Electrons vector model

A pictorial representation of the Tg-S mixing process follows from Fig. 6. Just as in normal n.m.r. or e.s.r. spectroscopy, precession can be represented by a vector model. When placed in an external magnetic field the two unpaired electrons of the radical pair 1 and 2 will precess... 

Figure 7. The occupation number densities as functions of wave vector for Na. The thick curves labeled (100), (110) and (111) represent the three principal directions within the first Brillouin zone, obtained by the FLAPW-GWA. The thin solid curve is obtained from an interacting electron-gas model [27]. The dash-dotted line represents the Fermi momentum. 

Holme, T. A., and Levine, R. D. (1988), An Algebraic Hamiltonian for Electronic Nuclear Degrees of Freedom Based on the Vector Model, Inti. J. Quant. Chem. 34, 457. 

Fig. 5. Vector model coupling of the spin angular momenta of two, three, and. four electrons...

Starting from the vector model of a precessing spin, the polarization properties of an electron beam with electrons of spin (l/2)h are explained and then formulated... 

The Stokes parameters for the polarization of an electron beam can be represented in a Cartesian basis which also provides a convenient pictorial view for the polarization state of an electron beam. Since the polarization of an ensemble of electrons requires the determination of spin projections along preselected directions, the classical vector model of a precessing spin will first be discussed. Here the spin is represented by a vector s of length 3/2 (in atomic units) which processes around a preselected direction, yielding as expectation values the projections (in atomic units, see Fig. 9.1)... 

Figure 9.1 Vector model of the electron spin. Using atomic units, the spin is represented by a vector s of length y/3/2 which precesses around the z-axis. By looking at the respective projections of the precessing spin vector, the model provides two important properties the projection onto the z-axis leads to a sharp value, ms = 1/2 in the case shown (ms = —1/2 for a precession around the negative z-axis), but no sharp values exist for the projections in the xy-plane, i.e., for the projections onto the x- or y-axis one finds with equal probability...

Figure 9.2 Illustration of an electron beam of six electrons with different spin polarization components. It is assumed that the electrons travel along the z-axis and the spin vectors shown are constant in a statistical average, i.e., they are repeated when the beam passes by. (a) Vector model of precessing spins (b) short notation showing only the corresponding...

The fundamental problem in the case of a rotating molecule (as a rule, for simplicity s sake, we will consider a diatomic or a linear polyatomic one) is that of the interaction between the electronic motion and rotation of the nuclei. For better clarity and in conformity with the style of our further presentation, we will apply the vector model approach. 

FIGURE 9.1 Vector model for intersystem crossing of radical pairs in high magnetic fields. Left, singlet state centre, superposition state right, state ITq). The labels 1 and 2 denote the electron spins of the two radicals. Further explanation, see text. 

FIGURE 9.2 Vector models (projections) illustrating pair substitution. The labels 1 and 2 denote the electron spin of the first and the second radical of the pairs. The observed proton is contained in the first radical. Its spin state, la) or IP), is displayed at the respective leftmost projection. The radical pairs are bom in the triplet state, and the product is formed from the singlet state c gives the singlet character. First radical pair RPi, positive g-value difference, zero hyperfine coupling constant second radical pair RP2, equal g values, positive hyperfine coupling constant. For the situations without pair substitution, the spin evolutions under the influence of the Zeeman and the hyperfine interaction have been separated for clarity. Further explanation, see text. 

The first improved theory addressing the weakly coupled, or non-adiabatic electronic excitation transfer was the semiclassical vector model proposed by Forster [15]. It was further developed and refined by Levinson [16], Kasha [17], and others [18], who sometimes referred to it as the molecular exciton theory . Notably, this was the first successful attempt to link the rate of electronic excitation transfer with readily available experimental parameters, such as the absorption spectrum of the... 

Finally, it must be remembered that the vector model gives only an approximate description of the electronic structure. According to (4) the... 

Fig. 10. The Dirac Vector Model showing the orientation of nuclear and electron spins.

Figure 5 Phase of a CIDNP net effect explained with vector models (projections). From left to right, starting state influence of theg-value difference influence of the nuclear spin state through the hyperfine coupling constant resulting population difference of the nuclear spin states in the product resulting CIDNP signal. The labels on the vector models denote the electron spin of radical 1 or 2. Further explanation, see text.

Within the exchange region the axis V in the vector model of Figure 3 practically coincides with the x axis of the coordinate system. As is obvious from the figure, the precession of q around "V therefore leaves unchanged the population difference between S) and T0> only electron spin polarization and phase correlation are mixed in this region. Under these circumstances (J Q), Eq. 31 is reduced to... 

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