For ensemble

In the mid 1970s, Ugi and co-workers developed a scheme based on treating reactions by means of matrices - reaction (R-) matrices [16, 17]. The representation of chemical structures by bond and electron (BE-) matrices was presented in Section 2.4. BE-matrices can be constructed not only for single molecules but also for ensembles of them, such as the starting materials of a reaction, e.g., formaldehyde (methanal) and hydrocyanic add as shown with the B E-matrix, B, in Figure 3-12. Figure 3-12 also shows the BE-matrix, E, of the reaction product, the cyanohydrin of formaldehyde. 

Figure 1 Convergence of the total energy and of the Hellmann-Feynman forces for ensembles of paramagnetic Fe atoms with 4 to 32 atoms. Part (a) shows the results of non-selfconsistent calculations performed with a fixed potential, part (b) the results of selfconsistent calculations. Full lines represent the RMM-DIIS (iterative diagonal-ization) results, broken lines the CGa (total-energy minimization) calculations. (4. text.

Fig. 2. Schematic representation of selected hydrocarbon chains in the crystalline and the amorphous regions of polyethylene, respectively. For ensembles of chains uniformly distributed around the dashed lines the deuterons from planar and conical distributions as indicated...

Gross, E. K. U., Oliveira, L. N., Kohn, W., 1988a, Rayleigh-Ritz Variational Principle for Ensembles of Fractionally Occupied States , Phys. Rev. A, 37, 2805. 

The first terms of the power series obtained by the multipole expansion of the Coulomb intermolecular potential account for dipole-dipole interactions prevailing in systems of polar molecules. As an adequate approximation for ensembles of... 

The sizes of the dendrimers have been determined by calculating the molecular volumes, as defined by the van der Waals radii of the atoms, and by calculating the radii of gyration for several configurations of the dendrimers, as obtained from a molecular dynamics simulation at room temperature. The solvent influence on the calculated radii was estimated by scaling the nonbonded interactions between the atoms. Molecular volumes and average radii for ensembles of 500 conformations of the BAB-dendr-(NH2)D dendrimers have been collected in Table 26.2. 

To prove inequality s/y 1 (for ensembles with well-separated constants, with... 

For ensemble of ions in vacuum we have the following, in accordance with the definition ... 

Nagy, A. (1995). Coordinate scaling and adiabatic connection formula for ensembles of fractionally occupied excited states, Ini. J. Quantum Chem. 56, 225-228. 

It has been suggested that weak interactions could be responsible for driving racemates towards homochirality via a deterministic process. However, it is difficult to deduce conclusions regarding the role played by these forces in chemical reactions for ensembles of molecules, since they induce a chiral bias of only 106 molecules per mole six orders of magnitude lower than the stochastic fluctuations present in a racemate. 

The basis for the formulation of BB-matrices for ensembles of molecules is an extension of the concept of isomerism to ensembles of molecules The cyanohydrin of formaldehyde 1 is isomeric with the ensemble of its components formaldehyde 2 and hydrocyanic acid 3. Both 1 and 2 + 3 have the same set of atoms and the same number of valence electrons. 

For ensembles of particles the light extinction is additive, as discussed previously. However, in certain cases simplifying assumptions can be made. For example, suppose there is a polydisperse aerosol having n d) particles of diameter d per unit volume. From Eq. 16.11... 

Simulation calculations on finite systems will entail some error associated with the submacroscopic size considered (Lebowitz and Percus, 1961a b 1963). For example, periodic boundary conditions will influence molecular correlations to some extent (Pratt and Haan, 1981a b). Support of a claim of accuracy would typically involve some practical investigation of the thermodynamic limit. A claim of preference for calculations in one ensemble over another is typically made first on the basis of convenience rather than on the basis of accuracy defined in some absolute way. Thus, advantages of practical accuracy for ensemble-specialized alternatives to Eq. (3.18) are not proven typically, and they are not necessary fundamentally. 

This is our generalization of the Hohenberg-Kohn theorem for ensembles of states characterized by a density matrix D. 

The density functional theory for ensembles is based on the generalized Rayleigh-Ritz variational principle [7]. The eigenvalue problem of the Hamiltonian H is given by... 

There are a number of model exchange-correlation functionals for the ground-state. How do they perform for ensemble states Recently, several local density functional approximations have been tested [24]. The Gunnarsson-Lundqvist-Wilkins (GLW) [26], the von Barth-Hedin (VBH)[25] and Ceperley-Alder [27] local density approximations parametrized by Perdew and Zunger [28] and Vosko, Wilk and Nusair (VWN) [29] are applied to calculate the first excitation energies of atoms. 

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