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Lindemann index 计算求助
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文献上称这个为Lindemann index ，也有叫做root-mean-square fluntuation,不知道大家在模拟统计过程中，有没有碰到与我类似的问题，
我计算时候，总是碰到开方里面出现负值！相当于出现“距离平方的平均值”小于“距离平均值的平方”，另外我现在也在怀疑是否我的原子体系模拟的不对，导致出现了这种情况？
有谁要是有类似经验，请多多指教啊
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$\langle r_{ij}^2\rangle$应该大于$\langle r_{ij}\rangle ^2.
说说你具体是怎么计算这两个量的，应该是你的计算哪里出了问题
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2# yahoohoo
我的程序段
系统稳定后，开始：
call distance(x,y,z,r)!统计出此刻的距离信息
do k=1,zn-1
do j=k+1,zn
r2(k,j)=r2(k,j)+r(k,j)**2 ！距离平方求和，开头置零
r1(k,j)=r1(k,j)+r(k,j)！距离求和，开头已置零
在一个温度上统计Tmcs-Emcs次之后，运行下面的
do 1200 k=1,zn-1
do 1200 j=k+1,zn
r1(k,j)=r1(k,j)/((Tmcs-Emcs)*1.0)！距离的统计平均
r2(k,j)=r2(k,j)/((Tmcs-Emcs)*1.0)！距离平方的统计平均
deta=deta+sqrt(r2(k,j)-r1(k,j)**2)/r1(k,j)！这就是lindemann index 公式右边的
1200continue
deta=2.0*deta/(n*(n-1.0))！最后计算这个温度下的这个值
这就是我的程序段，运行时，有时候可以运行一段时间，往往就会出现sqrt(r2(k,j)-r1(k,j)**2)这个地方有问题，开方里面出现负值！
请帮我看看，谢谢啦
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你的fortran程序思路很清楚，我没看出错误。你再检查下(1)计算两两距离时周期性边界条件是否正确；(2)NVTMC的温度是否太低，是否已经达到平衡；(3)可以计算下$C_v/k_B = \beta ^2 \langle E^2\rangle - \langle E \rangle ^2$，看比热是否是正的。
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4# yahoohoo
首先我统计的是一个团簇的lindemann index 信息，所以统计的范围就是整个团簇了，没有周期性边界条件
另外我也同时统计Cv的，值是正的，但是效果不好，曲线的走势不好。
似乎数学上说，平方的平均值应该大于或者等于平均的平方，如果是这样，那么应该与系统的高温还是低温或者平衡是没有关系，或者说不应该出现负值的情况，我就郁闷在这了。当然如果那样就是统计出来的物理信息不正确。
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4# yahoohoo
首先我统计的是一个团簇的lindemann index 信息，所以统计的范围就是整个团簇了，没有周期性边界条件
另外我也同时统计Cv的，值是正的，但是效果不好，曲线的走势不好。
似乎数学上说， ...
nantian112 发表于
不清楚你具体模拟的细节，但是只要你使用一定形状的盒子，那都会有PBC的问题。
如果你考虑S-L转变，那在转变点附近进行不同温度的NVT MC模拟，$C_V$对$T$的曲线上也可以反映出转变来，与你想要计算的Lindemann index也能相互验证。你所谓的走势不好我不清楚具体是什么意思？但似乎说明你计算的体系没有达到热力学平衡。这也正是我问你温度的原因。低温下平衡的时间很长，甚至有些算法根本达不到平衡。
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模拟的团簇也就总共几十个或者上百个原子，整个体系也就这么多，全部也就这么多，所以没有PBC的问题了，
具体原因我还要继续考虑了，谢谢你与我分享
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找到原因请于大家分享，以便后来人借鉴，呵呵
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惨了，又想了一天了，还没有解决
Powered byThe comparison of the two figures clearly shows the different behavior of the (TeF 6 ) 89 and (SF 6 ) 89 clusters. The tellurium cluster has a well-resolved jump in the heat capacity at the upper-temperature solid–solid transition, while the values of the heat capacity in the (SF 6 ) 89 clusters are of the same order of magnitude for the three transitions in the temperature region ? 0–90 K ? : At about 80 K is a liquid–solid transition as the Lindemann criterion shows, Fig. 5; at about 60 K there is a solid–solid transition, disordered cubic-to- monoclinic, a disc and at about 40 K, there is another solid–solid transition, disordered-to-ordered monoclinic. To distinguish between the liquid–solid and solid–solid transitions, we have computed the Lindemann index, ? lin , from ? lin ? N ? N 2 ? 1 ? i , j ( ? ? i ) ? 1 r i j ? r i j ? r i j , ? 6 ? where ? i r i j ? ? ? r i ( t ) ? ? ? r j ( t ) ? and ? . ? denotes time averaging. For bulk, ? lin у 0.1 corresponds to a melted phase. In free clusters, due to the surface, ? lin is size-dependent and is less than 0.1 for a fluid phase. More precise estimation of the melting temperature can be obtained from the slope of ? lin for a given cluster size, i.e., from ? ? lin / ? T . The plot in Fig. 5 shows that the sulphur cluster is completely melted at T ? 80 K while the tellurium cluster with the same number of molecules is solid, Fig. 6. The shift of the transition temperatures towards lower values for the case of the SF 6 cluster can be estimated from Fig. 4. The lower melting temperature is consistent with the softer potential in Fig. 1 ? b ? . Let us remember that changing the range and curvature at the minimum of the pair potential has a strong effect on the shape of the multidimensional surface, which in turn determines the dynamics of the system. 23 The simultaneous analysis of Figs. 3– 6 shows that the two solid–solid transformations ( ? ? 0.1) occur at ? 90 K and ? 60 K for a (TeF ) cluster and at ? 60 K and ? 40 K for an (SF 6 ) 89 cluster. In both cases the transition temperatures rescale from the sulphur cluster to its tellurium counterpart with a factor ? 1.5. The liquid–solid transformation of the (SF 6 ) 89 cluster at ? 80 K is discontinuous and we observe a clear picture of coexisting phases, Fig. 7. The upper-temperature solid–solid transformation ? 60 K for SF 6 and 90 K for TeF 6 ) is also discontinuous, the finite-system analogue of a first-order transition as well, and we consequently observe a temperature band of phase coexistence in the simulations, Figs. 8 and 9. This phase change is associated with a reconstruction of the lattice, as easily seen from the slope of ? lin ( T ) in the region above 90 K and below 85 K for (TeF 6 ) 89 , Fig. 6. Due to the smoother topography of its potential, the SF 6 cluster spends less time in each minimum associated with one of the phases. The second solid–solid transformation ? 60 K for TeF 6 and 40 K for SF 6 ) is continuous and is related to a spatial ordering of the molecular orientations. There is no reorganization of the lattice as the smooth variation in the Lindemann index shows. In this region there is no phase coexistence, so that this phase change fits the traditional theory of critical phenomena for second-order transitions, with a single minimum in the free energy. In order to detect coexisting solid structures, we have quenched the molecular dynamics ? MD ? trajectory in the upper temperature transition region, at ? 90 K for (TeF 6 ) 89 and at ? 60 K for (SF 6 ) 89 . Quenching is a means to associate arbitrary points in the phase space with the local minima in whose wells they lie on the PES. 24 We have used the conju- gate gradient method 25 to minimize the energy of the MD trajectory. At the end of every 250 fs interval, we have re- corded the quenched structures and have subsequently performed normal mode analysis. The plot of the quenched potential energy Fig. 8 indicates two well-resolved states in the sulfur cluster after quenching: One state, A 1 , has a total binding energy of approximately ? 12.95 eV and the other, state A 2 , lies at approximately ? 13.61 eV. The equilibrium constant for the transition between the phase A 1 and the phase A 2 is estimated from: K ? ? amount of time spent in A 1 )/ ? amount of time spent in A 2 ) ? 1.58, which indicates that the state with the ordered molecular axes is less favorable for this system, at T ? 58.5 K, the transition temperature of this simulation. The quenched potential energy of a tellurium cluster in the transition region is plotted in Fig. 9, for T ? 89.4 K. The system spends almost the same amount of time in the two solid phases: Cubic ? with orientational disorder ? and monoclinic ? with partial order ? . The evolution of the cluster, seen in animations, 26 shows that the orientational ordering transformations are initiated at the surface. The diagrams of Figs. 10–12 show three snapshots from an animation ? indicated with circles in Fig. 9 ? : Frames 21, 33, 60, taken at times 200, 450, and 780 ps, respectively. Frame 21 shows a quenched structure ordered along the z axis, but is orientationally disordered in the xy -plane, while Frame 60 shows a cubic, orientation-disordered structure. Frame 33 clearly demonstrates two coexisting structures. At temperatures below the second, lower solid–solid transition, all axes become orientation ordered. A better monitoring of the order gives the distribution of the mutual orientation of the molecules. This distribution is a function of the temperature, of course, and changes during the run, due to the dynamics. Figures 13–16 show the evolution of the orientation distribution in the transition region, where the systems transform from one structure to another. Finally, the vibrational spectrum, shown in Fig. 17, indicates that the system adopts two slightly different lattices along its MD trajectory, namely in the intervals around time steps 200 and 800. Due to the small lattice reconstruction, the cluster can transform between the two structures in an observable period of time. We must point out that a very small change of the temperature, about 0.1 K, is enough to bring the cluster into the upper- or the lower-energy phase for time intervals longer than the duration of a reasonable computation, 6 –10 ns. Only by careful tuning of the temperature was it possible to detect the coexistence of these two phases, and, thereby, to demonstrate the existence of two local minima in the free energy for this phase change. The thermal behavior of TeF 6 and SF 6 clusters consisting of the same number of octahedral molecules have been studied both analytically and numerically to find the reason for an apparent discrepancy in their phase behavior published in the literature. We show that the general properties of the transitions observed in the two systems are very simi- lar, i.e., both systems transform from a highly symmetrical state, a bcc structure with complete disorder of the molecular orientation axes of symmetry, to a lower-symmetry lattice, first with partially ordered ? aligned ? , and then, at lower temperatures or energies, with completely ordered molecular axes. For the finite-system analogue of a first-order transformation, we observe phase coexistence by inspecting the time evolution of the potential energy. A detailed analysis of the topography of PES reveals the reason for the phases of the SF 6 clusters being more difficult to distinguish than those of the TeF 6 cluster: The smoother landscape of the sulfur PES increases the rate constants for transitions between neighboring minima. This decreases the time the system spends in any specific phase. Because of that, one must perform very long runs, with a carefully se- lected step of integration ? t and with enough slightly different initial points in the phase space in order to avoid quasiperiodic orbits generated by truncation of the numbers in the computers. NATO Grant CLG SA PST.CLG.7 is ac- knowledged. This work has been funded by a Grant from National Science Foundation. Two of the authors ? A.P. and S.P. ? acknowledge partial support from the University of Sofia Scientific Fund, Grant No. 00. The complicated energy surface of a molecular cluster imposes specific requirements for performing reliable calculations. On one hand, Liouville’s theorem states that the phase space volume for Hamiltonian systems is incompress- ible. As a consequence, Hamiltonian systems do not have attractors in the usual sense of having bounded subsets to which regions of initial conditions of nonzero phase space volume converge asymptotically with increasing time. We consider systems in which the Hamiltonian has no explicit time dependence, H ? H ( p , q ) and the energy E tot of the system is conserved. A basic structural property of Hamiltonian’s equations is that they are symplectic . That is, if we consider three orbits that are infinitesimally displaced from each other, ( p ( t ), q ( t )), ( p ( t ) ? ? p ( t ), q ( t ) ? ? q ( t )) and ( p ( t ) ? ? p ? ( t ), q ( t ) ? ? q ? ( t )), where ? p , ? q , ? p ? , and ? q ? are infinitesimal n vectors, then the quantity ? p o ? q ? ? ? q o ? p ? , which is called the differential symplectic area, is independent of timeJoin ResearchGate to access over 30 million figures and 100+ million publications – all in one place.Copy referenceCopy captionEmbed figurePublished in
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If the system is unstable due to the competition between its subsystems, the dominance of attractive forces will cause the appearance of the back-bending phenomenon. As pointed out by Berry [33] , a rugged potential energy surface requires a slow enough change of the system's temperature to avoid trapping it in a metastable state. If this trapping happens, the system cannot escape from a local minimum for realistic computational times.
Full-text · Article · Jan 2012 · Physical Review EThe studied clusters include noble gases [2,3,678, mixed system [15], alkali halides [9,22, water [23] and metals [24], by using effective potentials. Also, for example, TeF6 [25,26], pure [27,28] and doped [29] sodium, and LiCl [30] by using ab initio methods. Some studies include the application of external fields [31] and others are performed in supercritical fluids [32]. ABSTRACT: A review of our work on phase transitions, coexistence and crystal growth dynamics in ionic nanoclusters is presented. The foundations and limitations of the proposed models are discussed and perspectives for extended treatments are given. Additionally, supported on a compilation of the asymptotic behaviour of the properties towards bulk conditions, new results concerned with the operational meaning of the thermodynamic limit are also presented. Some topics are complemented with link references to on-line animations that provide a visualisation of the focused behaviours. The simulations were carried out by molecular dynamics on KCl, NaCl, LiCl and NaI clusters. Full-text · Article · Jan 2010 ABSTRACT: Free methane clusters containing up to 250 molecules adopt local icosahedral-like structures at low temperatures (10-50 K). The long-range order is lost and the cluster state could be attributed to a liquid-like state even it is 'frozen'. Full-text · Article · Physical Review E Full-text · Article · Jan 2002 ABSTRACT: Small systems, notably clusters of tens or hundreds of atoms or molecules, exhibit forms almost precisely analogous to the phases of bulk systems. However their small sizes make these systems behave in ways quite different from their bulk counterparts. These differences can be elucidated and related to the behavior of bulk systems. Understanding these relationships gives us new insights into the traditional, classical bulk phase transitions, and shows us some unique properties of phases and phase equilibrium of nanoscale systems. To cite this article: R.S. Berry, C. R. Physique 3 (–326. Full-text · Article · Apr 2002 ABSTRACT: The glass transition in a quantum Lennard-Jones mixture is investigated by constant-volume path-integral simulations. Particles are assumed to be distinguishable, and the strength of quantum effects is varied by changing variant Planck's over 2pi from zero (the classical case) to one (corresponding to a highly quantum-mechanical regime). Quantum delocalization and zero point energy drastically reduce the sensitivity of structural and thermodynamic properties to the glass transition. Nevertheless, the glass transition temperature T(g) can be determined by analyzing the phase space mobility of path-integral centroids. At constant volume, the T(g) of the simulated model increases monotonically with increasing variant Planck's over 2pi. Low temperature tunneling centers are identified, and the quantum versus thermal character of each center is analyzed. The relation between these centers and soft quasilocalized harmonic vibrations is investigated. Periodic minimizations of the potential energy with respect to the positions of the particles are performed to determine the inherent structure of classical and quantum glassy samples. The geometries corresponding to these energy minima are found to be qualitatively similar in all cases. Systematic comparisons for ordered and disordered structures, harmonic and anharmonic dynamics, classical and quantum systems show that disorder, anharmonicity, and quantum effects are closely interlinked.Article · Jul 2002