- Email: [email protected]
Potential effects on instantaneous normal modes of liquids
Physica A 254 (1998) 257–271
Potential e ects on instantaneous normal modes of liquids a
b
Ten-Ming Wu a; ∗ , Wen-Jong Ma b , Shiow-Fon Tsay b
Institute of Physics, National Chiao-Tung University, Hsin Chu 300, Taiwan Department of Physics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan
Abstract We calculated the instantaneous-normal-mode (INM) densities of states for liquid Na, LennardJones (LJ) liquid and truncated LJ liquid with the same reduced density and temperature by both molecular-dynamics simulation and the vector-random-walk approach under two-body approximation. By comparing the calculated results for these three liquids, we give some physical insights in the relationship between the INM density of state and the two-particle interaction potential. The e ects of the short-range repulsive core, the long-range tail in the potential and the presence c 1998 Elsevier Science B.V. All rights reserved of attractive well are elucidated. PACS: 63.50.+x; 61.25.Bi; 61.25.Mv Keywords: Instantaneous normal modes; Pair potential; Random walk
1. Introduction The dynamics of liquids is much more complicated than that of crystalline solids, for the structure in a liquid state is highly disordered and varies with time. In general, the dynamics of a many-body system can be represented as the motion of a point in con guration space over the total potential energy surface, which composes of many local minima, barrier tops and saddle points [1,2]. The dynamics of crystalline solids are just small vibrations around the global minimum of the potential energy surface and well described by phonon modes. However, the dynamics of liquids may go around the whole con guration space via a series of barrier crossings. Recently, a phonon-like approach, which is named as instantaneous-normal-mode (INM) theory [3–9], has been proposed to analyze rigorously the short-time dynamics of liquids and gained a considerable interest. In the INM theory, the total potential energy surface around each con guration is expanded up to the quadratic terms of ∗
Corresponding author. Fax: (886) 03-5720728; e-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 0 2 9 - 6
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particle displacements and the square root of each eigenvalue of the force constant matrix, or the Hessian matrix, gives an INM frequency, which measures the local curvature of the potential surface along the direction of that eigenvector. After an ensemble average with respect to all con gurations, the INM density of states, which is analogous to the phonon density of states of a crystalline solid, is obtained. However, the major di erence between these two kinds of density of states is that the INM density of states has two lobes, one for the real-frequency INMs and the other for the imaginary-frequency INMs, which characterize the upward and downward local curvatures, respectively. With the INM density of states, many dynamic quantities of liquids are formulated and the formula are accurate within a period time from initial [10,11]. More recently, the INM approach has been extended to analyze the dynamics of glassy states and gives some interesting results [12,13]. Certainly, the INM density of states has a strong relation with the pair potential between two particles in a liquid both directly through the Hessian matrix and indirectly through the microstructures of liquid con gurations [14]. Although many predictions on liquid dynamics by the INM theory have been given, the relationship between the INM density of states and the pair potential is still not so obvious. Simple liquids are still the most proper physical systems for studying this relation. So far, for simple liquids, the INM density of states for the Lennard-Jones (LJ) systems [ 3 –9], the soft-potential systems [12,13] and metallic liquids [14,15] have been reported; however, there is no systematic comparison between them. In this paper, we calculate and compare the INM density of states for three di erent kinds of liquid, in which the pair potentials between two particles are the pseudopotential of a metallic liquid, the LJ potential and the truncated LJ (TLJ) potential, which is only the repulsive portion of the LJ potential [16,17]. For comparison among these three potentials, each potential is scaled with the depth of the potential well, , which corresponds to the minimum of the potential, and the distance in each liquid is scaled with the collision length, , which is the smallest separation between two particles with the pair potential to be zero. In order to avoid the di erences due to thermodynamic states, we choose the same reduced density, ∗ = 3 , and reduced temperature, T ∗ = kB T=, for these three kinds of liquid. So far, almost all of works for calculating the INM density of states are based on computer simulation to generate liquid con gurations and then numerical diagonalizations for the Hessian matrix. On the other hand, it is well known that there is an isomorphism between the calculation of the phonon density of states of a crystalline solid and the calculation of the transport properties of a random walker on that structure. Recently, to calculating analytically the INM density of states, Wu and Loring have developed a vector-random-work (VRW) approach [18], which is a generalization of the theory of Gochanour, Andersen and Fayer (GAF) for the transport theory of a random walker in a random medium [19]. In GAF theory, the particles have no internal state and the transfer rate of the random walker from site to site is a scalar number. Due to the vector nature of the displacements of particles in d dimensions, the particles in the random walk problem isomorphic for the INM density of states have d internal states and the transfer rate for the random walker from site to site is
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a d-dimensional matrix. With a two-body approximation for the self energy of the Green’s function in the VRW problem, the INM density of states is obtained by solving a set of self-consistent equations. For the LJ liquid with reduced density about 0:75, the INM density of states calculated with the VRW approach has been proven to be a good agreement with the computer simulation results. Later on, Wan and Stratt showed that the VRW approach in the two-body approximation is equivalent to a mean- eld theory for a renormalized pair potential [20]. In this paper, we use both the method of molecular-dynamics (MD) simulation and the VRW approach to calculate the INM densities of states of the three kinds of liquid mentioned above and analyze the results. In Section 2, we give the three pair potentials and the details about MD simulation for generating the liquid con gurations. In Section 3, we present the INM densities of state of these three liquids obtained through the simulation. The VRW approach for calculating the INM density of states is summarized in Section 4 and the calculated results for these three liquids are also given. In Section 5, we give the conclusion of this paper. 2. Potential and MD simulation In this paper, we assume that the total potential of a liquid system with N particles is a pairwise summation of a pair potential (r). How the pair potential of liquid Na is obtained is described in Ref. [14]. One example of the pair potential for liquid Na is shown in Fig. 1, in which the potential and the distance are scaled with and , respectively. A typical feature of the pair potential for liquid Na is the long oscillating tail which is known as the Friedel oscillation [21]. With the same and , the LJ potential is 12 6 : (1) − LJ (r) = 4 r r By cutting o the attractive part of the LJ potential and then lifting the truncated potential by , we obtain the TLJ potential ( LJ (r) + for r¡21=6 ; TLJ (r) = (2) 0 for r¿21=6 : So, the TLJ potential is purely repulsive. The LJ and the TLJ potentials are also shown in Fig. 1; however, a linear term A(r=) + B has been added to the LJ potential for ensuring continuity for both the potential and the force at the cuto distance rc for simulation consideration. From Fig. 1, it is clear that the pair potential for liquid Na has a softer repulsive core than that of the LJ potential. With the periodic boundary condition, we perform MD simulations with velocityVerlet algorithm [22] on three systems of 375 particles having each a mass m, interacting via the three pair potentials, respectively. The cuto distances rc are 3:5 for the LJ liquid and 3:7 for liquid Na. The densities and the temperatures T of the three
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Fig. 1. The pseudopotential of liquid Na (the solid line), the LJ potential (the dashed line) and the TLJ potential (the dotted line). Each potential is scaled with the depth of the potential well, , and the distance is scaled with the collision length, . The insert gure shows these three potentials in the extended region.
systems are chosen to be the same values so that the mean nearest-neighbor separation and the average kinetic energy of these three systems are also of the same values. Thus, any physical quantities which are di erent among the three systems must arise from the pair potential (r). Within the simulation errors, the chosen values of the reduced density and reduced temperature are ∗ = 0:972 and T ∗ = 0:836. For liquid Na, the simulated system corresponds to a realistic thermodynamic state at T = 378 K, which is just above the melting temperature of Na. For the LJ and the TLJ liquids, the simulated systems are in the liquid states near their melting temperatures, respectively. For each simulated system, we collect data of 100 con gurations. The radial distribution functions g(r) of the three simulated systems are shown in Fig 2. The radial distribution functions for the LJ and the TLJ liquids are almost identical. This is consistent with the theory of Chandle, Weeks and Anderson, which concluded with the statement that for high densities (3 & 0:65) the equilibrium structure of a LJ liquid is quantitatively dominated by the repulsive force of the potential [16,17]. In Section 4, we will need a longer range for the pair potential and the radial distribution function of liquid Na. Thus, we perform another MD simulation for liquid Na of 3000 particles in the same thermodynamic and simulation conditions, except for cuto rc = 7:28. The radial distribution function of liquid Na obtained by our simulation has been checked with the experimental data. Due to the softer repulsive core, two particles in liquid Na may be compressed further than in the LJ systems so that the function of g(r) penetrates further into shorter distances and has smaller value in the maximum.
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Fig. 2. The radial distribution functions of the three liquids with the same reduced density and reduced temperature: the solid line corresponds to liquid Na, the dashed line to LJ liquid and the open dots to TLJ liquid.
3. INM density of states For each instantaneous con guration of N particles, the 3N -dimensional Hessian matrix is calculated according to Eqs. (2.3) and (2.4) given in Ref. [18] and then diagonalized to yield normal-mode frequencies. Repeating the procedure for 100 con gurations and nally taking the average, we get a spread INM density spectrum. After the spread spectrum is smoothed with FFT program, the smoothed and normalized INM density of states D(!) is obtained. The D(!) spectra for the three simulated liquids are shown in Fig. 3. We have adopted the plotting convention in which the part for the imaginary-frequency INMs is plotted on the negative frequency axis and the frequency scale is !0 = (=m2 )1=2 . For liquid Na, the real-frequency density spectrum Ds (!) has an extremely short tail in the high-frequency end and a hump near 20!0 . Thus, the shape of Ds (!) for liquid Na is quite di erent from those of the LJ systems. The Ds (!) spectra of the two LJ systems have almost identical high-frequency tails, which gives strong evidence that the high-frequency INMs are solely determined by the repulsive force of the pair potential. As the tail of the LJ potential is cut o , Ds (!) in the low-frequency end decreases. This information suggest that the attractive force of the pair potential plays an important role on INMs in the low-frequency end than on INMs in the high-frequency end. This conclusion is consistent with the case for liquid Na, whose Ds (!) in the low-frequency end has a much larger value.
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Fig. 3. The INM densities of states D(!) of the three liquids: the solid line corresponds to liquid Na, the dashed line to LJ liquid and the dotted line to TLJ liquid. As usual, the real-frequency lobe is plotted along the positive frequency axis and the imaginary-frequency lobe is plotted along the negative frequency axis. Frequencies are scaled by !0 = (=m2 )1=2 .
For the imaginary-frequency lobe, the density spectrum Du (!) of the TLJ liquid has the broadest width and the largest maximum value among the three spectra. The Du (!) spectra of the LJ liquid and liquid Na have almost the same maxima values, but the width of the spectrum of the LJ liquid is broader than that of liquid Na. This information suggest that the imaginary-frequency density spectrum Du (!) has something to do with the interaction range of the pair potential and the shape of the attractive well of the potential. The detail needs further investigation. The three Ds (!) and Du (!) spectra shown in Fig. 3 have the general expression Di (|!|) = Ai |!| exp(−gi (|!|))
for i = s or u ;
(3)
where the linear |!| dependence is understood to be the Jocabian factor and Ai is the slope of Di (!) at zero frequency. The exponent function gi (!) of each lobe is de ned to be the logarithm of Ai |!|=Di (!). For the real-frequency lobe, the exponent functions gs (!) of the three liquids can be well tted with a three-term polynomial in frequency n ls ms ! ! 1 ! s gs (!) = − Bs + Cs ; ms ¿ls ¿ns ; (4) ns !s !s !s
where |!s | is the frequency at which Ds (!) attains its maximum value D s and ns is determined by the formula −1 As |!s | : (5) ns = ln D s
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Table 1 The tting parameters for the INM density spectra of the real-frequency and the imaginary-frequency lobes, respectively. The units of |!i |, D i and Ai (i = s or u) are !0 , !0−1 and !0−2 , respectively Parameter Au |!u | D u nu lu mu As |!s | D s ns Bs × 103 ls Cs × 103 ms
Liquid Na
LJ liquid
TLJ liquid
6:16 × 10−3 4:91 1:99 × 10−2 2:38 1:24 5:72
5:64 × 10−3 5:36 2:03 × 10−2 2:52 1:20 4:35
5:89 × 10−3 6:13 2:92 × 10−2 4:62 — —
6:57 × 10−3 12:00 4:42 × 10−2 1:73 4:01 7:17 1:86 11:23
5:77 × 10−3 11:03 3:03 × 10−2 1:35 20:74 3:98 37:3 5:06
5:19 × 10−3 13:43 2:58 × 10−2 1:00 11:25 3:39 31:60 4:91
The four parameters ls , ms , Bs and Cs in Eq. (4) are adjustable for tting. For the imaginary-frequency lobe, the exponent functions gu (!) of liquid Na and the LJ liquid can be tted with a binomial in frequency, which has the following form: ( lu mu ) ! ! 1 (mu − nu ) + (nu − lu ) (6) ; mu ¿lu ; gu (!) = nu (mu − lu ) !u !u where |!u | is the frequency at which Du (!) attains its maximum value D u and nu are determined by Eq. (5), in which As , |!s | and D s are replaced with Au , |!u | and D u , respectively. The binomial given in Eq. (6) has two adjustable parameters lu and mu . However, using Eq. (6) to t the exponent function gu (!) of the TLJ liquid, we found that the two adjustable parameters lu and mu are almost equal and thus the tting formula can be reduced to be n 1 ! u (7) gu (!) = : nu !u Ai , |!i | and D i (i = s or u) can be obtained directly from the smoothed density spectra shown in Fig. 3 and their values are listed in Table 1. The values for the adjustable parameters in the tting formula are also given in Table 1. With the parameters given in Table 1 and Eqs. (3)–(7), the calculated density spectra are shown by open circles in Fig. 4. For the tted results, ns in Eq. (4) for the three systems is smaller than 2. For liquid Na, due to the extremely short tail in the real-frequency lobe, ls and ms are much larger than those for the LJ systems. Also, the di erence between ms and ls for liquid Na is large due to the hump near 20!0 ; however, ms − ls for the LJ system decreases so that the region dominated by the second term in Eq. (4) shrinks.
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Fig. 4. The INM densities of states D(!) of liquid Na (a), the LJ liquid (b) and the TLJ liquid (c). In each part, the solid line is calculated by the MD-simulation method; the dashed line is calculated by the VRW approach with the two-body approximation; the open circles are calculated by Eqs. (3) – (7) with the parameters given in Table 1.
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Recently, using the soft-potential model to describe the potential energy surface, Zurcher and Keyes derive an analytic expression for gu (!) [23], which is also a binomial in frequency. The two orders of their binomial are 2 and 4, which correspond to the high- and low-temperature limits, respectively. However, in our tted results, for liquid Na and the LJ liquid, one order (mu ) of the binomial is larger than 4 and the other order (lu ) is smaller than 2. For the TLJ liquid, the two orders of the binomial approach to each other and the binomial reduces to a single term with power larger than 4.
4. Vector-random-walk (VRW) approach The details of the VRW approach are presented in Ref. [18] and we summarize the results here. The INM density of states D(!) may be obtained through the following formula: D(!) =
2|!| 00 p (−!2 ) ; 0
(8)
where p000 (−!2 ) is the imaginary part of the Green’s function p0 (−!2 ), which corresponds to the Laplace transform of the probability for the random walker returning to the original site and original internal state at time t. The real and imaginary parts of the Green’s function p0 ≡ p00 + ip000 satisfy the following pair of coupled equations: p00 = −!2 + 0 ; |p0 |2 p000 = −00 ; |p0 |2
(9) (10)
where |p0 |2 ≡ (p00 )2 + (p000 )2 . 0 and 00 are the real and imaginary parts of the self energy , which is a function of −!2 . Because the liquid is isotropic, the self energy under the two-body approximation [19] can be written as Z dr g(r)Tr[t(r) · (I3 + 2p0 t(r))−1 ] ; (11) = 3 where Tr[A] is the trace of matrix A and I3 is the 3-dimensional unit matrix. The matrix t(r) that appears here corresponds to the 3-dimensional transfer rate in the VRW problem and is related to the pair potential (r) through the following formula: 0 (r) 0 (r) 00 I3 + (r) − rˆrˆ ; (12) t(r) = r r where 0 (r) and 00 (r) denote the rst and second derivatives of (r) with respect to r, and rˆ is the unit vector along r. Hence, the self energy is an explicit function of p00 and p000 . The self energy given in Eq. (10) is equivalent to the mean- eld self
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energy in the Wen–Stratt theory [20], and can be expressed as Z = dr g(r) (r; p0 ) with the renormalized pair potential (r; p0 ) to be 2 0 (r)=r 1 00 (r) + : (r; p0 ) = 3 1 + 2p0 00 (r) 1 + 2p0 0 (r)=r
(13)
(14)
In terms of p00 and p000 , the real and imaginary parts of are expressed explicitly as Z 00 (r)[1 + 2p00 00 (r)] 0 0 00 (p0 ; p0 ) = dr g(r) 3 [1 + 2p00 00 (r)]2 + [2p000 00 (r)]2 2 0 (r)=r[1 + 2p00 0 (r)=r] + ; (15) [1 + 2p00 0 (r)=r]2 + [2p000 0 (r)=r]2
00
Z
[ 00 (r)]2 p000 3 [1 + 2p00 00 (r)]2 + [2p000 00 (r)]2 2[ 0 (r)=r]2 p000 + : [1 + 2p00 0 (r)=r]2 + [2p000 0 (r)=r]2
2 (p00 ; p000 ) = −
dr g(r)
(16)
After substituting Eqs. (15) and (16) into Eqs. (9) and (10), we get a pair of selfconsistent equations for p00 and p000 . By solving these self-consistent equations sequentially as described in Ref. [18], we obtain the INM density of states according to Eq. (8). With the pair potentials and g(r) as presented in Fig. 2, the INM density of states of the three liquids obtained by the VRW approach are shown with the dash lines in Fig. 4. However, we should remind the reader that the data of the pair potential and g(r) for liquid Na we used in the calculation are generated from the 3000-particle simulation. Apparently, the D(!) calculated with the VRW approach works better for the LJ liquid than for liquid Na. This is due to the fact that the collective motions are, in general, much stronger in metallic liquids than in the LJ liquids. Thus, the two-body approximation for the self energy is worse for liquid Na than for the LJ liquid. The fractions fu of the imaginary-frequency INMs in these three liquids obtained both from the method of MD simulation and from the VRW approach are given in Table 2. The values of fu predicted by the VRW approach are overestimated than the simulated values by about 77%, 43% and 20% for liquid Na, the LJ liquid and the TLJ liquid, respectively. Thus, under the two-body approximation, the VRW approach for calculating D(!) works better for the TLJ liquid than for the LJ liquid, since cutting out the attractive force of the LJ potential reduces the collective e ect in liquids. The normalized velocity autocorrelation function (VACF) Cvv (t), which describes the dynamics of a single particle in a liquid, is an important quantity to check the INM theory [10,24]. In terms of the INM density of states, including both Ds (!)
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Table 2 The fraction of the imaginary-frequency INMs, fu , and the 2nd and the 4-th moments of the density spectra, M2 and M4 , of these three liquids calculated by Eq. (19) with the INM density of states obtained from either MD simulation or the VRW approach. M2 calculated by Eq. (20) is also given Liquid Na fu
MD simulation VRW approach
M2
MD simulation VRW approach Eq. (20)
M4
MD simulation VRW approach
LJ Liquid
0.108 0.192 218.54 218.11 218.35 91270 113690
0.130 0.186 450.35 445.83 447.56 512380 575920
and Du (|!|), the expression for Cvv (t) is given as Z∞ Z∞ Cvv (t) = d! cos (!t)Ds (!) + d cosh (t)Du () : 0
TLJ Liquid 0.190 0.228 460.70 460.67 462.33 554570 640780
(17)
0
Then, the small-time expansion of Cvv (t) can be written as Cvv (t) = 1 −
M2 2 M4 4 t + t + ··· ; 2! 4!
(18)
where the coecients M2 and M4 are the second and the fourth moments of the density of state, Z∞ Z∞ n=2 n d n Du (); for n = 2; 4 : (19) Mn = d! ! Ds (!) + (−1) 0
0
Physically, M2 gives the Einstein frequency of the liquid, which measures the meansquare force on a particle or the average local curvature of the potential energy surface. In terms of the pair potential (r), M2 is given by Z M2 = dr g(r) { 00 (r) + 2 0 (r)=r} : (20) 3 M4 is a quantity related to the distribution function of three particles and the explicit formula is given in Ref. [18]. With the INM density of states obtained directly from the MD simulation, Cvv (t) calculated by Eq. (17) has been proved to be exact up to t 4 [8,10]; Consequently, M2 and M4 calculated with Eq. (19) are exact. On the other hand, with the density of states obtained from the VRW approach, M2 calculated with Eq. (19) is proved to be still exact; however, the calculated M4 is approximated due to the two-body approximation, which is equivalent to the superposition approximation for the three-particle distribution function [18]. Beyond the comparison of the overall shape and the fraction of imaginary-frequency INMs, the comparison of Cvv (t), M2 and M4 provide some more information for the potential e ects on the collective motions in a liquid. The VACF is calculated through
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Fig. 5. The VACFs Cvv (t) of liquid Na (a), the LJ liquid (b) and the TLJ liquid (c). In each part, the solid line is the simulation result; the dashed and the dotted lines are calculated by Eq. (17) with the INM density of states obtained from MD simulation and the VRW approach, respectively. Times are scaled according to t0 = !0−1 .
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Eq. (17) and the MD simulated results are given in Fig. 5; the data for M2 and M4 are given in Table 2. The time, within which the VACF predicted by the density of states agrees with the simulated VACF, is longer for liquid Na than for the LJ liquid. For these three liquids, M2 calculated either by Eq. (19) with the MD-simulated or the VRW-approach densities of states or by Eq. (20) agree well within numerical errors. The smaller values of M2 for the LJ liquid than that for the TLJ liquid is due to the addition of the extra linear term A(r=) + B to the LJ potential for MD simulation consideration. This linear term makes the repulsive core of each particle a little bigger and therefore the function of g(r) shifting outward a little. It seems that the repulsive force of the LJ potential dominates the contribution to the mean force on a particle in the density we consider. For liquid Na, M2 is much smaller than that of the LJ liquid, so that the decay of the VACF from t = 0 is much slower. For all of the three liquids, M4 calculated by the VRW-approach density of states is overestimated than the exact result, which is obtained with the MD-simulated density of states. The overestimation is more than 20% for liquid Na and more than 10% for the LJ systems. This proves again that the three-body e ect is more important in liquid Na than in the LJ liquid.
5. Conclusions The INMs of a liquid are a kind of collective motions of many particles. The determination of the relationship between the INM density of states and the pair interaction between two particles is an interesting problem, but by no means an obvious one. This relationship contains much information about liquid dynamics and many liquid-related quantities can be formulated and calculated using the INM density of states. In this paper, by comparing the densities of states of liquid Na, the LJ liquid and the TLJ liquid (with particle interaction to be the repulsive force of the LJ potential only), we give some physical insights on this relationship. We use both the simulation method and the VRW approach with two-body approximation to calculate the densities of states of these three liquids. The reduced densities and reduced temperatures of these three simulated liquids are chosen to be the same values. Therefore, any di erence among the densities of states of these three liquids must arise from the potential interaction. According to our results, for the real-frequency lobe, we are very sure that the repulsive core of the potential plays a dominant role on the high-frequency INMs and the tail of the potential has a signi cant e ect on the low-frequency INMs. The longer of the tail, the more of the low-frequency INMs. For the imaginary-frequency lobe, we conjecture that the shape of the density spectrum is related to the shape of the potential well and the interaction range of the pair potential. As the interaction range is extended, the fraction of the imaginary-frequency INMs shrinks. The detailed relation for the imaginary-frequency lobe needs further investigation. Through the density of states, we may de ne the exponent functions for the realand imaginary-frequency lobes. For the real-frequency lobe, the exponent functions of the three liquids we study in this paper can be tted with a three-term polynomial in
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frequency. For the imaginary-frequency lobe, the exponent functions of liquid Na and the LJ liquid can be tted with a binomial in frequency; however, we nd that the tting function for the exponent function of the TLJ liquid reduces to a single term of frequency with power larger than 4. Recently, Zurcher and Keyes have derived an analytic expression for the exponent function of the imaginary-frequency lobe, which is also a binomial in frequency but with orders di erent from what we get through data tting. Comparison of the density spectrum calculated either by the MD simulation or by the VRW approach under the two-body approximation shows that the collective motions in liquid Na are stronger than in the LJ liquid. Thus, the two-body approximation in the VRW approach works better for the LJ liquid than for liquid Na. However, for the VACF, the period within which the INM theory predicts accurately, extends longer for liquid Na than for the LJ liquid, since the fraction of the imaginary-frequency INMs of liquid Na is smaller. Finally, the INM density of states of a liquid mixture has been studied by Stassen and Gburski [25], and recently by Larsen, Goodyear and Stratt [26]. In their reports, the dependence of the INM density of states with the concentration of the mixture and the mass ratio of the two kinds of mixed particles are presented. However, they studied the LJ mixtures only. Our results in this paper and their results complement with each other for understanding the many-body INM density of states. Acknowledgements The authors acknowledge support from the National Science Council of Taiwan, R.O.C. by NSC 87-2112-M009-014 and NSC 87-2112-M110-007. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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[19] C.R. Gochanour, H.C. Andersen, M.D. Fayer, J. Chem. Phys. 70 (1979) 4254. [20] Y. Wan, R.M. Stratt, J. Chem. Phys. 100 (1994) 5123. [21] J.P. Hansen, I.R. McDonald, Theory of Simple Liquid, 2nd ed., Academic Press, New York, 1986, Chap. 11. [22] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Science, Oxford, 1991. [23] U. Zurcher, T. Keyes, Phys. Rev. E 55 (1997) 6917. [24] T.M. Wu, R.F. Loring, J. Chem. Phys. 99 (1993) 8936. [25] H. Stassen, Z.E. Gburski, Chem. Phys. Lett. 217 (1994) 325. [26] R.E. Larsen, G. Goodyear, R.M. Stratt, J. Chem. Phys. 104 (1996) 2987.
Potential e ects on instantaneous normal modes of liquids a
b
Ten-Ming Wu a; ∗ , Wen-Jong Ma b , Shiow-Fon Tsay b
Institute of Physics, National Chiao-Tung University, Hsin Chu 300, Taiwan Department of Physics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan
Abstract We calculated the instantaneous-normal-mode (INM) densities of states for liquid Na, LennardJones (LJ) liquid and truncated LJ liquid with the same reduced density and temperature by both molecular-dynamics simulation and the vector-random-walk approach under two-body approximation. By comparing the calculated results for these three liquids, we give some physical insights in the relationship between the INM density of state and the two-particle interaction potential. The e ects of the short-range repulsive core, the long-range tail in the potential and the presence c 1998 Elsevier Science B.V. All rights reserved of attractive well are elucidated. PACS: 63.50.+x; 61.25.Bi; 61.25.Mv Keywords: Instantaneous normal modes; Pair potential; Random walk
1. Introduction The dynamics of liquids is much more complicated than that of crystalline solids, for the structure in a liquid state is highly disordered and varies with time. In general, the dynamics of a many-body system can be represented as the motion of a point in con guration space over the total potential energy surface, which composes of many local minima, barrier tops and saddle points [1,2]. The dynamics of crystalline solids are just small vibrations around the global minimum of the potential energy surface and well described by phonon modes. However, the dynamics of liquids may go around the whole con guration space via a series of barrier crossings. Recently, a phonon-like approach, which is named as instantaneous-normal-mode (INM) theory [3–9], has been proposed to analyze rigorously the short-time dynamics of liquids and gained a considerable interest. In the INM theory, the total potential energy surface around each con guration is expanded up to the quadratic terms of ∗
Corresponding author. Fax: (886) 03-5720728; e-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 0 2 9 - 6
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particle displacements and the square root of each eigenvalue of the force constant matrix, or the Hessian matrix, gives an INM frequency, which measures the local curvature of the potential surface along the direction of that eigenvector. After an ensemble average with respect to all con gurations, the INM density of states, which is analogous to the phonon density of states of a crystalline solid, is obtained. However, the major di erence between these two kinds of density of states is that the INM density of states has two lobes, one for the real-frequency INMs and the other for the imaginary-frequency INMs, which characterize the upward and downward local curvatures, respectively. With the INM density of states, many dynamic quantities of liquids are formulated and the formula are accurate within a period time from initial [10,11]. More recently, the INM approach has been extended to analyze the dynamics of glassy states and gives some interesting results [12,13]. Certainly, the INM density of states has a strong relation with the pair potential between two particles in a liquid both directly through the Hessian matrix and indirectly through the microstructures of liquid con gurations [14]. Although many predictions on liquid dynamics by the INM theory have been given, the relationship between the INM density of states and the pair potential is still not so obvious. Simple liquids are still the most proper physical systems for studying this relation. So far, for simple liquids, the INM density of states for the Lennard-Jones (LJ) systems [ 3 –9], the soft-potential systems [12,13] and metallic liquids [14,15] have been reported; however, there is no systematic comparison between them. In this paper, we calculate and compare the INM density of states for three di erent kinds of liquid, in which the pair potentials between two particles are the pseudopotential of a metallic liquid, the LJ potential and the truncated LJ (TLJ) potential, which is only the repulsive portion of the LJ potential [16,17]. For comparison among these three potentials, each potential is scaled with the depth of the potential well, , which corresponds to the minimum of the potential, and the distance in each liquid is scaled with the collision length, , which is the smallest separation between two particles with the pair potential to be zero. In order to avoid the di erences due to thermodynamic states, we choose the same reduced density, ∗ = 3 , and reduced temperature, T ∗ = kB T=, for these three kinds of liquid. So far, almost all of works for calculating the INM density of states are based on computer simulation to generate liquid con gurations and then numerical diagonalizations for the Hessian matrix. On the other hand, it is well known that there is an isomorphism between the calculation of the phonon density of states of a crystalline solid and the calculation of the transport properties of a random walker on that structure. Recently, to calculating analytically the INM density of states, Wu and Loring have developed a vector-random-work (VRW) approach [18], which is a generalization of the theory of Gochanour, Andersen and Fayer (GAF) for the transport theory of a random walker in a random medium [19]. In GAF theory, the particles have no internal state and the transfer rate of the random walker from site to site is a scalar number. Due to the vector nature of the displacements of particles in d dimensions, the particles in the random walk problem isomorphic for the INM density of states have d internal states and the transfer rate for the random walker from site to site is
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a d-dimensional matrix. With a two-body approximation for the self energy of the Green’s function in the VRW problem, the INM density of states is obtained by solving a set of self-consistent equations. For the LJ liquid with reduced density about 0:75, the INM density of states calculated with the VRW approach has been proven to be a good agreement with the computer simulation results. Later on, Wan and Stratt showed that the VRW approach in the two-body approximation is equivalent to a mean- eld theory for a renormalized pair potential [20]. In this paper, we use both the method of molecular-dynamics (MD) simulation and the VRW approach to calculate the INM densities of states of the three kinds of liquid mentioned above and analyze the results. In Section 2, we give the three pair potentials and the details about MD simulation for generating the liquid con gurations. In Section 3, we present the INM densities of state of these three liquids obtained through the simulation. The VRW approach for calculating the INM density of states is summarized in Section 4 and the calculated results for these three liquids are also given. In Section 5, we give the conclusion of this paper. 2. Potential and MD simulation In this paper, we assume that the total potential of a liquid system with N particles is a pairwise summation of a pair potential (r). How the pair potential of liquid Na is obtained is described in Ref. [14]. One example of the pair potential for liquid Na is shown in Fig. 1, in which the potential and the distance are scaled with and , respectively. A typical feature of the pair potential for liquid Na is the long oscillating tail which is known as the Friedel oscillation [21]. With the same and , the LJ potential is 12 6 : (1) − LJ (r) = 4 r r By cutting o the attractive part of the LJ potential and then lifting the truncated potential by , we obtain the TLJ potential ( LJ (r) + for r¡21=6 ; TLJ (r) = (2) 0 for r¿21=6 : So, the TLJ potential is purely repulsive. The LJ and the TLJ potentials are also shown in Fig. 1; however, a linear term A(r=) + B has been added to the LJ potential for ensuring continuity for both the potential and the force at the cuto distance rc for simulation consideration. From Fig. 1, it is clear that the pair potential for liquid Na has a softer repulsive core than that of the LJ potential. With the periodic boundary condition, we perform MD simulations with velocityVerlet algorithm [22] on three systems of 375 particles having each a mass m, interacting via the three pair potentials, respectively. The cuto distances rc are 3:5 for the LJ liquid and 3:7 for liquid Na. The densities and the temperatures T of the three
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Fig. 1. The pseudopotential of liquid Na (the solid line), the LJ potential (the dashed line) and the TLJ potential (the dotted line). Each potential is scaled with the depth of the potential well, , and the distance is scaled with the collision length, . The insert gure shows these three potentials in the extended region.
systems are chosen to be the same values so that the mean nearest-neighbor separation and the average kinetic energy of these three systems are also of the same values. Thus, any physical quantities which are di erent among the three systems must arise from the pair potential (r). Within the simulation errors, the chosen values of the reduced density and reduced temperature are ∗ = 0:972 and T ∗ = 0:836. For liquid Na, the simulated system corresponds to a realistic thermodynamic state at T = 378 K, which is just above the melting temperature of Na. For the LJ and the TLJ liquids, the simulated systems are in the liquid states near their melting temperatures, respectively. For each simulated system, we collect data of 100 con gurations. The radial distribution functions g(r) of the three simulated systems are shown in Fig 2. The radial distribution functions for the LJ and the TLJ liquids are almost identical. This is consistent with the theory of Chandle, Weeks and Anderson, which concluded with the statement that for high densities (3 & 0:65) the equilibrium structure of a LJ liquid is quantitatively dominated by the repulsive force of the potential [16,17]. In Section 4, we will need a longer range for the pair potential and the radial distribution function of liquid Na. Thus, we perform another MD simulation for liquid Na of 3000 particles in the same thermodynamic and simulation conditions, except for cuto rc = 7:28. The radial distribution function of liquid Na obtained by our simulation has been checked with the experimental data. Due to the softer repulsive core, two particles in liquid Na may be compressed further than in the LJ systems so that the function of g(r) penetrates further into shorter distances and has smaller value in the maximum.
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Fig. 2. The radial distribution functions of the three liquids with the same reduced density and reduced temperature: the solid line corresponds to liquid Na, the dashed line to LJ liquid and the open dots to TLJ liquid.
3. INM density of states For each instantaneous con guration of N particles, the 3N -dimensional Hessian matrix is calculated according to Eqs. (2.3) and (2.4) given in Ref. [18] and then diagonalized to yield normal-mode frequencies. Repeating the procedure for 100 con gurations and nally taking the average, we get a spread INM density spectrum. After the spread spectrum is smoothed with FFT program, the smoothed and normalized INM density of states D(!) is obtained. The D(!) spectra for the three simulated liquids are shown in Fig. 3. We have adopted the plotting convention in which the part for the imaginary-frequency INMs is plotted on the negative frequency axis and the frequency scale is !0 = (=m2 )1=2 . For liquid Na, the real-frequency density spectrum Ds (!) has an extremely short tail in the high-frequency end and a hump near 20!0 . Thus, the shape of Ds (!) for liquid Na is quite di erent from those of the LJ systems. The Ds (!) spectra of the two LJ systems have almost identical high-frequency tails, which gives strong evidence that the high-frequency INMs are solely determined by the repulsive force of the pair potential. As the tail of the LJ potential is cut o , Ds (!) in the low-frequency end decreases. This information suggest that the attractive force of the pair potential plays an important role on INMs in the low-frequency end than on INMs in the high-frequency end. This conclusion is consistent with the case for liquid Na, whose Ds (!) in the low-frequency end has a much larger value.
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Fig. 3. The INM densities of states D(!) of the three liquids: the solid line corresponds to liquid Na, the dashed line to LJ liquid and the dotted line to TLJ liquid. As usual, the real-frequency lobe is plotted along the positive frequency axis and the imaginary-frequency lobe is plotted along the negative frequency axis. Frequencies are scaled by !0 = (=m2 )1=2 .
For the imaginary-frequency lobe, the density spectrum Du (!) of the TLJ liquid has the broadest width and the largest maximum value among the three spectra. The Du (!) spectra of the LJ liquid and liquid Na have almost the same maxima values, but the width of the spectrum of the LJ liquid is broader than that of liquid Na. This information suggest that the imaginary-frequency density spectrum Du (!) has something to do with the interaction range of the pair potential and the shape of the attractive well of the potential. The detail needs further investigation. The three Ds (!) and Du (!) spectra shown in Fig. 3 have the general expression Di (|!|) = Ai |!| exp(−gi (|!|))
for i = s or u ;
(3)
where the linear |!| dependence is understood to be the Jocabian factor and Ai is the slope of Di (!) at zero frequency. The exponent function gi (!) of each lobe is de ned to be the logarithm of Ai |!|=Di (!). For the real-frequency lobe, the exponent functions gs (!) of the three liquids can be well tted with a three-term polynomial in frequency n ls ms ! ! 1 ! s gs (!) = − Bs + Cs ; ms ¿ls ¿ns ; (4) ns !s !s !s
where |!s | is the frequency at which Ds (!) attains its maximum value D s and ns is determined by the formula −1 As |!s | : (5) ns = ln D s
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Table 1 The tting parameters for the INM density spectra of the real-frequency and the imaginary-frequency lobes, respectively. The units of |!i |, D i and Ai (i = s or u) are !0 , !0−1 and !0−2 , respectively Parameter Au |!u | D u nu lu mu As |!s | D s ns Bs × 103 ls Cs × 103 ms
Liquid Na
LJ liquid
TLJ liquid
6:16 × 10−3 4:91 1:99 × 10−2 2:38 1:24 5:72
5:64 × 10−3 5:36 2:03 × 10−2 2:52 1:20 4:35
5:89 × 10−3 6:13 2:92 × 10−2 4:62 — —
6:57 × 10−3 12:00 4:42 × 10−2 1:73 4:01 7:17 1:86 11:23
5:77 × 10−3 11:03 3:03 × 10−2 1:35 20:74 3:98 37:3 5:06
5:19 × 10−3 13:43 2:58 × 10−2 1:00 11:25 3:39 31:60 4:91
The four parameters ls , ms , Bs and Cs in Eq. (4) are adjustable for tting. For the imaginary-frequency lobe, the exponent functions gu (!) of liquid Na and the LJ liquid can be tted with a binomial in frequency, which has the following form: ( lu mu ) ! ! 1 (mu − nu ) + (nu − lu ) (6) ; mu ¿lu ; gu (!) = nu (mu − lu ) !u !u where |!u | is the frequency at which Du (!) attains its maximum value D u and nu are determined by Eq. (5), in which As , |!s | and D s are replaced with Au , |!u | and D u , respectively. The binomial given in Eq. (6) has two adjustable parameters lu and mu . However, using Eq. (6) to t the exponent function gu (!) of the TLJ liquid, we found that the two adjustable parameters lu and mu are almost equal and thus the tting formula can be reduced to be n 1 ! u (7) gu (!) = : nu !u Ai , |!i | and D i (i = s or u) can be obtained directly from the smoothed density spectra shown in Fig. 3 and their values are listed in Table 1. The values for the adjustable parameters in the tting formula are also given in Table 1. With the parameters given in Table 1 and Eqs. (3)–(7), the calculated density spectra are shown by open circles in Fig. 4. For the tted results, ns in Eq. (4) for the three systems is smaller than 2. For liquid Na, due to the extremely short tail in the real-frequency lobe, ls and ms are much larger than those for the LJ systems. Also, the di erence between ms and ls for liquid Na is large due to the hump near 20!0 ; however, ms − ls for the LJ system decreases so that the region dominated by the second term in Eq. (4) shrinks.
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Fig. 4. The INM densities of states D(!) of liquid Na (a), the LJ liquid (b) and the TLJ liquid (c). In each part, the solid line is calculated by the MD-simulation method; the dashed line is calculated by the VRW approach with the two-body approximation; the open circles are calculated by Eqs. (3) – (7) with the parameters given in Table 1.
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Recently, using the soft-potential model to describe the potential energy surface, Zurcher and Keyes derive an analytic expression for gu (!) [23], which is also a binomial in frequency. The two orders of their binomial are 2 and 4, which correspond to the high- and low-temperature limits, respectively. However, in our tted results, for liquid Na and the LJ liquid, one order (mu ) of the binomial is larger than 4 and the other order (lu ) is smaller than 2. For the TLJ liquid, the two orders of the binomial approach to each other and the binomial reduces to a single term with power larger than 4.
4. Vector-random-walk (VRW) approach The details of the VRW approach are presented in Ref. [18] and we summarize the results here. The INM density of states D(!) may be obtained through the following formula: D(!) =
2|!| 00 p (−!2 ) ; 0
(8)
where p000 (−!2 ) is the imaginary part of the Green’s function p0 (−!2 ), which corresponds to the Laplace transform of the probability for the random walker returning to the original site and original internal state at time t. The real and imaginary parts of the Green’s function p0 ≡ p00 + ip000 satisfy the following pair of coupled equations: p00 = −!2 + 0 ; |p0 |2 p000 = −00 ; |p0 |2
(9) (10)
where |p0 |2 ≡ (p00 )2 + (p000 )2 . 0 and 00 are the real and imaginary parts of the self energy , which is a function of −!2 . Because the liquid is isotropic, the self energy under the two-body approximation [19] can be written as Z dr g(r)Tr[t(r) · (I3 + 2p0 t(r))−1 ] ; (11) = 3 where Tr[A] is the trace of matrix A and I3 is the 3-dimensional unit matrix. The matrix t(r) that appears here corresponds to the 3-dimensional transfer rate in the VRW problem and is related to the pair potential (r) through the following formula: 0 (r) 0 (r) 00 I3 + (r) − rˆrˆ ; (12) t(r) = r r where 0 (r) and 00 (r) denote the rst and second derivatives of (r) with respect to r, and rˆ is the unit vector along r. Hence, the self energy is an explicit function of p00 and p000 . The self energy given in Eq. (10) is equivalent to the mean- eld self
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energy in the Wen–Stratt theory [20], and can be expressed as Z = dr g(r) (r; p0 ) with the renormalized pair potential (r; p0 ) to be 2 0 (r)=r 1 00 (r) + : (r; p0 ) = 3 1 + 2p0 00 (r) 1 + 2p0 0 (r)=r
(13)
(14)
In terms of p00 and p000 , the real and imaginary parts of are expressed explicitly as Z 00 (r)[1 + 2p00 00 (r)] 0 0 00 (p0 ; p0 ) = dr g(r) 3 [1 + 2p00 00 (r)]2 + [2p000 00 (r)]2 2 0 (r)=r[1 + 2p00 0 (r)=r] + ; (15) [1 + 2p00 0 (r)=r]2 + [2p000 0 (r)=r]2
00
Z
[ 00 (r)]2 p000 3 [1 + 2p00 00 (r)]2 + [2p000 00 (r)]2 2[ 0 (r)=r]2 p000 + : [1 + 2p00 0 (r)=r]2 + [2p000 0 (r)=r]2
2 (p00 ; p000 ) = −
dr g(r)
(16)
After substituting Eqs. (15) and (16) into Eqs. (9) and (10), we get a pair of selfconsistent equations for p00 and p000 . By solving these self-consistent equations sequentially as described in Ref. [18], we obtain the INM density of states according to Eq. (8). With the pair potentials and g(r) as presented in Fig. 2, the INM density of states of the three liquids obtained by the VRW approach are shown with the dash lines in Fig. 4. However, we should remind the reader that the data of the pair potential and g(r) for liquid Na we used in the calculation are generated from the 3000-particle simulation. Apparently, the D(!) calculated with the VRW approach works better for the LJ liquid than for liquid Na. This is due to the fact that the collective motions are, in general, much stronger in metallic liquids than in the LJ liquids. Thus, the two-body approximation for the self energy is worse for liquid Na than for the LJ liquid. The fractions fu of the imaginary-frequency INMs in these three liquids obtained both from the method of MD simulation and from the VRW approach are given in Table 2. The values of fu predicted by the VRW approach are overestimated than the simulated values by about 77%, 43% and 20% for liquid Na, the LJ liquid and the TLJ liquid, respectively. Thus, under the two-body approximation, the VRW approach for calculating D(!) works better for the TLJ liquid than for the LJ liquid, since cutting out the attractive force of the LJ potential reduces the collective e ect in liquids. The normalized velocity autocorrelation function (VACF) Cvv (t), which describes the dynamics of a single particle in a liquid, is an important quantity to check the INM theory [10,24]. In terms of the INM density of states, including both Ds (!)
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Table 2 The fraction of the imaginary-frequency INMs, fu , and the 2nd and the 4-th moments of the density spectra, M2 and M4 , of these three liquids calculated by Eq. (19) with the INM density of states obtained from either MD simulation or the VRW approach. M2 calculated by Eq. (20) is also given Liquid Na fu
MD simulation VRW approach
M2
MD simulation VRW approach Eq. (20)
M4
MD simulation VRW approach
LJ Liquid
0.108 0.192 218.54 218.11 218.35 91270 113690
0.130 0.186 450.35 445.83 447.56 512380 575920
and Du (|!|), the expression for Cvv (t) is given as Z∞ Z∞ Cvv (t) = d! cos (!t)Ds (!) + d cosh (t)Du () : 0
TLJ Liquid 0.190 0.228 460.70 460.67 462.33 554570 640780
(17)
0
Then, the small-time expansion of Cvv (t) can be written as Cvv (t) = 1 −
M2 2 M4 4 t + t + ··· ; 2! 4!
(18)
where the coecients M2 and M4 are the second and the fourth moments of the density of state, Z∞ Z∞ n=2 n d n Du (); for n = 2; 4 : (19) Mn = d! ! Ds (!) + (−1) 0
0
Physically, M2 gives the Einstein frequency of the liquid, which measures the meansquare force on a particle or the average local curvature of the potential energy surface. In terms of the pair potential (r), M2 is given by Z M2 = dr g(r) { 00 (r) + 2 0 (r)=r} : (20) 3 M4 is a quantity related to the distribution function of three particles and the explicit formula is given in Ref. [18]. With the INM density of states obtained directly from the MD simulation, Cvv (t) calculated by Eq. (17) has been proved to be exact up to t 4 [8,10]; Consequently, M2 and M4 calculated with Eq. (19) are exact. On the other hand, with the density of states obtained from the VRW approach, M2 calculated with Eq. (19) is proved to be still exact; however, the calculated M4 is approximated due to the two-body approximation, which is equivalent to the superposition approximation for the three-particle distribution function [18]. Beyond the comparison of the overall shape and the fraction of imaginary-frequency INMs, the comparison of Cvv (t), M2 and M4 provide some more information for the potential e ects on the collective motions in a liquid. The VACF is calculated through
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Fig. 5. The VACFs Cvv (t) of liquid Na (a), the LJ liquid (b) and the TLJ liquid (c). In each part, the solid line is the simulation result; the dashed and the dotted lines are calculated by Eq. (17) with the INM density of states obtained from MD simulation and the VRW approach, respectively. Times are scaled according to t0 = !0−1 .
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Eq. (17) and the MD simulated results are given in Fig. 5; the data for M2 and M4 are given in Table 2. The time, within which the VACF predicted by the density of states agrees with the simulated VACF, is longer for liquid Na than for the LJ liquid. For these three liquids, M2 calculated either by Eq. (19) with the MD-simulated or the VRW-approach densities of states or by Eq. (20) agree well within numerical errors. The smaller values of M2 for the LJ liquid than that for the TLJ liquid is due to the addition of the extra linear term A(r=) + B to the LJ potential for MD simulation consideration. This linear term makes the repulsive core of each particle a little bigger and therefore the function of g(r) shifting outward a little. It seems that the repulsive force of the LJ potential dominates the contribution to the mean force on a particle in the density we consider. For liquid Na, M2 is much smaller than that of the LJ liquid, so that the decay of the VACF from t = 0 is much slower. For all of the three liquids, M4 calculated by the VRW-approach density of states is overestimated than the exact result, which is obtained with the MD-simulated density of states. The overestimation is more than 20% for liquid Na and more than 10% for the LJ systems. This proves again that the three-body e ect is more important in liquid Na than in the LJ liquid.
5. Conclusions The INMs of a liquid are a kind of collective motions of many particles. The determination of the relationship between the INM density of states and the pair interaction between two particles is an interesting problem, but by no means an obvious one. This relationship contains much information about liquid dynamics and many liquid-related quantities can be formulated and calculated using the INM density of states. In this paper, by comparing the densities of states of liquid Na, the LJ liquid and the TLJ liquid (with particle interaction to be the repulsive force of the LJ potential only), we give some physical insights on this relationship. We use both the simulation method and the VRW approach with two-body approximation to calculate the densities of states of these three liquids. The reduced densities and reduced temperatures of these three simulated liquids are chosen to be the same values. Therefore, any di erence among the densities of states of these three liquids must arise from the potential interaction. According to our results, for the real-frequency lobe, we are very sure that the repulsive core of the potential plays a dominant role on the high-frequency INMs and the tail of the potential has a signi cant e ect on the low-frequency INMs. The longer of the tail, the more of the low-frequency INMs. For the imaginary-frequency lobe, we conjecture that the shape of the density spectrum is related to the shape of the potential well and the interaction range of the pair potential. As the interaction range is extended, the fraction of the imaginary-frequency INMs shrinks. The detailed relation for the imaginary-frequency lobe needs further investigation. Through the density of states, we may de ne the exponent functions for the realand imaginary-frequency lobes. For the real-frequency lobe, the exponent functions of the three liquids we study in this paper can be tted with a three-term polynomial in
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frequency. For the imaginary-frequency lobe, the exponent functions of liquid Na and the LJ liquid can be tted with a binomial in frequency; however, we nd that the tting function for the exponent function of the TLJ liquid reduces to a single term of frequency with power larger than 4. Recently, Zurcher and Keyes have derived an analytic expression for the exponent function of the imaginary-frequency lobe, which is also a binomial in frequency but with orders di erent from what we get through data tting. Comparison of the density spectrum calculated either by the MD simulation or by the VRW approach under the two-body approximation shows that the collective motions in liquid Na are stronger than in the LJ liquid. Thus, the two-body approximation in the VRW approach works better for the LJ liquid than for liquid Na. However, for the VACF, the period within which the INM theory predicts accurately, extends longer for liquid Na than for the LJ liquid, since the fraction of the imaginary-frequency INMs of liquid Na is smaller. Finally, the INM density of states of a liquid mixture has been studied by Stassen and Gburski [25], and recently by Larsen, Goodyear and Stratt [26]. In their reports, the dependence of the INM density of states with the concentration of the mixture and the mass ratio of the two kinds of mixed particles are presented. However, they studied the LJ mixtures only. Our results in this paper and their results complement with each other for understanding the many-body INM density of states. Acknowledgements The authors acknowledge support from the National Science Council of Taiwan, R.O.C. by NSC 87-2112-M009-014 and NSC 87-2112-M110-007. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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