Dyson series

The first case has already been considered section 2.0 the second case leads to a strong classical spin-orbit coupling, which is reflected in a Hamiltonian nature of the classical combined dynamics. In both situations the procedure is to find a suitable approximate Hamiltonian Hq( ) that propagates coherent states exactly along appropriate classical spin-orbit trajectories (x(l,),p(t),n(l,)). (For problems with only translational degrees of freedom this has been suggested in (Heller, 1975) and proven in (Combescure and Robert, 1997).) Then one treats the full Hamiltonian as a perturbation of the approximate one and calculates the full time evolution in quantum mechanical perturbation theory (via the Dyson series), i.e., one iterates the Duhamel formula... 

We know from quantum theory that the path-ordered exponent of an operator A can be expressed by the power series called the Dyson series ... 

The second theoretical tool needed is called a Dyson series. It comes from the realization that one may start from an initial guess for the Green s function and refine it by a series of terms which successively incorporate the real physical situation by way of a physical potential. The great discovery of Dyson was that such a series actually converges to the exact solution of the problem. One can therefore construct the interface Green s function starting from the sum of the two components, or the zero order Green s function ... 

The first order Green s function, from the Dyson series and taking into account the exponential decay of electron states, contains two discrete terms, which can be written in terms of the Bardeen matrix elements... 

In order to illustrate the magnitude of the computational difficulty for fhe case of sfrong fields, we recall that the formal solution of the TDSE on the interval (0,f) is obtained as a time-ordered infinite perturbation series, the Dyson series [3, 4, 29] ... 

There is immediate that having this Dyson series of Green function the wave function solution of the perturbed systems may be written according with the Huygens quantum principle as ... 

While this discussion based on electrostatics ignores the time dependence in PF(t) and omits the effects of magnetic fields associated with the light wave, it does anticipate some of our final results in this section regarding electric dipole and electric quadrupole contributions to the matrix elements . It yields no insight into magnetic multipole transitions or into the nature of the time-ordered integrals in the Dyson series expansion of Eq. 1.96. 

A simple repetition of the iteration procedure (2.20)-(2.22) results in divergence of higher order solutions. However, a perturbation theory series may be summed up so that all unbound diagrams are taken into account, just as is usually done for derivation of the Dyson equation [120]. As a result P satisfies the integral-differential equation... 

Fig. 1. Conformational energy diagram for the alanine dipeptide (adapted from Ramachandran et al., 1963). Energy contours are drawn at intervals of 1 kcal mol-1. The potential energy minima for p, ofR, and aL are labeled. The dependence of the sequential d (i, i + 1) distance (in A) on the 0 and 0 dihedral angles (Billeter etal., 1982) is shown as a set of contours labeled according to interproton distance at the right of the figure. The da (i, i + 1) distance depends only on 0 for trans peptide bonds (Wright et al., 1988) and is represented as a series of contours parallel to the 0 axis. Reproduced from Dyson and Wright (1991). Ann. Rev. Biophys. Chem. 20, 519-538, with permission from Annual Reviews.

Dyson introduces a series of mathematical and physical simplifications and then reaches his important conclusions. These involve the reciprocal relationship of three parameters when these are chosen, the model is completely defined. The parameters are ... 

A series of tri- and tetra-nuclear Ru clusters previously reported by the groups of Siiss-Fink [60] and of Dyson [61] as catalysts for the hydrogenation of benzene and other simple aromatics in biphasic media have later been shown to consist predominantly of active metallic particles [9, 10, 62]. 

Dyson, P. J. Geldbach, T. J. Moro, F. Taeschler, C. Zhao, D. In Ionic Liquids III Fundamentals, Progress, Challenges, and Opportunities, ACS Symposium Series 901,2005. 

Dyson, N. (1998), Chromatographic Integration Methods, Second Edition, RSC Chromatography Monographs Series (Ekl. R.M. Smith), Royal Society of Chemistry, Cambridge, U.K. 

Dyson, G. M., Advances in Chem. Series No. 4, 96 (1950). Searching the Older Chemical Literature. 

Details about ILs properties are covered in this book in the contributions by Seddon, Chiappe and Scott. However, two features deserve a comment for their possible consequences on reactivity and catalysis. First, depending on a delicate balance of entropie and enthalpic factors, including the polarity of the transition state structures with respect to regents, a reaction can be either speeded up or decelerated when carried out in an ionic liquid medium compared to a molecular solvent. An elegant study by Welton shows that in S-,2 reactions, primary, secondary and tertiary amines are more reactive as nucleophiles in ionic liquids, while halides react faster in conventional molecular solvents such as CH2CI2. In particular in a series of [Bmim] salts the order of nucleophilicity of halides is determined by the anion partner. To the same direction moves a kinetic study by Dyson on a cationic Ru(II) complex-catalysed hydrogenation of styrene in ILs, where it is clearly demonstrated that both the cation and the anion of the IL can inhibit or accelerate the formation of the active catalytic species. ... 

Dyson PJ, Geldbrach T, Moro F, Taeschler C, Zhao D (2003) In Rogers RD, Seddon KR (eds) Ionic liquids IIIB fundamentals, progress, challenges and opportunities, ACS Symposium Series 902, p 322, Washington, DC. ISBN 0-8412-3894-4... 

It is known that certain perturbation series in quantum mechanics and field theory are divergent, due to the instability of the vacuum state when the coupling constant changes its sign (Dyson s phenomenon [10]). In the case of the 1/n-expansion, such an argument is unknown to us, and therefore a numerical investigation of the asymptotic behaviour of at A 00 was performed. 

Approximations are made to Dyson s second equation [Eq. (123)] by truncating the infinite series for the renormalized vertex. The zero-order approximation... 

Rules for lUPAC Notation for Organic Compounds (Longmans, 1961) is based on the Dyson system and provides a linear cipher for all chemical substances of known structure which can be adopted for indexes. It supersedes A Proposed International Chemical Notation (1958). Some useful lists of symbols have also been published e.g. the Manual of Physico-chemical Symbols and Terminology (issued separately and in J. Amer. chem. Soc. 82, 5517-22 (1960), and the now out-of-date Symbols of Thermodynamical and Physico-chemical Quantities and Conventions Relating to their Use Chem. Ind. 15, 860-5 (1937) Analyst 62, 800-5 (1937)), and International Physico-chemical Symbols (Z. Elektrochem. 27, 5TI-yi (1921). Another volume in the Advances in Chemistry series. No. 14, Nomenclature for Terpene Hydrocarbons, describes the system approved by the ACS and accepted in part by lUPAC. 

---