Variational Formalism

SO that a is the electrostatic binding energy of the ion-pairs formed on the polymer backbone due to the adsorption of ions and Sa is the translational entropy of the adsorbed counterions along the backbone. 

Similarly, is divided into enthalpic and entropic contributions due to the excluded volume and electrostatic interactions and into entropic contributions because of small ions, solvent molecules, and the polyelectrolyte chain. Denoting these contributions by Ew) 1 i, Ss, and Sp, respectively, Ff is given by 

Explicit expressions for different constituents of Fp are presented in Table 6.2 in terms of densities and fields at the saddle point. 

For the sake of completeness, we present the procedure to obtain the variational free energy as presented in Ref [1] in the absence ofion-pair correlations. In Ref [l],ithas been assumed that the counterions from the polyelectrolyte are indistinguishable from the counterions from the salt. So, we start from a partition function similar to Eq. (6.124) with the solvent, counterions (from the polyelectrolyte and the salt). 

Although F inthe variational theory is the same as in SCFT (cf Eq. (6.128)), other contributions involving the free ions, the chain entropy, and so on (that means Ff) differ significantly in terms of computational details. In SCFT, Ff is computed after solving for fields experienced by different components in the system, which arise as a result of interactions of a particular component with the others. On the other hand, in variational calculations [1], a single polyelectrolyte chain, whose monomers interact with the excluded volume and the electrostatic interactions in the presence of the small ions is approximated by an effective Gaussian chain, whose conformational statistics depend on the different kinds of interactions in the system. To compute the equilibrium free energy, its variational ansatz is minimized with respect to the 

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In writing the Lagrangean density of quantum mechanics in the modulus-phase representation, Eq. (140), one notices a striking similarity between this Lagrangean density and that of potential fluid dynamics (fluid dynamics without vorticity) as represented in the work of Seliger and Whitham [325]. We recall briefly some parts of their work that are relevant, and then discuss the connections with quantum mechanics. The connection between fluid dynamics and quantum mechanics of an electron was already discussed by Madelung [326] and in Holland s book [324]. However, the discussion by Madelung refers to the equations only and does not address the variational formalism which we discuss here. 

Variational principles have turned out to be of great practical use in modem theory. They often provide a compact and general statement of theory, invariant or covariant under transformations of coordinates or functions, and can be used to formulate internally consistent computational algorithms. Symmetry properties are often most easily derived in a variational formalism. 

The variational formalism makes it possible to postulate a relativistic Lagrangian that is Lorentz invariant and reduces to Newtonian mechanics in the classical limit. Introducing a parameter m, the proper mass of a particle, or mass as measured in its own instantaneous rest frame, the Lagrangian for a free particle can be postulated to have the invariant form A = mulxiilx = — mc2. The canonical momentum is pf, = iiiuj, and the Lagrangian equation of motion is... 

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. 

Ion channel studies motivated Allen et al. [47] who have developed an elegant variational formalism to compute polarization charges induced on dielectric interfaces. They solved the variational problem with a steepest descent method and applied their formulation in molecular dynamics (MD) simulations of water permeation through nanopores in a polarizable membrane [48-50], Note that the functional chosen by Allen et al. [47] is not the only formalism that can be used. Polarization free energy functionals [51-53] are more appropriate for dynamical problems, such as macromolecule conformational changes and solvation [54-57],... 

For any orbital-functional model, an optimal effective (local) potential (OEP) can be constructed following a well-defined variational formalism [24,25]. If a Frechet derivative existed for the exchange-correlation energy E,lc for ground states, it would be obtained in an OEP calculation, while the minimum energy and corresponding reference state would coincide with OFT results. Thus numerically accurate OEP calculations test the locality hypothesis. 

Continues the quantum mechanical path integral modeling with the specialized Fe5mman-Kleinert variational formalism leading to a comprehensive understanding of atomic stability ... 

Comparison of Theories SCFT and Variational Formalism 325 Table 6.2 Comparison of contributions to Ff in SCFT and variational formalism. 

In this work, we have ignored one-loop corrections to the free energy within SCFT. However, one-loop corrections to the free energy coming from the density fluctuations of small ions, within the variational formalism, is given by... 

Figure 6.26 Comparison of SCFT and the variational formalism (with one-loop corrections) to illustrate the effect of correlations among small ions on the effective degree of ionization (/ ). Zp = -Zc = -1, R/l= 10, N = 100,= 0.1 M,Xps = 0.45, and 6 = 3.

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