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J. Phys. Chem. A 2007, 111, 4862-4868
Is Spin-Component Scaled Second-Order Møller-Plesset Perturbation Theory an Appropriate Method for the Study of Noncovalent Interactions in Molecules? Jens Antony and Stefan Grimme* Theoretische Organische Chemie, Organisch-Chemisches Institut der UniVersita¨t Mu¨nster, Corrensstrasse 40, D-48149 Mu¨nster, Germany ReceiVed: January 23, 2007; In Final Form: March 21, 2007
Testing of the spin-component scaled second-order Møller-Plesset (SCS-MP2) method for the computation of noncovalent interaction energies is done with a database of 165 biologically relevant complexes. The effects of the spin-scaling procedure (i.e., MP2 vs SCS-MP2), the basis set size, and the corrections for basis set superposition error (BSSE) are systematically examined. When using two-point basis set extrapolations for the correlation energy, augmentation of the atomic orbital basis with computationally costly diffuse functions is found to be obsolete. In general, SCS-MP2 also improves results for noncovalent interactions statistically on MP2, and significant outliers are removed. Moreover, it is shown that effects of BSSE and one-particle basis set incompleteness almost cancel each other in the case of triple-ζ sets (SCS-MP2/TZVPP or SCSMP2/cc-pVTZ without counterpoise correction), which opens a practical route to efficient computations for large systems. We recommend SCS-MP2 as the preferred quantum chemical wave function based method for the noncovalent interactions in large biologically relevant systems when reasonable coupled-cluster with single and double and perturbative triple excitations (CCSD(T)) calculations cannot be performed anymore. A comparison to MP2 and CCSD(T) interaction energies for n-alkane dimers, however, indicates (and this also holds to a lesser extent for hydrogen-bonded systems) limitations of SCS-MP2 when treating chemically “saturated” interactions. The different behavior of second-order perturbation theory for saturated and for stacked π-systems is discussed.
1. Introduction The fundamental importance of noncovalent (intermolecular) interactions to many fields of science cannot be overestimated. An understanding of real systems and their properties and of all condensed-phase matter must be rooted in the knowledge of noncovalent interactions. Consequently, they are nowadays considered as cornerstones in supramolecular chemistry, materials science, and biochemistry.1-4 Although a very detailed understanding on an atomic or a molecular level is still lacking, important progress has been achieved in recent years in the quantum mechanical description of the relevant forces.5 However, still problematic is the accurate account of the dispersion (van der Waals, vdW) part of the interactions for large systems. Dispersion interactions are ubiquitous, long-range attractive forces that act between separated molecules or fragments even in the absence of charges or permanent electric moments. They stem from many particle (electron correlation) effects that are complicated by the quantum mechanical wave-nature of matter.5 Because of their relatively long-ranged (R-6 dependence with intermolecular distance) character (as compared to other quantum mechanical effects), dispersion contributions to inter- and intramolecular energies become increasingly significant for larger systems that are relevant in supramolecular or nanochemistry.6,7 Coupled-cluster theory8 is currently the most accurate wavefunction-based method for calculating dispersion interactions. Because of the N7 scaling of the computational effort for the coupled-cluster with single and double and perturbative triple * To whom correspondence should be addressed. E-mail: grimmes@ uni-muenster.de; fax: (+49)251-83-36515.
excitations (CCSD(T)) method, where N characterizes the size of the electronic system, calculations on realistic complexes have to be performed with relatively small one-particle basis sets. Usually the basis set dependence of the correlation energy is estimated with lower-level electronic structure methods such as second-order Møller-Plesset perturbation theory (MP2).9-11 Within this standard procedure, systems containing up to 50 atoms have been treated. Beyond this limit, reasonable CCSD(T) calculations cannot be executed with present computational resources, and MP2 alone is not accurate enough. Furthermore, small basis sets make a treatment of the basis set superposition error (BSSE) necessary, for example, with the laborious counterpoise correction.12 The goal of describing macromolecular systems with, for example, several hundreds of atoms and with a consistent treatment of intramolecular dispersion requires alternative approaches. Recently,13 the capability of density functional theory (DFT) including dispersion corrections (DFT-D) to calculate intermolecular interaction energies was tested by comparison with CCSD(T) complete basis set (CBS) limit estimates for an extensive benchmark set provided by Jurecka et al.14 It consists mainly of DNA base pairs and amino acid pairs in a wide variety of realistic geometries and is expected to yield conclusive insight into the performance of quantum chemical methods. It was shown that the DFT-D interaction energies deviate on average by less than 1 kcal‚mol-1, or 10%, from the reference values, which thereby substantially improves the results of pure DFT. Despite this and other successes (see ref 6 for a recent review), the description of dispersion interactions in DFT is mostly
10.1021/jp070589p CCC: $37.00 © 2007 American Chemical Society Published on Web 05/17/2007
Study of Noncovalent Interactions in Molecules empirical (similar to many density functionals themselves), which may limit the accuracy of unconventional systems. The DFT-D approach cannot, for example, account for the influence of changes in the molecular polarizability on the strength of the vdW interaction, because the C6 coefficients are fixed (system independent) parameters.15 If in a series of structurally similar molecules of increasing size the HOMO-LUMO gap gets smaller, then DFT will account for the increase of the fragment polarizability (induction effects), but the dispersion correction to the intermolecular interaction energies remains (pairwise) additive. To take care of system-dependent vdW effects, we currently have to resort to wavefunction-based methods (for a recent DFT approach to this problem see refs 16 and 17). Because dispersion is an electron correlation effect, a level of at least second-order perturbation theory (e.g., MP2) has to be applied. Whereas it is rather successful in the case of saturated or hydrogen-bonded molecules, MP2 quite strongly overestimates dispersion in π-stacked systems (see, e.g., refs 18 and 19). A modification of MP2, in which the total MP2 correlation energy is partitioned into parallel- and antiparallel-spin components that are separately scaled, is denoted as SCS-MP2.20 In general, this method improves in almost all respects on standard MP2, and many successful applications in various areas of chemistry are documented.7,21-24 Preliminary studies on systems with important dispersive interactions already indicated significant improvements as compared to standard MP2.20,25,26 This has been recently confirmed by local SCS-LMP2 calculations of the benzene dimer.19 The SCS-LMP2 interaction energies for the parallel-displaced and sandwich structures are in excellent agreement with the best available literature values along the entire potential energy curves, whereas for the T-shaped structure SCS-LMP2 slightly underestimates the binding energy.19 Therefore, SCS-MP2 appears to be a suitable method for investigating larger π-systems. A variant of SCS-MP2 where only the (scaled) opposite-spin correlation energy is retained (termed SOS-MP2) has been proposed by Jung et al.27 This approach has been further developed to become asymptotically correct for vdW interactions (MOS-MP2).28 Although promising applications of these OSMP2 methods have recently been published for very large vdW systems,29 a comprehensive study of their performance for a wide variety of vdW complexes is still missing. Very recently, specially adjusted scaling parameters (denoted as SCSN) for weak and stacking interactions have been proposed by Hill and Platts.30 To get a more conclusive picture about the performance of MP2 and related SCS methods, the current work tests the SCSMP2 method with the database of 165 noncovalent complexes14 that has already been used in the aforementioned DFT-D study.13 The effects of the spin-scaling procedure (i.e., MP2 vs SCS-MP2), the basis set size, and the BSSE are systematically examined. We also tested SOS-MP227, but because it behaves qualitatively very similar to SCS-MP2 (in fact slightly worse on average), we find it inconvenient and not very enlightening to further expand the data presentation and discussion. Therefore, we concentrate on SCS-MP2 and MP2 and briefly present only a few representative statistical data for SOS-MP2 (and SCSNMP2) in section 3.4 (additionally, all data can be found in Supporting Information). It is shown here comprehensively, that for noncovalent interactions SCS-MP2 improves, in general (statistically), on MP2. Moreover, we find that effects of the BSSE and of the one-particle basis set almost cancel each other in the case of
J. Phys. Chem. A, Vol. 111, No. 22, 2007 4863 triple-ζ basis sets (SCS-MP2/TZVPP or SCS-MP2/cc-pVTZ without counterpoise correction), which opens a practical route to efficient computations for large systems. A comparison to MP2 and CCSD(T) interaction energies for n-alkane dimers,31 however, indicates limitations of SCS-MP2 when treating saturated molecules. 2. Technical Details All calculations were performed with the TURBOMOLE suite of programs,32,33 and the resolution of identity (RI) approach34,35 was used for the correlation energy. The Hartree-Fock calculations were done without approximation. Because the error introduced by the RI treatment is insignificant (
