Theorem Kramers

At this stage, we are ready to prove that Kramers theorem holds also for the total angular momentum F. We will do it by reductio ad absurdum. Then, let Itte) be the eigenvector of H with eigenvalue E,... 

II electronic states, 638-640 vibronic coupling, 628-631 triatomic molecules, 594-598 Hamiltonian equations, 612-615 pragmatic models, 620-621 Kramers doublets, geometric phase theory linear Jahn-Teller effect, 20-22 spin-orbit coupling, 20-22 Kramers-Kronig reciprocity, wave function analycity, 201 -205 Kramers theorem ... 

However, as mentioned above, T c)3) will be orthogonal to all the k states, and T ) is nonzero. This implies that the number of total states of the same eigenvalue E is (k + 1), which contradicts our initial hypothesis. Thus, we conclude that k must be even, and hence proved the generalized Kramers theorem for total angular momentum. The implication is that we can use double groups as a powerful means to study the molecular systems including the rotational spectra of molecules. In analyses of the symmetry of the rotational wave function for molecules, the three-dimensional (3D) rotation group SO(3) will be used. 

This is due to the so-called Kramers theorem, which establishes that all electronic energy levels containing an odd number of electrons are at least doubly degenerate. 

Exercise 13.4-6 It might appear that the last line of Table 13.1 contravenes Kramers theorem. Explain why this is not so. 

Exercise 13.4-6 It was shown in Exercise 13.4-5 that the dimension / of representations of type (c) is an even integer. Therefore, even though time reversal introduces no new degeneracies, l is always at least 2 and Kramers theorem is satisfied. 

Kramers theorem requires that all half-integer spin systems be at least doubly degenerate in the absence of a magnetic held. Next, note that the splitting of these levels by a magnetic held depends on its orientation relative to the axes of the ZFS tensor of the metal ion. The VTVH MCD saturation magnetization curve behavior reflects the difference in the population of these levels and their spin expectation values in a specific molecular direchon. This direction must be perpendicular to the polarizations of the transition (Mih where i / j are the two perpendicular polarizations... 

Rigorously, ORD and CD spectra are related through the Kronig-Kramers theorem, a well-known general relationship between refraction and absorption, i.e. nL - nR is determined by eL - % for A from zero to infinity [128], (The analogous relationship between refraction and reflection applies to cholesteric liquid crystals.) Hence, in order to maximize ORD in the transparent region, Cotton effects, associated with exciton coupling (both intramolecular and intermolecular), have... 

According to Kramers theorem, half-integer spin states like 85/2 will always be doubly degenerate as long as no strong external magnetic fields are applied. The 85/2 state splits into three Kramers doublets if the crystal field exhibits a symmetry lower than cubic. 

Several sophisticated physical studies of the Vis cluster have been published, for example I. Chiorescu, W Werns-dorfer, A. Muller et al., Butterfly Hysteresis Loop and Dissipative Spin Reversal in the S = 1/2, Vis Molecular Complex, Phys. Rev. Lett. 2000, 84, 3454-3457 B. Barbara, I. Chiorescu, W Wernsdorfer et al. The Vis Molecule, a Multi-Spin Two-Level System Adiabatic LZS Transi-tion with or without Dissipation and Kramers Theorem, Progr. Theor. Phys. Suppl. 2002, 145, 357-369. 

Radius of gyration of an ideal branched polymer (Kramers theorem)... 

The Kramers theorem is expressed in terms of this average over all possible ways of dividing the molecule into two parts ... 

The Kramers theorem effectively cuts a randomly branched polymer with N monomers into two parts, with N andN- Ni monomers. 

Substituting this average [Eq. (2.66)] into the Kramers theorem [Eq. (2.65)] recovers the classical result for the radius of gyration of an ideal linear chain [Eq. (2.54)]. In Section 6.4.6, we apply the Kramers theorem [Eq. (2.65)] to ideal randomly branched polymers. In this case the average is not only over different ways of dividing a molecule into two parts, but also over different branched molecules with the same degree of polymerization N. 

The radius of gyration of ideal branched polymers can be calculated using the Kramers theorem [Eq. (2.65)]. 

In Chapter 2 we have presented a proof of the Kramers theorem for branched molecules containing N monomers of size b, but no loops (Eq. 2.65). The mean-square radius of gyration of these molecules is... 

The Kramers theorem relates the ideal size of molecules to a purely structural property—the number of ways of dividing a molecule into two... 

The probability that a chosen bond divides a molecule into two branches—the first one with N monomers and the second one with N-Ni monomers is jv, (p)Wjv-a/, (p). Therefore, the Kramers theorem can be rewritten ... 

Substituting Eqs (6.87) and (6.88) into the Kramers theorem expression [Eq. (6.85)] gives the mean-square radius of gyration for an ideal randomly branched AA-mer ... 

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