Classical MD

2.2 Molecular Dynamics

2.2.1 Classical MD

InclassicalMD simulation,predefinedinter-atomic potential is used for the calculation of the force, and the second order differential equation is solved to obtain the evolution of the coordinates and velocities in real time [1, 2, 9],

~ F_I³

~ R_I´

= −~_∇IV³

~ R_I´

, (2.1)

M_Id²R~_I

dt² = ~F_I³ R~_I´

, (2.2)

The force (F~_I) is calculated from the negative gradient of interaction potentialV(_R).~ In eq. (2.2),M_I andR~_I ({~^r1,~^r2,~^r3, ...,~^rN}) are respectively mass and coordinates of the I^thatom, in a system ofNatoms/ions. The standard algorithm begins with the initial set of atomic coordinates of the system of interest (usually obtained from experimental data) and their velocities (usually generated from a Maxwellian distribution). The desired temperature of simulation can be achieved through different ways depending upon the statistical ensemble used. In the following sections, the essential ingredients for a MD simulation, namely integration scheme, periodic boundary condition, inter-atomic potentials, various aspects related to ensembles, are discussed in some detail.

CHAPTER 2. THEORETICAL BACKGROUND

2.2.1.1 Integration Scheme

Among different available integration schemes (such as Runge-Kutta, Euler etc.) velocity verlet algorithm is perhaps one of the most popular choice for MD. This algorithm is a modification of the simple ‘velocity’ verlet scheme, and handles positions, velocities, and accelerations of all the particles simultaneously. The algorithm is of the form of,

R(t~ +∆^t) =_R(t)~ ₊~^v(t)∆^t+1 2

~F(t)

M ∆^{t2 (2.3a)}

~^v(t+∆^t) =~^v(t)+ ∆^t 2M

³~_F(t+∆^t)+~_F(t)^´_{, (2.3b)} Elaborately,

• Move the velocity byhalf time-step using the initial velocity and acceleration,

~^v(t+∆^t

2 )=~^v(t)+~F(t)

2M∆^{t (2.4)}

• Move the positions as,

R(t~ +∆^t)=_R(t)~ ₊~^v(t+∆^t

2 )∆^{t (2.5)}

• Calculate the new accelerations~^a(t+∆t) from the new positionsR(t~ +∆^t)

• Move the velocities to full time step using,

~^v(t+∆^t)=~^v(t+∆^t 2 )+1

2~^a(t+∆^t)∆^{t (2.6)} As the new set of positions and velocities of the atoms are available, new iterations are performed to generate the trajectory.

2.2.1.2 Periodic Boundary Condition (PBC)

The number of atoms that can be handled within a reasonable computational expenditure lies typically between 1000-10000. But, as any realbulksystem consists of much larger number of atoms, of the order of 10²³, calculations limiting to this low number of atoms gives rise to spurious ‘surface effects’. This problem arises as large number of atoms resides at boundaries in comparison with the real bulk system. As a result these large number of surface atoms feels a completely different environment than the atoms residing at the central region.

Imposing periodic boundary condition (PBC) is an effective way to eliminate this so called ‘surface effects’ during simulation. This method allows an atom to re-enter from the opposite boundary of the simulation box in case it is leaving the simulation box from one boundary. In other words, one boundary of the simulation box, under PBC, always ‘feels’ the opposite boundary of the box. In practice, this phenomena is realized by considering adjacent ‘images’ of the simulation box in all direction as follows,

x⁰=x_box+n_xL_x (2.7a)

y⁰=y_box+n_yL_y (2.7b)

z⁰=z_box+n_zL_z (2.7c)

wherex_box, y_box,z_boxare position coordinates of the particle inside the simulation cell, n_x,n_y,n_z are integer numbers representing the number of images in each direction, L_x,L_y,L_zare the lattice vectors. In case the simulation box is non-orthogonal, thereal coordinates of the atoms are usually transformed into what is called ‘scaled’ coordinates through a matrix multiplication. This matrix is termed as ‘h-matrix’ and looks like,

h=









a_x b_x c_x 0 b_y c_y 0 0 c_z









(2.8)

where the matrix elements are different components of the lattice vectors,~^a,~band~^c of thesimulation box. The ‘scaled’ coordinates is obtained by,

~^s=h⁻¹~^{r (2.9)}

which will provide a orthogonal box with unit dimensions. The PBC can be applied in standard way, following equations (2.7) but for a cubic box of unit dimensions. Then all the coordinates are transformed back to real coordinates by,

~^r=h~^{s (2.10)}

2.2.1.3 Inter-atomic Potential

The accuracy of the results produced from classical MD majorly depends on the inter- atomic potential form and parameters used for the simulation. Some of the popular functional forms for inter-atomic potential available in the literature are, Lennard-Jones, Vashistha Rahman, Born Mayer etc.

CHAPTER 2. THEORETICAL BACKGROUND

Lennard-Jones potential is given as, V=4²[( σ

r_{i j})¹²−( σ

r_{i j})⁶] (2.11)

where²is the depth of the potential at the equilibrium distance for a pair of atom,σ is the diameter of atom and r_{i j} is the distance between two atoms. The first term in the bracket represents the Pauli repulsion originating from the electron cloud overlapping and the second term represents the attractive part originating from the instantaneous dipole-dipole interactions. Lennard-Jones potential is extensively used to simulate inert gas systems, such as, argon, but can be extended to simulate certain ionic solids by adding the Coulombic term to the potential.

The functional form of Born Mayer (Buckingham) potential[10] is given as, V=q_iq_j

r_{i j} +A_{i j}exp(r_{i j}

ρ )−D_{i j} r⁸_{i j} −C_{i j}

r⁶_{i j} (2.12)

where qis the charge, A_{i j} andρare respectively the strength and range of the overlap repulsive term, D_{i j}, C_{i j} are respectively the dipole-quadrupole interaction term and the dipole-dipole interaction term. This potential has widely been employed to simulate inorganic solids, molten salts and many fast ion conductors such as β-alumina,Li₃N, CaF₂etc.

The Vashistha Rahman potential [4] is given as, V= q_iq_j

r_{i j} +A_{i j}(σi+σj)^{ni j} rⁿ_{i j}^{i j} −P_{i j}

r⁴

i j

−C_{i j}

r⁶_{i j} (2.13)

where q is the charge,σ is ionic radius, A_{i j}, P_{i j},C_{i j} are respectively the overlap repulsive term, charge dipole interaction and the dispersion energy terms.n_{i j} is usually taken as 11, 9 and 7 forcation-cation,cation-anionandanion-anionpairs respectively.

This potential form was first proposed for the study of fast ion conductor A gI. Later on, it has been implemented to study several other fast ion conductors as well.

While calculating the interaction potentials under PBC, if all the ‘real atoms’ and the ‘images’ are considered, the computational expenditure will be too high and such calculations may be unnecessary as well. For ionic systems, the potential usually consists of two parts, theshort rangeterm and thelong range Couloumbicterm. To minimize the computational expenditure, often acut off distance (R_c) is incorporated beyond which the potential energy contributions from the short-range interactions are neglected. The implementation ofcut off distance is pretty straightforward for theshort rangeterm, and

is incorporated in such a way that an atom will either interact with another atom or it’s image, whichever is nearer – but not both. This is known asminimum image convention in which thecut off is usually chosen as half the box length (R_c≤L/2).