Long-range Asymptotics Research Articles

Excited-State Properties for Extended Systems: Efficient Hybrid Density Functional Methods.

Time-dependent density functional theory has become state-of-the-art for describing photophysical and photochemical processes in extended materials because of its affordable cost. The inclusion of exact exchange was shown to be essential for the correct description of the long-range asymptotics of electronic interactions and thus a well-balanced description of valence, Rydberg, and charge-transfer excitations. Several approaches for an efficient treatment of exact exchange have been established for the ground state, while implementations for excited-state properties are rare. Furthermore, the high computational costs required for excited-state properties in comparison to ground-state computations often hinder large-scale applications on periodic systems with hybrid functional accuracy. We therefore propose two approximate schemes for improving computational efficiency for the treatment of exact exchange. Within the auxiliary density matrix method (ADMM), exact exchange is estimated using a relatively small auxiliary basis and the introduced basis set incompleteness error is compensated by an exchange density functional correction term. Benchmark results for a test set of 35 molecules demonstrate that the mean absolute error introduced by ADMM is smaller than 0.3 pm for excited-state bond lengths and in the range of 0.02-0.04 eV for vertical excitation, adiabatic excitation, and fluorescence energies. Computational timings for a series of covalent-organic frameworks demonstrate that a speed-up of at least 1 order of magnitude can be achieved for excited-state geometry optimizations in comparison to conventional hybrid functionals. The second method is to use a semiempirical tight binding approximation for both Coulomb and exchange contributions to the excited-state kernel. This simplified Tamm-Dancoff approximation (sTDA) achieves an accuracy comparable to approximated hybrid density functional theory when referring to highly accurate coupled-cluster reference data. We find that excited-state bond lengths deviate by 1.1 pm on average and mean absolute errors in vertical excitation, adiabatic excitation, and fluorescence energies are in the range of 0.2-0.5 eV. In comparison to ADMM-approximated hybrid functional theory, sTDA accelerates the computation of broad-band excitation spectra by 1 order of magnitude, suggesting its potential use for large-scale screening purposes.

Accelerating the 3D reference interaction site model theory of molecular solvation with treecode summation and cut-offs.

The 3D reference interaction site model (3D-RISM) of molecular solvation is a powerful tool for computing the equilibrium thermodynamics and density distributions of solvents, such as water and co-ions, around solute molecules. However, 3D-RISM solutions can be expensive to calculate, especially for proteins and other large molecules where calculating the potential energy between solute and solvent requires more than half the computation time. To address this problem, we have developed and implemented treecode summation for long-range interactions and analytically corrected cut-offs for short-range interactions to accelerate the potential energy and long-range asymptotics calculations in non-periodic 3D-RISM in the AmberTools molecular modeling suite. For the largest single protein considered in this work, tubulin, the total computation time was reduced by a factor of 4. In addition, parallel calculations with these new methods scale almost linearly and the iterative solver remains the largest impediment to parallel scaling. To demonstrate the utility of our approach for large systems, we used 3D-RISM to calculate the solvation thermodynamics and density distribution of 7-ring microtubule, consisting of 910 tubulin dimers, over 1.2 million atoms.

Diagrammatics of tunable interactions in anisotropic colloids in rotating electric or magnetic fields: New kind of dipole-like interactions.

Anisotropic particles are widely presented in nature, from colloidal to bacterial systems, and control over their interactions is of crucial importance for many applications, from self-assembly of novel materials to microfluidics. Placed in rapidly rotating external electric fields, colloidal particles attain a tunable long-range and many-body part in their interactions. For spherical colloids, this approach has been shown to offer rich capabilities to construct the tunable interactions via designing the internal structure of particles and spatial hodographs of external rotating fields, but in the case of anisotropic particles, the interactions remain poorly understood. Here, we show that tunable interactions between anisotropic rod-like and spheroidal colloidal particles in rotating electric or magnetic fields can be calculated and analyzed with the diagrammatic technique we developed in the present work. With this technique, we considered an in-plane rotating electric field, obtained the long-range asymptotics of the anisotropic interactions, calculated the tunable interactions between particles rotating synchronously, and found conditions for rotator repulsion. We compared the mechanisms providing tunable interactions to those for orientational (Keesom), induction (Debye), and dispersion (London) interactions in molecular systems and found that the tunable interactions between anisotropic particles represent a novel kind of dipole-like interaction. The method can be directly generalized for magnetically induced interactions, 3D systems, and fields with spatial hodographs. The results provide significant advance in theoretical methods for tunable interactions in colloids and, therefore, are of broad interest in condensed matter, chemical physics, physical chemistry, materials science, and soft matter.

Detecting quantum critical points in the t-t' Fermi-Hubbard model via complex network theory

A considerable success in phenomenological description of text {high-T}_{text{c}} superconductors has been achieved within the paradigm of Quantum Critical Point (QCP)—a parental state of a variety of exotic phases that is characterized by dense entanglement and absence of well-defined quasiparticles. However, the microscopic origin of the critical regime in real materials remains an open question. On the other hand, there is a popular view that a single-band t-t' Hubbard model is the minimal model to catch the main relevant physics of superconducting compounds. Here, we suggest that emergence of the QCP is tightly connected with entanglement in real space and identify its location on the phase diagram of the hole-doped t-t' Hubbard model. To detect the QCP we study a weighted graph of inter-site quantum mutual information within a four-by-four plaquette that is solved by exact diagonalization. We demonstrate that some quantitative characteristics of such a graph, viewed as a complex network, exhibit peculiar behavior around a certain submanifold in the parametric space of the model. This method allows us to overcome difficulties caused by finite size effects and to identify precursors of the transition point even on a small lattice, where long-range asymptotics of correlation functions cannot be accessed.

Long-range asymptotics of exchange energy in the hydrogen molecule

The exchange energy, i.e., the splitting ΔE between gerade and ungerade states in the hydrogen molecule, has proven very difficult in numerical calculation at large internuclear distances R, while the known results are sparse and highly inaccurate. On the other hand, there are conflicting analytical results in the literature concerning its asymptotics. In this work, we develop a flexible and efficient numerical approach using explicitly correlated exponential functions and demonstrate highly accurate exchange energies for internuclear distances as large as 57.5 a.u. This approach may find further applications in calculations of inter-atomic interactions. In particular, our results support the asymptotics form ΔE ∼ R5/2e-2R, but with the leading coefficient being 2σ away from the analytically derived value.

Electric dipole image forces in three-layer systems: The classical electrostatic model.

General exact analytical expressions have been derived for the image force energy Wi(Z, φ) of a point dipole in a classical three-layer system composed of dispersionless media with arbitrary constant dielectric permittivities εi. Here, i = 1-3 is the layer number, and Z and φ are the dipole coordinate and orientation angle, respectively. It was found that the long-range asymptotics Wi(Z→∞,φ) in both covers (i = 1, 3) are reached unexpectedly far from the interlayer (i = 2). Another specific feature of the solution consists in that the interference of the fields created by polarization charges emerging at both interfaces leads to the appearance of a constant contribution inside the interlayer with a non-standard dependence on the dipole orientation angle φ. It was shown that by changing the dielectric constants of the structure components, one can realize two peculiar regimes of the Wi(Z, φ) behavior in the covers; namely, there arises either a potential barrier preventing adsorption or a well far from the interface, both being of a totally electrostatic origin, i.e., without involving the Pauli exchange repulsion, which is taken into account in the conventional theories of physical adsorption. The results obtained provide a fresh insight into the physics of adsorption in physical electronics, chemical physics, and electrochemistry.

Low-density expansions for the homogeneous dipolar Bose gas at zero temperature

The low-density expansions for the energy, chemical potential, and condensate depletion of the homogeneous dilute dipolar Bose gas are obtained by regularizing the dipole-dipole interaction at long distances. It is shown that the leading term, proportional to the density, allows a simple physical interpretation and consistently describes the thermodynamic stability of the system. The long-range asymptotics are obtained analytically for the normal and anomalous one-particle correlation functions and the pair distribution function. We discuss the properties of the two-body scattering with zero relative momentum for the dipole-dipole interaction, in particular, we derive the asymptotics of the wave function and a correction to the scattering length for small values of the dipolar range. We show how the density expansions can be derived within the Bogoliubov model of weakly interacting particles without any divergence from the assumption of universality of the expansions at low densities.

Singling Out Dynamic and Nondynamic Correlation.

The correlation part of the pair density is separated into two components, one of them being predominant at short electronic ranges and the other at long ranges. The analysis of the intracular part of these components permits to classify molecular systems according to the prevailing correlation: dynamic or nondynamic. The study of the long-range asymptotics reveals the key component of the pair density that is responsible for the description of London dispersion forces and a universal decay with the interelectronic distance. The natural range-separation, the identification of the dispersion forces, and the kind of predominant correlation type that arise from this analysis are expected to be important assets in the development of new electronic structure methods in wave function, density, and reduced density-matrix functional theories.

Finite-size corrections in numerical simulation of liquid water.

Monte Carlo (MC) simulations of the SPC/E liquid water model are performed at two numbers of molecules N = 100 and 512 and in canonical NVT, isobaric NPT, and grand canonical μVT ensembles. The molecular non-spherically symmetric pair distribution function g(r, Ω) (pdf) is accumulated in terms of projections onto rotational invariants. The precisely measured differences between N values and between ensembles are in very good agreement with the theoretical predictions for the expected finite-size corrections of different origins: (1) the canonical simulation is affected by explicit corrections due to the absence of density fluctuations. Beyond the well-known shift in the long-range asymptote, all projections exhibit rich short-range contributions. These corrections vanish exactly in the isobaric ensemble provided that the pdf is constructed using the rigorous definition. (2) In the presence of dielectric discontinuity with the external medium surrounding the central box and its replica within the Ewald treatment of the Coulombic interactions, the disagreement with the prediction of the literature, already noticed recently for dipolar fluids, is confirmed in the present site-site model. (3) The implicit corrections originate from the coupling between the environment around a given particle and that around its periodic images in the neighboring cells. The recent, powerful MC/HNC mixed integral equation, which offers a complete and exact description of the molecular correlations in the whole real and Fourier spaces, enables us to quantify the observed N-dependence in the pdf projections down to the sub 10-3 levels.

Fingerprints of different interaction mechanisms on the collective modes in complex (dusty) plasmas

In this paper, we discuss the relations between the exact shape of interparticle interactions in complex (dusty) plasmas and the dispersion relation of the longitudinal collective mode. Several representative repulsive potentials, predicted previously theoretically, are chosen, and the corresponding dispersion relations are calculated using the quasi-crystalline approximation. Both weakly coupled and strongly coupled regimes are considered. It is shown that the long-wavelength portions of the dispersion curves can be sensitive to the long-range asymptote of the interaction potential. This can be used to discriminate between different interaction mechanisms operational in complex plasmas experimentally. Main requirements are briefly discussed.

Electrostatic charge-charge and dipole-dipole interactions near the surface of a medium with screening non-locality (Review Article)

Interaction energy between two point charges, WQQ′, or two point dipoles, WPP′, located in a medium with a constant dielectric permittivity near the plane surface of a metallic or semiconducting substrate with the spatial dispersion of its dielectric function has been revisited. The calculations were made on the basis of the Green's function method for layered systems. Long-range lateral asymptotics were found. The non-local character of screening in the substrates was shown to substantially modify the dependences of WQQ′ and WPP′ on the distance between the objects concerned. Thus, the purported conventional electrostatic interactions between adsorbed atoms and molecules (modeled by point charges and point dipoles) should be reconsidered making allowance for the substrate polarization. In particular, this factor may significantly influence the structure of electrostatic dipole lattices arising near the surfaces of solids, as well as the kinetics of charge or dipole motion over the surface.

On the errors of local density (LDA) and generalized gradient (GGA) approximations to the Kohn-Sham potential and orbital energies.

In spite of the high quality of exchange-correlation energies Exc obtained with the generalized gradient approximations (GGAs) of density functional theory, their xc potentials vxc are strongly deficient, yielding upshifts of ca. 5 eV in the orbital energy spectrum (in the order of 50% of high-lying valence orbital energies). The GGAs share this deficiency with the local density approximation (LDA). We argue that this error is not caused by the incorrect long-range asymptotics of vxc or by self-interaction error. It arises from incorrect density dependencies of LDA and GGA exchange functionals leading to incorrect (too repulsive) functional derivatives (i.e., response parts of the potentials). The vxc potential is partitioned into the potential of the xc hole vxchole (twice the xc energy density ϵxc), which determines Exc, and the response potential vresp, which does not contribute to Exc explicitly. The substantial upshift of LDA/GGA orbital energies is due to a too repulsive LDA exchange response potential vxresp (LDA) in the bulk region. Retaining the LDA exchange hole potential plus the B88 gradient correction to it but replacing the response parts of these potentials by the model orbital-dependent response potential vxresp (GLLB) of Gritsenko et al. [Phys. Rev. A 51, 1944 (1995)], which has the proper step-wise form, improves the orbital energies by more than an order of magnitude. Examples are given for the prototype molecules: dihydrogen, dinitrogen, carbon monoxide, ethylene, formaldehyde, and formic acid.

Nuclear mass corrections to the Casimir-Polder interaction

We present a derivation of the finite nuclear mass corrections to the Casimir-Polder interaction between two atomic systems in the ground state. Equivalently, we show how the long-range asymptotics of the adiabatic correction is modified due to the finite speed of light. We show that in addition to the contribution resulting from the finite-mass correction to atomic polarizabilities, a further contribution exists.

On dynamical justification of quantum scattering cross section

We suggest a dynamical justification of quantum differential cross section in the context of long-time transition to stationary regime for the Schrödinger equation. The problem has been stated by Reed and Simon. Our approach is based on spherical incident waves produced by a harmonic source and the long-range asymptotics for the corresponding spherical limiting amplitudes. The main results are as follows: i) the convergence of spherical limiting amplitudes to the limit as the source goes away to infinity, and ii) the proof of the coincidence of the corresponding limit scattering cross section with the universally recognized formula. The main technical ingredients are the Agmon–Jensen–Kato's analytical theory of the Green function, Ikebe's uniqueness theorem for the Lippmann–Schwinger equation, and some refinement of classical long-range asymptotics for the Coulomb potentials.

Calculation of the molecular integrals with the range-separated correlation factor.

Explicitly correlated quantum chemical calculations require calculations of five types of two-electron integrals beyond the standard electron repulsion integrals. We present a novel scheme, which utilises general ideas of the McMurchie-Davidson technique, to compute these integrals when the so-called "range-separated" correlation factor is used. This correlation factor combines the well-known short range behaviour resulting from the electronic cusp condition, with the exact long-range asymptotics derived for the helium atom [Lesiuk, Jeziorski, and Moszynski, J. Chem. Phys. 139, 134102 (2013)]. Almost all steps of the presented procedure are formulated recursively, so that an efficient implementation and control of the precision are possible. Additionally, the present formulation is very flexible and general, and it allows for use of an arbitrary correlation factor in the electronic structure calculations with minor or no changes.

Long-range atom-wall interactions and mixing terms: Metastable hydrogen

We investigate the interaction of metastable 2S hydrogen atoms with a perfectly conducting wall, including parity-breaking S-P mixing terms (with full account of retardation). The neighboring 2P_1/2 and 2P_3/2 levels are found to have a profound effect on the transition from the short-range, nonrelativistic regime, to the retarded form of the Casimir-Polder interaction. The corresponding P state admixtures to the metastable 2S state are calculated. We find the long-range asymptotics of the retarded Casimir-Polder potentials and mixing amplitudes, for general excited states, including a fully quantum electrodynamic treatment of the dipole-quadrupole mixing term. The decay width of the metastable 2S state is roughly doubled even at a comparatively large distance of 918 atomic units (Bohr radii) from the perfect conductor. The magnitude of the calculated effects is compared to the unexplained Sokolov effect.

Extraction of site–site bridge functions and effective pair potentials from simulations of polar molecular liquids

We develop an efficient method to extract site-site bridge functions from molecular simulations. The method is based on the inverse solution of the reference site interaction model. Using the exact long-range asymptotics of site-site direct correlation functions defined by the site-site Ornstein-Zernike equations, we regularize the ill-posed inverse problem, and then calculate site-site bridge functions and effective pair potentials for ambient water, methanol, and ethanol. We have tested the proposed algorithm and checked its performance. Our study has revealed various peculiarities of the site-site bridge functions, such as long-range behavior, strong dependence on the electrostatic interactions. Using the obtained data, we have calculated thermodynamic properties of the solvents, namely, isothermal compressibility, internal energy, and Kirkwood-Buff integrals. The obtained values are in excellent agreement not only with molecular simulations but also with available experimental data. Further extensions of the method are discussed.

Hyperfine-frequency shifts of alkali-metal atoms during long-range collisions

Collisions with chemically inert atoms or molecules change the hyperfine coupling of an alkali-metal atom through the hyperfine-shift interaction. This interaction is responsible for the pressure shifts of the microwave resonances of alkali-metal atoms in buffer gases, is an important spin interaction in alkali-metal--noble-gas van der Waals molecules, and is anticipated to enable the magnetoassociation of ultracold molecules such as RbSr. An improved estimate is presented for the long-range asymptote of this interaction for Na, K, Rb, and Cs. To test the results, the change in hyperfine coupling due to a static electric field is estimated and reasonable agreement is found.

Extraction of atom–atom bridge and direct correlation functions from molecular simulations: A test for ambient water

We compute partial structure factors for SPC/E water under ambient conditions, using radial distribution functions derived from simulations and employing finite-size corrections. We calculate atom–atom direct correlation functions by an inverse solution to reference interaction site model integral equations with account of long-range asymptotics of the functions as additional constraints. We split the short- and long-range contributions to the direct functions in the Ewald type manner. Finally we reconstruct atom–atom bridge functions. The obtained direct correlation functions are compared with the data obtained by the hypernetted chain approximation.

Multipole effects in atom-surface interactions: A theoretical study with an application to He–α-quartz

We address some practically important and experimentally relevant questions concerning atom-surface interactions. For example, the quadrupole interaction of an atom with a dielectric macroscopic surface is obtained with full allowance for retardation and in closed form, valid for an arbitrary atom-surface distance. We also explore the relevance of higher-order multipole effects in atom-surface interactions, and we find that details in the dielectric response function of the surface material have to be resolved in order to accurately calculate the atom-surface potential. They lead to important effects at an intermediate range which cannot be explained by either short-range or long-range asymptotics. A numerical example (helium interacting with $\ensuremath{\alpha}$-quartz) is included in order to illustrate this aspect.