The third-order term in equation

For the iodo complex, reaction (37) is the only pathway available, k3 being zero, but the complex fra s-(PtL2MeCl) is also acidolysed by a mechanism corresponding to the third-order term in equation (35). Mechanism (38) was suggested48 for this reaction path. 

The third-order term in equation (40), p. 157, was suggested to correspond to a mechanism in which the intermediate (XI) is now attacked by a second bromine molecule, viz. 

For confidence in the accuracy of /cHa.b> it is important that the curvature in Fig. 11.2A corresponds to an increase of two-fold or greater in the slope. Too often, the third-order term in Equation 11.2 (/cha b) is inaccurate because it derives from a very small fraction of the total reaction flux, and needs confirmation. Data for the enolisation of acetone are reproduced in Fig. 11.3, which shows that substantial differences between observed rate constants and calculated values not including the third-order term are seen only at high concentration of the acid-base pair [2]. Concentrations of acid-base pairs are between 0 and 2 M, and there are no major specific salt or solvent effects. This is important as the high... 

Values of 2bs and k3bs are given in Table 20, and it may be seen that, whereas only the second-order term in equation (40) is of importance for substitution of tetramethyltin and tetraethyltin, yet the third-order term is of as equal importance as the second-order term for the substitution of tetraisopropyltin. The value of k2bs obtained in the latter case (0.35) is naturally rather different to the original value of 1.61 l.mole-1.sec-1 given in Table 19. However, the values of k2bs given in Table 19 for the /i-tetraalkyls are quite valid. 

In infrared spectroscopy of diatomic molecules, the vibrational motion is generally limited to the first two vibrational states of a diatomic molecule whereby the displacement of the bond is near the minimum (i.e. small values of s). As a result, it is reasonable as a first approximation to confine the anharmonicity to the third order term of Equation 6-37. The potential can be represented by a third order polynomial such that the first term is the same as in the harmonic oscillator model problem. 

The matrix elements n q n) are evaluated in Section 4.4. According to equations (4.45c) and (4.50e), the first and third terms on the right-hand side of (10.46) vanish. The matrix elements in the second and fourth terms are given by equations (4.48b) and (4.51c), respectively. Thus, the first-order correction in equation (10.46) is... 

Studies of the amine concentration rate dependence show that the reactions are strictly third-order in amine for DMSO <2%. For DMSO constants >10% the reactions show the classical behaviour usually found in base-catalysed SjvAr180. The specific solvent effects observed for small additions of the HBD co-solvent are consistent with the formation of the mixed aggregate, and a linear correlation was found between kA and [DMSO], shown by equation 36, which expresses that the third-order term is more affected by the small additions of DMSO than the fourth-order term. Equation 36 is valid for [DMSO] <2% (0.282 M). 

The third-order interaction energy, equation 4, contains contributions, which can be expressed in terms of the second-order election current density vector JBE induced by the fields,... 

In enzymes, the active site may possess acid and base groups intimately associated with the conjugate base and acid functions, respectively, of the complexed substrate the push-pull mechanism is possible but might not be a driving force. The halogenation of acetone in the presence of aqueous solutions of carboxylic acid buffers exhibits the rate law of Equation 11.2 where the third-order term, although small, has been shown to be significant and due to bifunctional concerted acid-base catalysis (Scheme 11.13) ... 

When (dpjdx j- P = 0, (32G/3n )TP 2 = 0 according to Equation (5.138). The entire term then becomes zero for the critical solution phase. The third-order term must be zero when the second-order term is zero, because otherwise some variations would result in negative values of the term. The condition to make the third term zero is that (d3G/dn])T P ri2 or the equivalent (82pi/3xi) is zero. If the fourth-order term is positive, then (d4G/dn4)T P is positive. Thus, the conditions of stability for the critical solution phase are... 

This equation makes it possible to calculate stresses for low velocity gradients to within third-order terms in the velocity gradient if one knows the moments to within second-order terms in the velocity gradients. Due to the approximations, used earlier in Chapter 2, the results are applicable for small extensions of the macromolecular coil and hence for low velocity gradients the results for the moments are valid to within second-order terms in the velocity gradients. 

Section III.C presents results of a study of certain third-order terms in the EOM equation that had previously been neglected in IP-EA calculations. It is found that some of these terms are reasonably small but not negligible, whereas the inclusion of others in the EOM equation can cause a complete breakdown of the traditional perturbative EOM method for nitrogen when using the standard choice of the primary operator space. 

Table I presents the results of EOM calculations of the three lowest IPs of nitrogen. Comparison of the first two columns of Table I demonstrates that there is a difference of 0.2 to 0.3 eV in the IPs when the EOM A matrix is symmetrized as by Simons, 21-order method I, and when the symmetrized form of the EOM equations, (21), 2j-order method II, is employed. The lack of symmetry in <0 (0,[//,0 ]) 0) in a 2 -order calculation arises from the inclusion of certain second-order A and terms, which contain the products of electron-electron interaction matrix elements with first-order double excitation correlation coefficients, and the neglect of other second-order A and A - terms, which involve second-order single excitation correlation coefficients multiplied by linear combinations of orbital energies. The discrepancies between the EOM 2 -order methods I and II are a measure of the importance of the terms due to single excitations in the ground-state wave function. In Section III.C, we consider the third-order terms not included in this primitive 2 -order EOM theory. The calculations imply although these terms are small, they are certainly not negligible. ...

The order of an integration method is the degree to which the Taylor series expansion. Equation (7.2), is truncated it is the lowest term that is not present in the expansion. The order may not always be apparent from the formulae used. For example, the highest-order derivative that appears in the Verlet formulae is the second, a(f), yet the Verlet algorithm is, in fact, a fourth-order method. This is because the third-order terms, which cancel when Equation (7.6) is added to Equation (7.7), are still implied in the expansion ... 

It can be observed that the expression for the paramagnetic contribution, equation (49), contains terms to third order in d, which do not appear in the relationship for the diamagnetic contribution (50). Since the total property (22) is invariant in the gauge transformation (45), the coefficients of the third-order terms, i.e., the last term of equation (49) must vanish. In fact, the sum rule... 

The possibility of evaluating the coupling coefficients using the analogy of multidimensional infrared (IR) spectroscopy [17] was also considered on the basis of the classical formulation shown in Equation 5.3. In the case of the third-order term aiq q2, the coupling coefficient 0C2 corresponds to the off-diagonal peak intensity of the power spectra, /(Qj, 111, - Here, the power spectra associated with multidimensional IR spectroscopy can be calculated via 2D Fourier transform of the following time correlation function ... 

The first two terms plus the first one inside the square brackets give Leist s expression 5.2.21. The third term in eqn. 5.2.26 gives the high order effect of the local velocity which arose from the perturbation of the ionic electrostatic potential. The second term in the brackets represents the higher order contribution of the asymmetric potentials and distribution functions from the continuity equation. The last term in eqn. 5.2.26 arises from in eqn. 5.2.25, which is due to the introduction of the first order term in U y into the continuity eqn. 5.2.12 and is equal to ajSc/(l + j) (V(2) + y The functions Si and Ti have been tabulated by Pitts for values of Ka in the range 0.02-0.50. 

TheEinEqs. 17.9 and 17.10 is an electric field vector. As such, the symmetry properties of the material will be important. In particular, it can be shown that the odd-order terms of these equations are independent of any symmetry considerations, but the even-order terms vanish in a centrosymmetric environment. That is, x is zero for any sample that has a center of inversion. This is always the case for bulk liquids, and it is true for most solids. Thus, symmetry and orientation effects should be crucial for discussions of SHG and related phenomena. We have noted earlier that crystal engineering has significant implications for many types of applications, and NLO is definitely one of them. The ability to rationally design or predict non-centrosymmetric crystals would be very valuable. Similarly, at the molecular level, p will be zero for molecules with a center of symmetry. No such restrictions apply to the third-order terms (x and y). 

---