Chapter 2. Theoretical background
2.3 Geometry optimisations, molecular dynamics and free energy simulations
Geometry optimisations on a 0 K Born-‐Oppenheimer PES are probably the most common way of using quantum chemical methods to model molecular behaviour. The nuclear motion is ignored and critical points on the surface are located by calculating the atomic forces of the nuclei (differentiation of the total energy with respect to atomic coordinates) and searching for chemically relevant critical points which are either minima or 1st order saddle points. The 0 K potential energy difference between
critical points can be regarded as an excellent approximation to conformational energies, reaction energies and transition states of reactions, obtainable from experiments. Corrections for the lack of nuclear motion can be obtained by solving approximations to Eq. 10, e.g. the harmonic approximation, which results in zero-‐ point energy corrections and vibrational frequencies which when combined with approximations for rotational and translational motion from statistical mechanics, yields corrections so that temperature-‐dependent enthalpies, entropies and free energies can be calculated.
However, there are situations where the dynamic behaviour of the nuclei should be accounted for from the beginning. Molecular dynamics simulations involve solving Newton’s equations of motion for the classical nuclei.
Newton’s second law relates the force, mass and acceleration of particles: F=ma The nuclear forces (and masses) of a molecule can thus be related to velocities and changes of nuclear positions according to:
(30) Solving Eq. 30 enables one to calculate how the nuclear positions change as a function of time. Assuming a molecule with an initial position vector ri containing all atomic
coordinates, the position vector a timestep later can be given by a Taylor expansion:2
(31) or equivalently: (32) Different algorithms have been developed to conveniently estimate the position vector a timestep later. One is the Verlet algorithm:49
(33)
(34) The Verlet algorithm is correct to third order (the third term of the Taylor expansion cancels out in the derivation of the algorithm) and easily allows the prediction of the next position using the information of the current and previous positions and the calculated forces. The accuracy of the algorithm is controlled by choosing smaller and smaller timesteps.2
A problem with the Verlet algorithm is that velocities are not part of the equations, creating difficulties in using the algorithm to generate ensembles at constant temperature as temperature is usually maintained by modifying the velocities of particles.
In the leapfrog algorithm, however, velocities appear explicitly:2
(35)
(36) Here the velocity and position updates are out of phase by half a time step. This
nevertheless leads to very accurate trajectories and is accurate to third order as the Verlet algorithm but is numerically more stable. The leapfrog method was used in the molecular dynamics simulations in this work.
Any computational method (from quantum mechanics or molecular mechanics) can be used to evaluate the forces in the Verlet and leapfrog algorithms and hence the time-‐dependent behaviour of molecular systems can be studied under the
assumption that the classical behaviour of the nuclei is an accurate enough
representation of their real quantum nature. This assumption is generally valid, with the lightest nucleus, hydrogen, being a borderline case.
Molecular dynamics simulations generate trajectories of an ensemble, the most straightforward one to calculate being the NVE ensemble which corresponds to the
a= dv dt = d2r dt2 = F m =− dE dr 1 m ri+1 =ri+ dr dt(∆t) + 1 2 d2r dt2(∆t) 2+ 1 6 d3r dt3(∆t) 3+... ri+1=ri+vi(∆t) + 1 2ai(∆t) 2+1 6bi(∆t) 3+... ri+1 = (2ri−ri−1) +ai(∆t)2 ai = Fi mi =− 1 mi dE dri ri+1 =ri+vi+1 2(∆t) vi+1 2 =vi− 1 2 +ai(∆t)
particle number, volume and energy being constant during the simulation. As the energy is fixed, although kinetic and potential energy will be interconverted, the temperature can fluctuate considerably but no thermal exchange with the
environment is allowed (in fact there is no environment). This corresponds well to the experimental situtation of a molecule in a vacuum. In condensed phase
experiments, the temperature is often held constant, and hence the corresponding ensemble to use in simulations is usually either NVT or NPT where in the former the volume and temperature are constant, while in the latter the pressure and
temperature are constant.
The NVT ensemble (which is used in this work) is also called the canonical ensemble. In order to make the temperature constant, the simulated system is usually coupled to a heat bath, or thermostat. The Nosé-‐Hoover method50-‐53 is an example of where
the thermostat becomes an integral part of the molecular system as an artifical dimensionless particle with a mass and velocity is introduced and equilibrates with the system and exchanges kinetic energy with all real particles of the system. By controlling the equations of motion of this dimensionless particle, any desired temperature can be set while preserving all the features of the Verlet/leapfrog algorithm and the NVT ensemble.
The free energy is the thermodynamical quantity directly related to the equilibrium constant and is related to the probability that a system will be in a particular state. The ability to determine free energy differences from molecular dynamics
simulations is highly useful as they can then be directly related to the thermodynamical data from experimental measurements.
The Helmholtz free energy of a system in the NVT ensemble is given by:
(37) where Q is the partition function of the system which is expressed classically as a double integral over all energy states which are functions of spatial and momentum coordinates:1
(38) It is very hard to determine free energy differences of a system by direct calculation of free energies by the equations above due to the large number of degrees of freedom that need to be sampled. However, useful methods to calculate free energy differences from molecular dynamics simulations nevertheless exist. The free energy perturbation expression:
(39) allows direct calculation of the free energy difference between state a and state b where all the ensemble averaging is based on state a.1 As long as state a and state b
are sufficiently similar (i.e. the perturbation is small), this is a very useful
approximation. It can additionally be expanded so that the free energy difference between a and b is calculated by dividing up the path connecting a and b into intermediate states, calculating the energy difference between each intermediate state and add everything up in the end.
A=−kBT lnQ
Q=
� �
e−E(p,r)/kBTdpdr
Another very useful method is the thermodynamic integration technique where the free energy difference can be determined by integration of the ensemble average of the potential energy gradient w.r.t. fixed values of the reaction coordinate. For certain reaction coordinates this becomes equal to integrating over the ensemble average of the force of the constrained reaction coordinate:54,55
(40) A reaction coordinate ξ, connecting the two states is thus identified and constrained. A number of simulations are then run, constraining different values of the reaction coordinate while sampling the force of the constraint f(ξ). Integrating over the average constrained force for each simulation (the more simulations the better) results in the free energy difference between state a and b. In this work, the reaction coordinate will be a geometric variable (although it can be any variable shared by state a and b).