Neighborhood kernel

In order to use the notional particles to estimate f x, we need a method to identify a finite sample of notional particles in the neighborhood of x on which to base our estimate. In transported PDF codes, this can be done by introducing a kernel function hw(s) centered at 5 = 0 with bandwidth W. For example, a so-called constant kernel function (Wand and lones 1995) can be employed ... 

Since the coordinates of the structure of intere.st are available. it is readily possible to calculate which atoms would occur within a particular chosen distance from each of the atoms in a kernel. Such atoms would form the neighborhood. In order to effect a pairing of all electrons, it may be necessary to attach some hydrogen atoms to atoms belonging to the neighborhood or a moiety that is part of the molecule, or both. This can extend the outer limits of the fragment somewhat. It was used in the leu -zervamicin example to be described. 

The fragment calculations for the hydrated hexadecapep-tide, leu -zervamicin, were performed by defining 19 kernels as the 16 peptide residues, two clusters of water molecules and a cluster of a water and an ethanol molecule. In this application, the neighborhoods were formed with atoms within 5 A of the kernels plus a few additions to assure that all electrons were paired. The kernels and their neighborhoods are listed in Table 1. The numbers refer to the peptide residues in the sequence that were given above in the description of the chemical content of the leu -zervamicin molecule. The symbols in Table 1 correspond to those found in the crystal... 

Table 1 Composition of the 19 Fragments, i.e., Kernels and Their Corresponding Neighborhoods, Calculation of the P Matrix for the Hydrated Hexadecapeptide, Leu -zervamicin ...

Kernel Neighborhood No. of Atoms No. of Basis No. of Atoms No. of Basis... 

The number of elements for the R matrix represented in the various boxes in Figure 1 varies considerably and may be judged by examination of Table I with respect to the atomic composition of the kernels and also their neighborhoods. Nevertheless each row and each column of both R and P... 

Figure 1 A schematic diagram of the R matrix that was obtained from the fragment calculations for hydrated leu -zervamicin. The numbers in the figure label the kernels as in Table 1. The elements in the dark squares are composed of functions of the coefficients of pairs of orbitals associated with the atoms of the particular kernel. The elements in the gray squares and hatched squares ate composed of functions of the coefficients of pairs of orbitals, one of which is associated with a kernel atom of the dark square in the same row and the other is associated with an atom of the labeled neighborhood. The contents of the dark square labeled by kernel 7 are shown in Figure 2 at higher resolution...

The difference studies shown in Figures 5 and 6 indicate that the electron density is well represented by the P matrix obtained from the fragment calculations. As has been noted, the method for developing a P matrix presented here differs from the one presented in Ref. 1. The P matrix formed here is more accurate than that formed from the R matrix in Ref. 1. It was not necessary to employ the Clinton equations here to improve the projector quality. In fact, use of the Clinton equations left the good quality of the P matrix unchanged. Even greater accuracy may accrue if more atoms were included in the neighborhoods of the kernels. 

The kernel-neighborhood fragment calculation affords the opportunity to make ab initio calculations for molecules whose complexity exceeds the capacity of available computer programs, particularly since the increase in computer time is essentially a linear function of the complexity. In addition, the fragment calculation may save time when employed as an alternative to a calculation made on a complex, full molecule. 

---