On the compositions of rotations

Donchev, Veliko; Mladenova, Clementina D.; Mladenov, Iväılo M.

doi:10.1063/1.4934315

Cited by 4 publications

(7 citation statements)

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“…It was shown that screening of charges for neutral molecules can vary from 1 el  (about 0.7) to a much weaker factor ~0.9. The exact value of the factor depends on the model of intramolecular polarizability which can not be defined unambiguously even in most sophisticated polarizable models [20][21][22][23][24][25].…”

Section: Discussionmentioning

confidence: 99%

“…The apparent reason why nonpolarizable force fields such as AMBER [5,6], CHARMM [7], GROMOS [8], OPLS [9,10] avoided scaling of the ion charges is that the * Usually, the treatment of solvation energy in terms of continuum models requires consideration of the molecular cavity ; it should be noticed that molecular cavities do not appear in the interaction terms that are related to dynamics, nor do they appear in the solvation energy terms of individual sites. 22 reduction of charges by itself results in completely wrong hydration free energies [7]. The reason is that the charging free energy of ions from MD simulations by itself does not account for the electronic polarization energy, which typically amounts to more than 50% of the total.…”

Section: Molecular Dynamics In Electronic Continuummentioning

confidence: 99%

“…A straightforward way to avoid such questions is to use polarizable force fields; a number of such force fields are currently being developed [15][16][17][18][19][20][21][22][23][24][25][26]. Polarizable models, however, still suffer from some unresolved problems, such as consistency and balance in treatment of intra-vs. intermolecular polarizations [27].…”

Section: Introductionmentioning

confidence: 99%

“…The computational efficiency of such models is also an issue. Achieving good sampling of relevant configurations [28] in large biological simulations can become computationally prohibitive if microscopic interactions are computationally expensive [20][21][22][23][24][25]. Clearly, the simple but computationally efficient nonpolarizable models have advantages over more sophisticated polarizable ones, provided they are accurate enough.…”

Section: Introductionmentioning

confidence: 99%

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Polarizable molecular interactions in condensed phase and their equivalent nonpolarizable models

Leontyev

Stuchebrukhov

2014

The Journal of Chemical Physics

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Earlier, using phenomenological approach, we showed that in some cases polarizable models of condensed phase systems can be reduced to nonpolarizable equivalent models with scaled charges. Examples of such systems include ionic liquids, TIPnP-type models of water, protein force fields, and others, where interactions and dynamics of inherently polarizable species can be accurately described by nonpolarizable models. To describe electrostatic interactions, the effective charges of simple ionic liquids are obtained by scaling the actual charges of ions by a factor of 1/ √ ε el , which is due to electronic polarization screening effect; the scaling factor of neutral species is more complicated. Here, using several theoretical models, we examine how exactly the scaling factors appear in theory, and how, and under what conditions, polarizable Hamiltonians are reduced to nonpolarizable ones. These models allow one to trace the origin of the scaling factors, determine their values, and obtain important insights on the nature of polarizable interactions in condensed matter systems. © 2014 AIP Publishing LLC.[http://dx

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Section: Discussionmentioning

confidence: 99%

Section: Molecular Dynamics In Electronic Continuummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Polarizable molecular interactions in condensed phase and their equivalent nonpolarizable models

Leontyev

Stuchebrukhov

2014

The Journal of Chemical Physics

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“…Within this formalism, the natural covering map SU(2) → SO(3, R) and its sections are written and studied. The technique developed in [5] is used in [6] to extend the composition law (6) in the cases when half-turns are involved or the result of the composition is a halfturn, i.e., when c 2 .c 1 = 1. To do that, a half-turn R(n, π) = O(n) is represented as a ray, i.e., by the set of all three dimensional non-zero vectors, collinear with the axis of rotation n. This is an alternative description of SO(3, R) and the corresponding composition laws are more intuitive and are computationally cheaper even than the quaternionic formalism [10] when it comes to the composition of rotations [11].…”

Section: Introductionmentioning

confidence: 99%

Vector-Parameter Forms of SU(1,1), SL(2,R) and Their Connection to SO(2,1)

Donchev¹,

Mladenova²,

Mladenov³

2016

GIQ

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The Cayley maps for the Lie algebras su(1, 1) and so(2, 1) converting them into the corresponding Lie groups SU(1, 1) and SO(2, 1) along their natural vector-parameterizations are examined. Using the isomorphism between SU(1, 1) and SL(2, R), the vector-parameterization of the latter is also established. The explicit form of the covering map SU(1, 1) → SO(2, 1) and its sections are presented. Using the so developed vector-parameter formalism, the composition law of SO(2, 1) in vector-parameter form is extended so that it covers compositions of all kinds of elements including also those that can not be parameterized properly by regular SO(2, 1) vectorparameters. The latter are characterized and it is shown that they can be represented by SU(1, 1) vector parameters with pseudo length equal to minus four. In all cases of compositions inside SO(2, 1), criteria for determination of their type (elliptic, parabolic, hyperbolic) have been presented. On the base of the vector-parameter formalism the problem of taking a square root in SO(2,1) is solved explicitly. Also, an analogue of Cartan's theorem about the decomposition of orthogonal matrix of order n into product of at most n reflections is formulated and proved for the subset of hyperbolic elements of the group of pseudo-orthogonal matrices from SO(2, 1).

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Visualizing rotations and composition of rotations with the Rodrigues vector

Valdenebro

2016

Eur. J. Phys.

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The purpose of this paper is to show that the mathematical treatment of three-dimensional rotations can be simplified, and its geometrical understanding improved, by using the Rodrigues' vector representation. We present a novel geometrical interpretation of the Rodrigues' vector. Based on this interpretation and simple geometrical considerations, we derive Euler-Rodrigues' formula, Cayley's rotation formula, and the composition law for finite rotations. The level of this discussion should be suitable for undergraduate physics or engineering courses where rotations are discussed.

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On the compositions of rotations

Cited by 4 publications

References 4 publications

Polarizable molecular interactions in condensed phase and their equivalent nonpolarizable models

Polarizable molecular interactions in condensed phase and their equivalent nonpolarizable models

Vector-Parameter Forms of SU(1,1), SL(2,R) and Their Connection to SO(2,1)

Visualizing rotations and composition of rotations with the Rodrigues vector

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