Quantum Mechanical Treatment Research Articles

Electric transition dipole moment in pre-Born–Oppenheimer molecular structure theory

This paper presents the calculation of the electric transition dipole moment in a pre-Born-Oppenheimer framework. Electrons and nuclei are treated equally in terms of the parametrization of the non-relativistic total wave function, which is written as a linear combination of basis functions constructed from explicitly correlated Gaussian functions and the global vector representation. The integrals of the electric transition dipole moment are derived corresponding to these basis functions in both the length and the velocity representation. The calculations are performed in laboratory-fixed Cartesian coordinates without relying on coordinates which separate the center of mass from the translationally invariant degrees of freedom. The effect of the overall motion is eliminated through translationally invariant integral expressions. The electric transition dipole moment is calculated between two rovibronic levels of the H2 molecule assignable to the lowest rovibrational states of the X (1)Σ(g)(+) and B (1)Σ(u)(+) electronic states in the clamped-nuclei framework. This is the first evaluation of this quantity in a full quantum mechanical treatment without relying on the Born-Oppenheimer approximation.

Quantum mechanical treatment of variable molecular composition: from 'alchemical' changes of state functions to rational compound design.

'Alchemical' interpolation paths, i.e. coupling systems along fictitious paths without realistic correspondence, are frequently used within materials and molecular modeling simulation protocols for the estimation of changes in state functions such as free energies. We discuss alchemical changes in the context of quantum chemistry, and present illustrative numerical results for the changes of HOMO eigenvalue of the He atom due to alchemical teleportation - the simultaneous annihilation and creation of nuclear charges at different locations. To demonstrate the predictive power of alchemical first order derivatives (Hellmann-Feynman) the covalent bond potential of hydrogen fluoride and hydrogen chloride is investigated, as well as the hydrogen bond in the water-water and water-hydrogen fluoride dimer, respectively. Based on converged electron densities for one configuration, the versatility of alchemical derivatives is exemplified for the screening of entire binding potentials with reasonable accuracy. Finally, we discuss new constraints for the identification of non-linear coupling potentials for which the energy's Hellmann-Feynman derivative will yield accurate predictions.

Reduced-density-matrix description for pump-probe optical phenomena in moving atomic systems

Linear and nonlinear (especially coherent) electromagnetic interactions of moving many-electron atoms are investigated using a reduced-density-matrix description, which is applied to electromagnetically induced transparency and related resonant pump-probe optical phenomena. External magnetic fields are included on an equal footing with the electromagnetic fields and spin-Zeeman interactions are taken into account. Complimentary time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations of the reduced-density-matrix description are self-consistently developed. The general nonperturbative and non-Markovian formulations provide a fundamental framework for systematic evaluations of corrections to the standard Born (lowest-order-perturbation) and Markov (short-memory-time) approximations. The macroscopic electromagnetic response is described semiclassically, employing a perturbation expansion of the reduced-density operator in powers of the classical electromagnetic field. Our primary results are compact Liouville-space operator expressions for the linear and general ($n\text{th}$-order) nonlinear macroscopic electromagnetic-response tensors, which can be evaluated for nonlocal and nonstationary optical media described by multilevel atomic-system representations. Interactions among atoms and with environmental photons are treated as line-broadening effects by means of a general Liouville-space self-energy operator, for which the tetradic-matrix elements are explicitly evaluated in the diagonal, lowest-order, and Markov approximations. The compact Liouville-space operator expressions that are derived for the macroscopic electromagnetic-response tensors are introduced into the dynamical description of the electromagnetic-field propagation. It is pointed out that a quantized-electromagnetic-field approach will be required for a fully self-consistent quantum-mechanical treatment of local-field effects and radiative corrections.

Ontological aspects of the Casimir Effect

The role of the vacuum, in the Casimir Effect, is a matter of some dispute: the Casimir force has been variously described as a phenomenon resulting “from the alteration, by the boundaries, of the zero-point electromagnetic energy” (Bordag, Mohideen, & Mostepanenko, 2001), or a “van der Waals force between the metal plates” that can be “computed without reference to zero point energies” (Jaffe, 2005). Neither of these descriptions is grounded in a consistently quantum mechanical treatment of matter interacting with the electromagnetic field. However, the Casimir Effect has been canonically described within the framework of macroscopic quantum electrodynamics (Philbin, 2010). On this general account, the force is seen to arise due to the coupling of fluctuating currents to the zero-point radiation, and it is in this restricted sense that the phenomenon requires the existence of zero-point fields. The conflicting descriptions of the Casimir Effect, on the other hand, appear to arise from ontologies in which an unwarranted metaphysical priority is assigned either to the matter or the fields, and this may have a direct bearing on the problem of the cosmological constant.

Transition path sampling with quantum/classical mechanics for reaction rates.

Predicting rates of biochemical reactions through molecular simulations poses a particular challenge for two reasons. First, the process involves bond formation and/or cleavage and thus requires a quantum mechanical (QM) treatment of the reaction center, which can be combined with a more efficient molecular mechanical (MM) description for the remainder of the system, resulting in a QM/MM approach. Second, reaction time scales are typically many orders of magnitude larger than the (sub-)nanosecond scale accessible by QM/MM simulations. Transition path sampling (TPS) allows to efficiently sample the space of dynamic trajectories from the reactant to the product state without an additional biasing potential. We outline here the application of TPS and QM/MM to calculate rates for biochemical reactions, by means of a simple toy system. In a step-by-step protocol, we specifically refer to our implementation within the MD suite Gromacs, which we have made available to the research community, and include practical advice on the choice of parameters.

Predicting the borderline between the classical and quantum regimes in 4He gas from the second virial coefficient

The second Virial coefficient in both classical and quantum regimes of 4He gas is investigated in the temperature range 4.2–10 K. Full quantum mechanical and classical treatments are undertaken to calculate this coefficient. The main input in computing the quantum coefficient is the “effective” phase shifts. These are determined within the framework of the Galitskii–Migdal–Feynman formalism, using two interatomic potentials. The borderline between the classical and quantum regimes is found to depend on the temperature, the number density, and the interparticle potential.

Quantum Mechanical Treatment of Optical Amplifiers Based on Population Inversion

Optical amplification in a population-inverted medium is quantum-mechanically described, using the Heisenberg picture. Unlike conventional approaches that use a photon rate equation or a phenomenological field equation, the time evolution of the field operator, the photon number operator, and their squares are directly derived from first principles. The results give a logical theoretical base for the traditional phenomenological expressions describing quantum-mechanical characteristics of optical amplifiers.

Recent advances toward a general purpose linear-scaling quantum force field.

ConspectusThere is need in the molecular simulation community to develop new quantum mechanical (QM) methods that can be routinely applied to the simulation of large molecular systems in complex, heterogeneous condensed phase environments. Although conventional methods, such as the hybrid quantum mechanical/molecular mechanical (QM/MM) method, are adequate for many problems, there remain other applications that demand a fully quantum mechanical approach. QM methods are generally required in applications that involve changes in electronic structure, such as when chemical bond formation or cleavage occurs, when molecules respond to one another through polarization or charge transfer, or when matter interacts with electromagnetic fields. A full QM treatment, rather than QM/MM, is necessary when these features present themselves over a wide spatial range that, in some cases, may span the entire system. Specific examples include the study of catalytic events that involve delocalized changes in chemical bonds, charge transfer, or extensive polarization of the macromolecular environment; drug discovery applications, where the wide range of nonstandard residues and protonation states are challenging to model with purely empirical MM force fields; and the interpretation of spectroscopic observables. Unfortunately, the enormous computational cost of conventional QM methods limit their practical application to small systems. Linear-scaling electronic structure methods (LSQMs) make possible the calculation of large systems but are still too computationally intensive to be applied with the degree of configurational sampling often required to make meaningful comparison with experiment.In this work, we present advances in the development of a quantum mechanical force field (QMFF) suitable for application to biological macromolecules and condensed phase simulations. QMFFs leverage the benefits provided by the LSQM and QM/MM approaches to produce a fully QM method that is able to simultaneously achieve very high accuracy and efficiency. The efficiency of the QMFF is made possible by partitioning the system into fragments and self-consistently solving for the fragment-localized molecular orbitals in the presence of the other fragment’s electron densities. Unlike a LSQM, the QMFF introduces empirical parameters that are tuned to obtain very accurate intermolecular forces. The speed and accuracy of our QMFF is demonstrated through a series of examples ranging from small molecule clusters to condensed phase simulation, and applications to drug docking and protein–protein interactions. In these examples, comparisons are made to conventional molecular mechanical models, semiempirical methods, ab initio Hamiltonians, and a hybrid QM/MM method. The comparisons demonstrate the superior accuracy of our QMFF relative to the other models; nonetheless, we stress that the overarching role of QMFFs is not to supplant these established computational methods for problems where their use is appropriate. The role of QMFFs within the toolbox of multiscale modeling methods is to extend the range of applications to include problems that demand a fully quantum mechanical treatment of a large system with extensive configurational sampling.

Raman spectrum of NaAlSi2O6 jadeite. A quantum mechanical simulation

The Raman spectrum of NaAlSi2O6 jadeite is simulated and compared with two recent experimental data sets. In one experiment, only 17 (out of 30 symmetry allowed) peaks and a qualitative estimate of the intensities are provided. In the second case, the digitalized spectrum is available, from which we have been able to extract 20 evident peaks and an estimate of the relative intensities. The present calculation is based on an ab initio quantum mechanical treatment. Using an all‐electron Gaussian‐type basis set, together with the hybrid B3LYP density functional, the full set of 30 active modes and their (polycrystalline and polarized) intensities are obtained. The simulated intensities (not available in a previous study of the same system) permit the two experimental spectra to be reconciled and explain why the missing peaks were not seen. This ultimately leads to excellent agreement between experiment and theory. By artificially varying the mass of the Na + and Al3 + cations in the simulations, which can be performed automatically and at essentially no computational cost, the vibrational modes to which these ions contribute are identified. We conclude that quantum mechanical simulation can be a very useful complementary tool for the interpretation of experimental Raman spectra. Copyright © 2014 John Wiley & Sons, Ltd.

Non-Markovian behavior of ultrafast coherent ionization dynamics in a crystal exposed to a seeded free-electron-laser pulse

We investigate the ionization dynamics of a crystal structure driven by ultrafast coherent x-ray pulses of moderate to high intensities for excitations where dipole-allowed, single-photon ionization dominates. Using a simple model of the crystal, we demonstrate that quantum coherences may already play an important role at moderate pulse intensities, leading to qualitatively novel features which cannot be described by rate equations. In particular, the ionization may exhibit a minimum as a function of the pulse duration, where the ionization drops to almost zero, although during the pulse a noticeable fraction of the electrons is promoted to unbound states. For higher intensities, the qualitative deviations between the coherent quantum-mechanical treatment and the rate description is even more pronounced. In particular, due to the presence of quantum-mechanical coherences, the full theory predicts, even for the single-photon transitions to a continuum of free-electron states, a Rabi-type behavior similar to what is known for two-level systems.

Non-equilibrium quantum dynamics with collisional correlations

We present a fully quantum mechanical treatment of collisional correlations beyond mean-field dynamics. Correlations are handled by an ensemble of mean-field states emerging from a stochastic propagation scheme. This is done by first propagating two-body correlations in time-dependent perturbation theory for a certain time span, reducing the coherently correlated state into an incoherent sum of two-particle-two-hole states, and then sampling this state into an ensemble of mean-field states according to the jump probability obtained from perturbation theory. The scheme is applied to a simple 1D test case calibrated to typical scales of molecules and clusters. Even for the small system size and low dimension, the scheme produces robust results. Occupation numbers and entropy show steady relaxation towards thermal equilibrium.

Some Puzzles and Unresolved Issues About Quantum Entanglement

Schrodinger (Proc Camb Philos Soc 31:555–563, 1935) averred that entanglement is the characteristic trait of quantum mechanics. The first part of this paper is simultaneously an exploration of Schrodinger’s claim and an investigation into the distinction between mere entanglement and genuine quantum entanglement. The typical discussion of these matters in the philosophical literature neglects the structure of the algebra of observables, implicitly assuming a tensor product structure of the simple Type I factor algebras used in ordinary Quantum Mechanics (QM). This limitation is overcome by adopting the algebraic approach to quantum physics, which allows a uniform treatment of ordinary QM, relativistic quantum field theory, and quantum statistical mechanics. The algebraic apparatus helps to distinguish several different criteria of quantum entanglement and to prove results about the relation of quantum entanglement to two additional ways of characterizing the classical versus quantum divide, viz. abelian versus non-abelian algebras of observables, and the ability versus inability to interrogate the system without disturbing it. Schrodinger’s claim is reassessed in the light of this discussion. The second part of the paper deals with the relativity-to-ambiguity threat: the entanglement of a state on a system algebra is entanglement of the state relative to a decomposition of the system algebra into subsystem algebras; a state may be entangled with respect to one decomposition but not another; hence, unless there is some principled way to choose a decomposition, entanglement is a radically ambiguous notion. The problem is illustrated in terms a Realist versus Pragmatist debate, the former claiming that the decomposition must correspond to real as opposed to virtual subsystems, while the latter claims that the real versus virtual distinction is bogus and that practical considerations can steer the choice of decomposition. This debate is applied to the fraught problem of measuring entanglement for indistinguishable particles. The paper ends with some (intentionally inflammatory) remarks about claims in the philosophical literature that entanglement undermines the separability or independence of subsystems while promoting holism.

Spin Polarization Oscillations without Spin Precession: Spin-Orbit Entangled Resonances in Quasi-One-Dimensional Spin Transport

``Ballistic spin resonance'' refers to the counterintuitive phenomenon in which spin-polarized electrons moving down a conducting wire lose their collective spin polarization at special values of an applied magnetic field. A quantum-mechanical treatment shows that electron spin-orbit coupling and quantum confinement are at the root of the phenomenon.

Time-dependent pseudo Jahn-Teller effect: Phonon-mediated long-time nonadiabatic relaxation

Our system under theoretical consideration is an impurity center in a solid. We are considering the time evolution of the center in a quasi-degenerate electronic state. Strict quantum mechanical treatment of non-adiabadicity of the state is used. The phonon continuum is taken into account in addition to the vibration responsible for the main vibronic interaction. To describe the dynamics of the excited state a master equation has been used. The theoretical considerations are illustrated by the calculations of the long-time evolution of vibrations of the center, influenced by the emission of phonons to the bulk.

The Angular Momenta Dipole Moments and Gyromagnetic Ratios of the Electron and the Proton

We had previously obtained analytical formulae for the dipole moments and angular momenta of rotating spherical bodies. The resulting formulae were applied to the Sun, the star 78 Virginis and the Earth. The agreement of the theoretical formulae with the actual real situations is indeed remarkable. In this note we apply the same formulae to the electron and the proton, using the classical values of the radii, so no quantum mechanical treatment is considered.

New Insights on Interactions of a Quantum Vibration with an Environment of Hydrogen-Bonded Groups from the Mixed Quantum-Classical Liouville Approach

How the energy released during ATP hydrolysis can perform work in proteins despite being weak and short-lived has been a long-standing open question in biology. To address this crisis in bioenergetics, A.S. Davydov proposed a quantum mechanical model by which amide I vibrational excitations couple to alpha-helical phonon modes in such a way as to facilitate energy storage and propagation. Because of the computational expense associated with fully quantum mechanical treatments of systems with many degrees of freedom, practical simulations of the Davydov-Scott model may be realized via a mixed quantum-classical approach. We take a novel mixed quantum-classical approach to this problem by implementing two methods for solving the mixed quantum-classical Liouville equation, a surface-hopping solution and an approximate, mean-field-like solution in which the quantum subsystem is described in terms of continuous variables. The time-scale of vibrational delocalization is investigated for a one-dimensional model of a protein alpha helix. Results from the surface-hopping model suggest that population transfer between distal sites may occur in the case of an asymmetric potential, although no transfer occurs if the vibrational excitation is treated by a mean-field description.

Improving frequency selection of driven pulses using derivative-based transition suppression

Many techniques in quantum control rely on frequency separation as a means for suppressing unwanted couplings. In its simplest form, the mechanism relies on the low bandwidth of control pulses of long duration. Here we perform a higher-order quantum-mechanical treatment that allows for higher precision and shorter times. In particular, we identify three kinds of off-resonant effects: i) simultaneous unwanted driven couplings (e.g. due to drive crosstalk), ii) additional (initially undriven) transitions such as those in an infinite ladder system, and iii) sideband frequencies of the driving waveform such as we find in corrections to the rotating wave approximation. With a framework that is applicable to all three cases, in addition to the known adiabatic error responsible for a shift of the energy levels we typically see in the spectroscopy of such systems, we derive error terms in a controlled expansion corresponding to higher order adiabatic effects and diabatic excitations. We show, by also expanding the driving waveform in a basis of different order derivatives of a trial function (typically a Gaussian) these different error terms can be corrected for in a systematic way hence strongly improving quantum control of systems with dense spectra.

Electronically coarse-grained molecular dynamics using quantum Drude oscillators

Standard molecular dynamics (MD) simulations generally make use of a basic description of intermolecular forces which consists of fixed, pairwise, atom-centred Coulomb, van der Waals and short-range repulsive terms. Important interactions such as many-body polarisation and many-body dispersion which are sensitive to changes in the environment are usually neglected, and their effects treated effectively within mean-field approximations to reproduce a single thermodynamic state point or physical environment. This leads to difficulties in modelling the complex interfaces of interest today where the behaviour may be quite different from the regime of parameterisation. Here, we describe the construction and properties of a Gaussian coarse-grained electronic structure, which naturally generates many-body polarisation and dispersion interactions. The electronic structure arises from a fully quantum mechanical treatment of a set of distributed quantum Drude oscillators (QDOs), harmonic atoms which interact with each other and other moieties via electrostatic (Coulomb) interactions; this coarse-grained approach is capable of describing many-body polarisation and dispersion but not short-range interactions which must be parametrised. We describe how on-the-fly forces due to this exchange-free Gaussian model may be generated with linear scale in the number of atoms in the system using an adiabatic path integral molecular dynamics for quantum Drude oscillators technique (APIMD-QDO). We demonstrate the applicability of the QDO approach to realistic systems via a study of the liquid–vapour interface of water.

Quantum versus classical foundation of statistical mechanics under experimentally realistic conditions

Focusing on isolated macroscopic systems, described in terms of either a quantum mechanical or a classical model, our two key questions are how far does an initial ensemble (usually far from equilibrium and largely unknown in detail) evolve towards a stationary long-time behavior (equilibration) and how far is this steady state in agreement with the microcanonical ensemble as predicted by statistical mechanics (thermalization). A recently developed quantum mechanical treatment of the problem is briefly summarized, putting particular emphasis on the realistic modeling of experimental measurements and nonequilibrium initial conditions. Within this framework, equilibration can be proven under very weak assumptions about those measurements and initial conditions, while thermalization still requires quite strong additional hypotheses. An analogous approach within the framework of classical mechanics is developed and compared with the quantum case. In particular, the assumptions to guarantee classical equilibration are now rather strong, while thermalization then follows under relatively weak additional conditions.

Formal Analogy between the Dirac Equation in Its Majorana Form and the Discrete-Velocity Version of the Boltzmann Kinetic Equation

We point out a formal analogy between the Dirac equation in Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. By a systematic analysis based on the theory of operator splitting, this analogy is shown to turn into a concrete and efficient computational method, providing a unified treatment of relativistic and nonrelativistic quantum mechanics. This might have potentially far-reaching implications for both classical and quantum computing, because it shows that, by splitting time along the three spatial directions, quantum information (Dirac-Majorana wave function) propagates in space-time as a classical statistical process (Boltzmann distribution).