Explicit using approximating functions

Appendix 2 Explicit calculation of compressible flow using approximating functions... 

The procedure which I have indicated proceeds by explicitly accumulating data on different types or classes of physical systems. Approximate functionals which have been developed for a given class of system need not automatically give adequate representations for different classes of systems. To describe this fact, it is useful to recognize explicit universality subclasses . Consider an example. It is feasible to imagine that the above procedures could result in the construction of functionals of the density which give highly accurate descrip-... 

Two commonly used approximations are the Hartree-Fock approach and density-functional theory (DFT). The Hartree-Fock approach approximates the exact solution of the Schrodinger equation using a series of equations that describe the wavefunc-tions of each individual electron. If these equations are solved explicitly during the calculation, the method is known as ab initio Hartree-Fock. The less expensive (i.e., less time-consuming) semi-empirical methods use preselected parameters for some of the integrals. DFT, on the other hand, uses the electronic density as the basic quantity, instead of a many-body electronic wavefunction. The advantage of this is that the density is a function of only three variables (instead of 3N variables), and is simpler to deal with both in concept and in practice. 

As was mentioned above, in KS-TDDFT the effects of electron exchange and Coulomb correlation are incorporated in the exchange-correlation potential vxaJ and kernel fxl- While the potential determines the KS orbitals (j)ia and the zero-order TDDFRT estimate (35) for excitation energies, the kernel determines the change of vxca with Eqs. 21, 22, 24. Though both vxca and are well defined in the theory, their exact explicit form as functionals of the density is not known. Rather accurate vxca potentials can be constructed numerically from the ab initio densities p for atoms [35-38] and molecules [39-42]. However, this requires tedious correlated ab initio calculations, usually with some type of configuration interaction (Cl) method. Therefore, approximations to vxcn and are to be used in TDDFT. 

Application of the functionals 3s and 3E in computer simulations hinges on approximations to Tpad[pA, Pb]- This functional is explicitly used in the evaluation of the total energy, whereas its functional derivative is one of the components of the KSCED effective potential. Unfortunately, the analytic form of this functional is not known except for some particular cases. Below, our attempts to approximate this functional and its functional derivative will be reviewed. We start with an overview of exact properties of Tpad[pA, Pb]-... 

The ITT approach formally exactly solves the Smoluchowski equation, following the transients dynamics into the stationary state. In this way the kinetic competition between Brownian motion and shearing, which arises from the stationary flux, is taken into account in the stationary distribution function. To compute it explicitly, but approximatively, using ideas based on MCT, MCT-ITT approximates the obtained averages by following the transient structural changes encoded in the transient density correlator. 

The theory also explains the value of 5. Unfortunately, that part of the story requires more sophisticated apparatus than we are prepared to discuss (operators in function space, Frechet derivatives, etc.). Instead we turn now to a concrete example of renormalization. The calculations are only approximate, but they can be done explicitly, using algebra instead of functional equations. 

A set of polynomial approximating functions are derived in Appendix 2 to be used with the explicit equations listed above. The critical pressure ratio just inside the pipe outlet is approximated by the polynomial ... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. K a more elaborate treatment is required, the suimnation for the partition functions must be carried out explicitly. An approximate account for vibrational anharmonidty can be obtained by using the harmonic form for the partition function (and resulting enthalpy and entropy... 

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