Grand canonical ensemble, density functional

The electronic ground-state energy /i TV, v for a given average number of electrons N and the external potential v of the BO approximation is also the functional of the grand-canonical ensemble density [123] ... 

The Fukui function is primarily associated with the response of the density function of a system to a change in number of electrons (N) under the constraint of a constant external potential [v(r)]. To probe the more global reactivity, indicators in the grand canonical ensemble are often obtained by replacing derivatives with respect to N, by derivatives with respect to the chemical potential /x. As a consequence, in the grand canonical ensemble, the local softness sir) replaces the Fukui function/(r). Both quantities are thus mutually related and can be written as follows ... 

Computationally, polydispersity is best handled within a grand canonical (GCE) or semi-grand canonical ensemble in which the density distribution p(a) is controlled by a conjugate chemical potential distribution p(cr). Use of such an ensemble is attractive because it allows p(a) to fluctuate as a whole, thereby sampling many different realizations of the disorder and hence reducing finite-size effects. Within such a framework, the case of variable polydispersity is considerably easier to tackle than fixed polydispersity The phase behavior is simply obtained as a function of the width of the prescribed p(cr) distribution. Perhaps for this reason, most simulation studies of phase behavior in polydisperse systems have focused on the variable case [90, 101-103]. 

Recently Haymet and Oxtoby and Klupsch ° independently develojjed related density functional theories of the liquid-solid interface. These are statistical mechanical theories that work with the grand canonical ensemble free energy Cl, which is a functional of the one-particle density p r),... 

N and /z have their usual meaning, or to the case where E is the electronic energy, N the number of electrons and p the electronic chemical potential. In either case Mermin showed that the grand potential is a unique functional of the density for a system at finite temperature. Also, the correct density for the system will give a minimum value of Q. Thus we have a DFT for finite temperature by taking a grand canonical ensemble of the system of interest and calculating its properties. 

In the density functional theoiy (DFT) the statistical mechanical grand canonical ensemble is utilized. The appropriate free energy quantity is the grand Helmholtz free energy, or grand potential functional, 2(r. This free energy functional is expressed in terms of the density... 

One can prove that in thermal equilibrium, in a grand canonical ensemble (i.e., volume, chemical potential, and temperature are fixed), the grand canonical free energy Q(p(r)) of a system can be written as a functional of the one-body density p(r) alone, which will depend on the position r in inhomogeneous systems. The density distribution peq(r) which minimizes the grand potential functional is the equilibrium density distribution. This statement is the basis of the equilibrium density functional theory (DFT) for classical fluids which has been used with great... 

Lynch GC, Pettitt BM (1997) Grand canonical ensemble molecular dynamics simulations Reformulation of extended system dynamics approaches. J Chem Phys 107 8594-8610 Madura JD, Pettitt BM, Calef DF (1988) Water under high pressure. Mol Phys 64 325 Mahoney MW, Jorgensen WL (2000) A five-site model for liquid water and the reproduction of the density anomaly by rigid, nonpolarizable potential functions. J ChemPhys 112 8910-8922 March RP, Eyring H (1964) Application of significant stmcture theory to water. J Phys Chem 68 221-228 Martin MG, Chen B, Siepman JI (1998) A novel Monte Carlo algorithm for polarizable force fields. 

Theoretical studies, like first-principles calculations, grand canonical ensemble Monte Carlo (GCMC) simulations, second order Moller-Plesset perturbation theory (MP2) calculations and density functional theory (DFT) calculations, have been utilized to investigate optimal structures and their properties. Combined experimental and theoretical data provide a window to the plan of design of these network structures and lead to a new direction to investigate porous networks. 

In PI work on homogeneous and isotropic fluids one utilizes the three basic statistical ensembles canonical, isothermal-isobaric, and grand canonical. In the canonical ensemble the partition function is given by the trace (Tr) of the density matrix operator... 

As a last application of ensemble theory to the quantum mechanical ideal gas, we obtain the equation of state for fermions and bosons. To this end, the most convenient approach is to use the grand canonical partition function and the momentum representation, in which the matrix elements of the density are diagonal. This gives for the canonical partition function... 

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