Катин МОЛЕЦУЛАР ДЫНАМИЦС ИН МУЛТИСЦАЛЕ МОДЕЛИНГ 2015

high-energy CL-20 molecule C6H6N12O12 and double-walled carbon natotube, respectively. There are minimize, maximize and close buttons in the title, which are typical for the Windows operating system. The menu bar consists of five menu items. They are file, view, type of atom, regime and relax. “File” menu item is responsible for the work with input and output data. “View” menu item allows to set a user-friendly display style and also to open the control panel for dynamic molecular images. By using the “Type of atom” item, user can chose a type of atom which should be added to the cluster with the help of molecular builder. This tool is switched by the “Regime” item. The visualization area is used for immediate display of a molecule and for the control of this process using a computer mouse. To control the program in the dynamic mode, there is a special Movie manager panel. More detailed program description one can find in the following article: Scientific Visualization, Vol. 7, P. 30-37 (2015). Note that all basic functions presented in ClustVis are also available in the more sophisticated visualizers.

Fig. 4. Image of the high-energy CL-20 molecule (hexanitrohexaazaisowurtzitane) visualized by the ClustVis. Spheres denote carbon,

hydrogen, nitrogen and oxygen atoms

Exercise 23. Please install any molecular visualizer on your computer. Try to visualize CL-20 molecule and try to define all its interatomic distances. After that try to visualize any dynamical process as a computer animation.

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Integrals of motion in MD simulation

Integrals of motion are the quantities that are conserved throughout the motion. Among them are total energy, momentum and angular momentum of the system. If the numerical scheme is used to solve the equations of motion, then the integrals of motion may change due to reducing the problem from continuum to discrete case. As a result, one can obtain the qualitatively wrong results. For example, the lifetime of thermally activated atomic cluster is strongly (exponentially) depends on its total energy. So, even a small error in energy leads to the incorrect calculation of lifetime.

Let us analytically prove, that the Störmer’s second-order method (and, therefore velocity Verlet algorithm, see Ex. 21) provides angular momentum M conservation.

M (t) mi [ri (t), vi (t)]

i

i

be equal to zero, formula (11) can be rewrited as:

On the other hand,

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presumed to be equal to zero, formulae (12) and (13) allow concluding that the angular moment does not change: M(t) M(t t) , q.e.d.

Exercise 24. Please prove analytically the conservation of momentum for all numerical schemes listed in the Table 1. Please demonstrate that some numerical schemes presented in Table 1 do not provide the angular momentum conservation.

Exercise 25. Please simulate the benzene molecule oscillations using the velocity Verlet algorithm. The initial nuclei positions correspond to equilibrium state (see Ex. 20). The initial velocity of one of the carbon nuclei is directed toward the molecule centre and is equal to 1 Å/fs. The other initial velocities are equal to zero. Please calculate seven integrals of motion: total energy, momentum (three projections), and angular momentum (three projections)

at each calculation time step. What can you say about the integrals of motion conservation in the velocity Verlet scheme?

Energy conservation during the MD simulation

As already noted, conservation of the total energy is the crucial point for the correct MD simulation of the isolated systems. However, numerical schemes often cannot provide the energy conservation due to una-

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voidable increasing numerical errors. To solve this problem velocity scaling correction should be applied. The main idea of this method is to rescale nuclei velocities at each time step or several time steps of the MD simulation. The rescaling is calculated as

i

where “corr” is the correction coefficient, k is an additional parameter, and E0 is the initial total energy. The value k = 1 provides complete correction, so, that the total energy is strictly constant. In practice, however, the smaller values of k (0 < k < 1) are used. Even a small velocities correction in the designed direction leads to the stability increasing of a given numerical algorithm (the energy conservation becomes acceptable). In the future we shall adopt k = 0.5.

Exercise 26. Please simulate the benzene molecule oscillations using the same initial conditions as in Ex. 25. Add the velocities rescaling module in your program (see formula (14)). Please estimate the accuracy of the total energy conservation for the different values of k, namely, k = 0 (see Ex. 25), k = 0.5, and k = 1.

Microcanonical temperature

The classical concept of the temperature is not applicable to the sufficiently small isolated molecular systems. Nevertheless, the concept of

“microcanonical temperature” Tm can be used. The Tm is the measure of the average kinetic nuclei energy and it is defined as

where angular brackets denote the averaging over a few thousands of MD steps, kB is the Boltzmann constant, X is the number of degrees of freedom. For the system containing N atoms X = 3N. If the molecular system in general does not move and rotate (i.e., total momentum and total angular momentum are equal to zero), then the number of degrees of freedom can be reduced by six. Therefore, X = 3N – 6. To simplify the further equations we assume kB = 1. So, the temperature will be measured in electron-volts (energy units). One should take into account the following: 1 electron-volt corresponds to ~11602 K.

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Exercise 27. Please express in Kelvins 0.3 eV and 120 meV. Please express in electron-volts 77, 300, and 1000 K.

Thermostat in the MD simulation

The classical concept of temperature is applicable to the molecular systems, which are in the thermal contact with the other “infinite” (i.e. sufficiently large) system named thermostat. The total energies of such systems are not constant. In other words, they fluctuate in accordance with the Boltzmann’s law. These fluctuations can play an important role in the processes simulated, thus, thermalized system cannot be regarded as isolated one. To simulate the interaction between the system considered and thermostat, several methods are developed. Here we briefly review some of them. The simplest method consists in changing the velocity of nucleus selected randomly to the value chosen in accordance with the Maxwell’s distribution. This changing is made at random time moments. In the other method a variable “viscosity” η (positive or negative) is introduced. This also leads to the changing of nuclei velocities. To realize this approach, the new term into the Newton’s equations is added

Often, viscosity provides invariability of the kinetic energy of the system. However, in this case the Maxwell’s distribution is violated. Note that better approaches allow taking into account the Maxwell’s distribution for the nuclei velocities under the viscosity changing. For more details, see, for example, the following article: Chemical Physics Letters, Vol. 439, P. 219 (2007).

Despite the efficiently of the approaches described above, they are seemed too artificial. More physically justified methods are based only on the direct simulations of the interaction with the thermostat. For example, if the buffer gas plays the thermostat role, then the collisions of its molecules with the system considered should be modeled. However, sometimes it is impossible to employ this approach (see Ex. 28).

Exercise 28. Let us consider the carbon nanotube as a cylinder with the diameter 10 Å and the length 50 Å.

A. Please estimate the number of carbon atoms in the nanotube.

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B.Imagine that the nanotube is placed into the argon buffer atmosphere at normal conditions (i.e., temperature is equal to 300 K, and pressure is equal to 0.1 MPa). Please estimate the mean time period between the two collisions of the nanotube with the argon atom.

C.Please estimate the mean time period between the two collisions of a given carbon atom with any argon atom. How many MD time steps does this period contain, if one time step is equal to 1 fs?

Computing the random numbers with designed distribution

To simulate the interaction of the system with the thermostat, it is often necessary to compute the random numbers with the designed distribution. Most of modern software packages have the possibility to return a uniformly distributed random numbers in the range [0, 1]. Using them one can get numbers with any distribution function p = f(x) via the following procedure. At first, any random uniformly distributed value x from the available range is selected. Next the value of p = f(x) is compared with the uniformly distributed random value rand in the range [0, 1]. If p is greater than rand, then x is accepted. Otherwise, the procedure is repeated for the new x until the acceptance will be reached. As a result, the obtained values of x are distributed in accordance with the formula p = f(x).

Exercise 29. Please make a function returning the nuclei velocity

temperature T = 300 K. Plot the normalized diagram for the 100 000 values returned, and plot the analytical density of distributions on the same figure. The heights of the diagram rows are proportional to the number of random values belonging to the corresponding range (the range from zero to 0.01 Å/fs with the step 0.0005 Å/fs are recommended).

Constant volume conditions

In the previous lecture we discussed how to provide the conservation of the total energy or temperature during the MD simulation. In any case, the other quantities should be constant or satisfy conditions that

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are more sophisticated. For example, imagine that the volume of the system considered is confined. In that case, the atoms should not leave the permissible space during the simulation. It can be achieved using two the most popular approaches. The first is based on the new term addition in the potential energy U function, which is equal to zero for atoms inside the permissible space and is very large for outside atoms. Exceeding of the boundaries leads to a significant increasing of U, thus, the system avoids these states. Another approach requires verification of the nuclei positions at each MD step. If any nucleus exceeds the permissible volume, the special correction is applied. For example, the velocity direction of the nucleus can be changed to send it back, toward the permissible space (so called “elastic reflection” from the boundary). In other simulation, exceeded nucleus can be “transferred” to the random position inside the permissible space (if it is not chemically-bonded with other nuclei) or excluded from consideration (if the conservation of the number of nuclei is not taken into account).

Periodic boundary conditions

If we simulate a part of the crystal or any other periodical structure, then the periodical boundary conditions can be applied. Periodicity means that a given cell considered is surrounded by the similar cells. So, for every atom located at the position ri, there are similar atoms located at the positions ri + n1t1 + n2t2 + n3t3, where n are the arbitrary integer values, and t are the translational vectors. If the cell size is sufficiently large, then in three-dimensional case only twenty-six neighbor cells (with n is equal to –1, 0 or 1) effect on the considered cell. To take into account this fact in the MD simulations, one should consider twenty-six “twins” for every atom in a given cell. If the pairwise nucleus-nucleus term is added in the potential energy function U, one should choose the nearest pair among the all “twins” of the atom. Note that this method is applicable for simple empirical potential functions, but it may fail if the high-level approaches are used.

Equilibrium and non-equilibrium problems

MD simulation is applicable to solve both equilibrium and nonequilibrium problems, but each case has its own specific features. Note

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that equilibrium problems are related to the systems located in the thermodynamically equilibrium state. Only reversible processes can proceed in such systems, and MD procedure provides the development of the system’s phase space to get sufficient statistics for thermodynamic averaging (see the next lecture for details). The long-run studies provides higher accuracy, and one can chose the time period required in accordance with the accuracy needed. On the other hand, non-equilibrium irreversible processes can also be modeled in the frame of MD approach. In this case, MD simulation of the process (chemical reaction, phase transition, etc.) is running until the specific event will be occurred. Below we consider both equilibrium and non-equilibrium problem. We shall start with the equilibrium one.

Statistical ensembles

In statistical mechanics the term “ensemble” is commonly used to define all possible states of the molecular system in equilibrium. Let us consider an isolated molecule that does not interact with the environment. So, its total energy is constant. Since there is no atomic exchange between the system and environment, the total number of atoms as well as chemical composition does not change. If the system volume is also constant, then the system can be described in the frame of microcanonical ensemble (or NVE-ensemble). There are different nuclei positions (system’s microstates) corresponding to the identical values of the number of nuclei (N), volume (V) and total energy (E). All these microstates can emerge during the MD simulation with different probabilities. If the system is in the heat equilibrium with the thermostat, then it is characterized by the temperature T. So, the system should be described in the frame of canonical ensemble (or NVT-ensemble). In that case, the total energy of the system fluctuates with accordance to Boltzmann’s distribution. If the nuclei exchange and chemical equilibrium with the environment are implemented, then the grand canonical ensemble (or μVT-ensemble, where μ is the chemical potential) takes place.

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Statistical averaging over the ensemble

Nuclei motion studied within the MD takes place during a very short period of time (a few femtoseconds) and cannot be directly observed. Measuring any physical quantity, we actually obtain the value averaged over a huge number of microstates that are included in this averaging with the different statistical weights. For example, if we regard the length of the oscillated hydrogen molecule, we obtain the averaged value. The lengths close to the equilibrium make a smaller contribution, because the molecule spends less time in the equilibrium position due to a larger velocity. To obtain the correct value for the molecule length within the MD simulation, we should compute the length in every few steps for a long time and then calculate the arithmetical mean. The different values with the correct statistical weights are automatically occurred among the averaged lengths.

The same is also applicable to any other physical quantity characterizing the molecular system. Molecular system can located in one of the many microstates corresponding to different values of the physical quantity considered. The number of microstates increases rapidly as the number of atoms N increases. In practice, it is incomputable even for relatively small systems containing about ten atoms. Since we cannot make averaging over the all-available system microstates, the correct statistical sampling is needed. The sequence of microstates obtained during the MD simulation can become this sampling.

Exercise 30. A. Imagine that we want to conduct a survey in Russian Federation to obtain the mean age and the mean salary of the citizens. Since the all territory of Russian Federation is very large, we choose the sampling of 2 500 respondents, as it is commonly done. Can we use all students at our University to obtain the representative results? B. Imagine that we want to conduct a survey in our University to obtain the mean age. The new sampling is proposed: fifty professors, fifty associate professors and fifty students. Can we use this sampling to obtain the representative results?

Exercise 31. Apply the velocity Verlet algorithm (10) to the hydrogen molecule assuming the Lennard-Jones potential (1) with the parameters derived in Ex. 4, B. The initial interatomic distance is 0.741 Å that corresponds to the absence of the interatomic forces. The initial nuclei velocities are directed to one another and are equal to v = 0.05 Å/fs.

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