Session Z25: Theoretical Methods and Applications

Rate description of Fokker-Planck processes with time-periodic parameters

The large time dynamics of periodically driven Fokker-Planck processes possessing several metastable states is investigated. At weak noise the rare transitions between these metastable states can be represented as a discrete Markov process characterized by time dependent rates. At large times the full Fokker-Planck process is completely specified by the transition probabilities of this discrete process and by two types of functions associated to each metastable state: so-called state specific probability densities and localizing functions. The localizing functions assign to the continuous states of the original Fokker-Planck process probabilities for the metastable states. The state specific probabilities allocate a time dependent probability density of continuous states to each metastable state. We specify these functions by their equations of motion and illustrate and validate the presented theory for a periodically forced bistable Brownian oscillator in a wide range of driving frequencies. P. Talkner, J. Luczka, Phys. Rev. E, 69, 046109 (2004). C. Kim, P. Talkner, E.K. Lee, P. Hanggi, Chem. Phys. (2009), doi:10.1016/j.chemphys.2009.10.027; arXiv:0908.1730.   

\textit{Ab initio} study of anharmonic vibrations for polymers

Energies of optically active $k=0$ phonons in extended systems of one-dimensional periodicity (polyethylene) are computed by taking account of the anharmonicity in the potential energy surfaces (PES) and the resulting phonon-phonon couplings explicitly. The electronic part of the calculations is based on Gaussian-basis-set crystalline orbital theory at the coupled-cluster singles and doubles, second-order M{\o}ller--Plesset perturbation (MP2), and Hartree--Fock levels, providing one-, two-, and three-dimensional slices of the PES, respectively, which are in turn expanded in the fourth-order Taylor series of normal coordinates. For the vibrational part, we employ the vibrational self-consistent-field, vibrational MP2 and vibrational truncated configuration-interaction (VCI) methods within the $\Gamma $ approximation that amounts to including only $k=0$ phonons. It is shown that inclusion of both electron correlation and anharmonicity is essential in achieving good agreement between computed and observed frequencies of optical phonons in polyethylene. The VCI calculations also identify quantitatively the frequency separation and intensity ratio of the Fermi doublets in the vibrational spectrum of polyethylene.   

A statistical estimation of the precision of reweighting-based simulations

Reweighting a sampled configuration plays a central role in decreasing the computational cost of a simulation involving ergodic and semi-ergodic systems and is the base of all biased sampling methods. A very simple application of this method is reweighting a primary ensemble, usually generated by a computationally cheap Hamiltonian, to a target ensemble, typically a more expensive Hamiltonian. However, the precision can be strongly affected by the distribution of the Hamiltonians' difference in each bin of conformation distribution. Theoretically, one should sample forever to have a complete distribution of energy in the ensemble, but sampling in energy space is much faster than sampling in conformation space. Using this idea and taking advantage of the Gaussian nature of the energy distribution, we study a way to make a good estimate of error in reweighting histogram procedures before running a long simulation on the system. We are applying these ideas on two polypeptide chains, ala 10 and fs21, to see how they work practically.   

Applications of the Projective Dynamics method to stochastically driven systems

A dynamic system which can be interpreted as continuously evolving along one coordinate can be discretized by dividing this coordinate into non-overlapping intervals, which cover the entire domain. We further impose the (sufficient) condition that only motion between adjacent intervals are permitted. A generalization of the Projective Dynamics method [1] then ensures that correct mean first passage times to an absorbing interval can be obtained by having correct transition rates. Furthermore the theoretical framework shows that the intervals can be chosen in an arbitrarily way, while keeping the above minor condition. Thus we project the dynamic system onto a master equation with the same mean first passage time. We present applications demonstrating that this procedure is in general applicable to a wide range of problems. We illustrate the application of the Projective Dynamics method to Brownian motion of particles in one and two dimensional smooth or rough energy landscapes. We also apply the method to the folding process of small linear polymer chains (with two types of atoms) subject to Brownian motion. We compare results of the mean first passage time obtained from the Projective Dynamics method with those of direct measurements. [1]Phys. Rev. Lett. 80, 3384(1998)]   

Understanding many-electron fermionic ground states as bosonic excited states

We consider the novel question of, ``Which state in the spectrum of a system of distinguishable particles is the ground state of the same system made up of indistinguishable fermions?'' We discuss how this can be approximated for many-electron systems and illustrate its importance in our new semiclassical method correlated Thomas-Fermi (CTF).   

Displaced path integral formulation for the momentum distribution of quantum particles

The proton momentum distribution, accessible by deep inelastic neutron scattering, is a very sensitive probe of the potential of mean force experienced by the protons in hydrogen-bonded systems. In this work we introduce a novel estimator for the end-to-end distribution of the Feynman paths, i.e. the Fourier transform of the momentum distribution. In this formulation, free particle and environmental contributions factorize. Moreover, the environmental contribution has a natural analogy to a free energy surface in statistical mechanics, facilitating the interpretation of experiments. The new formulation is not only conceptually but also computationally advantageous, because the displaced path distribution can be sampled accurately with thermodynamic integration techniques. We illustrate the method with applications to one-dimensional model systems and to an empirical water model.   

A Comparative Study of Mixed Quantum-Classical and Semiclassical Methods for an Electronic Spectroscopy Benchmark Model

We performed a comparative study of the reliability of different mixed quantum-classical and semiclassical approaches to calculating equilibrium one- and two-dimensional electronic spectra. These approaches include the popular second-order Cumulant expansion approximation, the Mixed Quantum-Classical Liouville method, and the Forward-Backward Semiclassical method. The comparison was performed within the framework of a benchmark system that can distinguish between these methods and for which the exact quantum results can be obtained [1]. More recently, this work has been extended to cases where the system is initially prepared in a non-equilibrium state, in order to study electronic pump-vibrational probe type spectroscopies.   

Effective interactions between pH-responsive particles

The DLVO potential is commonly used to describe the pair interactions between charged macroions in solution. It neglects effects due to responsive surface groups, by which the charge and surface entropy can fluctuate. Here, within the framework of density functional theory, we calculate the pair potential between reactive macromolecules. We compare with results found by molecular dynamics simulations of the restricted primitive model, including a short ranged binding potential between the ions and the macroions. Thereby, we extend DLVO-theory significantly, and even find conditions for a like-charge attraction and opposite-charge repulsion.   

The relation between collapsed states of a polymer chain and compact Lennard-Jones clusters

We present computational studies of compact states of small linear polymer chains consisting of hydrophobic and polar monomers. At sufficiently low temperatures, the polymer chain undergoes a transition from a fully extended to a compact state, and the dynamics of the collapse process is studied with a particular emphasis on the distribution of the end-to-end distances of the polymer. We show that in certain parameter regimes the distribution of end-to-end distances of the polymer chain near the ground state resembles the distribution of the pairwise distances of compact Lennard-Jones (non-polymeric) clusters.   

How long does it take to measure eternity? Solving problems in hours that would take brute-force simulations the age of the universe

We present two new rare event techniques that we use to determine transition times in stochastic systems. We call the first technique forward flux sampling in time (FFST). This algorithm is similar to forward flux sampling (FFS) but doesn't suffer from the main flaw of FFS: a choice between either a slow initial flux calculation or an approximate solution for the transition time. The second algorithm, called the barrier method, is significantly more efficient than FFS and FFST, especially for transitions with long-lived meta-stable states. These algorithms can be applied to any problem that can be written as a Markov (memory-free) process. The barrier method is useful in relatively low dimensional problems. We present results comparing these new algorithms with FFS on 1D exactly solvable systems.   

Tunable Fano resonance in a ferromagnetic diatomic molecular transistor

We investigate electron transport through a diatomic molecule parallelly coupled to ferromagnetic source and drain contacts. We utilize a model Hamiltonian involving a Hubbard term in which the contacts are modeled using recently developed complex source and sink potentials. The zero bias transmission spectrum for a symmetrically coupled system as a function of the Fermi energy acquires a Fano lineshape as the Hubbard interaction is turned on. For large values of U, the Fano lineshape broadens and shifts to higher energy values disappearing eventually. Meanwhile, the Breit-Wigner resonance located at the bonding resonance in the noninteracting limit survives but its position is shifted twice the coupling between the atoms in the molecule in the infinite U limit and its linewidth is reduced to half. We attribute this behaviour to the unavailability of one of the transmission channels.