Limiting degeneration

Cool flame behaviour is the result of highly limited degenerate branching, where the branching coefficient, a, is small, e.g. is of the order of 1.1, compared with being of the order of two or three in normal branching. 

Let the degree of freedom subject to limiting degeneration be denoted by the separation co-ordinate qf, whose libration limits coincide. The action variable corresponding to it,... 

The problem of finding this solution involves a mathematical difficulty. Returning to our example, the perturbation function contains in general terms which are linear in the eccentricity, that is in terms in Vj/.1 Now this can occur quite generally if the unperturbed system has limiting degeneration. Terms in 1/Vj/ then occur in... 

The restriction to simple limiting degeneration is not necessary. The corresponding considerations and calculations are also valid for limiting degeneration of arbitrary multiplicity. The appropriate expression for the generator S is... 

To summarise, we may state For an initial motion possessing limiting degeneration, the perturbed motion, selected in accordance with the quantum theory, has the same degree of periodicity s as the unperturbed motion. Its energy is... 

In 45 we had to leave unanswered the question whether, in the case of an accidentally degenerate initial motion, the motions singled out by the quantum theory have the same degree of periodicity as the initial one, when the work is carried to any degree of approximation. The method developed for limiting degeneration now enables us to answer this question. At the same time the restriction on the wp0 s given in 45 will be established by an independent method. 

The expansions of the cartesian co-ordinates as functions of the angle variables (to be calculated from (26), 22) must now be introduced, to provide a starting-point for the calculation of the perturbations. In this connection, however, there is one point to be borne in mind. In the unperturbed Kepler motion (without taking account of the variation in mass) only Jx is fixed by the quantum theory, whilst J2, i.e. the eccentricity, remains arbitrary in the relativistic Kepler motion, J2 is also to be quantised and, for a one-quantum orbit, J2=J1=A. We shall not take account quantitatively of the relativistic variation of mass, but we shall assume that the initial orbit of each electron is circular with limiting degeneration J1=A,... 

The unperturbed system consists therefore of two circular orbits of the same size. In addition to the double limiting degeneration due to the circular orbits, we have also a double intrinsic degeneration, arising from the fact that the planes of the two orbits are... 

Since the initial motion of the inner electron exhibits limiting degeneration, it is convenient to replace the variables w, m>2, J/, J2 by other variables. We therefore perform the canonical transformation... 

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. 

For high rotational levels, or for a moleeule like OFI, for whieh the spin-orbit splitting is small, even for low J, the pattern of rotational/fme-stnieture levels approaehes the Flund s ease (b) limit. In this situation, it is not meaningful to speak of the projeetion quantum number Rather, we first eonsider the rotational angular momentum N exelusive of the eleetron spin. This is then eoupled with the spin to yield levels with total angular momentum J = N + dand A - d. As before, there are two nearly degenerate pairs of levels assoeiated... 

The work, W, can range from 2ero, if the engine is completely ineffective, to the limiting negative value attained for reversible operation. If IT = 0, then the process degenerates to one of simple heat transfer, for which... 

At the dissociation limit the UHF wave function is essentially an equal mixture of a singlet and a triplet state, as discussed in Section 4.4. Removal of the triplet state by projection (PUHF) lowers the energy in the intermediate range, but has no effect when the bond is completely broken, since the singlet and triplet states are degenerate here. 

At the equilibrium inter-atomic distance R, two paired electrons of occupy the bonding orbital with a closed-shell low-spin singlet (S = 0). When the bond length is further increased, the chemical bond becomes weaker. The dissociation limit of corresponds to a diradical with two unpaired electrons localized at each atom (Fig. 1). In this case, the singlet (S spin-antiparaUel) and triplet (T spin-parallel) states are nearly degenerate. Different from such a pure diradical with... 

Fig. 1 A schematic illustration of the in-phase and out-of-phase combinations of the atomic orbitals into the bonding and antibonding molecular orbitals, respectively. The dissociation limit of a H molecule corresponds to a pure diradical with degenerate singlet and triplet states...

---