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Lev Kantorovich

Publications and source records attributed to Lev Kantorovich.

18 recordsLinked to original sources

From London to Morse via Binnig, Quate, and Gerber

In their landmark paper introducing the atomic force microscope [Phys. Rev. Lett. \textbf{56}, 930 (1986)], Binnig, Quate, and Gerber presciently anticipated that the technique would ultimately be capable of probing interactions running the gamut from weak van der Waals interactions to strong covalent bonding. They also highlighted that the tip-sample forces central to AFM are present, and often highly influential, in scanning tunnelling microscopy; indeed, this realisation directly inspired the invention of the force microscope. In this perspective for the \textit{Forty Years of AFM} special issue, we review selected aspects of two decades of work from our group at the University of Nottingham that span the force range highlighted by BQG and are united by a common, central theme: the probe as active participant rather than passive observer. Our selection of results also covers length- and correlation-scales from the microscopic right down to the single chemical bond limit, tracking a spectrum of interactions from van der Waals/Hamaker forces, through hydrogen bonding, to covalent bonds and, finally, atom-by-atom assembly of metal clusters via vertical tip-sample transfer. Echoing BQG's own observations on the prevalence of probe-sample forces in STM, we also discuss recent evidence that tip-induced heterogeneity underpins first-passage dynamics in molecular diffusion and highlight the challenges in acquiring non-invasive measurements of diffusion barriers for adsorbed molecules that are readily perturbed by the probe. We close with a perspective on machine learning's growing role in automating tip-driven atomic and molecular manipulation.

cond-mat.mes-hall

Photon-assisted stochastic resonance in nanojunctions

We study stochastic resonance in molecular junctions driven by a periodically-varying external field. This is done using the time-dependent Landauer-B{ü}ttiker formalism, which follows from exact analytical solutions to the Kadanoff-Baym equations describing the molecular junction subject to an arbitrary time-dependent bias. We focus on a double quantum dot nanojunction and compare the effects of the temperature with the fluctuating bias in the statically-driven case. We then consider the combined effect of AC-driving and white noise fluctuations on the rectified current through the nanojunction, and find a stochastic resonance effect, where at certain driving conditions the bias fluctuations enhance the current signal. The study is then extended to include the color noise in the applied bias, so that the combined effect of the color noise correlation time and driving frequency on stochastic resonance is investigated. We thereby demonstrate that photon-assisted transport can be optimized by a suitably tuned environment.

cond-mat.mes-hall

Global Optimization of Molybdenum Subnanoclusters on Graphene: a Consistent Approach Towards Catalytic Applications

The development of novel sub-nanometer clusters (SNCs) catalysts with superior catalytic performance depends on the precise control of clusters' atomistic sizes, shapes, and accurate deposition onto surfaces. The intrinsic complexity of the adsorption process complicates the ability to achieve an atomistic understanding of the most relevant structure-reactivity relationships hampering the rational design of novel catalytic materials. In most cases, existing computational approaches rely on just a few structures to draw conclusions on clusters' reactivity thereby neglecting the complexity of the existing energy landscapes thus leading to insufficient sampling and, most likely, unreliable predictions. Moreover, modelling of the actual experimental procedure that is responsible for the deposition of SNCs on surfaces is often not done even though in some cases this procedure may enhance the significance of certain (e.g., metastable) adsorption geometries. This study proposes a novel systematic approach that utilizes global search techniques, specifically, the particle swarm optimization (PSO) method, in conjunction with \textit{ab-initio} calculations, to simulate {\it all stages} in the beam experiments, from predicting the most relevant SNCs structures in the beam and on a surface, to their reactivity. To illustrate the main steps of our approach, we consider the deposition of Molybdenum SNC of 6 Mo atoms on a free-standing graphene surface, as well as their catalytic properties with respect to the CO molecule dissociation reaction. Even though our calculations are not exhaustive and serve only to produce an illustration of the method, they are still able to provide insight into the complicated energy landscape of Mo SNCs on graphene demonstrating the catalytic activity of Mo SNCs and the importance of performing statistical sampling of available configurations...

cond-mat.mtrl-sci

A comprehensive exploration of structural and electronic properties of Molybdenum clusters

Molybdenum clusters, characterised by their unique structure and intriguing catalytic properties, have gained significant attention in recent years. In several existing studies density functional theory (DFT) methods have been used to find the lowest energy Mo clusters and explore their electronic and magnetic structure. In all cases, with the exception of a single recent study, where a genetic algorithm was employed, initial geometries of the clusters, prior to geometry optimisation, were chosen using heuristic approaches based on symmetry considerations and known structures. DFT calculations were performed using different types of pseudopotentials, from hard to soft, and different types of basis sets. However, no comprehensive study has yet been done in which a DFT method with the best control on its precision would be complemented by a reliable global minimum search method to find the lowest energy Mo clusters. In this work, we employ a combination of a plane wave-based DFT method and \emph{ab initio} random structure searching (AIRSS) technique to find the lowest energy clusters of up to 10 Mo atoms. In each case, the search has been performed for clusters with different spin multiplicities, which enabled us to explore their magnetic structure. The results are compared for both hard and soft pseudopotentials stressing the importance of treating more electrons explicitly, in agreement with some of the previous studies. For most of the low-energy magnetic structures found, we investigate the distribution of their spin densities, and for all low energy clusters, we confirm their stability by calculating their phonon structure. Finally, free energies of the Mo clusters, within the quasi-harmonic approximation, are also calculated and discussed.

cond-mat.mtrl-sci

On-Surface Decarboxylation Coupling Facilitated by Lock-to-Unlock Variation of Molecules upon the Reaction

On-surface synthesis (OSS) involving relatively high energy barriers remains challenging due to a typical dilemma: firm molecular anchor is required to prevent molecular desorption upon the reaction, whereas sufficient lateral mobility is crucial for subsequent coupling and assembly. By locking the molecular precursors on the substrate then unlocking them during the reaction, we present a strategy to address this challenge. High-yield synthesis based on well-defined decarboxylation, intermediate transition and hexamerization is demonstrated, resulting in an extended and ordered network exclusively composed of the newly-synthesized macrocyclic compound. Thanks to the steric hindrance of its maleimide group, we attain a preferential selection of the coupling. This work unlocks a promising path to enrich the reaction types and improve the coupling selectivity hence the structual homogeneity of the final product for OSS.

physics.chem-ph

Subsurface Carbon-Induced Local Charge of Copper for On-Surface Displacement Reaction

Transition metal carbides have sparked unprecedented enthusiasm as high-performance catalysts in recent years. Still, the catalytic properties of copper (Cu) carbide remain unexplored. By introducing subsurface carbon (C) to Cu(111), displacement reaction of proton in carboxyl acid group with single Cu atom is demonstrated at the atomic scale and room temperature. Its occurrence is attributed to the C-doping induced local charge of surface Cu atoms (up to +0.30 e/atom), which accelerates the rate of on-surface deprotonation via reduction of the corresponding energy barrier, thus enabling the instant displacement of a proton with a Cu atom when the molecules land on the surface. Such well-defined and robust Cu$^{δ+}$ surface based on the subsurface C doping offers a novel catalytic platform for on-surface synthesis.

cond-mat.other

Describing non-Hermitian dynamics using a Generalized Three-Time NEGF for a Partition-free Molecular Junction with Electron-Phonon Coupling

In this paper we develop the Non-Equilibrium Green's Function (NEGF) formalism for a dissipative molecular junction that consists of a central molecular system with one-dimensional electronic transport coupled to a phonon environment and attached to multiple electronic leads. Our approach is partitionless - initial preparation of the system places the whole system in the correct canonical equilibrium state - and is valid for an external bias with arbitrary time dependence. Using path integrals as an intermediary tool, we apply a two-time Hubbard-Stratonovich transformation to the phonon influence functional with mixed real and imaginary times to obtain an exact expression for the electronic density matrix at the expense of introducing coloured Gaussian noises whose properties are rigorously derived from the environment action. This results in a unique stochastic Hamiltonian on each branch of the Konstantinov-Perel' contour (upper, lower, vertical) such that the time evolution operators in the Liouville equation no longer form a Hermitian conjugate pair, thus corresponding to non-Hermitian dynamics. To account for this we develop a generalized three-time NEGF which is sensitive to all branches of the contour, and relate it to the standard NEGF in the absence of phonons via a perturbative expansion of the noises. This approach is exact and fully general, describing the non-equilibrium driven dynamics from an initial thermal state while subject to inelastic scattering, and can be applied to non-Hermitian dynamics in general.

cond-mat.mes-hall

Efficient choice of coloured noises in stochastic dynamics of open quantum systems

The Stochastic Liouville-von Neumann (SLN) equation describes the dynamics of an open quantum system reduced density matrix coupled to a non-Markovian harmonic environment. The interaction with the environment is represented by complex coloured noises which drive the system, and whose correlation functions are set by the properties of the environment. We present a number of schemes capable of generating coloured noises of this kind that are built on a noise amplitude reduction procedure [Imai et al, Chem. Phys. 446, 134 (2015)], including two analytically optimised schemes. In doing so, we pay close attention to the properties of the correlation functions in Fourier space, which we derive in full. For some schemes the method of Wiener filtering for deconvolutions leads to the realisation that weakening causality in one of the noise correlation functions improves numerical convergence considerably, allowing us to introduce a well controlled method for doing so. We compare the ability of these schemes, along with an alternative optimised scheme [Schmitz and Stockburger, Eur. Phys. J.: Spec. Top. 227, 1929 (2019)], to reduce the growth in the mean and variance of the trace of the reduced density matrix, and their ability to extend the region in which the dynamics is stable and well converged for a range of temperatures. By numerically optimising an additional noise scaling freedom, we identify the scheme which performs best for the parameters used, improving convergence by orders of magnitude and increasing the time accessible by simulation.

quant-ph

Quantum interference and the time-dependent radiation of nanojunctions

Using the recently developed time-dependent Landauer-Büttiker formalism and Jefimenko's retarded solutions to the Maxwell equations, we show how to compute the time-dependent electromagnetic field produced by the charge and current densities in nanojunctions out of equilibrium. We then apply this formalism to a benzene ring junction, and show that geometry-dependent quantum interference effects can be used to control the magnetic field in the vicinity of the molecule. Then, treating the molecular junction as a quantum emitter, we demonstrate clear signatures of the local molecular geometry in the non-local radiated power.

cond-mat.mes-hall

Partition-free theory of time-dependent current correlations in nanojunctions in response to an arbitrary time-dependent bias

Working within the Nonequilibrium Green's Function (NEGF) formalism, a formula for the two-time current correlation function is derived for the case of transport through a nanojunction in response to an arbitrary time-dependent bias. The one-particle Hamiltonian and the Wide Band Limit Approximation (WBLA) are assumed, enabling us to extract all necessary Green's functions and self energies for the system, extending the analytic work presented previously [Ridley et al. Phys. Rev. B (2015)]. We show that our new expression for the two-time correlation function generalises the Büttiker theory of shot and thermal noise on the current through a nanojunction to the time-dependent bias case including the transient regime following the switch-on. Transient terms in the correlation function arise from an initial state that does not assume (as is usually done) that the system is initially uncoupled, i.e. our approach is partition-free. We show that when the bias loses its time-dependence, the long time-limit of the current correlation function depends on the time difference only, as in this case an ideal steady state is reached. This enables derivation of known results for the single frequency power spectrum and for the zero frequency limit of this power spectrum. In addition, we present a technique which for the first time facilitates fast calculations of the transient quantum noise, valid for arbitrary temperature, time and voltage scales. We apply this to the quantum dot and molecular wire systems for both DC and AC biases, and find a novel signature of the traversal time for electrons crossing the wire in the time-dependent cross-lead current correlations.

cond-mat.mes-hall

Calculation of the current response in a nanojunction for an arbitrary time-dependent bias: application to the molecular wire

Recently [Phys. Rev. B 91, 125433 (2015)] we derived a general formula for the time-dependent quantum electron current through a molecular junction subject to an arbitrary time-dependent bias within the Wide Band Limit Approximation (WBLA) and assuming a single particle Hamiltonian. Here we present an efficient numerical scheme for calculating the current and particle number. Using the Padé expansion of the Fermi function, it is shown that all frequency integrals occurring in the general formula for the current can be removed analytically. Furthermore, when the bias in the reservoirs is assumed to be sinusoidal it is possible to manipulate the general formula into a form containing only summations over special functions. To illustrate the method, we consider electron transport through a one-dimensional molecular wire coupled to two leads subject to out-of-phase biases. We also investigate finite size effects in the current response and particle number that results from the switch-on of such a bias.

cond-mat.mes-hall

Current through a multi-lead junction caused by applied bias with arbitrary time-dependence

We apply the Nonequilibrium Green's Function (NEGF) formalism to the problem of a multi-terminal nanojunction subject to an arbitrary time-dependent bias. In particular, we show that taking a generic one-particle system Hamiltonian within the wide band limit approximation (WBLA), it is possible to obtain a closed analytical expression for the current in each lead. Our formula reduces to the well-known result of Jauho et. al. [doi:10.1103/PhysRevB.50.5528] in the limit where the switch-on time is taken to the remote past, and to the result of Tuovinen et. al. [doi:10.1088/1742-6596/427/1/012014] when the bias is maintained at a constant value after the switch-on. As we use a partition-free approach, our formula contains both the long-time current and transient effects due to the sudden switch-on of the bias. Numerical calculations performed for the simple case of a single-level quantum dot coupled to two leads are performed for a sinusoidally-varying bias. At certain frequencies of the driving bias, we observe `ringing' oscillations of the current, whose dependence on the dot level, level width, oscillation amplitude and temperature is also investigated.

cond-mat.mes-hall

Pauli's Principle in Probe Microscopy

Exceptionally clear images of intramolecular structure can be attained in dynamic force microscopy through the combination of a passivated tip apex and operation in what has become known as the "Pauli exclusion regime" of the tip-sample interaction. We discuss, from an experimentalist's perspective, a number of aspects of the exclusion principle which underpin this ability to achieve submolecular resolution. Our particular focus is on the origins, history, and interpretation of Pauli's principle in the context of interatomic and intermolecular interactions.

cond-mat.mes-hall

Non-equilibrium statistical mechanics of classical nuclei interacting with the quantum electron gas

Kinetic equations governing time evolution of positions and momenta of atoms in extended systems are derived using quantum-classical ensembles within the Non-Equilibrium Statistical Operator Method (NESOM). Ions are treated classically, while their electrons quantum mechanically; however, the statistical operator is not factorised in any way and no simplifying assumptions are made concerning the electronic subsystem. Using this method, we derive kinetic equations of motion for the classical degrees of freedom (atoms) which account fully for the interaction and energy exchange with the quantum variables (electrons). Our equations, alongside the usual Newtonian-like terms normally associated with the Ehrenfest dynamics, contain additional terms, proportional to the atoms velocities, which can be associated with the electronic friction. Possible ways of calculating the friction forces which are shown to be given via complicated non-equilibrium correlation functions, are discussed. In particular, we demonstrate that the correlation functions are directly related to the thermodynamic Matsubara Green's functions, and this relationship allows for the diagrammatic methods to be used in treating electron-electron interaction perturbatively when calculating the correlation functions. This work also generalises previous attempts, mostly based on model systems, of introducing the electronic friction into Molecular Dynamics equations of atoms.

cond-mat.stat-mech

Arrow diagram method based on overlapping electronic groups: corrections to the linked AD theorem

Arrow diagram (AD) method (L. Kantorovich and B. Zapol, J. Chem. Phys. \textbf{96}, 8420 (1992); \emph{ibid}, 8427) provides a convenient means of systematic calculation of arbitrary matrix elements, $<Ψ|\hat{O}|Ψ>$, of symmetrical operators, $\hat{O}$, in quantum chemistry when the total system wavefunction $Ψ$ is represented as an antisymmetrised product of overlapping many-electron group functions, $Φ_{A}$, corresponding to each part (group) $A$ of the system: $Ψ=\hat{A}\prod_{A}Φ_{A}$. For extended (e.g. infinite) systems the calculation is somewhat difficult, however, as mean values of the operators require that each term of the diagram expansion is to be divided by the normalisation integral $S=<Ψ|Ψ>$, which is given by an AD expansion as well. A linked AD theorem suggested previously (L. Kantorovich, Int. J. Quant. Chem. \textbf{76}, 511 (2000)) to deal with this problem is reexamined in this paper using a simple Hartree-Fock problem of a one-dimensional ring of infinite size which is found to be analytically solvable. We find that corrections to the linked AD theorem are necessary in a general case of a finite overlap between different electronic groups. A general method of constructing these corrections in a form of a power series expansion with respect to overlap is suggested. It is illustrated on the ring model system.

physics.chem-ph

Strongly localised molecular orbitals for $α$-quartz

A previously proposed computational procedure for constructing a set of nonorthogonal strongly localised one-electron molecular orbitals (O. Danyliv, L. Kantorovich - physics/0401107) is applied to a perfect $α$-quartz crystal characterised by an intermediate type of chemical bonding. The orbitals are constructed by applying various localisation methods to canonical Hartree-Fock orbitals calculated for a succession of finite molecular clusters of increased size with appropriate boundary conditions. The calculated orbitals span the same occupied Fock space as the canonical HF solutions, but have an advantage of reflecting the true chemical nature of the bonding in the system. The applicability of several localisation techniques as well as of a number of possible choices of localisation regions (structure elements) are discussed for this system in detail.

physics.chem-ph

Calculation of electron density of periodic systems using non-orthogonal localised orbitals

Methods for calculating an electron density of a periodic crystal constructed using non-orthogonal localised orbitals are discussed. We demonstrate that an existing method based on the matrix expansion of the inverse of the overlap matrix into a power series can only be used when the orbitals are highly localised (e.g. ionic systems). In other cases including covalent crystals or those with an intermediate type of chemical bonding this method may be either numerically inefficient or fail altogether. Instead, we suggest an exact and numerically efficient method which can be used for orbitals of practically arbitrary localisation. Theory is illustrated by numerical calculations on a model system.

physics.comp-ph

Comparison of localization procedures for applications in crystal embedding

With the aim of future applications in quantum mechanical embedding in extended systems such as crystals, we suggest a simple and computationally efficient method which enables construction of a set of nonorthogonal highly localized one-electron orbitals for periodic nonmetallic crystals which reflect their chemical nature. The orbitals are also used to build up the Hartree-Fock (HF) electron density of the entire crystals. The simplicity of the method stems from the fact that it does not require usage and/or modification of periodic electronic structure codes, and is instead based on the HF calculation of a sequence of finite clusters with subsequent application of a localization procedure to transform the HF canonical molecular orbitals. Two extreme cases of chemical bonding, ionic (MgO crystal) and covalent (Si crystal), are considered for which a number of known localization schemes are applied and compared. With some modifications our method can also be applied to nonperiodic nonmetallic systems as well.

physics.comp-ph