National Academies Press: OpenBook
« Previous: Afterword
Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×

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).

Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×

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.

Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×

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).

Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×

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.

Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×

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

Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×

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.

Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×

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.

Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×

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

Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×

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.

Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×
Page 123
Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×
Page 124
Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×
Page 125
Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×
Page 126
Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×
Page 127
Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×
Page 128
Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×
Page 129
Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×
Page 130
Suggested Citation:"Glossary." National Research Council. 1995. Mathematical Challenges from Theoretical/Computational Chemistry. Washington, DC: The National Academies Press. doi: 10.17226/4886.
×
Page 131
Mathematical Challenges from Theoretical/Computational Chemistry Get This Book
×
Buy Paperback | $34.00 Buy Ebook | $27.99
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

Computational methods are rapidly becoming major tools of theoretical, pharmaceutical, materials, and biological chemists. Accordingly, the mathematical models and numerical analysis that underlie these methods have an increasingly important and direct role to play in the progress of many areas of chemistry. This book explores the research interface between computational chemistry and the mathematical sciences. In language that is aimed at non-specialists, it documents some prominent examples of past successful cross-fertilizations between the fields and explores the mathematical research opportunities in a broad cross-section of chemical research frontiers. It also discusses cultural differences between the two fields and makes recommendations for overcoming those differences and generally promoting this interdisciplinary work.

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    Switch between the Original Pages, where you can read the report as it appeared in print, and Text Pages for the web version, where you can highlight and search the text.

    « Back Next »
  6. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  7. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  8. ×

    View our suggested citation for this chapter.

    « Back Next »
  9. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!