S

An s orbital is an atomic orbital with l = 0. A consequence of it having no angular momentum and therefore no angular nodes is that it has nonzero probability amplitude at the nucleus. An s orbital has a spherical boundary surface centred on the nucleus (Figure S.1). If it belongs to a shell with quantum number n, it has n – 1 radial nodes.

A σ orbital is a molecular orbital that has cylindrical symmetry and therefore zero orbital angular momentum around the internuclear axis of the atoms it joins (Figure S.2). Two electrons in a σ orbital with paired spins constitute a σ bond. In valence bond theory, a σ bond arises from the pairing of electrons in atomic orbitals with cylindrical symmetry about the internuclear axis.

The salting-out effect is the reduction of the solubility of a gas or other nonelectrolyte in water by the addition of a salt. The salting-in effect is the opposite.

Scanning tunnelling microscopy (STM) is the study of surfaces with atomic resolution by making use of the ability of electrons to tunnel across a narrow gap. In one variety of the technique, a platinum, rhodium, or tungsten needle with a fine point is attached to a piezoelectric ceramic rod, which expands and contracts in response to an applied potential difference. Electrons tunnel across the gap between the needle tip and the surface and the resulting current is monitored. In the adaptation atomic force microscopy (AFM), the force between needle point and atoms on the surface is monitored and interpreted in terms of the variation in height of the surface. An advantage of AFM over STM is that the surface need not be electrically conducting.

This function is plotted in Figure S.4 and S.5. If the scattering factor is large, then the atom contributes strongly to the scattering of X-rays that gives rise to a diffraction pattern.

The Schoenflies system of notation for point groups is as follows:

• C_n. Groups that include only one n-fold axis.

• C_nv. Groups that include one n-fold axis and n vertical mirror planes, σ_v.

• C_nh. Groups that include one n-fold axis and a horizontal mirror plane, σ_h.

• S_n. Groups that include an n-fold axis of improper rotation. If n is odd, the groups are the same as C_nh.

• D_n. Groups that include n 2-fold axes perpendicular to an n-fold principal axis.

• D_nd. Groups that in addition to the elements characteristic of the group D_n also include n dihedral mirror planes, σ_d, bisecting the 2-fold axes.

• D_nh. Groups that include the elements of D_n together with a horizontal mirror plane, σ_h.

• T, O, I. These are the cubic groups. T is the group of the tetrahedron, O the group of the octahedron, and I the group of the icosahedron.

• T_h. The group of a tetrahedron in which h denotes the presence of a centre of inversion arising from the presence of three horizontal mirror planes.

• T_d. The group of a tetrahedron in which d denotes the presence of three S₄ axes of improper rotation arising from the presence of three diagonal mirror planes.

• O_h. An octahedral group in which the h denotes the presence of a centre of inversion arising from the presence of horizontal and vertical mirror planes.

• I_h. An icosahedral group in which the h denotes the presence of a centre of inversion arising from the presence of horizontal and vertical mirror planes.

See also the Hermann–Mauguin system. For the classification of molecules, use the charts in Figure S.6 and S.7.

The empirical Schultze–Hardy rule states that hydrophobic colloids are flocculated most efficiently by ions of opposite charge type and high charge number.

The Second law of thermodynamics has a variety of formulations. Two observation-based statements (Figure S.8) are

1. Kelvin statement: no process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.

2. Clausius statement: no process is possible in which the sole result is the transfer of energy from a cooler to a hotter body.

These formulations inspired the introduction of the entropy as a state function and the equivalent statement of the Second law in the form:

• 3

  A spontaneous process in an isolated system is accompanied by an increase in entropy.

A spontaneous process is one that can occur without work needing to be done in the surroundings to bring it about. This statement leaves open the possibility that in one region of the isolated system there might be a decrease in entropy, provided that elsewhere in the system the entropy increases in compensation.

The molecular interpretation of the thermodynamic statement is based on the identification of entropy with molecular disorder: an increase in disorder can stem from the chaotic dispersal of matter and the chaotic dispersal of energy. Thus, the Second law is an expression of the view that chaotic dispersal is natural and that the emergence of order is unnatural.

A selection rule is a specification of the allowed spectroscopic transitions of an atom or molecule. A gross selection rule is a statement about the general properties that a molecule must possess in order to undergo a certain class of transitions. Thus, the gross selection rule for rotational transitions is that a molecule must be polar, and for rotational Raman transitions a molecule must have an anisotropic polarizability. A specific selection rule is a statement about the changes that may occur in terms of quantum numbers or other labels of the states of an atom or molecule. The following specific selection rules are often encountered for transitions that occur by an electric dipole mechanism:

1. Atomic transitions: Δl = ±1; ΔL = 0, ±1; ΔS = 0; ΔJ = 0, ±1

2. Vibrational transitions (infrared and Raman): $Δv=±1$

3. Rotational transitions (microwave, branch structure of vibrational transition): ΔJ = 0, ±1

4. Rotational transitions (Raman): Δl = 0, ±2 for linear rotors; Δl = 0, ±1, ±2; ΔK = 0 for symmetric rotors

5. Electronic transitions: g → u and u → g (the Laporte selection rule).

In most cases these rules can be traced to the spin (S = 1) of the photon and the conservation of angular momentum. They are established by considering the relevant transition dipole moment and noting the criteria for it to be nonzero either by using the properties of the relevant mathematical functions or in terms of group theory. In the latter case, the transition dipole moment is zero unless the integrand spans the fully symmetric irreducible representation of the molecular point group.

A self-consistent field method (an SCF method) is an iterative computational procedure that continues until certain calculated properties remain unchanged in a cycle (to within a predetermined criterion). It is commonly applied to the numerical solution of the Schrödinger equation for many-electron atoms and molecules, when it is commonly referred to as a Hartree–Fock method (HF method, or HF-SCF method). The method typically begins by proposing the wavefunctions to be used in the calculation (such as various atomic orbitals with variable parameters or linear combinations with variable coefficients). The potential energy of all the electrons is calculated using the electron distribution characteristic of that basis and a numerical solution of the Schrödinger equation is found for one of the electrons. The procedure is continued for all the electrons in the molecule and a refined set of orbitals is obtained. The refined set of orbitals will in general differ from the original orbitals, and the procedure is continued until no significant change occurs in the course of one iteration. At that stage, the orbitals are self-consistent and are accepted as the best solution for the system.

In a semi-empirical method, certain integrals that occur in a numerical solution of the Schrödinger equation for a many-electron system are either ignored (according to various criteria) or approximated by empirical data. For instance, various integrals may be adjusted to optimize the calculation of enthalpies of formation.

In the separation of variables procedure, a partial differential equation in several variables is separated into a series of (possibly linked) ordinary differential equations in each variable. For example, a two-dimensional particle-in-a-box system for which the wavefunctions depend are functions of x and y, can be expressed as ψ(x,y) = ψ(x)ψ(y) and the Schrödinger equations for ψ(x) and ψ(y) solved separately. Similarly, the system of a proton and an electron can be separated into equations for the translational motion of the centre of mass of the joint system and another for the motion of a particle with a reduced mass around that centre of mass. Moreover, the latter motion, which is described by the wavefunction ψ(r,θ,ϕ) can be separated further into radial and angular factors: ψ(r,θ,ϕ) = ψ(r)ψ(θ)ψ(ϕ), each of which satisfies an ordinary differential equation; in this case the three solutions are linked and consequently the quantum numbers that arise have ranges of values that depend on each other (for instance, l = 0, 1,…, n − 1). The viability of the separation of variables procedure depends on the form of the potential energy of the system.

The shell of an atom consists of all the n ² orbitals with the same principal quantum number, n. A closed shell is a shell in which all the orbitals are doubly occupied by electrons. This definition is sometimes relaxed when d orbitals are available and incompletely occupied.

Shielding is the modification of a field. An important example is the role of inner-shell electrons in shielding the electric field due to the nucleus of an atom. Thus, an electron (the ‘test electron’) in a many-electron atom is repelled by all the other electrons, and its net attraction to the nucleus is thereby reduced. Note that the other electrons do not actually ‘shield’ the field of the nucleus: they simply provide an opposing contribution to the total experienced by the test electron. If the shielding electrons are distributed spherically, and the test electron is at a radius r, then it experiences a net repulsion only from the electrons within a sphere of radius r (Figure S.10). The net attraction to the nucleus is expressed by replacing the actual nuclear charge, Ze, by the effective nuclear charge, Z_effe. See penetration.

A singlet state is an atomic or molecular state with total electron spin quantum numbers S = 0, M_S = 0; a triplet state is one with S = 1, M_S = 0, ±1. The spin wavefunctions for two electrons in these states are as follows:

Figure S.11 shows the vector model representations of these combinations.

Singlet states of two electrons (and therefore with ‘paired’ spins) are commonly represented as ↓↑ and triplet states (with two ‘unpaired’ electrons) as ↑↑. An atom with a singlet term has one level (J = L) and one with a triplet term, provided L > 0, has three levels (J = L + 1, L, L − 1).

A triplet term of a given configuration is typically lower in energy than the singlet term of the same configuration. See Hund’s rules. Electric dipole transitions between singlets and triplets are forbidden. Transitions between them are enabled by spin–orbit coupling. See intersystem crossing and phosphorescence.

A smectic phase is a liquid-crystalline mesophase in which molecules are ordered in two dimensions but not in a third (Figure S.12).

Space quantization is the restriction of the component of an angular momentum to discrete values m_jħ, with m_j = j, j − 1,…, −j, and therefore in the vector model with the vector representing the angular momentum at discrete angles to the z-axis.

The sphalerite structure (or zinc-blende structure) takes its name from the structure of ZnS and is one of the reasonably common crystal types, especially when covalent bonding is important (Figure S.14).

The spherical harmonics occur widely in dealing with systems with spherical symmetry. Some are listed in Table S.1.

Spin is the intrinsic angular momentum of a particle. Each fundamental particle has a characteristic, unchangeable spin, with spin quantum number s. The magnitude of the spin is {s(s + 1)}^1/2ħ and its z-components are m_sħ, with m_s = s, s – 1,…, −s. The spin of nuclei is specified similarly, with quantum numbers I and m_I. Fundamental particles and nuclei with integer spin quantum number are bosons; those with half-integer spin quantum number are fermions. Particles with nonzero spin and nonzero mass have a magnetic moment; particles with spin quantum number greater than $1 2$ may also have an electric quadrupole moment.

Electrons and protons have s =  $1 2$ and therefore m_s = ± $1 2$ ⁠. Those with m_s = + $1 2$ are commonly denoted either ↑ or α; those with m_s = − $1 2$ are likewise denoted wither ↓ or β. See photon, for its spin properties. The total spin angular momentum of a many-electron atom is denoted S, M_S, with S obtained by using a Clebsch–Gordan series. See singlet and triplet states.

A spin echo is a pulse of magnetization that arises when individual nuclear magnetic moments come back into phase after having dispersed after an earlier pulse. To produce a spin echo, a pulse of magnetic field is applied in the x-direction and the magnetization vector is rotated through 90° into the xy-plane (Figure S.16). The individual spins fan out on account of their slightly different Larmor frequencies. After an interval, a 180° pulse is applied in the y-direction. Now the spins begin to regroup and reassemble into a magnetization vector that give a pulse, the echo, in a detector. The amplitude of the echo decays exponentially with a time constant equal to the transverse relaxation time, T₂.

The spin–orbit coupling constant increases strongly with atomic number, so it is most important in heavy atoms. Spin–orbit coupling is also responsible for the process of intersystem crossing and the breakdown of the selection rule ΔS = 0 in electronic transitions.

The coupling constant for nuclei A and B that are separated by N bonds is denoted ^N J_AB. Its magnitude depends on a variety of factors as well as the value of N. These factors include the angles that the intervening bonds make with each other (see Karplus equation) and the hybridization of the intervening atoms. The mechanism of coupling is a combination of the non-Coulombic interaction between electrons and the bond and spin correlation: see polarization mechanism.

A spontaneous process is a change that can occur without an external agency having to do work to bring it about. Spontaneity in thermodynamics is a tendency which might not be realized in practice due to kinetic considerations. A process is spontaneous if the total entropy change that accompanies it is positive: ΔS_total > 0. At constant temperature and pressure, a spontaneous process is accompanied by a decrease in Gibbs energy (of the system): ΔG < 0. See also Einstein coefficients.

Standard ambient temperature and pressure (SATP) is 298.15 K and 1 bar. Distinguish it from standard temperature and pressure (STP) which is 0 °C and 1 atm.

The Stark effect is the removal of the degeneracy of the rotational states of polar molecules by the application of an electric field. It can be used to determine the electric dipole moment of a molecule. Stark modulation is the modulation of the rotational energy levels of polar molecules by the application of an oscillating electric field and is used in the detection of rotational transitions.

The Stark–Einstein law states that one photon is absorbed by each molecule responsible for the primary photochemical process. The law fails when the intensity of radiation is very high.

Statistical thermodynamics is a theory of thermodynamic properties in terms of the average behaviour of large assemblies of molecules. The basic concept used to evaluate averages is the ensemble, a collection of hypothetical replications of the system, and the evaluation of the partition function, a function that contains all the thermodynamic information about the system. In certain idealized cases, the partition function can be calculated from computed or spectroscopic properties of the molecules that constitute the system. See partition function and ensemble.

The steady-state approximation of chemical kinetics is that the concentrations of all intermediates in a proposed reaction mechanism are low and constant throughout the main part of the reaction. By setting d[I]/dt = 0, where I is an intermediate, the differential equation for its net production is turned into an algebraic equation which can be used to formulate the overall rate law corresponding to the proposed mechanism.

In simple cases the sticking probability decreases with the extent of surface remaining uncovered and s ∝ 1−θ, where θ is the fractional coverage of the surface (Figure S.18).

The stoichiometric point (formerly the closely related equivalence point) of a titration is the point at which the stoichiometric quantity of titrant has been added to the analyte solution. It should be distinguished from the end point, the point at which an indicator changes colour.

See also diffusion.

In the stopped-flow technique, reagents are impelled into a mixing chamber as a piston in the chamber is withdrawn suddenly. The composition of the chamber is then monitored. The technique is used to study fast reactions.

The intensity of the {hkl} reflection is proportional to |F_hkl|². For the interpretation of diffraction intensities in terms of electron density in the unit cell, see Fourier synthesis and the phase problem.

The subshell of an atomic shell of principal quantum number n consists of all 2l + 1 orbitals of the same orbital quantum number l in that shell. The following notation is used:

A supercritical fluid is a fluid at a temperature higher than its critical temperature but with a density comparable to that of a normal liquid. A supercritical fluid has properties that are intermediate between those of a liquid and a gas.

A superfluid is a liquid phase of matter that flows without viscosity. The only known examples are helium-4 and helium-3. The transition to the superfluid phase occurs below the λ-transition temperature (Figure S.19).

The most common application of superposition in chemistry is in the construction of molecular orbitals as linear combinations of atomic orbitals (LCAO) and, in valence bond theory, of resonance hybrids.

Surface reconstruction is the modification of the structure of a solid at and close to its surface, perhaps due to the attachment of an adsorbate.

The svedberg (Sv) is a non-SI unit encountered in sedimentation studies of macromolecules: 1 Sv = 10⁻¹³ s. It is commonly used to report the value of the sedimentation constant, S.

A symmetry operation is an action that leaves an object apparently unchanged. Examples are rotations, reflections, and inversions. Translation is a symmetry operation for an infinite crystal. A symmetry element is the point, line, or plane with respect to which the operation is carried out. Examples are:

A symmetry number, σ, is a factor included in the high-temperature form of a partition function to correct for the limitation on the occupation of rotational states imposed by the Pauli principle. It is equal to the order of the rotational subgroup of the molecular point group (in effect, allowing for that number of indistinguishable orientations); for a regular tetrahedron (Figure S.20), σ = 12. The high-temperature rotational partition function of a homonuclear diatomic molecule is kT/hcB̃ if the Pauli principle is ignored but kT/σhcB̃ with σ = 2 when it is taken into account.

A symmetry species (in general, Γ) is the label of the irreducible representation of a molecular point group. The letters A and B are used to label one-dimensional irreducible representations: A is used if the character of the principal rotation is +1 and B is used if it is −1. The label E denotes a two-dimensional irreducible representation and T denotes a three-dimensional irreducible representation. In icosahedral point groups, G and H are used for four- and five-dimensional irreducible representations.

A symmetry-adapted linear combination (SALC) is a weighted sum of atomic orbitals that is a basis for an irreducible representation of the molecular point group (Figure S.21). Molecular orbitals are constructed from the overlap of an SALC with orbitals of the same symmetry species on other atoms. For instance, the SALC ψ_H1sA + ψ_H1sB in H₂O has A₁ symmetry in the C_2v molecular point group and is used to form molecular orbitals with ψ_O2s and ψ_O2pz, both of which have A₁ symmetry.

Synchrotron radiation is the high-energy electromagnetic radiation generated by electrons moving at high speed in a circle (and therefore ceaselessly accelerating). The radiation extends into the ultraviolet and X-ray regions and has a higher intensity than can be achieved by conventional sources.

In thermodynamics, a system is the object of interest, such as a gas or a reaction mixture. The surroundings are the rest of the universe, and often emulated by a large water bath. The classification of systems is based on the properties of the boundary that separates them from their surroundings (Figure S.22):

A boundary is diathermic if it allows energy to pass as heat but adiabatic if it does not.

In X-ray crystallography, a systematic absence is the absence of reflections that follow a simple rule relating to the values of the indices hkl (Figure S.23). For example, the diffraction pattern from a cubic I (bcc) lattice can be constructed from that from a cubic P lattice by striking out all reflections with odd values of h + k + l. For a cubic F (fcc) lattice, the only reflections present are those with h, k, and l all even or all odd.