Катин МОЛЕЦУЛАР ДЫНАМИЦС ИН МУЛТИСЦАЛЕ МОДЕЛИНГ 2015

Exercise 9. Propose another two-parametric fitting function instead of (4) providing the oscillation frequency calculated closer to the experimental value mentioned in Ex. 5.

Construction of the simplest empirical potential for hydrocarbons

In this lecture we try to construct the empirical potential for hydrocarbon molecules. Although this oversimplified potential is not valid for high-accuracy MD studies of real systems, it will be useful for understanding the crucial ideas of the MD approach.

Our main purpose is to compute the potential energy U, when the all nuclei positions are known (for example, one has the file containing Cartesian coordinates for each nucleus). The only two types of atoms (hydrogen and carbon) are considered. Let us assume that the U is a sum of the pairwise interaction terms, and the three-nucleus angular terms. The former are determined by distances between two corresponding atoms, and the latter are determined by corresponding valence angels (angles between atomic bonds). Let us also assume that the contributions of the angular terms are proportional to quadratic deviations of the corresponding bond angles from their “ideal” value of 180°. This assumption reflects the repulsion between two electronic densities that form covalent bond, and therefore the angle of 180° provides maximum distance (minimum repulsion) between the centers of bonds. Another assumption is that the two atoms form the covalent bond only if the corresponding interatomic distance is less than 1.8 Å (this value corresponds to the maximum covalent bond length; here we do not take into account weak non-covalent interactions). Moreover, the number of bonds that a given atom can form with other atoms is limited by its valence (one for hydrogen and four for carbon atom). Thus, for a given atom we should choose the nearest neighboring atoms for the covalent bond formation. As a result, we obtain

where Upair is a pair potential, which may be chosen in form (1) or in any other form (see Ex. 5), rij is the interatomic distance between i-th

and j-th atoms, aklm is the valence angle between the bonds connecting l- th atom with k-th and m-th atoms, and p is the fitting parameter adjust-

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ing the contribution of the angular terms to the total energy (here we adopt p = 0.0005). The first sum in formula (5) is calculated for all pairs of bonded atoms, whereas the second one is calculated for all valence angles of the molecule.

In practice, the computing of the potential energy U begins with the determination of the first coordination number for each atom (i.e., for each atom the number of the nearest neighbors Z should be obtained). After that, it is necessary to reduce the number of the neighbors to provide the Z-value for each atom not exceeding its valence. The reduced Z value corresponds to the number of terms that this atom introduces to the first sum in the formula (5) (one should avoid the double accounting of the same bond, i.e., i-j and j-i terms). This value is also determine the number of terms that a given atom introduces to the second sum in the formula (5): zero for Z = 0 or 1, one for Z = 2, three for Z = 3, six for Z = 4 (there are no values of Z greater than four due to the carbon valence is equal to four).

Exercise 10. Find the experimental value for the equilibrium bond length, binding energy and vibrational frequency for both C2 and CH molecules. The actual data is available on NIST Computational Chemistry Comparison and Benchmark DataBase (http://cccbdb.nist.gov). Using this data, obtain the values of aCC, aCH, bCC, and bCH parameters for C-C and C-H interactions describing by formula (1) (see also Ex. 4). Try to use another expression instead of formula (1) (see Ex. 5) providing the calculated oscillation frequency closer to the experimental value than formula (1).

Exercise 11. Implement the described algorithm (see formula (5)) as a computer procedure using any programming language (for example, C/C++, MATLAB, Fortran, Pascal, Basic, etc.). This procedure must use the text file containing Cartesian coordinates and types (hydrogen or carbon) for all atoms of the molecular system as the input data, and it must return the value of potential energy U. Double-precision computation is highly recommended. Use the results obtained in the Ex. 10 to calculate the pairwise energy Upair.

Drawbacks of the constructed potential function

The potential function proposed above does not reflect some crucial properties of hydrocarbon molecules. One of the serious drawback is

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that the bond orders are not taken into account. The order of the carboncarbon bond may be equal to one (as in the ethane molecule), two (as in the ethylene molecule), three (as in acetylene molecule), and even four (as in the carbon dimer). All these bond types have different lengths and energies. Another drawback is the pairwise interaction assumption, whereas the energy in principle cannot be expressed as a sum of pairwise terms. For example, the energy of the CH2 molecule cannot be reduced to double energy of the CH molecule (see Ex. 12).

More sophisticated empirical potential functions overcome all mentioned above drawbacks. For example, conventional potential functions for hydrocarbon systems proposed by Tersoff and Brenner take into account the local environment of each atom. Nevertheless, empirical potential functions cannot reach the high accuracy, since the bond-order model describes the electronic density distribution very inaccurate.

Exercise 12. Find the experimental values of the atomization energies for both CH2 and CH molecules. The actual data is available on the Internet (see reference in Ex. 10). Is the ratio of these energies equal to 2? Are the first-order bonds in the CH2 and CH molecules equivalent?

Exercise 13. Study the original articles written by Tersoff (Physical Review B, Vol. 37, P. 6991, 1988) and Brenner (Physical Review B, Vol. 42, P. 9458, 1990). These articles describe the well-known empirical potential functions. Please prepare the presentation about these potentials. Pay attention to different terms in these functions associated with the different types of physical interaction.

Exercise 14. Find any recent research paper describing a new empirical potential for hydrocarbons. Prepare the presentation about this potential. Please identify the main differences and improvements between this potential and the conventional Tersoff-Brenner potential.

Forces acting on the atoms

Since we can compute the potential energy U as a function of nuclei positions, we can also compute the force fi acting on the i-th atom. According to the general rule, the force can be determined as the partial derivative of the potential energy:

The partial derivative can be calculated numerically or analytically. Numerical calculation is easier to realize. One should compute the ratio

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of the small change of U associated with the small displacement δ of a given atom to the value of this displacement. However, the problem of the δ choice and the problem of checking the convergence are appeared. Moreover, numerical calculation needs more computer resources and provides lower accuracy. Thus, the analytical approach is preferable in any case, when it can be applied.

Exercise 15. It is necessary to calculate the forces acting on every atom of the C60 fullerene in the frame of simple potential model described above (see formulae (5) and (6)). They can be computed analytically or numerically. Please estimate the ratio of the computer time required for numerical calculation to the computer time required for analytical one. Take into account that in both cases the same machine is used. Note that in the fullerene each atom has the common bonds with three other atoms.

Exercise 16. Add a new procedure to your program developed in Ex. 11. This procedure should use the text file containing Cartesian coordinates and types (hydrogen or carbon) for all atoms of the molecular system as the input data, and it should return the array of forces acting on the all atoms (3N values for N atoms should be obtained). Doubleprecision computation is highly recommended. The force calculation should be realized in two ways: analytically and numerically. Please compare the values obtained in both cases and estimate the accuracy of the numerical approach.

Exercise 17. Study the original article written by Feynman (Physical Review, Vol. 56, P. 340, 1939). This article describes the analytical method for the forces calculation in the tight-binding scheme. Please prepare the presentation about this approach.

Structural relaxation

At zero temperature every physical system goes to the state corresponding to the local or global minimum of the potential energy U. So, if the U is given as a function of all nuclei positions, then one can obtain the optimum geometry of the equilibrium state by minimizing the U value. In the general case, for the arbitrary U the minima search is the unsolvable problem due to a huge dimension (the number of variables is equal to 3N, where N is the number of atoms). However, if the U function is sufficiently smooth, a set of optimization methods can be applied.

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The simplest method is that the atoms are “moved” along the forces applied, and their displacements are proportional to the acting forces. In other words, cyclic optimization process

is realized. The stopping criterion is that the forces become sufficiently small (for example, maximum force is less than 0.01 eV/Å). The step value should be selected reasonably to provide the quick convergence (we adopt step = 0.001).

Exercise 18. Add a new procedure to your program developed in Ex. 11 and Ex. 16. This procedure should use the text file containing Cartesian coordinates and types (hydrogen or carbon) for all atoms of the system as the input data, and it should return the optimized Cartesian coordinates. Double-precision computation is highly recommended.

Hessian matrix and the minimum criterion

Zero forces do not guarantee that the minimum of the potential function U is found. To verify the intended minimum, the analysis of second derivatives is required. The matrix of mixed second derivatives, known as Hessian, is defined as

The Hessian dimension is 3N 3N, where N is the number of atoms considered. After the diagonalization of this matrix, eigenvalues can be easily found. Zero eigenvalues indicate that the dimension of the problem can be reduced, since the potential energy U does not depend on some generalized system coordinates. So, in the absence of external forces neither translations nor rotations of the molecular system as a whole do not change its energy. Therefore, the system dimensionality can be reduced by six, and six eigenvalues should be equal to zero (note that the exact zero values cannot be reached due to the limited computer accuracy). One can “fix” the first atom in the coordinate origin, the second atom in the Ox axis, and the third atom in the Oxy plane. Thus, the six variables r1x, r1y, r1z, r2y, r2z, r3z will be equal to zero, and the dimensionality problem will be reduced by six.

If all eigenvalues of the Hessian are positive (and all forces are equal to zero), then the local or global minimum of U is reached. The negativity of at least one of the eigenvalues means that the atomic configura-

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tion is not optimized.

Exercise 19. Imagine that the benzene molecule C6H6 is placed in the external force field f(x, y, z) = –kxi (i is the ort of the Ox axis). Please define the reduced dimension of the Hessian matrix for this case.

Exercise 20. Please add a new procedure to your program developed in Ex. 11, 16, 18. This procedure should use the text file containing Cartesian coordinates and types (hydrogen or carbon) for all N atoms of the system as the input data, and it should return the 3N Hessian eigenvalues. Double-precision computation is highly recommended. Standard libraries for matrix diagonalization (such as eigen, imsl, etc.) should be used. Optimize the geometry of the benzene molecule and verify the positivity of all Hessian eigenvalues.

The main idea of MD methods

The atoms begin to move, if the molecular system is under the influence of internal and (may be) external forces. If the initial positions, initial velocities and forces acting on every atom at each instant of time are known, then the trajectories of motion can be calculated via the Newton’s equations integration. The simplest numerical integration scheme is based on the assumption of uniformly accelerated motion during the sufficiently small period of time δt:

So, if the positions and velocities are known at the current instant of time t, they can be computed at the next moment t + δt using formulae

(9). Choosing the time step δt, one should take into account that too large δt leads to the loss of accuracy, since the motion during the long time step cannot be regarded as uniformly accelerated motion. On the other hand, too small δt leads to the increasing of computational steps, thus, the additional computational resources are required. In any case, δt should be much less than the characteristic vibration period of the system. Thus, δt ~ 1 fs is usually a reasonable choice.

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Numerical schemes for Newton’s equations integration

Numerical scheme (9) illustrates the main idea of MD methods, but it is never used in practice due to its low accuracy. But it can be significantly improved by replacing the forces acting at the moment t with arithmetic mean of the forces acting at the moments t and t + δt:

x, y, z.

The obtained scheme (10) is known as velocity Verlet algorithm and is widely used in MD simulations. It has the higher order (second) compared with the (9) scheme (first). It means that if the forces are sufficiently smooth coordinate functions, then the errors obtained at each MD step are proportional to δt2 instead of δt. The other more sophisticated schemes are also developed, and some of them are listed in Table 1.

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Application the high-order schemes makes each MD step more laborious, but allows to reduce the total number of steps by increasing the magnitude of single time step δt. However, these high-order approaches are based on the Taylor series of the potential function U. Therefore, the potential function should be sufficiently smooth.

Fig. 3. Fracture on the potential energy function because of the electronic levels degeneration. Energies of two electronic levels (up) and the resulting energy (bottom)

In practice, the potential energy function often has fractures at the semi-empirical and ab initio levels of theory. These fractures are associated with the electronic states degenerations. To make the physics of this process clearer, let us consider two atoms separated by the distance r. Electron “chooses” the lowest energy state between the atom-

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localized and hybridized states. The most energetically favorable state depends on the r value. Thus, the atom-localized state minimizes the Coulomb attraction energy between electron and nucleus, whereas the hybridized state minimizes quantum electron energy since the latter is more spatially delocalized. It is clear that at a certain critical r the most energetically favorable state changes. This leads to the changing in the character of the dependence U(r). So, the fracture on the U as a function of the atomic positions can appear (see Fig. 3). Insufficient smoothness of the potential function U makes the high-order schemes inapplicable. That is why we recommend to use the Velocity Verlet algorithm (10) in the most cases.

Exercise 21. Demonstrate that the second-order Störmer’s method (the second line in the Table 1) is identical to the velocity Verlet method (the first line in the Table 1).

Exercise 22. Add a new procedure to your program developed in Ex. 11, 16, 18, 20. This procedure should use the text file containing Cartesian coordinates and velocities for all N atoms of the system as well as the time step δt and the total number of MD steps as the input data. The procedure should return the resulting coordinates and velocities after MD simulation. Double-precision computation is highly recommended. Velocity Verlet algorithm (10) should be implemented.

Molecular visualizing

Usually, for analyzing the results obtained within the MD simulations, visualization of the atomic movements is needed. Often a “simple observation” of the molecule system evolution in a “real-time” mode allows understanding of corresponding physical or chemical processes. There are many molecular visualizers. The most popular program packages are Avogadro, Spartan, ChemCraft, etc. To perform valid MD simulations the using of such programs is necessary. Here we present a brief description of the ClustVis package. It is the simplest freeware open source molecular visualizer. It can be downloaded from http://www.ntbm.info.

In the ClustVis the atom-bond notation is used to display molecules (spheres represent atoms and cylinders represent bonds). The bond is

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displayed if the distance between two atoms is lower than 1.8 Å (which corresponds to characteristic covalent bond). The number of displayed bonds is also limited by atomic valence. Molecule can be scaled and rotated around its geometric center. The program allows user to create/delete atoms and to change atom location, “moving” it by the use of computer mouse. Thus, the possibility to create new cluster configurations is provided. When the program is just started, every atom of a certain chemical element has its own default color. The color of every atom or any atom group can be changed. There is also a possibility to specify “sizes” of atoms and bonds (default radii of spheres and cylinders are 0.24 and 0.1 Å, respectively). Various tools for cluster geometry analysis are also included. It allows getting information about bond lengths and valence triand tetratomic angles.

The program can be used in one of two displaying modes: static and dynamic. The static mode allows working with molecules, which geometrical configuration remains constant over the time, while the dynamic mode is a tool for working with films obtained from MD simulation. Each frame of such film specifies some novel geometrical configuration. The dynamic mode is inherently an extension of the static mode because it allows the possibility to work with a film frame as it is a static molecule: rotate, scale, move atoms, change colors and sizes, get information about molecular geometry, using the tool for displaying bond lengths and valence angles. For working in the dynamic mode there is a special control panel which allows the user to start and to stop playing film, to make a single frame jogging, to specify the number of the required frame, and also to increase or to reduce the play speed.

ClustVis can get input data directly from any other program as four arrays containing Cartesian atomic coordinates and atomic chemical symbols. Reading input data from file is also available, common formats *.xyz and*.mol are permissible. To input a film, all frames should be written in the text file in a row. In this case, in the first string the number of atoms per frame must be specified. Each frame or entire film can be saved in the same format. The program has a simple user-friendly interface. It is controlled by the use of a computer mouse. The entire window space is divided into three parts. They are window title, menu bar and visualization area. Fig. 4 and Fig. 5 show the obtained image of the

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