Split shells

Slides Split-shell bearings hard particles embedded in soft bearing alloys micrograph of section through layered bearing shell skiers automobile tyres. 

Fig. 1. Electron-electron distribution functions for single-configuration He wavefunction (a) radial probability distribution P(ri2) (b) intracule function h(ri2)- In both graphs, the curve with largest maximum is for the closed-shell wavefunction that of intermediate maximum is for the split-shell wavefunction that of smallest maximum is for the wavefunction containing exp( —yri2).

The differences between the single-configuration wavefunctions are more clearly illustrated by comparing their plots of the intracule function h(ri2), also shown in Fig. 1. This plot reveals the absence of an electron-electron cusp for both the closed and split-shell functions, but shows that the inclusion of exp( —yri2) causes the distribution to have a minimum at ri2=0, forming a cusp (of the correct sign) at that point. This feature will be important for the description of phenomena that depend upon the coincidence probability. 

Details can be found in Bdlare (ref. 3). The experimental equipment consists of a cumene reservoir, a thermogravimetric analyzer (TGA) and a gas chromatograph (GC). The hdium-cumene mixture enters the TGA, a Cahn System 113DC with a Cahn 2000 Recording Electrobalance, a quartz tubular reactor, and an external split-shell furnace. The catalyst is placed in the sample pan of the microbalance inside the quartz reactor, kept at a controlled temperature in the center of the split-shell furnace. The incremental weight due to coke deposition on the catalyst is monitored by an IBM PC. The reactor exit stream is injected into a Varian 3700 GC using FID. 

Such basis sets are sometimes referred to as double zeta or split shell basis sets. At a slightly simpler level, only the valence shell functions may be split (split valence shell basis). Several basis sets of this sort, both exponential24 and gaussian2s have been proposed and used. 

As mentioned in the previous section, the 6-31G basis does not contain polarization functions on hydrogen. No extensive systematic study using such a basis set has been reported. However, results are available for a number of small molecules and they may be used to evaluate the effect that the addition of p functions on hydrogen has on calculated dipole moments. Neumann and Moskowitz33,34 have obtained near Hartree-Fock wave functions for water and formaldehyde. From these studies they conclude that the inclusion of polarization functions on hydrogen results in rather small changes in the calculated dipole moment. However, it should be pointed out, that they added p functions to a split shell basis set and so their conclusions may not apply directly to the 6-31G level of basis. 

Snyder (personal communication) examined some small molecules and found that the addition of polarization functions to a split shell basis set produces a considerable lowering of the calculated values. 

Though Eq. (20) is still formal, it has distinct advantages over say the infinite exact C.I. series (t) it is not an infinite series if the system has a finite number (N) of electrons (tt) it is in a very detailed form so that each term can be examined and systematic approximations can be applied, as we do below and (Hi) the I functions are in a closed form. They may contain interelectronic coordinates, split shells", etc., or each may be separately expanded in a C.I. series. 

The interpretation of type 5 wavefunction is somewhat difficult. We have said that it can be described as a two-configuration wavefunction. This is sometimes referred to as a split-shell wavefunction others refer to this wave-function as providing for in-out correlation. All agree that wavefunction type 5 takes account of some of the radial correlation, but ten Hoor has cautioned about the interpretation of the results in terms of the in—out terminology. ... 

M. J. ten Hoor,/. Chem. Educ., 68,197 (1991). On the Interpretation of Simple Closed-Shell and Split-Shell Function of Atomic Two-Electron Systems. 

The wavefunction described with the optimum a,y corresponds well with a somewhat perturbed lslj 2s configuration. Two of the electrons depend on the electron-nuclear distance with a values (screening parameters) that correspond to a partially screened interaction with the - -3-charged Li nucleus in a split-shell electron distribution. The third electron (that with pre-multiplying r) has an a value somewhat larger than for a hydrogenic 2s orbital, indicative of the fact that the inner-shell electrons do not completely shield the Li nucleus. The electron-electron a values all reflect the existence of electron-electron repulsion, with the effect most pronounced for the Is-ls interaction. All these observations are consistent with the notion that the exponentially correlated wavefunction gives an excellent zero-order description of the electronic structure of Li. 

In this trial function, f has the same value in each atomic orbital. This is not a necessary restriction. There is no physical reason for not choosing the more general trial function where orbitals with different are used. Symmetry requires that such a function be written 2 lulls 2) + ls"(l)ls (2)]2 F2[, (j) (2) - /3(l)cr(2)]. This type of function is called a split shell wavefunction. It gives a lower energy for He than does the function (7-26). However, for most quanrnm-chemical calculations split shells are not used, the gain in accuracy usually not being commensurate with the increased computational effort. 

The rnethod used constructs the one-electron Bloch orbitals as a linear combination of a double-C (split-shell) type atomic basis set, i.e., 10 and 18 contracted atomic orbitals are applied on atoms N and S, respectively. These atomic orbitals themselves are taken as contractions of Gaussian lobe functions using 25 and 46 primitive functions for atoms N and S, respectively. The exponents of the Gaussians as well as the contraction coefficients were taken from the paper of Roos et Based... 

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