Non degenerate ground state

The principle of tire unattainability of absolute zero in no way limits one s ingenuity in trying to obtain lower and lower thennodynamic temperatures. The third law, in its statistical interpretation, essentially asserts that the ground quantum level of a system is ultimately non-degenerate, that some energy difference As must exist between states, so that at equilibrium at 0 K the system is certainly in that non-degenerate ground state with zero entropy. However, the As may be very small and temperatures of the order of As/Zr (where k is the Boltzmaim constant, the gas constant per molecule) may be obtainable. 

The constant of integration is zero at zero temperature all the modes go to the unique non-degenerate ground state corresponding to the zero point energy. For this state S log(g) = log(l) = 0, a confmnation of the Third Law of Thennodynamics for the photon gas. 

For a correlated N-electron system with a non-degenerate ground state > the one-particle Green s function has the spectral representation (20,21) ... 

At least for the case of a non-degenerate ground state of a closed shell system, it is possible to delineate the standard Kohn-Sham procedure quite sharply. (The caveat is directed toward issues of degeneracy at the Fermi level, fractional occupation, continuous non-integer electron number, and the like. In many but of course not all works, these aspects of the theory seem to be... 

The electron density of a non-degenerate ground state system determines essentially all physical properties of the system. This statement of the Hohenberg-Kohn theorem of Density Functional Theory plays an exceptionally important role among all the fundamental relations of Molecular Physics. 

As a consequence of the Hohenberg-Kohn theorem [14], a non-degenerate ground state electron density p(r) determines the Hamiltonian H of the system within an additive constant, implying that the electron density p(r) also determines all ground state and all excited state properties of the system. 

The non-degenerate ground state electron density p/(r) over any subset d of manifold S3, S3 zd d, where subset d has non-zero volume on S3, determines uniquely... 

Non-Degenerate Ground States, S =. All but two of the species dealt with under this heading are d5 systems with 2S+(a 84) ground states. Such systems are therefore... 

Non-Degenerate Ground States, S >. This particular problem was first considered in general terms by Pryce (137), who pointed out that in some applications of perturbation... 

Exceptions to this behaviour are Eu2+ and Gd3+, which — as a consequence of their if electronic configuration — present an orbitally non-degenerate ground state. 

For a non-degenerate ground state, F and density operator p = ip (r)ip(r), the electron density is given by... 

Strong terminal M—H stretching modes are found in these adducts. The magnetic moments are indicative of d1 complexes with orbitally non-degenerate ground states. ESR spectra are observed at room temperature ([NbH2Cl2(dmpe)2] (g) = 1.96, (/ ), , -109 G, (a)P = 25.5 G, (a)H = ll.lG). 

High spin iron(III) complexes occur with some or all weak donor atoms. High spin iron(III) has one unpaired electron in each of the five d orbitals and every orbital can contribute to the overall spin density. The ground state is a sextuplet with an orbitally non degenerate ground state. The orbitals and their occupancy in various symmetries are reported in Fig. 5.1. There are no excited levels with the same spin multiplicity, since moving one electron in an excited d orbital requires spin pairing, and thus electron relaxation is not efficient. 

The relative long values of zs arises from the orbitally non-degenerate ground state of the chromium(III) ion, which makes Orbach relaxation inefficient. Electron relaxation is attributed to modulation of the ZFS of the 5 = 3h ground state (Table 5.6). 

The density functional methods were, in the views of many, legitimized by the introduction of the first Hohenberg-Kohn theorem [10]. The consequence of this celebrated theorem is that for a non-degenerate ground state and a given external potential, v(r), the electronic ground state energy can be expressed as... 

In contrast to polyacetylene, the other CPs shown in Scheme 1 have non-degenerate ground states (i.e. they do not possess two equivalent resonance forms), and thus, do not show evidence of soliton formation. In this instance, the oxidation of the CP is believed to result in the destabilization (raising of the energy) of the orbital from which the electron is removed. This orbital s energy is... 

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