Electron correlation methods convergence

In a nutshell (electron correlated methods) Complete electronic description, straightforward interpretation Highly accurate methods that can rival experiment for small organic molecules Very time consuming Cl methods are not size-extensive CC methods are non-variational Slow convergence Strongly basis set dependent Systematic improvement straightforward. 

As the accuracy of electron correlation methods is Hmited by the choice of the dimension of the active orbital space, the convergence behavior of the spin density with respect to the size of the active space must be studied to ensure that accurate ah initio spin densities are obtained. Although the CASSCF spin densities were quantitatively converged for medium-sized active orbital spaces, larger active spaces (more than 13 electrons correlated in 13 orbitals) were found to be unstable, that is, active space orbitals have been rotated out of the active space during the MCSCF procedure, while the spin densities started to diverge compared to the smaller sized CASSCF results. 

Cremer, D. and He, Z. (1996) Sixth-order Moller-Plesset perturbation theory on the convergence of the MPn series. J. Phys. Chem., 100, 6173-6188. Grimme, S., Goerigk, L., and Fink, R.F. (2012) Spin-component-scaled electron correlation methods. Wiley Interdiscip. Rev. Comput. Mol Sci., 2, 886-906. Schwabe, X. and Grimme, S. (2008) Xheoretical thermodynamics for... 

Things have moved on since the early papers given above. The development of Mpller-Plesset perturbation theory (Chapter 11) marked a turning point in treatments of electron correlation, and made such calculations feasible for molecules of moderate size. The Mpller-Plesset method is usually implemented up to MP4 but the convergence of the MPn series is sometimes unsatisfactory. The effect... 

It is usually observed that the CP correction for methods including electron correlation is larger and more sensitive to the size of the basis set, than that at the HE level. This is in line with the fact that the HE wave function converges much faster with respect to the size of the basis set tlian correlated wave functions. 

When natural orbitals are determined from a wave function which only includes a limited amount of electron correlation (i.e. not full Cl), the convergence property is not rigorously guaranteed, but since most practical methods recover 80-90% of the total electron correlation, the occupation numbers provide a good guideline for how important a given orbital is. This is the reason why natural orbitals are often used for evaluating which orbitals should be included in an MCSCF wave function (Section 4.6). 

We need to look at the convergence as a function of basis set and amount of electron correlation (Figure 4.2). For the former we will use the correlation consistent basis sets of double, triple, quadruple, quintuple and, when possible, sextuple quality (Section 5.4.5), while the sensitivity to electron correlation will be sampled by the HF, MP2 and CCSD(T) methods (Sections 3.2, 4.8 and 4.9). Table 11.1 shows how the geometry changes as a function of basis set at the HF level of theory. 

The HF level as usual overestimates the polarity, in this case leading to an incorrect direction of the dipole moment. The MP perturbation series oscillates, and it is clear that the MP4 result is far from converged. The CCSD(T) method apparently recovers the most important part of the electron correlation, as compared to the full CCSDT result. However, even with the aug-cc-pV5Z basis sets, there is still a discrepancy of 0.01 D relative to the experimental value. 

Establishing a hierarchy of rapidly converging, generally applicable, systematic approximations of exact electronic wave functions is the holy grail of electronic structure theory [1]. The basis of these approximations is the Hartree-Fock (HF) method, which defines a simple noncorrelated reference wave function consisting of a single Slater determinant (an antisymmetrized product of orbitals). To introduce electron correlation into the description, the wave function is expanded as a combination of the reference and excited Slater determinants obtained by promotion of one, two, or more electrons into vacant virtual orbitals. The approximate wave functions thus defined are characterized by the manner of the expansion (linear, nonlinear), the maximum excitation rank, and by the size of one-electron basis used to represent the orbitals. 

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