Parr formulation

Chattaraj, Cedillo, and Parr formulated the variational principle for the hardness. The function i) that minimizes the functional J/LgI = //g( 7(. g( )drdr subject to the condition that J i)dr= 1 is the Fukui function /(/), and the value of the functional at the minimum, i.e., j/[4, is the chemical hardness of the system. Recently, Ayers and Parr used the basic DFT variational principles in the development of variational principles for Important DFT-based descriptors of chemical reactivity, the Fukui function and the local softness. ... 

The next step is the identification of the concept of chemical hardness, 17, with the second derivative of the energy with respect to the number of electrons, formulated by Parr and Pearson [14]... 

These relations highlight the fact that the formalism of DFT-based chemical reactivity built by Parr and coworkers, captures the essence of the pre DFT formulation of reactivity under frontier molecular orbital theory (FMO). Berkowitz showed that similar to FMO, DFT could also explain the orientation or stereoselectivity of a reaction [12]. In addition, DFT-based reactivity parameters are augmented by more global terms expressed in the softness. 

Just like Sanderson s electronegativity equalization principle, the Hard and Soft Acids and Bases principle was originally introduced without strong theoretical basis. Nevertheless, it was used widely from its formulation on. The principle states that hard acids prefer to coordinate with hard bases and soft acids with soft bases [82], In 1983, Parr and Pearson provided a definition for the chemical hardness [25]... 

Parr and collaborators [8-12] showed how Fukui s frontier-orbital concept could be grounded in a rigorous many-electron theory, density-functional theory (DFT) [8,16-18], They used the ensemble formulation of DFT to introduce the expectation value Jf of the total electron number as a continuous variable. They then defined the Fukui functions... 

Nalewajski [19-21] formulated a theory of static reactivity kernels prior to but closely related to that of Berkowitz and Parr [15] which started out from a second-order perturbative treatment of the total energy. He identified an external static potential of unspecified origin with the chemical stimulus and made the first connection between softness kernels and the total energy. His result is, in our notation,... 

Cell suspension tests can Indicate tdilch type of surfactant could be potentially incompatible after long periods of storage. Three basic types of surfactants exist anionic, cationic, and non-ionic. Parr (27) surmised that anionic surfactants are the least phytotoxic, but the results of Ernst (28) do not confirm that all anionic surfactants are harmless. Surfactants with short, aliphatic, carbon chains may also be more toxic than those with larger molecular groups (29). The surfactant s influence iii vitro may exaggerate the effect in dry, wettable powder formulations, and thus provide an accelerated test of compatablllty. 

Reports about the fate and behavior of any pesticide applied In different kinds of formulations to plants are rather limited. This Is true even more so for the other constituents of formulations Including surfactants. McWhorter (j ) has estimated that considerable quantities of surfactants are applied to plants with pesticides. The general assumption Is that these constituents present lesser environmental hazards than the pesticides. Parr ( ) has discussed some aspects of surfactant toxicology. 

Peart, J. Orban, J.C. McGlynn, P. Redmon, M.P. Sargeant, C.M. Byron, P.R. MDI electrostatics valve and formulation interactions that really make a difference. In Respiratory Drug Delivery VIII, Dalby, R.N., Byron, P.R., Peart, J., Parr, S.J., Eds. Davis Horwood International Raleigh, NC, 2002 223-230. 

The essential feature of the o—n separation is that an effective Hamiltonian can be defined for the n electrons in the field of the nuclei and the a core. As was pointed out by Sinanoglu 7>, this separation can be derived under conditions more general than the Lykos-Parr assumption. A slightly different formulation of the o—n separation can be obtained by the methods of second quantization 8>. 

P.C. Evangelista, C. Parr, and C. Revais, Control of formulation and optimization of self-flow castables based on pure calcium aluminates, Refract. Appl. News, 7(2) 14-18 (2002). 

The pyrido[l,2-a]pyrazinium cation has been compared with the quinolizinium ion and the other monoazaquinolizinium cations, using the Pariser-Parr-Pople method to calculate electronic structures and absorption spectra. The ultraviolet spectrum of pyrido[l,2-a]pyrazinium bromide in water shows maxima at 232, 276, 287, 322, and 336 nm. Ultraviolet spectra have been recorded for various substituted compounds and their iV-oxides. The bands at 1685 and 1655 cm in the infrared spectrum (KBr disc) of the oxo compound 5 (R = H) confirm its formulation as the 0X0 form rather than as the corresponding hydroxy tautomer. Similarly the dioxo derivative 6 exists as such, in preference to the hydroxy form 5 (R = OH). The presence of a two-proton singlet at 6 6.1 in the PMR spectrum (in trifluoroacetic acid) of compound 6 is strong evidence in favor of this formulation. Ultraviolet spectral data are consistent with structures 6 and 5 (R = H). ... 

The FF data carry the same information as local softnesses, which therefore should also serve as valid quantitative measures for chemical reactivity, e.g., in metals. As shown by Yang and Parr [11,25] in the ensemble formulation of the DFT the local softness s(f) also measures the local fluctuations in the electronic density p(f), as it varies with f. This has direct implications for catalytic activity on transition metals, for which fluctuations in charge were shown to be very important [66],... 

A MODIFIED PARISER-PARR-POPLE FORMULATION FOR THE STUDY OF LOW LYING EXCITATION ENERGIES OF POLYENES... 

A modified Pariser-Parr-Pople formulation is described as a simpler approach to the study of polyene low lying excitation energies. The result of its application to polyenes of up to 20 carbon atoms is presented and compared with previous result obtained with double excitation configuration interaction. It is found that the proper ordering of the 2 Ag and i Bu excitation energies as well as other important characteristics revealed in previous studies are reproduced in this calculation. 

The conventional formulation of Pariser-Parr-Popple (PPP) for the ir electron systems leads to the standard one-electron Hamiltonian represented by the following matrix elements... 

Having a viable EN quantum formulation, its natural companion named chemical hardness, /, can be immediately introduced (Parr Pearson, 1983)... 

Given the form of the linear response kernel of Eq. (4.198) (Ayers, 2001 Sablon et al., 2010 Yang et al., 2012) one has the problem to formulate the local form of the softness kernel 5(r,r ) fulfilling the Berkowitz-Ghosh-Parr relationship (4.197) (Berkowitz et al., 1985 Berkowitz Parr, 1988) within conceptual DFT. To this aim one starts with rearranging the Eq. (4.198) as an integral form along the chemical reaction path (Putz Chattaraj, 2013)... 

Many years after, a new emerging form of quantum mechanics, the DFT, appears as the modem quantum frame in which a chemical system (an atom, an ion, a radical, a molecule or several molecules) can be treated in a state of interaction (Parr Yang, 1989). In this modem context, the cornerstone EN definition of Parr as the minus of the chemical potential ( ) of a system in a grand canonical ensemble at zero temperature (7) was formulated (in atomic units), see (Parretal., 1978),j = -/r asinEq. (3.1), when the ground state energy E is assumed to be a smooth function of the total number of electrons N. 

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