Line integral techniques derivation

A common method for solving partial differential equations (PDEs) is known as the method of lines. Here, finite difference approximations for spatial derivatives are used to convert a PDE model to a large set of ordinary differential equations, which are then solved using any of the ODE integration techniques discussed earlier. 

However, in the second set of data, reporting scans of the PES for a limited set of small molecules, it appears that the geometries obtained are satisfactory. Moreover, the nature of the technique used for the determination of Exc, namely the use of a "senior" Exe functional, or the use of the virial theorem, as well as the use of a line integration (not reported here), leads to quite similar geometries. This point is in accord with a similar conclusion obtained by van Gisbergen et ol. in their frequency-dependent polarizabilities [75] they choose to use a "mixed scheme" where a different approximation for fxc and Vxc were used, whereas fxc is the functional derivative of the exchange-correlation potential Vxc, with respect to the time-dependent density. 

In this appendix we present a discussion of a few mathematical techniques frequently utilized in thermodynamics. We treat several topics in the analysis of real functions of several real variables. We assume that the functions considered have the continuity properties necessary for the operations performed upon them to be meaningful. In Sec. A-1, we discuss some of the properties of partial derivatives. In Sec. A-2 we define homogeneous functions and derive a useful relation. In Sec. A-3, we treat linear differential forms. Line integrals are discussed in Sec. A-4. 

After risks have been estimated, available information must be integrated and interpreted to form conclusions about risks to the assessment endpoints. Risk descriptions include an evaluation of the lines of evidence supporting or refuting the risk estimate(s) and an interpretation of the adverse effects on the assessment end point. Confidence in the conclusions of a risk assessment may be increased by using several lines of evidence to interpret and compare risk estimates. These lines of evidence may be derived from different sources or by different techniques relevant to adverse effects on the assessment end points, such as quotient estimates, modeling results, field experiments,... 

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