Energy minimum principle

The fact that each of the two variables, S and U may be expressed as a function of the other, indicates that the extremum principle could likewise be stated in terms of either entropy or energy. The alternative to the maximum entropy principle is a minimum energy principle, valid at constant entropy, as graphically illustrated for a one-component system in figure 1, below. 

In the sequence of orbital energies shown above the 4s orbitals have a lower energy than the 3d orbitals and so they will be filled first in keeping with the minimum energy principle. For example, the electron configuration of the outer 10 electrons of calcium (atomic number Z = 20) is 3s 3p 3d 4s. In the filling of the electron orbitals for elements 21 to 29, there are two irregularities, one at 24 (chromium) and one at 29 (copper). Each of these elements contains one 4s electron instead of two. The reason... 

The minimum energy principle in the closed system is derived from the First and Second Laws of Thermodynamics. Since iQ = TdS, we have... 

Thus the minimum energy principle is also given for the continuum. 

In principle, we could find the minimum-energy crystal lattice from electronic structure calculations, determine the appropriate A-body interaction potential in the presence of lattice defects, and use molecular dynamics methods to calculate ab initio dynamic macroscale material properties. Some of the problems associated with this approach are considered by Wallace [1]. Because of these problems it is useful to establish a bridge between the micro-... 

Classical electrostatic modeling based on the Coulomb equation demonstrated that the model system chosen could account for as much as 85% of the effect of the protein electric field on the reactants. Several preliminary computations were, moreover, required to establish the correct H-bond pattern of the catalytic water molecule (WAT in Fig. 2.6). Actually, in the crystal structure of Cdc42-GAP complex [60] the resolution of 2.10 A did not enable determination of the positions of the hydrogen atoms. Thus, in principle, the catalytic water molecule could establish several different H-bond patterns with the amino acids of the protein-active site. Both classical and quantum mechanical calculations showed that WAT, in its minimum-energy conformation,... 

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 transformation from spheres to cyhnders is a peculiar example for the self-adjustment of the molecular conformation. The switching shape can be regarded as an example for the principle of quasi equivalency established by A. Klug for the self-assembly of biomolecules and viruses [145] for the sake of an improved intermolecular packing, the molecules adopt a conformation different from the minimum energy one. This also demonstrates that shape control does not mean a fully constrained structure. Similar to biomolecules, the combination of flexible macromolecules and self-assembly principles is a powerful strategy for preparation of molecules with well-defined but switchable shape [23]. 

What is the lowest possible energy for the harmonic oscillator defined in Eq. (5.10) Using classical mechanics, the answer is quite simple it is the equilibrium state with x 0, zero kinetic energy and potential energy E0. The quantum mechanical answer cannot be quite so simple because of the Heisenberg uncertainty principle, which says (roughly) that the position and momentum of a particle cannot both be known with arbitrary precision. Because the classical minimum energy state specifies both the momentum and position of the oscillator exactly (as zero), it is not a valid quantum... 

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