The Variational Theory

The free energy P has six contributions [1, 13] Fi,F2,F2,F4,Fs, and Fg related, respectively, to (i) entropy of mobility of the adsorbed counterions and coions along the polymer backbone, (ii) translational entropy of the unadsorbed counterions and coions (including salt ions) that are free to move within the volume Q, (hi) 

The entropic contribution arising from the various distributions of the adsorbed counterions and coions is determined as follows. We note that for the general case of both mono- and divalent salts being present, there are N monomers. Mi adsorbed monovalent counterions (Na ), M2—M3 adsorbed divalent counterions (Ba ) with no coion (Cl ) condensation, and M3 ion-triplets ( monomer-Ba -Cl ) in the system. Therefore, N—Mi—M2 monomers remain with their bare charge uncompensated. Consequently, the partition function is 

To determine the translational entropy of the unadsorbed ions that are distributed in the bulk volume Q, we count mobile ions of various species as N—Mi + i + monovalent counterions (Na ), H2+—M2 divalent counterions (Ba ), and ni + + 2tt2 + — M3 monovalent colons (Cl ). Therefore, the partition function related 

Using F2 = —keTln Z2 and after some calculations we arrive at 

The free energy contribution from the correlations of all dissociated ions is given by the Debye-Hiickel electrostatic free energy. 

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Equation (ASA. 110) represents the canonical fonn T= constant) of the variational theory. Minimization at constant energy yields the analogous microcanonical version. It is clear that, in general, this is only an approximation to the general theory, although this point has sometimes been overlooked. One may also define a free energy... 

We also adopt the above combination rule (Eq. [6]) for the general case of exp-6 mixtures that include polar species. Moreover, in this case, we calculate the polar free energy contribution Afj using the effective hard sphere diameter creff of the variational theory. 

THL.30. I. Prigogine, Sur la theorie variationnelle des phenomenes irreversibles (On the variational theory of irreversible phenomena), Bull. Cl. Sci. Acad. Roy. Belg. 40, 471 83 (1954). 

Improving the foundation of manufacturing science in our current manufacturing practices should be the primary basis for moving away from the corrective action crisis to continuous improvement. Knowledge of the variation theory is, therefore, an essential element of manufacturing science. It requires an in-depth understanding of a process or system (15) ... 

Over the years, several computational methods have been developed. The variational theory can be used either without using experimental data other than the fundamental constants (i.e., ab initio methods) or by using empirical data to reduce the needed amount of numerical work (i.e., semiempirical data methods). There are various levels of sophistication in both ab initio [HF(IGLO), DFT GIAO-MP2, GIAO-CCSD(T)] and semiempirical methods. In the ab initio methods, various kinds of basic sets can be employed, while in the semiempirical methods, different choices of empirical parameters and parametric functions exist. The reader is referred to reviews of the subject.18,77... 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

The nonrelativistic Schrodinger theory is readily extended to systems of N interacting electrons. The variational theory of finite A-electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived. 

The Hohenberg-Kohn theory of /V-clcctron ground states is based on consideration of the spin-indexed density function. Much earlier in the development of quantum mechanics, Thomas-Fermi theory [402, 108] (TFT) was formulated as exactly such a density-dependent formalism, justified as a semiclassical statistical theory [231, 232], Since Hohenberg-Kohn theory establishes the existence of an exact universal functional Fs [p] for ground states, it apparently implies the existence of an exact ground-state Thomas-Fermi theory. The variational theory that might support such a conclusion is considered here. 

It should be noted that these equations are to be solved for each position of the centroid q. The frequency in Eq. (2.27) is the same as the effective frequency obtained for the optimized LHO reference system using the path-integral centroid density version of the Gibbs-Bogoliubov variational method [1, pp. 303-307 2, pp. 86-96], Correspondingly, Eqs. (2.27) and (2.28) are exactly the same as those in the quadratic effective potential theory [1,21-23], The derivation above does not make use of the variational principle but, instead, is the result of the vertex renormalization procedure. The diagrammatic analysis thus provides a method of systematic identification and evaluation of the corrections to the variational theory [3],... 

If we follow the style of the variational theory, the above discussions are equivalent to a process to obtain Euler s equation by taking a variation of the... 

D. Steigmann, E. Baesu, R.R. Rudd, J. Belak, and M. McElfresh, On the variational theory of cell-membrane equilibria. Interface Free Bound. 5, 357-366 (2003). 

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