Molecular Dynamics

Molecular dynamics allows the study of thermodynamic and dynamic properties of a many body system over a set time period. There are two main groups of molecular dynamics simulations, classical molecular dynamics (MD) and ab initio molecular dynamics (AIMD), which describe the interatomic interactions in different ways. In MD, simulation particles are treated as classical objects and potentials are used to describe interatomic interactions, whereas in AIMD interatomic interactions are calculated from first principles. Both the classical and ab initio methods aim to describe the interatomic interactions sufficiently to approximate the movement of ions at equilibrium. However, due to differences in the way the forces are calculated MD is suitable for dynamic studies of large systems made up of many thousands of atoms for nanosecond timescales while the computational cost of AIMD becomes unfeasible for systems and timescales of this size.97

2.12.1 Classical Molecular Dynamics

In MD, molecules are treated as classical objects obeying the laws of classical mechanics which describe the position, velocity and acceleration of each atom as they vary with time.98,99 _For the system of interest a set of initial conditions are specified which define the positions and velocities of all particles, and an interaction potential which derives the forces between particles. The trajectory of a system is followed by solving the differential equations from Newton’s equation of motion, (𝐅𝑖 = 𝑚𝑖𝑎𝑖), where 𝑎𝑖 is the acceleration of each particle:98

𝐅𝑖 = 𝑚𝑖

𝑑2_𝐫

𝑖(𝑡)

𝑑𝑡2 𝐄𝐪. 𝟐. 𝟓𝟕

Where 𝐅 is the force acting upon the particle 𝑖 at position 𝐫 at time 𝑡 and 𝑚 is the mass of the particle.

45 The force can also be expressed as the gradient of the potential energy of the system:

𝐅𝑖 = −∇𝑖𝑉 𝐄𝐪. 𝟐. 𝟓𝟖

combining equations 𝐄𝐪. 𝟐. 𝟓𝟕 and 𝐄𝐪. 𝟐. 𝟓𝟖 relates the derivative of the potential energy to changes in the position of atoms as a function of time:

−𝑑𝑉 𝑑𝐫𝑖

= 𝑚𝑖

𝑑2𝐫𝑖(𝑡)

𝑑𝑡2 𝐄𝐪. 𝟐. 𝟓𝟗

New atomic positions from time 𝑡 to 𝑡 + 𝛿𝑡 are progated through the solving of integrators such as the Verlet algorithm.100–103 _{The time step, 𝛿𝑡, has to be chosen carefully in order to} guarantee the stability of the integrator and reduce drift in the system’s energy. Interaction models or potentials, such as the Lennard-Jones pair potential,104 _{describe the interaction a} systems constituents. The pressure, temperature or the number or particles of a simulation are controlled using a statistical ensemble. For example, in the micro canonical (NVE) ensemble the number of atoms (N), the volume (V) of the simulation cell, and the energy (E) are kept constant, essentially representing an isolated system. These three parts, the integrator, the interaction model and the ensemble, define a MD simulation and are varied to describe the system precisely over the timescale of a simulation.

In systems where temperature control is essential a thermostat can be used such as in an NVT ensemble.97 _{The Nosé-Hoover thermostat}105,106 _{is an algorithm for constant temperature} simulations used across MD and AIMD simulations. The thermostat introduces a fictitious ‘heat bath’ term with an associated mass, 𝑄. At the correct value of 𝑄, the thermal interaction between the heat reservoir and the dynamic system maintains the temperature of the system. A 𝑄 value which is too high would result in slow energy flow between the system and the reservoir while a 𝑄 value which is too low results in rapid temperature fluctuations. The Nosé -Hoover thermostat is commonly used as one of the most accurate and efficient methods for constant-temperature molecular dynamic simulations.107

The use of potentials instead of describing particle interactions means that MD is an ideal method for examining the dynamics within a system of millions of atoms for timescales in the nanosecond range without an excessive computational cost. However, there are limitations to the atom-atom interactions which can be simulated using potentials and as electron configurations are undefined any changes in bonding and reactions between molecules cannot be easily simulated.103,108

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2.12.2 Ab Initio Molecular Dynamics

Ab initio molecular dynamics (AIMD) unifies molecular dynamics and electronic structure theory by computing the forces acting on the nuclei from electronic structure calculations performed “on-the-fly” as the dynamic trajectory is generated. One of the advantages of this method over the classical molecular dynamics approach is that electrons are explicitly included, meaning that it is possible to see how chemical processes occur over the timescale of the calculation. However, the introduction of the first principles (ab initio) basis for calculations comes at significant computational cost.

The computational cost of calculating the ground state energy of each step reduces a AIMD calculation to a few hundred atoms and a few picoseconds of length compared to the million atom, nanosecond long classical molecular dynamics calculations.109 _{A shorter timescale raises} concerns that a chemical system is not fully equilibrated, resulting in a highly variable energy, temperature or pressure.110 _{In the case of the total energy of a system, an erratic energy or a} system where the energy drifts across the timescale of a simulation can be controlled with a well-chosen SCF convergence criteria.111 _{In contrast to 0 K static gas phase DFT calculations, a} carefully parameterised thermostat and barostat can maintain a constant temperature and pressure.97

The periodic nature of AIMD calculations and the relatively small periodic box compared to classical MD means that self-interaction error can be a concern.112 _{Self-interaction error is} where a molecule or atom in one periodic box will interact with itself in an adjacent box. The size of simulation box must be large enough that the distance between a molecule and its periodic image minimises any self-interaction.113 _{In the case of bulk solid calculations or} material slabs, a vacuum between periodic boxes can also be used.97

In spite of these concerns with AIMD, for the calculations laid out in the subsequent chapters, it is the appropriate computational method, and the theory behind all AIMD techniques is the time-dependent Schrödinger equation. In principle the use of the time-dependent Schrödinger equation describes how a molecular system evolves over time:

𝐻̂Ψ = 𝒾ℏ𝜕Ψ

∂𝑡 𝐄𝐪. 𝟐. 𝟔𝟎

Where 𝐻̂ is the standard Hamiltonian given in 𝐄𝐪. 𝟐. 𝟏𝟐. There are three approaches to AIMD, Ehrenfest molecular dynamics (EMD)97_{, Car-Parrinello molecular dynamics (CPMD)}114 _and Born-Oppenheimer molecular dynamics (BOMD).54

47 EMD is a mixed approach where the classical equations of motion for electrons are solved simultaneously with the Schrödinger equation for the electrons. The energy of a molecular system is minimised in the first time step, the time scale and time step are determined by the dynamics of the electrons and thus EMD needs a very short time-step to allow the integration of the electronic equations of motion. This confines the MD trajectory to much smaller time scales than BOMD which depends on nuclear motion.

CPMD also calculates the forces at each time step using DFT, however the electrons and nuclei are considered to be moving at the same time, and the electrons are kept close to the ground state by use of “fictitious dynamics” which keeps the electrons in a low electronic fictitious temperature oscillation around a constant value. The evolution of a trajectory according to CPMD does not require an energy minimum to be reached in each time step.

Born-Oppenheimer molecular dynamics (BOMD)54_{, is the AIMD method used for the} calculations presented in this thesis, and in the CP2K package.115 _{It uses the assumptions of} the Born-Oppenheimer Approximation, described in Section 2.5. BOMD includes the electronic structure in dynamics simulations by solving the static electronic structure problem at each trajectory step using the set of fixed nuclear positions at that time step.

The BOMD method can be written: 97 𝑚𝑖

𝑑2𝑹𝐼(𝑡)

𝑑𝑡2 = −𝛻𝑖𝑚𝑖𝑛𝛹{⟨𝛹|𝐻̂𝑒𝑙|𝛹⟩} 𝐄𝐪. 𝟐. 𝟔𝟏

for the nuclear, 𝐑𝐼, degrees of freedom, where 𝐻̂𝑒𝑙 is the electronic Hamiltonian (𝐄𝐪. 𝟐. 𝟏𝟓)

and 𝑚𝑖 is the mass of electron 𝑖. An energy minimum has to be reached in each time step using

Kohn-Sham DFT, described in Section 2.10.3, to ensure accurate results.97 _{The electronic} structure is solved from the time-independent Schrödinger equation, 𝐄𝐪. 𝟐. 𝟓, while the nuclei positions are propagated according to 𝐄𝐪. 𝟐. 𝟔𝟏. Therefore, the time-dependence of the electronic structure is a consequence of nuclear motion. As the energy must be minimised at each time step the efficiency of the BOMD approach depends on how effectively ground state energy can be reached.

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