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WIREs Comput Mol Sci
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Spin‐component‐scaled electron correlation methods

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Abstract Spin‐component‐scaled (SCS) electron correlation methods for electronic structure theory are reviewed. The methods can be derived theoretically by applying special conditions to the underlying wave functions in perturbation theory. They are based on the insight that low‐order wave function expansions treat the correlation effects of electron pairs with opposite spin (OS) and same spin (SS) differently because of their different treatment at the underlying Hartree–Fock level. Physically, this is related to the different average inter‐electronic distances in the SS and OS electron pairs. The overview starts with the original SCS‐MP2 method and discusses its strengths and weaknesses and various ways to parameterize the scaling factors. Extensions to coupled‐cluster and excited state methods as well the connection to virtual‐orbital dependent density functional approaches are highlighted. The performance of various SCS methods in large thermochemical benchmarks and for excitation energies is discussed in comparison with other common electronic structure methods. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods

Left: Ratio of OS and SS MP2 correlation energies versus the size of the triples correction in CCSD(T). Right: Same but versus the D1 diagnostic for nondynamic correlation effects in CCSD. The molecules are CH4, H2O, ethene, ethane, F2, N2, O3, S3, FOOF, and C4H4. All values have been obtained with the cc‐pVTZ47 basis set for PBE048,49/cc‐pVTZ optimized structures. The straight lines represent linear regressions with a correlation coefficient of about 0.6 in both cases.

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MD(a) and MAD (b) of various CIS(D), CC2, and TD‐DFT methods for the dye benchmark set. Values taken from Ref 94.

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Chemical structures of the dye benchmark set.94,119

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WTMADs of various MP2 variants and two double‐hybrid density functionals in kcal/mol for the basic properties (a), the reaction energies (b), the noncovalent interactions (c) and the complete GMTKN30 set (d). ”TZ level” stands for (aug‐)def2‐TZVPP; ”QZ level” for (aug‐)def2‐QZVP. CBS extrapolations are based on these two levels. The numbers show values at the CBS limit for the wave function based methods and at the quadruple‐ζ level for the double hybrids. Values were taken from Ref 111.

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Potential energy curve for the argon dimer with MP2/aug‐cc‐p5Z computed OS and SS correlation energy contributions. The OS and SS parts asymptotically become equal (very similar values are found at about >1.5 × Re) but are significantly different in the equilibrium region already for this simple system.

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SS and OS scaling parameters of different SCS variants and the line given by Eq. (9) along which the spin‐component scaled MP2 energy stays roughly constant. For the neon atom and the N2 molecule at equilibrium bond distance and at a 20% stretched geometry, the third‐order S2‐MP contour lines of E[3] evaluated with the cc‐pVTZ basis are also shown. The inner and outer ellipses correspond to 1 and 2 mEh above the corresponding minimum value, respectively.

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Schematic description of the contributions of exchange and correlation to electronic energies that are relevant for the SCS‐MP2 method. At the Hartree level, only the classical electrostatic interactions of the electrons are considered. Including the Pauli principle leads to Fermi correlation of SS electrons in HF theory, whereas the OS electron pairs remain uncorrelated. This biased starting is corrected by the two SCS factors. Finally, an overall more accurate (balanced) correlation energy than standard MP2 with respect to the full configuration interaction (FCI) limit is obtained. Note that, although the absolute SCS‐MP2 correlation energy is similar to (or even smaller than) MP2, on average it nevertheless yields more accurate chemically relevant relative energies. (Reprinted with permission from Ref 54. Copyright 2008 American Chemical Society.)

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Left: Ratio of OS and SS MP2 correlation energies for alkanes and polyenes of different lengths. Right: Recovered basis set correlation energy with respect to CCSD(T) for the same systems.

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