Angular Momentum Correlation Function Research Articles

Galactic angular momenta and angular momentum couplings in the large-scale structure

In this paper, we revisit the aquisition of angular momentum of galaxies by tidal shearing and compute the angular momentum variance sigma_L^2 as well as the angular momentum correlation function C_L(r) from a peak-restricted Gaussian random process. This stochastic process describing the initial conditions treats both the tidal shear as well as the inertia as dynamical fields and explicitly accounts for the discreteness of the inertia field. We describe the way in which the correlations in angular momentum result from an interplay of long-ranged correlations in the tidal shear, and short ranged correlations in the inertia field and which reflects the correlation between the eigensystems of these two symmetric tensors. We propose a new form of the angular momentum correlation function which is able to distinguish between parallel and antiparallel alignment of angular momentum vectors, and comment on implications of intrinsic alignments for weak lensing measurements. We confirm the scaling L/M propto. M^{2/3} and find the angular momentum distribution of Milky Way-sized haloes to be correlated on scales of ~1 Mpc/h. The correlation function can be well fitted by an empirical relation of the form C_L(r) propto. exp(-[r/r_0]^beta).

GALACTIC ANGULAR MOMENTA AND ANGULAR MOMENTUM CORRELATIONS IN THE COSMOLOGICAL LARGE-SCALE STRUCTURE

I review the theory of angular momentum acquisition of galaxies by tidal torquing, the resulting angular momentum distribution and the angular momentum correlation function, and discuss the implications of angular momentum alignments for weak lensing measurements. Starting from linear models for tidal torquing, I summarize perturbative approaches and the results from n-body simulations of cosmic structure formation. Then I discuss the validity of decompositions of the tidal shear and inertia fields, the effects of angular momentum biasing, the applicability of parametrized angular momentum correlation models and the consequences of angular momentum correlations for shape alignments. I compile the results of observations of shape alignments in recent galaxy surveys as well as in n-body simulations. Finally, I review the contamination of weak lensing surveys by spin-induced shape alignments and methods for suppressing this contamination.

Solvation shell dynamics studied by molecular dynamics simulation in relation to the translational and rotational dynamics of supercritical water and benzene

The solvation shell dynamics of supercritical water is analyzed by molecular dynamics simulation with emphasis on its relationship to the translational and rotational dynamics. The relaxation times of the solvation number (tau S), the velocity autocorrelation function (tau D), the angular momentum correlation function (tau J), and the second-order reorientational correlation function (tau 2R) are studied at a supercritical temperature of 400 degrees C over a wide density region of 0.01-1.5 g cm(-3). The relaxation times are decomposed into those conditioned by the solvation number n, and the effect of the short-ranged structure is examined in terms of its probability Pn of occurrence. In the low to medium-density range of 0.01-0.4 g cm(-3), the time scales of water dynamics are in the following sequence: tau D>tau S approximately or > tau J approximately or > tau 2R. This means that the rotation in supercritical water is of the "in-shell" type while the translational diffusion is not. The comparison to supercritical benzene is also performed and the effect of hydrogen bonding is examined. The water diffusion is not of the in-shell type up to the ambient density of 1.0 g cm(-3), which corresponds to the absence of the transition from the collision to the Brownian picture, whereas such transition is present in the case of benzene. The absence of the transition in water comes from the fast reorganization of the hydrogen bonds and the enhanced mobility of the solvation shell in supercritical conditions.

Relaxation of rotational angular momentum of polar diatomic molecules in simple liquids

The relaxation processes of rotational angular momentum of polar diatomic molecules diluted in simple liquids are analyzed by applying a non-Markovian relaxation theory to the study of the binary time autocorrelation function of the angular momentum. This non-Markovian theory was previously applied to the study of the infrared and Raman spectroscopy, and also to the analysis of the rotational energy relaxation processes. We have obtained non-Markovian evolution equations for the two-time $j$-level angular momentum correlation components involved in the angular momentum correlation function. In these equations, the time-dependent angular momentum transfer rates and the pure orientational angular transfer rates are given in terms of the binary time autocorrelation function of the diatomic-solvent anisotropic interaction. The non-Markovian evolution equations converge to Markovian ones in the long time limit, reaching the angular momentum transfer rates in the usual time-independent form. Alternative time scales for the angular relaxation processes, relative to the individual rotational processes as well as to the global decay correlations, are introduced and analyzed. The theory is applied to the study of the angular momentum relaxation processes of HCl diluted in liquid $\mathrm{S}{\mathrm{F}}_{6}$, a system for which rotational energy relaxation and infrared and Raman spectroscopy was previously analyzed in the scope of the same theory.

Molecular dynamics simulations of the orientational motion of liquid chloroform at 298 K and 220K

Molecular dynamics simulations of liquid chloroform at 298 K and 220 K, using established optimized potential parameters and run on single-processor workstations, are employed to compute the first- and second-order Legendre polynomial, single-molecule orientational correlation functions describing the tumbling motion of the carbon-hydrogen bond direction. The results of the simulated Raman correlation functions compare well with their experimental counterparts from previously reported Fourier-transformed band contours of two totally symmetric vibration-rotation modes of the liquid. Simulated angular momentum, angular velocity, and intermolecular torque correlation functions are shown, and helped to characterize the insignificance of free orientational mobility within the system. Perturbation effects from rotation-vibration interaction and collision-induced intensities are present, but their effects on the correlation results are not sufficiently high to be of major significance.

Effect of collisions on the orientational relaxation of photofragments

Effect of collisions is investigated on the photofragment anisotropy decay. We restrict ourselves to studying linear fragments, and no rotational predissociation is assumed. Photoproducts are produced with a nonequilibrium rotational distribution, basically due to the applied torque and the parent molecule rotation. A kinetic equation, describing rotational relaxation of linear fragments under nonequilibrium conditions, is derived and solved for angular momentum correlation functions (CFs), rotational energy CFs, and also orientational CFs (OCFs). The characteristic decay times for the angular momentum and rotational energy CFs are shown to be insensitive to the mechanism of the photofragmentation. On the contrary, OCF of the second rank, that completely determines anisotropy of the photoproduct emission, is demonstrated to be very sensitive to peculiarities of the dissociation process and collision dynamics. This is confirmed by comparison of the calculated photoproduct anisotropies with experimental [M. Volk, S. Gnanakaran, E. Gooding, Y. Kholodenko, N. Pugliano, R.M. Hochstrasser, J. Phys. Chem. A 101 (1997) 638] and simulated [I. Benjamin, U. Banin, S. Ruhman, J. Chem. Phys. 98 (1993) 8337] ones. The fragment ensemble is proved to retain some memory about its initial nonequilibrium distribution. This pertains not only to the short time behavior for the anisotropy, but also for its long time decay. Therefore, the study of the polarization response of the ensemble of photoproducts allows one to know about features of the photofragmentation dynamics, and also to get information about peculiarities of collisions of fragments with buffer medium species.

A stochastic cage model for linear solutes

A stochastic cage model describing a linear solute reorienting under the action of a librational potential due to neighboring solvent molecules is presented. The fluctuations of the cage structure are taken into account by means of a suitable distribution of the librational frequency. Moreover, a detailed description of the cage dynamics is introduced by considering both the cage rotation and its restructuring through randomizations of the librational frequency and of the equilibrium orientation of the solute. With a suitable choice of the basis functions for the representation of the time evolution operator, the cage model can be solved numerically in order to compute different types of dynamical observables: angular momentum and orientational correlation functions, frequency dependent dielectric permittivity, and far-infrared spectra. Typical behavior of such observables in normal liquids is recovered from the cage model, thus demonstrating its capability of describing experimental observations at quite different time scales.

Response to ‘‘Comment on ‘Long-time tails in angular momentum correlations’ ’’ [J. Chem. Phys. 104, 7363 (1996)

~Received 29 December 1995; accepted 22 January 1996!@S0021-9606~96!02216-4#In their comment on our paper, Cichocki and Felderhofpoint out that our conclusion about the shape independenceof the long-time tail in the angular momentum correlationfunction must be incorrect because it contradicts a ‘‘rigor-ous’’ result that they have obtained previously. Moreover,they stress that we have incorrectly stated that their theoreti-cal results are limited to the case that the orientation of theparticle is fixed.To start with the second point: It is obvious from Ref. 1,that Cichocki and Felderhof do not consider the case of largeamplitude reorientational motion. In fact, they say as muchin the preamble to the derivation in Ref. 1 where they statethat they consider the case of ‘‘small amplitude motion.’’ Inour paper, we referred to this case as the limit that the ori-entation is ‘‘fixed.’’ Admittedly, it would have been moreaccurate to speak about an orientation that is ‘‘essentiallyfixed,’’ rather than literally fixed. We thought this to be un-necessarily verbose, because truly fixed orientations areunrealistic—in this limit the angular velocity autocorrelationfunction ~AVACF! does not exist.Let us then compare the two cases of real interest,namely the Cichocki–Felderhof limit of small amplitude an-gular motion and the limit of large amplitude reorientationthat we considered in our paper.Cichocki and Felderhof point out that for small ampli-tude motion, the AVACF does depend on the shape of theparticle, and we completely agree. More interestingly, wewere happy to notice that the theoretical predictions ofCichocki and Felderhof for this limit are in excellent agree-ment with the numerical results that for nonspherical objectsunder conditions where the angular displacement is negli-gible. For small amplitude motion, the shape and frequencydependent friction coefficient is proportional to the memoryfunction of the AVACF and gives a shape dependent tail.Thus, so far as the calculation of the friction coefficient isconcerned, the particle is fixed ~‘‘essentially fixed’’ in ourpresent terminology!. The calculation described in Ref. 1neglects the effect of the particle’s orientational motion onthe frequency dependent rotational friction coefficient. Oursimulations show quite convincingly, that for a particle thatis ‘‘essentially fixed,’’ a shape dependent decay of theAVACF is indeed observed. It is reassuring that our model,which uses a relatively crude lattice representation of a non-spherical object, is in essentially quantitative agreement withthe relevant theoretical predictions. This gives us great en-couragement that we can reliably extend our simulations tothe case of concentrated suspensions of non-spherical ob-jects.Next, let us consider the case discussed in Ref. 2, viz.the effect of reorientation on the dynamics of the particle.Reorientation is important because, on a sufficiently longtimescale, i.e., in the truly asymptotic regime, reorientation isalways important. In Ref. 2 we stressed that the theory con-tained in Ref. 1 assumes that reorientation is negligible andis therefore not appropriate ~let alone ‘‘rigorous’’! in thislimit. However, the microscopic theory contained in Ref. 4should apply in the limit of large-amplitude reorientation. Itpredicts that reorientation changes the decay of the AVACFin such a way that it becomes identical to the result for aspherical object with the same moment of inertia. A carefulmicroscopic analysis produces kinetic equations that, in theBrownian limit, reduce to the Navier–Stokes equations ~plusappropriate boundary conditions!. Not all the mode-couplingor kinetic theory treatments, which have been proposed, givethis limit correctly.

Nonlinearly coupled generalized Fokker-Planck equation for rotational relaxation.

We have investigated the stochastic dynamics of a one dimensional rotor with ${\mathit{C}}_{3}$ symmetry and zero barrier to rotation about the symmetry axis. Angular momentum correlation functions derived from various stochastic dynamics models were compared to the corresponding correlation functions obtained from a molecular dynamics simulation [G. Widmalm, R. W. Pastor, and T. E. Bull, J. Chem. Phys. 94, 4097 (1991)]. None of the existing classical models agrees with the simulation; and we have shown in general that no linearly coupled generalized Langevin equation with Gaussian random noise can reproduce the simulation results. A quantum stochastic dynamics model [T. E. Bull, Chem. Phys. 143, 381 (1990)] extrapolated to the classical limit does, however, agree with the computer simulations. But this model is limited to very small molecules because the matrices involved become prohibitively large for even moderately sized molecules. In order to address some of these limitations, we have constructed a nonlinearly coupled rotor-bath model for the rotor. The form of the nonlinear coupling between the rotor and bath is determined by the symmetry of the rotor. A classical nonlinearly coupled generalized Langevin equation and its corresponding nonlinearly coupled Fokker-Planck equation were derived from this microscopic rotor-bath model using the projection operator formalism. In the limit of white noise, these equations reduce to the standard equations derived with linear coupling. With colored noise, however, the linearly and nonlinearly coupled equations are distinct. Angular momentum correlation functions calculated with this nonlinearly coupled Fokker-Planck equation are in excellent agreement with the simulations both in terms of the short time Gaussian decay and long time exponential tail and in terms of the magnitudes of the correlation functions. Collision operators derived from this model should therefore provide a more accurate connection between experimentally measured quantities and the underlying microscopic dynamics.

Extended fokker-planck model for molecular rotations in liquids

Extended Fokker-Planck (FP) equations are generalized stochastic equations which describe the evolution of a set of coordinates, loosely referred to as solute, coupled with a relevant set of solvent variables. The applicational range of multidimensional stochastic operators is wide. In this paper we focus attention on model “liquid” systems. We discuss effect of various kinds of cage degrees of freedom on the relaxation of orientational and angular momentum correlation functions of a probe molecule, for a model liquid composed of planar rotating dipoles interacting via a cosine potential. The results of numerically exact and asymptotic solutions of FP equations are used to interpret the dynamic behaviour of polar liquids in the frequency regions of dielectric relaxation and far-infrared absorption.

Molecular dynamics computer simulation of non-linear optical effects; electric polarisation due to optical rectification in a circularly polarised laser

The technique of field applied molecular dynamics computer simulation is used to investigate the picosecond molecular dynamics of static and dynamic electric polarisation due to a laser field. A pulse of intense, circularly polarised, electromagnetic radiation is applied to a sample of chiral molecules and the molecular dynamics analysed statistically with time correlation functions computed in the presence of the laser. The effect of the laser on the molecular dynamics was animated and analysed visually. These dynamics measure the interaction between the imaginary part of the dynamic electronic polarisability and the conjugate product Π A of the laser. When the laser frequency is tuned to the THz region (corresponding to picosecond time scales of the natural molecular dynamics) the computer simulation shows the presence of frequency dependent magnetisation, measured by the t→∞ level of the component of the angular momentum correlation function in the axis of propagation of the laser. In general, the details of the molecular dynamic response are very sensitive to the frequency and intensity of the laser.

Local scaling analysis of state-to-state rotational energy-transfer rates in N2 from direct measurements

An analysis of recent rotational energy state-to-state rates data in N2 is achieved by using an exact local equation valid within the energy corrected sudden (ECS) frame. This equation connects the rates belonging to each group of five cross-shaped elements characterized by the rotational quantum numbers {(J,J′), (J,J′+2), (J, J′−2), (J−2,J′), (J+2,J′)}. Such an analysis is free of any assumption on the dependence of the basis rates on the rotational energy. It allows one to perform a study of the local validity of the scaling law and a more direct estimation of the scaling length. Previous ECS studies, only based on the inversion–fitting procedure of line-broadening coefficients, are compared with the present results. It is shown that, in spite of good agreement of these last phenomenological approaches with stimulated Raman profiles for compressed N2 gas, they may be unadapted to describe other physical properties. This point is clearly illustrated by the behavior of the rotational angular momentum correlation function. The origin of such a discrepancy is evidenced and a convenient modification of the adiabaticity correcting factor is introduced. Finally, excellent consistency with available experimental data is obtained.

Dynamics of an ammonium ion in water: Molecular dynamics simulation

The translational and rotational dynamics of an aqueous ammonium ion are examined using an NVE molecular dynamics computer simulation and a rigid model of water. The linear momentum, angular momentum and reorientational autocorrelation functions of the ammonium ion have been measured. From these, the memory functions for the linear and angular momentum correlation functions are calculated. Reorientation of the ammonium ion about its center of mass shows unambiguous departure from Debye rotational behavior. The translational and rotational diffusion constants have been calculated.

Ionic dynamics in computer simulated molten LiNO3. II. Tumbling and spinning motions of nitrate ions

The microdynamics of orientational motion (the tumbling of the symmetry axis and the spinning about the axis) of nitrate ions has been investigated in a molecular dynamics simulation of molten LiNO3. Simple Coulomb potential with Born type repulsion has been adopted and all degrees of freedom of vibrating nitrate ions have been included in the simulation. The relaxation of the orientational correlation functions is studied in the cumulant expansion scheme, and a correlation function which reflects only the spinning motion is separated by the method of Lynden-Bell et al. It is found that the angular momentum correlation function for the perpendicular component C⊥J(t) has a pronounced negative portion, which is clear evidence of librations in a temporary solvent cage. C∥J(t) has a shallow negative lobe. The spinning motion is less hindered and the reorientation is faster. Both the tumbling and the spinning motions are found to be predominantly Markovian. The difference between the tumbling and the spinning motions is examined from the magnitude of the torque impulse as compared to the average angular momentum. The coupling between the environmental structure and the rotational behavior of a nitrate ion has been examined through selectively sampled correlation functions. It is found that the rotational diffusion caused by the tumbling motion is slow and that by the spinning motion is fast if the coordination number remains low.

Ionic dynamics in computer simulated molten LiNO3. I. Translational and reorientational motion

In order to study the microdynamics of translational and orientational motion of ions in liquid electrolytes, a molecular dynamics simulation of molten LiNO3, which includes all degrees of freedom of vibrating nitrate ions, has been performed. Simple Coulomb pair potentials with Born-type repulsions have been adopted in the simulation. The calculated properties include: translational velocity and force correlation functions, rotational correlation functions, and angular momentum and torque correlation functions. The nature of the ionic motion is analyzed in terms of quasioscillation and quasilibration. Local structures surrounding nitrate ions, which can be characterized by, e.g., the coordination number, are found to persist for a long time (∼1 ps). The coupling between the environmental structure and the translational and orientational behaviors of a given nitrate ion has been examined through selectively sampled correlation functions. We found that the change of coordination number enhances the translational diffusion and the rotational diffusion is slow if the coordination number remains low. This shows that the dynamics are very sensitive to the exact shape of the potential and the dynamics itself. Results of the simulation are compared with the experimental diffusion constants and dynamical behaviors revealed by Raman spectroscopy. While the interionic interaction assumed has been found to describe quite well the static structure of molten LiNO3, simulated translational diffusion and rotational relaxation seem to be somewhat too fast, suggesting too shallow an attractive well in the potential we have assumed.

Correlation functions from depolarized Raman and Rayleigh spectra of N2at high density and 150 K

Orientational correlation functions derived from the rotational and rotovibrational Raman spectra of N2 at 150 K and 585 amagat are compared with the results of a molecular dynamics simulation available in the literature. The angular momentum correlation function is also estimated from the experimental memory function and compared with the one obtained by simulation. Mean square torques, intermolecular correlations and interaction induced contributions are also discussed.

Crystal- and magnetic-field effects on nuclear-spin—lattice relaxation in solidH2

A previous study of nuclear-spin---lattice relaxation in solid mixtures of orthohydrogen and parahydrogen is extended to include the crystal- and magnetic-field effects which have been observed experimentally at low $o$-${\mathrm{H}}_{2}$ concentrations $c$. The dynamical rotational angular momentum correlation functions which determine the spin-lattice relaxation rates are calculated as functions of $c$ and crystal- and magnetic-field strength using the self-consistent Sung-Arnold method to treat the electric quadrupole-quadrupole interactions and a rotating-frame transformation to treat the crystal and magnetic fields. Numerical solutions for these correlation functions are then used to obtain the relaxation times ${T}_{1}$ and ${T}_{2}$ as functions of $c$, applied-magnetic-field strength, and crystal-field energy. The results obtained for ${T}_{1}$ are in excellent agreement with the recent experiments of Buzerak, Chan, and Meyer. On the other hand, there is a major qualitative and quantitative discrepancy between the predictions obtained here for ${T}_{2}$ and the experimental results of Buzerak et al. The reason for this discrepancy is not understood. In addition, the dependences of the spectral density functions and nuclear free-induction decay on $o$-${\mathrm{H}}_{2}$ concentration and magnetic- and crystal-field strenghts are calculated.

Ultrasonic attenuation in a quadrupolar solid

A general expression for the high-temperature ultrasonic attenuation in a quadrupolar solid is derived. The four-point correlation function occurring in this expression is then factored into products of two-point functions so that the attenuation is proportional to the convolution of two of the usual two-point angular momentum correlation functions. Using first-principles results for these functions, which were previously obtained for solid ${\mathrm{H}}_{2}$, the field-dependent or resonant part of the ultrasonic attenuation is then calculated as a function of ortho-${\mathrm{H}}_{2}$ concentration $c$. The result of this calculation indicates that the resonant attenuation for solid ${\mathrm{H}}_{2}$ has a magnitude such that it could be observable by current techniques. A new kind of ultrasonic magnetic resonance is thus predicted.

Cross-correlation functions for angular momentum and orientation

A general expression is derived that relates the generalized second moment of a rotational spectral band to an angular momentum−orientation time−correlation function for the ensemble of molecules. It is demonstrated that previous equations relating these spectral transforms to angular momentum correlation functions are recovered in the limit of short times. The extended diffusion models proposed by Gordon are used to calculate cross−correlation functions as a function of time, and the results are compared with several approximations, including the decoupling assumption widely used in spin−rotation relaxation. The cross correlations that appear in the spin−rotation problem are considered in detail. In contrast to the behavior of the spectral transform cross−correlation functions, it is explicitly demonstrated that the spin−rotation correlations for the extended diffusion models vanish after one collision.

Spectral density functions for aJ=1quadrupolar solid: Application to solidH2

Integral equations for the dynamical two-point angular momentum correlation functions of a $J=1$ quadrupolar solid are derived by the use of a diagrammatic technique which was previously developed for dense magnetic systems. For comparison to experiments on solid mixtures of ortho and para hydrogen, a simple scheme, valid in the high-concentration limit, is used to impurity average these equations. The averaged equations are then solved numerically to obtain the spectral functions for solid ${\mathrm{H}}_{2}$ self-consistently for the first time. The resulting spectral functions are then used to compute the nuclear spin-lattice relaxation time of solid ${\mathrm{H}}_{2}$ as a function of the ortho-molecule concentration and this is shown to agree well with experiments at 10 K and over a concentration range of $0.5\ensuremath{\le}C\ensuremath{\le}1$. Finally, a formula is derived which expresses the differential inelastic-neutron-scattering cross section in solid ${\mathrm{H}}_{2}$ in terms of the calculated spectral functions. If the neutron wave vector and scattering angle are specified, one can then predict the scattering cross section from the numerical results for the spectral functions.