Scaling electron correlation methods

Grimme, S. Goaigk, L. Fink, R. F. Spin-component-scaled electron correlation methods, WlREs Comput. Mol. Sci. 2012,2, 886-906. 

Linear-scaling electron-correlation methods were developed by 1) combining the Pulay-Saebo local correlation variant [136] with integral-direct techniques [131] and consequently exploiting the spatial locahty of the electron-correlation effect 2) Laplace-transform techniques suggested in [137] and apphed in [129]. 

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... 

Note that one particularly attractive feature of Eq. (7.60) is that if the particular electron correlation method has available analytic derivatives, so too must Ssac-e.c.m., since derivatives for the latter will be simply determined as appropriately scaled sums of the e.c.m. and HF derivatives. Geometry optimization, and indeed the entire calculation, can essentially be carried out for exactly the cost of the e.c.m. 

The second consideration in choosing a method is the level of electron correlation. A range of methods from no electron correlation (Hartree-Fock methods) to full configuration interaction is available however, the more extensive the electron correlation, the more computationally demanding the calculations become. Some electron correlation methods, such as the Mpller-Plesset method, can scale as N5 where N is the number of electrons.45 One can imagine that such methods become impractical for larger model systems. 

Schutz, M. Low-order scaling local electron correlation methods. III. Linear scaling local perturbative triples correction (T), J. Chem. Phys. 2000,113,9986-10001. 

Linear Scaling Techniques Semi-Empirical Methods 3.9.1 Neglect of Diatomic Differential Overlap Approximation (NDDO) 3.9.2 Intermediate Neglect of Differential Overlap Approximation (INDO) 80 81 82 83 4.13 Locahzed Orbital Methods 4.14 Summary of Electron Correlation Methods 4.15 Excited States References 5 Basis Sets 144 144 147 148 150... 

Second-order Moller-Plesset perturbation theory (MP2) is the computationally least expensive and most popular ab initio electron correlation method [4,15]. Except for transition metal compounds, MP2 equilibrium geometries are of comparable accuracy to DFT. However, MP2 captures long-range correlation effects (like dispersion) which are lacking in present-day density functionals. The computational cost of MP2 calculations is dominated by the integral transformation from the atomic orbital (AO) to the molecular orbital (MO) basis which scales as 0(N5) with the system size. This four-index transformation can be avoided by introduction of the RI integral approximation which requires just the transformation of three-index quantities and reduces the prefactor without significant loss in accuracy [36,37]. This makes RI-MP2 the most efficient alternative for small- to medium-sized molecular systems for which DFT fails. 

Two relevant topics have been ignored completely in this short chapter the treatment of electron correlation with more sophisticated methods than DFT (that remains unsatisfactory from many points of view) and the related subject of excited states. Wave function-based methods for the calculation of electron correlation, like the perturbative Moller-Plesset (MP) expansion or the coupled cluster approximation, have registered an impressive advancement in the molecular context. The computational cost increases with the molecular size (as the fifth power in the most favorable cases), especially for molecules with low symmetry. That increase was the main disadvantage of these electron correlation methods, and it limited their application to tiny molecules. This scaling problem has been improved dramatically by modern reformulation of the theory by localized molecular orbitals, and now a much more favorable scaling is possible with the appropriate approximations. Linear scaling with such low prefactors has been achieved with MP schemes that the... 

Low-order scaling local electron correlation methods. I. Linear scaling local MP2 ... 

The work of Schutz, Hetzer and Wemer " begins a series of papers exploring the development of local electron correlation methods with low-order scaling by presenting a linear scaling local MP2 . They describe a novel multipole approximation based on a sphtting of the Coulomb operator into two terms... 

The accurate treatment of transition metal sandwich compounds is a considerable challenge for ab initio quantum chemistry. For example, it took about a decade and numerous large-scale calculations using different electron correlation methods until recently the metal-ring distance of ferrocene was obtained by Park and Almloef (1991) in satisfactory agreement with the experimental value. Studies on the lanthanide and actinide sandwich compounds appear to be even more complicated. This is due to the larger effects of relativity and electron correlation. 

Local electron-correlation methods are ab-initio wavefunction-based electronic-structure methods that exploit the short-range nature of dynamic correlation effects and in this way allow linear scaling 0 N) in the electron-correlation calculations [128,129,131-135] to be attained. 0 N) methods are applied to the treatment of extended molecular systems at a very high level of accuracy and rehabihty as CPU time, memory and disk requirements scale hneaily with increasing molecular size N. 

Electron Correlation Methods. IV. Linear Scaling Local Coupled-Cluster (LCCSD). 

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