Approximate linear dependence

This system is based on the observations (Pauling 1947) that a linear relation between single-bond radius and atomic number holds for bonds of constant hybrid character, and that for an element the single-bond radius is (at least approximately) linearly dependent on the d character of the dsp hybrid bond orbitals. 

The enthalpy of the R02 + RH reaction is determined by the strengths of disrupted and newly formed bonds AH= Z>R H—Droo—h- For the values of O—H BDEs in hydroperoxides, see the earlier discussion on page 41. The dissociation energies of the C—H bonds of hydrocarbons depend on their structure and vary in the range 300 - 440 kJ mol-1 (see Chapter 7). The approximate linear dependence (Polany-Semenov relationship) between activation energy E and enthalpy of reaction AH was observed with different E0 values for hydrogen atom abstraction from aliphatic (R1 ), olefinic (R2H), and alkylaromatic (R3H) hydrocarbons [119] ... 

The increase in bandgap energy AEG shows an approximately linear dependence on 1/iu... 

A pitch is made for a renewed, rigorous and systematic implementation of the GW method of Hedin and Lundquist for extended, periodic systems. Building on previous accurate Hartree-Fock calculations with Slater orbital basis set expansions, in which extensive use was made of Fourier transform methods, it is advocated to use a mixed Slater-orbital/plane-wave basis. Earher studies showed the amehoration of approximate linear dependence problems, while such a basis set also holds various physical and anal3ftical advantages. The basic formahsm and its realization with Fourier transform expressions is explained. Modem needs of materials by precise design, assisted by the enormous advances in computational capabilities, should make such a program viable, attractive and necessary. 

Approximate linear dependence of AO-based sets is always a numerical problem, especially in 3D extended systems. Slater functions are no exceptions. We studied and recommended the use of mixed Slater/plane-wave (AO-PW) basis sets [15]. It offers a good compromise of local accuracy (nuclear cusps can be correctly described), global flexibility (nodes in /ik) outside primitive unit cell can be correct) and reduced PW expansion lengths. It seems also beneficial for GW calculations that need low-lying excited bands (not available with AO bases), yet limited numbers of PWs. Computationally the AOs and PWs mix perfectly mixed AO-PW matrix elements are even easier to calculate than those involving AO-AO combinations. 

The approximate linear dependence (Polany-Semenov relationship) between activation energy E and enthalpy of reaction AH was observed with different E0 values for hydrogen atom abstraction from aliphatic (R1 ), olefinic (R2H), and alkylaromatic (R3H) hydrocarbons [119] ... 

However, if one accepts the hypothesis that only microbially promoted processes are occurring, it becomes necessary to explain the approximately linear dependence of the oxidation rate on 02 partial pressure as a conveniently exact effect on metabolic reaction rates. Ross and Bartlett (76) suggested that oxidation of Mn2+ is initiated by bacteria but that subsequent reaction is dominated by abiotic autocatalytic processes, once catalytically active colloids and particles are formed in sufficient numbers. 

J.O. Nordling, J.S. Faulkner, Approximate Linear Dependence, Scaling and Operator Convergence in the Harmonic Oscillator Problem, J. Mol. Spectros. 12 (1964) 171. 

Under these circumstances, the height of the transient will depend on the light intensity and the decay constant again only on the electron concentration at the surface. As an example of this behaviour, consider the photocurrent transients for depopulation of a set of surface states on p-GaP, which are shown in Fig. 80 [152]. The statistics are poor but show an approximate linear dependence of the reverse transient on light intensity, 0, and also... 

The shape of the polish rate vs. concentration curves in Figures 7.18 and 7.21 indicate that at low etchant concentrations, the polish rate is approximately linearly dependent on the etchant... 

In Pina s work, 2,3-butanedione is trapped inside hemicarcerand 63 to give the corresponding hemicarceplex, and the reactions with several electron donors are investigated in order to elucidate the effect of the encapsulation on the rates of the electron transfer to the triplet excited state of the guest. The determination of phosphorescence quenching lifetimes of free and incarcerated 2,3-butanedione reveals that quenching for uncomplexed 2,3-butanedione occurs at the diffusion-controlled limits for any donor quencher (e.g., amines), whereas in the case of the encapsulated substrate the rate constants are lower (ranging between 3.5 x 10 and 4 X 10 s ) and display an approximately linear dependence upon the oxidation potential of the external quencher. In particular, rate constants decrease with the... 

The choice of basis sets is a particularly important technical issue for this type of calculation. Specifically, the small, highly contracted basis sets such as used in Ref. [16] generally are insufficient to calculate the real parts of dielectric functions and, consequently, the energy loss functions. We started from Partridge s 16sllp set [17], contracted the seven tightest s-functions and the four tightest p-functions, removed the most diffuse s and the two most diffuse p-functions to avoid approximate linear dependencies, and added a full set of three d-functions with exponents equal to those of the... 

Inhibition of self-diffusion becomes apparent at timescales A short compared jD, where d is the characteristic pore diameter, because a fraction of molecules is always close to the walls, and their diffusion is hindered. This fraction depends on the surface-to-volume ratio (S/V) of the sample and the observation time A of the diffusion. This simple model leads to a fraction of 2DAf 5/V molecules being restricted in their motion. An analytical derivation [Mitl, Mit2, Mit6, Mit7] confirms an approximately linear dependence of the effective diffusion coefficient Deff(A) on... 

For a Newtonian fluid [ike. water the exponent ti is given by I independent of the rotation speed. This is indeed observed experimentally by an approximately linear dependence of the velocity on the radial position (Fig. 10.1.6, top right). For a shear-thinning fluid... 

Since b values for simple electron-transfer-controlled processes are approximately of the correct magnitude at 298 K, taking P — 0.5, it is clear that the temperature factor in the experimental behavior must be entering the electrochemical Arrhenius expression in more or less the conventional way, i.e., as a (kT) term. However, since b is often found to be independent of 7, it is clear that there must be another compensating temperature-dependent effect, namely an approximately linear dependence of a or j8 on temperature in the Tafel slope, b = RT/a T)F. The experimental results for a variety of reactions, summarized in Section III, show that this is a general effect. Reduction of C2H5NO2 is an exception while reduction of other nitro compounds takes place with substantial potential dependence of a ... 

This type of problem occurs for all fitting methods. The identification of linear combinations that are ill-determined is almost always important in a statement of what has been learned from an experiment. A more resistant fitting method is not necessarily more helpful in dealing with cases of exact or approximate linear dependence. 

It is important to realize that this approximately linear dependence of the phase angle on the offset is an intrinsic property of the experiment and has nothing to do with other mechanisms which can give rise to phase shifts such as instrumental delays in filters or inappropriate choice of the time origin in the Fourier transform process. In fact, since all these effects are approximately linear in x, they are usually lumped together and treated as one effect. Since the choice of the time origin is arbitrary and is entirely under the experimenter s control, all contributions to such frequency dependent phase shifts can be cancelled by an appropriate choice of the time origin and this is done quite often in FT of very wide lines. 

Influence of miscibility on thermal conductivity and diffusivity was studied [Agari, 1993, 1994]. In the blend of low molecular weight PS with coumarone-indene resin, which showed miscibility over all blend compositions, the thermal conductivity was approximately linearly dependent on composition [Agari, 1993]. Thermal conductivity, thermal diffusivity and heat capacity of PMMA/PC blends were studied with respect to temperature and blend composition. The specific... 

Despite the approximately linear dependence of both parameters on the reaction order, as can be seen from Figure 4-44, their range of values is rather restricted. In the sense of chemical process safety it is therefore absolutely justifiable to postulate a safety assessment criterion independent of the reaction order ... 

The relaxation time constant defined by Eqn. 47 and the steady-state distribution of charges q(V, oo)/0nuK (which is equal to a/( + )) represent two expressions for a and in terms of measurable experimental quantities. Therefore it is possible to evaluate the rate constants at any given potential V. The results showed that the rate constants are exponential functions of and that the energy differences governing the rate constants are. to a first approximation, linearly dependent on the membrane potential. 

CoUinearity of data refers to approximate linear dependence among variables. 

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