Uri Peskin

The solution of the time‐dependent Schrödinger equation by the (t,t’) method: Theory, computational algorithm and applications

A new powerful computational method is introduced for the solution of the time dependent Schrodinger equation with time‐dependent Hamiltonians (not necessarily time‐periodic). The method is based on the use of the Floquet‐type operator in an extended Hilbert space, which was introduced by H. Sambe [Phys. Rev. A 7, 2203 (1973)] for time periodic Hamiltonians, and was extended by J. Howland [Math Ann. 207, 315 (1974)] for general time dependent Hamiltonians. The new proposed computational algorithm avoids the need to introduce the time ordering operator when the time‐dependent Schrodinger equation is integrated. Therefore it enables one to obtain the solution of the time‐dependent Schrodinger equation by using computational techniques that were originally developed for cases where the Hamiltonian is time independent. A time‐independent expression for state‐to‐state transition probabilities is derived by using the analytical time dependence of the time evolution operator in the generalized Hilbert space. Ill...

The solution of the time dependent Schrödinger equation by the (t,t’) method: The use of global polynomial propagators for time dependent Hamiltonians

Using the (t,t’) method as introduced in Ref. [J. Chem. Phys. 99, 4590 (1993)] computational techniques which originally were developed for time independent Hamiltonians can be used for propagating an initial state for explicitly time dependent Hamiltonians. The present paper presents a time dependent integrator of the Schrodinger equation based on a Chebychev expansion, of the operator U(x,t’,t0→t), and the Fourier pseudospectral method for calculating spatial derivatives [(∂2/∂x2),(∂/∂t’)]. Illustrative numerical examples for harmonic and Morse oscillators interacting with CW and short pulsed laser fields are given.

On the relation between unimolecular reaction rates and overlapping resonances

Unimolecular decay processes are studied in the regime of overlapping resonances with the goal of elucidating how unimolecular reaction rates depend on resonances widths (the imaginary part of the Siegert eigenvalues). As illustrated analytically for one‐dimensional models and numerically for a more general random matrix version of Feshbach’s optical model, transition state theory (TST, Rice–Ramsperger–Kassel–Marcus) provides the correct average unimolecular decay rate whether the resonances are overlapping or not. For all studied cases, the explicit ‘‘universal’’ dependence of the TST average rate on the average resonance width (for a given energy, or an energy interval) is that of a saturation curve: in the regime of nonoverlapping resonances (i.e., weak coupling) the standard relation ‘‘unimolecular decay rate=resonance width /ℏ’’ holds, but as the resonance overlap increases (strong coupling) the rate saturates, becoming practically independent of the average resonance width in the strong overlapping ...

Transient resonance structures in electron tunneling through water

The mechanism of electrons tunneling through a narrow water barrier between two Pt(100) metal surfaces is studied. Assuming an adiabatic picture in which the water configuration is static on the time scale of the electron motion, the tunneling probabilities are found to increase nonmonotonically as a function of incident electron energy. A numerical investigation of single electron scattering wave functions suggests that the tunneling is enhanced by resonances, associated with molecular cavities in which the electron is trapped between repulsive oxygen cores. The lifetimes of these resonances are calculated using a novel filter diagonalization scheme, based on a converging high-order perturbative expansion of the single-electron Green’s function, and are found to be of order ⩽10 fs. The possibility that transient resonance supporting structures contribute to the enhancement of tunneling through water is discussed.

An introduction to the formulation of steady-state transport through molecular junctions

A basic theoretical introduction is given for the phenomenon of electronic transport through molecular junctions. The electrode–molecule–electrode system is represented using a model Hamiltonian framework based on separation between the molecular and the electrode single-particle subspaces, using projection operators. The Landauer formulation of the steady-state current through the junction is introduced and the transmission function is derived from basic single-particle quantum scattering theory. Detailed implementations to a generic tight-binding model demonstrate the typical characteristics of the transmission function, and resonant transport through discrete quantum molecular states is analysed in detail. An alternative formulation based on the time-dependent Liouville–von Neumann equation leads to a quantum kinetic representation of the current in terms of rate constants for electron hopping between the molecule and the electrodes. The generalization of this approach to inelastic transport is discussed.

Final state-selected spectra in unimolecular reactions: A transition-state- based random matrix model for overlapping resonances

Final state-selected spectra in unimolecular decomposition are obtained by a random matrix version of Feshbach’s optical model. The number of final states which are independently coupled to the molecular quasibound states is identified with the number of states at the dividing surface of transition state theory ~TST!. The coupling of the transition state to the molecular complex is modeled via a universal random matrix effective Hamiltonian which is characterized by its resonance eigenstates and provides the correct average unimolecular decay rate. The transition from nonoverlapping resonances which are associated with isolated Lorentzian spectral peaks, to overlapping resonances, associated with more complex spectra, is characterized in terms of deviations from a x 2 -like distribution of the resonance widths and the approach to a random phase-distribution of the resonance scattering amplitudes. The evolution of the system from a tight transition state to reaction products is treated explicitly as a scattering process where specific dynamics can be incorporated. Comparisons with recently measured final state-selected spectra and rotational distributions for the unimolecular reaction of NO 2 show that the present model provides a useful new approach for understanding and interpreting experimental results which are dominated by overlapping resonances.

Transient dynamics in molecular junctions: Coherent bichromophoric molecular electron pumps

The possibility of using single molecule junctions as electron pumps for energy conversion and storage is considered. It is argued that the small dimensions of these systems enable to make use of unique intra-molecular quantum coherences in order to pump electrons between two leads and to overcome relaxation processes which tend to suppress the pumping efficiency. In particular, we demonstrate that a selective transient excitation of one chromophore in a bi-chromophoric donor-bridge-acceptor molecular junction model yields currents which transfer charge (electron and holes) unevenly to the two leads in the absence of a bias potential. The utility of this mechanism for charge pumping in steady state conditions is proposed.

Partial widths by asymptotic analysis of the complex scaled resonance wave function

The complex scaled square‐integrable resonance wave function describing the scattering of a particle at a distance r from a target with internal state energies and wave functions denoted ej and χj (x) is given by ∑jχj(x)φj(r), where the φj(r)’s are the channel functions. The partial widths Γj (i.e., the decay rates into the channels open for dissociation) are obtained by calculating ‖φj(r)(kj/m)1/2 exp[−ikjr exp(iθ)]‖2 as r→∞, where exp(iθ) is the complex scaling factor, m is the reduced mass of the two scattered entities, and kj=[2m(Eres −ej)]1/2. Eres is the complex resonance eigenvalues of the complex scaled Hamiltonian H(x,r exp(iθ)). The wave function is determined either from a propagation plus matching technique or using a basis of particle‐in‐a‐box functions. The former procedure is applicable even in the limit of zero rotation angle. Illustrative examples are given for a two‐channel model Hamiltonian studied previously by Noro and Taylor, and by Bacic and Simons, and for a Hamiltonian which descr...

Coherently controlled molecular junctions

Within a generic model, we discuss the possibility of coherent control of charge fluxes in unbiased molecular junctions. The control is induced by resonances between the Rabi frequency due to a pumping laser field and internal characteristic frequencies of pre-designed molecular donor-bridge-acceptor complexes. Two models are considered: a coherently controlled molecular charge pump and a molecular switch. The study generalizes previous consideration of light induced current [M. Galperin and A. Nitzan, Phys. Rev. Lett. 95, 206802 (2005)] and of a molecular electron pump [R. Volkovich and U. Peskin, Phys. Rev. B 83, 033403 (2011)] and accounts for the coherently driven charge transport in an unbiased molecular junction with symmetric coupling to leads. Numerical examples demonstrate the feasibility of the control mechanism for realistic junctions parameters.

Electronic transport through molecular junctions with nonrigid molecule-leads coupling

The Landauer-type formulation of current through a molecular junction with electronic-nuclear coupling introduced by Troisi et al. [J. Chem. Phys. 118, 6072 (2003)] is generalized to account for the dependence of the molecule-leads coupling terms on the nuclear coordinates. Although this electronic-nuclear coupling is external to the molecule there is no need to extend the molecular subspace when projection operators are employed for calculations of the current through the junction. A test case of a conductor with vibrating contacts to the leads is studied numerically. It is demonstrated that contact vibrations lead to inelastic contributions to the current and to characteristic features in the I-V curve and its derivatives, similar to the ones observed for internal (molecular) electronic-nuclear coupling.