Vectors representation

Figure Bl.16.4. Part A is the vector representations of the. S state, an intennediate state, and the Jq state of a radical pair. Part B is the radical reaction scheme for CIDNP.

Vector representation of momenta and vector coupling approximations 7.1.2.1 Angular momenta and magnetic moments... 

Figure 1.5 (a) Vector representation displaying a greater number of spins aligned... 

Following is a pictorial vector representation of a doublet. It shows the evolution of the components of a signal. Draw vector positions in the fourth and fifth frames, along with their directions of rotation. 

No. Since the direction of precession of the vectors is reversed during the second half of the sequence, they continue to diverge from each other. The effect of a spin-echo experiment on spin-spin coupling in a first-order homonuclear spin system is shown in the following vector representation ... 

Figure 5.42 Pulse sequence and vector representation of a NOESY experiment.

When an NMR experiment is performed, the application of a RFpulse orthogonal to the axis of the applied magnetic field perturbs the Boltzmann distribution, thereby producing an observable event that is governed by the Bloch equations [3]. Using a vector representation, the... 

A brief comment on dimensionality is in order at this point. As used here, the number of dimensions is taken equal to the number of subscripts on the data matrix. Thus, an optical or mass or nuclear spectrum is one-dimensional, but if different samples or sampling times are involved it is considered two-dimensional, as in GC-MS. In this context, we treat the vector representation of a spectrum or a multielement analysis as single dimension, though it is frequently viewed as "a point in hyperspace."... 

A brief introduction to the types of molecular representations typically encountered in MSA is presented at the beginning of Subheading 2. followed in Subheading 2.1. by a discussion of similarity measures based on chemical-graph representations. Although graph-based representations are the most familiar to chemists, their use has been somewhat limited in similarity studies due to the difficulty of evaluating the appropriate similarity measures. This section is followed by a discussion of similarity measures based on finite vector representations, the most ubiquitous types of representations. In these cases, the vector components can be of four types ... 

In the traditional molecular-fragment approach the coefficients of the vector components are usually binary- or positive integer-valued. A continuous vector representation can be constructed in the following way. Choose a set of/molecular fragments, or whole molecules, as a molecular basis for representing chemistry space... 

The only common factor is that the charge-current 4-tensor transforms in the same way. The vector representation develops a time-like component under Lorentz transformation, while the tensor representation does not. However, the underlying equations in both cases are the Maxwell-Heaviside equations, which transform covariantly in both cases and obviously in the same way for both vector and tensor representations. 

Figure 4.14 Electric vector representation of completely depolarized, partially polarized and completely polarized radiation.

Fig. 21S. Determination of the absolute configuration of a non-centrosymmetric structure by using anomalous scattering. Left—Scattering by anomalously scattering atom JP and by the rest of the molecule, E. Centre—Representation of amplitudes and phases ot waves. Right—Corresponding vector representation (scale of amplitudes doubled).

To give a concrete example of the general vector representation, the substantial-derivative operator can be expanded in cylindrical coordinates as... 

The matrix-algebraic representation (9.20a-e) of Euclidean geometrical relationships has both conceptual and notational drawbacks. On the conceptual side, the introduction of an arbitrary Cartesian axis system (or alternatively, of an arbitrarily chosen set of basis vectors ) to provide vector representations v of geometric points V seems to detract from the intrinsic geometrical properties of the points themselves. On the notational side, typographical resources are strained by the need to carefully distinguish various types of... 

Fig. 8.19. Vector representation of a H-13C HMQC experiment. The first 90° pulse along y rotates the equilibrium magnetization of the proton spin, /H, from the z axis to the x axis. After a time /d = 1/2/Hx, the antiphase coherence 2/J1/ t (see Appendix IX) is at its maximum. A 90° pulse on carbon along y then transforms the antiphase coherence into a MQ (multiple quantum) coherence (the 2/J1/ component is shown). During t the MQ evolves (with a 180 refocusing pulse on proton in the middle), until a further 90 pulse on carbon along x transforms the —2/ / component (shown at its maximum for clarity) into a 2/ /f antiphase coherence. After the time fd, in-phase coherence of the proton spin develops. The latter is detected during h. Its initial intensity is modulated by the carbon Larmor frequency during t (if proton refocusing has been used), thus originating a proton-carbon cross peak.

Fig. IX. 1. Product operators for a two-spin system. The density matrix form is shown along with the vector representation (adapted from [1]).

Notice that the latter nine operators are formed by all the possible products (hence the name of the formalism), two at time, of the three spin angular momentum operators of spin / and the three spin angular momentum operators of spin K. Vector representations of all these operators are shown in Fig. IX.2. 

In order to illuminate both the phase problem and its solution, I will represent structure factors as vectors on a two-dimensional plane of complex numbers of the form a + ib, where i is the imaginary number (—1)1/2. This allows me to show geometrically how to compute phases. I will begin by introducing complex numbers and their representation as points having coordinates (a,b) on the complex plane. Then I will show how to represent structure factors as vectors on the same plane. Because we will now start thinking of the structure factor as a vector, I will hereafter write it in boldface (FM,Z) instead of the italics used for simple variables and functions. Finally, I will use the vector representation of structure factors to explain a few common methods of obtaining phases. 

In Chapter 4, Section HI.G, I mentioned Friedel s law, that lhkl = h k i-It will be helpful for later discussions to look at the vector representations of pairs of structure factors Fhkl and F h k l, which are called Friedel pairs. Even though hkl and l h k l are equal, Fhkl and F h k l are not. The structure factors of Friedel pairs have opposite phases, as shown in Fig. 6.3. 

We say that z forms a basis for A,or that z belongs to Ai, or that z transforms according to the totally symmetric representation Ai. The s orbitals have spherical symmetry and so always belong to IY This is taken to be understood and is not stated explicitly in character tables. Rx, Ry, Rz tell us how rotations about x, y, and z transform (see Section 4.6). Table 4.5 is in fact only a partial character table, which includes only the vector representations. When we allow for the existence of electron spin, the state function ip(x y z) is replaced by f(x y z)x(ms), where x(ms) describes the electron spin. There are two ways of dealing with this complication. In the first one, the introduction of a new... 

The character tables in Appendix A3 include the spinor representations of the common point groups. Double group characters are not given explicitly but, if required, these may be derived very easily. The extra classes in the double group are given by Opechowski s rules. The character of if in these new classes in vector representations is the same as that of R but in spinor representations x(ff) = — (R). The bases of spinor representations will be described in Section 12.8. 

The IRs of G comprise the vector representations, which are the IRs of G, and new representations called the spinor or double group representations, which correspond to half-integral j. The double group G contains twice as many elements as G but not twice as many classes g, and g,- are in different classes in G except when g,- is a proper or improper BB rotation (that is, a rotation about a binary axis that is normal to another binary axis), in which case g, and gt are in the same class and (gj, (xg,) are necessarily zero in spinor... 

Example 12.6-2 The classes of Td are E 4C3 3C2 6S4 6binary rotations are BB rotations. The six dihedral planes occur in three pairs of perpendicular improper BB rotations so both 3C2 and 6rrd are irregular classes. There are therefore Nv 5 vector representations and Ns = 3 spinor representations. 

The PFs gi gy] are a set ofg2 complex numbers, which by convention are all chosen to be square roots of unity. (For vector representations the PFs are all unity.) PFs have the following properties (Altmann (1977)) ... 

From the definition of a class X(gi) = gk gi gt - V gk G G (with repetitions deleted) it follows that for vector representations... 

Exercise 12.6-1 D3h= 2C3 3C2 crh 2S3 3crv, where crh = /C2, aV = IC2", C2-L(C2, C2")- Thus the improper binary axis C2 is normal to the 3C2 proper binary axes and the three improper C2" binary axes. There are, therefore, three irregular classes crh, 3 C2, and 3oy. There are six classes in all and therefore six vector representations (Nv = 6). There are three regular classes and therefore three spinor representations, each of which is doubly degenerate since J]3=1 2 = 22 + 22 + 22 = 12 = g. 

D3 = E 2C3 3C2. There are no BB rotations so that the groups both consist of three regular classes. There are therefore three vector representations (/Vv = Nc) and three spinor representations (NS = NTC = NC). The dimensions of the Nv vector representations are /V = 1 1 2 (because g 6) and of the Ns spinor representations also /s = 1... 

The dashed line separates the vector representations, for which j is an integer, from the spinor representations, which correspond to half-integer values of j. 

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