Intermediate Case Situations

No Rydberg state ever exactly corresponds to a pure Hund s case, except at n = 00 or 3, but rather to a situation in an intermediate Hund s case, and it is necessary to choose, to describe it, a basis set corresponding to any one of the pure Hund s cases. The energy terms to be added in each Hund s case are different depending on this choice of basis set. To clarify these choices, consider first the simple case of atoms. 

In molecules, starting from a case (d) basis set, the part of the electronic energy that must be added corresponds to the energy difference between the states of different A due to the different Coulomb and exchange interactions with the core [see Section 3.3.2], a difference which can be expressed, like the example shown by Herzberg and Jungen (1972) for a p-complex, as a function of the difference between the quantum defects of a and 7r. 

In case (e), one has the sum of the effects of cases (d) and (c) and this is why, starting from a case (e) basis set, the neglected part of the electronic energy corresponds to the energy differences between all of the case (a) components (see Eq. 17 of Lefebvre-Brion, 1995). 

The spherical-1 n/), spherical ion-core limit resembles the electronic structure of an alkali atom. It corresponds to an electronic Hamiltonian, Hel(° in which the 2/ + 1 A-components (—/ A /) (eigenfunctions of both l2 and lz) in a case (a) or (b) basis set are exactly degenerate or, in a case (d) basis set, the 2/ + 1, —/ Ir /, Ir = N — N+, / -components (eigenstates of Hel(°) + HROT) that belong to the same value of N+ are exactly degenerate. 

11n contrast to A, where the basis states with well defined parity (see Section 3.2.2) have the form 21/2[ A) — A)], for Ir, each of the signed-l/j I,Ir) states has parity (— 

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Chemists refer to the bond in a molecule like sodium chloride as ionic , meaning that its electron pair resides entirely on chlorine. At the other extreme is the covalent bond in the hydrogen molecule, where the electron pair is shared equally between the two hydrogens. Intermediate cases, such as the bond in hydrogen fluoride which is clearly polarized toward fluorine, are generally referred to as polar covalent bonds (rather than partially ionic bonds). Are these situations really all different or do they instead represent different degrees of the same thing ... 

A unique situation is encountered if Fe-M6ssbauer spectroscopy is applied for the study of spin-state transitions in iron complexes. The half-life of the excited state of the Fe nucleus involved in the Mossbauer experiment is tj/2 = 0.977 X 10 s which is related to the decay constant k by tj/2 = ln2/fe. The lifetime t = l//c is therefore = 1.410 x 10 s which value is just at the centre of the range estimated for the spin-state lifetime Tl = I/Zclh- Thus both the situations discussed above are expected to appear under suitable conditions in the Mossbauer spectra. The quantity of importance is here the nuclear Larmor precession frequency co . If the spin-state lifetime Tl = 1/feLH is long relative to the nuclear precession time l/co , i.e. Tl > l/o) , individual and sharp resonance lines for the two spin states are observed. On the other hand, if the spin-state lifetime is short and thus < l/o) , averaged spectra with intermediate values of quadrupole splitting A q and isomer shift 5 are found. For the intermediate case where Tl 1/cl , broadened and asymmetric resonance lines are obtained. These may be the subject of a lineshape analysis that will eventually produce values of rate constants for the dynamic spin-state inter-conversion process. The rate constants extracted from the spectra will be necessarily of the order of 10 -10 s"F... 

As is apparent from Fig. 1, the dianions of monoalkyl phosphates normally resist hydrolysis. However, for leaving groups whose conjugate acids exhibit a pKa < 5 in water, hydrolysis of the dianion becomes faster than that of the monoanion. Fig. 2 shows a pH profile characteristic of this situation. Whereas the hydrolysis rate of 2,4,6-trichlorophenyl phosphate (pKa of the phenol 6.1) still shows the typical monoanion preference as seen for methyl phosphates (Fig. 1), the dianion of 2,4-dinitrophenyl phosphate (pKa of the phenol 4.09) is hydrolyzed far faster than the monoanion 2-chloro-4-nitrophenyl phosphate represents an intermediate case (pKa of the phenol 5.45)6S). 

If the situation for Q and T is disappointing owing to their connections with C only, the position of the term R is intermediate between the two extremes (i) the R electrons belong purely to C (ii) the R electrons are shared equally between C and A. The former extreme is intuitively acceptable only if the C—A bonds are purely ionic, in which case the MO term anti-bonding loses much of its original significance. In (ii) and intermediate cases, the only excuse for excluding R from Eqn. 11.19 would be explicitly to assume Eqn. 11.13. 

The first two terms describe the chemical shifts for nuclei i and j and the last, the nuclear spin-spin interaction. Nuclear spins are coupled via intervening electrons. The coupling constant A, sometimes represented by J in the literature, is expressed in cycles per second (cps). Analyses of observed spectra for S,-, 5,- and A,-, are quite straightforward in situations where po 5i — 5, Aij but can become quite tedious where i 0 5 — 5,- and A,-,-are comparable. va is the resonance frequency expressed in cps. The latter case is the so-called intermediate coupling situation and is discussed in detail by Pople et al. (108). 

The V)f(q) curves in Fig. 20.13 indeed reach constant values for q — °° which are (approximately) equal to Kv This result also follows directly from evaluating Eq. 20-51 for q —> °°. Since tanh(g — °°) = 1, the second term in the denominator approaches zero for q — °°. Finally, Fig. 20.13 nicely demonstrates how p(maximum value (1 + Kr) for q — o°. Note that for large Kt values, this transition extends from q 1 to fairly large q values. Thus, for these cases flux enhancement remains -dependent in situations where tT is much smaller than tw. Thus, our first attempt to characterize the intermediate case with tT /w should rather be replaced by ... 

We have discussed above the influence of reagent mobility on the kinetics of the electron tunneling reaction in two extreme situations, a < a and A > Rz. In the intermediate case, a a 4, i r(the mixed mechanism of reagent mobility), the equations for the kinetics of electron tunneling reactions also have a sufficiently simple form only in two extreme situations small, t r = Rz/D, and large, t > r, observation times [32], If the initial spatial distribution of reagents is random and n(0) N, the kinetics of electron tunneling reactions is described by eqn. (35) at l z, and by the equation... 

The main argument for making MIP CEC is to combine the selectivity of the MIPs with the high separation efficiency of CEC. This argument appears to fail, however, if the adsorption isotherm of the MIP is nonlinear, which seems to be the rule. In the case of nonlinear isotherms, the peak shapes depend mainly on the isotherm, particularly so if the separation system is otherwise very efficient (has low theoretical plate height, see Fig. 1). In the case of ionized analytes the situation is more complex. If an ionized analyte is not adsorbed at all on the MIP, then it is separated only due to electrophoresis, and its peak will not be widened due to the nonlinear effect. In this case, however, the MIP is merely behaving like an inert porous material. In intermediate cases an ionized analyte may participate in both separation mechanisms and for this case we do not have exact predictions of the peak shape. 

Case 9 Intermediate cases. There are two major difficulties in describing the supported oxide or, more generally, the precursor (c.g. oxychlorides, salts of various oxides or other compounds) which undergo activation. One is the fact that a same precursor may be involved in different types of interactions according to the nature of the support. The second is that several of the situations described above may coexist in a given catalyst. M0O3 offers an excellent illustration of these two difficulties. 

I conclude that in such cases the fact of state autonomy can be explained in terms of class interest, even if the autonomously made state decisions cannot. A class may have the ability to take the political power that option is within its feasible set. Yet it may have some weakness that makes abstention a superior option. 1 have been arguing that the autonomy of the state is not made less substantial by the fact that the class keeps out of politics rather than being kept out of it. We are, in fact, dealing with an intermediate case between two "normal" situations. At one extreme is the situation in whidi no class would be able to dethrone the government, because the latter has superior means of coercion at its disposal. At the other extreme we have the situation in which the economically dominant class has nothing to fear from taking power, and txmsequently takes it. Marx was concerned with the paradoxical case in which a dominant class has the ability, but not the inclination, to concentrate the formal powers of decision in its own hands. 

With a large and small molecule linjit, there must also be an intermediate case. There are a number of possible manifestations of the intermediate case, so one interesting choice is discussed, 6) There may be situations in which the weakly coupled states 14>w I are sufficiently dense that on the time scales of real experiments, the 4>w behave, as an effective continum for decay of the zeroth... 

In the case of a 5 f3 configuration and 4I9/2 ground term, the accessible final states are 3H4, 3H5 and 3F2, and their relative final intensities in an intermediate coupling situation are predicted to be 2.137 0.187 0.612. Since the separation of the 3H4 and 3F2 state should be > 1 eV, at least two bands are expected in the P.E. spectrum of an f3 ionization. Unfortunately the quality of the data on U(t -C5H5)3-THF does not allow a definite conclusion as to the number of f-bands. 

In this paper we would like to discuss the theoretical approaches developed for the description of coil-globule transition of polyelectrolyte macromolecules in these two limiting situations and in the intermediate case when both of the contributions (due to the electrostatic repulsion and due to the translational entropy of counterions) are of the same order. We will discuss in detail mainly the theories constructed or developed by members of the Moscow group. 

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