CISD wave function

We have carried out several applications showing the promise of this procedure [63,64], as well as addressed the question of the size-consistency and size-extensivity [65-67], to which we wish to turn our attention again in this paper. Finally, we have also extended the idea of externally corrected (ec) SR CCSD methods [68-70] (see also Refs. [21,24]) to the MR case, introducing the (N, M)-CCSB method [71], which exploits an Preference (A R) CISD wave functions as a source of higher-than-pair clusters in an M-reference SU CCSD method. Both the CMS and (N, M)-CCSD allow us to avoid the undesirable intruder states, while providing very encouraging results. 

If the CISD wave functions for two identical molecules are multiplied, to give the wave function for the pair of molecules, there are terms in the resulting wave function in which both molecules are doubly excited. Since these terms represent quadruple excitations from the HF configuration, they are not included in the CISD wave function for two identical molecules at infinity. Consequently, the CISD energy for a pair of identical molecules is higher than twice the CISD energy for an individual molecule. 

In the last 20 years, Cl calculations based on a single reference function have lost favor among practitioners. The principal shortcoming of these approaches is that they do not satisfy the property of size-consistency, which means that the Cl energy does not scale properly with the size of the system [112]. It is fairly easy to see why this is so. Consider two beryllium atoms, separated by a distance sufficiently large that the true physical interaction between the atoms vanishes. In a CISD description of this system, contributions to the wave function are excluded in which two electrons on each beryllium atom are in virtual orbitals, since these correspond to quadruply excited determinants and would require a method such as CISDQ or CISDTQ for their inclusion. However, the CISD wave function for a single beryllium atom contains all determinants with two electrons in virtual orbitals. Since Cl methods involve... 

Our recently developed reduced multireference (RMR) CCSD method [16, 21, 22, 23, 24, 25] represents such a combined approach. In essence, this is a version of the so-called externally corrected CCSD method [26, 27, 28, 29, 30, 31, 32, 33, 34] that uses a low dimensional MR CISD as an external source. Thus, rather than neglecting higher-than-pair cluster amplitudes, as is done in standard CCSD, it uses approximate values for triply and quadruply excited cluster amplitudes that are extracted by the cluster analysis from the MR CISD wave function. The latter is based on a small active space, yet large enough to allow proper dissociation, and thus a proper account of dynamic correlation. It is the objective of this paper to review this approach in more detail and to illustrate its performance on a few examples. 

At the ab-initio level, the most obvious possibility is offered by CAS SCF or CAS FCI (i.e., Cl within the CAS or, equivalently, CAS SCF without the orbital reoptimization based on RHF orbitals, cf. [33, 34]) wave functions based on the smallest possible active-space that warrants the correct description of the dissociation channel at hand. This option was also suggested by Stolarczyk [29], although we are not aware of any concrete implementation. Our testing proved to be very encouraging [33, 34], particularly for open shell systems, in which case we employed the spin-adapted CCSD based on the unitary group approach (UGA) [16, 36]. Even in the case of triple bond breaking, the applicability of the CCSD approximation can be significantly extended, as will be shown in Sect. 4. Most recently, we have explored the MR CISD wave function as an external source, as described in the next section. 

In general, the MR CISD wave function based on an M-dimensional reference space has the form... 

To summarize, the RMR CCSD method involves the following three steps (i) We choose a suitable reference space and compute the corresponding MR CISD. Next, (ii) we compute r3(0) and r4(0) clusters by cluster analyzing the MR CISD wave function of step (i), and finally (iii) we use these amplitudes to generate and solve ecCCSD equations. The details of the actual implementation of the RMR CCSD method for various types of reference spaces can be found in our earlier papers [21, 22, 23, 24, 25]. 

The other possibility is to focus on the MR CISD wave function and exploit the Tj(0) and r20) clusters it provides to account for the dynamic correlation due to disconnected triples and quadruples that are absent in the MR CISD wave function. This approach, recently proposed and tested by Meissner and Gra-bowski [42], may thus be characterized as a CC-ansatz-based Davidson-type correction to MR CISD. The duplication of contributions from higher-than-doubly excited configurations that arise in MR CISD as well as through the CC exponential ansatz is avoided by a suitable projection onto the orthogonal complement to the MR CISD N-electron space. The results are very encouraging, particularly in view of their affordability, though somewhat inferior to RMR CCSD. 

We can thus conveniently exploit the Cl-type wave functions as a source of approximate three- and four-body amplitudes. This is precisely the basis of the so-called reduced MR (RMR) CC method [216,218,219,221]. Modest-size MR CISD wave functions are nowadays computationally very affordable, and their cluster analysis provides us with a relatively small subset of the most important three- and four-body cluster amplitudes, which can be used to correct the standard CCSD equations. Moreover, such amplitudes implicitly account for higher than four-body amplitudes as well, as long as they are present in the MR CISD wave function. In this way, we were able to properly describe even the difficult triple-bond breaking in the nitrogen molecule [217]. Amplitude-type corrections are even more useful in the MR SU CCSD approach (see below). Very similar results are obtained with the energy-correcting CCSD, in which case we employ the MR CISD wave function in the asymmetric energy formula [220,221]. 

With the GMS-based SU CCSD method, we were able to carry out a series of test calculations for model systems that allow a comparison with full Cl results, considering GMSs of as high a dimension as 14. These results are most promissing. Moreover, we have formulated a generalization of the RMR CCSD method, resulting in the so-called (M, N)-CCSD approach [226] that employs an M-reference MR CISD wave functions as a source of higher-than-pair clusters in an Al-reference MR SU CCSD (clearly, we require that M S N). In this way, the effect of intruders can be taken care of via external corrections, which are even more essential at the MR level than in the SR theory, because, in contrast to... 

The objective of this study is to explore the possibilities offered by the AL version of RMR CCSD, since in the RMR-type approaches one obtains the three and four body clusters from the corresponding MR CISD wave function. 

Recently, both valence bond (VB) and complete active space (CAS) SCF and CAS FCI wave functions were employed as a source of T3 and T< cluster amplitudes (11,13) (see also Ref. 12). The most satisfactory approach, however, that was developed very recently, relies on the MR CISD wave function, based on a relatively small model space. This approach is referred to as the reduced MR (RMR) CCSD method (15-17). 

As already pointed out in Ref. 13, the externally corrected CCSD is equivalent to (truncated) CCSDTQ with zero-iteration on and T4 amplitudes that are in turn obtained from some external sources. Depending on the source of these amplitudes, we usually deal with only a proper subset of all possible T3 and T4 amplitudes. This subset is fixed in the externally corrected CCSD calculations. The RMR CCSD is then a special case of the general externally corrected CCSD in which the MR CISD wave function is used as the external source. The RMR CCSD method represents in fact a multireference approach in the sense that it is uniquely defined by the choice of the reference space and the fact that the RMR CCSD wave function involves the same number of connected cluster amplitudes as the corresponding genuine MR CCSD, such as the state-universal CCSD employing the same reference space. 

The split-amplitude strategy represents the total amplitudes as the sum of an a priori known approximate value, obtained from some external source, and an unknown correcting term. Assuming, further, that the known amplitudes represent a good approximation to the true ones, the unknown corrections can be obtained to a high degree of accuracy from a set of linear equations. The results of this article show that when a proper reference space is used, the connected clusters obtained from the MR CISD wave function represent indeed a very good approximation, and the almost linear versions of the RMR CCSD method performs very well. 

The simplest way of truncating the complete set of electronic configurations is to include those Slater determinants in the Cl expansion that differ by one, two, three, and so on spin orbitals compared to some reference configuration Oq. Truncated Cl expansions then are linear combinations of the reference determinant and aU singly, doubly, triply, and so on excited configurations. For instance, the Cl singles doubles (CISD) wave functions can be written as... 

Because the CISD wave function is a variation function, the variation theorem assures us that the CISD energy cannot be less than the true energy. The CISD method is therefore variational. 

Make a rough estimate of the total number of determinants in the MR-CISD wave function for a system with 74 electrons, 154 orbitals and a CAS(2,2)CI reference wave function. Calculate the percentage of 2h-2p excitations in the MR-CISD wave function (neglect the Ih, Ip, Ih-lp, 2h and 2p excitations, they give rise to a very small number of determinants). 

Consider first the calculations at equilibrium. For the monomer, the CISD wave function is dominated by the Hartree-Fock determinant and recovers as much as 94.5% of the correlation... 

Table 11 The CISD monomer correlation energy (in mEh) for one to eight noninteracting water molecules, calculated in the cc-pVDZ basis with and without the Davidson correction applied. The weight of the Hartree-Fock configuration in the CISD wave function IVhf is also listed. The calculations have been carried out for the C2 water molecule with R = / ref and R = IR ct where Rra = 1.84345ao- The HOH bond angle is fixed at 110.565°. At Rnt and 2R,et, the Hartree-Fock energies are —76.0240 and —75.5877 Eh, respectively, and the t CI correlation energies are —217.8 and —364.0 mEh, respectively...

Fig. 11.2. The weight of the Hartree-Fock determinant in the FCI wave function (grey line) and in the CISD wave function (black line) as a function of the number of noninteracting water monomers in the cc-pVDZ basis. The plot on the left corresponds to the molecular equilibrium geometry the plot on the right represents a situation where the OH bonds have been stretched to twice the equilibrium bond distance. For details on the calculations, see Table 11.2.

Comparing the two plots in Figure 11.2, we find that the Hartree-Fock configuration plays a more dominant role at the CISD level than at the FCI level, in particular for the stretched molecules. This behaviour occurs because the CISD wave function lacks the ability to describe properly the independent excitations in the noninteracting monomers. 

In conclusion, the CISD wave function is unable to mend the deficiencies of the Hartree-Fock solutions for the allyl radical. Of course, we may further reduce the discrepancies between the UHF- and RHF-based solutions by including in the Cl expansion excitations higher than doubles. Quite apart from being very expensive, such a solution is not very attractive since the symmetry-broken solutions would probably persist for rather high excitation levels. A more attractive and fundamental solution to the symmetry problem would be to select the most important configurations for the description of the allyl radical and to carry out an MCSCF calculation where the orbitals and configurations are simultaneously optimized. As discussed in the next chapter, such a procedure yields a wave function that exhibits none of the unphysical behaviour characterizing the Hartree-Fock-based solutions. 

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