A divide and conquer real-space approach for all-electron molecular electrostatic potentials and interaction energies

A computational scheme to perform accurate numerical calculations of electrostatic potentials and interaction energies for molecular systems has been developed and implemented. Molecular electron and energy densities are divided into overlapping atom-centered atomic contributions and a three-dimensional molecular remainder. The steep nuclear cusps are included in the atom-centered functions making the three-dimensional remainder smooth enough to be accurately represented with a tractable amount of grid points. The one-dimensional radial functions of the atom-centered contributions as well as the three-dimensional remainder are expanded using finite element functions. The electrostatic potential is calculated by integrating the Coulomb potential for each separate density contribution, using our tensorial finite element method for the three-dimensional remainder. We also provide algorithms to compute accurate electron-electron and electron-nuclear interactions numerically using the proposed partitioning. The methods have been tested on all-electron densities of 18 reasonable large molecules containing elements up to Zn. The accuracy of the calculated Coulomb interaction energies is in the range of 10(-3) to 10(-6) E(h) when using an equidistant grid with a step length of 0.05 a(0).     

Towards a force field based on density fitting

Total intermolecular interaction energies are determined with a first version of the Gaussian electrostatic model (GEM-0), a force field based on a density fitting approach using s-type Gaussian functions. The total interaction energy is computed in the spirit of the sum of interacting fragment ab initio (SIBFA) force field by separately evaluating each one of its components: electrostatic (Coulomb), exchange repulsion, polarization, and charge transfer intermolecular interaction energies, in order to reproduce reference constrained space orbital variation (CSOV) energy decomposition calculations at the B3LYP/aug-cc-pVTZ level. The use of an auxiliary basis set restricted to spherical Gaussian functions facilitates the rotation of the fitted densities of rigid fragments and enables a fast and accurate density fitting evaluation of Coulomb and exchange-repulsion energy, the latter using the overlap model introduced by Wheatley and Price [Mol. Phys. 69, 50718 (1990)]. The SIBFA energy scheme for polarization and charge transfer has been implemented using the electric fields and electrostatic potentials generated by the fitted densities. GEM-0 has been tested on ten stationary points of the water dimer potential energy surface and on three water clusters (n = 16,20,64). The results show very good agreement with density functional theory calculations, reproducing the individual CSOV energy contributions for a given interaction as well as the B3LYP total interaction energies with errors below kBT at room temperature. Preliminary results for Coulomb and exchange-repulsion energies of metal cation complexes and coupled cluster singles doubles electron densities are discussed.     

A self-consistent coulomb bath model using density fitting

A self-consistent Coulomb bath model is presented to provide an accurate and efficient way of performing calculations for interfragment electrostatic and polarization interactions. In this method, a condensed-phase system is partitioned into molecular fragment blocks. Each fragment is embedded in the Coulomb bath due to other fragments. Importantly, the present Coulomb bath is represented using a density fitting method in which the electron densities of molecular fragments are fitted using an atom-centered auxiliary basis set of Gaussian type. The Coulomb bath is incorporated into an effective Hamiltonian for each fragment, with which the electron density is optimized through an iterative double self-consistent field (DSCF) procedure to realize the mutual many-body polarization effects. In this work, the accuracy of interfragment interaction energies enumerated using the Coulomb bath is tested, showing a good agreement with the exact results from an energy decomposition analysis. The qualitative features of many-body polarization effects are visualized by electron density difference plots. It is also shown that the present DSCF method can yield fast and robust convergence with near-linear scaling in performance with increase in system size.     

Gaussian multipoles in practice: Electrostatic energies for intermolecular potentials

A method is presented for calculating the total electrostatic interaction energies between molecules from ab initio monomer wave functions. This approach differs from existing methods, such as Stone's distributed multipole analysis (DMA), in including the short-range penetration energy as well as the long-range multipolar energy. The monomer charge densities are expressed as distributed series of atom-centered functions which we call Gaussian multipoles; these are analogous to the distributed point multipoles used in DMA. Our procedure has been encoded in the GMUL program. Calculations have been performed on the formamide/formaldehyde complex, a model system for N? H …? O hydrogen bonding in biological molecules, and also on guanidinium/benzene, modeling amino/aromatic interactions in proteins. We find that the penetration energy can be significant, especially in its contribution to the variation of the electrostatic energy with interaction geometry. A hybrid method, which uses Gaussian multipoles for short-range atom pair interactions and point multipoles for long-range ones, allows the electrostatic energies, including penetration, to be calculated at a much reduced cost. We also note that the penetration energy may provide the best route to an atom–atom anisotropic model for the exchange-repulsion energy in intermolecular potentials. © 1994 by John Wiley & Sons, Inc.     

On basis set superposition error corrected stabilization energies for large n-body clusters

In this contribution, we propose an approximate basis set superposition error (BSSE) correction scheme for the site-site function counterpoise and for the Valiron-Mayer function counterpoise correction of second order to account for the basis set superposition error in clusters with a large number of subunits. The accuracy of the proposed scheme has been investigated for a water cluster series at the CCSD(T), CCSD, MP2, and self-consistent field levels of theory using Dunning's correlation consistent basis sets. The BSSE corrected stabilization energies for a series of water clusters are presented. A study regarding the possible savings with respect to computational resources has been carried out as well as a monitoring of the basis set dependence of the approximate BSSE corrections.     

Accurate ab initio binding energies of alkaline earth metal clusters

The effects of basis set superposition error (BSSE) and core-correlation on the electronic binding energies of alkaline earth metal clusters Y(n) (Y = Be, Mg, Ca; n = 2-4) at the Moller-Plesset second-order perturbation theory (MP2) and the single and double coupled cluster method with perturbative triples correction (CCSD(T)) levels are examined using the correlation consistent basis sets cc-pVXZ and cc-pCVXZ (X = D, T, Q, 5). It is found that, while BSSE has a negligible effect for valence-electron-only-correlated calculations for most basis sets, its magnitude becomes more pronounced for all-electron-correlated calculations, including core electrons. By utilizing the negligible effect of BSSE on the binding energies for valence-electron-only-correlated calculations, in combination with the negligible core-correlation effect at the CCSD(T) level, accurate binding energies of these clusters up to pentamers (octamers in the case of the Be clusters) are estimated via the basis set extrapolation of ab initio CCSD(T) correlation energies of the monomer and cluster with only the cc-pVDZ and cc-pVTZ sets, using the basis set and correlation-dependent extrapolation formula recently devised. A comparison between the CCSD(T) and density functional theory (DFT) binding energies is made to identify the most appropriate DFT method for the study of these clusters.     

水团簇(H2O)n中多体作用强度与水分子间距的关系

使用高精度从头算方法(含基组重叠误差校正)计算了水团簇(H₂O)_n (n = 8, 10, 16, 20, 22, 24)中的所有二体、三体和四体作用能,分析了水团簇中的多体效应.研究表明,二体作用对体系总作用能的贡献高达70%以上,三体作用对总作用能的贡献可高达25%,四体作用在总作用能中所占比例不超过3%,五体及以上多体作用能在总作用能中所占比例更小,不超过0.5%.本文研究还表明,两个水分子间距小于0.68 nm的二体作用、三个和四个水分子中最近的两个水分子间距小于0.31 nm的三体和四体作用对体系总作用能的贡献高达99.4%.因此,以生物体系为对象的分子模拟方法应该具备准确地模拟两个水分子间距小于0.68 nm的二体作用、三个和四个水分子中最近的两个分子间距小于0.31 nm的三体和四体作用的能力.     

Base stacking in cytosine dimer. A comparison of correlated ab initio calculations with three empirical potential models and density functional theory calculations

Ab initio MP2/6-31G* interaction energies were calculated for more than 80 geometries of stacked cytosine dimer. Diffuse polarization functions were used to properly cover the dispersion energy. The results of ab initio calculations were compared with those obtained from three electrostatic empirical potential models, constructed as the sum of a Lennard-Jones potential (covering dispersion and repulsion contributions) and the electrostatic term. Point charges and point multipoles of the electrostatic term were also obtained at the MP2/6-31G* level of theory. The point charge MEP model (atomic charges derived from molecular electrostatic potential) satisfactorily reproduced the ab initio data. Addition of π-charges localized below and above the cytosine plane did not affect the calculated energies. The model employing the distributed multipole analysis gave worse agreement with the ab initio data than the MEP approach. The MP2 MEP charges were also derived using larger sets of atomic orbitals: cc-pVDZ, 6-311 + G(2d, p), and aug-cc-pVDZ. Differences between interaction energies calculated using these three sets of point charges and the MP2/6-31G* charges were smaller than 0.8 kcal/mol. The correlated ab initio calculations were also compared with the density functional theory (DFT) method. DFT calculations well reproduced the electrostatic part of interaction energy. They also covered some nonelectrostatic short-range effects which were not reproduced by the empirical potentials. The DFT method does not include the dispersion energy. This energy, approximated by an empirical term, was therefore added to the DFT interaction energy. The resulting interaction energy exhibited an artifact secondary minimum for a 3.9-4.0 vertical separation of bases. This defect is inherent in the DFT functionals, because it is not observed for the Hartree-Fock + dispersion interaction energy.© 1996 John Wiley & Sons, Inc.     

Theoretical studies for structures and energetics of Rgn-N2O (Rg=He, Ne, Ar) clusters

Minimum-energy structures of the Rg(2)-N(2)O (Rg=He, Ne, Ar) clusters have been determined with ab initio MP2 optimization, whereas the minimum-energy structures of the Rg(n)-N(2)O clusters with n = 3-7 have been obtained with the pairwise additive potentials. Interaction energies and nonadditive three-body effects of the Rg(2)-N(2)O ternary complex have been calculated using supermolecule method at MP4 and CCSD(T) levels. It was found from the calculations that there are two minima corresponding to one distorted tetrahedral structure and one planar structure for the ternary complex. The nonadditive three-body effects were found to be small for Rg(2)-N(2)O complexes. Our calculations also indicated that, for He(n)-N(2)O and Ne(n)-N(2)O clusters, the first six He and Ne atoms form the first solvation ring around the middle nitrogen of the N(2)O monomer, while for Ar(n)-N(2)O, the first five Ar atoms form the first solvation ring.     

Toward a new approach for determination of solute's charge distribution to analyze interatomic electrostatic interactions in quantum mechanical/molecular mechanical simulations

We present an alternative approach to determine "density-dependent property"-derived charges for molecules in the condensed phase. In the case of a solution, it is essential to take into consideration the electron polarization of molecules in the active site of this system. The solute and solvent molecules in this site have to be described by a quantum mechanical technique and the others are allowed to be treated by a molecular mechanical method (QM/MM scheme). For calculations based on this scheme, using the forces and interaction energy as density-dependent property our charges from interaction energy and forces (CHIEF) approach can provide the atom-centered charges on the solute atoms. These charges reproduce well the electrostatic potentials around the solvent molecules and present properly the picture of the electron density of the QM subsystem in the solution system. Thus, the CHIEF charges can be considered as the atomic charges under the conditions of the QM/MM simulation, and then enable one to analyze electrostatic interactions between atoms in the QM and MM regions. This approach would give a view of the QM nuclei and electrons different from the conventional methods.     

Potential-Derived point-charge model study of electrostatic interaction energies in some hydrogen-bonded systems

Mulliken's atomic charges (MC ) and potential derived (PD ) point charges obtained from STO -3G wave functions are used to study the electrostatic interaction energies for a series of representative hydrogenbonded complexes. The results of the above-mentioned models are compared with the more accurate results of segmental multipole moment (SMM ) expansion, and it is shown that the PD model is superior to the Mc model. The results of PD model are shown to be well correlated with the results of SMM expansion technique. Results of our calculations using 6-31G and 6-31G** PD charges are also reported here. Electrostatic interaction energies obtained using 6-31G** PD charges are compared with the 6-31G** SCF interaction energies available for the nine hydrogen-bonded dimers of ammonia, water, and hydrogen fluoride and a good con-elation between the two is shown. The interrelationship between the results of different basis sets are also examined for the PD point-charge model. The electrostatic interaction energies obtained using STO -3G PD model are shown to be well correlated to the results of 6-31G and 6-31G** PD models.     

Theoretical study of the potential energy surface for the interaction of cisplatin and their aquated species with water

In this work, a systematic study of the interaction of neutral cisplatin ([Pt(NH(3))(2)Cl(2)]) and their charged aquated species ([Pt(NH(3))(2)Cl(H(2)O)](+) and [Pt(NH(3))(2)(H(2)O)(2)](2+)) with water was carried out. The potential energy surface (PES) was analyzed by considering 35 spatial orientations for the interacting species. The calculations were performed at various levels of theory including Moller-Plesset fourth order perturbation theory and density functional theory (DFT-B3LYP) using extended basis sets. Lennard-Jones (12-6) plus Coulomb classical potential was also used to assess the repulsion-dispersion and electrostatic contributions. The effect of atomic charges on the interaction energies is discussed using Mulliken, charges from electrostatic potential grid method and natural bond orbital schemes. The outcomes show that the electrostatic term plays a primary role on the calculation of interaction energies, with the absolute values of atomic charges from different approaches significantly affecting the overall interaction. Unusual results were revealed by basis set superposition error calculations for the structures located on the platinum-water PES.     

A theoretical study of the interactions between N,N-dimethylformamide and xylenes

The ab initio and density functional (DFT) methods were performed on binary systems of N,N-dimethylformamide (DMF) with xylenes (o-, or m-, or p-xylene), and seven stable configurations were obtained with no imaginary frequencies. To obtain the interaction energies of these complexes, single-point energy calculations with basis set superposition error (BSSE) correction were carried out at B3LYP/6-31G* and MP2/6-31G* levels. The structures, Chelpg (charges from electrostatic potentials using a grid-based method) charge distribution and bond characteristics of the mentioned complexes were calculated. The results indicated the presence of double C–H···O hydrogen bonds between DMF and xylenes in these complexes and the interaction energies of hydrogen bonding between DMF and xylene systems decreased in the following sequence: DMF–o-xylene: a1 > DMF–m-xylene: b1 > DMF–p-xylene: c1.     

Magnitude and nature of interactions in benzene-X (X=ethylene and acetylene) in the gas phase: significantly different CH/pi interaction of acetylene as compared with those of ethylene and methane

The accurate interaction energies of the CH/pi interaction in the benzene-X clusters (X = ethylene and acetylene) were experimentally and theoretically determined. Two-color multiphoton ionization spectroscopy was applied, and the binding energies in the neutral ground state of the clusters were evaluated from the dissociation threshold measurements of the cluster cations. The experimental binding energies of the clusters (D0) were 1.4+/-0.2 and 2.7+/-0.2 kcal/mol, respectively. Estimated CCSD(T) interaction energies for the clusters at the basis set limit (De) were 2.2 and 2.8 kcal/mol, respectively. Calculated D0 values (1.7 and 2.4 kcal/mol, respectively) are close to the experimental values. Large electron correlation contributions (Ecorr=-3.6 and -2.8 kcal/mol, respectively) show that dispersion is the major source of the attraction in both clusters. The electrostatic interaction in the ethylene cluster is very small (-0.38 kcal/mol), as in the case of the benzene-methane cluster, whereas the electrostatic interaction in the acetylene cluster is large (-1.70 kcal/mol). The shifts of the S1-S0 transition also suggest that the ethylene cluster is a van der Waals-type cluster, but the acetylene cluster is a pi-hydrogen-bonded cluster. The nature of the CH/pi interaction of the "activated" alkyne C-H bond is significantly different from that of the "nonactivated" (or typical) alkane and alkene C-H bonds.     

Numerical fitting of molecular properties to Hermite Gaussians

A procedure is presented to fit gridded molecular properties to auxiliary basis sets (ABSs) of Hermite Gaussians, analogous to the density fitting (DF) method (Dunlap; et al. J. Chem. Phys. 1979, 71, 4993). In this procedure, the ab initio calculated properties (density, electrostatic potential, and/or electric field) are fitted via a linear- or nonlinear-least-squares procedure to auxiliary basis sets (ABS). The calculated fitting coefficients from the numerical grids are shown to be more robust than analytic density fitting due to the neglect of the core contributions. The fitting coefficients are tested by calculating intermolecular Coulomb and exchange interactions for a set of dimers. It is shown that the numerical instabilities observed in DF are caused by the attempt of the ABS to fit the core contributions. In addition, this new approach allows us to reduce the number of functions required to obtain an accurate fit. This results in decreased computational cost, which is shown by calculating the Coulomb energy of a 4096 water box in periodic boundary conditions. Using atom centered Hermite Gaussians, this calculation is only 1 order of magnitude slower than conventional atom-centered point charges.     

Calculation of electrostatic interaction energies in molecular dimers from atomic multipole moments obtained by different methods of electron density partitioning

Accurate and fast evaluation of electrostatic interactions in molecular systems is still one of the most challenging tasks in the rapidly advancing field of macromolecular chemistry, including molecular recognition, protein modeling and drug design. One of the most convenient and accurate approaches is based on a Buckingham-type approximation that uses the multipole moment expansion of molecular/atomic charge distributions. In the mid-1980s it was shown that the pseudoatom model commonly used in experimental X-ray charge density studies can be easily combined with the Buckingham-type approach for calculation of electrostatic interactions, plus atom-atom potentials for evaluation of the total interaction energies in molecular systems. While many such studies have been reported, little attention has been paid to the accuracy of evaluation of the purely electrostatic interactions as errors may be absorbed in the semiempirical atom-atom potentials that have to be used to account for exchange repulsion and dispersion forces. This study is aimed at the evaluation of the accuracy of the calculation of electrostatic interaction energies with the Buckingham approach. To eliminate experimental uncertainties, the atomic moments are based on theoretical single-molecule electron densities calculated at various levels of theory. The electrostatic interaction energies for a total of 11 dimers of alpha-glycine, N-acetylglycine and L-(+)-lactic acid structures calculated according to Buckingham with pseudoatom, stockholder and atoms-in-molecules moments are compared with those evaluated with the Morokuma-Ziegler energy decomposition scheme. For alpha-glycine a comparison with direct "pixel-by-pixel" integration method, recently developed Gavezzotti, is also made. It is found that the theoretical pseudoatom moments combined with the Buckingham model do predict the correct relative electrostatic interactions energies, although the absolute interaction energies are underestimated in some cases. The good agreement between electrostatic interaction energies computed with Morokuma-Ziegler partitioning, Gavezzotti's method, and the Buckingham approach with atoms-in-molecules moments demonstrates that reliable and accurate evaluation of electrostatic interactions in molecular systems of considerable complexity is now feasible.     

Using multipole point charge distributions to provide the electrostatic potential in the variational explicit polarization (X-Pol) potential

The equations defining the variational explicit polarization (X-Pol) potential introduced in earlier work are modified in the present work so that multipole point charge distributions are used instead of Mulliken charges to polarize the monomers that comprise the system. In addition, when computing the electrostatic interaction between a monomer whose molecular orbitals are being optimized and a monomer whose electron density is being used to polarize the first monomer, the electron densities of both monomers are represented by atom-centered multipole point charge distributions. In the original formulation of the variational X-Pol potential, the continuous electron density of the monomer being optimized interacts with external Mulliken charges, but this corresponds to the monopole truncation in a multipole expansion scheme in the computation of the Fock matrix elements of the given monomer. The formulation of the variational X-Pol potential introduced in this work (which we are calling the “multipole variational X-Pol potential”) represents the electron density of the monomer whose wave function is being variationally optimized in the same way that it represents the electron densities of external monomers when computing the Coulomb interactions between them.     

Intermolecular coulomb couplings from ab initio electrostatic potentials: application to optical transitions of strongly coupled pigments in photosynthetic antennae and reaction centers

An accurate and numerically efficient method for the calculation of intermolecular Coulomb couplings between charge densities of electronic states and between transition densities of electronic excitations is presented. The coupling of transition densities yields the F?rster type excitation energy transfer coupling, and from the charge density coupling, a shift in molecular excitation energies results. Starting from an ab initio calculation of the charge and transition densities, atomic partial charges are determined such as to fit the resulting electrostatic potentials of the different states and the transition. The different intermolecular couplings are then obtained from the Coulomb couplings between the respective atomic partial charges. The excitation energy transfer couplings obtained in the present TrEsp (transition charge from electrostatic potential) method are compared with couplings obtained from the simple point-dipole and extended dipole approximations and with those from the ab initio transition density cube method of Krüger, Scholes, and Fleming. The present method is of the same accuracy as the latter but computationally more efficient. The method is applied to study strongly coupled pigments in the light-harvesting complexes of green sulfur bacteria (FMO), purple bacteria (LH2), and higher plants (LHC-II) and the "special pairs" of bacterial reaction centers and reaction centers of photosystems I and II. For the pigment dimers in the antennae, it is found that the mutual orientation of the pigments is optimized for maximum excitonic coupling. A driving force for this orientation is the Coulomb coupling between ground-state charge densities. In the case of excitonic couplings in the "special pairs", a breakdown of the point-dipole approximation is found for all three reaction centers, but the extended dipole approximation works surprisingly well, if the extent of the transition dipole is chosen larger than assumed previously. For the "special pairs", a large shift in local transition energies is found due to charge density coupling.     

可快速计算水团簇三体作用强度的新方法

将水分子视为由2个O—H键偶极构成, 再将水分子间的三体作用视为长程诱导作用和短程校正之和, 使用Thole模型计算长程诱导作用, 通过同时考虑不同水分子间的置换和同一个水分子中2个键偶极间的置换计算短程校正, 从而提出了一个可快速计算水团簇三体作用强度的新方法. 根据已报道的12347个水三聚体的结构和CCSD(T)三体作用能, 确定了该方法所需参数. 将该方法和所确定的参数应用于67个水团簇体系, 计算这些体系的三体作用能, 并与CCSD(T), MP2, M06-2X方法的计算结果进行比较. 结果表明, 相对于CCSD(T)方法的总三体作用能, 本文方法的均方根偏差(RMSD)仅为3.32 kJ/mol, 平均相对偏差(MRD)仅为2.43%; 对较大水团簇体系, 该方法计算精度稍优于MP2方法, 明显优于M06-2X方法, 并且更快捷高效.