G-density functional theory

See Yang, W. Parr R.G. Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989, for a full discussion. 

Most of the formal mathematical structure of conventional density functional theory transfers to g-density functional theories without change [1-5]. As in density functional theory, the key for practical implementations is the variational principle. 

It is clear from the preceding sections that the powerful A-representable constraints from the orbital representation do not extend to the spatial representation. This suggests reformulating the variational principle in g-density functional theory in the orbital representation. 

To motivate the form of orbital-based g-density functional theory, it is useful to start with the familiar case of 1-density functional theory, where the orbital representation is well established [55]. 

Nonetheless, Eq. (95) is perhaps the most natural generalization of the Kohn-Sham formulation to g-density functional theory. Indeed, Ziesche s first papers on 2-density functional theory feature an algorithm based on Eq. (95), although he did not write his equations in the potential functional formulation [1, 4]. The early work of Gonis and co-workers [68, 69] is also of this form. 

Equation (95) reveals an interesting link between g-density functional theory and g-matrix functional theory. Consider rewriting Eq. (95) in a form analogous to Eq. (83),... 

It is fair to say that neither of these two approaches works especially well N-representability conditions in the spatial representation are virtually unknown and the orbital-resolved computational methods are promising, but untested. It is interesting to note that one of the most common computational algorithms (cf. Eq. (96)) can be viewed as a density-matrix optimization, although most authors consider only a weak A -representability constraint on the occupation numbers of the g-matrix [1, 4, 69]. Additional A-representability constraints could, of course, be added, but it seems unlikely that the resulting g-density functional theory approach would be more efficient than direct methods based on semidefinite programming [33, 35-37]. 

Chattaraj, P. K., Parr, R. G. Density Functional Theory of Chemical Hardness. Vol. 80,... 

The series of 10 chapters that constitute Part 3 of the book deals mainly with the use of adsorption as a means of characterizing carbons. Thus, the first three chapters in this section complement each other in the use of gas-solid or liquid-solid adsorption to characterize the porous texture and/or the surface chemistry of carbons. Porous texture characterization based on gas adsorption is addressed in Chapter 11 in a very comprehensive manner and includes a description of a number of classical and advanced tools (e.g., density functional theory and Monte Carlo simulations) for the characterization of porosity in carbons. Chapter 12 illustrates the use of adsorption at the liquid-solid interface as a means to characterize both pore texture and surface chemistry. The authon propose these methods (calorimetry, adsorption from solution) to characterize carbons for use in such processes as liquid purification or liquid-solid heterogeneous catalysis, for example. Next, the surface chemical characterization of carbons is comprehensively treated in Chapter 13, which discusses topics such as hydrophilicity and functional groups in carbon as well as the amphoteric characteristics and electrokinetic phenomena on carbon surfaces. 

Kohn, W., Becke, A.D. and Parr, R.G. Density functional theory of electronic stracture, J. Phys. Chem., 100 (1996), 12974-12980. 

Cramer, C. J. Truhlar, D. G. Density functional theory for transition metals and transition metal chemistry. Phys. Chem. Chem. Phys. 2009,11, 10757-10816. 

If the necessary computational resources were available, the direct simulation of PI triplet structure factors would be the best way to obtain accurate results. This route would also provide a means to fix C3 functions viaEq. (134) for CM3 (exact), and ET3 (approximate), which could be utilized in other contexts (e.g., density functional theories). However, this PI route does not seem feasible today if one wishes to obtain complete descriptions of the functions, and the use of seems... 

The hexadecapeptide of this article was treated within the context of the ab initio Hartree-Fock approximation. However, the concept of extracting kernel matrices from fragments smaller than a full molecule would be applicable within the context of any method based upon a molecular orbital representation, e.g., density functional theory. 

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