STAR operator

Figure 19 Schematic effect of the STAR operator on 2JCH and 3,/CH couplings. The vicinal component of magnetization in the long-range response that is two-bond coupled to a protonated carbon experiences modulation, which serves as a pseudo-evolution for this coupling. In contrast, the vicinal component of magnetization in the long-range response that is three-bond coupled to a protonated carbon does not exhibit a F, skew. Homonuclear modulation during the evolution period f, is still present, as the full experiment is not a constant-time experiment.

Figure 20 Timing diagram of the suggested 2y,3y-HMBC experiment, including a LPJF3 for efficient 1JCH suppression. The sequence is virtually identical to the CIGAR-HMBC pulse sequence. The STAR operator is also a constant-time variable element. In this fashion, scalable F, modulation can be specifically introduced for 2JCH cross-peaks into the spectrum independently of the digitization employed in the second frequency domain.

The Yang-Mills functional [17] is defined by the integration of the wedge product F A F, where denotes the Hodge dual-star operator... 

HMBC pulse sequence. The relative strengths of the 10 gradients are as follows (i) dual-stage, low-pass 7-filter (G1-G3) 10, -6.63,-3.37 (ii) STAR operator (G4-G7)... 

Fig. 19. Pulse sequence for the J, J-HMBC experiment described by Krishnamurthy ei This experiment represents the most refined version of the accordion-optimized experiments to be developed thus far and allows the differentiation of Jxh from Jxh long-range correlations to protonated heteroatoms ( - C and N). The experiment further modifies the concept of the constant time variable delay used in the IMPEACH-MBC and CIGAR-HMBC experiments to even more selectively manipulate various components of magnetization. This is done using the pulse sequence operator given the acronym STAR (Selectively Tailored Accordion F Refocusing) (.see also Fig. 20). Differentiation of various components of heteronuclear long-range magnetization is accomplished within the STAR operator, with the balance of the pulse sequence similar to that of the IMPEACH-MBC and CIGAR-HMBC experiments.

The STAR operator used in the J, J-HMBC sequence is shown in Fig. 20. The STAR operator utilizes four variable delays, D, which are incremented from zero in concert with the decrementation of vd, A1 and A2, the sum of which is held constant, and vd, which is decremented as a function of the accordion-optimization range selected in the usual fashion from 7i ax to Tmin- Hence, the overall duration of the variable delay interval is constant. Within the STAR operator, the first variable delay, Al, is incremented from zero to y caic x fi ax in steps of Jsc-,x c x ti, while conversely A2 is decremented from a starting value of Vsciiie X fimax to zero in steps of Jscaie X t. Thus, while the sum of A1 -1- A2 is constant, each of the delays are variable and a function of the evolution time, t. ... 

The Fi modulation of Jcih2 by the STAR operator is manifest as shown schematically in Fig. 21. The extent of F modulation is a function of the value of /scale selected in performing the experiment. The long-range correlation, in... 

There have been no reported applications of the J, J-HMBC experiment published to date. There has, however, been one report of a COSY-type artefact observed in -J. J-HMBC spectra of a cyclopentafurnanone. The COSY-type responses observed are displaced in Fj as a function of the choice of Jscaie-Removing the bipolar gradients flanking the BIRD pulse in the A2 interval of the STAR operator and superimposing a CYCLOPS phase cycle on the BIRD pulse completely suppresses the COSY-type response artefacts associated with the -J, J-HMBC experiment. ... 

Fig. 21. Schematic representation of the differentiation of "Jch from Jch long-range correlation responses in the "J, J-HMBC experiment as a function of the operation of the STAR operator shown in Fig. 20.

Fig. 22. Aliphatic region of the 6 to 10 Hz optimized -J, J-HMBC spectrum of strychnine, 14, with = 16. Responses in the aliphatic region enclosed in boxes are -Jen long-range correlation responses. The two- and three-bond long-range correlations of the HI la resonance to Cl2 and C13 via two- and three-bonds, respectively, is shown in the expansion to the left. As expected, the Jch correlation to CI2 is modulated in F as a function of the STAR operator, while the J( h correlation to the Cl3 is unaffected by the operation of the STAR operator.

Fig. 8.28 The (/, -HMBC experiment is the most sophisticated accordion-optimized long-range heteronuclear shift correlation experiment reported to date [148]. The experiment uses a pulse sequence operator known as a STAR (selectively tailored F, accordion refocusing) to selectively manipulate two-bond and three-bond long-range correlations to protonated carbon or nitrogen resonances. A. STAR operator used in the (/, J-HMBC experiment. The experiment takes advantage of the ability of a BIRD(x,x,x) pulse to refocus the one-bond heteronuclear coupling of a protonated carbon. By doing this, the coupling to this proton...

Fig. 8.29 Schematic representation of expecta- of what would be expected from the function of tions for two-bond (left, e.g. the Hlla—C12) the STAR operator. Experimental verification of and three-bond long-range couplings (right, these anticipated results is shown in Fig. 8.30. e.g. the Hlla—C13) long-range couplings in (Reproduced with permission - Academic st chnine (2). The staggered F skew is typical Press).

In this Appendix we clarify the cause for postulating symmetry relation (6.18). For this aim we introduce a formal operation which can be named the duality transformation and which is well known in multilinear algebra as the Hodge star operation, or Hodge dual [132]. In the RDM theory an equivalent transformation was applied in [19, 133], without recognizing it as a Hodge dual. The following simple example helps to explain this notion in the more familiar terms of many-electron state vectors. 

Table 5.1 Embedding of the Csu point group in the Longuet-Higgins group. The symmetry elements of the point group act on the electrons. They are identified as the product of nuclear permutations, inversion of all particles (star operation), and bodily rotations of aU particles (Q operators) along particular directions...

Then, by injecting this scaling factor into expression 11.36 of the basic quantity, one obtains the general solution under the form of a convolution equation, also called convolution product and notated as a multiplication by means of a star operator (dyadic operator) ... 

For the calculation of the S(q), one has to take into account that at concentrations c cJtaj, only a limited interpenetration of the star coronae occurs. Therefore, the effective interaction range on which repulsion between unperturbed central regions of the stars operates exceeds rc(c) and is on the order of the overall star size Hjtar(c)- Similar to the situation in dilute solution, this repulsion should be described in terms of soft repulsive potential operating on the distance between star centers d [Nflcf-. ... 

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