Direct product decomposition

Now, consider the subgroup C3 of D3 , (since no out-of-plane bending is possible for a triatomic system, and hence the subgroup C3 may be used for the discussion). Then, Eq. (E. 15) contains only four symmetry types of electronic operators Iia, Iia, h, and hy. The direct product decompositions for C3 may then be shown (see Table 57 of [28]) to assume the fomi... 

Term in Assignment of the direct product Decomposition in O according to... 

J. F. Stanton, J. Gauss, J. D. Watts, and R. J. Bartlett, ]. Chem. Phys., 94, 4334 (1991). A Direct Product Decomposition Approach for Symmetry Exploitation in Many-Body Methods. I. Energy Calculations. 

Open-Shell Analytic Gradients Implementation of the Direct Product Decomposition Approach in Energy Gradient Calculations. 

Direct Product Decomposition Approach for Symmetry Exploitation in Many-Body Methods. 1. Energy Calculations. 

In Table 1 all point groups are classified according to the possibility or not to be described by a direct product decomposition ... 

Intramolecular isotope effects were studied in the systems N2-HD, CO-HD, 02-HD and C02-HD (20). Product decomposition directly associated with rupture of OH or OD bonds was not observed in these reactions. Isotope effects in decomposition processes which gave OH + or OD+ from reactions of 02+ with HD and COH+ or COD + from... 

The examples used above to illustrate the features of the software were kept deliberately simple. The utility of the symbolic software becomes appreciated when larger problems are attacked. For example, the direct product of S3 (order 6) and S4 (isomorphic to the tetrahedral point group) is of order 144, and has 15 classes and representations. The list of classes and the character table each require nearly a full page of lineprinter printout. When asked for, the correlation tables and decomposition of products of representations are evaluated and displayed on the screen within one or two seconds. Table VII shows the results of decomposing the products of two pairs of representations in this product group. 

Table VII. Display of the decomposition of products of representations within the direct product group S3-DP-S4.

Lede J, Lapicque F, Villermaux J (1983) Production of hydrogen by direct thermal decomposition of water. Int J Hydrogen Energy 8 675-679... 

This reducible representation (the occupancy of two e orbitals in the anion gives rise to more than one state, so the direct product e x e contains more than one symmetry component) can be decomposed into pure symmetry components (labels T are used to denote the irreducible symmetries) by using the decomposition formula given in Appendix E ... 

The direct production of SOz in a reaction such as (73b) was suggested by the independence of the measured S02 yields on the oxygen concentration. NOz is not produced in significant yields (<5%) in the N03-CH3SH reaction (Dlugokencky and Howard, 1988), ruling out a simple oxygen atom transfer or decomposition of the adduct to N02 and other products. 

Each of these configurations, except d1 and d9, will give rise to terms under the action of HER as a perturbation. The terms, of course, bear the labels of the cubic group, here Oh. The terms which arise are determined qualitatively by the decomposition of the group theoretical direct product of the electrons (or holes) involved into irreducible representations of the cubic groups. A t2s or eR orbital set more than half-full is treated as the equivalent number of holes, and a filled one is ignored. Then, for instance... 

Table 66 Decomposition of the direct product F F of irreducible representations in some double groups... 

The complete mechanism either for the mercury sensitized or for the direct photochemical decomposition of ammonia is complex and there exists reasonable doubt about some of the steps even after many years of intensive work. Under static experimental conditions hydrogen and nitrogen are virtually the only products and they are formed in the ratio of 3 1. Other products must, therefore, be of negligible importance under ordinary conditions. 

It was then shown by M. G. Levi1 that the yield is not diminished by allowing the temperature to rise as high as 30° C., and is almost independent of the cathode material used, but he found that a new smooth platinum anode gives a better yield than an old one which has a somewhat rough surface. A patent process of 1904 claimed the use of an undivided cell with the addition of hydrofluoric acid, under which circumstances E. Miiller2 had shown that the yield of potassium persulphate was equal to that of the ammonium salt. This rendered possible the direct production of the potassium salt without relying on the intermediate formation of ammonium persulphate and subsequent decomposition with potassium chloride. 

Possible point group decompositions in direct products ... 

Several studies are available addressing the mechanism and kinetics of the MSR reaction over Cu-based catalysts [115-121], There is agreement nowadays that C02 is a direct product of the MSR reaction and not of a sequence of methanol decomposition and WGS reactions. The main source of CO is the rWGS reaction taking place as a secondary reaction after MSR. Frank et al. [121] presented a... 

For a convenient decomposition of y(jc), we need two conditions on the region of interest of the input variables. First, x is assumed to be a direct product of one-dimensional regions, which we write as... 

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