Degenerate ground states

Exact values of critical exponents are more difficult to obtain, because variational bounds do not give estimations of the exponents. Then the result presented by M. Hoffmann-Ostenhof et al. [64] for the two-electron atom in the infinite mass approximation is the only result we know for /V-body problems with N > 1. They proved that there exists a minimum (critical) charge where the ground state degenerates with the continuum, there is a normalized wave function at the critical charge, and the critical exponent of the energy is a = 1. 

Hund s rules Rules which describe the electronic configuration of degenerate orbitals in the ground state. The electronic configuration will have the maximum number of unpaired... 

The one-dimensional cases discussed above illustrate many of die qualitative features of quantum mechanics, and their relative simplicity makes them quite easy to study. Motion in more than one dimension and (especially) that of more than one particle is considerably more complicated, but many of the general features of these systems can be understood from simple considerations. Wliile one relatively connnon feature of multidimensional problems in quantum mechanics is degeneracy, it turns out that the ground state must be non-degenerate. To prove this, simply assume the opposite to be true, i.e. 

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. 

Figure 5. A cut across the ground state (GS) and the excited state (ES) potential surfaces of the H4 system. The parameter Qp is the phase preserving nuclear coordinate connecting the H(lll) with the transition state between H(I) and H(1I) (Fig, 4). Keeping the phase of the electronic wave function constant, this coordinate leads from the ground to the excited state. At a certain point, the two surfaces must touch. At the crossing point, the wave function is degenerate.

Figure 20, The potential surface near the degeneracy point of a degenerate E state that distorts along two coordinates and Q. The parameter is the stabilization energy of the ground state (the depth of the moat ), [Adapted from [70]].

The search for a conical intersection is also successful. The predicted structure is at the left. The predicted energies of the two states—the ground state and the first excited state—differ by about 0.00014 Hartrees, confirming that they are degenerate at these points on the two potential energy surfaces. ... 

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