Second-order variation

It is necessary to calibrate the 14C time scale for greater dating accuracy. However, the second-order variations are at least as important as the first-order constancy of atmospheric 14C. For example, they provide a record of prehistoric solar variations, changes in the Earth s dipole moment and an insight into the fate of C02 from fossil fuel combustion. Improved techniques are needed that will enable the precise measurement of small cellulose samples from single tree rings. The tandem accelerator mass spectrometer (TAMS) may fill this need. 

Fukui functions and other response properties can also be derived from the one-electron Kohn-Sham orbitals of the unperturbed system [14]. Following Equation 12.9, Fukui functions can be connected and estimated within the molecular orbital picture as well. Under frozen orbital approximation (FOA of Fukui) and neglecting the second-order variations in the electron density, the Fukui function can be approximated as follows [15] ... 

The sum is over all classical paths connecting R1 and R in time t, S is the classical action along such paths, and Det denotes the determinant which ensures unitarity of the propagator K up to second-order variations in S. The classical action is a solution of the Hamilton-Jacobi equation,... 

Second order variational equations and the strong maximum principle (with M.M. Denn). Chem. Eng. ScL 20,373-384 (1965). 

The stability of the critical phase can be discussed most easily in terms of the Helmholtz energy and the condition expressed in Equation (5.154). By use of the method used in Section 5.15, the second-order variation at constant temperature is expressed as... 

As opposed to a conical intersection, f (Qo) = AE(Qo) > 0 at the FC point. However with quasidiabatic states fiiDo) = 0. As a consequence, the second-order variation of the adiabatic energy difference satisfies... 

Substituting Eqs. (5.25) into Eq. (5.24), subtracting the optimal values from both sides, and neglecting second-order variations yields... 

Truncation of Taylor expansion (Eq. (8)) after the second-order variation may be shown to be legitimate because the third-order derivative is often small [117], which is, however, not always true [56]. Application of EEP gives ... 

We must, therefore, look at the analogue of the second derivative in our derivation the second-order variation induced in the total energy by a variation in the MO coefficients. 

To provide some idea of the magnitude of second-order variations from simple behavior, Figures 5.32 and 5.33 illustrate the calculated spectra of two additional systems (—CH—CH2— and —CH2—CH2—). The first-order spectra appear at the top AvU > 10), while increasing amounts of second-order complexity are encountered as we move toward the bottom AvU approaches zero). 

The matrix A — B defines the second-order variation of the energy function and is often referred to as the Hessian matrix. The double-commutator form of the Hessian matrix allows these second-order terms to be expressed as a quadratic form. 

To apply the stability criterion (8.1.8) we need the second-order variation of S which, from (8.1.9), is found to be... 

Since the quantities /of interest to us form exact differentials, the second-order variation in (G.0.5) is invariant under an exchange of the indices i and j. We need not proceed further into the variational calculus here because in this book we use the variations 8/and 8 /merely as a notational convenience relations (G.0.4) and (G.0.5) define the notation we use. 

Second Order Variation in Charge—First Order in... 

If the above two equations are substituted in the continuity and Navier-Stokes equations and inviscid flow is assumed, after several steps of derivations, time-averaged second-order variation is given as follows [1] ... 

Let X be the difference between any fluctuating thermodynamic parameter and its mean (equilibrium) value. Eg. X = T - (T), whence X) = 0. The probability of X taking a value within [X,X + dX] is expressed by Equation 7, but the arbitrary second-order variation 6 S has to be replaced by a second-order derivative of S with respect to Xy which transforms Equation 7 into... 

The constant Vq may be set equal to zero. In a point of equilibrium, the first derivatives are equal to zero, while those higher than the second-order variations are neglected (this is the meaning of the term small molecular vibrations ). 

The stability criteria which we have discussed in the preceding sections of this chapter are local or differential criteria since they are formulated in terms of second-order variations and their time derivatives. On the other hand, we have seen in Chapter 6 that it would be very helpful to gain knowledge about the global behaviour of a network for classifying its properties and deciding whether it may... 

Therefore, the stability of the original phase is defined with respect to the second-order variations of s. If the second-order variations are zero, third-order variations must also be zero, and then fourth-order variations should be negative. 

For AU to be positive, since %N N/N is positive, the second-order variations with respect to s and c have to he positive. In case the second-order variations are zero, higher-order even variations should be positive. In other words, the first nonvanishing terms of order two or higher-even order should be positive. Therefore, stability of the single-phase state demands that... 

Here and are linear variation coefficients. Second-order variation of the MCDF energy (Eq. 1.7) with respect to the parameters Tpe and configuration mixing coefficients Cjk leads to the Newton-Raphson (NR) equations for second-order MCDF SCF,... 

Through these relations, the change of the thermodynamic potentials AF, AG or AH due to a fluctuation can be related to the entropy production Aj S. The system is stable to all fluctuations that result in Aj 5 < 0, because they do not correspond to the spont eous evolution of a system due to irreversible processes. From the above relations it is clear how one could also characterize stability of the equilibrium state by stating that the system is stable to fluctuations for which AF > 0, AG > 0 or AH > 0. For fluctuations in the equilibrium state, these conditions can be written more explicitly in terms of the second-order variations 6 V > 0,5 G > 0 and d H > 0, which in turn can be expressed using the. second-order derivatives of these potentials. The conditions for stability obtained in this way are identical to those obtained in Chapter 12. 

The validity of (A18.1.1) depends on the validity of local equilibrium. In Chapter 12 we have seen that the second-order variation of entropy 5 5 is negative because quantities such as the molar heat capacity Cy, isothermal compressibility k, and positive. This condition... 

Alternatively, the eigenvalues may be obtained from the zeroth- and second-order variation equations. We can now use the simplified fourth-order energy functional to define a set of equations to solve in our finite basis for the second-order wave function. 

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