Second-order term

The second contribution to the molecular g-tensor is given by the second-order perturbation theory. According to Table 4.6 the relevant Hamiltonian terms are  

This formula contains two types of integral. The first one is 

It can be expressed (with the help of the Slater rules) in terms of spin-orbitals as follows 

An analogous expression to the above holds true for The content of the second integral is 

The Slater rules predict that the one-electron term survives only for mono-excited configurations whereas the two-electron term survives also for biex-cited configurations. However, the perturbation formula for Ag b involves the product of the first-type (one electron only) and the second-type integrals so that the biexcited configurations cannot contribute. This allows to consider the whole second term as if it were constructed of an effective one-electron operator 

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After the application of Green s theorem to the second order term in Equation (2.81) we get the weak form of the residual statement as... 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

Sufficient conditions are that any local move away from the optimal point ti gives rise to an increase in the objective function. Expand F in a Taylor series locally around the candidate point ti up to second-order terms ... 

The variance equation with second order terms is ... 

Using the first and second order terms in the variance equation gives exactly the same answer. For different conditions, say where one variable is not dominating the situation as above for the load, then the use of the variance equation with second order terms will be more effective. 

In nonpolar solvents, the observed rate of bromination is frequently found to be described as a sum of the first two terms in the general expression. The second-order term... 

This is the general linear equation of motion for an almost planar and rough one-dimensional phase boundary. The fourth-order term in the spatial derivative acts as a stabilizer just like the second-order term, and is not really crucial here. 

Another example is a second-order term containing the concentration of an acid and a base, say... 

The second-order terms give the magnetizability. The first term is known as the diamagnetic part and it is particularly easy to calculate since it is just the expectation value of the second moment operators. The second term is called the paramagnetic part. 

The second-order term, the magnetizability, has two components. The derivative expression (10.31) is... 

Kessler, P., Comfit, rend. 240, 1058, Calcul approch du terme du second ordre en th<5orie des perturbations." Approximate calculation of the second-order term in perturbation theory. 

Friedly (F4) expanded the theoretical analysis of Hart and McClure and included second-order perturbation terms. His analysis shows that the linear response of the combustion zone (i.e., the acoustic admittance) is not sign-ficantly altered by the incorporation of second-order perturbation terms. However, the second-order perturbation terms predict changes in the propellant burning rate (i.e., transition from the linear to nonlinear behavior) consistent with experimental observation. The analysis including second-order terms also shows that second-harmonic frequency oscillations of the combustion chamber can become important. 

In the modified Ridd mechanism for region B the deprotonation of the A-nitroso-anilinium ion Ar —NH2NO in Scheme 3-23 is rapid, and therefore does not influence the overall rate. However, the second-order term in the rate equation for region C (Scheme 3-25) is consistent with a mechanism in which the deprotonation of the A-nitrosoanilinium ion (Scheme 3-24) and of the C-nitrosoanilinium dication (Scheme 3-22) belongs to the rate-determining part of the reaction. 

Neglecting second order terms and noting that sin = dz/dl ... 

It is readily seen that second-order terms of (Al.lb) do not contribute to ju(0), correcting only the (i (0) value. In order to calculate this correction, one has to substitute (A1.2) into (Al.lb) and to reduce the similar terms in (2.82). This leads to the following expression for fi ... 

It will be seen that the second-order treatment leads to results which deviate more from the correct values than do those given by the first-order treatment alone. This is due in part to the fact that the second-order energy was derived without considerar-tion of the resonance phenomenon, and is probably in error for that reason. The third-order energy is also no doubt appreciable. It can be concluded from table 3 that the first-order perturbation calculation in problems of this type will usually lead to rather good results, and that in general the second-order term need not be evaluated. 

The contributions of the second order terms in for the splitting in ESR is usually neglected since they are very small, and in feet they correspond to the NMR lines detected in some ESR experiments (5). However, the analysis of the second order expressions is important since it allows for the calculation of the indirect nuclear spin-spin couplings in NMR spectroscoi. These spin-spin couplings are usually calcdated via a closed shell polarization propagator (138-140), so that, the approach described here would allow for the same calculations to be performed within the electron Hopagator theory for open shell systems. 

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

The system is non-Newtonian and viscosity is a function of temperature and shear rate. A constitutive equation including a second order term for the logarithm of shear rate was used. 

The final step in the formulation of the model [4-6] is to recognize that the second-order term, say, must be a quadratic function of the angular... 

It is apparent that the existence of a second-order term in the rate expression does not of itself offer any proof of associative or dissociative activation, for there are two possible alternative mechanisms compatible. These are ... 

Alternative (ii) corresponds to the [Re(CO)4X]2 case, equations (41) and (42) above. However, it was here favoured largely because no second-order term was observed for the Re(CO)5X and Re(CO)4LX substitution. In the case of Mo(CO)5Py, expected to be closely similar to Mo (CO)4dipy, a second-order dependence has been observed. 

We now substitute eqns 2, 3, and 4 into eqn 1, eliminate second order terms in AC, multiply on the left by ct, and separate the resulting equation into two as follows. We find, first of all ... 

Relaxing the Einstein convention, sums over repeated Greek indices a = x,y,z are made explicit in this Section, to avoid misunderstanding whenever two couples of repeated indices a and / , with a < (3, appear in a formula, compare fo r (92) hereafter. Introducing a basis set x of atomic functions, for the second-order term one defines the expansion... 

Further degradation of the information encoded in the electron beam takes place in the recording step since the signal is proportional to the square modulus of the image wave-function, i.e. neglecting small second order terms ... 

In many applications there is no second-order term in the perturbed Hamiltonian operator so that zero. In such cases each unperturbed... 

Thus, the first-order perturbation to the eigenvalue is zero. The second-order term E is evaluated using equations (9.34), (9.50), and (4.50), giving the result... 

The Gauss-Newton method arises when the second order terms on the right hand side of Equation 5.20 are ignored. As seen, the Hessian matrix used in Equation 5.11 contains only first derivatives of the model equations f(x,k). Leaving out the second derivative containing terms may be justified by the fact that these terms contain the residuals e, as factors. These residuals are expected to be small quantities. 

The Gauss-Newton method is directly related to Newton s method. The main difference between the two is that Newton s method requires the computation of second order derivatives as they arise from the direct differentiation of the objective function with respect to k. These second order terms are avoided when the Gauss-Newton method is used since the model equations are first linearized and then substituted into the objective function. The latter constitutes a key advantage of the Gauss-Newton method compared to Newton s method, which also exhibits quadratic convergence. 

It is cumbersome to write the partial fraction with complex numbers. With complex conjugate poles, we commonly combine the two first order terms into a second order term. With notations that we will introduce formally in Chapter 3, we can write the second order term as... 

In establishing the relationship between time-domain and Laplace-domain, we use only first and second order differential equations. That s because we are working strictly with linearized systems. As we have seen in partial fraction expansion, any function can be "broken up" into first order terms. Terms of complex roots can be combined together to form a second order term. 

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