Conjugate equations

Compounds similar to the above may also be obtained by an intramolecular Heck-type reaction of the bromo compound 103. Treatment of 103 with palladium(ll) acetate gives a mixture of two isomers, of which the major is 104 (with the more extended conjugation) (Equation 20) <1995JOC2312>. 

Multiply the free-particle equation (4) on the left by k and the conjugate equation by [Pg.199]

Reactions of Mi4-anions with aldehydes resulted in facile 1,2-additions leading to a-metallated alcohols (equations 15716 24 and 1589). When the aldehyde was highly conjugated (equation 158), the nucleophilic addition did not occur and a SET reaction took place (see Section V.E). In the case of tin, the reaction was used for the synthesis of a-stannylalcohol (equation 159)4b and then applied to the preparation of a-alkoxystannanes (equations 160 and 161)104,171 and a -iodoslann anes1 °2.172. 

Then, using the eigenvalue equation of the coherent state and its conjugate equation, one obtains after using the fact that the coherent state is normalized... 

C. Blue light excites the Soret band of Chi a, which rapidly becomes deexcited to the lower excited singlet (see Fig. 4.9). Therefore, the energy available is given by the red fluorescence of Chi a (A,max = 666 nm see Fig. 5.3). Consequently, the first compound (A,max = 580 nm) cannot be excited by Chi a. For the pigment with 10 double bonds in conjugation, Equation 4.2 indicates that the absorption maximum occurs at... 

Studies of ring-substituted cyclopropyl peracetates did not reveal any significant rate enhancement attributable to cyclopropyl conjugation (equation 28) ". ... 

The intensity of the beam diffracted by all the atoms of the unit cell in a direction predicted by the Bragg law is proportional simply to the square of the amplitude of the resultant beam, and [Fp is obtained by multiplying the expression given for F in Eq. (4-11) by its complex conjugate. Equation (4-11) is therefore a very important relation in x-ray crystallography, since it permits a calculation of the intensity of any hkl reflection from a knowledge of the atomic positions. 

Studies of the absorption and fluorescence spectra of some bi-benzimidazoles have demonstrated that there is hydrogen bonding in (27a, b) as well as considerable interannular conjugation (Equation... 

To demonstrate the natural unification of the gravitational and electromagnetic aspects of the quaternion field equation (46a) and its conjugate equation (46b), we follow this procedure. Multiply (46a) on the right with the conjugated quaternion solution q, and the conjugated equation (46b) on the left with q-r... 

Thus we have seen that the factorized quaternion field equations (46a) (or their conjugated equations (46b)) -the irreducible form of electromagnetism... 

Solving Eq. (4.48) and its complex-conjugate equation = cosO — i sin0 for sin0 and cos0, we can represent these trigonometric functions in complex exponential form ... 

The degree to which each of the effects will impact the respective isomer will depend not only on the amine substituents but also on the specific structure of the ketone involved. For example, and rather surprisingly, it has been shown that A - -strain can be sufficiently greater that A -strain so as to favor the less substituted enamine, even when the substituent is an alkoxycarbonyl group capable of adding stability to the more substituted form through extended conjugation (equation Further, it has been... 

It should be pointed out that the process suggested by equation (75) has been recently subjected to criticism and revision The electrons of amines can also influence the spectrum of a distant chromophore. In Table 16 the absorption bands of some compounds (90-93) are shown in which long range conjugation (equation 77) possibly takes place, resulting in a displacement towards lower wavelength and smaller Smax values as compared with the model compound 94 . 

The time-reversal symmetry of the crystalline Hamiltonian introduces an additional energy-level degeneracy.Let the Hamiltonian operator H be real. The transition in the time-dependent Schrodinger equation to a complex-conjugate equation with simultaneous time-inversion substitution... 

The relation to the random phase approximation (RPA) in Section 10.3 becomes clear, when one considers also the complex conjugate equation of Eq. (11.24)... 

The complex-conjugate equation simply ensures (cf. Section 6.1) that the matrix of Lagrangian multipliers is Hermitian. 

Hiis equation arises from the e " terms, a Hermitian-conjugate equation coming from terms in e . From an assumed first approximation (e.g. d = 0), the equations may be solved for d, and the revised approximation will define G(d) for starting the next iteration. 

In what sense is it possible to say that 3/9/ behaves like a 1-electron operator in the derivation of (12.6.1)7 Continue the derivation of the time-dependent analogue (12.6.7) of the HP equations, giving a proof that the matrix of Lagrangian multipliers is Hermitian. [Hint Write down a complex-conjugate equation, take appropriate scalar products, and eliminate the time.]... 

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