Theory, defined

The theory of atoms in molecules defines chemical properties such as bonds between atoms and atomic charges on the basis of the topology of the electron density p, characterized in terms of p itself, its gradient Vp, and the Laplacian of the electron density V p. The theory defines an atom as the region of space enclosed by a zero-/lMx surface the surface such that Vp n=0, indicating that there is no component of the gradient of the electron density perpendicular to the surface (n is a normal vector). The nucleus within the atom is a local maximum of the electron density. 

The impact theory defines uniquely the spectral transformation in the limit of weak collisions. Expanding in a series over J — J the integrand of Eq. (6.4) one can obtain at T = /coq%j 1... 

At the microscopic level, the Arrhenius theory defines acids as substances which, when dissolved in water, yield the hydronium ion (H30+) or H+(aq). Bases are defined as substances which, when dissolved in water, yield the hydroxide ion (OH). Acids and bases may be strong (as in strong electrolytes), dissociating completely in water, or weak (as in weak electrolytes), partially dissociating in water. (We will see the more useful Brpnsted-Lowry definitions of acids and bases in Chapter 15.) Strong acids include ... 

The limitations of the Arrhenius theory of acids and bases are overcome by a more general theory, called the Bronsted-Lowry theory. This theory was proposed independently, in 1923, by Johannes Br0nsted, a Danish chemist, and Thomas Lowry, an English chemist. It recognizes an acid-base reaction as a chemical equilibrium, having both a forward reaction and a reverse reaction that involve the transfer of a proton. The Bronsted-Lowry theory defines acids and bases as follows ... 

This theory assumes the definition of (i) the frame of discernment Q consisting of the exhaustive and exclusive h3rpothesis and (ii) the reference set 2 of all the disjunctions of the elements of fi. The Evidence theory defines the basic belief assignment (bba) function as an elementary mass function m 2 —> [0,1] verif3dng for all elements A of 2 ... 

The example candidate for the topological field theory defining the l.h.s. of the non-Abelian Stokes theorem could be given by the (classical) action... 

A nice feature of (Euclidean) two-dimensional Yang-Mills gauge theory defined by the action... 

According to the Arrhenius theory, acids (HA) are substances that dissociate in water to produce H + (aq). Bases (MOH) are substances that dissociate to yield OH aq). The more general Bransted-Lowry theory defines an acid as a proton donor, a base as a proton acceptor, and an acid-base reaction as a proton-transfer reaction. Examples of Bronsted-Lowry acids are HC1, NH4+, and HSO4- examples of Bronsted-Lowry bases are OH-, F-, and NH3. 

If the gauge connection is Abelian, then the term eA A A vanishes by the antisymmetry of the wedge product. This means that = dAv /. This is an example of an Abelian gauge theory, defined according to that vanishing of commutators between gauge potentials. 

According to Bardeen, first order perturbation theory defines the current between two electrodes (the sample and tip) as... 

Using the projection-operator formalism of Feshbach [ 115,116], an implicit variational solution for the coefficients cIJiS in can be incorporated into an equivalent partitioned equation for the channel orbital functions. This is a multichannel variant of the logic used to derive the correlation potential operator vc in orbital-functional theory. Define a projection operator Q such that... 

Noether s theorem will be proved here for a classical relativistic theory defined by a generic field , which may have spinor or tensor indices. The Lagrangian density (, 9/x) is assumed to be Lorentz invariant and to depend only on scalar forms defined by spinor or tensor fields. It is assumed that coordinate displacements are described by Jacobi s theorem S(d4x) = d4x 9/xgeneral variation of the action integral, evaluated over a closed space-time region 2, is... 

The second-order exchange energy in the SRS theory, defined as Ef = (2) (2) eXCn sRS- poi, separates naturally into two contributions exchange-induction and... 

For compounds containing X = N, O, or S in place of C, some trivial extensions of Hiickel theory define the on-site and off-site integrals c/.x and (3XY in terms of the corresponding integrals a,c and (3Cc for carbon (called a and (3 further above), through numerical "fudge" (i.e., adjustable) parameters hx and kXy (both typically between 0.5 and 2)... 

Bronsted-Lowry theory of acids and bases the theory defining an acid as a substance from which a hydrogen ion can be removed and a base as a substance that can remove a hydrogen ion from an acid (10.1)... 

The saddle point between isomer B and isomer A is represented by yj = and the saddle point between isomer B and isomer A is represented by y = q g As in the case of double-well potentials, the MRRKM theory defines isomerization separatrix surfaces for the three isomer states B, A and B by... 

The MRRKM theory defines the separatrix with respect to the reaction coordinate y by the relation... 

The renormalisation constants Z, Z2, Z3 and dm have to be understood as functions of the finite physical charge e and mass m of the electrons which can be constructed order by order in the perturbation series. It is important to notice that these constants are uniquely determined by vacuum QED without any external potential. They do not depend on the specific external potential present. If one bases the perturbation expansion on the Lagrangian (A.49) all Greens and n-point functions of the theory (defined in terms of the physical fields tj/ and T ) are finite. 

An estimate of the magnitude of the salt effect can be obtained from the Debye-Hiickel theory. Defining... 

Chain Branching. A branched molecule is more compact, having greater density and lower [iq] than its linear counterpart. The Zimm-Stock-mayer theory defines the g factor for a polymer as the ratio of [iq] for the branched polymer to [iq] of the linear polymer, at the same molecular weight, with s being the shape factor (—0.75). 

A second nonlinear fluorescence quenching data treatment method developed by Ventry and Ryan may also be used to extract conditional stability constants, and ligand concentrations from titrations of FA with Cu (23). The model designed is a modification of the original Stem-Volmer theory defined by equation 7 which accounts for either static or dynamic quenching of fluorescent species. 

It may surprise you to find that, despite the reason that acids and bases are found in so many of the materials around us, there are several different theories defining what acids and bases are. The real difference between the theories has to do with how broad or limited our definitions of acids and bases should be. For example, should only compounds that ionize to release hydroxide ions (OH ) be considered bases, or should other compounds that neutralize acids without releasing hydroxide ions be included in our definition Examine the following theories. 

There are two issues that have to be addressed before one can use Eqs. (25) or (28) in practical calculations. First of all, the exact MMCC corrections SgA) and < qCCSD, Eqs. (25) and (28), respectively, have the form of long many-body expansions involving all n-tuply excited configurations with n == i/ia + I, ., /V, where N is the number of correlated electrons in a system. Thus, in order to propose the computationally inexpensive MMCC methods, we have to truncate the many-body expansions for SgA> or excitation level This leads to the so-called MMCC( i, mB) schemes [11-15,24,33,34,39,48,120,121], The CR-CCSD(T) and CR-CCSD(TQ) methods [11-14,24,33,34], reviewed and tested in this work, are the MMCC( u, mB) schemes with mA = 2 and mB = 3 (the CR-CCSD(T) case) or 4 (the CR-CCSD(TQ) case). Second of all, the wave function % that enters the exact Eqs. (25) or (28) is a full Cl ground state, which we usually do not know (if we knew the exact ko> state, we would not have to perform any calculations ). Thus, in order to propose the computationally tractable approaches based on the MMCC theory defined by Eqs. (25) and (28), we must approximate fi o) in some way as well. The CR-CCSD(T) and CR-CCSD(TQ) methods employ the low-order MBPT-like expressions to define fi o) [11-14,24,33,34],... 

Educational measurement theory defines large-scale assessment as a technical activity. Consequently, each aspect of an assessment situation is treated as a variable more or less within the control of the assessment designer or administrator. Composition theory, however, treats writing as a complex of activities and influences, most of which cannot be cleanly isolated for analysis or evaluation. (Lynne, 4)... 

Because diffusion in the liquid phase is much slower than in the gas phase, the liquid film presents most of the resistance to mass transfer. Gas-liquid film theory defines the mass transfer rate (MTR) in terms of the liquid film, as follows... 

Hydrodynamic theory defines the behavior of shock waves at interfaces between elements of exqilosive trains in terms of curves that relate the shock velocity to the particle velocity, called Hugoniot curves. Assuming that mass and momentum are conserved across the shock front, one can write... 

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