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Pure Appl. Chem., Vol. 71, No. 10, pp. 1919-1981, 1999

Glossary of terms used in theoretical organic chemistry

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Half-electron (HE) model - An approach to treating open-shell molecular systems by a closed-shell formalism which utilizes the similarity between the SCF equations (see Hartee-Fock method) and those for a fictitious closed-shell system in which the odd electron is replaced by two half- electrons. The model is applied for calculating energies of radicals by use of semiempirical quantum mechanical methods. DEWAR, HASHNALL, and VENIER (1968).

Hamiltonian - A differential operator of the total energy. For any normalized wavefunction y, the energy is the expectation value of the Hamiltonian operator.

E = y*Hy dt

Hard and Soft Acid and Base (HSAB) principle - A structure-reactivity concept formulated in terms of acid-base interaction. According to this principle soft acids react faster and form stronger bonds with soft bases, whereas hard acids react faster and form stronger bonds with hard bases, everything else being assumed aproximately equal. Hardness is associated with low polarizability, high electronegativity and energy low-lying HOMO (bases) or energy high-lying LUMO (acids). Softness is related to high polarizability, low electronegativity, energy high-lying HOMO (bases) and energy low lying LUMO (acids). The soft-soft preference is characteristic mostly of the reactions controlled by the orbital interaction, and the hard- hard preference relates to reactions in which electrostatic factors prevail. PEARSON (1963); PEARSON (1990); CHATTARAJ and SCHLEYER (1994).
See also Absolute hardness.

Harmonic approximation - The approximation of the full nuclear potential of a molecular system in its equilibrium geometry and in the vicinity of the respective minimum on the potential energy surface by the function

V = (1/2) 2V/qiqj

where qi are mass-weighted cartesian displacements of nuclei relative to their equilibrium positions. The approximation allows one to describe vibrational motion in terms of independent vibrational modes normal modes) each of which is governed by a simple one-dimensional harmonic potential.

Hartree-Fock limit - The lowest energy that would be obtained via the SCF procedure (see Hartree-Fock method) if there were no restrictions on the sorts of function that molecular orbitals could adopt.

Hartree-Fock (Self-Consistent Field, SCF) method - Method for determination of the spatial orbitals yi of the many-electron determinantal wavefunction (see Slater determinant) based on reducing coupled nonlinear differential equations for the optimum forms of the molecular orbitals by use of the variational method (see variational principle). The Hartree-Fock hamiltonian operator is defined in terms of these orbitals through the operators of coulomb and exchange repulsion. The general procedure for solving the Hartree-Fock equations is to make the orbitals self-consistent with the potential field they generate. It is achieved through an iterative trial-and-error computational process, for which reason the entire procedure is called the self-consistent field method. In the case of open-shell systems one should distinguish between the spin-restricted Hartree-Fock (RHF) method and spin-unrestricted Hartree-Fock (UHF) method. In the former approach a single set of molecular orbitals is preset, some being doubly occupied and some being singly occupied with an electron of spin. In the UHF approach different spatial orbitals are assigned to electrons with a and b spins and the orbitals yi doubly occupied in the RHF method are replaced by two distinct orbitals yi(a) and yi(b). HEHRE, RADOM, SCHLEYER, and POPLE (1985).

Heat of atomization - The negative value of the heat of formation. It is equal to the total bond energy of a compound.

Heat of formation - The heat absorbed or released upon formation of a compound from the elements in their standard states (i.e. the most stable form of the elements under standard temperature and pressure, 25oC and 1 atm).

Hessian matrix (synonymous with force constant matrix) - The matrix of second derivatives of energy. At any local minimum, all eigenvalues of the Hessian matrix are positive. At a first-order saddle point, only one eigenvalue is negative and all others are positive.

Heterolytic bond cleavage - A bond-dissociation reaction of the type R-X R-+ X+ resulting in the splitting of the electron pair of the ruptured bond in a way which leaves the pair with one of the fragments formed upon dissociation.

High-spin state - When the separation between highest occupied and lowest unoccupied molecular orbitals is not large, two alternative electronic states may be considered for the two uppermost electrons. The state with the largest number of unpaired electrons is called the high spin state. In the cases where the one-electron energy needed to promote an electron to the LUMO is smaller than the Coulomb and exchange repulsion energies required to pair up two electrons in the HOMO, the high-spin state is the ground state.

For compounds with more than two unpaired electrons, several spin states may exist, but in octahedral (or tetrahedral) transition metal complexes (e.g., organometallic compounds), the high degeneracy of the MOs results in only one low-spin and one high-spin state.

Whether a given compound is in its low-spin or high-spin state can be usually determined from magnetic susceptibility or ESR experiments, although in some cases both states have similar energies and are thermally populated, giving rise to a peculiar temperature dependence of the compound’s magnetic behavior.
See also Hund’s rule.

Highest Occupied Molecular Orbital (HOMO) - see Frontier orbitals.

Homoaromaticity - The aromatic stabilization of cyclic conjugated systems with 4n + 2 p-electrons is partly preserved when one or more methylene (or other saturated) groups are inserted into the ring. The molecules and ions thus formed are regarded as homoaromatic structures. Examples are cyclobutenyl 1 and cyclooctatrienyl (homotropylium) 2 cations. Trishomocyclopropenyl cation 3 represents a trishomoaromatic structure.

Homodesmotic reaction - A subclass of isodesmic reactions in which reactants and products contain equal numbers of carbon atoms in corresponding states of hybridization; moreover, there is matching of the carbon-hydrogen bonds in terms of the number of hydrogen atoms joined to the individual carbon atoms. To achieve all this matching, one should significantly extend the number and types of reference molecules. In the aliphatic series of hydrocarbons these are propane, isobutane and neopentanes as well as propene and isobutene; for aromatics - buta-1,3-diene, 3-methylidenepenta-1,4-diene (2-vinylbutadiene) and 3,4-bismethylidenehexa-1,5-diene (2,3-divinylbutadiene). Thus to assess strain energy of cyclopropane and aromatic stabilization of benzene the following homodesmotic reactions are to be respectively analyzed .

c-(CH2)3 + 3 CH3-CH3 3 CH3CH2CH3

DH0exp = -26.5 kcal/mol (110.9 kJ/mol)

C6H6 + 3 CH2=CH2 3 CH2=CH-CH=CH2 (trans)

DH0calc (MP2/6-31G**) = 23.9 kcal/mol (100.0 kJ/mol)

Due to closer matching of the hybridization states of the atoms of reactants and products as compared to isodesmic reactions, the homodesmotic reactions give more accurate estimates of the intrinsic strain and the cyclic delocalization. The definition may be extended to molecules with heteroatoms. GEORGE, TRACHTMAN, BOCK, and BRETT (1975).

Homolytic bond cleavage - The bond-dissociation reaction of the type R - X R. + X. resulting in the splitting of the electron pair of the ruptured bond in a way that leaves by one electron on each of the fragments formed upon dissociation.

Hot ground state photoreaction - A photochemical diabatic reaction which occurs as the jumps from the potential energy surface of the excited state of a molecule to the ground state surface with excess of thermal energy, large enough to overcome the potential energy barrier between the reactant and the product.

Hot state reaction - A reaction proceeding from an ensemble of molecular entities possessing a higher average vibrational, rotational or translation energy than they would at thermal equilibrium with the surrounding medium. IUPAC PHOTOCHEMICAL GLOSSARY (1988).
See also Hot ground state photoreaction.

Hückel molecular orbital (HMO) theory - The simplest molecular orbital theory of p-conjugated molecular systems. It uses the following approximations: p-electron approximation; LCAO representation of the p-molecular orbitals; neglect of electron-electron and nuclear-nuclear repulsions (in fact, the assumption that these cancel). The diagonal elements of the effective Hamiltonian, coulombic integrals, and the off-diagonal elements, resonance integrals, (accounted for only directly bonded atoms) are chosen as empirical parameters, all overlap integrals being neglected. HEILBRONNER and BOCK (1976); HÜCKEL (1931).

Hückel resonance energy - see Resonance energy, various type of.

Hückel rule - (also known as the 4n + 2 rule). The electron-counting rule which states that those monocyclic p-conjugated molecules and ions are stable which contain (4n + 2) p-electrons, where n = 0, 1, 2.. .While being originally formulated on the basis of the simple Hückel molecular orbital theory, the rule is not limited by the conditions of this theory and proved to be valid also in the framework of the self-consistent-field approximation with electron correlation taken into account. From the structural point of view, the rule has certain limitations when applied to polycyclic systems and heterocycles with highly electronegative or highly electropositive heteroatoms.

Hund's rule - A rule for building up the electronic configuration of atoms and molecules: where a species possesses degenerate orbitals one electron is placed into each of these before two electrons are placed in any one of the degenerate set, and the electrons in the singly occupied orbitals have parallel spins. Thus, the ground state electron configuration of the nitrogen atom is represented by N(1s22s22px2py2pz). Apart from this rule (called sometimes Hund's multiplicity rule), two other rules of determining ground electronic states from the magnitudes of the total orbital angular momentum are referred to as Hund rules. IUPAC PHOTOCHEMICAL GLOSSARY (1988); BORDEN, IWAMURA, and BERSON (1994).

Hybrid orbital - An atomic orbital derived through hybridization of atomic orbitals with different angular momentum quantum numbers located at a given atom.

Hybrid Quantum Mechanics/Molecular Mechanics (QM/MM) methods - A procedure for the teratment of large molecular systems in such a way that part of the system is treated explicitly by quantum mechanical (QM) techniques, while the rest of the system is approximated by a classical or molecular mechanics (MM) force field. GAO (1998), TOMASI and POMELLI (1998).

Hybridization - The vectorial type mixing of atomic orbitals with different angular momentum quantum numbers on the same atomic center. Mathematically, the hybridization corresponds to an orthogonal transformation of a given basis set (e.g. c1=2s, c2=2px, c3=2py, c4=2pz) to an equivalent basis set {ll}. In simple cases, it may involve the mixing of two (2s, 2px), three (2s, 2px, 2py) and four (2s, 2px, 2py, 2pz) AOs, the respective types of hybrid orbitals being referred to as sp, sp2 and sp3 AOs. For inequivalent ligands or unequal bond lengths more general spl hybrid orbitals have the form:

hl(q) = N (s + l1/2pq)

where pq is a normalized p orbital pointing in the direction q, N is a normalization constant, and l is generally noninteger.

Hydrogen bond - A particular type of multicenter (three center - four electron bond) X-H ...Y in which the central hydrogen atom covalently linked to an electronegative atom X (C,N,O,S...) forms an additional weaker bond with atom Y (N,O,S...) in the direction of its lone electron pair orbital. The energy of hydrogen bonds, which is usually in the range of 3 - 15 kcal/mol (12 - 65 kJ/mol), results from the electrostatic interaction and also from the orbital interaction of the antibonding s*(XH) MO of the molecule acting as the hydrogen donor and the nonbonding lone electron pair MO nY of the hydrogen acceptor molecule. KOLLMAN and ALLEN (1972); PIMENTEL and McCLELLAN (1960); PERRIN and NIELSON (1997).

Hydrophobic effect - The tendency of nonpolar species to aggregate in water solution so as to decrease the hydrocarbon - water interfacial area. The effect is a principal factor determining the structures of proteins and nucleic acids, the binding of substrates to enzymes, and the binding of antigens to antibodies. BRESLOW (1991).

Hyperconjugation - The effect of donation of electron density from the symmetry adapted s-orbitals of saturated groups to vacant p*-orbitals of conjugated fragments of a molecular entity; this builds p-character into bonds that nominally possess only s-character.
See also Negative hyperconjugation.

Hypercoordination - A property of main-group atoms in molecular entities to acquire coordination numbers greater than four (which would comply with the Lewis octet rule). Hypercoordination may be associated with hypervalency, but usually is referred to peculiar atomic centers in the electron-deficient species with multicenter s-bonding, in which the bonding power of a pair of electrons is spread over more than two atoms. An example of a hypercoordinated atom is the five-coordinate carbon atom in the cation methanium, where three C-H bonds may be regarded as normal two center - two electron bonds and the bonding in the remaining CH2 fragment is governed by the three-center, two-electron bond.

A particular case of a hypercoordinated atom is the hydrogen atom included into a hydrogen bond. OLAH, PRAKASH, WILLIAMS, FIELD, and WADE (1987)

Hyperfine coupling - The interaction between the spin magnetic moment of an unpaired electron and the nuclear spin magnetic moments resulting in the splitting of the a (spin up) and b (spin down) energy levels in an external magnetic field and, thus, in the multiplet pattern of the ESR spectra of radical-like species and transition metal compounds.

Two main contributions to the hyperfine coupling are usually considered, Fermi contact and dipolar interactions. The contact interaction is isotropic and related to the unpaired spin density at the nucleus, |Yo|2. The dipolar interaction is anisotropic, and related to r-3, where r is the distance between the atom holding the unpaired electron and the nucleus with non-zero spin.

Hyperpolarizability (of nth order) - The energy of a molecule in an external electrostatic field can be expanded as

E = Eo - miFi - (1/2)aijFiFj - (1/6)bijkFiFjFk - (1/24)gijklFiFjFkFl - ...

where Eo is the unperturbed energy, Fi is the component of the field in the i direction, mi is the permanent dipole moment, aij is the polarizability tensor, and bijk and gijkl are the first and second hyperpolarizability tensors, respectively. b is a third order symmetric tensor that measures the second order response of the molecular electric dipole moment to the action of an external electric field and thus often referred to as dipole hyperpolarizability.BÖTTCHER (1973), HURST, DUPUIS, and CLEMENTI (1988).

Hypervalency - The ability of an atom in a molecular entity to expand its valence shell beyond the limits of the Lewis octet rule. Hypervalent compounds are common for the second and subsequent row elements in groups 15-18 of the periodic table. A description of the hypervalent bonding implies a transfer of the electrons from the central (hypervalent) atom to the nonbonding molecular orbitals which it forms with (usually more electronegative) ligands. A typical example of the hypervalent bond is a linear three-center, four-electron bond, e.g. that of Fap-P-Fap fragment of PF5. MUSHER (1969); REED and SCHLEYER (1990).

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> Abstract
> General remarks
> Arrangement

> Fundamental physical constants used in the glossary
> References
> Appendix. Glossary of acronyms of terms used in theoretical organic chemistry

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