Generalized Solvent Coordinate
5.4 C om putational Strategies
5.4.1 E quilibration from Initial C onfiguration
Before going into the details of how such experiments can be simulated through MD calculations to obtain computationally the solvent response function, a few important technical details are described. Before an attempt may be made to obtain any physical data from an MD trajectory, the system has to be in its equilibrium configuration.
The first part of any MD calculation, then, is its equilibration to the minimum free energy state since any arbitrarily chosen initial configuration is not necessarily a probable member of the equilibrium set of configurations. The choice of the initial starting configuration is an important technical point and the initial configuration should be such that non-physical, large and impulsive forces are avoided since the solutions of the MD equations are liable to diverge when the forces are abnormally large.
One technique is to use a crystalline starting configuration. Since such a config
uration may be too ordered for liquid phase simulation, in the reported calculations the initial configuration is obtained using a “skew start” method[28] that ensures that there is at least a minimum separation between the molecular centers of mass without placing them at the vertices of a crystalline structure. The initial orientations of the individual molecules are typically chosen to be random and the initial velocities are obtained from a Maxwell-Boltzmann distribution.
Since the initial configuration is unlikely to belong to the set of minimum free energy configurations, propagation of the MD equations in an ensemble at a given N (number of particles), V (volume) and E (energy) (NVE) is going to take the system to, on the average, lower potential energy configurations and thereby increasing its kinetic energy. To maintain a generally constant temperature during the actual simu
lation, the approach to equilibrium is accompanied by periodic scaling of the transla
tional and rotational velocities[20, p. 171]. The starting configuration is propagated forward in time with the MD equations and every few time-steps the translational and rotational velocities of the molecules are scaled to the predetermined applied
temperature, at which the simulation is desired to occur, using
| i V i 6T = i m { v ! ) = i / ( w2>. (5.6) This scheme allows for the dissipation of the excess kinetic energy that would build up when the starting configuration does not belong to the set of the m inim um energy states. This procedure, when continued for a few thousand time-steps, allows the system to reach thermal equilibrium. This approach has proved successful in practice. One simple measure of the system’s being in equilibrium is obtained from monitoring the total potential and kinetic energy of the system after scaling has been stopped. If the potential and kinetic energies remain constant, within some fluctuations, without any systematic deviation, the system may be considered to be at equilibrium.
Once the system has reached equilibrium, the scaling is stopped and the system is then allowed to exclusively follow Newtonian dynamics (x { t ) = — ^§j£) within the predetermined ensemble (here, NVE). Since the positions, velocities and orientations are available for all the molecules during the whole trajectory, any physical property of interest may then be calculated.
5.4.2 Solvent S tru ctu re
Before any quantities particular to solvent dynamics are calculated, it must be en
sured that the potential used and the calculational strategy employed (such as PBC or reaction field methods) result in a good description of the solvent. By a good description it is meant that the MD trajectories yield accurate structural and dy
namical information for the solvent. To this end, solvent radial distribution functions (RDFs) are usually calculated and compared to previously known computational or experimental data. The RDF is defined as
ga/3(r) = Vr (5(|r-t-rl a - r 2^|)), (5.7)
where, a and 0 represent the types of atoms and r iQ and r 2p refer to the position of the particle a or 0 on molecule 1 or 2. The average is taken over all the molecules in the system. The RDF is calculated from the pair distances between atoms for a select number of configurations obtained from the MD trajectory. The radial distribution function is a quantity of central importance in liquids because they provide informa
tion about the local short-range order around a centred molecule[29]. The histogram of pair distances is normalized to obtain the RDFs. The normalization is such that, for a single component system, if p = number density, then
r* *
/ pg(r)r dr = N, (5.8)
Jo
where, N is the number of particles in a sphere of radius R c. For water, RDFs of particular interest are those for O-O, 0-H and H-H ones. These quantities axe compared with the experimental quantities to determine the extent to which the po
tential energy function and the calculational scheme yield a correct description of the solvent. Other quantities usually calculated are the velocity correlation function, density profiles at various places within the simulation volume, and the orientational distribution functions. In some studies it has also been attempted to obtain the dipole correlation function ( M( 0) M( t ) ) and the dielectric dispersion curve from such calculations [13,15]. The dipole correlation function is usually the most difficult quan
tity to obtain accurately through simulations.
5.4.3 Solvent D y n a m ics from M D trajectories
Once the system of solute and solvent molecules has been equilibrated and once the structural and dynamical quantities have been calculated to show that the system is a good approximation to that being simulated, the solvent dynamics quantities can be calculated with more confidence. Starting from the calculation for an equilibrated system, two methods may be used to obtain the computational approximation to the solvent response function S( t) described in equation 5.5. Using the fluctuation- dissipation theorem, it can be shown[15, 30] that, in the linear response regime, the
correlation of the fluctuations in the electrostatic potential V (f) created by the solvent at the solute,
m o w ® )
(JK (O )^(O ))’ ^ '
is equal to the solvent response function S(t) that would be obtained from the result of a sudden change in the charge of the solute. It is assumed here that the non
polar interaction between the solute and the solvent is the same regardless of the solute’s net charge. Similar relationships exist between other equilibrium correlation functions and solvent response functions obtained from other step function changes of the electrical properties of the solute. V(t) is simply the electrostatic potential created by the solvent at the solute that can be calculated at every time-step in the trajectory and
SV(t) = (V(t)) - V (t). (5.10)
It is also possible to directly obtain S{t) from MD calculations by simulating the step function change in the electronic property of the solute that is analogous to the experimental change in the solute that would be caused by the interaction of the laser with the solute. This step function change can be simulated by suddenly changing, in an equilibrated system, the electrical property whose effect is being probed. Some of the changes usually probed include the change in the charge distribution (e. g., dipole) in the solute that occurs when the solute is electronically excited or a change in the total charge of the solute that would occur upon photo-ionization. Upon this change, the solvent molecules are no longer in equilibrium with the new electrostatic state of the solute since they were in equilibrium with the ground (or other initial) state of the solute. The solvent molecules change their orientational and translational positions to return to thermal equilibrium with the solute. It is this change that is monitored and analyzed via equation 5.5 to obtain the response function. Computationally, after changing the electrical property in question the MD calculation is continued to obtain the time-dependent solute-solvent interaction energy. The solvent response function
is calculated analogously to experiment,
Ejntit) - Eint{oo)
Eint{0) - Eint{o o )' (5.11)
Eint(t) includes the above defined V{t) along with E u { t) , the Lennard-Jones (or van der Waals) contribution to the solute-solvent interaction energy. Rigorously, the equality between C (t) and S(t) is correct only when V(t) is used in equation 5.11 instead of Eint(t). As a practical matter, for the solute used, E u { t ) is less than 5%
of the total solute-solvent interaction energy and its inclusion in the above equation does not change the resulting solvent response function. This calculation is performed for several different initial configurations of the equilibrated ground state to obtain a representative collection of initial configurations and to compare with C(t) that is, by definition, a configurational average. Comparison of C(t) with (S(t)) also provides a test of the linear response approximation. It is important to note that as the perturbed system returns to thermal equilibrium, there is some increase of kinetic energy and in the temperature of the system (this phenomenon has been referred to as “local heating.”) The system should be large enough to absorb this increase without any change in the observables or the lifetime of the observables should be short enough such that the “local heating” has no effect on it.
Having obtained all the analytical machinery for the propagation of the dynamical equations and for data analysis, we now define the potential function we use to govern the dynamics. A good potential is, of course, essential to obtain good dynamical information from the MD trajectories.