Nudged elastic band and free energy calculations
In this exercise we will study the $S_N2$ nucleophilic substitution reaction:
$$ \text{Cl}^- + \text{CH}_3\text{Cl} \longleftrightarrow \text{Cl}\text{CH}_3 + \text{Cl}^-. $$
For the energy and force evaluation, we will use the Parameterization Method 6 (PM6), which is a relatively inexpensive semiempirical model for electronic structure evaluation. For accurate reaction characterizations, ab initio methods, such as DFT or higher level, should be used instead.
NEB activation barrier
In order to find the activation barrier for the reaction we will use the NEB method. The NEB method provides a way to find the minimum energy path (MEP) between two different conformations of the system which correspond to local potential energy minima. The MEP is a one-dimensional path on the $3N$-dimensional potential energy landscape for which every point along its path is a potential energy minimum in all directions perpendicular to the path. Any MEP passes through at least one saddle point and the energy of the highest saddle point is the peak of the activation barrier of the reaction.
Geometry optimizations
To start the NEB calculation, we first need two different geometries at local energy minima, which we can obtain by running geometry optimization calculations. Use the following CP2K input script to obtain the first geometry:
- geo.inp
&GLOBAL PRINT_LEVEL LOW PROJECT ch3cl RUN_TYPE GEO_OPT # Geometry optimization calculation &END GLOBAL &MOTION &GEO_OPT # Parameters for GEO_OPT convergence MAX_FORCE 1.0E-4 MAX_ITER 2000 OPTIMIZER BFGS &BFGS TRUST_RADIUS [bohr] 0.1 &END &END GEO_OPT &END MOTION &FORCE_EVAL METHOD Quickstep # Quickstep - Electronic structure methods &DFT CHARGE -1 # There is a negatively charged anion &QS METHOD PM6 # Parametrization Method 6 &SE &END SE &END QS &SCF # Convergence parameters for force evaluation SCF_GUESS ATOMIC EPS_SCF 1.0E-5 MAX_SCF 50 &OUTER_SCF EPS_SCF 1.0E-7 MAX_SCF 500 &END &END SCF &POISSON # POISSON solver for non-periodic calculation PERIODIC NONE PSOLVER WAVELET &END &END DFT &SUBSYS &CELL ABC 10.0 10.0 10.0 PERIODIC NONE &END CELL &COORD C -4.03963494 0.97427857 -0.29785096 H -3.97152206 2.11080568 -0.35500445 H -3.95108814 0.19729141 0.53163699 H -3.95810737 0.47922964 -1.32151084 Cl -5.77885435 1.07089312 -0.04609427 Cl -1.77974793 0.99422072 -0.29785096 &END COORD &END SUBSYS &END FORCE_EVAL
- Run the geometry optimization calculation
- Modify the initial geometry guess in the input and run a second optimization to obtain the other local minimum geometry (The other
Clneeds to make a covalent bond with theC)
Another possibility is to change the PROJECT name in the input file which will result in different names for the output files.
NEB
Now we are ready to start the NEB calculation. Obtain the two optimized geometries from the GEO_OPT trajectories (the last step in the corresponding *-pos-1.xyz file) and place them in a separate folder in files named init.xyz and final.xyz, respectively.
Now, the NEB calculation can be run by the following input script:
- neb.inp
&GLOBAL PRINT_LEVEL LOW PROJECT ch3cl RUN_TYPE BAND # Nudged elastic band calculation &END GLOBAL &MOTION &BAND NUMBER_OF_REPLICA 10 # Number of "replica" geometries along the path K_SPRING 0.05 &OPTIMIZE_BAND OPT_TYPE DIIS &DIIS MAX_STEPS 1000 &END &END BAND_TYPE CI-NEB # Climbing-image NEB &CI_NEB NSTEPS_IT 5 # First take 5 normal steps, then start CI &END &REPLICA COORD_FILE_NAME init.xyz # Filename of the initial coordinate file &END &REPLICA COORD_FILE_NAME final.xyz # Filename of the final coordinate file &END &PROGRAM_RUN_INFO INITIAL_CONFIGURATION_INFO &END &END BAND &END MOTION &FORCE_EVAL METHOD Quickstep &DFT ... same as GEO_OPT ... &END DFT &SUBSYS &CELL ABC 10.0 10.0 10.0 PERIODIC NONE &END CELL &TOPOLOGY COORD_FILE_NAME init.xyz # Filename of the initial coordinate file COORDINATE xyz &END TOPOLOGY &END SUBSYS &END FORCE_EVAL
In the main output of the NEB calculation, you will find a number of the following sections:
*******************************************************************************
BAND TYPE = CI-NEB
BAND TYPE OPTIMIZATION = SD
STEP NUMBER = 1
NUMBER OF NEB REPLICA = 10
DISTANCES REP = 0.252266 0.229810 0.225933 0.227113
0.201788 0.138731 0.138676 0.204817
0.252328
ENERGIES [au] = -24.815004 -24.813368 -24.808500 -24.803002
-24.800607 -24.802583 -24.805589 -24.809114
-24.813372 -24.815004
BAND TOTAL ENERGY [au] = -248.08548199708440
*******************************************************************************
These sections show, for every replica geometry along the NEB trajectory, the distance to its neighbors and its energy. The last of these sections corresponds to the converged NEB trajectory.
- Run the NEB calculation
- Find the activation barrier of the reaction in eV.
Free energy surface
Sampling the free energy surface (FES) of a chemical system is a convenient method to explore various stable conformations and possible reaction pathways. To calculate the FES for complicated systems, advanced sampling methods (such as umbrella sampling, metadynamics, parallel tempering, …) have to be used. However, for our simple $S_N2$ reaction, we will use unbiased Molecular Dynamics (MD).
The FES is a projection of the high-dimensional free energy landscape onto a small number, usually two, dimensions. These two dimensions are called collective variables (CV) and they must be chosen such that various stable conformations can be distinguished and the reaction pathways can be adequately described by the FES. For complex systems, the choice of CVs is a non-trivial task. Fortunately, for our simple system, the choice is simple: we take the distances of the two Cl anions from the central C as the CVs.
To help the simulation sample the parts of the FES which we care about, we will include restraints in the MD run, which prevent the Cl anions from going too far away from the molecule.
The following CP2K input script runs our MD calculation and prints out the CV values for every step:
- md.inp
&GLOBAL PRINT_LEVEL LOW PROJECT ch3cl RUN_TYPE MD # Molecular Dynamics &END GLOBAL &MOTION &MD # MD parameters ENSEMBLE NVT STEPS 200000 TIMESTEP 0.5 TEMPERATURE 1000.0 &THERMOSTAT TYPE NOSE &NOSE TIMECON 100. &END NOSE &END THERMOSTAT &END MD &CONSTRAINT # Restraints to keep Cl from going too far &COLLECTIVE INTERMOLECULAR COLVAR 1 TARGET 1.8 &RESTRAINT K 0.005 &END &END &COLLECTIVE INTERMOLECULAR COLVAR 2 TARGET 1.8 &RESTRAINT K 0.005 &END &END &END &FREE_ENERGY # Section to print out the values of CVs every step &METADYN DO_HILLS .FALSE. &METAVAR SCALE 0.2 COLVAR 1 &END &METAVAR SCALE 0.2 COLVAR 2 &END &PRINT &COLVAR COMMON_ITERATION_LEVELS 3 &EACH MD 1 &END &END COLVAR &END PRINT &END METADYN &END FREE_ENERGY &END MOTION &FORCE_EVAL METHOD Quickstep &DFT ... Same as before ... &END DFT &SUBSYS &CELL ABC 10.0 10.0 10.0 PERIODIC NONE &END CELL &COORD C -4.03963494 0.97427857 -0.29785096 H -3.97152206 2.11080568 -0.35500445 H -3.95108814 0.19729141 0.53163699 H -3.95810737 0.47922964 -1.32151084 Cl -5.77885435 1.07089312 -0.04609427 Cl -1.77974793 0.99422072 -0.29785096 &END COORD &COLVAR # CV definitions &DISTANCE ATOMS 1 5 &END &END COLVAR &COLVAR &DISTANCE ATOMS 1 6 &END &END COLVAR &END SUBSYS &END FORCE_EVAL
Free energy is expressed by
$$ F(s) = -k T \log(P(s)),$$
where $s$ is the set of CVs and $P(s)$ is the probability that the system has the set of CV values $s$.
The following Python script can be used to calculate the FES from the ch3f-COLVAR.metadynLog file produced by the MD run. (Don't forget to change the temperature in the Python script!)
import numpy as np
import matplotlib.pyplot as plt
bohr_2_angstrom = 0.529177
kb = 8.6173303e-5 # eV * K^-1
temperature = 1000.0 # Change the temperature according to your MD simulations!
colvar_path = "./ch3f-COLVAR.metadynLog"
# Load the colvar file
colvar_raw = np.loadtxt(colvar_path)
# Extract the two CVs
d1 = colvar_raw[:, 1] * bohr_2_angstrom
d2 = colvar_raw[:, 2] * bohr_2_angstrom
# Create a 2d histogram corresponding to the CV occurances
cv_hist = np.histogram2d(d1, d2, bins=50)
# probability from the histogram
prob = cv_hist[0]/len(d1)
# Free energy surface
fes = -kb * temperature * np.log(prob)
# Save the image
extent = (np.min(cv_hist[1]), np.max(cv_hist[1]), np.min(cv_hist[2]), np.max(cv_hist[2]))
plt.figure(figsize=(8, 8))
plt.imshow(fes.T, extent=extent, aspect='auto', origin='lower', cmap='hsv')
cbar = plt.colorbar()
cbar.set_label("Free energy [eV]")
plt.xlabel("d1 [ang]")
plt.ylabel("d2 [ang]")
plt.savefig("./fes.png", dpi=200)
plt.close()
Here is an example output for a temperature of 1000K. We clearly see the two local minima corresponding to one of the chlorine atoms being covalently bonded (distance 1.8 Å) while the other one is around a distance of 2.5 Å.
- Run the MD calculation for 400K, 800K, 1200K and 1600K. (These calculations can a take a while.)
- Create the corresponding FES plots and discuss the temperature dependence.
- In general, how does potential energy differ from free energy? For our reaction, what are the activation barriers from the different free energy surfaces? How and why do they differ from the NEB barrier?
plot.py:24: RuntimeWarning: divide by zero encountered in log fes = -kb * temperature * np.log(prob)
Don't worry about this, the script still works as expected and produces the plot in the file fes.png.