Formula decoupling

The reduced matrix element for a two-electron system is given by the decoupling formula, which adopts a special form ... 

The reduced matrix element (free of projections of angular momenta) is evaluable according to the following decoupling formula... 

Let us apply the decoupling formula to several important cases. ... 

The decoupling formula for the reduced matrix element yields... 

The tensor ranks entering the decoupling formula for the evaluation of the reduced matrix elements are assigned as follows... 

The decoupling formula for the reduced matrix element yields (a) the first type (pair-interaction) reduced matrix elements, which are straightforward... 

The general formula and the individual cases as presented in Eq. (97) indicate that indeed the number of conical intersections in a given snb-space and the number of possible sign flips within this sub-sub-Hilbert space are interrelated, similar to a spin J with respect to its magnetic components Mj. In other words, each decoupled sub-space is now characterized by a spin quantum number J that connects between the number of conical intersections in this system and the topological effects which characterize it. 

The decoupled set of equations in system (20) can be solved for all the Qi and associated velocities V<.i by closed-form formulas that depend on the eigenvalues [71]. The harmonic position and velocity vectors at time nAt can then obtained from the expressions ... 

The pulse sequence which is used to record CH COSY Involves the H- C polarisation transfer which is the basis of the DEPT sequence and which Increases the sensitivity by a factor of up to four. Consequently, a CH COSY experiment does not require any more sample than a H broadband decoupled C NMR spectrum. The result is a two-dimensional CH correlation, in which the C shift is mapped on to the abscissa and the H shift is mapped on to the ordinate (or vice versa). The C and //shifts of the //and C nuclei which are bonded to one another are read as coordinates of the cross signal as shown in the CH COSY stacked plot (Fig. 2.14b) and the associated contour plots of the a-plnene (Fig. 2.14a and c). To evaluate them, one need only read off the coordinates of the correlation signals. In Fig. 2.14c, for example, the protons with shifts Sh= 1.16 (proton A) and 2.34 (proton B of an AB system) are bonded to the C atom at c = 31.5. Formula 1 shows all of the C//connectivities (C//bonds) of a-pinene which can be read from Fig. 2.14. 

In the //broadband decoupled C NMR spectrum, 15 carbon signals can be identified, in agreement with the molecular formula which indicates a sesquiterpene. The DEPT experiments show that the compound contains four quaternary C atoms, three CH units, seven CH units and a CH3... 

Simple analytic formulas for a for the decoupled 2s-2p channels are derivable for any value of L and for any combination of the masses of the point charge, m0, and of the two particles, nit and m2, constituting the hydrogen-like atom [76] ... 

The compound for this problem is a reagent commonly used in organic synthesis its molecular formula is C6H15SiCl. Provided is the H, 13C, DEPT, and 2vSi (proton-decoupled) spectra. 

Determine the structure of the phosphorus-containing compound whose molecular formula is C 9H18PBr from the H, l3C, DEPT, and 3IP (both proton-coupled and proton-decoupled). 

These simple formulas get considerably involved when all of the n protons coupled to silicon are not equivalent, have different couplings with silicon, 7(SiI F ), and exhibit homonuclear 1H—1H couplings, 7(1 V q). Formulas describing the enhancement were derived numerically by Schenker and von Philipsbom108 and analytically by Blechta and Schraml109. Repeated analysis confirmed the formulas110. The derived formulas for the decoupled variants are ... 

Let s consider how we might solve this structure, given only the 13C NMR spectrum and the molecular formula. As we have seen in Figures 13-41 and 13-42, the signal at 173 ppm is appropriate for a carbonyl carbon. The off-resonance-decoupled spectrum shows a singlet at 173 ppm, implying that no hydrogens are bonded to the carbonyl carbon. 

Off-resonance-decoupled and broadband-decoupled spectra of 5-valerolactone, molecular formula... 

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