Cumulant formalism

Much of the recent literature on RDM reconstruction functionals is couched in terms of cumulant decompositions [13, 27-38]. Insofar as the p-RDM represents a quantum mechanical probability distribution for p-electron subsystems of an M-electron supersystem, the RDM cumulant formalism bears much similarity to the cumulant formalism of classical statistical mechanics, as formalized long ago by by Kubo [39]. (Quantum mechanics introduces important differences, however, as we shall discuss.) Within the cumulant formalism, the p-RDM is decomposed into connected and unconnected contributions, with the latter obtained in a known way from the lower-order -RDMs, q < p. The connected part defines the pth-order RDM cumulant (p-RDMC). In contrast to the p-RDM, the p-RDMC is an extensive quantity, meaning that it is additively separable in the case of a composite system composed of noninteracting subsystems. (The p-RDM is multiphcatively separable in such cases [28, 32]). The implication is that the RDMCs, and the connected equations that they satisfy, behave correctly in the limit of noninteracting subsystems by construction, whereas a 2-RDM obtained by approximate solution of the CSE may fail to preserve extensivity, or in other words may not be size-consistent [40, 42]. 

The combinatorial point of view is reminiscent of the classical cumulant formalism developed by Kubo [39], and indeed the structure of Eqs. (25) and (28) is essentially the same as the equations that define the classical cumulants, up to the use of an antisymmetrized product in the present context. In further analogy to the classical cumulants, the p-RDMC is identically zero if simultaneous p-electron correlations are negligible. In that case, the p-RDM is precisely an antisymmetrized product of lower-order RDMs. 

Compared with the Gram-Charlier temperature factor of Eq. (2.31), the entire series now occurs in the exponent, so, in the cumulant formalism, terms are added to the exponent of the harmonic temperature factor P0(H) = exp — fijkhjhk. ... 

Unlike the Gram-Charlier and cumulant formalisms, the OPP model has a physical rather than a statistical basis. It assumes that each atom vibrates in a potential well K(u), determined by the interaction with the other atoms in the crystal, without any correlation between vibrations of adjacent atoms. 

One can identify two major categories of uncertainty in EIA data (scientific) uncertainty inherited in input data (e.g., incomplete or irrelevant baseline information, project characteristics, the misidentification of sources of impacts, as well as secondary, and cumulative impacts) and in impact prediction based on these data (lack of scientific evidence on the nature of affected objects and impacts, the misidentification of source-pathway-receptor relationships, model errors, misuse of proxy data from the analogous contexts) and decision (societal) uncertainty resulting from, e.g., inadequate scoping of impacts, imperfection of impact evaluation (e.g., insufficient provisions for public participation), human factor in formal decision-making (e.g., subjectivity, bias, any kind of pressure on a decision-maker), lack of strategic plans and policies and possible implications of nearby developments (Demidova, 2002). 

These held operators are sometimes termed probe variables because they function as dummy placeholders in the formal differentiahons that follow but do not appear in the hnal expressions for the cumulants, which are obtained formally in the limit that/,/t —> 0. 

A further point is of interest in the formal discussion of the canonical transformation theory. So far we have assumed that the reference function is fixed and have considered only solving for the amplitudes in the excitation operator. We may also consider optimization of the reference function itself in the presence of the excitation operator A. This consideration is useful in understanding the nature of the cumulant decomposition in the canonical transformation theory. 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

The OPP formalism, though based on the assumption of independent motion, has the advantage of assigning a physical meaning to the terms in the expansion. By equating the OPP terms to the corresponding ones in the statistical expansions, the quasimoments and cumulants can be related to the parameters of the potential model, and their temperature dependence can be predicted. 

Formally, in the two steps of the DMS-process (dibromocyclopropanation and reaction with alkyllithium) a carbon atom is inserted between the two centers of a double bond. The reaction may be extended to the preparation of still higher cumulated bond systems as well as to numerous other — including functionalized — allenic systems which cannot or only with much effort be prepared by other routes. The examples shown here serve illustrative purposes only, for more extensive coverage of the literature the reader is referred to the various reviews and monographs which have appeared recently [66, 69, 71, 72]. 

A carbocation is strongly stabilized by an X substituent (Figure 7.1a) through a -type interaction which also involves partial delocalization of the nonbonded electron pair of X to the formally electron-deficient center. At the same time, the LUMO is elevated, reducing the reactivity of the electron-deficient center toward attack by nucleophiles. The effects of substitution are cumulative. Thus, the more X -type substituents there are, the more thermodynamically stable is the cation and the less reactive it is as a Lewis acid. As an extreme example, guanidinium ion, which may be written as [C(NH2)3]+, is stable in water. Species of the type [— ( ) ]1 are common intermediates in acyl hydrolysis reactions. Even cations stabilized by fluorine have been reported and recently studied theoretically [127]. 

Trisilaallene 67 was synthesized as the first stable compound with a formally sp-hybridized silicon atom and also the first cumulative Si = Si doubly bonded... 

The time-local approach is based on the Hashitsume-Shibata-Takahashi identity and is also denoted as time-convolutionless formalism [43], partial time ordering prescription (POP) [40-42], or Tokuyama-Mori approach [46]. This can be derived formally from a second-order cumulant expansion of the time-ordered exponential function and yields a resummation of the COP expression [40,42]. Sometimes the approach is also called the time-dependent Redfield theory [47]. As was shown by Gzyl [48] the time-convolutionless formulation of Shibata et al. [10,11] is equivalent to the antecedent version by Fulinski and Kramarczyk [49, 50]. Using the Hashitsume-Shibata-Takahashi identity whose derivation is reviewed in the appendix, one yields in second-order in the system-bath coupling [51]... 

Because analytical results can be accumulated over long periods on easily accessible disks — a big improvement on laboratory notebooks — it is easier to evaluate the performance of a method or process over a period of time. It is easy to determine the frequency of defects in performance, both by inspection and by more formal methods. One of these which is being used more widely is the cusum procedure. A reference value, k, is selected typically it is the mean value of the expected result. Then sequentially the cumulative sum, Si is calculated... 

This equation formally defines a cumulant /q ([t) as a coefficient in the series expansion of X (9, t). It too is easily found from its generating function as... 

Examples of such compounds are [(i7-Cp)MoS]4 and [(RS)4Fe4S4]2. There are also compounds of S, Se, and Te, formally in high oxidation states, that have cumulated multiple bonds. 

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