Shell-pairs

Pople-Nesbet Equations. The set of equations describing the best Unrestricted Single Determinant Wavefunction within the LCAO Approximation. These reduce the Roothaan-Hall Equations for Closed Shell (paired electron) systems. 

Note that in fact cyclobutadiene does not have degenerate, singly-occupied molecular orbitals, as a Jahn-Teller type (actually a pseudo-Jahn-Teller) distortion lowers its symmetry from square to rectangular and leads to a closed-shell paired-electron molecule [4]. 

To manufacture the shells, pairs of molds, for the body and the cap, are dipped into an aqueous gelatin solution (25-30% w/w), which is maintained at about 50°C in a jacketed heating pan. As the pins are withdrawn, they are rotated to distribute the gelatin evenly and blasted with cool air to set the film. Drying is carried out by... 

The front face of the MD PRISM is shown below. As the arrows indicate, the MD-PRISM algorithm consists of a set of highly interrelated pathways from shell-pair data to the desired brakets. 

The first step (To), the generation of [0](m) integrals from shell-pair data, is common to both the MD- and HGP-PRISM algorithms. We will discuss To in detail in Section 4.3. 

Inspection of (6) reveals that it describes the interaction of two charge distributions, < >a(r)b(r) and ( (r), and our first task is to collect information about all such charge distributions in the molecule. Because brakets are formed in classes, rather than individually, it is convenient to compile data for shell-pairs (rather than basis-function-pairs) and this shell-pair dataset is central to any modem integral program. To generate all of the desired brakets, we will later loop over all pairs of shell-pairs, that is, over all shell-quartets. 

Once we have compiled a list of models for all of the significant shell-pairs, they are sorted by "type", i.e. by the angular momenta of the component shells and by the degree of contraction of the (modeled) shell-pair. Thus, all uncontracted ss-pairs are stored consecutively, followed by all doubly-contracted... 

The last factor in (75), which scales it according to the angular momentum (a + b) of the shell-pair, is termed the principal scaling and is included only if the MD-PRISM is used. For reasons which will become clearer below, its presence reduces the Flop- and Mop-costs of the algorithm. 

The total computational effort involved in setting up the shell-pair data increases linearly with the size N of the basis. For tasks such as large Direct SCF calculations [44,45], it is entirely negligible compared with the subsequent work for less computationally demanding tasks, such as finding potential-derived atomic charges [98], it typically constitutes 10% of the job time. 

Given a sorted list of significant shell-pairs, we can construct all potentially important shell-quartets [99] by pairing the shell-pairs with one another. For the sake of vectorization, we deal with batches of shell-quartets of the same type and we utilize the memory which is available as effectively as possible in order to maximize the sizes of such batches [100]. 

Not every pair of shell-pairs, however, is necessarily accepted as a worthwhile shell-quartet. Although the shell-pair database has been carefully screened and contains no negligible shell-pairs, there are several ways in which a pair of significant shell-pairs may yield a shell-quartet which can be neglected... 

Unfortunately, in general, it is not easy to vectorize the shell-quartet selection process because of the conditional nature of quartet acceptance. In the special case where one desires the electrostatic potential on a large grid, Johnson et al. have circumvented this problem by the so-called fixed shell-pair scheme [98] which is completely vectorizable. However, this approach is not directly applicable to the general case. 

In computations of the electrostatic potential on a grid, Gaussian 92 calls the subroutine CalcOG to compute the [0] and this routine accounts for roughly 50% of the total CPU time. As a result of the use of the fixed shell pair scheme [98] and very careful optimization, CalcOG runs at approximately 180 MFlops on the Cray Y-MP. 

We note, too, that things only improve under the fixed shell-pair scheme which is used in electrostatic grid calculations because the scalings become loop-invariants [98]. 

Call ListS2 to form list of significant shell-pairs... 

Call SortS2 to sort shell-pair data by type... 

Call CutoS2 to compute cutoff parameters for each shell-pair... 

Form petite list of bra shell-pairs of current type Loop over KKet values... 

Pairs of shells are required for most PRISM calculations but, for example, if overlap integrals are being calculated as outlined in Section 3.2.1, the "shell-pairs" are constructed by pairing each shell with a "dummy" (p = 0) shell. 

For the higher numbers of valence-shell pairs, 7, 8 and 9, there are only a few non-transition element complexes known. In the case of IF7 the structure appears to be a pentagonal bipyramid, contrary to the entry in Table 4-1. However, with these higher numbers, the predictions of preferred arrangements necessarily become less certain because the repulsive energy of the set of electron pairs does not have a pronounced minimum for any one configuration and atom-atom interactions assume greater importance. 

All electrons not in the conduction band will be in filled shells, paired, and not magnetically active. 

Predicted and measured parallel efficiencies on a Linux cluster for two-electron integral computation for alkanes employing the cc-pVTZ basis set and using static distribution of shell quartets (a) and shell pairs (b). ... 

We demonstrated in the previous section that a static distribution of shell pairs in the computation of the two-electron integrals causes a significant performance degradation for large process counts, resulting in a reduced degree of parallelism. For instance, for the butane molecule using the cc-pVTZ... 

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