Conjugated, definition

Division by complex numbers calls upon using the complex conjugate definition... 

The two exponential tenns are complex conjugates of one another, so that all structure amplitudes must be real and their phases can therefore be only zero or n. (Nearly 40% of all known structures belong to monoclinic space group Pl c. The systematic absences of (OlcO) reflections when A is odd and of (liOl) reflections when / is odd identify this space group and show tiiat it is centrosyimnetric.) Even in the absence of a definitive set of systematic absences it is still possible to infer the (probable) presence of a centre of synnnetry. A J C Wilson [21] first observed that the probability distribution of the magnitudes of the structure amplitudes would be different if the amplitudes were constrained to be real from that if they could be complex. Wilson and co-workers established a procedure by which the frequencies of suitably scaled values of F could be compared with the tlieoretical distributions for centrosymmetric and noncentrosymmetric structures. (Note that Wilson named the statistical distributions centric and acentric. These were not intended to be synonyms for centrosyimnetric and noncentrosynnnetric, but they have come to be used that way.)... 

From the definition of Hermitian conjugate and Eq. (B.5), one then gets... 

By substituting back into the definition of a , we get the solution set for the energy spectrum Ei. In ethylene there are two elements on the diagonal, xu and X22, leading to Ei and 2- In larger conjugated n systems, there will be more. 

This method of diene formation with definite E and Z structures has wide synthetic applications [518], particularly for the syntheses of natural products with conjugated polyene structures. Bombykol and its isomers (650 and 651) have been prepared by this method[5l9]. The synthesis of chlorothricolide is... 

By definition, the vector of conjugate momenta corresponding to the vector of generalized internal coordinates q is... 

Buffers are solutions that tend to resist changes in their pH as acid or base is added. Typically, a buffer system is composed of a weak acid and its conjugate base. A solution of a weak acid that has a pH nearly equal to its by definition contains an amount of the conjugate base nearly equivalent to the weak acid. Note that in this region, the titration curve is relatively flat (Figure 2.15). Addition of H then has little effect because it is absorbed by the following reaction ... 

Although I used the example of ethene, where n =2, the same consideration, apply to ZDO calculations on all conjugated molecules. All overlap matrices are real symmetric, positive definite and so have eigenvalues > 0. 

For commonly encountered conjugated systems like butadiene and benzene, the ad hoc assignment of new parameters is usually preferred as it is simpler than the computationaly more demanding PPP method. For less common conjugated systems the PPP approach is more elegant and has the definite advantage that the common user does not need to worry about assigning new parameters. If the system of interest contains... 

According to the Hiickel criteria for aromaticity, a molecule must be cyclic, conjugated (that is, be nearly planar and have ap orbital on each carbon) and have 4n + 2 tt electrons. Nothing in this definition says that the number of p orbitals and the number of nr elections in those orbitals must be the same. In fact, they can he different. The 4n + 2 rule is broadly applicable to many kinds of molecules and ions, not just to neutral hydrocarbons. For example, both the cydopentadienyl anion and the cycloheptatrienyl cation are aromatic. 

Conjugated polymers are centrosymmetric systems where excited states have definite parity of even (A,) or odd (B ) and electric dipole transitions are allowed only between states of opposite parity. The ground state of conjugated polymers is an even parity singlet state, written as the 1A... PM spectroscopy is a linear technique probing dipole allowed one-photon transitions. Non linear spectroscopies complement these measurements as they can couple to dipole-forbidden trail-... 

The absoiption spectra of these three materials are shown in the bottom panel of Figure 9-16. From these spectra it becomes clear that the m-LPPP shows the longest effective conjugation length 23 the best resolution of vibronic progression, and the steepest onset of absorption [231. Therefore, one would assume the m-LPPP to be a material of the highest chemical definition. This is indeed con-... 

The values of Hn and E are zero for water, by virtue of the constants 1.74 and 2.60. In these definitions, pKa refers to the acid ionization constant of the conjugate acid of the nucleophile, and E° to the standard electrode potential for the two-electron half-reaction ... 

The products of proton transfer in aqueous solution may also react with water. For example, the CIST ion produced when HCN loses a proton to water can accept a proton from a water molecule and form HCN again. Therefore, according to the Bronsted definition, CN is a base it is called the conjugate base of the acid HCN. In general, a conjugate base of an acid is the species left when the acid donates a proton ... 

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