The electrostatic surface term: (I) periodic systems

The authors propose a new approach to understand the electrostatic surface contributions to the interactions of large but finite periodic distributions of charges. They present a simple method to derive and interpret the surface contribution to any electrostatic field produced by a periodic distribution of charges. They discuss the physical and mathematical interpretations of this term. They present several examples and physical details associated with the calculation of the surface term. Finally, they provide a simple derivation of the surface contribution to the virial. This term does not disappear even if tinfoil boundary conditions are applied.     

Laplace-transformed diagonal Dyson correction to quasiparticle energies in periodic systems

We present a method to self-consistently evaluate quasiparticle energies of periodic systems within the diagonal approximation for solving Dyson's equation. Our method is based on the Laplace transform of the second-order M?ller-Plesset perturbation (MP2) theory kernel in the atomic basis formulation. The overhead computational cost of evaluating the fully self-consistent diagonal correction over the MP2 band energy calculation is negligible. We present numerical benchmark results for the band structure of trans-polyacetylene and compare it with MP2 and other approaches.     

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.     

A symbolic algorithm for periodic tridiagonal systems of equations

In the current paper, we present a novel symbolic algorithm for solving periodic tridiagonal linear systems without imposing any restrictive conditions. The computational cost of the algorithm is less than or almost equal to those of three well-known algorithms given by Chawla and Khazal (Int. J. Comput. Math. 79(4):473–484, 2002) and by El-Mikkawy (Appl. Math. Comput. 161:691–696, 2005), respectively. In addition, the solution of periodic anti-tridiagonal linear systems is also discussed. Two numerical experiments are provided in order to illustrate the performance and effectiveness of our algorithm. All of the experiments were performed on a computer with aid of programs written in MATLAB.     

Efficient algorithm of the transcorrelated method for periodic systems

The transcorrelated (TC) method is one of the promising wave-function-based approaches for the first-principles electronic structure calculations. In this method, the many-body wave function is approximated as the Jastrow-Slater type and one-electron orbitals in the Slater determinant are optimized with a one-body self-consistent-field equation such as that in the Hartree-Fock (HF) method. Although the TC method has yielded good results for both molecules and solids, its computational cost in solid-state calculations, being of order O(N(k)(3)N(b)(3)) with N(k) and N(b) the respective numbers of k-points and bands, has for some years hindered its wide application in condensed matter physics. Although an efficient algorithm was proposed for a Gaussian basis set, that algorithm is not applicable to a plane-wave basis that is suited to and widely used in solid-state calculations. In this paper, we present a new efficient algorithm of the TC method for the plane-wave basis or an arbitrary basis function set expanded in terms of plane waves, with which the computational cost of the TC method scales as O(N(k)(2)N(b) (2)). This is the same as that of the HF method. We applied the TC method with the new algorithm to obtain converged band structure and cell parameters of some semiconductors.     

The inclusion of electrostatic hydration energies in molecular mechanics calculations

Summary The problem of including solvent effects in molecular mechanics calculations is discussed. It is argued that the neglect of charge-solvent (solvation) interactions can introduce significant errors. The finite difference Poisson-Boltzmann (FDPB) method for calculating electrostatic interactions is summarized and is used as a basis for introducing a new pairwise energy term which accounts for charge-solvent interactions. This term acts between all pairs of atoms usually considered in molecular mechanics calculations and can be easily incorporated into existing force fields. As an example, a parameterization is developed for the CHARMm force field and the results compared to the predictions of the FDPB method. An approach to the realistic incorporation of solvent screening into force fields is also outlined.     

Comparison of treecodes for computing electrostatic potentials in charged particle systems with disjoint targets and sources

In molecular simulations, it is sometimes necessary to compute the electrostatic potential at M target sites due to a disjoint set of N charged source particles. Direct summation requires O(MN) operations, which is prohibitively expensive when M and N are large. Here, we consider two alternative tree‐based methods that reduce the cost. The standard particle‐cluster treecode partitions the N sources into an octree and applies a far‐field approximation, whereas a recently developed cluster‐particle treecode instead partitions the M targets into an octree and applies a near‐field approximation. We compare the two treecodes with direct summation and document their accuracy, CPU run time, and memory usage. We find that the particle‐cluster treecode is faster when N > M, that is, when the sources outnumber the targets, and conversely, the cluster‐particle treecode is faster when M > N, that is, when the targets outnumber the sources. Hence, the two treecodes provide useful tools for computing electrostatic potentials in charged particle systems with disjoint targets and sources. © 2013 Wiley Periodicals, Inc.     

Calculated structures,relative energies and electrostatic potentials of some tetraaza cyclic systems

A computational analysis of two tetraazapentalenes, a tetraazacyclooctatetraene, and some of their dinitro and difurazan derivatives is reported. Optimized structures were obtained at the 3-21G level with the ab initio SCF GAUSSIAN-82 procedure, and these were then used to compute STO-5G molecular electrostatic potentials. The tetraazapentalenes were found to have a high degree of electron delocalization, and to be much more stable than the tetraazacyclooctatetraene, which has localized double bonds. However the difurazan derivatives show a complete reversal of these relative stabilities. In all of the molecules, there are strong negative electrostatic potentials associated with the doubly coordinated nitrogens, indicative of basic character. These become weaker as other portions of the system become more electron withdrawing. A buildup of positive potential is observed above the bond between the triply coordinated bridgehead nitrogens in the tetraazapentalenes; this should be an attractive site for the initial approach of a nucleophile. The positive buildup becomes progressively stronger in going to the dinitro and the difurazan derivatives.     

An approach to computing electrostatic charges for molecules

We present an approach for deriving net atomic charges from ab initio quantum mechanical calculations using a least squares fit of the quantum mechanically calculated electrostatic potential to that of the partial charge model. Our computational approach is similar to those presented by Momany [J. Phys. Chem., 82 , 592 (1978)], Smit, Derissen, and van Duijneveldt [Mol. Phys., 37 , 521 (1979)], and Cox and Williams [J. Comput. Chem., 2 , 304 (1981)], but differs in the approach to choosing the positions for evaluating the potential. In this article, we present applications to the molecules H₂O, CH₃OH, (CH₃)₂O, H₂CO, NH₃, (CH₃O)₂PO, deoxyribose, ribose, adenine, 9-CH₃ adenine, thymine, 1-CH₃ thymine, guanine, 9-CH₃ guanine, cytosine, 1-CH₃ cytosine, uracil, and 1-CH₃ uracil. We also address the question of inclusion of “lone pairs,” their location and charge.     

Box length search algorithm for molecular simulation of systems containing periodic structures

We have developed a box length search algorithm to efficiently find the appropriate box dimensions for constant-volume molecular simulation of periodic structures. The algorithm works by finding the box lengths that equalize the pressure in each direction while maintaining constant total volume. Maintaining the volume at a fixed value ensures that quantitative comparisons can be made between simulation and experimental, theoretical or other simulation results for systems that are incompressible or nearly incompressible. We test the algorithm on a system of phase-separated block copolymers that has a preferred box length in one dimension. We also describe and test a Monte Carlo algorithm that allows the box lengths to change while maintaining constant volume. We find that the box length search algorithm converges at least two orders of magnitude more quickly than the variable box length Monte Carlo method. Although the box length search algorithm is not ergodic, it successfully finds the box length that minimizes the free energy of the system. We verify this by examining the free energy as determined by the Monte Carlo simulation.     

The importance of excluded solvent volume effects in computing hydration free energies

Continuum dielectric methods such as the Born equation have been widely used to compute the electrostatic component of the solvation free energy, DeltaG(solv)(elec), because they do not need to include solvent molecules explicitly and are thus far less costly compared to molecular simulations. All of these methods can be derived from Gauss Law of Maxwell's equations, which yields an analytical solution for the solvation free energy, DeltaG(Born), when the solute is spherical. However, in Maxwell's equations, the solvent is assumed to be a structureless continuum, whereas in reality, the near-solute solvent molecules are highly structured unlike far-solute bulk solvent. Since we have recently reformulated Gauss Law of Maxwell's equations to incorporate the near-solute solvent structure by considering excluded solvent volume effects, we have used it in this work to derive an analytical solution for the hydration free energy of an ion. In contrast to continuum solvent models, which assume that the normalized induced solvent electric dipole density P(n) is constant, P(n) mimics that observed from simulations. The analytical formula for the ionic hydration free energy shows that the Born radius, which has been used as an adjustable parameter to fit experimental hydration free energies, is no longer ill defined but is related to the radius and polarizability of the water molecule, the hydration number, and the first peak position of the solute-solvent radial distribution function. The resulting DeltaG(solv)(elec) values are shown to be close to the respective experimental numbers.     

Fragment-based quantum mechanical methods for periodic systems with Ewald summation and mean image charge convention for long-range electrostatic interactions

We describe an Ewald-summation method to incorporate long-range electrostatic interactions into fragment-based electronic structure methods for periodic systems. The present method is an extension of the particle-mesh Ewald technique for combined quantum mechanical and molecular mechanical (QM/MM) calculations, and it has been implemented into the explicit polarization (X-Pol) potential to illustrate the computational details. As in the QM/MM-Ewald method, the X-Pol-Ewald approach is a linear-scaling electrostatic method, in which the short-range electrostatic interactions are determined explicitly in real space and the long-range Ewald pair potential is incorporated into the Fock matrix as a correction. To avoid the time-consuming Fock matrix update during the self-consistent field procedure, a mean image charge (MIC) approximation is introduced, in which the running average with a user-chosen correlation time is used to represent the long-range electrostatic correction as an average effect. Test simulations on liquid water show that the present X-Pol-Ewald method takes about 25% more CPU time than the usual X-Pol method using spherical cutoff, whereas the use of the MIC approximation reduces the extra costs for long-range electrostatic interactions by 15%. The present X-Pol-Ewald method provides a general procedure for incorporating long-range electrostatic effects into fragment-based electronic structure methods for treating biomolecular and condensed-phase systems under periodic boundary conditions.     

New possibilities of Möller-Plesset perturbation theory for calculating correlation energies of closed-shell systems

A new scheme for generating and selecting configurational wave functions (CWF) including the correlation effects is suggested. Standard Möller-Plesset perturbation theory (SMP) is modified in such a way that the newly constructed CWF and the Hartree-Fock reference determinant are the eigenfunctions of a zero-order Hamiltonian. Possessing all advantages of SMP, the calculation scheme allows for more than 80% of the empirical correlation energy in the second order. The results for diatomic hydrides are comparable in accuracy to those obtained by the configuration interaction method.     

Meaningful activation energies for complex systems

It is known that the application of an inappropriate kinetic method to the thermal analysis of complex systems can lead to misleading results. To avoid this problem, the new parameter instantaneous mean activation energy is introduced and the Ozawa-Flynn-Wall method is assessed as a means of obtaining it. It is concluded that good results can generally be obtained by this method, provided that the various reactions occurring in the complex system are of the same type. The pyrolysis of coal is considered as a possible application of the method.

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Zusammenfassung Es ist bekannt, da\ die Anwendung einer ungeeigneten kinetischen Methode bei der thermischen Analyse komplexer Systeme zu falschen Ergebnissen führen kann. Um dieses Problem zu vermeiden, wird der neue Parameter momentane mittlere Aktivierungsenergie eingeführt. Die Ozawa-Flynn-Wall-Methode wird als zur Bestimmung dieses Parameters geeignet angesehen. Es wird der Schlu\ gezogen, da\ mit dieser Methode im allgemeinen gute Ergebnisse zu erhalten sein sollten, vorausgesetzt, da\ die in komplexen Systemen verlaufenden verschiedenen Reaktionen vom gleichen Typ sind. Die Pyrolyse von Kohle wird als mögliche Anwendung dieser Methode angesehen.

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The author wishes to thank the British Gas Corporation for permission to publish this work.     

Multipolar electrostatics for proteins: Atom–atom electrostatic energies in crambin

Accurate electrostatics necessitates the use of multipole moments centered on nuclei or extra point charges centered away from the nuclei. Here, we follow the former alternative and investigate the convergence behavior of atom‐atom electrostatic interactions in the pilot protein crambin. Amino acids are cut out from a Protein Data Bank structure of crambin, as single amino acids, di, or tripeptides, and are then capped with a peptide bond at each side. The atoms in the amino acids are defined through Quantum Chemical Topology (QCT) as finite volume electron density fragments. Atom‐atom electrostatic energies are computed by means of a multipole expansion with regular spherical harmonics, up to a total interaction rank of L = ?_A+ ?_B + 1 = 10. The minimum internuclear distance in the convergent region of all the 15 possible types of atom‐atom interactions in crambin that were calculated based on single amino acids are close to the values calculated from di and tripeptides. Values obtained at B3LYP/aug‐cc‐pVTZ and MP2/aug‐cc‐pVTZ levels are only slightly larger than those calculated at HF/6‐31G(d,p) level. This convergence behavior is transferable to the well‐known amyloid beta polypeptide Aβ_1–42. Moreover, for a selected central atom, the influence of its neighbors on its multipole moments is investigated, and how far away this influence can be ignored is also determined. Finally, the convergence behavior of AMBER becomes closer to that of QCT with increasing internuclear distance. © 2013 Wiley Periodicals, Inc.     

An efficient simulation technique for electrostatic free energies with applications to azurin

We present an efficient technique for Monte Carlo simulation of electrostatic free energy changes in biomolecular systems. It is a development of a recent method for the study of the influence of electrostatic interactions on the ion binding properties and redox potentials of biomolecules. The electrolyte solution is described by the primitive model, in which ions are treated as hard charged spheres and the solvent is replaced by a structureless continuum. The protein is kept fixed in the center of a spherical simulation cell, and the dielectric constant has the solvent value throughout the cell. By a multiparticle perturbation approach, it is possible to obtain a number of free energy changes within one simulation only. The usefulness of the method is illustrated with a study of the copper binding electron-transport protein azurin (from Alcaligenes denitrificans). The change in acidity of the histidine residues upon changing the redox state of the copper ion is calculated. The theoretical predictions agree well with available experiments. © 1995 by John Wiley & Sons, Inc.     

Meaningful activation energies for complex systems II

The validity of the Friedman method is assessed for systems of overlapping reactions. By means of mathematical analysis and numerical examples it is shown that, in the case of competitive reactions, the method gives the true value of the instantaneous mean activation energy. However, some error may be incurred if this method is applied to systems of independent reactions. The relative accuracy of the Friedman and Ozawa-Flynn-Wall methods is discussed in respect of complex systems of reactions.

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Zusammenfassung Die Gültigkeit der Friedman-Methode wird für Systeme von überlappenden Reaktionen einer Betrachtung unterzogen. Durch mathematische Analyse und numerische Beispiele wird gezeigt, daß im Falle kompetitiver Reaktionen die Methode den richtigen Wert der jeweiligen mittleren Aktivierungsenergie ergibt. Fehler können jedoch auftreten, wenn diese Methode auf Systeme voneinander unabhängiger Reaktionen angewandt wird. Die relative Genauigkeit der Methoden von Friedman und Ozawa-Flynn-Wall wird mit Hinsicht auf komplexe Reaktionssysteme diskutiert.

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The author wishes to thank the British Gas Corporation for permission to publish this work.     

Hyperviral extension for energies of parameter-dependent systems

The energy of a linearly perturbed enclosed system is calculated by way of the hypervirial methodology in the semiclassical limit. Solutions of this equation are compared with the exact as well as with WKB approximate results for the harmonic oscillator model.     

A calculation of exciton energies in periodic systems with helical symmetry: Application to a hydrogen fluoride chain

Ab initio Hartree–Fock crystal orbital results were used as input for the calculation of exciton energies in the Tamm–Dancoff and random-phase approximation for polymers with helical symmetry. The calculations were applied to a hydrogen fluoride chain. We show that the Tamm–Dancoff method is a good approximation to the random-phase theory. Furthermore, the influence of the band–band interaction in the exciton calculation has been investigated.     

Wannier-type atomic orbitals for periodic systems

The results of an application of Wannier-type atomic orbitals for calculations of local properties of electronic structure for periodic systems (atomic charges and covalencies, bond orders and total valencies), published earlier by the authors, are summarized. The numerical results are given for bulk crystals with the perovskite structure ATiO₃, A=Sr,Ba,Pb and LaMnO₃ (the Bloch functions are calculated in LCAO basis), for bulk MgO crystal (the Bloch functions are calculated both in LCAO and PW basis) and for the two periodic slab models of surfaces (001) of MgO and (110) of rutile TiO₂.