Local energy minimum principle

Local energy minimum principle. Li, Nunes, and Vanderbilt formulated the first density matrix method. An equivalent method was also proposed by Daw at the same time. The method is based on the McWeeny transformation ... 

This local energy minimum principle has been successfully tested and applied to tight-binding calculations where the one-electron Hamiltonian is independent of the density matrix, and also to self-consistent DFT calculations. ... 

Local energy minimum principle. The multi-minimum problem was overcome by a energy functional of Kim, Mauri, and Galli, who made the generalization to allow the number of the localized orbitals to exceed the number of occupied states. The functional is defined for an V-electron system described by a nonself-consistent Hamiltonian h and a chemical potential p. In terms of a set of M (possibly linearly dependent) orbitals i =, ...,M, where M > N/I, it is... 

The MEP in resonance states of many-electron atoms (molecules) complicates things not only computationally but also conceptually and formally. For example, the resonance state is in the continuous spectrum, with an infinity of lower states of the same symmetry. Its localized part, and its energy Eg, do not obey rigorously a variational minimum principle to all orders. Instead, in a variational calculation, what one expects is a correct convergence to a local energy minimum, secured by the anticipated localization of the state. In addition, there are complications from possible... 

An intermediate Is a molecule or ion that represents a localized energy minimum—an energy barrier must be overcome before the intermediate forms something more stable. An intermediate can In principle be isolated (although in practice its high energy can make this difficult). 

Viewed in terms of quantum computational principles, lipid selectivity is the interaction of molecules embedded in a many-dimensional (Hilbert) state space. In this view, geometry-dependent orbital surfaces are not 3D manifolds but sets of fluctuating hypersurfaces, each with an attached probability value, existing in linear superposition [33,3], The molecular interactions are thus a massively parallel search for a local energy minimum. These features would appear to fulfill the requirements for a quantum computing system. In terms of Atmar s... 

Absolute energy minimum principle. Yang developed two absolute energy minimum principles for first-principle linear-scaling electronic structure calculations. One is with a normalization constraint and the other without any constraint. The density matrix is represented by a set of nonorthogonal localized orbitals [i/] and an auxiliary matrix X which at the minimum becomes the (l)-inverse of the overlap matrix S of the localized orbitals. The number of localized orbitals is allowed to exceed the number of occupied orbitals. 

The Poynting theorem for the true vacuum can be developed as in Eqs. (258)-(262). The true vacuum energy (355) comes from the vacuum current in Eq. (350), which is transformed into a matter current by a minimal prescription as discussed already. This matter current in principle provides an electromotive force in a circuit. It is to be noted that the local Higgs maximum occurs at A = 0 [46], so the local Higgs minimum occurs below the zero value of A. 

According to recent investigations into the structure of OAp, there exists a Unear chain of 0 ions paraUel to the c-axis, with each one followed by a vacancy (Alberius Helming et al., 1999) (Figure 10.11b). Calculations using density-functional theory with local-density approximation (DFT-LDA) and first-principles pseudopotentials (Calderin et al., 2003) postulated the existence of a hexagonal c-empty structure Caio(P04)6D2 vvith a stable total-energy minimum. Hence, the defect apatite structure appears to be insensitive to the choice of anion (OH , F , ) in the c-axis column that is, if these are removed completely, the structure... 

A classical mechanical system at equilibrium is at rest in a (local) minimum on the PES and will have a potential energy given by the minimum value, E ". A (real) quantum system will not, in general, be able to attain such an absolute potential energy minimum. This is due to the Heisenberg uncertainty principle, which prescribes that the uncertainty in the position of a particle. Ax, is related to the uncertainty in the momentum, Ap, of the particle ... 

In the early sixties, it was shown by Roothaan [ 1 ] and Lowdin [2] that the symmetry adapted solution of the Hartree-Fock equations (i.e. belonging to an irreducible representation of the symmetry group of the Hamiltonian) corresponds to a specific extreme value of the total energy. A basic fact is to know whether this value is associated with the global minimum or a local minimum, maximum or even a saddle point of the energy. Thus, in principle, there may be some symmetry breaking solutions whose energy is lower than that of a symmetry adapted solution. 

The conclusion that the local hardness is given entirely by the variable parts of the kinetic energy is very logical. It is the kinetic energy increase which limits the distribution of electron density in all systems with fixed nuclei. Since the equilibrium state of atoms and molecules is characterized by minimum energy, they will also be marked by maximum kinetic energy because of the virial theorem. This will put them in agreement with the principles of maximum hardness, for which much evidence exists. 

The pathway characterized with the lower energy barrier is expected to be the preferred reaction channel, especially when the addition leads to the same product. Following the local HSAB principle, one has to look at the softness matching criteria, and the minimum of sAi —. BJ and sAj —. B will determine the preferred site of attack. 

These two paths are normally associated with different barrier heights introducing, thus, a regio-selectivity in the cycloadditive process. The path associated with the lower energy barrier should be preferred, and the corresponding cycloadduct will be dominant. Now, direct application of HSAB at the local level is not possible here, because it has to be satisfied for both the termini simultaneously. A softness matching criteria, thus, needs to be defined for the multisite interaction that measures the extent of the fulfillment of local HSAB principle. A quantity (A.v) can, thus, be defined to measure the softness matching criteria for the two paths in a least square sense, and the minimum value of this quantity should be preferable [27] ... 

In principle, geometry optimization carried out in the absence of symmetry, i.e., in Ci symmetry, must result in a local minimum. On the other hand, imposition of symmetry may result in a geometry which is not a local minimum. For example, optimization of ammonia constrained to a planar trigonal geometry (Dbi, symmetry) will result in a geometry which is an energy maximum in one dimension. Indeed,... 

A second relation involving a can in principle be derived when we know the current value of the dark energy equation of state parameter wbe- Indeed, since the potential does not possess a local minimum, the field never stops, so that it never behaves exactly as a cosmological constant. Moreover, even if its equation of state parameter w decays (without ever reaching) toward —1, the rate at which this transition occurs depends on the steepness of the potential the steeper the potential, the slowest w goes toward —1. In particular, one finds, at the epoch Qq 0.7,... 

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