All-order calculations

Valence removal energies for cesium from an all-order calculation (see Ref [44]). Units a.u. 

A typical application of the relativistic SD equations is given in Table 6, where we compare MBPT and SD calculations of energies (relative to the ionization threshold) of 2si/2, 2pi/2, and 2ps/2 levels of lithium and singly ionized beryllium. The all-order calculations include partial waves through /max = 7. The second-order MBPT calculation is carried out in a large (n = 100) basis set and includes partial waves up to Imax = 12 The third-order values and i/extra sre calculated with n = 40 spline basis set and /max = 7. Breit and reduced mass (RM) and mass polarization (MP) corrections values are taken from [54]. The SD value of the 2p3/2 — 2pi/2 fine structure interval for Li is 0.00000156 a.u. compared with the measured value 0.000001534(2) a.u.. The corresponding theoretical and experimental values for Be" " are 0.00003001 a.u. and 0.00002998(3) a.u.. The tiny differences between the SD energies and experiment on the last line of Table 6 are probably dominated by the incomplete treatment of triple excitations. 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

For comparison with the usual second-order perturbation in the spin-orbit coupling, we assume that the first order calculation has taken all first-order effects into account as in Eq.(l 1). The second-order perturbation due to the interaction operator W is given by... 

Integral intensities were obtained after dead-time corrections, background subtraction and normalization to averaged monitor counts. The Lp correction was applied in the usual way. Since the polarization ratio was not measured at BW5 so far, 90% linear horizontally polarized radiation was assumed for all scans. Calculations show that even a change in the beam polarization of 10% would effect the intensities of the highest order reflections of less than 1.5%. 

It is inappropriate in this survey to attempt to summarise in a short space the results of all the above treatments, but virtually all the calculations indicate that in M(Cp)2 species the 7r(ej) metal-ligand interaction is dominant in the metal to ring bonding. However, debate has largely been concentrated on two points, in the first place the extent of the validity (or otherwise) of Koopmans theorem, and, further, the question of the correct energetic ordering (5 < o < n or o < 5 < n) of the mainly metal tf-type orbitals. 

Green and Pimblott (1991) criticize the truncated distributions of Mozumder (1971) and of Dodelet and Freeman (1975) used to calculate the free-ion yield in a multiple ion-pair case. In place of the truncated distribution used by the earlier authors, Green and Pimblott introduce the marginal distribution for all ordered pairs, which is statistically the correct one (see Sect. 9.3 for a description of this distribution). 

It is possible to place costings against each of these factors. Of course, the actual numbers will vary according to a producer s individual circumstances, but the orders of magnitude involved can be demonstrated. For all the calculations a plant producing 600 metric tonnes of chlorine per day (tpd) with an on-stream time of 350 days per year has been assumed. The entire chlorine production is assumed to be dedicated to the vinyl chain. 

The CASSCF/CASPT2 calculations were performed with an active space including the five nd, the (n + l)s, the three (n+ l)p orbitals, and a second set of nd orbitals to account for the double shell effect. The importance of including a second 3d shell in the active space was detected in an early study of the electronic spectrum of the nickel atom [2]. This had already been suggested from MRCI results [1]. The results obtained by RT at about the same time indicated that such effects are effectively accounted for when a method is used that includes cluster corrections to all orders, like the QCI method used by them [3]. This result will hold true also for the less approximate coupled cluster method CCSD(T). 

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