Effects of charge-dependent vibrational frequencies and anharmonicities in transport through molecules

As a step towards a more realistic modeling of vibrations in single-molecule devices, we investigate the effects of charge-dependent vibrational frequencies and anharmonic potentials on electronic transport. For weak phonon relaxation, we find that in both cases vibrational steps split into a multitude of substeps. This effectively leads to a bias-dependent broadening of vibrational features in current-voltage and conductance characteristics, which provides a fingerprint of nonequilibrium vibrations whenever other broadening mechanisms are secondary. In the case of an asymmetric molecule-lead coupling, we observe that frequency differences can also cause negative differential conductance.Comment: 4+ pages, 3 figures; accepted for publication in Phys. Rev. B. (Brief Reports

Nonlinear First-Order Partial Differential Equations by the Characteristic Method applied to The Hamilton-Jacobi Equation

This master's thesis has two main goals. First, to give a rigorous presentation of the method of characteristics without losing the important intuitive aspects that accompany it. Second, to use the method of characteristics to present the Hamilton-Jacobi equation, solve it in a reasonable manner and then introduce some of its most important properties. These topics will be accompanied by enlightening examples. In the first part we develop the method of characteristics by transforming a first-order nonlinear PDE to a system of first-order ODE:s. We start by defining the notion of complete integrals as a sort of singleton solution to our PDE, and then combining these to a single family; an envelope encompassing the full solution. We then assume that the PDE itself can be written in a moving coordinate frame, and using this concept make an important assumption regarding the nature of the underlying curves at each moment. The full solution is finally achieved by weaving these curves to form the solution surface. To complete the theory, proper care needs to be taken of the boundary to make it compatible with our notion of curves. Lastly, all of the theory will be combined to make sure that the method actually produces well-defined local solutions. With the method of characteristics developed, we have a look at the Hamilton-Jacobi equation. We will give sufficient conditions on when this initial-value problem can be solved with the Hopf-Lax formula. Based on this formula, a notion of weak solution will be given with its uniqueness proof. The Hamilton-Jacobi initial-value problem will be approached with the tools of variational calculus and convex analysis. These tools will be used to intimately link the Hamiltonian and Lagrangian by the means of the Legendre transform. The Hopf-Lax formula will then be constructed with the aim of solving the Hamilton-Jacobi initial-value problem. The formula is shown to have a useful functional identity as well as being Lipschitz continuous. Finally the uniqueness of the solution will be achieved by assuming semiconcavity from the initial function, or uniform convexity from the Hamiltonian. The final chapter gives an insight as to how the developed theory can be further generalized and used. We will refer to some bibliography containing an abundance of further reading on semiconcave functions, optimal control theory and the Hamilton-Jacobi-Bellman equations

Carleman estimates and absence of embedded eigenvalues

Let L be a Schroedinger operator with potential W in L^{(n+1)/2}. We prove that there is no embedded eigenvalue. The main tool is an Lp Carleman type estimate, which builds on delicate dispersive estimates established in a previous paper. The arguments extend to variable coefficient operators with long range potentials and with gradient potentials.Comment: 26 page

Coarse-grained interaction potentials for polyaromatic hydrocarbons

Using Kohn-Sham density functional theory (KS-DFT), we have studied the interaction between various polyaromatic hydrocarbon molecules. The systems range from mono-cyclic benzene up to hexabenzocoronene (hbc). For several conventional exchange-correlation functionals potential energy curves of interaction of the $\pi$-$\pi$ stacking hbc dimer are reported. It is found that all pure local density or generalized gradient approximated functionals yield qualitatively incorrect predictions regarding structure and interaction. Inclusion of a non-local, atom-centered correction to the KS-Hamiltonian enables quantitative predictions. The computed potential energy surfaces of interaction yield parameters for a coarse-grained potential, which can be employed to study discotic liquid-crystalline mesophases of derived polyaromatic macromolecules

Understanding molecular representations in machine learning: The role of uniqueness and target similarity

The predictive accuracy of Machine Learning (ML) models of molecular properties depends on the choice of the molecular representation. Based on the postulates of quantum mechanics, we introduce a hierarchy of representations which meet uniqueness and target similarity criteria. To systematically control target similarity, we rely on interatomic many body expansions, as implemented in universal force-fields, including Bonding, Angular, and higher order terms (BA). Addition of higher order contributions systematically increases similarity to the true potential energy and predictive accuracy of the resulting ML models. We report numerical evidence for the performance of BAML models trained on molecular properties pre-calculated at electron-correlated and density functional theory level of theory for thousands of small organic molecules. Properties studied include enthalpies and free energies of atomization, heatcapacity, zero-point vibrational energies, dipole-moment, polarizability, HOMO/LUMO energies and gap, ionization potential, electron affinity, and electronic excitations. After training, BAML predicts energies or electronic properties of out-of-sample molecules with unprecedented accuracy and speed

Pair tunneling through single molecules

By a polaronic energy shift, the effective charging energy of molecules can become negative, favoring ground states with even numbers of electrons. Here, we show that charge transport through such molecules near ground-state degeneracies is dominated by tunneling of electron pairs which coexists with (featureless) single-electron cotunneling. Due to the restricted phase space for pair tunneling, the current-voltage characteristics exhibits striking differences from the conventional Coulomb blockade. In asymmetric junctions, pair tunneling can be used for gate-controlled current rectification and switching.Comment: 4+ pages, 4 figures; minor changes, version published in Phys. Rev. Let

Vibrational absorption sidebands in the Coulomb blockade regime of single-molecule transistors

Current-driven vibrational non-equilibrium induces vibrational sidebands in single-molecule transistors which arise from tunneling processes accompanied by absorption of vibrational quanta. Unlike conventional sidebands, these absorption sidebands occur in a regime where the current is nominally Coulomb blockaded. Here, we develop a detailed and analytical theory of absorption sidebands, including current-voltage characteristics as well as shot noise. We discuss the relation of our predictions to recent experiments.Comment: 7 pages, 6 figures; revised discussion of relation to experimen

Toward transferable interatomic van der Waals interactions without electrons: The role of multipole electrostatics and many-body dispersion

We estimate polarizabilities of atoms in molecules without electron density, using a Voronoi tesselation approach instead of conventional density partitioning schemes. The resulting atomic dispersion coefficients are calculated, as well as many-body dispersion effects on intermolecular potential energies. We also estimate contributions from multipole electrostatics and compare them to dispersion. We assess the performance of the resulting intermolecular interaction model from dispersion and electrostatics for more than 1,300 neutral and charged, small organic molecular dimers. Applications to water clusters, the benzene crystal, the anti-cancer drug ellipticine---intercalated between two Watson-Crick DNA base pairs, as well as six macro-molecular host-guest complexes highlight the potential of this method and help to identify points of future improvement. The mean absolute error made by the combination of static electrostatics with many-body dispersion reduces at larger distances, while it plateaus for two-body dispersion, in conflict with the common assumption that the simple $1/R^6$ correction will yield proper dissociative tails. Overall, the method achieves an accuracy well within conventional molecular force fields while exhibiting a simple parametrization protocol.Comment: 13 pages, 8 figure

Current-induced nonequilibrium vibrations in single-molecule devices

Finite-bias electron transport through single molecules generally induces nonequilibrium molecular vibrations (phonons). By a mapping to a Fokker-Planck equation, we obtain analytical scaling forms for the nonequilibrium phonon distribution in the limit of weak electron-phonon coupling $\lambda$ within a minimal model. Remarkably, the width of the phonon distribution diverges as $\sim\lambda^{-\alpha}$ when the coupling decreases, with voltage-dependent, non-integer exponents $\alpha$. This implies a breakdown of perturbation theory in the electron-phonon coupling for fully developed nonequilibrium. We also discuss possible experimental implications of this result such as current-induced dissociation of molecules.Comment: 7 pages, 4 figures; revised and extended version published in Phys. Rev.