The Current Approximation Function

In this chapter, the current approximation function Q, defined in Chap. 3, (3.25), will be used extensively. Note also that since this function is a linear combination of the array argument (for example, C as in Q(C,n. H)), the function of a weighted sum of two arrays, such as the arrays u and v (to be met later), the following holds (a being some scalar factor) ... 

In this chapter, the current approximation function Q, defined in Chap. 3, Eq. (3.25), will be used extensively. Note also that since this function is a linear... 

One can now approximate the current autocorrelation function in the diffusive limit. When the time integration is performed, the above expression reduces to... 

The above treatment includes the current approximation on an unequal grid, and the subroutine U DERIV can compute it. It is, however, a little unwieldy, and a simpler interface to it is also mentioned in the same Appendix, function GU, which only requires the three arguments (C, x, n). Similarly, the function CU computes Co from a given concentration profile and a known current,... 

Once we have the subroutine U DERIV shown above, it is simple to construct a more convenient function to calculate the current approximation, if that is all we want (that is, if we do not want the coefficients that make it up). The function GU does the job, calling the more complicated U DERIV to do the hard work. 

U DERIV can be used to compute Co, given the current (as in chronopoten-tiometry) and the concentration profile. As for equal intervals, the current approximation formula on n points is adapted, by the function CU. 

The unresolved problem yet is the determination of a rigorously exact functional. For this reason all of the currently used functionals are approximate. Furthermore, one of the main problem of DFT methods is that Van der Waals dispersion contribution is not explicitly considered in the equations, which leads to some severe errors for the description of the zeolitic guest-host interaction. We will give further details on this point later in the discussion. 

If one compares the attempts reviewed in sec. 3.2 to base majiy-electron quantum mechanics on the two-particle density matrix, i.e. a 2-particle density matrix functional theory with the current density functional theory one realizes that for the former the functional is exactly known, while the full n-representability condition is unknown. For DFT on the other hand, the functional is unknown, but the n representability does not cause problems. Why should one take incomplete information on n-representability as more serious as lack of information on the exact functional Possibly there was just more reluctance in the two-particle-density matrix functional community to be satisfied with approximate n-representability conditions than in the density functional community to accept approximate density functionals, and that this different attitude was decisive for the historical development. 

During the iterative solution of the LCAO equations, at each iteration, the diagonalisation of the current approximation to the self-consistent Hartree-Fock matrix generates such a partition of the total function space, i.e. a current (non-self-consistent) set of occupied orbitals and a set of current victuals a current occupied space and a current virtual space. These current spaces share some of the properties of the final self-consistent spaces in particular the current single-determinant is invariant against linear transformations within the current occupied space. 

D FT approach is the Xa method, which uses only the exchange part in a local density approximation (LDA, local value of the electron density rather than integration over space) [49, 50]. The currently available functionals for approximate D FT calculations can, in most cases, provide excellent accuracy for problems involving transition metal compounds. Therefore, DFThas replaced semi-empirical MO calculations in most areas of inorganic chemistry. 

A special problem where the time-dependent density-functional theory could be useful is that of calculating the polarizability and hyperpolarizability (Section 12). It turned out that although accurate results could be achieved for smaller molecules (partly, however, requiring a careful choice of the approximate density functional), severe problems could turn up (but did not always) when considering extended systems. It might mean that the current density functionals are lacking an explicit dependence on the polarization, but further studies are needed in order to clarify this point. 

Based on the stationality condition of the above functional, an iterative solution procedure can be constructed. Let us assume, that /, L/ is the current approximation and df/ isthe... 

In Eq. [41] we input the current approximation for the function on a given level, which is what is done in the solver as it progresses toward the exact solution, UK The coarsest grid level is labeled by the smallest k. It is a good exercise to show that Eq. [41] is required on levels two or more removed from the finest level. Notice that the defect correction on level k requires information only from the next-finer k + 1) level. 

This step interpolates the change in the coarse-grid function during iterations onto the next-finer level and corrects the current approximation there. 

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