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Mathematical Challenges from Theoretical/Computational Chemistry GLOSSARY The following entries represent words, phrases, abbreviations, and acronyms that appear in this report. In many cases, definitions have been given in the text itself, but not with each occurrence. This collection is not offered as an exhaustive mathematics-chemistry bilingual dictionary, but rather as a representative guide to the kinds of technical terms often used by the respective communities and an aid to the reader. It contains examples of multiple meanings that can degrade communication between well-intentioned, but differently trained, professionals. Even without reading the main text, a perusal of this Glossary may convey a sense of the linguistic barriers that occasionally inhibit effective collaboration. A Ab initio method— Usually, quantum chemical computation procedures that explicitly include all electrons and utilize their full Hamiltonian operator; occasionally used to describe pseudopotential methods or other approaches with minimal semi-empirical input. ACS— American Chemical Society. Adiabatic— Sufficiently slow variation of the externally controllable parameters of a quantum mechanical system so that the quantum eigenstate occupation probabilities remain unchanged over time. Adjacency matrix— For an n-atom molecule, an n × n matrix with unity as the i, jth entry if atoms i and j share a covalent bond; otherwise it is zero. Advection— Similar in spirit to ''convection,'' but referring to forced flow under more general circumstances. Affinity— Capacity for binding between two molecular units that leads to a more or less stable chemical combination. Alkaloids— Nitrogen-containing organic compounds of natural origin that act as bases in solution; examples are nicotine, quinine, morphine, and lysergic acid. Alkane— Saturated hydrocarbon (i.e., containing only single bonds between pairs of carbon atoms and carbon-hydrogen pairs). In the strict usage this refers to molecules in which the skeleton of carbon-carbon single bonds has the form of a Cayley tree and so exhibits the chemical formula CnH2n+2; occasionally it refers to saturated hydrocarbons with cyclic skeletal closures (cycloalkanes) that have lower relative hydrogen content. Amino acid— Organic acid containing an amino (nitrogen-hydrogen) group and a carboxyl group, usually in neighboring locations; the modular building blocks of polypeptides and proteins. AMS— American Mathematical Society. Anionic site— Location in a protein, gel, catalytic solid, etc., that bears an excess negative charge of one or more fundamental units (the electron charge).
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Mathematical Challenges from Theoretical/Computational Chemistry APS— American Physical Society. Assay— Experimental procedure to determine the magnitude of a property of a substance. Usually used in a biological or biochemical context. B Band reduction— Algorithmic process applied to matrices to result in all nonzero terms being close to the diagonal. Bar-coding— Familiar supermarket technology applied to efficient automated screening of compounds in the pharmaceutical industry: physical labels for packages with patterns of black and white parallel bars that encode text and numeric information. Bechgaard salts— Organoselenium ionic compounds exhibiting low-temperature superconductivity. Biological assay— An experimental procedure to test molecules for their ability to elicit a biochemical or pharmacological response. Born-Oppenheimer approximation, potential surface— Approximate separation of variables in atomic and molecular quantum mechanics justified by the large ratio of nuclear to electron masses; also the potential energy surface resulting therefrom, as a function of nuclear positions. Breit-Pauli relativistic corrections— An approximate correction for relativistic effects, used as a perturbation to solutions of the intrinsically nonrelativistic Schrödinger wave equation. Buckyball— Nickname for fullerene (see below). Usually applied to the prototypical C60 molecule. C Catalytic antibodies— Proteins central to the immune response that have been artificially endowed with catalytic properties. Catenane— Chemical compound consisting of two or more ring molecules that are unconnected by chemical bonds but linked by topological entanglement. Cayley tree— Linear graph (vertices linked by bonds) that is connected but contains no closed polygons. Chirality— Geometric or topological property of a molecule or other structured object distinguishing it from its mirror image (i.e., "handedness"). Clique detection— Computational procedure to identify the common subgraph(s) of a pair of graphs, especially the maximal common subgraph. Cluster analysis, clustering— The grouping of chemical compounds, usually on the basis of many distinct attributes, so that members of any one grouping resemble one another, but are clearly different from those of all other groupings. COLUMBUS— Software package for quantum chemical calculations. Combinatorial synthesis (combinatorial chemistry)— A procedure for the simultaneous synthesis of a collection of related molecules, usually accomplished on a polymeric support.
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Mathematical Challenges from Theoretical/Computational Chemistry Condensed phase— A state of matter in which the constituent particles (atoms or molecules) are densely packed in space and in constant interaction with neighbors; usually refers to liquids, liquid crystals, glasses, crystalline solids, or quasicrystals. Conformation, conformation space— Geometric specification of the spatial arrangement of a molecule, possibly flexible; the space of all possible arrangements. Cut-and-projection method— Technique for generating quasi-crystal structures by isolating slabs from higher-dimensional periodic structures and projecting them into three-dimensional space. D DARC— Commercial software for managing chemical information databases. Daylight— Chemical database software. Density matrix— Hermitian matrix composed as the direct product of a wavefunction and its complex conjugate. "Reduced density matrices" are obtained by integrating over some of the variables in the full density matrix. Distance geometry— Reconstruction of the full three-dimensional shape of a protein or other biopolymer from a given set of intramolecular distances, which in some applications are known only between limits. DNA plasmid— An autonomously replicating circular DNA molecule. Docking— The fitting and binding of small molecules (ligands) at the active sites of biomacromolecules (e.g., enzymes and DNA). Dopamine receptor— The protein on the cell surface that recognizes the neurotransmitter dopamine and turns on cell responses as a consequence of this recognition. E Electrophoresis— An experimental technique to fractionate (separate) charged molecules that relies on differential migration velocities through an obstructive medium (gel), under the influence of an applied electric field. Enantiomer— Mirror image counterpart of a chiral molecule. Ergodicity— Capacity of a dynamical system spontaneously to sample all of its phase space. Euler's gamma function— Analytic extension of the factorial function from the positive integers to the complex plane. EVB— An approach to an approximate solution of the electronic Schrödinger equation, using an empirical valence bond wave function. F FCC crystal— Face-centered cubic crystal, one of the close-packed arrangements of rigid spheres in three-dimensional space. Fermi resonance— Coupling between normal modes of vibration due to anharmonic potential energy contributions, when a resonance condition is satisfied (rational ratio of normal mode frequencies).
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Mathematical Challenges from Theoretical/Computational Chemistry Fermion— Particle with half-integer spin, and therefore exhibiting wavefunction antisymmetry and Fermi statistics. FFT— Fast Fourier transform. Fluxional molecule— Flexible molecule, capable of substantially changing its conformation (shape) under the prevailing temperature and surrounding-medium conditions. Fullerene— Stable molecule consisting entirely of carbon atoms arranged at the vertices of a convex polyhedron. G GAMESS— Software package for quantum chemical calculations. GAUSSIAN— Software package for quantum chemical calculations that emphasizes use of Gaussian basis functions. H Hamiltonian— In classical mechanics, the sum of kinetic and potential energy functions (i.e., the total energy); in quantum mechanics, the corresponding linear Hermitian operator. Harmonic analysis— A mathematical theory that aims to decompose complex phenomena into the superposition of simpler phenomena (such as wave forms). Hartree-Fock approximation— Replacement of an exact wavefunction with an antisymmetrized product of single-particle orbitals (i.e., a Slater determinant). Hermitian operator— Linear operator whose matrix elements exhibit the property that reflection across the diagonal is equivalent to complex conjugation. Hessian matrix— Symmetric matrix of second partial derivatives, often used in optimization routines for many types of objective functions. A frequently encountered application involves searching for extremes on potential energy surfaces. Heteropolymer— A polymer formed from nonequivalent monomer units. High-performance computers— A phrase adopted to refer to emerging computers of a variety of designs (parallel, vector, and others) that reflects the convergence of many of these design concepts. Homotopic paths— Paths that share the same end points and can be continuously deformed into one another while those end points are kept fixed. HONDO— Software package for quantum chemical calculations. Hückel theory— An approximate theory of molecular electronic structure that uses a minimal basis of atomic orbitals, and simplified Coulomb and exchange integrals. Human genome (project)— The 3 × 109 base pairs that constitute the entire human genetic heritage; the Human Genome Project aims for a complete mapping of the human genome, identifying and sequencing regions of chromosomes that code for individual proteins.
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Mathematical Challenges from Theoretical/Computational Chemistry Hydrogen bond— Weak noncovalent interaction (mainly electrostatic) bond between a significantly positive hydrogen atom and an electronegative atom (usually nitrogen, oxygen, or fluorine); consequently, an important structural element in water, proteins, and other biopolymers, as well as in the recognition (specific binding) of small molecules by biopolymers. I Interval analysis— A mathematical technique that involves computation of strict bounds to bracket the global minimum of a function. Irreducible representation— Group representation (e.g., as matrices) that does not admit of decomposition into a direct product of elements of lower dimension. J JPBM— Joint Policy Board for Mathematics. K Koopmans's "theorem"— Asserts that orbital eigenvalues of the Hartree-Fock approximation provide good estimates of the vertical (fixed nucleus) ionization energies of atoms or molecules. L Langevin equation, dynamics— An approach for analysis and simulation of molecular motion in which the molecular forces in Newton's classical equations are modified by additional terms (a frictional kernel and stochastic forces) that attempt to describe in a simple and computationally feasible way the effects of surroundings (e.g., solvent medium) on motion of the molecule of interest. Lie group— A differentiable manifold that has a group structure on its elements, with the property that the group operations (multiplication and inversion) are continuous. A standard and useful example is the set of nonsingular real n × n matrices. Other examples are the group of rigid motions in Euclidean space and the group of equivalence classes of transformations that agree on some neighborhood of the identity. The last was the original object of study by Sophus Lie around 1890. Ligand— A molecule or other chemical grouping attached to a larger molecular structure. In the case of a small molecule that binds to a biomacromolecule, the latter is frequently referred to as the "receptor." M MAA— Mathematical Association of America. MDI— Commercial software for chemical information databases (Molecular Design Incorporated, San Leandro, California). MELD— Software package for quantum chemical calculations. Molecular dynamics (MD)— Computer simulation technique for many-body systems that relies on numerical solution of classical equations of motion for atoms or molecules and evaluates thermodynamic, kinetic, and structural properties as time averages. Molecular mechanics— The field encompassing molecular statics, or the construction of appropriate force fields for representing molecular systems and the associated potential energy minimizations. Molecular modeling— The art of representing and analyzing molecular systems by using mathematical concepts and techniques, numerical computation, graphics, etc. Monte Carlo methods— Numerical evaluation procedures relying on probability and the repeated use
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Mathematical Challenges from Theoretical/Computational Chemistry of random number generators. N NMR— Nuclear magnetic resonance. Nodal cells— Connected regions of the configuration space for a set of electrons, over which their wavefunction maintains constancy of sign. NOE— Nuclear Overhauser effect. Normal modes— Independent harmonic motions of a mechanical system that possesses a quadratic potential energy function; alternatively, such motions of small amplitude on a more complicated surface that can be locally approximated as quadratic. N-representability— Attribute of candidates for reduced density matrices (in quantum mechanics) or for molecular distribution functions (in classical statistical mechanics) that they correspond to the contraction of some legitimate N-body wavefunction or distribution function. NSF— National Science Foundation. Nucleotide— Hydrolysis product (monomeric unit) of nucleic acids. n-valent vertices— Vertex of a linear graph at which n bonds (edges) are attached. P Padé approximant— Rational function approximation to an analytic function, usually determined by matching power series coefficients. Parallel computer— A computer with multiple processors working in parallel on portions of one computation. Path integral— An appropriately weighted sum over a family of line integrals connecting fixed end points. Applications include descriptions of Brownian motion and quantum transition probabilities. Pauli exclusion principle— The requirement for electrons (or other fermions) that the wavefunction for a collection of such particles is antisymmetric under interchange of the position and spin coordinates of any pair of identical particles. In the case of noninteracting or uncorrelated particles, it leads to the requirement that at most, one particle can be in each single-particle state. Pharmacophore— The chemical identity and geometrical arrangement of key substituents in a molecule that confer biochemical or pharmacological effects. Phase— In quantum mechanics, the phase angle of a complex wavefunction or order parameter; in statistical mechanics, the state of aggregation of matter (e.g., crystal, liquid, vapor, quasicrystal, liquid crystal); in dynamics, the position-momentum specification. Phase space— The joint space of configurational coordinates and their conjugate momenta for a classical dynamical system. Pisot numbers— Root of a polynomial with integer coefficients, all of whose other roots have absolute values less than unity.
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Mathematical Challenges from Theoretical/Computational Chemistry Polyacetylene— Linear polymer bearing delocalized electrons, formed from acetylene (C2H2). Polyvinylcarbazole— Photoconductive polymer that has been used as an active material in xerographic applications. Potential energy surface— Surface (hypersurface) in (3N + 1)-dimensional space expressing the potential energy of interaction within an N-particle system, as a function of the 3N position coordinates; see Born-Oppenheimer approximation. Prokaryote— A single-cell organism that has no distinct nucleus. Protein folding— Spontaneous development, under ambient conditions, of the natural three-dimensional shape of a protein molecule that facilitates its biological function. Pseudorandom number generator— Numerical algorithm to approximate an ideal generator of statistically independent, uniformly distributed, random numbers. Q QSAR— Quantitative structure-activity relationship. A statistical method that relates biological potency to physical properties of molecules. Quantum chaos— Dynamical evolution of a quantum mechanical system that displays irregular motions and extreme sensitivity to initial conditions. Quasicrystals— Solids exhibiting long-range orientational coherence of local atomic coordination geometry but no crystallographic periodicity or unit cell. Quasi-ergodic— Property of a dynamical system that it eventually visits a well-defined neighborhood of any kinematically accessible point in its phase space. R Racemates— Mixtures containing equal amounts of enantiomeric (mirror-image stereoisomer) molecules. Raman spectroscopy— Technique of molecular spectroscopy enabled by the coupling of vibrations to electronic polarizability. Random number generator— A random number generator is a computer procedure that scrambles the bits of a current number or set of numbers in such a way that the result appears to be randomly distributed among the set of possible numbers and to be independent of the previously generated numbers. The lagged-Fibonacci random number generator, for instance, uses a shift operation and a binary operation on n-tuples from a finite set (usually the integers mod m). Reaction coordinate— Configurational coordinate measuring distance across a transition state saddle point on the potential surface of a chemically reactive system. Residue— A chemical unit, such as an amino acid, of a large molecule. REU— The Research Experience for Undergraduates program of the National Science Foundation. Ribosome— Cellular structure that is the usual site of protein synthesis in living organisms.
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Mathematical Challenges from Theoretical/Computational Chemistry Riemann zeta function— Sum of the inverse sth powers of the positive integers, viewed as an analytic function of s. S Salem numbers— Root of a polynomial with integer coefficients, all of whose other roots have absolute values equal to or less than unity. Scalable parallel algorithms or parallel computers— Those that can continue to perform efficiently as both the problem size and the number of processors increase. Schur's lemma— Relation between irreducible representations of a finite group. Semiclassical approximation— Treatment of dynamics in which quantum effects are regarded as a weak perturbation on classical mechanics; often implemented formally as the limit at which Planck's constant approaches zero. Sequence space— The set of possible ordered monomer sequences for a biopolymer. For proteins this is the "primary structure," the amino acid sequence along the molecular backbone; for DNA it is the sequence gene-encoding bases (adenine, cytosine, guanine, thymine). Sequential computers (uniprocessors)— Computers that operate on individual pieces of data in sequence. Contrast with vector processing and parallel computers. SIAM— Society for Industrial and Applied Mathematics. Slater determinant— Determinant whose elements are n distinct orbitals for n distinct electrons. Solitons— Stable pulse or particle-like solutions of some nonlinear wave equations. Spin glass— A large collection of spins whose interactions frustrate simple ordering and thus create many deep potential energy minima that are nearly degenerate. Statistical mechanics— The study of the collective behavior of large numbers of interacting particles. Properties of interest include those describing time-dependent, irreversible process. The basic principles of this discipline were laid down in the nineteenth century by Ludwig Boltzmann, James Clerk Maxwell, and Josiah Willard Gibbs. Stereoisomers— Molecules that differ only by mirror inversion of bonds at chiral centers (usually carbon atoms). Stochastic dynamics— A framework for expressing the dynamics of a molecular system that includes stochastic elements (e.g., random forces that mimic effects of the environment); see Molecular dynamics and Langevin equation. Supercoiling— The interwinding of double helical DNA upon itself; also called supertwisting. Symplectic integrator— Class of numerical algorithms for integrating classical many-body equations of motion, that exactly preserve phase space volume and other classical invariants of motion. T Terpenes— Unsaturated (double-bond-containing) hydrocarbons with composition C10H16; typically
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Mathematical Challenges from Theoretical/Computational Chemistry aromatic oils of natural origin. Topological (geometric) phase— Phase factor appearing in a nuclear wavefunction that arises from, and varies continuously around, a conical intersection of two electronic potential surfaces. Transition state— Saddle point on a potential energy surface separating chemical reactants from products. Tridiagonalization— Transformation of a symmetric matrix so that the only nonzero elements are those along the principal diagonal and its two immediately flanking diagonals. U Ultrametric, ultrametricity— Distance (metric) structure applied to a set of objects arrayed on a Cayley tree; for a given pair of objects it is the minimum number of steps along branches of the tree necessary to effect connection. Unrestricted Hartree-Fock approximation— Use of an approximate wavefunction consisting of a sum of Slater determinants, wherein distinct spatial orbitals are employed for distinct spin directions. Upwinding schemes— Techniques for solving advection problems that take into account force direction to produce more accurate results. V Vectorization— Adaptation of a computer program to take advantage of vector processing capacity. Vector processing— A mode of computing (using "vector computers") that relies heavily on the concurrent processing of the elements of (possibly large) linear arrays. V-representability— Special form of N-Representability (see above) guaranteeing that the density matrix or molecular distribution function of interest corresponds to an N-body system with additive pair interactions (specifically Coulomb interactions in the atomic or molecular quantum cases). W Wavelets— Basis functions selected to represent efficiently certain types of frequently encountered pattern elements, such as discontinuities. Y Yield— Fraction (normally expressed as a percentage) of the chemical reactants that proceed to the desired product substance. Z Zeolite— Solid silicates with open crystal structures permitting action as ion exchange media, catalysts, or molecular sieves.
Representative terms from entire chapter: