Half-electron approximation

Within some programs, the ROMPn methods do not support analytic gradients. Thus, the fastest way to run the calculation is as a single point energy calculation with a geometry from another method. If a geometry optimization must be done at this level of theory, a non-gradient-based method such as the Fletcher-Powell optimization should be used. 

When it has been shown that the errors introduced by spin contamination are unacceptable, restricted open-shell calculations are often the best way to obtain a reliable wave function. 

Another approach is to run an unrestricted calculation and then project out the spin contamination after the wave function has been obtained (PUHF, PMP2). This gives a correction to the energy but does not affect the wave function. Spin projection nearly always improves ah initio results, but may seriously harm the accuracy of DFT results. 

A spin projected result does not give the energy obtained by using a restricted open-shell calculation. This is because the unrestricted orbitals were optimized to describe the contaminated state, rather than the spin-projected state. In cases of very-high-spin contamination, the spin projection may fail, resulting in an increase in spin contamination. 

A similar effect is obtained by using the spin-constrained UHF method (SUHF). In this method, the spin contamination error in a UHF wave function is constrained by the use of a Lagrangian multiplier. This removes the spin contamination completely as the multiplier goes to infinity. In practice, small positive values remove most of the spin contamination. 

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Semiempirical programs often use the half-electron approximation for radical calculations. The half-electron method is a mathematical technique for treating a singly occupied orbital in an RHF calculation. This results in consistent total energy at the expense of having an approximate wave function and orbital energies. Since a single-determinant calculation is used, there is no spin contamination. 

An advantage of the half-electron technique is its simplicity. HyperChem can carry it out with only minor modifications of the usual calculation. A disadvantage is that forces may not be accurate because of the half electron approximation. 

In contrast to the great number of calculations concerning the all-carbon Diels-Alder reaction [39], there are only a few theoretical studies on the hetero Diels-Alder reaction [41,42,45 - 53 ]. The general mechanism of the Diels-Alder reaction is still in discussion however, in most cases a concerted reaction is assumed,but there is also evidence for a two-step path. The ab initio calculations carried out for the butadiene/ethene system by Houk, Ortega, Bernardi und Gajewski gave a symmetrical transition structure only using the semiempirical AM1/CI method (half electron approximation) an unsymmetrical diradicaloid intermediate was found [40]. 

The origin of these effects has been debated. One possibility is the Peierls instability [57], which is discussed elsewhere in this book In a one-dimensional system with a half-filled band and electron-photon coupling, the total energy is decreased by relaxing the atomic positions so that the unit cell is doubled and a gap opens in the conduction band at the Brillouin zone boundary. However, this is again within an independent electron approximation, and electron correlations should not be neglected. They certainly are important in polyenes, and the fact that the lowest-lying excited state in polyenes is a totally symmetric (Ag) state instead of an antisymmetric (Bu) state, as expected from independent electron models, is a consequence... 

A UHF scheme of the MINDO/3 method [59] and a simpler half-electron scheme based on the RHF approximation [60] have been developed with the aim to describe radical reactions and structures. A detailed list of various areas where the MINDO/3 method yields good results is given in the review article [54]. 

To first order, the dispersion (a-a) interaction is independent of the structure in a condensed medium and should be approximately pairwise additive. Qualitatively, this is because the dispersion interaction results from a small perturbation of electronic motions so that many such perturbations can add without serious mutual interaction. Because of this simplification and its ubiquity in colloid and surface science, dispersion forces have received the most significant attention in the past half-century. The way dispersion forces lead to long-range interactions is discussed in Section VI-3 below. Before we present this discussion, it is useful to recast the key equations in cgs/esu units and SI units in Tables VI-2 and VI-3. 

Amorphous Silicon. Amorphous alloys made of thin films of hydrogenated siUcon (a-Si H) are an alternative to crystalline siUcon devices. Amorphous siUcon ahoy devices have demonstrated smah-area laboratory device efficiencies above 13%, but a-Si H materials exhibit an inherent dynamic effect cahed the Staebler-Wronski effect in which electron—hole recombination, via photogeneration or junction currents, creates electricahy active defects that reduce the light-to-electricity efficiency of a-Si H devices. Quasi-steady-state efficiencies are typicahy reached outdoors after a few weeks of exposure as photoinduced defect generation is balanced by thermally activated defect annihilation. Commercial single-junction devices have initial efficiencies of ca 7.5%, photoinduced losses of ca 20 rel %, and stabilized efficiencies of ca 6%. These stabilized efficiencies are approximately half those of commercial crystalline shicon PV modules. In the future, initial module efficiencies up to 12.5% and photoinduced losses of ca 10 rel % are projected, suggesting stabilized module aperture-area efficiencies above 11%. 

About half of epoxide resin production is used for surface coating applications, with the rest divided approximately equally between electronic applications (particularly for printed circuit boards and encapsulation), the building sector and miscellaneous uses. In tonnage terms consumption of epoxide-fibre laminates is only about one-tenth that of polyester laminates, but in terms of value it is much greater. 

The departure of the Eh values from a smooth trend is somewhat over half as large as that of the dissociation energy values in the last row of Table VI. Until the London energy calculations are refined to eliminate the dipole-dipole approximation and other uncertainties, it is not possible to say whether that effect accounts for the entire anomaly or not. In any event a substantial portion of the anomaly may be ascribed to the correlation of the motion of the unshared electron pairs in. the valence shell. 

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