Four-component relativistic theory for nuclear magnetic shielding constants: the orbital decomposition approach

The authors present a scheme to simplify four-component relativistic calculations of nuclear magnetic shielding constants. The central idea is to decompose each first order orbital into two terms, one is magnetically balanced and directly leads to the diamagnetic term, and the other is, to leading order of relativity, kinetically balanced and can therefore simply be represented in the basis of unperturbed positive energy states. As a matrix formulation, the present approach is far simpler than other operator theories. Combined with the Dirac-Kohn-Sham ansatz, the nuclear magnetic shielding constants for the Kr, Xe, and Rn atoms as well as the HBr and HI molecules are calculated, and the results compare favorably with those of other schemes.     

Electrostatic energy in the effective fragment potential method: theory and application to benzene dimer

Evaluation of the electrostatic energy within the effective fragment potential (EFP) method is presented. The performance of two variants of the distributed multipole analysis (DMA) together with two different models for estimating the charge penetration energies was studied using six homonuclear dimers. The importance of damping the higher order multipole terms, i.e. charge dipole, was also investigated. Damping corrections recover more than 70% of the charge penetration energy in all dimers, whereas higher order damping introduces only minor improvement. Electrostatic energies calculated by the numerical DMA are less accurate than those calculated by the analytic DMA. Analysis of bonding in the benzene dimer shows that EFP with inclusion of the electrostatic damping term performs very well compared to the high-level coupled cluster singles, doubles, and perturbative triples method. The largest error of 0.4 kcal/mol occurs for the sandwich dimer configuration. This error is about half the size of the corresponding error in second order perturbation theory. Thus, EFP in the current implementation is an accurate and computationally inexpensive method for calculating interaction energies in weakly bonded molecular complexes.     

Methanol-water mixtures: a microsolvation study using the effective fragment potential method

The formation process of methanol-water mixtures, (MeOH/H2O)n, n = 2, 3,..., 8 is studied at the molecular level using the general effective fragment potential (EFP2) method and second-order perturbation theory (MP2). Extensive Monte Carlo/simulated annealing global optimizations were used to locate global minimum structures for each n, for both homo and hetero clusters. Mixing at the microscopic level was investigated, and some general conclusions about the microsolvation behavior of these mixtures are presented. For all of these clusters, incomplete mixing is observed at the molecular level. Low-energy (MeOH/H2O)n clusters retain much of their initial structure in the global minima of the mixed clusters.     

Charge transfer interaction in the effective fragment potential method

An approximate formula is derived and implemented in the general effective fragment potential (EFP2) method to model the intermolecular charge transfer interaction. This formula is based on second order intermolecular perturbation theory and utilizes canonical molecular orbitals and Fock matrices obtained with preparative self-consistent field calculations. It predicts charge transfer energies that are in reasonable agreement with the reduced variational space energy decomposition analysis. The formulas for the charge transfer gradients with respect to EFP translational and rotational displacements are also derived and implemented.     

Analytic gradient and molecular dynamics simulations using the fragment molecular orbital method combined with effective potentials

The completely analytic energy gradients are derived and implemented for the two-body fragment molecular orbital (FMO2) method combined with the model core potentials (MCP) and effective fragment potentials (EFP). The many-body terms in EFP require solving coupled-perturbed Hartree-Fock equations, which are derived and implemented. The molecular dynamics (MD) simulations are performed using the FMO2/MCP method for the capped alanine decamer and with the FMO2/EFP method for the zwitterionic conformer of glycine tetramer immersed in the water layer of 6.0 Å (135 water molecules). The results of the MD simulations using the FMO2/EFP and FMO2/MCP gradients show that the total energy is conserved at the time steps less than 1 fs.     

Multiconfiguration self-consistent-field theory based upon the fragment molecular orbital method

The fragment molecular orbital (FMO) method was combined with the multiconfiguration self-consistent-field (MCSCF) theory. One- and two-layer approaches were developed, the former involving all dimer MCSCF calculations and the latter limiting MCSCF calculations to a small part of the system. The accuracy of the two methods was tested using the six electrons in six orbitals complete active space type of MCSCF and singlet spin state for phenol+(H(2)O)(n), n=16,32,64 (6-31G( *) and 6-311G( *) basis sets); alpha helices and beta strands of phenylalanine-(alanine)(n), n=4,8,16 (6-31G( *)). Both double-zeta and triple-zeta quality basis sets with polarization were found to have very similar accuracy. The error in the correlation energy was at most 0.000 88 a.u., the error in the gradient of the correlation energy was at most 6.x10(-5) a.u./bohr and the error in the correlation correction to the dipole moment was at most 0.018 D. In addition, vertical singlet-triplet electron excitation energies were computed for phenol+(H(2)O)(n), (n=16,32,64), 6-31G( *), and the errors were found to be at most 0.02 eV. Approximately linear scaling was observed for the FMO-based MCSCF methods. As an example, an FMO-based MCSCF calculation with 1262 basis functions took 98 min on one 3.0 GHz Pentium4 node with 1 Gbyte RAM.     

Ab initio calculation of magnetic susceptibilities and screening constants using gauge invariant Gaussian orbitals

The magnetic susceptibility and screening constant tensors are calculated using an ab initio finite perturbation SCF method, with gauge invariant Gaussian orbitals. The isoelectronic BH BeH^?, and CH⁺ molecules have been studied. The calculated values for BH are relatively different from those obtained by other methods. The CH⁺ molecule seems to exhibit a stronger temperature independent paramagnetism than BH, while the BeH^? molecule would be diamagnetic. The screening constants of the heavy atom of these molecules present a similar variation.     

Benzene-pyridine interactions predicted by the effective fragment potential method

The accurate representation of nitrogen-containing heterocycles is essential for modeling biological systems. In this study, the general effective fragment potential (EFP2) method is used to model dimers of benzene and pyridine, complexes for which high-level theoretical data -including large basis spin-component-scaled second-order perturbation theory (SCS-MP2), symmetry-adapted perturbation theory (SAPT), and coupled cluster with singles, doubles, and perturbative triples (CCSD(T))-are available. An extensive comparison of potential energy curves and components of the interaction energy is presented for sandwich, T-shaped, parallel displaced, and hydrogen-bonded structures of these dimers. EFP2 and CCSD(T) potential energy curves for the sandwich, T-shaped, and hydrogen-bonded dimers have an average root-mean-square deviation (RMSD) of 0.49 kcal/mol; EFP2 and SCS-MP2 curves for the parallel displaced dimers have an average RMSD of 0.52 kcal/mol. Additionally, results are presented from an EFP2 Monte Carlo/simulated annealing (MC/SA) computation to sample the potential energy surface of the benzene-pyridine and pyridine dimers.     

Energy decomposition analysis in solution based on the fragment molecular orbital method

We develop the pair interaction energy decomposition analysis (PIEDA) in solution by combining the fragment molecular orbital (FMO) method with the polarizable continuum model (PCM). The solvent screening of the electrostatic interaction and the desolvation penalty in complex formation are described by this approach from ab initio calculations of fragments and their pairs. The applications to the complex of solvated sodium and chlorine ions, as well as to lysine and aspartic acid, show how the analysis helps reveal the physical picture. The PIEDA/PCM method is also applied to a small protein chignolin (PDB: 1UAO), and the solvent screening of the pair interactions is discussed.     

Time-dependent density functional theory based upon the fragment molecular orbital method

Time-dependent density functional theory (TDDFT) was combined with the two-body fragment molecular orbital method (FMO2). In this FMO2-TDDFT scheme, the system is divided into fragments, and the electron density for fragments is determined self-consistently. Consequently, only one main fragment of interest and several fragment pairs including it are calculated by TDDFT. To demonstrate the accuracy of FMO2-TDDFT, we computed several low-lying singlet and triplet excited states of solvated phenol and polyalanine using our method and the standard TDDFT for the full system. The BLYP functional with the long-range correction (LC-BLYP) was employed with the 6-31G(*) basis set (some tests were also performed with 6-311G(*), as well as with B3LYP and time-dependent Hartree-Fock). Typically, FMO2-TDDFT reproduced the full TDDFT excitation energies within 0.1 eV, and for one excited state the error was about 0.2 eV. Beside the accurate reproduction of the TDDFT excitation energies, we also automatically get an excitation energy decomposition analysis, which provides the contributions of individual fragments. Finally, the efficiency of our approach was exemplified on the LC-BLYP6-31G(*) calculation of the lowest singlet excitation of the photoactive yellow protein which consists of 1931 atoms, and the obtained value of 3.1 eV is in agreement with the experimental value of 2.8 eV.     

Gradients of the polarization energy in the effective fragment potential method

The effective fragment potential (EFP) method is an ab initio based polarizable classical method in which the intermolecular interaction parameters are obtained from preparative ab initio calculations on isolated molecules. The polarization energy in the EFP method is modeled with asymmetric anisotropic dipole polarizability tensors located at the centroids of localized bond and lone pair orbitals of the molecules. Analytic expressions for the translational and rotational gradients (forces and torques) of the EFP polarization energy have been derived and implemented. Periodic boundary conditions (the minimum image convention) and switching functions have also been implemented for the polarization energy, as well as for other EFP interaction terms. With these improvements, molecular dynamics simulations can be performed with the EFP method for various chemical systems.     

The role of orbital products in the optimized effective potential method

The orbital products of occupied and virtual orbitals are employed as an expansion basis for the charge density generating the local potential in the optimized effective potential method thus avoiding the use of auxiliary basis sets. The high computational cost arising from the quadratic increase of the dimension of this product basis with system size can be greatly reduced by elimination of the linearly dependent products according to a procedure suggested by Beebe and Linderberg [Int. J. Quantum Chem. 12, 683 (1977)]. Numerical results from this approach show a very good agreement with those obtained from balancing the auxiliary basis for the expansion of the local potential with the orbital basis set.     

Noncovalent interactions in extended systems described by the effective fragment potential method: theory and application to nucleobase oligomers

The implementation of the effective fragment potential (EFP) method within the Q-CHEM electronic structure package is presented. The EFP method is used to study noncovalent π-π and hydrogen-bonding interactions in DNA strands. Since EFP is a computationally inexpensive alternative to high-level ab initio calculations, it is possible to go beyond the dimers of nucleic acid bases and to investigate the asymptotic behavior of different components of the total interaction energy. The calculations demonstrated that the dispersion energy is a leading component in π-stacked oligomers of all sizes. Exchange-repulsion energy also plays an important role. The contribution of polarization is small in these systems, whereas the magnitude of electrostatics varies. Pairwise fragment interactions (i.e., the sum of dimer binding energies) were found to be a good approximation for the oligomer energy.     

An interface between the universal force field and the effective fragment potential method

In order to properly describe reactions in heterogeneous catalyst systems, the reactants, solvent, and bulk effects of the surface must be taken into account. Embedded-cluster QM (quantum mechanics)/MM (molecular mechanics) methods can treat reactions on surfaces (the gas-surface interface), and the effective fragment potential method (EFP) can accurately treat the solvent effects on reactions (the gas-liquid interface). In order to create a QM/MM/EFP hybrid method for treatment of heterogeneous catalytic systems in the presence of a solvent (the liquid-surface interface), an EFP-MM interaction potential has been developed. Example calculations on small clusters of silica and water have been carried out.     

Predicting exchange coupling constants in frustrated molecular magnets using density functional theory

We study the Heisenberg exchange couplings in polynuclear transition-metal clusters with strong spin frustration using a variety of theoretical techniques. We present results for a trinuclear Cr(III) molecule, a tetranuclear Fe(III) complex, and an octanuclear Fe(III) molecular magnet. We explore the physics of the exchange couplings in these systems using standard broken-symmetry (BS) techniques and a more recently developed constrained density functional theory (C-DFT) approach. The calculations show that the expected picture of localized spin moments on the metal centers is appropriate, and in each case C-DFT predicts coupling constant values in good agreement with experiment. Furthermore, we demonstrate that all of the C-DFT spin states for a given cluster can be reasonably described by a single Heisenberg Hamiltonian. These findings are significant in part because standard BS calculations are in conflict with the experiments on a number of key points. For example, BS-DFT predicts a doublet (rather than quartet) ground state for the Cr(III) cluster while for the Fe(III) complexes BS-DFT predicts some of the exchange couplings to be ferromagnetic whereas the experimentally derived couplings are all antiferromagnetic. Furthermore, for BS-DFT the best-fit exchange parameters can depend significantly on the set of spin configurations chosen. For example, by choosing configurations with Ms closer to Ms(max) the BS-DFT couplings can typically be made somewhat closer to the C-DFT and experimental results. Thus, in these cases, our results consistently support the experimental findings.     

Optimizing conical intersections of solvated molecules: the combined spin-flip density functional theory/effective fragment potential method

Solvent effects on a potential energy surface crossing are investigated by optimizing a conical intersection (CI) in solution. To this end, the analytic energy gradient has been derived and implemented for the collinear spin-flip density functional theory (SFDFT) combined with the effective fragment potential (EFP) solvent model. The new method is applied to the azomethane-water cluster and the chromophore of green fluorescent protein in aqueous solution. These applications illustrate not only dramatic changes in the CI geometries but also strong stabilization of the CI in a polar solvent. Furthermore, the CI geometries obtained by the hybrid SFDFT/EFP scheme reproduce those by the full SFDFT, indicating that the SFDFT/EFP method is an efficient and promising approach for understanding nonadiabatic processes in solution.     

Fast fragments: the development of a parallel effective fragment potential method

The Effective Fragment Potential (EFP) method for solvation decreases the cost of a fully quantum mechanical calculation by dividing a chemical system into an ab initio region that contains the solute plus some number of solvent molecules, if desired, and an "effective fragment" region that contains the remaining solvent molecules. Interactions introduced with this fragment region (for example, Coulomb and polarization interactions) are added as one-electron terms to the total system Hamiltonian. As larger systems and dynamics are just starting to be studied with the EFP method, more needs to be done to decrease the calculation time of the method. This article considers parallelization of both the EFP fragment-fragment and mixed quantum mechanics (QM)-EFP interaction energy and gradient computation within the GAMESS suite of programs. The iteratively self-consistent polarization term is treated with a new algorithm that makes use of nonblocking communication to obtain better scalability. Results show that reasonable speedup is achieved with a variety of sizes of water clusters and number of processors.     

Gauge invariant calculations of nuclear magnetic shielding constants using the continuous transformation of the origin of the current density approach. II. Density functional and coupled cluster theory

The quantum mechanical current density induced in a molecule by an external magnetic field is invariant to translations of the coordinate system. This fundamental symmetry is exploited to formally annihilate the diamagnetic contribution to the current density via the approach of "continuous transformation of the origin of the current density-diamagnetic zero" (CTOCD-DZ). The relationships obtained by this method for the magnetic shielding at the nuclei are intrinsically independent of the origin of the coordinate system for any approximate computational scheme relying on the algebraic approximation. The authors report for the first time an extended series of origin-independent estimates of nuclear magnetic shielding constants using the CTOCD-DZ approach at the level of density functional theory (DFT) with four different types of functionals and unrelaxed coupled cluster singles and doubles linear response (CCSD-LR) theory. The results obtained indicate that in the case of DFT the procedure employed is competitive with currently adopted computational methods allowing for basis sets of gauge-including atomic orbitals, whereas larger differences between CTOCD-DZ and common origin CCSD-LR results are observed due to the incomplete fulfillment of hypervirial relations in standard CCSD-LR theory. It was found furthermore that the unrelaxed CCSD-LR calculations predict larger correlation corrections for the shielding constants of almost all nonhydrogen atoms in their set of molecules than the usual relaxed energy derivative CCSD calculations. Finally the results confirm the excellent performance of Keal and Tozer's third functional, in particular, for the multiply bonded systems with a lot of electron correlation, but find also that the simple local density functional gives even better results for the few singly bonded molecules in their study where correlation effects are small.     

Modeling pi-pi interactions with the effective fragment potential method: the benzene dimer and substituents

This study compares the results of the general effective fragment potential (EFP2) method to the results of a previous combined coupled cluster with single, double, and perturbative triple excitations [CCSD(T)] and symmetry-adapted perturbation theory (SAPT) study [Sinnokrot and Sherrill, J. Am. Chem. Soc., 2004, 126, 7690] on substituent effects in pi-pi interactions. EFP2 is found to accurately model the binding energies of the benzene-benzene, benzene-phenol, benzene-toluene, benzene-fluorobenzene, and benzene-benzonitrile dimers, as compared with high-level methods [Sinnokrot and Sherrill, J. Am. Chem. Soc., 2004, 126, 7690], but at a fraction of the computational cost of CCSD(T). In addition, an EFP-based Monte Carlo/simulated annealing study was undertaken to examine the potential energy surface of the substituted dimers.     

Calculation of exchange coupling constants in solid state transition metal compounds using localized atomic orbital basis sets

The application of theoretical methods based on density functional theory using hybrid functionals and localized, atomic orbital type basis sets is shown to provide good estimates for exchange coupling constants in non-metallic, solid state transition metal compounds with relatively complex crystal structures. The accuracy of the calculated exchange coupling constants is similar to that previously obtained for dinuclear and polynuclear molecular compounds. As an application of this procedure, the magnetic properties of the high-temperature phase of CuGeO₃, the recently synthesized silver copper oxide Ag₂Cu₂O₃, and the family of M[N(CN)₂]₂ (M=Cr(II), Mn(II), Fe(II), Co(II), Ni(II) and Cu(II)) compounds are analyzed via the computation of their most relevant exchange coupling constants.