Pascal’s rule

Since gm is related to the orbital angular momentum vector L the second term is often written in terms of this operator. The first term is the Langevin term it is the expectation value of a one-electron operator 2 (xt2 + jV), and can therefore be obtained directly from the ground-state wavefunction. It has formed the basis for a quite successful additivity scheme, Pascal s rules, which work well except for the conjugated hydrocarbons where non-additivity is ascribed to ring currents. Nevertheless, as it depends on the square of the electron co-ordinates, xL will be sensitive to the basis set used in variational calculations. 

Molecule Theory Experiment Pascal s rule (theory)... 

The above study of saturated hydrocarbons illustrates how theoretical calculations may be used to extract information unattainable by experiment alone. From this study, we also conclude that caution should be exercised in applying Pascal s rule to highly strained ring systems even when no jt-electrons are present. In the next section, we shall turn our attention to molecular systems containing the more loosely attached -electrons. 

It is evident from the results in Table 7 that the molecular susceptibilities obtained from the local values in Table 8 are in good agreement with experimental results. These local values can now be used to predict the molecular susceptibility of nonstrained, nonaromatic compounds on which measurements are not available. In conclusion we have extended the well-known Pascal s rules for bulk susceptibility to the individual diagonal elements in the molecular magnetic susceptibility tensor. Of course, the diagonal sums of the values given in Table 8 also allow a prediction of the bulk susceptibilities. 

Many more mean values have been reported than full tensors or anisotropies (see, for example, refs 104— 109). Very early in the development of this field it was found that these mean values were additive, being sums of atomic terms plus some so-called constitutive corrections (see, for example, refs 110 and 111 and Chapter 6 of ref 106). Any deviation from these (Pascal s"°) rules was said to be a measure of magnetic exaltation (see section III.A.4, which follows). 

Patterns of this third class in fact demonstrate a complex form of scale-invariance by their self-similarity, in the infinite time limit, different magnifications observed at the same resolution are indistinguishable. The pattern generated by rule R90, for example, matches that of the successive lines in Pascal s triangle ai t) is given by the coefficient of in the expansion of (1 - - xY modulo-tv/o (see figure 3.2). 

Something of scientific interest (among numerous other applications, Pascal s triangle is used to predict nuclear magnetic resonance [NMR] multiplet splitting patterns) has been created from the repeated application of a simple rule. 

Figures 1.9a and b demonstrate the effect of proton broadband decoupling in the 13C NMR spectrum of a mixture of ethanol and hexadeuterioethanol. The CH3 and CH2 signals of ethanol appear as intense singlets upon proton broadband decoupling while the CD3 and CD2 resonances of the deuteriated compound still display their septet and quintet fine structure deuterium nuclei are not affected by lH decoupling because their Larmor frequencies are far removed from those of protons further, the nuclear spin quantum number of deuterium is ID = 1 in keeping with the general multiplicity rule (2 nxh+ 1, Section 1.4), triplets, quintets and septets are observed for CD, CD2 and CD3 groups, respectively. The relative intensities in these multiplets do not follow Pascal s triangle (1 1 1 triplet for CD 1 3 4 3 1 quintet for CD2 1 3 6 7 6 3 1 septet for CD3).

Pascal s triangle The (n+1) rule only applies in systems where the... 

We can easily verify that the intensity ratios of multiplets derived fixrm the n -I-1 Rule follow the entries in the mathematical mnemonic device called Pascal s triangle (Fig. 3.33). Each entry in the triangle is the sum of the two entries above it and to its immediate left and right. Notice that the intensities of the outer peaks of a multiplet such as a septet are so small compared to the inner peaks that they are often obscured in the baseline of the spectram. Figure 3.27 is an example of this phenomenon. 

IT = 0, 1,. . . , n in the binomial expansion. A device known as Pascal s triangle is the favorite method for doing this (Table LVIII). The rule for forming the values of the coefficients should be clear to the reader. 

Figure 14,31 Pascal s triangle, used in conjunction with the n + 1 rule.

As the term Av/J decreases, the observed spectra become second-order and are more complicated. The splitting patterns may no longer be predictable by the n + 1 rule, and the intensities of the peaks in the multiplets cannot be approximated by using Pascal s triangle. This trend may be illustrated by examining the H NMR spectra for 1-butanol (4) in Figures 8.28 and 8.29 and pentane (5) in Figure 8.30. 

There is an important lesson to be learned from the fact that fewer peaks are observed than would be expected from the +1 rule When multiplets of six or more are predicted, one or more of the outermost absorptions of the multiplet is frequently too weak (see Pascal s triangle. Figure 8.27) to appear under the normal operating conditions for the spectrometer. This fact must be taken into account when using spin-spin splitting patterns to interpret the spectrum of an unknown compound. 

If the one s in the first column are replaced with two s, and if the 2 and columns are filled according to the rule by which Pascal s triangle is built up, then we find ... 

The appearance of the peak depends on the number of neighbouring hydrogens. This can be calculated using the n +1 rule, where n is the number of equivalent neighbouring hydrogens. The multiplicity and relative intensities of the peaks can be obtained from Pascal s triangle (shown below). 

The relative intensities of the components in a multiplet signal are given by Pascal s triangle, which is based on the coefficients of the expansion of (a+b)". For proton spectra, the n+1 rule and Pascal s triangle lead to the midti-plicities and relative intensities for an observed resonance signal when there are n adjacent identical nuclei as shown in Table 5. 

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