Exact Interaction Energy Research Articles

An FFT based approach to account for elastic interactions in OkMC: Application to dislocation loops in iron

Object kinetic Monte Carlo (OkMC) is a fundamental tool for modeling defect evolution in volumes and times far beyond atomistic models. The elastic interaction between defects is classically considered using a dipolar approximation but this approach is limited to simple cases and can be inaccurate for large and close interacting defects. In this work a novel framework is proposed to include exact elastic interactions between defects in OkMC valid for any type of defect and anisotropic media. In this method, the elastic interaction energy of a defect is computed by volume integration of its elastic strain multiplied by the stress created by all the other defects, being both fields obtained numerically using a FFT solver. The resulting interaction energies reproduce analytical elastic solutions and show the limited accuracy of dipole approaches for close and large defects.The OkMC framework proposed is used to simulate the evolution in space and time of self-interstitial atoms and dislocation loops in iron. It is found that including the anisotropy has a quantitative effect in the evolution of all the type of defects studied. Regarding dislocation loops, it is observed that using the exact interaction energy results in higher interactions than using the dipole approximation for close loops.

Prediction of many-electron wavefunctions using atomic potentials: Refinements and extensions to transition metals and large systems.

For a given many-electron molecule, it is possible to define a corresponding one-electron Schrödinger equation, using potentials derived from simple atomic densities, whose solution predicts fairly accurate molecular orbitals for single-determinant and multideterminant wavefunctions for the molecule. The energy is not predicted and must be evaluated by calculating Coulomb and exchange interactions over the predicted orbitals. Transferable potentials for first-row atoms and transition metal oxides that can be used without modification in different molecules are reported. For improved accuracy, molecular wavefunctions can be refined by slightly scaling nuclear charges and by introducing potentials optimized for functional groups. The accuracy is further improved by a single diagonalization of the Fock matrix constructed from the predicted orbitals. For a test set of 20 molecules representing different bonding environments, the transferable potentials with scaling give wavefunctions with energies that deviate from exact self-consistent field or configuration interaction energies by less than 0.05 eV and 0.02 eV per bond or valence electron pair, respectively. On diagonalization of the Fock matrix, the corresponding errors are reduced by a factor of three to less than 0.016 eV and 0.006 eV, respectively. Applications to the ground and excited states of a Ti18O36 nanoparticle and chlorophyll-a are reported.

Prediction of many-electron wavefunctions using atomic potentials.

For a given many-electron molecule, it is possible to define a corresponding one-electron Schrödinger equation, using potentials derived from simple atomic densities, whose solution predicts fairly accurate molecular orbitals for single- and multi-determinant wavefunctions for the molecule. The energy is not predicted and must be evaluated by calculating Coulomb and exchange interactions over the predicted orbitals. Potentials are found by minimizing the energy of predicted wavefunctions. There exist slightly less accurate average potentials for first-row atoms that can be used without modification in different molecules. For a test set of molecules representing different bonding environments, these average potentials give wavefunctions with energies that deviate from exact self-consistent field or configuration interaction energies by less than 0.08 eV and 0.03 eV per bond or valence electron pair, respectively.

Application of electronic structure methods to coupled Drude oscillators

The configuration interaction, perturbation theory, coupled cluster doubles, and random phase approximation methods are applied to a system of two quantum Drude oscillators interacting through dipole–dipole coupling. It is found that the random phase approximation gives the exact excitation energies and exact interaction energy of this system even when allowing only excitations into the first excited levels of the oscillators. In contrast, to obtain the exact results from configuration interaction or coupled cluster treatments of the model requires inclusion of excitations into all accessible excited configurations.

Pair Densities in Density Functional Theory

The exact interaction energy of a many-electron system is determined by the electron pair density, which is not well-approximated in standard Kohn--Sham density functional models. Here we study the (complicated but well-defined) exact universal map from density to pair density. We survey how many common functionals, including the most basic version of the local density approximation (Dirac exchange with no correlation contribution), arise from particular approximations of this map. We develop an algorithm to compute the map numerically and apply it to one-parameter families $\{\alpha\rho(\alpha x)\}_{\alpha>0}$ of one-dimensional homogeneous and inhomogeneous single-particle densities. We observe that the pair density develops remarkable multiscale patterns which strongly depend on both the particle number and the “width” $\alpha^{-1}$ of the single-particle density. The simulation results are confirmed by rigorous asymptotic results in the limiting regimes $\alpha\gg$ and $\alpha\ll1$. For one-dimensional homogeneous systems, we show that the whole spectrum of patterns is reproduced surprisingly well by a simple asymptotics-based ansatz which slowly smoothens out the “strictly correlated” $\alpha=0$ pair density while slowly turning on the $\alpha=\infty$ “exchange” terms as $\alpha$ increases. Our findings lend theoretical support to the celebrated semi-empirical idea [A. D. Becke, J. Chem. Phys., 98 (1993), pp. 5648--5652] to mix in a fractional amount of exchange, albeit not to assuming the mixing to be additive and taking the fraction to be a system-independent constant.

Intergeminal Correction to the Antisymmetrized Product of Strongly Orthogonal Geminals Derived from the Extended Random Phase Approximation.

We present a correction to the antisymmetrized product of strongly orthogonalized geminals (APSG) approach accounting for intergeminal correlation energy that APSG lacks. The correction is based on the fluctuation dissipation theorem formulated for geminals with transition density matrices obtained from the recently formulated extended random phase approximation. We show that the proposed intergeminal correlation correction greatly improves upon APSG energies by accounting for short- and long-range dynamical correlation. For covalently bonded molecules the potential energy curves are in good agreement with the exact results in the entire range of bond breaking. Also the description of weakly interacting systems is superior to that of APSG. In particular, we show that the proposed intergeminal correlation energy reduces to the correct form of the dispersion energy and asymptotically yields exact interaction energy of the helium dimer.

How well can polarization models of pairwise nonadditive forces describe liquid water?

Properties of liquid water have been computed using a near-exact rigid-monomer two-body (pairwise-additive) force field and the same field supplemented by a simple, non-empirical polarization model of pairwise nonadditive many-body forces. The inclusion of nonadditive polarization forces leads to a dramatic decrease, sometimes by an order of magnitude, of the deviations of water properties computed using classical molecular dynamics from experiment results. The remaining deviations are typically of the order of 10%. The model correctly predicts the temperature dependence of the properties except for the density of supercooled water. This good performance is achieved despite the known failure of the polarization model in reproducing trimer nonadditive interaction energies, confirmed here by showing that for a random set of trimers with all O-O separations smaller than 3.4 Å, selected from simulation snapshots, the average error of the model relative to accurate ab initio values is 71%. However, the errors gradually decrease for larger trimers, more abundant in liquid, and one can estimate that the polarization model should reproduce the exact liquid interaction energy to within about 6%. Although this accuracy is consistent with the observed performance of the polarization model, it does not explain the dramatic improvements over the two-body model. These improvements are due to the restructuring of liquid into tetrahedral arrangements instigated by the nonadditive polarization forces. The deviations of our predictions from experiments are generally also consistent with the estimated contributions from leading neglected effects other than the exchange nonadditive forces: the monomer flexibility and quantum nuclear motion effects.

Water at the Potassium Channel Gate: Quantum Calculations Show an Oscillation in Water Structure

Quantum calculations on the open gate and the immediately surrounding region of the Kv1.2 voltage gated potassium channel show that water adopts two conformations there, one when the ion is present, and another with no ion present. An energy minimum at the gate with no ion allows an ion to enter from the solution; from the gate it repels an ion in the channel pore cavity into the selectivity filter (essentially the “knock-on” mechanism). The pore cavity minimum can then be occupied by the ion moving from the gate minimum, leaving it available for anion from bulk, so the cycle continues. The arrangement of water, and the ion hydration, defines the gate energy minimum, and this in turn depends on the highly conserved prolines, especially P407 (2A79 pdb numbering), in the PVPV sequence in the gate. Absent an energy minimum at the gate an ion entering from solution would be repelled back into solution by the ion in the cavity. Furthermore, the ion remains hydrated by 8 water molecules, at least 1 more than in bulk, as it passes through the gate, based on optimizations with the ion at three different positions: 1) at the gate position nearest P407 2) 2A above 3) 2A below, this position. The open gate maintains very nearly the same protein atom positions throughout. Additional calculations will be required to determine the exact interaction energy with an ion in the cavity, and the amount of water that is present in the cavity. Optimizations of the system configuration were done at HF/4-22GSP level, with 692 atoms (693 with the ion). Energy was determined using both B3LYP/6-31G∗ and bvp86/6-31G∗, but the system was too large for free energy determination.

MODE SUMMATION APPROACH TO CASIMIR EFFECT BETWEEN TWO OBJECTS

In the last few years, several approaches have been developed to compute the exact Casimir interaction energy between two nonplanar objects, all lead to the same functional form, which is called the TGTG formula. In this paper, we explore the TGTG formula from the perspective of mode summation approach. Both scalar fields and electromagnetic fields are considered. In this approach, one has to first solve the equation of motion to find a wave basis for each object. The two T's in the TGTG formula are [Formula: see text]-matrices representing the Lippmann–Schwinger T-operators, one for each of the objects. Each [Formula: see text]-matrix can be found by matching the boundary conditions imposed on the object, and it is independent of the other object. However, it depends on whether the object is interacting with an object outside it, or an object inside it. The two G's in the TGTG formula are the translation matrices, relating the wave basis of an object to the wave basis of the other object. These translation matrices only depend on the wave basis chosen for each object, and they are independent of the boundary conditions on the objects. After discussing the general theory, we apply the prescription to derive the explicit formulas for the Casimir energies for the sphere–sphere, sphere–plane, cylinder–cylinder and cylinder–plane interactions. First the [Formula: see text]-matrices for a plane, a sphere and a cylinder are derived for the following cases: the object is imposed with Dirichlet, Neumann or general Robin boundary conditions; the object is semitransparent; and the object is a magnetodielectric object immersed in a magnetodielectric media. Then the operator approach developed by R. C. Wittman [IEEE Trans. Antennas Propag.36, 1078 (1988)] is used to derive the translation matrices. From these, the explicit TGTG formula for each of the scenarios can be written down. On the one hand, we have summarized all the TGTG formulas that have been derived so far for the sphere–sphere, cylinder–cylinder, sphere–plane and cylinder–plane configurations. On the other hand, we provide the TGTG formulas for some scenarios that have not been considered before.

Formation of matter-wave soliton molecules

We propose an experiment for forming matter-wave soliton molecules. In the proposed setup, we show that if two solitons are initially prepared in phase and with a sufficiently small separation and relative velocity, a bound pair will always form. This is verified by direct numerical simulation of the Gross–Pitaevskii equation and by the derivation of the exact interaction energy of two solitons, which takes the form of a Morse potential. This interaction potential depends not only on the separation but also on the relative phase of the solitons and is essential for an analytical treatment of a host of other problems, such as the soliton gas and the Toda lattice of solitons.

Lifshitz theory of atom-wall interaction with applications to quantum reflection

The Casimir-Polder interaction of an atom with a metallic wall is investigated in the framework of the Lifshitz theory. It is demonstrated that in some temperature (separation) region the Casimir-Polder entropy takes negative values and goes to zero when the temperature vanishes. This result is obtained both for an ideal metal wall and for real metal walls. Simple analytical representations for the Casimir-Polder free energy and force are also obtained. These results are used to make a comparison between the phenomenological potential used in the theoretical description of quantum reflection and exact atom-wall interaction energy, as given by the Lifshitz theory. Computations are performed for the atom of metastable ${\mathrm{He}}^{*}$ interacting with metal (Au) and dielectric (Si) walls. It is shown that the relative differences between the exact and phenomenological interaction energies are smaller in the case of a metallic wall. This is explained by the effect of negative entropy which occurs only for a metal wall. More accurate atom-wall interaction energies computed here can be used for the interpretation of measurement data in the experiments on quantum reflection.

Exploring the new three-dimensionalab initiointeraction energy surface of the Ar–HF complex: Rovibrational calculations for Ar–HF and Ar–DF with vibrationally excited diatoms

Several features and the performance of the recently published [P. Jankowski and M. Ziolkowski, Mol. Phys. 104, 2293 (2006)] three-dimensional intermolecular potential energy surface for the Ar-HF complex have been investigated. This full-dimensional surface has been obtained using the method of the local expansion of the exact interaction energy surface [P. Jankowski, J. Chem. Phys. 121, 1655 (2004)] in the Taylor series with respect to intramolecular coordinates. The interaction energies have been calculated with the coupled-cluster supermolecular method with single, double, and noniterative triple excitations. The convergence of the interaction energy with respect to the size of the basis set is discussed. The two-dimensional surfaces resulting from averaging of the full-dimensional surface over the intramolecular vibration of HF have been obtained and directly compared to the empirical H6(4,3,2) set of surfaces proposed by Hutson [J. Chem. Phys. 96, 6752 (1992)]. A very good agreement has been observed. The averaged potentials have been used to calculate the rovibrational energy levels of the Ar-HF and Ar-DF complexes and compared to the experimental data. The accuracy of rovibrational calculations achieved with the new surface is much better than with any of the ab initio surfaces available so far. Predictions of the rovibrational energy levels and spectroscopic constants have also been done for Ar-HF with HF in the v=4,5 vibrational states, and for Ar-DF and DF in the v=3,4 states. The full-dimensional surface studied in this paper is the first ab initio surface which is fully compatible with the empirical H6(4,3,2) surface proposed by Hutson.

Long range behavior of high‐rank topological multipole moments

The construction of a high-rank multipolar force field (for peptides) is a complex task, leading to several intermediate questions in need of a clear answer. Here we focus on the convergence of the (electrostatic) multipolar expansion at medium and long range. Using molecular electron densities, quantum chemical topology (QCT) defines the atoms as finite volumes, each endowed with multipole moments. The terms in the multipole expansion are grouped according to powers of the internuclear distance, R(-L). Given two atom types at a given distance, we determine which rank (L) is necessary for the electrostatic energy to converge to the exact interaction energy within a certain error. With this information, the rank of the expansion for each interaction can be adapted to the required accuracy and the available computing power.

Exact zero-point interaction energy between cylinders

We calculate the exact Casimir interaction energy between two perfectly conducting, very long, eccentric cylindrical shells using a mode summation technique. Several limiting cases of the exact formula for the Casimir energy corresponding to this configuration are studied both analytically and numerically. These include concentric cylinders, cylinder-plane and eccentric cylinders, for small and large separations between the surfaces. For small separations we recover the proximity approximation, while for large separations we find a weak logarithmic decay of the Casimir interaction energy, typical of cylindrical geometries.

Approximate generation of full-dimensional ab initio van der Waals surfaces for high-resolution spectroscopy.

A method for the generation of highly accurate, nearly-exact, full-dimensional interaction energy surfaces for weakly interacting subsystems is proposed. The method is based on the local expansion of the exact interaction energy surface in the Taylor series with respect to intramolecular coordinates. It is shown that without any significant loss of accuracy this expansion can be limited to a few low-order terms. This leads to significant savings in computations of the full-dimensional interaction energy surfaces. Also a method for the direct calculation of the interaction energy surface of reduced dimensionality, corresponding to averaging over the intramolecular vibrations, without explicit knowledge of the full-dimensional surface, is presented. The main ideas and computational features of the proposed scheme are comprehensively tested for the Ar-HF system.

Two electrons in one-dimensional nanorings: Exact solutions and interaction energies

Exact series solutions for two electrons in one-dimensional nanorings have been obtained by expanding the Coulomb potential into power series and solving the corresponding equations in different regions. Electronic structures and interaction energies of two electrons in nanorings with different sizes have been calculated. The quantum-size effect of interaction energies and energy levels and the size-dependent magnetic oscillations of two-electron spectra have been revealed.

Exact non-periodic interaction energy for plane-wave charge distributions

If two non-periodic charge densities are described using a sum of plane waves, the evaluation of their interaction energies faces the problem of periodic images. The current paper solves this problem by treating the interaction with the full periodic charge density as the limiting case of an interaction with a spatially damped periodic charge density. The problem of cancelling infinities is avoided by cancelling the spurious interactions using a multipole expansion before the damping is removed. The result is a sum of the well-known terms involving the Fourier coefficients of the Coulomb potential and three additional terms. It is exact if the charge densities are contained within a sphere inscribed into the unit cell. The computational cost of evaluating the interaction energy is linear in the number of plane waves used.

The escaped charge problem in solvation continuum models

In the present article, we provide an analytical proof of the accuracy shown by integral equation formalism (IEF) continuum solvation method in dealing with the solute electronic charge that quantum mechanically penetrates outside the cavity embedding it. In particular we show that the approximation of the solute–solvent interaction energy obtained by this method is always an upper bound of the exact electrostatic interaction energy and that a lower bound of the exact energy can also be defined. A numerical example for which both approximated bounds and exact energies can be computed is presented.

Ab initio basis set and correlation limit interaction energies for He–He, He–H 2, and H–H 2

An ab initio search for the exact nonrelativistic interaction energies for He–He, He–H 2, and H–H 2 systems was made by estimating the basis set limit CCSD(T) interaction energies from the extrapolation of counterpoise (CP) interaction energies with aug-cc-pV5Z and aug-cc-pV6Z basis sets by 1/( X−1) 3 ( X=5,6) and correcting for the difference between the FCI and CCSD(T) energies. While the result for He 2 is in perfect agreement with the exact quantum Monte Carlo result, the result for H 3 suggests the barrier height for colinear H+H 2→H 2+H reaction is close to 9.60 kcal/mol, ∼0.01 kcal/mol lower than the best result reported so far

Atom–atom partitioning of intramolecular and intermolecular Coulomb energy

An atom–atom partitioning of the (super)molecular Coulomb energy is proposed on the basis of the topological partitioning of the electron density. Atom–atom contributions to the molecular intra- and intermolecular Coulomb energy are computed exactly, i.e., via a double integration over atomic basins, and by means of the spherical tensor multipole expansion, up to rank L=lA+lB+1=5. The convergence of the multipole expansion is able to reproduce the exact interaction energy with an accuracy of 0.1–2.3 kJ/mol at L=5 for atom pairs, each atom belonging to a different molecule constituting a van der Waals complex, and for nonbonded atom–atom interactions in single molecules. The atom–atom contributions do not show a significant basis set dependence (3%) provided electron correlation and polarization basis functions are included. The proposed atom–atom Coulomb interaction energy can be used both with post-Hartree–Fock wave functions and experimental charge densities in principle. The Coulomb interaction energy between two molecules in a van der Waals complex can be computed by summing the additive atom–atom contributions between the molecules. Our method is able to extract from the supermolecule wave function an estimate of the molecular interaction energy in a complex, without invoking the reference state of free noninteracting molecules. We provide computational details of this method and apply it to (C2H2)2; (HF)2; (H2O)2; butane; 1,3,5-hexatriene; acrolein and urocanic acid, thereby covering a cross section of hydrogen bonds, and covalent bonds with and without charge transfer.