Gordon decomposition

Jerry A. Boatz (North Dakota State University) and Mark S. Gordon Decomposition of Normal Coordinate Vibrational Frequencies. ... 

The starting point for this generalization is the Gordon decomposition, in which the total current is split into the paramagnetic (orbital) component yp, a gauge term proportional to the scalar density Ps, and the curl of the magnetization density m. 

On the other hand, it should be pointed out that the relativistic current density can be recast in a form similar to that of the non-relativistic one through the Gordon decomposition [61,38]. Also, the non-relativistic current density depends explicitly on the external vector potential, whereas the relativistic one appears not to. However, also for the relativistic current density an explicit dependence can be brought out by considering the modification of the coupling of large and small components induced by the external vector potential. Finite basis set calculations rely on the kinetic balance relation [62,63]... 

It is impossible to develop a current-dependent relativistic exchange-correlation functional, which is computationally tractable and reduces to spin-density functional theory in the weakly relativistic limit. One reason is that there is no local approximation to such a functional since j vanishes for any homogeneous system. This means that the relativistic electron gas cannot serve as a starting point. More insight is gained from a Gordon decomposition of the current density j (see e.g. Refs. [7, 24]), which shows that j consists of an orbital part and the curl of a magnetisation density fh. 

The relation to the spin density can be made more explicit by invoking a Gordon decomposition of the current density to produce expressions for charge- and spin-related currents [392,397]. Although we have already encountered the Gordon decomposition for the 4-current in section 8.8.1, Appendix F considers explicitly the decomposition of its spatial components, that is, of the current density, in standard notation. From Appendix F, we take the result for the many-electron case,... 

Since the Gordon decomposition is required for the spatial components of the 4-current in section 8.8, we explicitly derive it for the current density in standard notation here (and follow the derivation presented in the appendix of Ref. [394], which bears some similarities to the derivations around Eqs. (5.39) and (5.40) in chapter 5). 

In practice, one typically employs so-called relativistic SDFT (R-SDFT), in which a Gordon decomposition of the four-current is performed to separate orbital and spin degrees of freedom, and only the spin magnetization is maintained as a fundamental variable. This reduced RDFT is not Lorentz-invariant (which is not essential in solid-state physics and quantum chemistry, where a preferred frame is provided by the laboratory) and does not account for orbital magnetism (except, possibly, induced by spin-orbit coupling, which can be treated as a perturbation). Relativistic CDFT (R-CDFT) can overcome both limitations, but is less frequently used since it is more complicated to implement, and less is known about approximate four-current functionals. 

F. E. Btinckman and G. Gordon, Proceedings of the International Symposium on Decomposition of Organometallic Compounds to Eefractoy Ceramics, Metals, and Metal Alloys, Dayton, Ohio, 1967. 

Adam, L.C. Gordon, G. (1999) Hypochlorite ion decomposition effects of temperature, ionic strength, and chloride ion. Inorganic Chemistry, 38, 1299-1304. 

Tomiyasu, H Fukutomi, H Gordon, G. Kinetics and mechanism of ozone decomposition in basic aqueous soiution. inorganic Chemistry, 1985 24 (19), 2962-2966. 

Gordon and Campbell [86], for instance, examined the exothermic decomposition of potassium perchlorate mixtures with carbon within the temperature range 300-360°C, while Grodzinski [87] studied the thermal decomposition of mixtures of various combustibles with potassium perchlorate. 

This most important reaction—the oxidation of hydrazine—has not yet been investigated fully. Work on the subject has consisted mainly of Studies of the kinetics of the process in dilute aqueous solutions. Gordon [51], studying the kinetics of decomposition of hydrazine and hydrogen peroxide, found that the reaction rate depends to a great extent on the pH. Its peak value is reached at pH = 10-11. 

The exothermic decomposition of the mixtures of potassium perchlorate with charcoal at 300-360°C was studied by Gordon and Campbell [19]. 

Tomiyasu H, Fukutomi H, Gordon G (1985) Kinetics and Mechanisms of Ozone Decomposition in Basic Aqueous Solutions, Inorganic Chemistry 24 2962-2985. 

In the interpretation based on reactions (13)-(22) the observed increase in C2H6/C4Hio is due to combination of (thermalized) ethyl radicals with H atoms. The concentration of H atoms increases at lower pressures as a result of increased decomposition of the hot ethyl radicals and as a result the ratio C2H6/C4Hi0 also increases. It is of interest that Bradley et al. had indications that some propane was formed, which, if true, would suggest that C2H6 and H did interact to some extent at least. With improved analytical techniques Heller and Gordon (56) have been able to measure the small amounts of propane formed under their experimental conditions. 

In other studies it was found that a maximum of HOC1 decomposition exists at pH = 6.89. For the third-order reaction (7.14), catalytic activity of the chloride ion was suggested for hypochlorite decomposition and the stabilising effect of higher pH was quantified in the pH range 9-14 (Adam and Gordon 1999). 

The thermal decomposition of magnesium hydroxide was studied by Gordon and Kingery [18]. Their results, given in Figure 5.12, show... 

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