Application of vibration-transit theory to distinct dynamic response for a monatomic liquid
Duane C. Wallace, Eric D. Chisolm, Giulia De Lorenzi-Venneri
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p LA-UR-13-20814
Application of vibration-transit theory to distinct dynamicresponse for a monatomic liquid
Duane C. Wallace, Eric D. Chisolm, and Giulia De Lorenzi-Venneri
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Dated: July 13, 2018)
Abstract
We examine the distinct part of the density autocorrelation function F d ( q, t ), also called theintermediate scattering function, from the point of view of the vibration-transit (V-T) theory ofmonatomic liquid dynamics. A similar study has been reported for the self part, and we study theself and distinct parts separately because their damping processes are not simply related. We beginwith the perfect vibrational system, which provides precise definitions of the liquid correlations, andprovides the vibrational approximation F dvib ( q, t ) at all q and t . Two independent liquid correlationsare defined, motional and structural, and these are decorrelated sequentially, with a crossover time t c ( q ). This is done by two independent decorrelation processes: the first, vibrational dephasing, isnaturally present in F dvib ( q, t ) and operates to damp the motional correlation; the second, transit-induced decorrelation, is invoked to enhance the damping of motional correlation, and then to dampthe structural correlation. A microscopic model is made for the “transit drift,” the averaged transitmotion that damps motional correlation on 0 ≤ t ≤ t c ( q ). Following the previously developed self-decorrelation theory, a microscopic model is also made for the “transit random walk,” which dampsthe structural correlation on t ≥ t c ( q ). The complete model incorporates a property common toboth self and distinct decorrelation: simple exponential decay following a delay period, where thedelay is t c ( q ), the time required for the random walk to emerge from the drift. Our final result isan accurate expression for F d ( q, t ) for all q through the first peak in S d ( q ). (A modification will berequired at q where S d ( q ) converges to zero.) The theory is calibrated and tested using moleculardynamics (MD) calculations for liquid Na at 395 K; however, the theory itself does not depend onMD, and we consider other means for calibrating it. PACS numbers: 05.20.Jj, 63.50.+x, 61.20.Lc, 61.12.Bt . INTRODUCTION Our goal in developing V-T theory is to apply the established techniques of many bodyphysics to the mechanical problem of the motion of atoms in a monatomic liquid. The manybody formulation begins with an approximate Hamiltonian H composed of a completeorthogonal set of excitations, bosons or fermions, whose exact statistical mechanics is known. H is complemented with an interaction Hamiltonian H , expressing the key effect missingfrom H , and often but not always treated as a perturbation. The approach was developedto treat the wide variety of physical behaviors observed in condensed matter, and is wellillustrated in the monographs of Pines [1, 2] and Kittel. [3] The principles are evident inBoltzmann’s theory for a gas of freely moving atoms which interact via collisions, [4] andin the theory of Born and coworkers for a crystal of harmonic phonons interacting viaanharmonicity. [5, 6]In the absence of such a many body formulation, liquid dynamics theory has been ad-vanced by a series of conceptual developments. An important early step was learning howto construct physically realistic interatomic potentials for nearly-free-electron metals frompseudopotential perturbation theory. [7–10] It was shown that MD calculations using thesepotentials give an excellent account of experimental data for elemental condensed systems,for example: for thermodynamic properties of crystals [11] and liquids, [12] and for theliquid structure factor [13, 14] and dynamic structure factor. [14–16] Ab initio
MD was in-troduced, [17–19] and has since become the method of choice for reliable calculations onmany liquid types. The resulting physical picture of liquids as nuclei moving on the groundstate adiabatic potential, and subject to conventional statistical mechanical averaging, isthe basis of our theoretical work. Formal derivation of the corresponding condensed matterHamiltonian is reviewed in Ch. 1 of Ref. 20.The conceptual contribution of V-T theory is to classify the valleys that comprise theliquid’s many-body potential surface as random or symmetric. The random valleys allhave the same statistical mechanical averages, and together they dominate the potentialsurface in the thermodynamic limit. [21] Originally a hypothesis, this classification hasbeen numerically verified for various representative systems. [21–26] The atomic motionis vibrations in one (any) random valley, interspersed by transits, which carry the systembetween random valleys. Accordingly, we define the extended random valley as the harmonic2xtension to infinity of any random valley, and take the Hamiltonian H vib for motion in oneextended random valley as our zeroth order liquid Hamiltonian. To study any particularfunction, we first calculate the function assuming the Hamiltonian is H vib , and then weexamine the (often small) remainder to see how models of transit motion can account for it.(Studies of liquid motion in terms of hops between potential valleys date back at least toStillinger and Weber [27] and Zwanzig, [28] but without the benefit of the V-T classification,which makes clear the choice of initial Hamiltonian.)Transits have been observed in MD calculations for liquid Na and Ar, at very low tem-peratures where single transits are well resolved. [29] These transits occur in the highlycorrelated motion of a small local group of atoms. At and above melting, transits proceedat a high rate throughout the liquid. In its initial formulation, V-T theory incorporated anempirical melting-entropy constant to represent the transit contribution to the entropy ofelemental liquids. [21] This formulation requires extension in two ways: in order to treat all thermodynamic properties, the transit entropy theory must be replaced by a free energytheory; and in order to make a purely liquid theory, all sensitivity to the nature of the crys-tal or the melting process must be removed. These extensions were carried out through ananalysis of the temperature dependence of experimental entropy for elemental liquids, [30]plus a statistical mechanics free energy model calibrated to the entropy results. [31] A priori density functional theory (DFT) calculations then verified the theory to high accuracy forthermodynamic properties of liquid Na and Cu. [32]In nonequilibrium problems, V-T theory has achieved success in two applications todynamic response. First, it was found that an a priori calculation of the Brillouin peakdispersion curve, based on the vibrational motion alone, is in essentially perfect agreementwith MD calculations and with experimental inelastic scattering data for liquid Na. [33]Second, in comparison with the benchmark theories of generalized hydrodynamics and modecoupling, a near- a priori theory of self dynamic response was found to have significantlyimproved analytic properties and modestly improved accuracy. [34] The purpose of thepresent work is to test the viability of V-T theory for the distinct part of the densityautocorrelation function.Hansen and McDonald [35] define time correlation functions related to self and distinctcontributions to dynamic response, and discuss the self dynamic structure factor, includingits Gaussian approximation. These authors also point out it is possible in principle to3easure separately the self and distinct parts. We have not found studies dedicated tothe distinct part, presumably because the self part and the total function cover the entiretheoretical problem. However, the distinct part must be studied separately, because thestrong difference between self and distinct correlations implies a similar difference in theirdecorrelation processes.Formulation of dynamic response theory is a quintessential quantum mechanics problem,analyzed for crystals by Maradudin and Fein, [36] Ambegaokar, Conway and Baym, [37]Cowley, [38, 39] and Ashcroft and Mermin, [40] and for crystals and liquids by Lovesey [41]and Glyde. [42] However, our focus is the elemental liquids in general, so we omit thefew quantum liquids and work in classical statistical mechanics. We study the densityautocorrelation function F ( q, t ), also called the intermediate scattering function, for allwave vectors q and time t . Expansions in powers of scattering events are inefficient, and wemust work with the full theory, correct to all powers of q (see Appendix N of Ref. 40). Weapply the primitive-lattice harmonic crystal analysis from the above references to a systemmoving in an extended random valley, then extract the classical limit to obtain the zerothorder liquid function F vib ( q, t ) (details may be found in Ref. 43). In the present paper webegin with the distinct part F dvib ( q, t ), and introduce the decorrelating effects of transits todamp F dvib ( q, t ) to the function representing V-T theory, F dV T ( q, t ).Sec. II sets out the “standard plan” for constructing the theoretical function F dV T ( q, t ).The perfect vibrational system provides equations for F dvib ( q, t ) and for the motional andstructural correlations. Transits are responsible for all damping beyond the dephasing al-ready contained in F dvib ( q, t ). The transit contribution to F dV T ( q,
0) is assigned to the struc-tural correlation and is calibrated from MD. A crossover time t c ( q ) is defined, such thatmotional correlation damps to zero on 0 ≤ t ≤ t c , and structural correlation damps to zeroon t ≥ t c .In Sec. III, the massively averaged motion due specifically to transits on 0 ≤ t ≤ t c ismodeled as the transit drift, and a theory is made for its contribution to motional decorrela-tion. In Sec. IV, the transit random walk theory from self dynamic response [34] is extendedto the structural decorrelation. The agreement of theory with MD is at the remarkable levelof 0 . | F dMD ( q, | , for all t , and all q for which the standard plan applies. Sec. V examinesthe validity of the standard decorrelation plan as function of q .In Sec. VI, we review our main accomplishment: an expression F dV T ( q, t ) that includes the4hysical ideas behind V-T theory. We also summarize the physical nature of the standardplan and the current status of transit modeling, and identify the path to making the presenttheory fully a priori .In this work, we frequently use MD results to validate or parametrize our theory. Ourcomputational system represents liquid Na at 395 K, a bit above the melting temperature T m = 371 K. The system is a cube containing N = 500 atoms, with periodic boundaryconditions. The MD time step is 7 fs. The interatomic potential is based on pseudopotentialtheory, [7] and has produced excellent agreement with a wide range of experimental data. [20] II. OUTLINE OF THE THEORY
We study the distinct autocorrelation function, [44] defined by F d ( q, t ) = 1 N X K = L (cid:10) e − i q · ( r K ( t ) − r L (0)) (cid:11) , (1)for a system of atoms K = 1 , ..., N , located at r K ( t ) at time t . The average may be evaluatedanalytically for simple systems (like the vibrational system below) or numerically for a singleMD system, on an equilibrium trajectory, and includes an average over the star of q whenperiodic boundary conditions are used (a star is the set of all wave vectors related by thecubic point group). The function measures static pair configurational correlations in termsof the structure factor S d ( q ) = F d ( q, t > A. Perfect vibrational system
To evaluate Eq. (1) for the vibrational system, we write r K ( t ) = R K + u K ( t ) , (2)where R K is the random valley equilibrium position (structural site) and u K ( t ) is the mo-tional displacement. The vibrational average is then [43] F dvib ( q, t ) = 1 N X K = L e − i q · R KL e − W K ( q ) e − W L ( q ) e h q · u K ( t ) q · u L (0) i vib , (3)5here R KL = R K − R L , W K ( q ) is the Debye-Waller factor, W K ( q ) = 12 (cid:10) ( q · u K ) (cid:11) vib , (4)and the motional time-correlation functions are given by h q · u K ( t ) q · u L (0) i vib = kTM X λ ( q · w Kλ ) ( q · w Lλ ) cos ω λ tω λ . (5)The normal modes are labeled λ = 1 , ..., N −
3, the three zero-frequency modes beingomitted, mode λ has frequency ω λ , and eigenvector λ has Cartesian vector w Kλ at atom K .The right side of Eq. (3) is also averaged over each q star.To calibrate the vibrational Hamiltonian for liquid Na, we quench the Na computationalsystem to a random structure, [21–26] and there evaluate the equilibrium positions andthe vibrational frequencies and eigenvectors. [22–24] These Na parameters are then usedto evaluate the vibrational functions in this paper, and all other vibrational functions forliquid Na. Any one random structure is suitable for this purpose because of the statisticalsimilarity of the random valleys in the thermodynamic limit.Equation (3) for F dvib ( q, t ) makes use of three data sets, the structural positions R K ,the Debye-Waller factors in Eq. (4), and the motional time correlation functions in Eq. (5).These data sets are all strongly coupled inside the sums in Eq. (3), and that equation admitsof no acceptable decoupling approximation. The key to analysis is in the time dependence. F dvib ( q, t ) is subject to natural vibrational decorrelation, or vibrational dephasing, of thecos ω λ t factors in Eq. (5), as t increases from zero. This process is always present, and hasthe effect of reducing the time correlation functions to zero as t → ∞ . What remains of F dvib ( q, t ) as t → ∞ is therefore given by F dvib ( q, ∞ ) = 1 N X K = L e − i q · R KL e − W K ( q ) e − W L ( q ) . (6)Because the Debye-Waller factors are positive and increasing with q , Eq. (4), F dvib ( q, ∞ ) → q increases. F dvib ( q, t ) accounts for all correlations in a perfect vibrational system. These correlationscan be classified as motional and structural. We shall follow this intuitive notation, in amodification designed for V-T theory.a) Motional correlation is that contained in the time correlation functions with K = L in Eq. (3). Motional correlation is in the normal mode motion and resides in theeigenvectors, Eq. (5). 6) For vibrational motion in a single random valley, each atom K remains within a smallvolume around its equilibrium position R K . This constraint describes the structuralcorrelation, as the term is used here, and it is contained in Eq. (6). B. Transit contribution to the structure factor
Fig. 1 compares the distinct functions F dvib ( q,
0) and F dMD ( q, q in our system. Fluctuations are larger in the vibrationalcurve because it is calculated for a single random valley while the MD data averages overrandom valleys. The difference A ( q ) = F dMD ( q, − F dvib ( q,
0) (7)is formally identified in V-T theory as the transit contribution. -101230 1 2 3 q (1/ a ) F d ( q , ) a) FIG. 1. Circles are F dvib ( q,
0) at allowed q (large circles identify the test q from Table I) and theline is F dMD ( q, As the figure shows, the vibrational contribution alone produces the correct peak struc-ture, and it correctly locates the first peak. It also gets the width of the peak correct andprovides about two thirds of its height. In fact, the transit contribution A ( q ) is a smallcorrection everywhere except the first peak tip, where it is large and negative. (We founda similar situation with the Brillouin peak dispersion curve [33]: the vibrational part alone7ets the location of the peak in S ( q, ω ) but not its value.) While we do not yet have amicroscopic theory of how transits affect initial correlations, the fact that A ( q ) is large onlyat the nearest neighbor distance suggests that it is structural in the sense defined above andis thus contained in F dvib ( q, ∞ ). Accordingly we write A ( q ) = C ( q ) F dvib ( q, ∞ ) , (8)where C ( q ) has no dependence on { R K } .The formation of excess correlation by transits is one matter, but the transit-inducedprocess that cause those correlations to decay is another. We do have a microscopic theoryfor those processes, and we begin to lay the groundwork for that theory in the next section. TABLE I. MD and vibrational data for nine q chosen as test cases in the present study. Allfunctions are defined in text and equations. q ( a − ) F dMD ( q, F dvib ( q, A ( q ) F dvib ( q, ∞ ) t c ( q ) (ps) s ( q, t c )( a ) B ( q ) q -regime0.29711 -0.9730 -0.9747 0.0017 -0.9388 0.2937 0.761 1.286 Brillouin peak0.70726 -0.8943 -0.8878 -0.0065 -0.7118 0.2595 0.774 1.265 Brillouin peak1.01482 0.9423 0.8787 0.0636 0.4921 0.3547 0.658 1.111 first peak1.09165 1.6937 2.1415 -0.4478 1.6819 0.3336 0.613 1.405 first peak1.10505 1.5504 2.6187 -1.0683 2.1902 0.3222 0.528 1.436 first peak1.14429 1.0201 0.9820 0.0381 0.6619 0.2787 0.844 1.564 first peak1.50523 -0.3540 -0.3929 0.0389 -0.1791 0.2790 0.583 1.134 large q q q C. Standard decorrelation plan
We chose a representative set of q values to use in developing the present theory. These q are listed in Table I, along with the theoretically important functions for each q . The nine q in Table I are among the 17 q for which self-decorrelation calculations were done. [34]8he vibrational density autocorrelation function, normalized to MD data at t = 0, is F dvib ( q, t ) + A ( q ). Our object is to construct a theory for the transit-induced decorrelationof this function.Fig. 2 shows curves of F dMD ( q, t ) and F dvib ( q, t ) + A ( q ) for q = 1 .
09, at the tip of the firstpeak. The curves agree at t = 0, but the MD curve damps faster, and falls increasingly belowthe vibrational curve as t increases. Recall the vibrational curve contains natural vibrationaldecorrelation, which makes the curve converge eventually to the constant F dvib ( q, ∞ ) + A ( q ).Through the convergence process, from around 0 . t → ∞ limit. The featureis due to very-slowly-damped lowest frequency normal modes. The same vibrational excessis present in all the F dvib ( q, t ) curves, and in the curves of the self autocorrelation function F svib ( q, t ) as well. Its appearance in F svib ( q, t ) was noted previously. [34] Exceptionally, wenote that the vibrational excess does not appear at sufficiently large q , in F dvib ( q, t ) nor in F svib ( q, t ), because the functions converge to zero before the excess develops. F d ( q , t ) t (ps) q = 1.09 a t c FIG. 2. Circles are F dMD ( q, t ), dashed line is F dvib ( q, t )+ A ( q ), and solid line is F dvib ( q, ∞ )+ A ( q ). t c isthe time when MD crosses the horizontal solid line. The vibrational excess is F dvib ( q, t ) − F dvib ( q, ∞ )at t ≥ t c . In fact, the vibrational excess cannot be present in the liquid state, because the long-timenormal-mode correlation cannot survive in the presence of transits. The vibrational excessmust therefore be damped out of F dvib ( q, t ) by transit-induced motional decorrelation. Thesituation is shown graphically in Fig. 3. The constant F dvib ( q, ∞ ) + A ( q ) is the value of thenormalized vibrational autocorrelation function with all motional correlation damped and9 " %&’! ! ( " )*+ %&’ ! (,-%&(" %&’.( " )*+ %&’!(,-%&( " )*+ %&’!/(,-%&(" )*+ %&’.(,-%&(" %&’!( . !/ !" !" !"!" !"! !" FIG. 3. The same curves that are plotted in Fig. 2 during the motional decorrelation period( t ≤ t c ). The decrease in the vibrational curve is due to vibrational dephasing. Transit-induceddecorrelation is supposed to damp the vibrational curve to the MD curve. all structural correlation present. The figure suggests, and we shall adopt, a simplifyingapproximation for the complete decorrelation process: make the transit-induced motionaland structural decorrelations sequential, the first ending and the second beginning at thesame (crossover) point. Then with the crossover time denoted t c ( q ), the theory must satisfy F dV T ( q, t c ) = F dvib ( q, ∞ ) + A ( q ) . (9)As for t c ( q ) itself, we shall calibrate it with the help of MD data by F dMD ( q, t c ) = F dvib ( q, ∞ ) + A ( q ) , (10)as shown in Fig. 3. The standard decorrelation plan is then described as follows. • Introduce transit-induced motional decorrelation to damp the vibrational curve towardthe MD curve on 0 ≤ t ≤ t c ( q ). • Introduce transit-induced structural decorrelation to damp the line ( F dvib ( q, ∞ ) + A ( q ))toward the MD curve on t ≥ t c ( q ).In the process of this study we shall find those q for which the standard plan applies, andshall summarize the results in Sec. V. While such a qualitative conclusion can be drawn withsome reliability, quantitative behavior differs with q . In this work, one should be mindfulthat every q measures a different correlation.10 II. THEORY FOR TRANSIT-INDUCED MOTIONAL DECORRELATIONA. Transit drift
For the present construction, we start with a perfect vibrational system, for which theequilibrium trajectory has perfect vibrational configurations. Vibrational dephasing operatesin F dvib ( q, t ), and transits will contribute additional damping of the motional correlations.The decorrelation must begin from zero at any time chosen for the start of the calculation.This condition is satisfied by the vibrational dephasing, but it must be made an initialcondition for the transit-induced decorrelation. x time transit δ R x a) ! " !!" ) " (cid:1) ! *+ FIG. 4. Representation of Cartesian coordinates for a transiting atom. a) shows x vs t for one(any) of a local group of atoms that move together in a single transit (from Ref. 29). b) is a planarmodel of the motion in a).
While the transits observed in MD calculations occur in correlated groups, [29] the presentstatistical mechanics theory, Eq. (1), requires only the separate motion of one atom at atime. In the MD transits, the single atom motion shows a simple uniform behavior in graphsof the atomic Cartesian coordinates as functions of time. A schematic representation of onesuch graph is shown in Fig. 4a. A model representation of the same motion in the x − y plane is shown in Fig. 4b. The motion in 3-d is minimally described as follows.a) Before transit, the atom is in motion on the vibrational surface, approximately asphere, about equilibrium position R .b) After transit, the atom is in motion on the vibrational surface, approximately the same11phere, about equilibrium position R + δ R .c) The transit itself is merely the crossing of the boundary between two potential energyvalleys, and is essentially instantaneous.Because the transit is instantaneous, the complete motion as described is consistent withevery equilibrium configuration being a perfect vibrational configuration.Two transit parameters, previously calibrated for liquid Na at 395 K, are needed. δR isthe mean change in the equilibrium position of one atom in one transit, and is evaluatedfrom the transits observed in MD. [29] ν is the mean transit rate for one atom, and isevaluated by fitting a transit random walk to MD data for the self diffusion coefficient. [34]The results, in the form needed here, are12 δR = 0 . a ,ν − = 0 .
26 ps , (11)where ν − is the mean period between successive transits for one atom. For comparison, thenearest-neighbor distance is 7 . a , [23] and the mean vibrational period is 0 .
40 ps. [45]Consider a single atom trajectory containing one transit similar to that depicted in Fig. 4,in three dimensions. We ask for the motion contribution resulting specifically from thetransit. The transit location is the midpoint of δ R , and for all transits with a given δ R the collective motion has cylindrical symmetry about δ R . Hence the mean transit-inducedmotion, s ( t ), is along δ R , from the transit location, to the ultimate mean position δ R away.Consider now the entire system. The motions s ( t ) are proceeding throughout the systemat a constant rate. For each spatial direction, the s ( t ) collectively produce a steady statemotion characterized in the mean per atom by a constant velocity. At any time, thesemotions are uniformly distributed over angles. The total motion, measured per atom, isreferred to as the transit drift.The last theoretical issue is the state of the transit drift at the endpoint t c of the motionaldecorrelation period. We expect a single transit per atom, operating along with the naturaldephasing, to damp the motional correlation to zero. This takes a time ν − , hence provides atheoretical prediction for t c ( q ). The drift magnitude s ( t c ) achieved in time t c is close to δR .12he theory therefore makes three qualitative predictions about the microscopic process: t c ( q ) ≈ ν − ,s ( t c ) ≈ δR, (12) s ( t ) ∝ t. For the theory of transit-induced motional decorrelation, we write F dV T ( q, t ) = F dvtr ( q, t ) + A ( q ) , ≤ t ≤ t c ( q ) , (13)where F dvtr ( q, t ) is F dvib ( q, t ), Eq. (3), modified to include the transit-induced motion: F dvtr ( q, t ) = 1 N X K = L e − i q · R KL (cid:10)(cid:10) e − i q · ( u K ( t )+ s K ( t ) − u L (0)) (cid:11) tr (cid:11) vib , (14)where subscript tr denotes a transit property. Here the displacements u K ( t ) and u L (0)express normal-mode motion, while s K ( t ) represents the additional motion due to transits.We now decouple the s K ( t ), by evaluating them as the steady-state average s ( t ), with theinitial condition s (0) = 0. Then the factor e − i q · s ( t ) can be separately averaged over anglesand removed from the sums. The remaining factor is just F dvib ( q, t ), so that F dvtr ( q, t ) = F dvib ( q, t ) χ dtr ( q, t ) , (15)where the decorrelation factor (damping factor) is χ dtr ( q, t ) = sin qs ( t ) qs ( t ) , ≤ t ≤ t c ( q ) . (16)From Eqs. (15) and (16), the two motional decorrelation processes, natural and transit-induced, operate independently, and they satisfy the condition of zero decorrelation at t = 0.From Eqs. (13) and (15), the theory with motional decorrelation is F dV T ( q, t ) = F dvib ( q, t ) χ dtr ( q, t ) + A ( q ) , ≤ t ≤ t c ( q ) . (17)We have χ dtr ( q,
0) = 1, and the endpoint magnitude is calibrated by comparing Eq. (17) withEq. (9), to find χ dtr ( q, t c ) = F dvib ( q, ∞ ) F dvib ( q, t c ) . (18)Then with Eq. (16), χ dtr ( q, t c ) can be solved for s ( t c ), and the third line in Eq. (12) prescribes s ( t ) = tt c s ( t c ) . (19)13 . Comparison of theory with MD Comparison of F dV T ( q, t ) with F dMD ( q, t ) is shown in Fig. 5a for q = 1 .
09 and 0 ≤ t ≤ t c .On the intended theoretical range, 0 ≤ t ≤ t c , the agreement of V-T with MD is excellent.The time extension shows the theory is accurate to well beyond t c , revealing the possibilityof an improved theory in which the crossover is dispersed over a time interval. The samegraphs for q = 1 . , .
11 and 1 .
14, all on the first peak, are similar to Fig. 5a, with maximumV-T error up to 0 .
012 on 0 ≤ t ≤ t c . q = 1.09 a t/t c F d ( q , t ) a) -0.98-0.94-0.90 0 1 2 q = 0.30 a t/t c F d ( q , t ) b) -0.90-0.70-0.50 0 1 2 q = 0.71 a t/t c F d ( q , t ) c) FIG. 5. For F d ( q, t ): Comparison of V-T theory (solid line) with MD (circles) on 0 ≤ t ≤ t c .Theory is intended to apply on 0 ≤ t ≤ t c . a) is representative of all q in the first peak regime; b)and c) are characteristic of the Brillouin peak regime. Comparison of F dV T ( q, t ) with F dMD ( q, t ) is shown for q = 0 .
30 and 0 .
71 in Figs. 5b and5c, respectively. For q = 0 .
30 on 0 ≤ t ≤ t c , the maximum V-T error is 0 . q = 0 . ≤ t ≤ t c , the maximum V-T error is 0 . t > t c , for q = 0 .
30 and 0 .
71, the theoryprovides some transit-induced damping of the Brillouin peak oscillation. However, as thefigures show, the damping is still insufficient at t > t c .14 V. THEORY FOR TRANSIT-INDUCED STRUCTURAL DECORRELATIONA. Transit random walk
In the period 0 ≤ t ≤ t c , each atom undergoes one instantaneous transit, the “firsttransit”, and the subsequent motion specifically due to the transit is averaged to the steady-state transit drift. The drift has time and distance scales matching the normal mode motion,hence is efficient in decorrelating this motion, and does so until the decorrelation is completeat t c . At the same time, the drift has no effect on structural correlation, because it does notnoticeably alter the structure.At t c ≤ t ≤ t c , each atom undergoes its second transit after t = 0. Here and at alllater times, the mean transit effect is still the drift, which persists as the same steady flow,while the motional correlation remains zero. But the net drift of each atom accumulates,surpassing the time and distance scales of the normal mode motion, at which point the driftconstitutes motion of the atomic equilibrium positions. This motion will damp the systemstructural correlation.The analysis now is specifically for t ≥ t c . When structural decorrelation begins, at t c ,Eqs. (8) and (9) combine to yield F dV T ( q, t c ) = [1 + C ( q )] F dvib ( q, ∞ ) . (20)The structural information is explicit in the set { R K } contained in F dvib ( q, ∞ ). When theequilibrium positions begin to move, R K becomes R K ( t ), with R K ( t c ) = R K (0). Thetheoretical time dependence is then F dV T ( q, t ) = [1 + C ( q )] 1 N X K = L (cid:10) e − i q · ( R K ( t ) − R L (0)) (cid:11) tr × e − W K ( q ) e − W L ( q ) . (21)Denote the average h· · · i tr by D KL ( t ), suppressing the q dependence. Consider a timeinterval δt so small that atom K is very unlikely to transit more than once in δt . Then in δt , D KL ( t ) changes by δD KL ( t ) = (cid:10)(cid:2) e − i q · R K ( t + δt ) − e − i q · R K ( t ) (cid:3) e i q · R L (0) (cid:11) tr . (22)15n δt , atom K does not transit, or atom K transits once with probability νδt . If atom K does not transit, R K ( t + δt ) = R K ( t ). If atom K does transit, R K ( t + δt ) = R K ( t ) + δ R K ,where δ R K has no time dependence. Then Eq. (22) becomes δD KL ( t ) = (cid:10) [ e − i q · δ R K − e − i q · ( R K ( t ) − R L (0)) (cid:11) tr νδt. (23)In a first approximation we can assume | δ R K | = δR , the same for all transits, while thedirection of δ R K is uniformly distributed and uncorrelated with R K ( t ) − R L (0) in Eq. (23).In that case, the square bracket in Eq. (23) can be separately averaged over angles of δ R togive δD KL ( t ) δt ≈ − γ ( q ) D KL ( t ) , (24)where γ ( q ) = ν (cid:20) − sin qδRqδR (cid:21) . (25)The above derivation was presented in our self decorrelation study, [34] where γ ( q ) is foundto provide accurate structural damping. Here, however, Eq. (24) contains an error due to theneglect of directional correlation in Eq. (23) between the vectors δ R K and R K ( t ) − R L (0).This correlation is especially strong for the first transit and for atoms K, L being nearestneighbors at t = 0. The result is an anisotropic contribution to the random walk. Thecorresponding damping coefficient is G ( q ), which we write in the form G ( q ) = γ ( q ) + δG ( q ) , (26)where δG ( q ) represents the anisotropic contribution to the random walk decorrelation. ThenEq. (24) becomes δD KL ( t ) δt = − G ( q ) D KL ( t ) . (27)The equation integrates to D KL ( t ) = e − G ( q )( t − t c ) D KL ( t c ) . (28)Putting this into Eq. (21) gives the structural decorrelation theory, F dV T ( q, t ) = [1 + C ( q )] F dvib ( q, ∞ ) e − G ( q )( t − t c ( q )) , t ≥ t c ( q ) . (29)Equation (17) for t ≤ t c and Eq. (29) for t ≥ t c form our complete theory for the timeevolution of F d ( q, t ). 16 . Comparison of theory with MD Here we shall compare the full theory with MD, for t ≥
0. The functions needed tocalibrate F dV T ( q, t ) are listed in Tables I and II. G ( q ) is determined by fitting a straight lineto log F dMD ( q, t ) vs t on t > t c ( q ), where Eq. (29) is supposed to be valid. γ ( q ) is from Table Iof Ref. 34. For all test q for which the standard plan applies, the deviations ∆ F d ( q, t ) aregraphed in Fig. 6, where ∆ F d ( q, t ) = F dV T ( q, t ) − F dMD ( q, t ) . (30) -0.03-0.02-0.010.000.010.02 0 0.2 0.4 0.6 0.8 1 ∆ F d ( q , t ) t/t c a) t/t c ∆ F d ( q , t ) b) FIG. 6. (Color online) Deviation of V-T theory from MD, Eq. (30), vs t/t c , for the seven q wherethe standard plan is valid, see Table II. a) motional decorrelation, b) structural decorrelation, andthe dotted portions show the MD long time tail (see Fig. 9). Consider the first peak regime. The challenge here is the function A ( q ), which is verylarge (and negative) at the tip of the first peak, q = 1 .
09 and 1 .
11, while it has the characterof a small systematic correction at all other q , including the two additional q in the firstpeak regime (see Table I, also Fig. 1). The significant result is that the theory accountsuniformly well for the entire decorrelation process for all four test q in the first peak regime.The example of q = 1 .
09 is shown in Fig. 7. For the other three q , the overall shape issimilar to Fig. 7, and the comparison of theory with MD is qualitatively the same (Fig. 6).Two details are specific to the first peak regime. First, δG/γ is small and negative at thetip of the first peak ( q = 1 .
09 and 1 . δG/γ is small and positive on the leading edge( q = 1 . δG/γ is negligible on the trailing edge ( q = 1 . .01.02.0 0 4 8 12 q = 1.09 a t/t c F d ( q , t ) FIG. 7. For F d ( q, t ): Comparison of the complete V-T theory (line) with MD (circles), for a q representative of the first peak regime. Vertical line is at t c .TABLE II. Data required for the structural decorrelation, at t ≥ t c ( q ). γ ( q ) is from Table I ofRef. 34. q ( a − ) 1 + C ( q ) γ ( q )(ps − ) δG ( q ) /γ ( q )0.29711 0.9982 0.1733 -0.009 a a a a a In applying the theory, we set δG/γ = 0 for this q . will be discussed in Sec. VI. Second, while the slope of theory is not controlled at t c , so thata slope discontinuity is to be expected, there is no significant discontinuity in the first peakregime (Fig. 7). Note the time scale t/t c is an approximate count of the number of transitsper atom from t = 0.Success of the standard plan in the first peak regime encourages application of the sameprocedure to the Brillouin peak regime. The challenge here is the Brillouin peak oscillation.The oscillation itself is present in both F dMD ( q, t ) and F dvib ( q, t ), but it requires an additionaldamping in the vibrational function to agree with the MD function at t > t c . However,18n the standard plan, the oscillation is completely removed in F dV T ( q, t ), via Eq. (29), sothe entire oscillation in F dMD ( q, t ) appears as an error in F dV T ( q, t ) at t > t c . This is nota problem for the standard plan, since the oscillation amplitude is generally less than ourtheoretical error.Comparison of theory with MD in the Brillouin peak regime is shown in Fig. 8a and 8bfor q = 0 .
30 and 0 .
71, respectively. In spite of missing the oscillation, the standard planremains accurate to . .
01 in the Brillouin peak regime (Fig. 6b). It is significant that wecan set δG ( q ) = 0 for both test q in the Brillouin peak regime (see Table II). -1.2-0.8-0.4 0 10 20 q = 0.30 a F d ( q , t ) a) t/t c -1.00-0.90-0.800 1 2 3 4 -0.8-0.40.00 10 20 q = 0.71 a b) t/t c F d ( q , t ) -0.90-0.70-0.50-0.300 1 2 3 4 FIG. 8. For F d ( q, t ): Comparison of the complete V-T theory (line) with MD (circles), for two q in the Brillouin peak regime. Inset shows crossover and the Brillouin peak oscillation. The F MD ( q, t ) data exhibit a long time tail at q = 1 .
01 and 1 .
11, shown in Fig. 9 at q = 1 .
11 (see also Fig. 6b). We are investigating the source of this feature. Fig. 9 also showsthe good agreement of theory with MD as the curves approach zero, before the tail begins. q = 1.11 a t/t c F d ( q , t ) FIG. 9. For F d ( q, t ): Long-time data showing the MD tail at t/t c > In the large q regime, at q beyond the first peak, three q were chosen for study, located19espectively at the first minimum, second maximum, and second minimum of F dMD ( q,
0) (seeFig. 1 and Table I). At q = 1 .
51, shown in Fig. 10, it is perhaps surprising that the standardplan still works well. Both motional and structural decorrelation are accurately accountedfor, the theoretical slope discontinuity at t c is insignificant, and we are able to set δG ( q ) = 0in the structural damping. The deviation ∆ F d ( q, t ) in Fig. 6 is well below 0 . -0.4-0.20.00 2 4 6 q = 1.51 a t/t c F d ( q , t ) FIG. 10. For F d ( q, t ): Comparison of the complete V-T theory (line) with MD (circles), for q = 1 . a − , at the minimum after the first peak. Vertical line is at t c . As q increases further, two important changes occur. First, F dvib ( q, ∞ ) goes to zero,as noted following Eq. (6), and as seen in the last three entries of Table I. Second, thedecorrelation process becomes faster, with t c ( q ) falling below the first-transit period of ν − ,again as seen in Table I. These changes mark the ultimate convergence of the distinctautocorrelation function toward zero as q increases. Both trends are also found in the selfautocorrelation function as q increases toward the free particle limit. [34] As a result, at q = 2 .
00 and 2 .
51, the total process resembles a single motional decorrelation of an unusualshape. The indication is that q & .
00 does not measure structural correlation, and measuresmotional correlation in a way different from what we have seen at smaller q . V. RANGE OF APPLICABILITY OF THE STANDARD PLANA. Exclusion band
There are several q -segments where F dMD ( q,
0) and/or F dvib ( q,
0) are of small magnitude,say . .
1, where analysis is difficult for two reasons. First, the values of the MD andvibrational autocorrelation functions are small enough to be compromised by finite- N error,at our present N = 500. Second, the MD and vibrational functions cross zero at different20 -values, a phase shift not treated in the standard plan. It makes sense to leave these effectsuntil the dominant part of the theory is developed. B. First-peak regime
This is where F dMD ( q,
0) and F dvib ( q,
0) are positive in the first peak. This regime providedthe test case from which the standard plan was developed. Here the F dMD ( q, t ) curves haveuniform character across the q -range, and so do the F dvib ( q, t ) curves. Since distinct and selfcontributions are both positive, no cancellation complications arise. C. Brillouin-peak regime
This is where the Brillouin peak oscillation is apparent in F dMD ( q, t ); in our system it runsfrom q = 0 .
13, the smallest allowed q , to somewhat beyond q = 0 .
80. Here, F dMD ( q,
0) and F dvib ( q,
0) are around −
1; on this scale the standard plan is accurate, as shown in Fig. 6, andmay be applied.In this regime, the self and distinct autocorrelation functions nearly cancel for MD data,and also for vibrational data. The Brillouin peak oscillation appears entirely in the distinctfunction, and contains important physics not addressed in the standard plan, namely theinelastic scattering cross section as function of q and ω . For our system, the Brillouin peakoscillation appears in F dvib ( q, t ) with the same frequency as it appears in F dMD ( q, t ), for all q in the regime. Hence F dvib ( q, t ) already contains the Brillouin peak dispersion curve, a priori and to very high accuracy. [33] What is still required for a complete theory of dynamicresponse is an extremely accurate decorrelation theory for the Brillouin peak oscillation. D. Large- q regime Beyond the first peak, near the first and second minima and the second maximum, where | F dMD ( q, | is large enough to justify an exploratory investigation, the character of thedecorrelation process changes under two influences: The motional decorrelation becomesfaster, and F dvib ( q, ∞ ) goes to zero (see Table I). At q = 1 .
51, the standard plan still appliesand is still accurate, but a practiced eye will see the beginning of change. At q = 2 .
00 and21 .
51, the entire process has the appearance of a single motional decorrelation, an appearancewe expect to remain as q increases further, and the MD and vibrational autocorrelationfunctions converge to zero (see the discussion at the end of Sec. V). The standard plan doesnot apply at these q . VI. SUMMARY AND CONCLUSION
This work is an investigation into the validity of V-T theory for the distinct density au-tocorrelation function of a monatomic liquid. Our main result is Eqs. (17) and (29), whichtogether accurately model F d ( q, t ) over the broad range of q for which the standard decorre-lation plan applies. That result depends on a theoretical notion, the transit drift introducedin Sec. IIIA, which is the basis of our theories of motional and structural decorrelation in F d ( q, t ). In this study, the vibrational contribution is parametrized by appeal to a realisticmany-body potential for Na, and the various transit parameters are determined from MDsimulations using that potential, but we emphasize that this is not necessary: once oneunderstands the physical meaning behind the parameters, their values may be acquired bythe method of one’s choice. With that in mind, we summarize the physical considerationsthat led to the model and give the parameters their meaning.At the outset, the two classes of configurational correlation, motional and structural, areprecisely defined by the perfect vibrational system. The motional correlation is measured bythe time correlation functions in Eq. (5), and the motional decorrelation (due to vibrationsand transits) carries these functions from their t = 0 values to zero. In the process, F dvib ( q, t )goes from F dvib ( q,
0) to F dvib ( q, ∞ ) according to Eqs. (3) and (6). The structural correlation ismeasured by the set of equilibrium positions { R K } , and the Debye-Waller factors { W K ( q ) } ,in Eq. (6) for F dvib ( q, ∞ ). The R K remain fixed during the motional damping, then the R K begin to move and proceed to damp F dvib ( q, ∞ ) to zero. This outline of the process isdefined entirely in terms of the theoretical function F dvib ( q, t ), the definition applying to allmonatomic liquids.The three functions A ( q ), t c ( q ), and G ( q ) that enable calculation of the detailed timedependence are transit properties. In physical meaning, A ( q ) is the transit contribution tothe static structure factor. A ( q ) is supposed to be mainly structural, hence is modeled as amultiple of F dvib ( q, ∞ ). The crossover time t c ( q ) is the time required to damp the motional22orrelation to zero, under the simultaneous actions of natural dephasing and transit-induceddecorrelation. The physical key to t c ( q ) is the condition that long-time motional correlationdue to low-lying normal modes is damped out by transits in the liquid. Finally, the structuraldamping coefficient G ( q ) expresses the transit random walk, and an a priori zeroth-ordertheory for G ( q ) is already in place. [34] All these properties are attributed to the same transitmotion in every monatomic liquid.The transit drift is the massively averaged motion resulting specifically from transitsin the liquid state, and present in addition to the normal mode motion. In Sec. III, inconstructing a model equation for the drift-induced motional decorrelation, in the period0 ≤ t ≤ t c ( q ), three predictions are made about characteristic properties of the drift. Thepredictions are expressed in terms of the transit parameters ν and δR , previously calibratedindependently of any study of F ( q, t ), with values given in Eq. (11). The predictions arelisted in Eq. (12).a) The first prediction is t c ( q ) ≈ ν − . This is qualitatively correct. Specifically, ν − =0 .
26 ps, while from Table I the average t c ( q ) for seven q in the standard plan is0 . ± .
10 ps.b) The second prediction is s ( q, t c ) ≈ δR . This is qualitatively correct. Specifically, δR = 0 . a , while from Table I the average s ( q, t c ) for seven q in the standard planis 0 . ± . a .c) The third prediction is s ( q, t ) ∝ t for 0 ≤ t ≤ t c ( q ). The linear t dependence is usedin evaluating the motional decorrelation, Eq. (19), and agreement of theory with MDin Fig. 5 verifies the t dependence within the observed errors.The level of agreement is excellent for nonequilibrium data; further, the systematic characterof the V-T theory errors shown in Fig. 6a indicates that the motional decorrelation theoryof Eqs. (16) and (17) can still be improved.The transit drift is a steady state motion, hence is always present in the liquid. However,in order to address the structural decorrelation at t > t c , it is necessary to resolve thedrift into its effective motion of the equilibrium positions. This resolution yields the transitrandom walk. Indeed, the time required after t = 0 for the random walk to come intoeffect provides the physical explanation for the delay to t c ( q ) of the onset of structuraldecorrelation, as it appears in Eq. (29). 23imple exponential decay, beginning after a delay period, is a hallmark of MD data for selfand distinct density autocorrelation functions. In self decorrelation, the theoretical dampingcoefficient γ ( q ) is in excellent agreement with MD data for all q . The result constitutes aunification within V-T theory: One cannot calculate the density autocorrelation dampingcoefficient γ ( q ) from the self diffusion coefficient D , [35, 46] but one can calculate both γ ( q )and D from the same random walk of the equilibrium positions.Finally, we shall make a highly approximate model for the anistropic damping coefficient δG ( q ). Consider a representative pair of atoms K, L which are nearest neighbors at t = 0.Because the atomic dynamics enforces a minimum distance between atoms, the first-transitdrift of atom K will not be isotropically distributed about R K (0), but will move preferentiallyaway from R L (0). The transit random walk will not develop about R K (0), but about R K (0) + δ s K , where δ s K is a small displacement in the direction of R KL (0). The picturecan be developed from Eq. (14). The resulting δG ( q ) will be positive for q on the first-peakleading edge, and negative at the first-peak tip. This is what we see in Sec. IV. Moreover,since the effect appears only for nearest neighbors, δG/γ will be significant only for q in thefirst peak regime. Even for q in the first peak, δG/γ is small because the correlation is weakafter the first transit. All these details of δG/γ are confirmed by the last column of Table II.The theoretical challenge now is a more quantitative understanding of δG/γ .Our expressions for F d ( q, t ) rely on the vibrational parameters { R K , ω λ , w Kλ } and threetransit functions A ( q ) , t c ( q ), and G ( q ). In this study, the vibrational parameters were de-termined from a single random valley of the Na pseudopotential, and the transit functionswere determined from combined MD and vibrational data. This is done in order to showthe extreme accuracy of the calibrated theory; we believe this high-accuracy capability givescredence to the formulation. On the other hand, the vibrational parameters can be deter-mined from first principles: examine a random valley on a potential surface calculated fromDFT. The remaining functions are determined by transit dynamics and can in principle becalculated from a transit model. (The current transit model already provides reasonablyaccurate estimates ν − for t c ( q ), and γ ( q ) for G ( q ).) An improved transit model, in addi-tion to providing better estimates for t c ( q ) and G ( q ), would ideally (a) describe the transitdecorrelation processes that dominate for q outside the standard plan and (b) explain the24rigin of the initial transit-based correlations, given by A ( q ). [1] D. Pines, The Many-Body Problem (Addison-Wesley, 1997).[2] D. Pines,
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