Externally corrected energy

The results for N2 are summarized in Table 3. The bond is streched from 2.15 a.u. up to 3.00 a.u. Opposite to the previous molecules, even at the equilibrium geometry CCSD does not perform as well as the other approaches. For the [6e/6o] active space, it differs from FCI by 12.9 mHa, while CAS-SDCI does it by 8.3 mHa, (SCf CAS-SDCI by 3.9 mHa and ec-CCSD by 2.0 mHa or 1.8 mHa, depending on the external correcting source. Again, one may observe that the minimal dressing of CAS-SDCI not only makes it size-extensive, but also improves the absolute value of the yielded energy. 

The above presented data clearly demonstrate the usefulness of the ec CC approaches at both the SR and MR levels. While in the SR case the energy is fully determined by the one- and two-body clusters, and the truncation of the CC chain of equations at the CCSD level can be made exact by accounting for the three- and four-body clusters, the MR case is much more demanding, since the higher-than-pair clusters appear already in the effective Hamiltonian. An introduction of the external corrections is thus... 

The purpose of the present paper is to review the most essential elements of the excited-state MMCC theory and various approximate methods that result from it, including the aforementioned CR-EOMCCSD(T) [49,51,52,59] and externally corrected MMCC ]47-50, 52] approaches. In the discussion of approximate methods, we focus on the MMCC corrections to EOMCCSD energies due to triple excitations, since these corrections lead to the most practical computational schemes. Although some of the excited-state MMCC methods have already been described in our earlier reviews [49, 50, 52], it is important that we update our earlier work by the highly promising new developments that have not been mentioned before. For example, since the last review ]52], we have successfully extended the CR-EOMCCSD(T) methods to excited states of radicals and other open-shell systems ]59]. We have also developed a new type of the externally cor-... 

An interesting alternative to the externally corrected MMCC methods, discussed in Section 3.1.1, is offered by the CR-EOMCCSD(T) approach [49, 51,52,59]. The CR-EOMCCSD(T) method can be viewed as an extension of the ground-state CR-CCSD(T) approach of Refs. [61,62], which overcomes the failures of the standard CCSD(T) approximations when diradicals [76,104,105] and potential energy surfaces involving single bond breaking and single bond insertion [49,50,52,60-62,65,67,69,70,72,73] are examined, to excited states. 

For these and other reasons, much attention was given to the so-called state-selective or state-specific (SS) MR CC approaches. These are basically of two types (i) essentially SR CCSD methods that employ MR CC Ansatz to select a subset of important higher-than-pair clusters that are then incorporated either in a standard way [163,164], or implicitly [109-117], or via the so-called externally corrected (ec) approaches of either the amplimde [214-219] or energy [220,221] type, and (ii) those actually exploiting Bloch equations, but focusing on one state at a time [222]. The energy-correcting ec CC approaches [220,221] are in fact very closely related to the renormalized CCSD(T) method of Kowalski and Piecuch mentioned earlier [146,147]. 

The basic idea of the externally corrected CCSD methods relies on the fact that the electronic Hamiltonian, defining standard ah initio models, involves at most two body terms, so that the correlation energy is fully determined by one (Ti) and two (T2) body cluster amplitudes, while the subset of CC equations determining these amplitudes involves at most three (T3) and four (T4) body connected clusters. In order to decouple this subset of singly and doubly projected CC equations from the rest of the CC chain, one simply neglects all higher than pair cluster amplitudes by setting... 

The univocal relationship between the external potential applied to the electronic system and the electronic density is assured by the Hohenberg-Kohn theorems (Hohenbeig Kohn, 1964). Besides, one of the theorem also state the inequality between the energy as a functional density of a random electrorric state E p and the corrected energy of the fundamental electronic state of the system E p. ... 

Kim et al. [39], Velasquez-Orta et al. [40], and Lanthier et al. [41] all found that the number of recovered electrons in actual systems is significantly lower than that expected from substrate oxidation with 100% coulombic efficiency. We also observed that this was true for our experimental data. Therefore, a correction factor,/, is defined to represent the fraction of electrons obtainable for external electrical energy production. The formation of the reduced form of the mediator, M, is given by ... 

While all the above listed methods rely solely on the CC formalism (and, in some cases, on finite order MBPT), the so-called externally corrected (ec) CCSD methods 4,16,17) try to simultaneously exploit the information from some independent source, which is capable of handling the nondynamic correlation, is readily available, and requires only modest computational effort. The essence of these ecCCSD methods stems from the fact that the CC energy, at whatever level of truncation, is fully determined by the one- and two-body clusters via the asymmetric energy formula... 

There are a few other correction procedures that may be considered as extrapolation schemes. The Scaled External Correlation (SEC) and Scaled All Correlation (SAC) metiiods scale tire correlation energy by a factor such that calculated dissociation energy agrees with the experimental value. 

If the particular energy source in question is already taxed (say at a five percent rate), then part of the external cost is already internalized. The true cost of fuel would remain the same (P X 1.04 X 1.10) but the size of the additional tax needed to correct for the externality would be smaller (about five percent instead of ten percent). 

The most simple way to accomplish this objective is to correct the external field operator post factum, as was repeatedly done in magnetic resonance theory, e.g. in [39]. Unfortunately this method is inapplicable to systems with an unrestricted energy spectrum. Neither can one use the method utilizing the Landau-Teller formula for an equidistant energy spectrum of the harmonic oscillator. In this simplest case one need correct... 

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