Reflection invariant

The symmetry elements will leave certain classes of reflections invariant, or F(H ) = F(H). Examination of Eq. (B.ll) shows that unless F(H) = 0, exp ( — 27nH-s) must be equal to 1, or... 

C-3 of 1 is correctly specified by Mislow and Siege] as r, C-3 of both 2 and ent-2 are incorrectly specified as R. If this were the CIP specification then the system would indeed have to be criticized but the system 3 4 (see Section 1.1.5.3.2.) gives precedence to stereogenicity over local symmetry in fact, it demands a reflection-invariant descriptor, i.e., r, as stated in the formulas. 

As can be seen in the formulae of both endo-substituted enantiomers of renzapride shown below, if only the relative configuration is considered then the formulae are reflection invariant, whilst the absolute configuration is inverted by a mirror reflection. 

Reflection invariance provides further constraints, the total wave function of the two colliding particles retaining reflection symmetry through the collision plane. Since we have already assumed that electron spin plays no role in the collision, this means that the reflection symmetry of the excited P state is the same as that for the original S state, i.e. symmetric. Thus the coefficient of the antisymmetric py) orbital in the expansion of the excited state must vanish. From (8.6) we see that this requires /ii = —/i-i. In general reflection symmetry requires that the elements of the density matrix satisfy the condition (Blum and Kleinpoppen, 1979)... 

Hermiticity imposes further constraints. For / = 1 there are only five independent multipoles, e.g. (T ), (T/), T ), T ), Tq), with (T/) imaginary and the components of the alignment tensor real (Blum, 1981). When reflection invariance holds in the collision plane the state multipoles can be related to the orientation 0 and alignment parameters A, first introduced by Happer (1972) and Fano and Macek (1973), through... 

Reflection invariance with respect to the scattering plane (i.e. parity conservation) yields the further restriction that... 

Universal scaling reflects invariance of thcj system under spatial dilata tlons. (Chaps. 8,10.)... 

I, u Descriptors for the specification of relative configuration. A pair of stereo-genic units has the relative configuration I (for like) if the descriptor pairs are RR, SS, RRe, SSi, ReRe, SiSi, MM, PP, RM, SP, ReM, or SiP. The pair is specified as u (unlike) if they have descriptor pairs RS, RSi, ReS, ReSi, MP, RP, SM, ReP, and SiM [86]. Reflection invariant descriptors (r, s, re, si, p, and m) may be substituted in place of the reflection variant descriptors above. Note the use of lower case / and u letters, implying a reflection invariant relationship. 

Prochiral Tetrahedral atoms having heterotopic ligands, or heterotopic faces of trigonal atoms, may be described as being prochiral. Note that it is inappropriate to describe an entire molecule as being prochiral [105]. Heterotopic faces are described using Re, Si if reflection variant, and re, si if reflection invariant [91]. If the CIP priority of the three ligands is clockwise, the face... 

Another convention, used in biochemistry to specify the hydrogen atoms of a prochiral methylene, replaces a hydrogen with a deuterium. If such replacement results in the R configuration, the ligand position is pro-R. If the S configuration is obtained, it is pro-S [105]. If reflection invariant, the descriptors are pro-r and pros. 

Lower case descriptors re, si) are used for the rare cases that are reflection invariant [91] ... 

For examples of reflection invariant stereogenic centers and faces, see dia-stereotopic, and pseudoasymmetric atom. 

Reflection variant, reflection invariant The terms used to describe an object and its relationship with its mirror image. If the two are identical, the object is reflection invariant. If the object is enantiomorphous to its mirror image, it is reflection variant. 

The quantum number v embodies the set of nuclear dynamic states with their labels (see below) and /c stands for the electronic quantum state. Thus, the nuclear wave function is always determined relatively to particular electronic states which, in turn, must be correlated to the (point) symmetries of the system. This stationary wave function may define, for particular cases, a class of geometric elements having an invariant center of mass. Actually, the (equivalence) class of configurations are those for which symmetry operations leave invariant this center of mass. This framework shares the discrete symmetries, such as permutation and space reflection invariances that are properties of the molecular eigenstates. There exists, then, a specific geometric framework pok- At this point, the expectation value ofH ,. taken with respect to the universal wave function is stationary to any geometric variation. 

In all of the following we assume that the Hamiltonian is invariant to these transformations. For example, in the absence of electromagnetic forces the Hamiltonian is a quadratic function of the momentum—hence for time reversal invariance H(, p) = H q —p). In addition, if the potential is only a function of the distances between particles, H is translationally invariant, reflection invariant, and has even parity. Because po(F) po(F) will be invariant to all... 

The last term is not reflection invariant and secures the lowest energy for right-handedness, which is therefore the lowest energy state. 

The reflection-invariant relative topicity of approach of reactants is defined as like (Ik) and unlike (ul) if the corresponding descriptor pairs are Re, Re or R, Re, and Re, Si or R, Si, respectively. The descriptor pair notations (Ik and ul) of reactants disclose related steric courses of reactions more often than do the relative configurations of their products, for which the configurational notation I = R, R and u = R, S is proposed. [13]... 

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