Theory Formalism

Mathematical theory of labeled colored graphs is exclusively used to formalize the structure and substructure search problem. There is almost a one-to-one correspondence between the terms used in graph theory and the ones used in chemical structure theory. Formally a graph G can be given by Eq. (1), where V is the set of graph vertices and H the set of edges. 

Decision Theory Formalism in the Behavioral Sciences.46—A formal method for examining decisions is developed through the use of a utility matrix [ tj] similar to the payoff matrix of game theory. In this matrix the rows correspond to the various possible acts of a decision-maker (e.g., to invest money in enterprises AltA2, -, Am) and the columns to various states, i.e., circumstances (e.g., possible levels of development of each enterprise) under which the acts are performed. The element y gives the utility (return or value) for using act At when state sf prevails. 

To apply the perturbation-theory formalism we can first separate H into diagonal (unperturbed) and off-diagonal (perturbation) matrices,... 

The Perturbation Theory Formalism. The quantum mechanical model employed here is the conventional time-dependent pertur-... 

The theory formally can be extrapolated to the limit of monomers and thus can be applied to describe nonpolymeric glass-forming liquids. 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

The z axis is the internuclear axis, and the z axes on atoms a and b point toward each other. Without using the full group-theory formalism, we can write down by inspection the following symmetry-adapted orbitals, which... 

A transition state theory formalism (Equation 9.10) was used to calculate the lsO KIE from the reaction coordinate frequency (rrc), defined as the mode that converts the 0—0 vibration into a translation, and the pseudo-equilibrium constant for forming the transition state (KTs). In the expression for the latter (Equation 9.11), the one vibration that becomes the reaction coordinate has been removed. 

Recently, the response theory formalism was applied for the calculation of singlet-triplet spectra in polyenes and aromatic hydrocarbons [83,133,134,132]. In this section we shall discuss mainly the vertical Tn <— So transitions probability in polyenes. The minimum level of accuracy includes CAS calculations of the ground state correlating all x electrons followed by linear and quadratic multi-configuration response calculations. Because the ethene molecule is the simplest representative of unsaturated hydrocarbons, different types of active spaces and AO basis sets were tested more thoroughly for this molecule in order to study the convergence of the S-T transition probability with respect to the level of accuracy. 

In crystalline organic solids such as anthracene, in which the mobility is on the order of 1 cm2/Vs, the transport mechanism may still be explained by a band theory formalism. But for most organic solids (especially if disordered), the mobility values are far below the lower limit value and a hopping transport mechanism seems to be more appropriate. Organic polymers are classic examples of hopping transport materials. In poly-iV-... 

Although group theory formally is a branch of abstract mathematics its development has been linked intimately with concepts and operations related to the symmetry of physical objects and structures. 

Mahapatra, U. S. Datta, B. Mukheijee, D. A size-consistent state-specific multireference coupled cluster theory Formal developments and molecular applications, J. Chem. Phys. 1999,110, 6171-6188. 

I was inspired too by Linus Pauling (1901-94), another polymath with humanistic concerns. His Nature of the Chemical Bond (1939) brought a new perspective to theories of molecular structure, and refuted the implication of a popular examination question of the time, Is inorganic chemistry a largely closed and finished subject Pauling s resonance theory, formally based on the quantum-mechanical valence-bond (VB) method for... 

The problem was reconsidered in 1966 by Edwards, who introduced a field theory formalism in order to treat this problem, and who used self-consistent methods to calculate the screening effect. 

Linear response theory expression Alternatively, the spin-spin coupling constant can be expressed using the linear response theory formalism. Let us write the electronic energy of the system perturbed by the nuclear magnetic dipole moments M/f in the form E = E(Mjf, A), where A are the variational parameters of the wave function. A may represent orbital rotation parameters for the SCF wave function, or orbital rotation parameters and coefficients of the configuration interaction expansion for the MCSCF... 

Thu.s the sixteen symmetry transformations of the square fall into ten classes. Since operations within the same class will turn out to be eciuivalent in the group theory formalism we may drop the superscripts and write the fifteen operations as E, 2Ci, C = C4 ), 2C2, 2Cj, rotation axes, and the subscript d means diagonal. It is worth noting that the identity is always in a class by itself, since for all A, ... 

Tran, F. Weber. J. Wesolowski. T.A. Theoretical study of the benzene dimer by the density-functional-theory formalism based on electron-density partitioning. Helv. Chim. Acta 2001, 84 (6), 1489-1503. 

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