Universal variation

Levy, M., 1979, Universal Variational Functionals of Electron Densities, First Order Density Matrices, and Natural Spin Orbitals and Solution of the v-Representability Problem , Proc. Natl. Acad. Sci. USA, 16, 6062. 

In 1979, an elegant proof of the existence was provided by Levy [10]. He demonstrated that the universal variational functional for the electron-electron repulsion energy of an A -representable trial 1-RDM can be obtained by searching all antisymmetric wavefunctions that yield a fixed D. It was shown that the functional does not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus the v-representability is not required, only Al-representability. As a result, the 1-RDM functional theories of preceding works were unified. A year later, Valone [19] extended Levy s pure-state constrained search to include all ensemble representable 1-RDMs. He demonstrated that no new constraints are needed in the occupation-number variation of the energy functional. Diverse con-strained-search density functionals by Lieb [20, 21] also afforded insight into this issue. He proved independently that the constrained minimizations exist. 

M. Levy, Universal variational functionals of electron-densities, Ist-order density-matrices, and natural spin-orbitals and solution of the v-representability problem. Pmc. Natl. Acad. Sci. U.S.A. 76(12), 6062-6065 (1979). 

According to the actual formulation of the Hohenberg-Kohn theorem 44], there exists a universal variational functional F(fi) of trial electron densities p, such that if o is the ground-state energy and Po(r) is the ground-state electron density that belongs to the given external potential V(r) specified in Eq. 15, then. 

W. Kohn and L. H. Sham, Phys. Rev. A, 140, 1133 (1965). Self-Consistent Equations Including Exchange and Correlation Effects. M. Levy, Proc. Natl. Acad. Sci. U.S.A., 76, 6062 (1979). Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the v-Representability Problem. R. G. Parr and W. Yang, Eds., Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1988. J. Labanowski and J. Andzelm, Eds., Density Functional Methods in Chemistry, Springer Verlag, Heidelberg, 1991. L.J. Bartolotti and K. Flurchick, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1995, Vol. 7, pp. 187-216. An Introduction to Density Functional Theory. 

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

C. Lanczos, The Variational Principles of Mechanics , University of Toronto Press, Toronto, 1970... 

Variational methods - theoretically the variational approach offers the most powerful procedure for the generation of a computational grid subject to a multiplicity of constraints such as smoothness, uniformity, adaptivity, etc. which cannot be achieved using the simpler algebraic or differential techniques. However, the development of practical variational mesh generation techniques is complicated and a universally applicable procedure is not yet available. 

UFF (universal force field) a molecular mechanics force field unrestricted (spin unrestricted) calculation in which particles of different spins are described by different spatial functions VTST (variational transition state theory) method for predicting rate constants... 

Because variations in accurate isotope ratio measurements typically concern only a few parts per 1000 by mass and there are no universal absolute ratios, it is necessary to define some standards. For this purpose, samples of standard substances are produced and made available at two major centers IAEA (International Atomic Energy Authority, U.K.) and NIST (National Institute for Standards and Technology, U.S.). Standards from other sources are also available. These primary standards can be used as such, or alternative standards can be employed if the primary ones are not available. However, any alternative standards need to be related accurately to the primary ones (see formulae below). For example, the material PDB (PeeDee belemnite), used particularly as a standard for the ratio of isotopes, is no longer readily available, and a new standard, VPDB,... 

Whereas there is no universally accepted specification for marketed natural gas, standards addressed in the United States are Hsted in Table 6 (8). In addition to these specifications, the combustion behavior of natural gases is frequently characteri2ed by several parameters that aid in assessing the influence of compositional variations on the performance of a gas burner or burner configuration. The parameters of flash-back and blow-off limits help to define the operational limits of a burner with respect to flow rates. The yeUow-tip index helps to define the conditions under which components of the natural gas do not undergo complete combustion, and the characteristic blue flame of natural gas burners begins to show yellow at the flame tip. These... 

Figure 2 Variation of the gas transfer veloeity with wind speed. The units of transfer veloeity are equivalent to the number of em of the overlying air eolumn entering the water per hour (Taken from Bigg," with permission of Cambridge University Press)...

Part I of the paper develops an exact variational principle for the ground-state energy, in which the density (r) is the variable function (i.e. the one allowed to vary). The authors introduce a universal functional F[n(r)] which applies to all electronic systems in their ground states no matter what the external potential is. This functional is used to develop a variational principle. 

Numerous individual rotary bit designs are available from a number of manufacturers. All of them are designed to give optimum performance in various formation types. There is no universal agreement on this subject variations in operating practices, type of equipment used or hole conditions require an experimental approach. It has been noted in development drilling that those operators who consistently drill the fastest wells usually employ several types of bits. 

Jackshafts Some machine-trains use an extended or spacer shaft, called a jackshaft, to connect the driver and a driven unit. This type of shaft may use any combination of flexible coupling, universal joint, or splined coupling to provide the flexibility required making the connection. Typically, this type of intermediate drive is used either to absorb torsional variations during speed changes or to accommodate misalignment between the two machine-train components. 

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