Density-functional theory-symmetry-adapted intermolecular perturbation theory with density fitting: A new efficient method to study intermolecular interaction energies

Intermolecular electrostatic energies using density fitting

A method is presented to calculate the electron-electron and nuclear-electron intermolecular Coulomb interaction energy between two molecules by separately fitting the unperturbed molecular electron density of each monomer. This method is based on the variational Coulomb fitting method which relies on the expansion of the ab initio molecular electron density in site-centered auxiliary basis sets. By expanding the electron density of each monomer in this way the integral expressions for the intermolecular electrostatic calculations are simplified, lowering the operation count as well as the memory usage. Furthermore, this method allows the calculation of intermolecular Coulomb interactions with any level of theory from which a one-electron density matrix can be obtained. Our implementation is initially tested by calculating molecular properties with the density fitting method using three different auxiliary basis sets and comparing them to results obtained from ab initio calculations. These properties include dipoles for a series of molecules, as well as the molecular electrostatic potential and electric field for water. Subsequently, the intermolecular electrostatic energy is tested by calculating ten stationary points on the water dimer potential-energy surface. Results are presented for electron densities obtained at four different levels of theory using two different basis sets, fitted with three auxiliary basis sets. Additionally, a one-dimensional electrostatic energy surface scan is performed for four different systems (H2O dimer, Mg2+-H2O, Cu+-H2O, and n-methyl-formamide dimer). Our results show a very good agreement with ab initio calculations for all properties as well as interaction energies.

London dispersion forces by range-separated hybrid density functional with second order perturbational corrections: The case of rare gas complexes

A satisfactory account of the van der Waals (vdW) (London dispersion) forces is, in general not possible by the Kohn-Sham method using standard local, semilocal generalized gradient approximation (GGA), or meta-GGA density functionals. The recently proposed range-separated hybrid (RSH) approach, supplemented by second order perturbational corrections (MP2) to include long-range dynamic correlation effects, offers a physically consistent, seamless description of dispersion forces. It is based on a rigorous generalization of the Kohn-Sham method, where long-range exchange and correlation effects are treated by wave function methods, while short-range electron exchange and correlation are handled by local or semilocal functionals. The method is tested on a series of rare gas dimers in comparison with standard wave function theory and density functional theory approaches. In contrast to the most successful exchange correlation functionals, which describe at best the vdW minimum, the RSH+MP2 approach is valid also in the asymptotic region and the potential curve displays the correct 1/R(6) behavior at large internuclear separations. In contrast to usual MP2 calculations, the basis set superposition error is considerably reduced, making RSH+MP2 an ideal tool for exploring the potential energy surface of weakly bound molecular complexes.

Stabilisation energy of C6H6⋯C6X6(X = F, Cl, Br, I, CN) complexes: complete basis set limit calculations at MP2 and CCSD(T) levels

Stabilisation energies of stacked structures of C(6)H(6)...C(6)X(6) (X = F, Cl, Br, CN) complexes were determined at the CCSD(T) complete basis set (CBS) limit level. These energies were constructed from MP2/CBS stabilisation energies and a CCSD(T) correction term determined with a medium basis set (6-31G**). The former energies were extrapolated using the two-point formula of Helgaker et al. from aug-cc-pVDZ and aug-cc-pVTZ Hartree-Fock energies and MP2 correlation energies. The CCSD(T) correction term is systematically repulsive. The final CCSD(T)/CBS stabilisation energies are large, considerably larger than previously calculated and increase in the series as follows: hexafluorobenzene (6.3 kcal mol(-1)), hexachlorobenzene (8.8 kcal mol(-1)), hexabromobenzene (8.1 kcal mol(-1)) and hexacyanobenzene (11.0 kcal mol(-1)). MP2/SDD** relativistic calculations performed for all complexes mentioned and also for benzene[dot dot dot]hexaiodobenzene have clearly shown that due to relativistic effects the stabilisation energy of the hexaiodobenzene complex is lower than that of hexabromobenzene complex. The decomposition of the total interaction energy to physically defined energy components was made by using the symmetry adapted perturbation treatment (SAPT). The main stabilisation contribution for all complexes investigated is due to London dispersion energy, with the induction term being smaller. Electrostatic and induction terms which are attractive are compensated by their exchange counterparts. The stacked motif in the complexes studied is very stable and might thus be valuable as a supramolecular synthon.

How accurate is the density functional theory combined with symmetry-adapted perturbation theory approach for CH–π and π–π interactions? A comparison to supermolecular calculations for the acetylene–benzene dimer

Five different orientations of the acetylene-benzene dimer including the T-shaped global minimum structure are used to assess the accuracy of the density functional theory combined with symmetry adapted perturbation theory (DFT-SAPT) approach in its density-fitting implementation (DF-DFT-SAPT) for the study of CH-pi and pi-pi interactions. The results are compared with the outcome of counterpoise corrected supermolecular calculations employing second-order Møller-Plesset (MP2), spin-component scaled MP2 (SCS-MP2) and single and double excitation coupled cluster theory including perturbative triple excitations (CCSD(T)). For all considered orientations MP2 predicts much deeper potential energy curves with considerably shifted minima compared to CCSD(T) and DFT-SAPT. In spite of being an improvement over the results of MP2, SCS-MP2 tends to underestimate the well depth while DFT-SAPT, employing an asymptotically corrected hybrid exchange-correlation potential in conjunction with the adiabatic local density approximation for the exchange-correlation kernel, is found to be in excellent agreement with CCSD(T). Furthermore, DFT-SAPT provides a detailed understanding of the importance of the electrostatic, induction and dispersion contributions to the total interaction energy and their repulsive exchange corrections.

Density functional theory with dispersion corrections for supramolecular structures, aggregates, and complexes of (bio)organic molecules

Kohn-Sham density functional theory (KS-DFT) is nowadays the most widely used quantum chemical method for electronic structure calculations in chemistry and physics. Its further application in e.g. supramolecular chemistry or biochemistry has mainly been hampered by the inability of almost all current density functionals to describe the ubiquitous attractive long-range van der Waals (dispersion) interactions. We review here methods to overcome this defect, and describe in detail a very successful correction that is based on damped -C(6).R(-6) potentials (DFT-D). As examples we consider the non-covalent inter- and intra-molecular interactions in unsaturated organic molecules (so-called pi-pi stacking in benzenes and dyes), in biologically relevant systems (nucleic acid bases/pairs, proteins, and 'folding' models), between fluorinated molecules, between curved aromatics (corannulene and carbon nanotubes) and small molecules, and for the encapsulation of methane in water clusters. In selected cases we partition the interaction energies into the most relevant contributions from exchange-repulsion, electrostatics, and dispersion in order to provide qualitative insight into the binding character.

One-electron properties and electrostatic interaction energies from the expectation value expression and wave function of singles and doubles coupled cluster theory

One-electron density matrices resulting from the explicitly connected commutator expansion of the expectation value were implemented at the singles and doubles coupled cluster (CCSD) level. In the proposed approach the one-electron density matrix is obtained at a little extra cost in comparison to the calculation of the CCSD correlation energy. Therefore, in terms of the computational time the new method is significantly less demanding than the conventional linear-response CCSD theory which requires additionally an expensive calculation of the left-hand solution of the CCSD equations. The quality of the new density matrices was investigated by computing a set of one-electron properties for a series of molecules of varying sizes and comparing the results with data obtained using the full configuration interaction method or higher level coupled cluster theory. It has been found that the results obtained using the new approach are of the same quality as those predicted by the linear-response CCSD method. The novel one-electron density matrices have also been applied to study the energy of the electrostatic interaction for a number of van der Waals complexes, including the benzene and azulene dimers.

Third-order interactions in symmetry-adapted perturbation theory

We present an extension of many-body symmetry-adapted perturbation theory (SAPT) by including all third-order polarization and exchange contributions obtained with the neglect of intramonomer correlation effects. The third-order polarization energy, which naturally decomposes into the induction, dispersion, and mixed, induction-dispersion components, is significantly quenched at short range by electron exchange effects. We propose a decomposition of the total third-order exchange energy into the exchange-induction, exchange-dispersion, and exchange-induction-dispersion contributions which provide the quenching for the corresponding individual polarization contributions. All components of the third-order energy have been expressed in terms of molecular integrals and orbital energies. The obtained formulas, valid for both dimer- and monomer-centered basis sets, have been implemented within the general closed-shell many-electron SAPT program. Test calculations for several small dimers have been performed and their results are presented. For dispersion-bound dimers, the inclusion of the third-order effects eliminates the need for a hybrid SAPT approach, involving supermolecular Hartree-Fock calculations. For dimers consisting of strongly polar monomers, the hybrid approach remains more accurate. It is shown that, due to the extent of the quenching, the third-order polarization effects should be included only together with their exchange counterparts. Furthermore, the latter have to be calculated exactly, rather than estimated by scaling the second-order values.

Aliphatic C−H/π Interactions: Methane−Benzene, Methane−Phenol, and Methane−Indole Complexes

Noncovalent C-H/pi interactions are prevalent in biochemistry and are important in molecular recognition. In this work, we present potential energy curves for methane-benzene, methane-phenol, and methane-indole complexes as prototypes for interactions between C-H bonds and the aromatic components of phenylalanine, tyrosine, and tryptophan. Second-order perturbation theory (MP2) is used in conjunction with the aug-cc-pVDZ and aug-cc-pVTZ basis sets to determine the counterpoise-corrected interaction energy for selected complex configurations. Using corrections for higher-order electron correlation determined with coupled-cluster theory through perturbative triples [CCSD(T)] in the aug-cc-pVDZ basis set, we estimate, through an additive approximation, results at the very accurate CCSD(T)/aug-cc-pVTZ level of theory. Symmetry-adapted perturbation theory (SAPT) is employed to determine the physically significant components of the total interaction energy for each complex.

Potential Energy Surface for the Benzene Dimer and Perturbational Analysis of π−π Interactions

We present a complete 6-dimensional potential energy surface for the benzene dimer obtained using symmetry-adapted perturbation theory (SAPT) of intermolecular interactions based on Kohn-Sham's description of monomers. Ab initio calculations were performed for 491 dimer geometries in a triple-zeta-quality basis set supplemented by bond functions. An accurate analytic fit to the ab initio results has been developed and low-energy stationary points on the potential energy surface have been found. We have determined that there are three minima on the surface. Two of them, the tilted T-shape and the parallel-displaced, are nearly isoenergetic with interaction energies of -2.77 and -2.74 kcal/mol, respectively. The third minimum, a twisted edge-to-edge conformation, is significantly less attractive, with the interaction energy equal to -1.82 kcal/mol. Both the T-shape and sandwich geometries, sometimes assumed to be minima, are shown to be only saddle points. The potential energy surface is extremely flat between the two lowest minima, the barrier being only 0.10 kcal/mol above the global minimum. The second-virial coefficient obtained with the new potential agrees well with experimental results over a wide range of temperatures. The SAPT approach rigorously decomposes the interaction energy into physical components. The relative importance of these components has been analyzed.

High-Accuracy Quantum Mechanical Studies of π−π Interactions in Benzene Dimers

Although supramolecular chemistry and noncovalent interactions are playing an increasingly important role in modern chemical research, a detailed understanding of prototype noncovalent interactions remains lacking. In particular, pi-pi interactions, which are ubiquitous in biological systems, are not fully understood in terms of their strength, geometrical dependence, substituent effects, or fundamental physical nature. However, state-of-the-art quantum chemical methods are beginning to provide answers to these questions. Coupled-cluster theory through perturbative triple excitations in conjunction with large basis sets and extrapolations to the complete basis set limit have provided definitive results for the binding energy of several configurations of the benzene dimer, and benchmark-quality ab initio potential curves are being used to calibrate new density functional and force-field models for pi-pi interactions. Studies of substituted benzene dimers indicate flaws in the conventional wisdom about substituent effects in pi-pi interactions. Three-body and four-body interactions in benzene clusters have also been examined.

Interaction potential for water dimer from symmetry-adapted perturbation theory based on density functional description of monomers

A new six-dimensional interaction potential for the water dimer has been obtained by fitting interaction energies computed at 2510 geometries using a variant of symmetry-adapted perturbation theory (SAPT) based on density functional theory (DFT) description of monomers, referred to as SAPT(DFT). The stationary points, second virial coefficient, vibration-rotation-tunneling spectrum, and structure of liquid water computed with the new potential are in very good agreement with experiment and advanced ab initio calculations, confirming the high level of accuracy provided by SAPT(DFT).

Comparison of overlap-based models for approximating the exchange-repulsion energy

Different ways of approximating the exchange-repulsion energy with a classical potential function have been investigated by fitting various expressions to the exact exchange-repulsion energy for a large set of molecular dimers. The expressions involve either the orbital overlap or the electron-density overlap. For comparison, the parameter-free exchange-repulsion model of the effective fragment potential (EFP) is also evaluated. The results show that exchange-repulsion energy is nearly proportional to both the orbital overlap and the density overlap. For accurate results, a distance-dependent correction is needed in both cases. If few parameters are desired, orbital overlap is superior to density overlap, but the fit to density overlap can be significantly improved by introducing more parameters. The EFP performs well, except for delocalized pi systems. However, an overlap expression with a few parameters seems to be slightly more accurate and considerably easier to approximate.

Second-order correction to perfect pairing: An inexpensive electronic structure method for the treatment of strong electron-electron correlations

We have formulated a second-order perturbative correction for perfect-pairing wave functions [PP2] based on similarity-transformed perturbation techniques in coupled cluster theory. The perfect-pairing approximation is used to obtain a simple reference wave function which can qualitatively describe bond breaking, diradicals, and other highly correlated systems, and the perturbative correction accounts for the dynamical correlation. An efficient implementation of this correction using the resolution of the identity approximation enables PP2 to be computed at a cost only a few times larger than that of canonical MP2 for systems with hundreds of active electrons and tens of heavy atoms. PP2 significantly improves on MP2 predictions in various systems with a challenging electronic structure.

Distributed polarizabilities obtained using a constrained density-fitting algorithm

A computationally efficient method for obtaining distributed polarizabilities of arbitrary rank using a constrained density-fitting algorithm is demonstrated on the hydrogen, carbon dioxide, formamide, and N-methylpropanamide molecules. A description of the molecular polarization in terms of local polarizabilities without charge-flow terms is obtained when the nonlocal components of the polarizability tensor are transformed away using the localization method of Le Sueur and Stone [Mol. Phys. 83, 293 (1994)]. The resulting local polarizabilities are shown to be stable with respect to basis set used, exhibiting none of the artifacts of earlier basis-space partitioning methods. We also investigate the transferability of the resulting local polarizability models.

Calculation of intermolecular interactions in the benzene dimer using coupled-cluster and local electron correlation methods

Potential energy curves for the parallel-displaced, T-shaped and sandwich structures of the benzene dimer are computed with density fitted local second-order Møller-Plesset perturbation theory (DF-LMP2) as well as with the spin-component scaled (SCS) variant of DF-LMP2. While DF-LMP2 strongly overestimates the dispersion interaction, in common with canonical MP2, the DF-SCS-LMP2 interaction energies are in excellent agreement with the best available literature values along the entire potential energy curves. The DF-SCS-LMP2 dissociation energies for the three structures are also compared with new complete basis set estimates of the interaction energies obtained from accurate coupled cluster (CCSD(T)) and DF-SCS-MP2 calculations. Since LMP2 is essentially free of basis set superposition errors, counterpoise corrections are not required. As a result, DF-SCS-LMP2 is computationally inexpensive and represents an attractive method for the study of larger pi-stacked systems such as truncated sections of DNA.

Van der Waals Complexes of Polar Aromatic Molecules: Unexpected Structures for Dimers of Azulene

Full geometry optimizations at the dispersion corrected DFT-BLYP/TZV2P level of theory have been performed for dimers of azulene that may serve as a model system for the van der Waals complexes of polar pi systems. The structures and binding energies for 11 dimers are investigated in detail. The DFT-D interaction energies have been successfully checked against results from the accurate SCS-MP2/aug-cc-pVTZ approach. Out of the nine investigated stacked complexes, eight have binding energies larger than 7.4 kcal/mol (SCS-MP2) that exceed the value of 7.1 kcal/mol for the best naphthalene dimer. T-shaped arrangements (CH...pi) are significantly less stable. Two out of the three best structures have an antiparallel alignment of the monomer dipole moments in the complex, although the best ones with a parallel orientation are only about 0.5 kcal/mol less strongly bound which points to a minor importance of dipole-dipole interactions to binding. Quite surprisingly, the energetically lowest structure (DeltaE = -9.2 kcal/mol) corresponds to a situation where the two seven-membered rings are located almost on top of each other (7-7) and the long molecular axes are rotated against each other by 130 degrees. The 7-7 structural motif is found also in other energetically low-lying structures, and the expected 5-7 (two-side) arrangement is less strongly bound by about 2 kcal/mol. This can be explained by the electrostatic potential of azulene that only partially reflects the charge separation according to the common 4n + 2 pi electron rule. General rules for predicting stable van der Waals complexes of polar pi systems are discussed.

Local correlation potentials from Brueckner coupled-cluster theory

Local correlation potentials have been obtained from the nonlocal Brueckner coupled-cluster correlation potentials for the rare-gas atoms He, Ne, and Ar and the CO molecule. It is shown that the local correlation potential can mainly be expressed as a sum of two components: a "pure" correlation part and a relaxation contribution. While the total correlation potentials show an oscillating behavior near the nuclei, indicating the atomic shell structure, their components decrease rather monotonously, with a step structure in case of Ne and Ar. By looking at the determinantal overlap and one-electron properties it has been found that the orbitals obtained from these local potentials form a determinant which very well corresponds with the Brueckner determinant. Thus the previously found closeness between the Hartree-Fock determinant and the exchange-only Kohn-Sham determinant [Della Sala and Gorling, J. Chem. Phys. 115, 5718 (2001)] is confirmed also for the correlated case.