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## Full-Text Articles in Physics

The Schrödinger Equation With Friction From The Quantum Trajectory Perspective, Sophya V. Garashchuk, V. Dixit, B. Gu, J. Mazzuca

#### The Schrödinger Equation With Friction From The Quantum Trajectory Perspective, Sophya V. Garashchuk, V. Dixit, B. Gu, J. Mazzuca

*Faculty Publications*

Similarity of equations of motion for the classical and quantum trajectories is used to introduce afriction term dependent on the wavefunction phase into the time-dependent Schrödingerequation. The term describes irreversible energy loss by the quantum system. The force offriction is proportional to the velocity of a quantum trajectory. The resulting Schrödinger equationis nonlinear, conserves wavefunction normalization, and evolves an arbitrary wavefunction into the ground state of the system (of appropriate symmetry if applicable). Decrease in energy is proportional to the average kinetic energy of the quantum trajectory ensemble. Dynamics in the high friction regime is suitable for simple models of ...

Incorporation Of Quantum Effects For Selected Degrees Of Freedom Into The Trajectory-Based Dynamics Using Spatial Domains, Sophya V. Garashchuk, M. V. Volkov

#### Incorporation Of Quantum Effects For Selected Degrees Of Freedom Into The Trajectory-Based Dynamics Using Spatial Domains, Sophya V. Garashchuk, M. V. Volkov

*Faculty Publications*

The approach of defining quantum corrections on nuclear dynamics of molecular systems incorporated approximately into selected degrees of freedom, is described. The approach is based on the Madelung-de-Broglie-Bohm formulation of time-dependent quantum mechanics which represents a wavefunction in terms of an ensemble of trajectories. The trajectories follow classical laws of motion except that the quantum potential, dependent on the wavefunction amplitude and its derivatives, is added to the external, classical potential. In this framework the quantum potential, determined approximately for practical reasons, is included only into the “quantum” degrees of freedom describing light particles such as protons, while neglecting with ...

Efficient Quantum Trajectory Representation Of Wavefunctions Evolving In Imaginary Time, Sophya V. Garashchuk, J. Mazzuca, T. Vazhappilly

#### Efficient Quantum Trajectory Representation Of Wavefunctions Evolving In Imaginary Time, Sophya V. Garashchuk, J. Mazzuca, T. Vazhappilly

*Faculty Publications*

The Boltzmann evolution of a wavefunction can be recast as imaginary-time dynamics of the quantum trajectory ensemble. The quantum effects arise from the momentum-dependent quantum potential – computed approximately to be practical in high-dimensional systems – influencing the trajectories in addition to the external classical potential [S. Garashchuk, J. Chem. Phys.132, 014112 (2010)]. For a nodelesswavefunction represented as ψ(*x*, *t*) = exp ( −*S*(*x*, *t*)/ℏ) with the trajectory momenta defined by ∇*S*(*x*, *t*), analysis of the Lagrangian and Eulerian evolution shows that for bound potentials the former is more accurate while the latter is more practical because the Lagrangian ...

Modeling The Noble Metal/Tio_{2} (110) Interface With Hybrid Dft Functionals: A Periodic Electrostatic Embedded Cluster Model Study, Salai Cheettu Ammal, Andreas Heyden

#### Modeling The Noble Metal/Tio2 (110) Interface With Hybrid Dft Functionals: A Periodic Electrostatic Embedded Cluster Model Study, Salai Cheettu Ammal, Andreas Heyden

*Faculty Publications*

The interaction of Aun and Ptn (n=2,3) clusters with the stoichiometric and partially reduced rutile TiO2 (110) surfaces has been investigated using periodic slab and periodic electrostatic embedded cluster models. Compared to Au clusters, Pt clusters interact strongly with both stoichiometric and reduced TiO2 (110) surfaces and are able to enhance the reducibility of the TiO2 (110) surface, i.e., reduce the oxygen vacancy formation energy. The focus of this study is the effect of Hartree–Fock exchange on the description of the strength of chemical bonds at the interface of Au/Pt clusters and the TiO2 (110 ...

Quantum Trajectory Dynamics In Imaginary Time With The Momentum-Dependent Quantum Potential, Sophya V. Garashchuk

#### Quantum Trajectory Dynamics In Imaginary Time With The Momentum-Dependent Quantum Potential, Sophya V. Garashchuk

*Faculty Publications*

The quantum trajectory dynamics is extended to the wave function evolution in imaginary time. For a nodelesswave function a simple exponential form leads to the classical-like equations of motion of trajectories, representing the wave function, in the presence of the momentum-dependent quantum potential in addition to the external potential. For a Gaussian wave function this quantum potential is a time-dependent constant, generating zero quantum force yet contributing to the total energy. For anharmonic potentials the momentum-dependent quantum potential is cheaply estimated from the global Least-squares Fit to the trajectory momenta in the Taylor basis. Wave functions with nodes are described ...

Wavepacket Approach To The Cumulative Reaction Probability Within The Flux Operator Formalism, Sophya V. Garashchuk, T. Vazhappilly

#### Wavepacket Approach To The Cumulative Reaction Probability Within The Flux Operator Formalism, Sophya V. Garashchuk, T. Vazhappilly

*Faculty Publications*

Expressions for the singular flux operator eigenfunctions and eigenvalues are given in terms of the Diracδ-function representable as a localized Gaussian wavepacket. This functional form enables computation of the cumulative reaction probability N(E) from the wavepacket time-correlation functions. The Gaussian based form of the flux eigenfunctions, which is not tied to a finite basis of a quantum-mechanical calculation, is particularly useful for approximate calculation of N(E) with the trajectory based wavepacket propagation techniques. Numerical illustration is given for the Eckart barrier using the conventional quantum-mechanical propagation and the quantum trajectory dynamics with the approximate quantum potential. N(E ...

Stable Long-Time Semiclassical Description Of Zero-Point Energy In High-Dimensional Molecular Systems, Sophya V. Garashchuk, V. A. Rassolov

#### Stable Long-Time Semiclassical Description Of Zero-Point Energy In High-Dimensional Molecular Systems, Sophya V. Garashchuk, V. A. Rassolov

*Faculty Publications*

Semiclassical implementation of the quantum trajectory formalism [J. Chem. Phys.120, 1181 (2004)] is further developed to give a stable long-time description of zero-point energy in anharmonic systems of high dimensionality. The method is based on a numerically cheap linearized quantum force approach; stabilizing terms compensating for the linearization errors are added into the time-evolution equations for the classical and nonclassical components of the momentum operator. The wave function normalization and energy are rigorously conserved. Numerical tests are performed for model systems of up to 40 degrees of freedom.

Computation Of Correlation Functions And Wave Function Projections In The Context Of Quantum Trajectory Dynamics, Sophya V. Garashchuk

#### Computation Of Correlation Functions And Wave Function Projections In The Context Of Quantum Trajectory Dynamics, Sophya V. Garashchuk

*Faculty Publications*

The de Broglie-Bohm formulation of the Schrödinger equation implies conservation of the wave functionprobability density associated with each quantum trajectory in closed systems. This conservation property greatly simplifies numerical implementations of the quantum trajectory dynamics and increases its accuracy. The reconstruction of a wave function, however, becomes expensive or inaccurate as it requires fitting or interpolation procedures. In this paper we present a method of computing wave packet correlation functions and wave function projections, which typically contain all the desired information about dynamics, without the full knowledge of the wave function by making quadratic expansions of the wave function phase ...

Semiclassical Nonadiabatic Dynamics Based On Quantum Trajectories For The *O( ^{3}P,^{1}D)* +

*H*System, Sophya V. Garashchuk, V. A. Rassolov, G. C. Schatz

_{2}#### Semiclassical Nonadiabatic Dynamics Based On Quantum Trajectories For The O(3P,1D) + H2 System, Sophya V. Garashchuk, V. A. Rassolov, G. C. Schatz

*Faculty Publications*

The O(P3,D1)+H2→OH+Hreaction is studied using trajectory dynamics within the approximate quantum potential approach. Calculations of the wave-packet reaction probabilities are performed for four coupled electronic states for total angular momentum J=0 using a mixed coordinate/polar representation of the wave function. Semiclassical dynamics is based on a single set of trajectories evolving on an effective potential-energy surface and in the presence of the approximate quantum potential. Population functions associated with each trajectory are computed for each electronic state. The effective surface is a linear combination of the electronic states with the contributions of individual ...

Semiclassical Nonadiabatic Dynamics Using A Mixed Wave-Function Representation, Sophya V. Garashchuk, V. A. Rassolov, G. C. Schatz

#### Semiclassical Nonadiabatic Dynamics Using A Mixed Wave-Function Representation, Sophya V. Garashchuk, V. A. Rassolov, G. C. Schatz

*Faculty Publications*

Nonadiabaticeffects in quantum dynamics are described using a mixed polar/coordinate space representation of the wave function. The polar part evolves on dynamically determined potential surfaces that have diabatic and adiabatic potentials as limiting cases of weak localized and strong extended diabatic couplings. The coordinate space part, generalized to a matrix form, describes transitions between the surfaces. Choice of the effective potentials for the polar part and partitioning of the wave function enables one to represent the total wave function in terms of smooth components that can be accurately propagated semiclassically using the approximate quantum potential and small basis sets ...

Semiclassical Nonadiabatic Dynamics With Quantum Trajectories, Vitaly A. Rassolov, Sophya V. Garashchuk

#### Semiclassical Nonadiabatic Dynamics With Quantum Trajectories, Vitaly A. Rassolov, Sophya V. Garashchuk

*Faculty Publications*

Dynamics based on quantum trajectories with approximate quantum potential is generalized to nonadiabatic systems and its semiclassical properties are discussed. The formulation uses the mixed polar-coordinate space representation of a wave function. The polar part describes the overall time evolution of the wave-function components semiclassically using the single-surface approximate quantum potential. The coordinate part represents a complex“population” amplitude, which in case of localized coupling can be solved for quantum mechanically in an efficient manner. In the high-energy regime this is accomplished by using a small basis determined by the coupling between surfaces. An illustration is given for a typical ...

Energy Conserving Approximations To The Quantum Potential: Dynamics With Linearized Quantum Force, Sophya V. Garashchuk, V. A. Rassolov

#### Energy Conserving Approximations To The Quantum Potential: Dynamics With Linearized Quantum Force, Sophya V. Garashchuk, V. A. Rassolov

*Faculty Publications*

Solution of the Schrödinger equation within the de Broglie–Bohm formulation is based on propagation of trajectories in the presence of a nonlocal quantum potential. We present a new strategy for defining approximate quantum potentials within a restricted trial function by performing the optimal fit to the log-derivatives of the wave function density. This procedure results in the energy-conserving dynamics for a closed system. For one particular form of the trial function leading to the linear quantum force, the optimization problem is solved analytically in terms of the first and second moments of the weighted trajectory distribution. This approach gives ...

Modified Quantum Trajectory Dynamics Using A Mixed Wave Function Representation, Sophya V. Garashchuk, V. A. Rassolov

#### Modified Quantum Trajectory Dynamics Using A Mixed Wave Function Representation, Sophya V. Garashchuk, V. A. Rassolov

*Faculty Publications*

Dynamics of quantum trajectories provides an efficient framework for description of various quantum effects in large systems, but it is unstable near the wave function density nodes where the quantum potential becomes singular. A mixed coordinate space/polar representation of the wave function is used to circumvent this problem. The resulting modified trajectory dynamics associated with the polar representation is nonsingular and smooth. The interference structure and the nodes of the wave function density are described, in principle, exactly in the coordinate representation. The approximate version of this approach is consistent with the semiclassical linearized quantum force method [S. Garashchuk ...

Bohmian Dynamics On Subspaces Using Linearized Quantum Force, V. A. Rassolov, Sophya V. Garashchuk

#### Bohmian Dynamics On Subspaces Using Linearized Quantum Force, V. A. Rassolov, Sophya V. Garashchuk

*Faculty Publications*

In the de Broglie–Bohm formulation of quantum mechanics the time-dependent Schrödinger equation is solved in terms of quantum trajectories evolving under the influence of quantum and classical potentials. For a practical implementation that scales favorably with system size and is accurate for semiclassical systems, we use approximate quantum potentials. Recently, we have shown that optimization of the nonclassical component of the momentum operator in terms of fitting functions leads to the energy-conserving approximate quantum potential. In particular, linear fitting functions give the exact time evolution of a Gaussian wave packet in a locally quadratic potential and can describe the ...

Geminal Model Chemistry Ii. Perturbative Corrections, V. A. Rassolov, F. Xu, Sophya V. Garashchuk

#### Geminal Model Chemistry Ii. Perturbative Corrections, V. A. Rassolov, F. Xu, Sophya V. Garashchuk

*Faculty Publications*

We introduce and investigate a chemical model based on perturbative corrections to the product of singlet-type strongly orthogonal geminals wave function. Two specific points are addressed (i) Overall chemical accuracy of such a model with perturbative corrections at a leading order; (ii) Quality of strong orthogonality approximation of geminals in diverse chemical systems. We use the Epstein–Nesbet form of perturbation theory and show that its known shortcomings disappear when it is used with the reference Hamiltonian based on strongly orthogonal geminals. Application of this model to various chemical systems reveals that strongly orthogonal geminals are well suited for chemical ...

Semiclassical Dynamics With Quantum Trajectories: Formulation And Comparison With The Semiclassical Initial Value Representation Propagator, Sophya V. Garashchuk, V. A. Rassolov

#### Semiclassical Dynamics With Quantum Trajectories: Formulation And Comparison With The Semiclassical Initial Value Representation Propagator, Sophya V. Garashchuk, V. A. Rassolov

*Faculty Publications*

We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare ...

Semiclassical Application Of The Mo/Ller Operators In Reactive Scattering, Sophya V. Garashchuk, J. C. Light

#### Semiclassical Application Of The Mo/Ller Operators In Reactive Scattering, Sophya V. Garashchuk, J. C. Light

*Faculty Publications*

Mo/ller operators in the formulation of reaction probabilities in terms of wave packet correlation functions allow us to define the wave packets in the interaction region rather than in the asymptotic region of the potential surface. We combine Mo/ller operators with the semiclassical propagator of Herman and Kluk. This does not involve further approximations and can be used with any initial value representation (IVR) semiclassical propagator. Time propagation in asymptotic regions of the potential due to Mo/ller operators reduces the oscillations of the propagator integrand and improves convergence of the results with respect to the number of ...

Quasirandom Distributed Gaussian Bases For Bound Problems, Sophya V. Garashchuk, J. C. Light

#### Quasirandom Distributed Gaussian Bases For Bound Problems, Sophya V. Garashchuk, J. C. Light

*Faculty Publications*

We introduce quasirandom distributed Gaussian bases (QDGB) that are well suited for bound problems. The positions of the basis functions are chosen quasirandomly while their widths and density are functions of the potential. The basis function overlap and kinetic energy matrix elements are analytical. The potential energy matrix elements are accurately evaluated using few-point quadratures, since the Gaussian basis functions are localized. The resulting QDGB can be easily constructed and is shown to be accurate and efficient for eigenvalue calculation for several multidimensional model vibrational problems. As more demanding examples, we used a 2D QDGB-DVR basis to calculate the lowest ...

Simplified Calculation Of The Stability Matrix For Semiclassical Propagation, Sophya V. Garashchuk, J. C. Light

#### Simplified Calculation Of The Stability Matrix For Semiclassical Propagation, Sophya V. Garashchuk, J. C. Light

*Faculty Publications*

We present a simple method of calculation of the stability (monodromy) matrix that enters the widely used semiclassical propagator of Herman and Kluk and almost all other semiclassical propagators. The method is based on the unitarity of classical propagation and does not involve any approximations. The number of auxiliary differential equations per trajectory scales linearly rather than quadratically with the system size. Just the first derivatives of the potential surface are needed. The method is illustrated on the collinear H3 system.

Cumulative Reaction Probability In Terms Of Reactant-Product Wave Packet Correlation Functions, Sophya V. Garashchuk, D. J. Tannor

#### Cumulative Reaction Probability In Terms Of Reactant-Product Wave Packet Correlation Functions, Sophya V. Garashchuk, D. J. Tannor

*Faculty Publications*

We present new expressions for the cumulative reaction probability (N(E)), cast in terms of time-correlation functions of reactant and product wave packets. The derivation begins with a standard trace expression for the cumulative reaction probability, expressed in terms of the reactive scattering matrix elements in an asymptotic internal basis. By combining the property of invariance of the trace with a wave packet correlation function formulation of reactive scattering, we obtain an expression for N(E) in terms of the correlation matrices of incoming and outgoing wave packets which are *arbitrary* in the internal coordinates. This formulation, like other recent ...

Π-Systems As Lithium/Hydrogen Bond Acceptors: Some Theoretical Observations, Salai Cheettu Ammal, P. Venuvanalingam

#### Π-Systems As Lithium/Hydrogen Bond Acceptors: Some Theoretical Observations, Salai Cheettu Ammal, P. Venuvanalingam

*Faculty Publications*

*Ab initio* calculations at the Hartree–Fock and correlated levels and density functional theory calculations have been performed with 6-31++G(d,p) and 6-311++G(d,p)basis sets on LiF and HF complexes of benzene, ethylene, and acetylene. Complex binding energies have been corrected for basis set superposition error, and zero point energy corrections have been done on Hartree–Fock binding energies. Computed results indicate that the complexes exist in different conformations and among them those with π-lithium and π-hydrogen bonds are the most stable. π-lithium bonds are stronger than π-hydrogen bonds. The computed binding energies and geometry ...

Correlation Function Formulation For The State Selected Total Reaction Probability, Sophya V. Garashchuk, D. J. Tannor

#### Correlation Function Formulation For The State Selected Total Reaction Probability, Sophya V. Garashchuk, D. J. Tannor

*Faculty Publications*

A correlation function formulation for the state-selected total reaction probability, Nα(E), is suggested. A wave packet, correlating with a specific set of internal reactant quantum numbers, α, is propagated forward in time until bifurcation is complete at which time the nonreactive portion of the amplitude is discarded. The autocorrelation function of the remaining amplitude is then computed and Fourier transformed to obtain a reactivity spectrum. Dividing by the corresponding spectrum of the original, unfiltered, wave packet normalizes the reactivity spectrum, yielding the total reaction probability from the internal state, α. The procedure requires negligible storage and just one time-energy ...

Lithium Bonding Interaction In *H _{2}*Cy⋯Lif

*(Y*=O,S

*)*Complexes: A Theoretical Probe, Salai Cheettu Ammal, P. Venuvanalingam, S. Pal

#### Lithium Bonding Interaction In H2Cy⋯Lif (Y=O,S) Complexes: A Theoretical Probe, Salai Cheettu Ammal, P. Venuvanalingam, S. Pal

*Faculty Publications*

*Ab initio* calculations at 6-31++G(d,p) level have been done on H2CY⋯LiF (Y=O,S) complexes choosing ten possible orientations in each complex. The effect of correlation on complex binding energies has been studied via single point MP2 (full) calculations done on 6-31++G(d,p) geometry. Binding energies have been corrected for basis set superposition error. Frequency calculations confirm that H2CO⋯LiF and H2CS⋯LiF complexes have three and two stable forms, respectively. The most stable form in each complex has been found to have a strong lithium bonding interaction and a secondary hydrogen bondinginteraction. NBO ...

Semiclassical Approach To The Hydrogen-Exchange Reaction Reactive And Transition-State Dynamics, Sophya V. Garashchuk, F. Grossmann, D. Tannor

#### Semiclassical Approach To The Hydrogen-Exchange Reaction Reactive And Transition-State Dynamics, Sophya V. Garashchuk, F. Grossmann, D. Tannor

*Faculty Publications*

Scattering matrix elements and symmetric transition-state resonances for the collinear H _{2} + H → H + H _{2} reaction are obtained using a time-dependent approach. The correlation function between reactant channel wavepackets and product channel wavepackets is used to determine the *S*-matrix elements. In a similar fashion, autocorrelation functions are used to extract the positions and widths of transition-state resonances. The time propagation of the wavepackets is performed by the improved semiclassical frozen Gaussian method of Herman and Kluk, which is an initial value, uniformly converged method. The agreement between the quantum and semiclassical results is far better than that obtained ...