Complex-conjugate

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.)... 

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

This is written in terms of zt-, the moduli of the roots Zk-, since the roots are either real or come in mutually complex conjugate pairs. In any case, this constant term can be absorbed in the polynomial P t) in Eq. (15).]... 

By comparing Eq. (C.6) with Eqs. (C.2) and (C.3), the time-reversal operator can be expressed as a product of an unitary and a complex conjugate operators as follows... 

By Max Bom s postulate, the produet of /(a ) and its complex conjugate r / (A ) times an infinitesimal volume element d x is proportional to the probability that a paitiele will be in the volume element d x... 

In the notation of Eq. (9-29), 4 i(ri) = 1 If the U orbitals are normalized, then the spinorbitals 1 ja(l), etc. are normalized because a and P are normalized. If we take just the expanded determinant for two electrons without 1 / V2, the normalization constant, and (omitting complex conjugate notation for the moment) integrate over all space... 

It is the net intensity, not the electric field, which concerns us. We previously used the fact that intensity is proportional to E to evaluate i. Using complex numbers to represent E requires one slight modification of this procedure. In the present case we must multiply E by its complex conjugate -obtained by replacing / 1 by to evaluate intensity ... 

Multiplying both sides by J/, the complex conjugate of p obtained Ifom it by replacing all i (equal to — 1) by —i, and integrating over all space gives... 

The UV spectrum of a complex conjugated molecule is usually observed to consist of a few broad band systems, often with fine structure, which may be sharpened up in non-polar solvents. Such a spectrum can often be shown to be more complex than it superficially appears, by investigation of the magnetic circular dichroism (MCD) spectrum, or by introduction of dissymmetry and running the optical rotatory dispersion (ORD) or circular dichroism (CD) spectrum. These techniques will frequently separate and distinguish overlapping bands of different symmetry properties <71PMH(3)397). 

---