Convergence toward exact solution

Coupled cluster, configuration interaction, and other correlated ab initio methods Small organic and inorganic molecules About ten atoms Structures and binding energies High computational effort, benchmark quality results, systematic (but slow) convergence towards exact solution... 

Although for brevity s sake the 3-MCSE has not been considered here, it may be convenient to mention it in these final comments. This equation, which depends on the 1-CSE, does not have a unique solution. Indeed, this equation is satisfied not only by the FCI 3-RDM but also by the Hartree-Fock one. Alcoba [48] performed a series of calculations with the 3-MCSE for the beryllium iso-electronic series. Alcoba took as initial data a set of RDMs that corresponded to a state that had already some correlation and whose energy was below the Hartree-Fock s one. The results of these calculations showed that there was a smooth although very slow convergence toward the exact solution. For larger systems the situation will probably be similar to the 4-MCSE one and a strict... 

When = 0.90 this gives x2 = xCd = 3.1(10-4) while Eq. (A14) gives Xj = xHg = 0.9922. This is in agreement with the more exact computer calculations whose results are shown in Fig. 31. The activity coefficients in the metal-rich liquid in equilibrium with a solid solution with = 0.9 and at 673°K obtained from a computer calculation also agree to within 3% with the approximate values listed above. Thus the relative stability of CdTe(s) compared to HgTe(s) is a major factor in the tie-lines converging toward the Hg corner. The smallness of Q2/k5, which is determined in part by the large value of 244 for the activity coefficient of the CdTe liquid species, also enters and is less transparent. 

As pointed out in Secs. 2.3 and 4.2, wave function-based quantum chemical calculations can be set up such that a smooth and often monotonic convergence is established toward the exact result (i.e., toward the exact solution of the nonrelativistic electronic Schrodinger equation) (Eq. (2)). From an understanding of this convergence, we may improve upon our calculations in a systematic manner. Moreover, we may provide reliable estimates of the exact solution by means of extrapolation. 

The HF model is a kind of branching point, where either additional approximations can be invoked, leading to semi-empirical methods, or it can be improved by adding additional determinants, thereby generating models that can be made to converge towards the exact solution of the electronic Schrddinger equation. ... 

The hierarchy of coupled-cluster models provides a clear route towards the exact solution of the Schrodinger equation, but the slow basis-set convergence limits the accuracy sometimes even for small molecules. The way to overcome this problem is to combine the coupled-cluster model with the explicitly correlated approach. It can be done, in principle, for any model within the coupled-cluster hierarchy. The main task of this work is, however, the implementation of explicitly correlated CCSD model, hence the discussion will be focused on this particular model. 

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