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### Tutorial for Thermalized Drude oscillators in LAMMPS

This tutorial explains how to use Drude oscillators in LAMMPS to simulate polarizable systems using the USER-DRUDE package. As an illustration, the input files for a simulation of 250 phenol molecules are documented. First of all, LAMMPS has to be compiled with the USER-DRUDE package activated. Then, the data file and input scripts have to be modified to include the Drude dipoles and how to handle them.

Overview of Drude induced dipoles

Polarizable atoms acquire an induced electric dipole moment under the action of an external electric field, for example the electric field created by the surrounding particles. Drude oscillators represent these dipoles by two fixed charges: the core (DC) and the Drude particle (DP) bound by a harmonic potential. The Drude particle can be thought of as the electron cloud whose center can be displaced from the position of the corresponding nucleus.

The sum of the masses of a core-Drude pair should be the mass of the initial (unsplit) atom, \(m_C + m_D = m\). The sum of their charges should be the charge of the initial (unsplit) atom, \(q_C + q_D = q\). A harmonic potential between the core and Drude partners should be present, with force constant \(k_D\) and an equilibrium distance of zero. The (half-)stiffness of the harmonic bond \(K_D = k_D/2\) and the Drude charge \(q_D\) are related to the atom polarizability \(\alpha\) by

\begin{equation} K_D = \frac 1 2\, \frac {q_D^2} \alpha \end{equation}

Ideally, the mass of the Drude particle should be small, and the stiffness of the harmonic bond should be large, so that the Drude particle remains close ot the core. The values of Drude mass, Drude charge, and force constant can be chosen following different strategies, as in the following examples of polarizable force fields:

- Lamoureux and Roux suggest adopting a global half-stiffness, \(K_D\) = 500 kcal/(mol Ang \({}^2\)) - which corresponds to a force constant \(k_D\) = 4184 kJ/(mol Ang \({}^2\)) - for all types of core-Drude bond, a global mass \(m_D\) = 0.4 g/mol (or u) for all types of Drude particles, and to calculate the Drude charges for individual atom types from the atom polarizabilities using equation (1). This choice is followed in the polarizable CHARMM force field.
- Alternately Schroeder and Steinhauser suggest adopting a global charge \(q_D\) = -1.0e and a global mass \(m_D\) = 0.1 g/mol (or u) for all Drude particles, and to calculate the force constant for each type of core-Drude bond from equation (1). The timesteps used by these authors are between 0.5 and 2 fs, with the degrees of freedom of the Drude oscillators kept cold at 1 K.
- In both these force fields hydrogen atoms are treated as non-polarizable.

The motion of of the Drude particles can be calculated by minimizing the energy of the induced dipoles at each timestep, by an interative, self-consistent procedure. The Drude particles can be massless and therefore do not contribute to the kinetic energy. However, the relaxed method is computationall slow. An extended-lagrangian method can be used to calculate the positions of the Drude particles, but this requires them to have mass. It is important in this case to decouple the degrees of freedom associated with the Drude oscillators from those of the normal atoms. Thermalizing the Drude dipoles at temperatures comparable to the rest of the simulation leads to several problems (kinetic energy transfer, very short timestep, etc.), which can be remediated by the "cold Drude" technique (Lamoureux and Roux).

Two closely related models are used to represent polarization through "charges on a spring": the core-shell model and the Drude model. Although the basic idea is the same, the core-shell model is normally used for ionic/crystalline materials, whereas the Drude model is normally used for molecular systems and fluid states. In ionic crystals the symmetry around each ion and the distance between them are such that the core-shell model is sufficiently stable. But to be applicable to molecular/covalent systems the Drude model includes two important features:

1. The possibility to thermostat the additional degrees of freedom associated with the induced dipoles at very low temperature, in terms of the reduced coordinates of the Drude particles with respect to their cores. This makes the trajectory close to that of relaxed induced dipoles.
2. The Drude dipoles on covalently bonded atoms interact too strongly due to the short distances, so an atom may capture the Drude particle (shell) of a neighbor, or the induced dipoles within the same molecule may align too much. To avoid this, damping at short of the interactions between the point charges composing the induced dipole can be done by Thole functions.

Preparation of the data file

The data file is similar to a standard LAMMPS data file for atom_style full. The DPs and the harmonic bonds connecting them to their DC should appear in the data file as normal atoms and bonds.

You can use the polarizer tool (Python script distributed with the USER-DRUDE package) to convert a non-polarizable data file (here data.102494.lmp) to a polarizable data file (data-p.lmp)

    polarizer -q -f phenol.dff data.102494.lmp data-p.lmp

This will automatically insert the new atoms and bonds. The masses and charges of DCs and DPs are computed from phenol.dff, as well as the DC-DP bond constants. The file phenol.dff contains the polarizabilities of the atom types and the mass of the Drude particles, for instance:

    # units: kJ/mol, A, deg
    # kforce is in the form k/2 r_D^2
    # type  m_D/u   q_D/e    k_D   alpha/A3  thole
    OH      0.4    -1.0    4184.0   0.63     0.67
    CA      0.4    -1.0    4184.0   1.36     2.51
    CAI     0.4    -1.0    4184.0   1.09     2.51

The hydrogen atoms are absent from this file, so they will be treated as non-polarizable atoms. In the non-polarizable data file data.102494.lmp, atom names corresponding to the atom type numbers have to be specified as comments at the end of lines of the Masses section. You probably need to edit it to add these names. It should look like

    Masses

    1 12.011 # CAI
    2 12.011 # CA
    3 15.999 # OH
    4 1.008  # HA
    5 1.008  # HO

Basic input file

The atom style should be set to (or derive from) full, so that you can define atomic charges and molecular bonds, angles, dihedrals...

The polarizer tool also outputs certain lines related to the input script (the use of these lines will be explained below). In order for LAMMPS to recognize that you are using Drude oscillators, you should use the fix drude. The command is

    fix DRUDE all drude C C C N N D D D

The N, C, D following the drude keyword have the following meaning: There is one tag for each atom type. This tag is C for DCs, D for DPs and N for non-polarizable atoms. Here the atom types 1 to 3 (C and O atoms) are DC, atom types 4 and 5 (H atoms) are non-polarizable and the atom types 6 to 8 are the newly created DPs.

By recognizing the fix drude, LAMMPS will find and store matching DC-DP pairs and will treat DP as equivalent to their DC in the special bonds relations. It may be necessary to extend the space for storing such special relations. In this case extra space should be reserved by using the extra keyword of the special_bonds command. With our phenol, there is 1 more special neighbor for which space is required. Otherwise LAMMPS crashes and gives the required value.

    special_bonds lj/coul 0.0 0.0 0.5 extra 1

Let us assume we want to run a simple NVT simulation at 300 K. Note that Drude oscillators need to be thermalized at a low temperature in order to approximate a self-consistent field (SCF), therefore it is not possible to simulate an NVE ensemble with this package. Since dipoles are approximated by a charged DC-DP pair, the pair_style must include Coulomb interactions, for instance lj/cut/coul/long with kspace_style pppm. For example, with a cutoff of 10. and a precision 1.e-4:

    pair_style lj/cut/coul/long 10.0
    kspace_style pppm 1.0e-4

As compared to the non-polarizable input file, pair_coeff lines need to be added for the DPs. Since the DPs have no Lennard-Jones interactions, their epsilon is 0. so the only pair_coeff line that needs to be added is

    pair_coeff * 6* 0.0 0.0 # All-DPs

Now for the thermalization, the simplest choice is to use the fix langevin/drude.

    fix LANG all langevin/drude 300. 100 12435 1. 20 13977

This applies a Langevin thermostat at temperature 300. to the centers of mass of the DC-DP pairs, with relaxation time 100 and with random seed 12345. This fix applies also a Langevin thermostat at temperature 1. to the relative motion of the DPs around their DCs, with relaxation time 20 and random seed 13977. Only the DCs and non-polarizable atoms need to be in this fix's group. LAMMPS will thermostate the DPs together with their DC. For this, ghost atoms need to know their velocities. Thus you need to add the following command:

    comm_modify vel yes

In order to avoid that the center of mass of the whole system drifts due to the random forces of the Langevin thermostat on DCs, you can add the zero yes option at the end of the fix line.

If the fix shake is used to constrain the C-H bonds, it should be invoked after the fix langevin/drude for more accuracy.

    fix SHAKE ATOMS shake 0.0001 20 0 t 4 5

NOTE: The group of the fix shake must not include the DPs. If the group ATOMS is defined by non-DPs atom types, you could use

Since the fix langevin/drude does not perform time integration (just modification of forces but no position/velocity updates), the fix nve should be used in conjunction.

    fix NVE all nve

Finally, do not forget to update the atom type elements if you use them in a dump_modify ... element ... command, by adding the element type of the DPs. Here for instance

    dump DUMP all custom 10 dump.lammpstrj id mol type element x y z ix iy iz
    dump_modify DUMP element C C O H H D D D

The input file should now be ready for use!

You will notice that the global temperature thermo_temp computed by LAMMPS is not 300. K as wanted. This is because LAMMPS treats DPs as standard atoms in his default compute. If you want to output the temperatures of the DC-DP pair centers of mass and of the DPs relative to their DCs, you should use the compute temp_drude

    compute TDRUDE all temp/drude

And then output the correct temperatures of the Drude oscillators using thermo_style custom with respectively c_TDRUDE[1] and c_TDRUDE[2]. These should be close to 300.0 and 1.0 on average.

    thermo_style custom step temp c_TDRUDE[1] c_TDRUDE[2]

Thole screening

Dipolar interactions represented by point charges on springs may not be stable, for example if the atomic polarizability is too high for instance, a DP can escape from its DC and be captured by another DC, which makes the force and energy diverge and the simulation crash. Even without reaching this extreme case, the correlation between nearby dipoles on the same molecule may be exagerated. Often, special bond relations prevent bonded neighboring atoms to see the charge of each other's DP, so that the problem does not always appear. It is possible to use screened dipole dipole interactions by using the pair_style thole. This is implemented as a correction to the Coulomb pair_styles, which dampens at short distance the interactions between the charges representing the induced dipoles. It is to be used as hybrid/overlay with any standard coul pair style. In our example, we would use

    pair_style hybrid/overlay lj/cut/coul/long 10.0 thole 2.6 10.0

This tells LAMMPS that we are using two pair_styles. The first one is as above (lj/cut/coul/long 10.0). The second one is a thole pair_style with default screening factor 2.6 (Noskov) and cutoff 10.0.

Since hybrid/overlay does not support mixing rules, the interaction coefficients of all the pairs of atom types with i < j should be explicitly defined. The output of the polarizer script can be used to complete the pair_coeff section of the input file. In our example, this will look like:

    pair_coeff    1    1 lj/cut/coul/long    0.0700   3.550
    pair_coeff    1    2 lj/cut/coul/long    0.0700   3.550
    pair_coeff    1    3 lj/cut/coul/long    0.1091   3.310
    pair_coeff    1    4 lj/cut/coul/long    0.0458   2.985
    pair_coeff    2    2 lj/cut/coul/long    0.0700   3.550
    pair_coeff    2    3 lj/cut/coul/long    0.1091   3.310
    pair_coeff    2    4 lj/cut/coul/long    0.0458   2.985
    pair_coeff    3    3 lj/cut/coul/long    0.1700   3.070
    pair_coeff    3    4 lj/cut/coul/long    0.0714   2.745
    pair_coeff    4    4 lj/cut/coul/long    0.0300   2.420
    pair_coeff    *    5 lj/cut/coul/long    0.0000   0.000
    pair_coeff    *   6* lj/cut/coul/long    0.0000   0.000
    pair_coeff    1    1 thole   1.090   2.510
    pair_coeff    1    2 thole   1.218   2.510
    pair_coeff    1    3 thole   0.829   1.590
    pair_coeff    1    6 thole   1.090   2.510
    pair_coeff    1    7 thole   1.218   2.510
    pair_coeff    1    8 thole   0.829   1.590
    pair_coeff    2    2 thole   1.360   2.510
    pair_coeff    2    3 thole   0.926   1.590
    pair_coeff    2    6 thole   1.218   2.510
    pair_coeff    2    7 thole   1.360   2.510
    pair_coeff    2    8 thole   0.926   1.590
    pair_coeff    3    3 thole   0.630   0.670
    pair_coeff    3    6 thole   0.829   1.590
    pair_coeff    3    7 thole   0.926   1.590
    pair_coeff    3    8 thole   0.630   0.670
    pair_coeff    6    6 thole   1.090   2.510
    pair_coeff    6    7 thole   1.218   2.510
    pair_coeff    6    8 thole   0.829   1.590
    pair_coeff    7    7 thole   1.360   2.510
    pair_coeff    7    8 thole   0.926   1.590
    pair_coeff    8    8 thole   0.630   0.670

For the thole pair style the coefficients are

1. the atom polarizability in units of cubic length
2. the screening factor of the Thole function (optional, default value specified by the pair_style command)
3. the cutoff (optional, default value defined by the pair_style command)

The special neighbors have charge-charge and charge-dipole interactions screened by the coul factors of the special_bonds command (0.0, 0.0, and 0.5 in the example above). Without using the pair_style thole, dipole-dipole interactions are screened by the same factor. By using the pair_style thole, dipole-dipole interactions are screened by Thole's function, whatever their special relationship (except within each DC-DP pair of course). Consider for example 1-2 neighbors: using the pair_style thole, their dipoles will see each other (despite the coul factor being 0.) and the interactions between these dipoles will be damped by Thole's function.

Thermostats and barostats

Using a Nose-Hoover barostat with the langevin/drude thermostat is straightforward using fix nph instead of nve. For example:

    fix NPH all nph iso 1. 1. 500

It is also possible to use a Nose-Hoover instead of a Langevin thermostat. This requires to use fix drude/transform just before and after the time intergation fixes. The fix drude/transform/direct converts the atomic masses, positions, velocities and forces into a reduced representation, where the DCs transform into the centers of mass of the DC-DP pairs and the DPs transform into their relative position with respect to their DC. The fix drude/transform/inverse performs the reverse transformation. For a NVT simulation, with the DCs and atoms at 300 K and the DPs at 1 K relative to their DC one would use

    fix DIRECT all drude/transform/direct
    fix NVT1 ATOMS nvt temp 300. 300. 100
    fix NVT2 DRUDES nvt temp 1. 1. 20
    fix INVERSE all drude/transform/inverse

For our phenol example, the groups would be defined as

    group ATOMS  type 1 2 3 4 5 # DCs and non-polarizable atoms
    group CORES  type 1 2 3     # DCs
    group DRUDES type 6 7 8     # DPs

Note that with the fixes drude/transform, it is not required to specify comm_modify vel yes because the fixes do it anyway (several times and for the forces also). To avoid the flying ice cube artifact (Lamoureux), where the atoms progressively freeze and the center of mass of the whole system drifts faster and faster, the fix momentum can be used. For instance:

    fix MOMENTUM all momentum 100 linear 1 1 1

It is a bit more tricky to run a NPT simulation with Nose-Hoover barostat and thermostat. First, the volume should be integrated only once. So the fix for DCs and atoms should be npt while the fix for DPs should be nvt (or vice versa). Second, the fix npt computes a global pressure and thus a global temperature whatever the fix group. We do want the pressure to correspond to the whole system, but we want the temperature to correspond to the fix group only. We must then use the fix_modify command for this. In the end, the block of instructions for thermostating and barostating will look like

    compute TATOMS ATOMS temp
    fix DIRECT all drude/transform/direct
    fix NPT ATOMS npt temp 300. 300. 100 iso 1. 1. 500
    fix_modify NPT temp TATOMS press thermo_press
    fix NVT DRUDES nvt temp 1. 1. 20
    fix INVERSE all drude/transform/inverse

Rigid bodies

You may want to simulate molecules as rigid bodies (but polarizable). Common cases are water models such as SWM4-NDP, which is a kind of polarizable TIP4P water. The rigid bodies and the DPs should be integrated separately, even with the Langevin thermostat. Let us review the different thermostats and ensemble combinations.

NVT ensemble using Langevin thermostat:

    comm_modify vel yes
    fix LANG all langevin/drude 300. 100 12435 1. 20 13977
    fix RIGID ATOMS rigid/nve/small molecule
    fix NVE DRUDES nve

NVT ensemble using Nose-Hoover thermostat:

    fix DIRECT all drude/transform/direct
    fix RIGID ATOMS rigid/nvt/small molecule temp 300. 300. 100
    fix NVT DRUDES nvt temp 1. 1. 20
    fix INVERSE all drude/transform/inverse

NPT ensemble with Langevin thermostat:

    comm_modify vel yes
    fix LANG all langevin/drude 300. 100 12435 1. 20 13977
    fix RIGID ATOMS rigid/nph/small molecule iso 1. 1. 500
    fix NVE DRUDES nve

NPT ensemble using Nose-Hoover thermostat:

    compute TATOM ATOMS temp
    fix DIRECT all drude/transform/direct
    fix RIGID ATOMS rigid/npt/small molecule temp 300. 300. 100 iso 1. 1. 500
    fix_modify RIGID temp TATOM press thermo_press
    fix NVT DRUDES nvt temp 1. 1. 20
    fix INVERSE all drude/transform/inverse

(Lamoureux) Lamoureux and Roux, J Chem Phys, 119, 3025-3039 (2003)

(Schroeder) Schroeder and Steinhauser, J Chem Phys, 133, 154511 (2010).

(Jiang) Jiang, Hardy, Phillips, MacKerell, Schulten, and Roux, J Phys Chem Lett, 2, 87-92 (2011).

(Thole) Chem Phys, 59, 341 (1981).

(Noskov) Noskov, Lamoureux and Roux, J Phys Chem B, 109, 6705 (2005).

(SWM4-NDP) Lamoureux, Harder, Vorobyov, Roux, MacKerell, Chem Phys Let, 418, 245-249 (2006)