypothesis, in his famous treatise The Sceptical Chymist, that matter is composed of clusters of particles and that chemical change results from the rearrangement of the clusters. Boyle argued that matter's basic elements consisted of various sorts and sizes of particles, called "corpuscles", which were capable of arranging themselves into groups. In 1789, William Higgins published views on what he called combinations of "ultimate" particles, which foreshadowed the concept of valency bonds. If, for example, according to Higgins, the force between the ultimate particle of oxygen and the ultimate particle of nitrogen were 6, then the strength of the force would be divided accordingly, and similarly for the other combinations of ultimate particles.

Amedeo Avogadro created the word "molecule". His 1811 paper "Essay on Determining the Relative Masses of the Elementary Molecules of Bodies", he essentially states, i.e. according to Partington's A Short History of Chemistry, that:

The smallest particles of gases are not necessarily simple atoms, but are made up of a certain number of these atoms united by attraction to form a single molecule.

In coordination with these concepts, in 1833 the French chemist Marc Antoine Auguste Gaudin presented a clear account of Avogadro's hypothesis, regarding atomic weights, by making use of "volume diagrams", which clearly show both semi-correct molecular geometries, such as a linear water molecule, and correct molecular formulas, such as H₂O:

Marc Antoine Auguste Gaudin's volume diagrams of molecules in the gas phase (1833)

In 1917, an unknown American undergraduate chemical engineer named Linus Pauling was learning the Dalton hook-and-eye bonding method, which was the mainstream description of bonds between atoms at the time. Pauling, however, was not satisfied with this method and looked to the newly emerging field of quantum physics for a new method. In 1926, French physicist Jean Perrin received the Nobel Prize in physics for proving, conclusively, the existence of molecules. He did this by calculating the Avogadro constant using three different methods, all involving liquid phase systems. First, he used a gamboge soap-like emulsion, second by doing experimental work on Brownian motion, and third by confirming Einstein's theory of particle rotation in the liquid phase.

In 1927, the physicists Fritz London and Walter Heitler applied the new quantum mechanics to the deal with the saturable, nondynamic forces of attraction and repulsion, i.e., exchange forces, of the hydrogen molecule. Their valence bond treatment of this problem, in their joint paper, was a landmark in that it brought chemistry under quantum mechanics. Their work was an influence on Pauling, who had just received his doctorate and visited Heitler and London in Zürich on a Guggenheim Fellowship.

Subsequently, in 1931, building on the work of Heitler and London and on theories found in Lewis' famous article, Pauling published his ground-breaking article "The Nature of the Chemical Bond" in which he used quantum mechanics to calculate properties and structures of molecules, such as angles between bonds and rotation about bonds. On these concepts, Pauling developed hybridization theory to account for bonds in molecules such as CH₄, in which four sp³ hybridised orbitals are overlapped by hydrogen's 1s orbital, yielding four sigma (σ) bonds. The four bonds are of the same length and strength, which yields a molecular structure as shown below:

A schematic presentation of hybrid orbitals overlapping hydrogens's orbitals

Molecular science

The science of molecules is called molecular chemistry or molecular physics, depending on whether the focus is on chemistry or physics. Molecular chemistry deals with the laws governing the interaction between molecules that results in the formation and breakage of chemical bonds, while molecular physics deals with the laws governing their structure and properties. In practice, however, this distinction is vague. In molecular sciences, a molecule consists of a stable system (bound state) composed of two or more atoms. Polyatomic ions may sometimes be usefully thought of as electrically charged molecules. The term unstable molecule is used for very reactive species, i.e., short-lived assemblies (resonances) of electrons and nuclei, such as radicals, molecular ions, Rydberg molecules, transition states, van der Waals complexes, or systems of colliding atoms as in Bose–Einstein condensate.

Prevalence

Molecules as components of matter are common. They also make up most of the oceans and atmosphere. Most organic substances are molecules. The substances of life are almost entirely molecules, e.g. proteins, the amino acids of which they are composed, the nucleic acids (DNA and RNA), sugars, carbohydrates, fats, and vitamins. The nutrient minerals are generally ionic compounds, thus they are not molecules, e.g. iron sulfate.

However, the majority of familiar solid substances on Earth are made partly or completely of crystals or ionic compounds, which are not made of molecules. These include all of the minerals that make up the substance of the Earth, sand, clay, pebbles, rocks, boulders, bedrock, the molten interior, and the core of the Earth. All of these contain many chemical bonds, but are not made of identifiable molecules.

No typical molecule can be defined for salts nor for covalent crystals, although these are often composed of repeating unit cells that extend either in a plane, e.g. graphene; or three-dimensionally e.g. diamond, quartz, sodium chloride. The theme of repeated unit-cellular-structure also holds for most metals which are condensed phases with metallic bonding. Thus solid metals are not made of molecules. In glasses, which are solids that exist in a vitreous disordered state, the atoms are held together by chemical bonds with no presence of any definable molecule, nor any of the regularity of repeating unit-cellular-structure that characterizes salts, covalent crystals, and metals.

Bonding

Molecules are generally held together by covalent bonding. Several non-metallic elements exist only as molecules in the environment either in compounds or as homonuclear molecules, not as free atoms: for example, hydrogen.

While some people say a metallic crystal can be considered a single giant molecule held together by metallic bonding, others point out that metals behave very differently than molecules.

Covalent

A covalent bond forming H₂ (right) where two hydrogen atoms share the two electrons

Main article: Covalent bonding

A covalent bond is a chemical bond that involves the sharing of electron pairs between atoms. These electron pairs are termed shared pairs or bonding pairs, and the stable balance of attractive and repulsive forces between atoms, when they share electrons, is termed covalent bonding.

Ionic

Main article: Ionic bonding

Sodium and fluorine undergoing a redox reaction to form sodium fluoride. Sodium loses its outer electron to give it a stable electron configuration, and this electron enters the fluorine atom exothermically.

Ionic bonding is a type of chemical bond that involves the electrostatic attraction between oppositely charged ions, and is the primary interaction occurring in ionic compounds. The ions are atoms that have lost one or more electrons (termed cations) and atoms that have gained one or more electrons (termed anions). This transfer of electrons is termed electrovalence in contrast to covalence. In the simplest case, the cation is a metal atom and the anion is a nonmetal atom, but these ions can be of a more complicated nature, e.g. molecular ions like NH₄⁺ or SO₄²⁻. At normal temperatures and pressures, ionic bonding mostly creates solids (or occasionally liquids) without separate identifiable molecules, but the vaporization/sublimation of such materials does produce separate molecules where electrons are still transferred fully enough for the bonds to be considered ionic rather than covalent.

Molecular size

Most molecules are far too small to be seen with the naked eye, although molecules of many polymers can reach macroscopic sizes, including biopolymers such as DNA. Molecules commonly used as building blocks for organic synthesis have a dimension of a few angstroms (Å) to several dozen Å, or around one billionth of a meter. Single molecules cannot usually be observed by light (as noted above), but small molecules and even the outlines of individual atoms may be traced in some circumstances by use of an atomic force microscope. Some of the largest molecules are macromolecules or supermolecules.

The smallest molecule is the diatomic hydrogen (H₂), with a bond length of 0.74 Å.

Effective molecular radius is the size a molecule displays in solution.  The table of permselectivity for different substances contains examples.

Molecular formulas

Chemical formula types

Main article: Chemical formula

The chemical formula for a molecule uses one line of chemical element symbols, numbers, and sometimes also other symbols, such as parentheses, dashes, brackets, and plus (+) and minus (−) signs. These are limited to one typographic line of symbols, which may include subscripts and superscripts.

A compound's empirical formula is a very simple type of chemical formula. It is the simplest integer ratio of the chemical elements that constitute it. For example, water is always composed of a 2:1 ratio of hydrogen to oxygen atoms, and ethanol (ethyl alcohol) is always composed of carbon, hydrogen, and oxygen in a 2:6:1 ratio. However, this does not determine the kind of molecule uniquely – dimethyl ether has the same ratios as ethanol, for instance. Molecules with the same atoms in different arrangements are called isomers. Also carbohydrates, for example, have the same ratio (carbon:hydrogen:oxygen= 1:2:1) (and thus the same empirical formula) but different total numbers of atoms in the molecule.

The molecular formula reflects the exact number of atoms that compose the molecule and so characterizes different molecules. However different isomers can have the same atomic composition while being different molecules.

The empirical formula is often the same as the molecular formula but not always. For example, the molecule acetylene has molecular formula C₂H₂, but the simplest integer ratio of elements is CH.

The molecular mass can be calculated from the chemical formula and is typically expressed in daltons, which are equal to 1/12 of the mass of a neutral carbon-12 (¹²C isotope) atom. For network solids, the term formula unit is used in stoichiometric calculations.

Structural formula

3D (left and center) and 2D (right) representations of the terpenoid molecule atisane

Main article: Structural formula

For molecules with a complicated 3-dimensional structure, especially involving atoms bonded to four different substituents, a simple molecular formula or even semi-structural chemical formula may not be enough to completely specify the molecule. In this case, a graphical type of formula called a structural formula may be needed. Structural formulas may in turn be represented with a one-dimensional chemical name, but such chemical nomenclature requires many words and terms which are not part of chemical formulas.

Molecular geometry

Main article: Molecular geometry

Structure and STM image of a "cyanostar" dendrimer molecule

Molecules have fixed equilibrium geometries—bond lengths and angles— about which they continuously oscillate through vibrational and rotational motions. A pure substance is composed of molecules with the same average geometrical structure. The chemical formula and the structure of a molecule are the two important factors that determine its properties, particularly its reactivity. Isomers share a chemical formula but normally have very different properties because of their different structures. Stereoisomers, a particular type of isomer, may have very similar physico-chemical properties and at the same time different biochemical activities.

Molecular spectroscopy

Main article: Spectroscopy

Hydrogen can be removed from individual H₂TPP molecules by applying excess voltage to the tip of a scanning tunneling microscope (STM, a); this removal alters the current–voltage (I–V) curves of TPP molecules, measured using the same STM tip, from diode like (red curve in b) to resistor like (green curve). Image (c) shows a row of TPP, H₂TPP and TPP molecules. While scanning image (d), excess voltage was applied to H₂TPP at the black dot, which instantly removed hydrogen, as shown in the bottom part of (d) and in the rescan image (e). Such manipulations can be used in single-molecule electronics.

Molecular spectroscopy deals with the response (spectrum) of molecules interacting with probing signals of known energy (or frequency, according to the Planck relation). Molecules have quantized energy levels that can be analyzed by detecting the molecule's energy exchange through absorbance or emission. Spectroscopy does not generally refer to diffraction studies where particles such as neutrons, electrons, or high energy X-rays interact with a regular arrangement of molecules (as in a crystal).

Microwave spectroscopy commonly measures changes in the rotation of molecules, and can be used to identify molecules in outer space. Infrared spectroscopy measures the vibration of molecules, including stretching, bending or twisting motions. It is commonly used to identify the kinds of bonds or functional groups in molecules. Changes in the arrangements of electrons yield absorption or emission lines in ultraviolet, visible or near infrared light, and result in colour. Nuclear resonance spectroscopy measures the environment of particular nuclei in the molecule, and can be used to characterise the numbers of atoms in different positions in a molecule.

Theoretical aspects

The study of molecules by molecular physics and theoretical chemistry is largely based on quantum mechanics and is essential for the understanding of the chemical bond. The simplest of molecules is the hydrogen molecule-ion, H₂⁺, and the simplest of all the chemical bonds is the one-electron bond. H₂⁺ is composed of two positively charged protons and one negatively charged electron, which means that the Schrödinger equation for the system can be solved more easily due to the lack of electron–electron repulsion. With the development of fast digital computers, approximate solutions for more complicated molecules became possible and are one of the main aspects of computational chemistry.

When trying to define rigorously whether an arrangement of atoms is sufficiently stable to be considered a molecule, IUPAC suggests that it "must correspond to a depression on the potential energy surface that is deep enough to confine at least one vibrational state". This definition does not depend on the nature of the interaction between the atoms, but only on the strength of the interaction. In fact, it includes weakly bound species that would not traditionally be considered molecules, such as the helium dimer, He₂, which has one vibrational bound state and is so loosely bound that it is only likely to be observed at very low temperatures.

Whether or not an arrangement of atoms is sufficiently stable to be considered a molecule is inherently an operational definition. Philosophically, therefore, a molecule is not a fundamental entity (in contrast, for instance, to an elementary particle); rather, the concept of a molecule is the chemist's way of making a useful statement about the strengths of atomic-scale interactions in the world that we observe.