scholarly journals Deriving the exact nonadiabatic quantum propagator in the mapping variable representation

2016 ◽  
Vol 195 ◽  
pp. 269-289 ◽  
Author(s):  
Timothy J. H. Hele ◽  
Nandini Ananth
Keyword(s):  
Degrees Of Freedom ◽  
Reaction Rates ◽  
Time Correlation ◽  
Electron Transfer Reaction ◽  
Classical Dynamics ◽  
Nonadiabatic Dynamics ◽  
Path Integral Representation ◽  
Quantum Propagator ◽  
Large Systems ◽  
Variable Representation

We derive an exact quantum propagator for nonadiabatic dynamics in multi-state systems using the mapping variable representation, where classical-like Cartesian variables are used to represent both continuous nuclear degrees of freedom and discrete electronic states. The resulting Liouvillian is a Moyal series that, when suitably approximated, can allow for the use of classical dynamics to efficiently model large systems. We demonstrate that different truncations of the exact Liouvillian lead to existing approximate semiclassical and mixed quantum–classical methods and we derive an associated error term for each method. Furthermore, by combining the imaginary-time path-integral representation of the Boltzmann operator with the exact Liouvillian, we obtain an analytic expression for thermal quantum real-time correlation functions. These results provide a rigorous theoretical foundation for the development of accurate and efficient classical-like dynamics to compute observables such as electron transfer reaction rates in complex quantized systems.

Quantum simulations

2017 ◽  
Author(s):  
Michael P. Allen ◽  
Dominic J. Tildesley
Keyword(s):  
Ab Initio ◽  
Quantum Monte Carlo ◽  
Degrees Of Freedom ◽  
Small Region ◽  
Time Correlation ◽  
Ab Initio Molecular Dynamics ◽  
Classical Dynamics ◽  
Integral Methods ◽  
Quantum Time ◽  
Low Mass

This chapter covers the introduction of quantum mechanics into computer simulation methods. The chapter begins by explaining how electronic degrees of freedom may be handled in an ab initio fashion and how the resulting forces are included in the classical dynamics of the nuclei. The technique for combining the ab initio molecular dynamics of a small region, with classical dynamics or molecular mechanics applied to the surrounding environment, is explained. There is a section on handling quantum degrees of freedom, such as low-mass nuclei, by discretized path integral methods, complete with practical code examples. The problem of calculating quantum time correlation functions is addressed. Ground-state quantum Monte Carlo methods are explained, and the chapter concludes with a forward look to the future development of such techniques particularly to systems that include excited electronic states.

scholarly journals Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 518 ◽  
Author(s):  
Alessandro Sergi ◽  
Gabriel Hanna ◽  
Roberto Grimaudo ◽  
Antonino Messina
Keyword(s):  
Constant Temperature ◽  
Degrees Of Freedom ◽  
Open Quantum Systems ◽  
Hamiltonian Dynamics ◽  
Quantum Systems ◽  
Classical Dynamics ◽  
Open Quantum ◽  
Time Translation ◽  
Translation Symmetry ◽  
Lie Brackets

Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets augmented by dissipative terms. Quasi-Lie brackets possess the unique feature that, while conserving the energy (which the Noether theorem links to time-translation symmetry), they violate the time-translation symmetry of their algebra. This fact can be heuristically understood in terms of the dynamics of the open quantum subsystem. We then describe an example in which a quantum subsystem is embedded in a bath of classical spins, which are described by non-canonical coordinates. In this case, it has been shown that an off-diagonal open-bath geometric phase enters into the propagation of the quantum-classical dynamics. Next, we discuss how non-Hamiltonian dynamics may be employed to generate the constant-temperature evolution of phase space degrees of freedom coupled to the quantum subsystem. Constant-temperature dynamics may be generated by either a classical Langevin stochastic process or a Nosé–Hoover deterministic thermostat. These two approaches are not equivalent but have different advantages and drawbacks. In all cases, the calculation of the operator-valued quasi-probability function allows one to compute time-dependent statistical averages of observables. This may be accomplished in practice using a hybrid Molecular Dynamics/Monte Carlo algorithms, which we outline herein.

Quantum statistical physics: Functional integration formalism

2021 ◽  
pp. 64-89
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Phase Space ◽  
Integral Representation ◽  
Density Matrix ◽  
Thermal Equilibrium ◽  
Degrees Of Freedom ◽  
Functional Integral ◽  
Complex Variables ◽  
Path Integral Representation ◽  
Canonical Formulation ◽  
Grand Canonical

The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.

scholarly journals The interaction of electric charges

1930 ◽  
Vol 126 (803) ◽  
pp. 696-728 ◽  
Keyword(s):  
Quantum Theory ◽  
Gauge Transformation ◽  
Degrees Of Freedom ◽  
Frame Of Reference ◽  
Degree Of Freedom ◽  
Classical Dynamics ◽  
Proper Distance ◽  
Special Degree ◽  
Odd Degree ◽  
Absolute Quantity

1. The present investigation is a sequel to two earlier papers in these 'Proceedings.' In II a theory of the electron was proposed which led to a prediction of the value of the constant 2π e 2 / hc . The theory involved an appeal to the analogies of classical dynamics, which frequently prove useful though precarious; it has been my purpose to substitute a more satisfactory geometrical basis. In a problem of this kind, concerned with the whole question of the significance of the methods of quantum theory, it is unlikely that finality can have been reached even at the second attempt; but I think that the progress is now sufficient to justify publication. According to II the value of hc /2π e 2 was 136. I remarked that, as it represented the number of degrees of freedom of a system, small mistakes were unlikely; nevertheless I appear to have made such a mistake, and the new prediction is 137 (16). The 136 symmetrical degrees of freedom are a generalisation of rotations and translations in space; but it is characteristic of a pair of electrons that they possess one special degree of freedom unlike the others which has no analogue in the theory of a single electron, viz., an alteration of the proper distance between them; and whereas the 136 rotations are relative to the frame of reference employed, the odd degree of freedom represents alteration of an absolute quantity (the interval). The mistake in the earlier theory was not so much in overlooking this degree of freedom (for it there appeared as the rotation which "interchanges the identity of the two electrons") as in not recognising its distinctness from the others. No one of the 136 relativity transformations can play the part of this non-relativity or gauge transformation.

Homogeneous variables in classical dynamics

1933 ◽  
Vol 29 (3) ◽  
pp. 389-400 ◽  
Author(s):  
P. A. M. Dirac
Keyword(s):  
Degrees Of Freedom ◽  
Scalar Potential ◽  
Rest Mass ◽  
Classical Mechanics ◽  
Hamiltonian Function ◽  
Lagrangian Function ◽  
Practical Case ◽  
Classical Dynamics ◽  
Hamiltonian Method ◽  
Independent Functions

The well-known methods of classical mechanics, based on the use of a Lagrangian or Hamiltonian function, are adequate for the treatment of nearly all dynamical systems met with in practice. There are, however, a few exceptional cases to which the ordinary methods are not immediately applicable. For example, the ordinary Hamiltonian method cannot be used when the momenta pr, defined in terms of the Lagrangian function L by the usual formulae pr = ∂L/∂qr, are not independent functions of the velocities. A practical case of this kind is provided by the electromagnetic field, considered as a dynamical system with an infinite number of degrees of freedom, since the momentum conjugate to the scalar potential at any point vanishes identically. Again, for the very simple example of the relativistic motion of a particle of zero rest-mass in field-free space, the Lagrangian function vanishes and the usual Lagrangian method is not applicable.

scholarly journals Ultrafast photoactivation of C─H bonds inside water-soluble nanocages

2019 ◽  
Vol 5 (2) ◽  
pp. eaav4806 ◽  
Author(s):  
Ankita Das ◽  
Imon Mandal ◽  
Ravindra Venkatramani ◽  
Jyotishman Dasgupta
Keyword(s):  
Charge Transfer ◽  
Degrees Of Freedom ◽  
Bond Cleavage ◽  
Electron Transfer Reaction ◽  
Water Soluble ◽  
Charge Transfer Complexes ◽  
Chemical Bonds ◽  
Proton Coupled Electron Transfer ◽  
Selective Reaction ◽  
Carbon Centered Radical

Light energy absorbed by molecules can be harnessed to activate chemical bonds with extraordinary speed. However, excitation energy redistribution within various molecular degrees of freedom prohibits bond-selective chemistry. Inspired by enzymes, we devised a new photocatalytic scheme that preorganizes and polarizes target chemical bonds inside water-soluble cationic nanocavities to engineer selective functionalization. Specifically, we present a route to photoactivate weakly polarized sp3C─H bonds in water via host-guest charge transfer and control its reactivity with aerial O2. Electron-rich aromatic hydrocarbons self-organize inside redox complementary supramolecular cavities to form photoactivatable host-guest charge transfer complexes in water. An ultrafast C─H bond cleavage within ~10 to 400 ps is triggered by visible-light excitation, through a cage-assisted and solvent water–assisted proton-coupled electron transfer reaction. The confinement prolongs the lifetime of the carbon-centered radical to enable a facile yet selective reaction with molecular O2leading to photocatalytic turnover of oxidized products in water.

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