Term dependence

Excitations similar to giant resonances are observed for much deeper-shell excitation, and are more aptly named shape resonances, because the spatial overlap between the initial and final states is less complete than for giant resonances. An example is 3d — f in Ba, which is similar in character, but not identical, to Ad — / in the same element in particular, the smaller spatial overlap means that the exchange integral has a less dramatic influence on the spectrum, which is why, when studying solids (see chapter 11), deep shell excitation is preferred if the resonance is to be used as a probe of final state density in this way, one avoids most of the term dependence. 

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One problem with this treatment is that it neglects higher-order terms depending on higher moments of i/au that become undefined for slowly decaying interaction potentials (see Problem III-9). 

The sign of the last term depends on the parity of the system. Note that in the first and last term (in fact, determinants), the spin-orbit functions alternate, while in all others there are two pairs of adjacent atoms with the same spin functions. We denote the determinants in which the spin functions alternate as the alternant spin functions (ASF), as they turn out to be important reference terms. 

Each of these individual terms can be simplified if we recognise that terms dependent upon electrons other than those in the operator can be separated out. For example, the first term in the expansion, Equation (2.77), is ... 

The function/c is a smoothing function with the value 1 up to some distance Yy (typically chosen to include just the first neighbour shell) and then smoothly tapers to zero at the cutoff distance, by is the bond-order term, which incorporates an angular term dependent upon the bond angle 6yk- The Tersoff pofenfial is more broadly applicable than the Stillinger-Weber potential, but does contain more parameters. 

Note espeeially the eenter temi in Eq. (5-3). Physieal eoupling of the masses leads to a term dependent on both x and X2- Equation (5-3) as written is not separable into an equation only in x and an equation only in X2-... 

FiaaHy, an analysis of the energies of adsorption on many practical polar and nonpolar adsorbents has shown not only that the magnitude of the Op term depends directly upon the polarizabiUty a, but also that the sum of all of the nonspecific terms taken together, ie, + Lp, iacreases... 

For density functional based QM-MM methods the electronic energy terms depend explicitly on the electron density, p(r), of the atoms in the quantum region [46]. The... 

Essentially the same as Holland s formula, except buoyancy term depends on /A power. 

The OELs are expressed in terms of concentration, such as ppm (volume/ volume) or mg/m . For gases and vapors it is stated in terms independent of temperature and air pressure variables, in mL/m, or parts per million by volume in air, and in terms dependent on those variables in milligrams of substance per cubic meter of air at a temperature of 20 "C and a pressure of 01.3 kPa,... 

As an alternative to the above Eq. (9.167), it is easier to add to the higher pressure level a term depending on the difference between the considered levels, as shown in Table 9.15. 

Molecules that are composed of atoms having a maximum valency of 4 (as essentially all organic molecules) are with a few exceptions found to have rotational profiles showing at most three minima. The first three terms in the Fourier series eq. (2.9) are sufficient for qualitatively reproducing such profiles. Force fields which are aimed at large systems often limit the Fourier series to only one term, depending on the bond type (e.g. single bonds only have cos (3u ) and double bonds only cos (2u))). 

When we are dealing with electrolytes, two species of particles (positive and negative ions) are added to or removed from a solution at the same time. In the case of a uni-divalent solute, three particles arc added or removed at the same time. Since the cratic term depends only on the numbers of particles of various species that have been mixed, electrolytes that are completely dissociated in solution must be classified. according to their valence types—uni-univalent, di-divalent, and so on. Then in any very dilute solution the correct assertion to make is that the cratic term will have the same value for all electrolytes of the same valence type. 

In this connection it is important to consider the AF of the process depicted in Fig. 95. For each successive ion taken from the metal into a very dilute solution, work equal to Y must be done. Thus Y provides a unitary term and, at the same time, there will be in AF a cratic term depending on the concentration of the solution. At the electrode in the other half-cell, either positive ions are being removed from solution, when... 

In eq. 8, the rate of polymerization is shown as being half order in initiator (T). This is only true for initiators that decompose to two radicals both of which begin chains. The form of this term depends on the particular initiator and the initiation mechanism. The equation takes a slightly different form in the case of thermal initiation (S), redox initiation, diradical initiation, etc. Side reactions also cause a departure from ideal behavior. 

With reference to Eq. (26), an effect of temperature is expected since both 0M and the perturbation terms depend on temperature. In particular, the effect can be written as a temperature coefficient ... 

Equations (37) and (38), along with Eqs. (29) and (30), define the electrochemical oxidation process of a conducting polymer film controlled by conformational relaxation and diffusion processes in the polymeric structure. It must be remarked that if the initial potential is more anodic than Es, then the term depending on the cathodic overpotential vanishes and the oxidation process becomes only diffusion controlled. So the most usual oxidation processes studied in conducting polymers, which are controlled by diffusion of counter-ions in the polymer, can be considered as a particular case of a more general model of oxidation under conformational relaxation control. The addition of relaxation and diffusion components provides a complete description of the shapes of chronocoulograms and chronoamperograms in any experimental condition ... 

The energy of the bond for this function too is essentially resonance energy. Dickinson2 introduced an additional term, dependent on two additional parameters, in order to take polarization into account. He wrote for the (not yet normalized) variation function... 

It is essential that the solution be sufficiently dilute to behave ideally, a condition which is difficult to meet in practice. Ordinarily the dilutions required are beyond those at which the concentration gradient measurement by the refractive index method may be applied with accuracy. Corrections for nonideality are particularly difficult to introduce in a satisfactory manner owing to the fact that nonideality terms depend on the molecular weight distribution, and the molecular weight distribution (as well as the concentration) varies over the length of the cell. Largely as a consequence of this circumstance, the sedimentation equilibrium method has been far less successful in application to random-coil polymers than to the comparatively compact proteins, for which deviations from ideality are much less severe. 

In the second state the two terms depending on the l-and 2-HRDM compensate their errors to a large extent but nevertheless the hole -electron positive energy is too low and a global lowering of this state energy results. 

The state mixing term, the first in the r.h.s., usually dominates, at least in the presence of avoided crossings. Its determination reduces to a simple problem of interpolation of the Hu matrix elements, according to eq.(16). The second term corresponds, for large R, to the electron translation factor (see for instance [38]). This term depends on the choice of the reference frame that is, for baricentric frames, it depends on the isotopic masses. It contains the Gn matrix, which may be determined by numerical differentiation of the quasi-diabatic wavefunctions [16] this calculation is more demanding, especially in the case of many internal coordinates. It is therefore interesting to adopt the approximation ... 

However, such an equality is impossible, since by changing one of arguments, for example, R, the first term varies while the second one remains the same, and correspondingly the sum of these terms cannot be equal to zero for arbitrary values of R and 0. Therefore, we have to conclude that neither term depends on the coordinates and each is constant. This fact constitutes the key point of the method of separation of variables, allowing us to describe the function C/ as a product of two functions, each of them depending on one coordinate only. For convenience, let us represent this constant in the form +m, where m is called a constant of separation. Thus, instead of Laplace s equation we have two ordinary differential equations of second order ... 

We see that the expression for the current consists of two terms. The first term depends on time and coincides completely with Eq. (11.14) for transient diffusion to a flat electrode. The second term is time invariant. The first term is predominant initially, at short times t, where diffusion follows the same laws as for a flat electrode. During this period the diffusion-layer thickness is still small compared to radius a. At longer times t the first term decreases and the relative importance of the current given by the second term increases. At very long times t, the current tends not to zero as in the case of linear diffusion without stirring (when is large) but to a constant value. For the characteristic time required to attain this steady state (i.e., the time when the second term becomes equal to the first), we can write... 

Equation (1.12) describes an ORR rate dependence on potential that derives from two dijferent redox potentials, one affecting the exponential part of the expression and the other affecting the pre-exponential part. The term depending on — q2/h o reflects the lowering of the activation energy at an active metal site by an increase in cathode overpotential, whereas the term depending on ( pt(H20)/Pt-OHads describes the fraction of active metal sites, (1/Z + 1), at some value of . Equation (1.12) suggests... 

In a fluid model the correct calculation of the source terms of electron impact collisions (e.g. ionization) is important. These source terms depend on the EEDF. In the 2D model described here, the source terms as well as the electron transport coefficients are related to the average electron energy and the composition of the gas by first calculating the EEDF for a number of values of the electric field (by solving the Boltzmann equation in the two-term approximation) and constructing a lookup table. 

The first term on the left-hand side is a function only of the variable x and remains constant when y and z change but x does not. Similarly, the second term is a function only of y and does not change in value when x and z change but y does not. The third term depends only on z and keeps a constant value when only x and y change. However, the sum of these three terms is always equal to the constant energy E for all choices of x, y, z. Thus, each of the three independent terms must be equal to a constant... 

The first term is the starting phase that depends on the starting position Xo. The second term is linear in v, which is exactly what we are looking for. The third term is linear in the acceleration a, and subsequent terms depend on higher order derivatives of the coordinate. 

AB and CD denoted the center positions of the bonds formed by atoms A and B and C and D respectively. La and Ly represent the lone pair positions. As we will see, this formulation takes into account bonds and lone pairs hybridation, each one of the term depending of an overlap functional. N0CC(AB) and Nocc(CD) are the electron occupation numbers of the AB and BC bonds. Therefore, Nocc is equal to 2 for usual bonds and lone pairs. R, i .ci) is the distance between the barycenters of the AB and CD bonds. 

The first-order transfer and exit rate constants can be replaced by nonlinear terms dependent on the amount or concentration of drug in a particular compartment. For instance, saturable metabolism of drug in compartment 1 (the central compartment) would result in the Michaelis-Menten equation... 

FIGURE 2.1 Energy of the 0-0 vibrational transition in the principal electronic absorption spectrum of violaxanthin (l Ag-—>1 BU+), recorded in different organic solvents, versus the polarizability term, dependent on the refraction index of the solvent (n). The dashed line corresponds to the position of the absorption band for violaxanthin embedded into the liposomes formed with DMPC (Gruszecki and Sielewiesiuk, 1990) and the arrow corresponds to the polarizability term of the hydrophobic core of the membrane (n = 1.44). 

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