Energy

The system energy, or Hamiltonian, consists of a sum of potential energy terms,

$$ \mathcal{H}_{sys} = U_1 + U_2 + … $$

The energy terms are specified in energy at the top level input and evaluated in the order given. For example:

energy:
    - isobaric: {P/atm: 1}
    - sasa: {molarity: 0.2, radius: 1.4 }
    - confine: {type: sphere, radius: 10, molecules: [water]}
    - nonbonded:
        default: # applied to all atoms
        - lennardjones: {mixing: LB}
        - coulomb: {type: plain, epsr: 1}
        Na CH:   # overwrite specific atom pairs
        - wca: { mixing: LB }

    - maxenergy: 100
    - ...

The keyword maxenergy can be used to skip further energy evaluation if a term returns a large energy change (in kT), which will likely lead to rejection. The default value is infinity.

Energies in MC may contain implicit degrees of freedom, i.e. be temperature-dependent, effective potentials. This is inconsequential for sampling density of states, but care should be taken when interpreting derived functions such as energies, entropies, pressure etc.

Infinite and NaN Energies

In case one or more potential energy terms of the system Hamiltonian returns infinite or NaN energies, a set of conditions exists to evaluate the acceptance of the proposed move:

  • always reject if new energy is NaN (i.e. division by zero)
  • always accept if energy change is from NaN to finite energy
  • always accept if the energy difference is NaN (i.e. from infinity to minus infinity)

These conditions should be carefully considered if equilibrating a system far from equilibrium.

External Pressure

This adds the following pressure term (see i.e. Frenkel and Smith, Chapter 5.4) to the Hamiltonian, appropriate for MC moves in $\ln V$:

$$ U = PV - k_BT\left ( N + 1 \right ) \ln V $$

where $N$ is the total number of molecules and atomic species.

isobaric Description
P/unit External pressure where unit can be mM, atm, Pa, bar, kT

Nonbonded Interactions

This term loops over pairs of atoms, $i$, and $j$, summing a given pair-wise additive potential, $u_{ij}$,

$$ U = \sum_{i=0}^{N-1}\sum_{j=i+1}^N u_{ij}(\textbf{r}_j-\textbf{r}_i)$$

The most general method is nonbonded where potentials can be arbitrarily mixed and customized for specific particle combinations.

Example:

- nonbonded:
    default: # default pair potential
        - lennardjones: {mixing: LB}
        - coulomb: {type: fanourgakis, epsr: 1.0, cutoff: 12}
    Ow Ca: # custom potential for atom type "Ow" and atom type "Ca"
        - wca: {mixing: LB}

Below is a description of possible nonbonded methods. For simple potentials, the hard coded variants are often the fastest option. For better performance, it is recommended to use nonbonded_splined in place of the more robust nonbonded method.

energy $u_{ij}$
nonbonded Any combination of pair potentials (slower, but exact)
nonbonded_exact An alias for nonbonded
nonbonded_splined Any combination of pair potentials (splined)
nonbonded_cached Any combination of pair potentials (splined, only intergroup!)
nonbonded_coulomblj coulomb+lennardjones (hard coded)
nonbonded_coulombwca coulomb+wca (hard coded)
nonbonded_pm coulomb+hardsphere (fixed type=plain, cutoff$=\infty$)
nonbonded_pmwca coulomb+wca (fixed type=plain, cutoff$=\infty$)

Mass Center Cutoffs

For cutoff based pair-potentials working between large molecules, it can be efficient to use mass center cutoffs between molecular groups, thus skipping all pair-interactions. A single cutoff can be used between all molecules (default), or specified for specific combinations:

- nonbonded:
    cutoff_g2g:
      default: 40.0
      protein polymer: 20.0

If default is omitted, only the specified pairs are subject to the cutoffs. Finally, cutoff_g2g: 40.0 is allowed for a uniform cutoff between all groups.

Spline Options

The nonbonded_splined method internally splines the potential in an automatically determined interval [rmin,rmax] determined by the following policies:

  • rmin is decreased towards zero until the potential reaches u_at_rmin.
  • rmax is increased until the potential reaches u_at_rmax.

If above the interval, zero is returned. If below the interval, the exact energy (or infinity) is returned. For details about the splines for each pair, use to_disk and/or maximize the verbosity level (--verbosity) when running faunus.

Keyword Description
utol=1e-3 Spline precision
u_at_rmin=20 Energy threshold at short separations (kT)
u_at_rmax=1e-6 Energy threshold at long separations (kT)
to_disk=False Create datafiles w. exact and splined potentials
hardsphere=False Use hardsphere repulsion below rmin

Note: Anisotropic pair-potentials cannot be splined. This also applies to non-shifted electrostatic potentials such as plain and un-shifted yukawa.

Parallel summation

Depending on how Faunus was compiled, parallel, nonbonded summation schemes may be available. Activate with:

- nonbonded:
    summation_policy: parallel
    ...

where parallel uses C++ internal threading; openmp uses OpenMP; and serial skip parallel summation (default). A warning will be issued if the desired scheme is unavailable. For the openmp policy, you may control the number of threads with the environmental variable OMP_NUM_THREADS. Summation policies other than serial may require substantial memory for systems with many particles.

Electrostatics

coulomb Description
type Coulomb type, see below
cutoff Spherical cutoff, $R_c$ (Å) after which the potential is zero
epsr Relative dielectric constant of the medium
utol=0.005/lB Error tolerence for splining; default value depends on the Bjerrum length, lB
debyelength=$\infty$ Debye length (Å) if using ewald, poisson, yukawa

This is a multipurpose potential that handles several electrostatic methods. Beyond a spherical real-space cutoff, $R_c$, the potential is zero while if below,

$$ \tilde{u}^{(zz)}_{ij}(\bar{r}) = \frac{e^2 z_i z_j }{ 4\pi\epsilon_0\epsilon_r |\bar{r}| }\mathcal{S}(q) $$

where $\bar{r} = \bar{r}_j - \bar{r}_i$, and tilde indicate that a short-range function $\mathcal{S}(q=|\bar{r}|/R_c)$ is used to trucate the interactions. The available short-range functions are:

coulomb types Keywords $\mathcal{S}(q)$
plain 1
ewald alpha $\frac{1}{2}\text{erfc}\left(\alpha R_c q + \frac{\kappa}{2\alpha}\right)\text{exp}\left(2\kappa R_c q\right) + \frac{1}{2}\text{erfc}\left(\alpha R_c q - \frac{\kappa}{2\alpha}\right)$
reactionfield epsrf $1+\frac{\epsilon_{RF}-\epsilon_r}{2\epsilon_{RF}+\epsilon_r}q^3-3\frac{\epsilon_{RF}}{2\epsilon_{RF}+\epsilon_r}q$
poisson C=3, D=3 $(1-\acute{q})^{D+1}\sum_{c=0}^{C-1}\frac{C-c}{C}{D-1+c\choose c}\acute{q}^c$
qpotential order $\prod_{n=1}^{\text{order}}(1-q^n)$
fanourgakis $1-\frac{7}{4}q+\frac{21}{4}q^5-7q^6+\frac{5}{2}q^7$
fennell alpha $\text{erfc}(\alpha R_cq)-q\text{erfc}(\alpha R_c)+(q-1)q\left(\text{erfc}(\alpha R_c)+\frac{2\alpha R_c}{\sqrt{\pi}}\text{exp}(-\alpha^2R_c^2)\right)$
zerodipole alpha $\text{erfc}(\alpha R_cq)-q\text{erfc}(\alpha R_c)+\frac{(q^2-1)}{2}q\left(\text{erfc}(\alpha R_c)+\frac{2\alpha R_c}{\sqrt{\pi}}\text{exp}(-\alpha^2R_c^2)\right)$
zahn alpha $\text{erfc}(\alpha R_c q)-(q-1)q\left(\text{erfc}(\alpha R_c)+\frac{2\alpha R_c}{\sqrt{\pi}}\text{exp}(-\alpha^2R_c^2)\right)$
wolf alpha $\text{erfc}(\alpha R_cq)-\text{erfc}(\alpha R_c)q$
yukawa debyelength, shift=false As plain with screening or, if shifted, poisson with C=1 and D=1

Internally $\mathcal{S}(q)$ is splined whereby all types evaluate at similar speed. For the poisson potential,

$$ \acute{q} = \frac{1-\exp\left(2\kappa R_c q\right)}{1-\exp\left(2\kappa R_c\right)} $$

which as the inverse Debye length, $\kappa\to 0$ gives $\acute{q}=q$. The poisson scheme can generate a number of other truncated pair-potentials found in the litterature, depending on C and D. Thus, for an infinite Debye length, the following holds:

C D Equivalent to
1 -1 Plain Coulomb within cutoff, zero outside
1 0 Undamped Wolf
1 1 Levitt / Undamped Fenell
1 2 Kale
1 3 McCann
2 1 Undamped Fukuda
2 2 Markland
3 3 Stenqvist
4 3 Fanourgakis

Debye Screening Length

A background screening due to implicit ions can be added by specifying the keyword debyelength to the schemes

  • yukawa
  • ewald
  • poisson

The yukawa scheme has simple exponential screening and, like plain, an infinite cutoff. If shift: true is passed to the yukawa scheme, the potential is shifted to give zero potential and force at the now finite cutoff distance (simply an alias for poisson with C=1 and D=1). The list below shows alternative ways to specify the background electrolyte, and will automatically deduce the salt stoichiometry based on valencies:

    debyelength: 30.0, epsr: 79.8       # assuming 1:1 salt, e.g. NaCl
    molarity: 0.02                      # 0.02 M 1:1 salt, e.g. NaCl
    molarity: 0.01, valencies: [2,3,-2] # 0.01 M Ca₂Al₂(SO₄)₅

Multipoles

If the type coulomb is replaced with multipole then the electrostatic energy will in addition to monopole-monopole interactions include contributions from monopole-dipole, and dipole-dipole interactions. Multipolar properties of each particle is specified in the Topology. The zahn and fennell approaches have undefined dipolar self-energies and are therefore not recommended for such systems.

The ion-dipole interaction is described by

$$ \tilde{u}^{(z\mu)}_{ij}(\bar{r}) = -\frac{ez_i\left(\mu_j\cdot \hat{r}\right) }{|\bar{r}|^2} \left( \mathcal{S}(q) - q\mathcal{S}^{\prime}(q) \right) $$

where $\hat{r} = \bar{r}/|\bar{r}|$, and the dipole-dipole interaction by

$$ \tilde{u}^{\mu\mu}_{ij}(\bar{r}) = -\left ( \frac{3 ( \boldsymbol{\mu}_i \cdot \hat{r} ) \left(\boldsymbol{\mu}_j\cdot\hat{r}\right) - \boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j }{|\bar{r}|^3}\right) \left( \mathcal{S}(q) - q\mathcal{S}^{\prime}(q) + \frac{q^2}{3}\mathcal{S}^{\prime\prime}(q) \right) - \frac{\left(\boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j\right)}{|\bar{r}|^3}\frac{q^2}{3}\mathcal{S}^{\prime\prime}(q). $$

Warning:

  • The zahn and fennell approaches have undefined dipolar self-energies (see next section) and are therefore not recommended for dipolar systems.

Self-energies

When using coulomb or multipole, an electrostatic self-energy term is automatically added to the Hamiltonian. The monopole and dipole contributions are evaluated according to

$$ U_{self} = -\frac{1}{2}\sum_i^N\sum_{\ast\in{z,\mu}} \lim_{|\bar{r}_{ii}|\to 0}\left( u^{(\ast\ast)}_{ii}(\bar{r}_{ii})

  • \tilde{u}^{(\ast\ast)}_{ii}(\bar{r}_{ii}) \right ) $$

where no tilde indicates that $\mathcal{S}(q)\equiv 1$ for any $q$.

Ewald Summation

If type is ewald, terms from reciprocal space and surface energies are automatically added (in addition to the previously mentioned self- and real space-energy) to the Hamiltonian which activates the additional keywords:

type=ewald Description
ncutoff Reciprocal-space cutoff (unitless)
epss=0 Dielectric constant of surroundings, $\varepsilon_{surf}$ (0=tinfoil)
ewaldscheme=PBC Periodic (PBC) or isotropic periodic (IPBC) boundary conditions
spherical_sum=true Spherical/ellipsoidal summation in reciprocal space; cubic if false.
debyelength=$\infty$ Debye length (Å)

The added energy terms are:

$$ U_{\text{reciprocal}} = \frac{2\pi f}{V} \sum_{ {\bf k} \ne {\bf 0}} A_k \vert Q^{q\mu} \vert^2 $$

$$ U_{\text{surface}} = \frac{1}{4\pi\varepsilon_0\varepsilon_r}\frac{ 2\pi }{ (2\varepsilon_{surf} + 1) V } \left( \left|\sum_{j}q_j\bar{r}_j\right|^2 + 2 \sum_j q_i \bar{r}_j \cdot \sum_j \boldsymbol{\mu}_j + \left| \sum_j \boldsymbol{\mu}_j \right|^2 \right ) $$

where

$$ f = \frac{1}{4\pi\varepsilon_0\varepsilon_r} \quad\quad V=L_xL_yL_z $$

$$ A_k = \frac{e^{-( k^2 + \kappa^2 )/4\alpha^2}}{k^2} \quad \quad Q^{q\mu} = Q^{q} + Q^{\mu} $$

$$ Q^{q} = \sum_{j}q_je^{i({\bf k}\cdot {\bf r}_j)} \quad Q^{\mu} = \sum_{j}i({\boldsymbol{\mu}}_j\cdot {\bf k}) e^{i({\bf k}\cdot {\bf r}_j)} $$

$$ \bar{k} = 2\pi\left( \frac{n_x}{L_x} , \frac{n_y}{L_y} ,\frac{n_z}{L_z} \right)\quad \bar{n} \in \mathbb{Z}^3 $$

Like many other electrostatic methods, the Ewald scheme also adds a self-energy term as described above. In the case of isotropic periodic boundaries (ipbc=true), the orientational degeneracy of the periodic unit cell is exploited to mimic an isotropic environment, reducing the number of wave-vectors to one fourth compared with 3D PBC Ewald. For point charges, IPBC introduce the modification,

$$ Q^q = \sum_j q_j \prod_{\alpha\in{x,y,z}} \cos \left( \frac{2\pi}{L_{\alpha}} n_{\alpha} r_{\alpha,j} \right) $$

while for point dipoles (currently unavailable),

$$ Q^{\mu} = \sum_j \bar{\mu}_j \cdot \nabla_j \left( \prod_{ \alpha \in { x,y,z } } \cos \left ( \frac{2\pi}{L_{\alpha}} n_{\alpha} \bar{r}_{\alpha,j} \right ) \right ) $$

Mean-Field Correction

For cuboidal slit geometries, a correcting mean-field, external potential, $\varphi(z)$, from charges outside the box can be iteratively generated by averaging the charge density, $\rho(z)$, in $dz$-thick slices along $z$. This correction assumes that all charges interact with a plain Coulomb potential and that a cubic cutoff is used via the minimum image convention.

To enable the correction, use the akesson keyword at the top level of energy:

akesson Keywords
molecules Array of molecules to operate on
epsr Relative dielectric constant
nstep Number of energy evalutations between updating $\rho(z)$
dz=0.2 $z$ resolution (Å)
nphi=10 Multiple of nstep in between updating $\varphi(z)$
file=mfcorr.dat File with $\rho(z)$ to either load or save
fixed=false If true, assume that file is converged. No further updating and faster.

The density is updated every nstep energy calls, while the external potential can be updated slower (nphi) since it affects the ensemble. A reasonable value of nstep is system dependent and can be a rather large value. Updating the external potential on the fly leads to energy drifts that decrease for consecutive runs. Production runs should always be performed with fixed=true and a well converged $\rho(z)$.

At the end of simulation, file is overwritten unless fixed=true.

Pair Potentials

In addition to the Coulombic pair-potentials described above, a number of other pair-potentials can be used. Through the C++ API or the custom potential explained below, it is easy to add new potentials.

Charge-Nonpolar

The energy when the field from a point charge, $z_i$, induces a dipole in a polarizable particle of unit-less excess polarizability, $\alpha_j=\left ( \frac{\epsilon_j-\epsilon_r}{\epsilon_j+2\epsilon_r}\right ) a_j^3$, is

$$ \beta u_{ij} = -\frac{\lambda_B z_i^2 \alpha_j}{2r_{ij}^4} $$

where $a_j$ is the radius of the non-polar particle and $\alpha_j$ is set in the atom topology, alphax. For non-polar particles in a polar medium, $\alpha_i$ is a negative number. For more information, see J. Israelachvili’s book, Chapter 5.

ionalpha Description
epsr Relative dielectric constant of medium

Charge-polarizability products for each pair of species is evaluated once during construction and based on the defined atom types.

Cosine Attraction

An attractive potential used for coarse grained lipids and with the form,

$$ \beta u(r) = -\epsilon \cos^2 \left ( \frac{\pi(r-r_c)}{2w_c} \right ) $$

for $r_c\leq r \leq r_c+w_c$. For $r<r_c$, $\beta u=-\epsilon$, while zero for $r>r_c+w_c$.

cos2 Description
eps Depth, $\epsilon$ (kJ/mol)
rc Width, $r_c$ (Å)
wc Decay range, $w_c$ (Å)

Assorted Short Ranged Potentials

The potentials below are often used to keep particles apart and/or to introduce stickiness. The atomic interaction parameters, e.g., $\sigma_i$ and $\epsilon_i$, are taken from the topology.

Type Atomic parameters $u(r)$ (non-zero part)
hardsphere sigma $\infty$ for $r < \sigma_{ij}$
hertz sigma, eps $\epsilon_{ij} \left ( 1-r / \sigma_{ij}\right )^{5/2}$ for $r<\sigma_{ij}$
lennardjones sigma, eps $4\epsilon_{ij} \left ( (\sigma_{ij}/r_{ij})^{12} - (\sigma_{ij}/r_{ij})^6\right )$
squarewell sigma, eps $-\epsilon_{ij}$ for $r<\sigma_{ij}$
wca sigma, eps $u_{ij}^{\text{LJ}} + \epsilon_{ij}$ for $r < 2^{1/6}\sigma_{ij}$

If several potentials are used together and different values for the coefficients are desired, an aliasing of the parameters’ names can be introduced. For example by specifying sigma: sigma_hs, the potential uses the atomic value sigma_hs instead of sigma, as shown in example below. To avoid possible conflicts of parameters’ names with future keywords of Faunus, we recommend following naming scheme: property_pot, where property is either sigma or eps and pot stands for the potential abbreviation, i.e, hs, hz, lj, sw, and wca.

Mixing (combination) rules can be specified to automatically parametrize heterogeneous interactions. If not described otherwise, the same rule is applied to all atomic parameters used by the potential. No meaningful defaults are defined yet, hence always specify the mixing rule explicitly, e.g., arithmetic for hardsphere.

Rule Description Formula
arithmetic arithmetic mean $a_{ij} = \frac 12 \left( a_{ii} + a_{jj} \right)$
geometric geometric mean $a_{ij} = \sqrt{a_{ii} a_{jj}}$
lorentz_berthelot Lorentz-Berthelot arithmetic for sigma, geometric for eps

For convenience, the abbreviation LB can be used instead of lorentz_berthelot.

Custom parameter values can be specified to override the mixing rule for a given pair, as shown in the example bellow.

- lennardjones:
    mixing: LB
    custom:
      - Na Cl: {eps: 0.2, sigma: 2}
      - K Cl: { ... }
- hertz:
    mixing: LB
    eps: eps_hz
    custom:
      - Na Cl: {eps_hz: 0.2, sigma: 2}
- hardsphere:
    mixing: arithmetic
    sigma: sigma_hs
    custom:
      - Na Cl: {sigma_hs: 2}

SASA (pair potential)

This calculates the surface area of two intersecting particles or radii $R$ and $r$ to estimate an energy based on transfer-free-energies (TFE) and surface tension. The total surface area is calculated as

$$ A = 4\pi \left ( R^2 + r^2 \right ) - 2\pi \left ( Rh_1 + rh_2 \right ) $$

where $h_1$ and $h_2$ are the heights of the spherical caps comprising the lens formed by the overlapping spheres. For complete overlap, or when far apart, the full area of the bigger sphere or the sum of both spheres are returned. The pair-energy is calculated as:

$$ u_{ij} = A \left ( \gamma_{ij} + c_s \varepsilon_{\text{tfe},ij} \right ) $$

where $\gamma_{ij}$ and $\varepsilon_{\text{tfe},ij}$ are the arithmetic means of tension and tfe provided in the atomlist.

Note that SASA is strictly not additive and this pair-potential is merely a poor-mans way of approximately taking into account ion-specificity and hydrophobic/hydrophilic interactions. Faunus offers also a full, albeit yet experimental implementation of [Solvent Accessible Surface Area] energy.

sasa Description
molarity Molar concentration of co-solute, $c_s$
radius=1.4 Probe radius for SASA calculation (Å)
shift=true Shift to zero at large separations

Custom

This takes a user-defined expression and a list of constants to produce a runtime, custom pair-potential. While perhaps not as computationally efficient as hard-coded potentials, it is a convenient way to access alien potentials. Used in combination with nonbonded_splined there is no overhead since all potentials are splined.

custom Description
function Mathematical expression for the potential (units of kT)
constants User-defined constants
cutoff Spherical cutoff distance

The following illustrates how to define a Yukawa potential:

custom:
    function: lB * q1 * q2 / r * exp( -r/D ) # in kT
    constants:
        lB: 7.1  # Bjerrum length
        D: 30    # Debye length

The function is passed using the efficient ExprTk library and a rich set of mathematical functions and logic is available. In addition to user-defined constants, the following symbols are defined:

symbol Description
e0 Vacuum permittivity [C²/J/m]
inf ∞ (infinity)
kB Boltzmann constant [J/K]
kT Boltzmann constant × temperature [J]
Nav Avogadro's number [1/mol]
pi π (pi)
q1,q2 Particle charges [e]
r Particle-particle separation [Å]
Rc Spherical cut-off [Å]
s1,s2 Particle sigma [Å]
T Temperature [K]

Custom External Potential

This applies a custom external potential to atoms or molecular mass centra using the ExprTk library syntax.

customexternal Description
molecules Array of molecules to operate on
com=false Operate on mass-center instead of individual atoms?
function Mathematical expression for the potential (units of kT)
constants User-defined constants

In addition to user-defined constants, the following symbols are available:

symbol Description
e0 Vacuum permittivity [C²/J/m]
inf ∞ (infinity)
kB Boltzmann constant [J/K]
kT Boltzmann constant × temperature [J]
Nav Avogadro's number [1/mol]
pi π (pi)
q Particle charge [e]
s Particle sigma [Å]
x,y,z Particle positions [Å]
T Temperature [K]

If com=true, charge refers to the molecular net-charge, and x,y,z the mass-center coordinates. The following illustrates how to confine molecules in a spherical shell of radius, r, and thickness dr:

customexternal:
    molecules: [water]
    com: true
    constants: {radius: 15, dr: 3}
    function: >
        var r2 := x^2 + y^2 + z^2;
        if ( r2 < radius^2 )
           1000 * ( radius-sqrt(r2) )^2;
        else if ( r2 > (radius+dr)^2 )
           1000 * ( radius+dr-sqrt(r2) )^2;
        else
           0;

Gouy Chapman

By setting function=gouychapman, an electric potential from a uniformly, charged plane in a 1:1 salt solution is added; see e.g. the book Colloidal Domain by Evans and Wennerström, 1999. If a surface potential, $\varphi_0$ is specified,

$$ \rho = \sqrt{\frac{2 c_0}{\pi \lambda_B} } \sinh ( \beta e \varphi_0 / 2 ) $$ while if instead a surface charge density, $\rho$, is given, $$ \beta e \varphi_0 = 2\mbox{asinh} \left ( \rho \sqrt{\frac{\pi \lambda_B} {2 c_0}} \right ) $$ where $\lambda_B$ is the Bjerrum length. With $\Gamma_0 = \tanh{ \beta e \varphi_0 / 4 }$ the final, non-linearized external potential is: $$ \beta e \phi_i = 2 \ln \left ( \frac{1+\Gamma_0e^{-\kappa r_{z,i}}}{1-\Gamma_0 e^{-\kappa r_{z,i}}} \right ) $$ where $z_i$ is the particle charge; $e$ is the electron unit charge; $\kappa$ is the inverse Debye length; and $r_{z,i}$ is the distance from the charged $xy$-plane which is always placed at the minimum $z$-value of the simulation container (normally a slit geometry). Fluctuations of the simulation cell dimensions are respected.

The following parameters should be given under constants; the keywords rho, rhoinv, and phi0 are mutually exclusive.

constants Description
molarity Molar 1:1 salt concentration (mol/l)
epsr Relative dielectric constant
rho Charge per area (1/eŲ)
rhoinv Area per charge (eŲ) if rho nor phi0 are given
phi0 Unitless surface potential, $\beta e \varphi_0$, if rho or rhoinv not given
linearise=false Use linearised Poisson-Boltzmann approximation?

Bonded Interactions

Bonds and angular potentials are added via the keyword bondlist either directly in a molecule definition (topology) for intra-molecular bonds, or in energy->bonded where the latter can be used to add inter-molecular bonds:

moleculelist:
    - water: # TIP3P
        structure: "water.xyz"
        bondlist: # index relative to molecule
            - harmonic: { index: [0,1], k: 5024, req: 0.9572 }
            - harmonic: { index: [0,2], k: 5024, req: 0.9572 }
            - harmonic_torsion: { index: [1,0,2], k: 628, aeq: 104.52 }
energy:
    - bonded:
        bondlist: # absolute index; can be between molecules
           - harmonic: { index: [56,921], k: 10, req: 15 }

$\mu V T$ ensembles and Widom insertion are currently unsupported for molecules with bonds.

The following shows the possible bonded potential types:

Harmonic

harmonic Harmonic bond
k Harmonic spring constant (kJ/mol/Ų)
req Equilibrium distance (Å)
index Array with exactly two indices (relative to molecule)

$$ u(r) = \frac{1}{2}k(r-r_{\mathrm{eq}})^2 $$

Finite Extensible Nonlinear Elastic

fene Finite Extensible Nonlinear Elastic Potential
k Bond stiffness (kJ/mol/Ų)
rmax Maximum separation, $r_m$ (Å)
index Array with exactly two indices (relative to molecule)

Finite extensible nonlinear elastic potential long range repulsive potential.

$$ u(r) = \begin{cases} -\frac{1}{2} k r_{\mathrm{max}}^2 \ln \left [ 1-(r/r_{\mathrm{max}})^2 \right ], & \text{if } r < r_{\mathrm{max}} \ \infty, & \text{if } r \geq r_{\mathrm{max}} \end{cases} $$

It is recommended to only use the potential if the initial configuration is near equilibrium, which prevalently depends on the value of rmax. Should one insist on conducting simulations far from equilibrium, a large displacement parameter is recommended to reach finite energies.

Finite Extensible Nonlinear Elastic + WCA

fene+wca Finite Extensible Nonlinear Elastic Potential + WCA
k Bond stiffness (kJ/mol/Ų)
rmax Maximum separation, $r_m$ (Å)
eps=0 Epsilon energy scaling (kJ/mol)
sigma=0 Particle diameter (Å)
index Array with exactly two indices (relative to molecule)

Finite extensible nonlinear elastic potential long range repulsive potential combined with the short ranged Weeks-Chandler-Andersen (wca) repulsive potential. This potential is particularly useful in combination with the nonbonded_cached energy.

$$ u(r) = \begin{cases} -\frac{1}{2} k r_{\mathrm{max}}^2 \ln \left [ 1-(r/r_{\mathrm{max}})^2 \right ] + u_{\mathrm{wca}}, & \text{if } 0 < r \leq 2^{1/6}\sigma \ -\frac{1}{2} k r_{\mathrm{max}}^2 \ln \left [ 1-(r/r_{\mathrm{max}})^2 \right ], & \text{if } 2^{1/6}\sigma < r < r_{\mathrm{max}} \ \infty, & \text{if } r \geq r_{\mathrm{max}} \end{cases} $$

It is recommended to only use this potential if the initial configuration is near equilibrium, which prevalently depends on the value of rmax. Should one insist on conducting simulations far from equilibrium, a large displacement parameter is recommended to reach finite energies.

Harmonic torsion

harmonic_torsion Harmonic torsion
k Harmonic spring constant (kJ/mol/rad²)
aeq Equilibrium angle $\alpha_{\mathrm{eq}}$ (deg)
index Array with exactly three indices (relative to molecule)

$$ u(\alpha) = \frac{1}{2}k(\alpha - \alpha_{\mathrm{eq}})^2 $$ where $\alpha$ is the angle between vector 1→0 and 1→2 (numbers refer to the position in index).

Cosine based torsion (GROMOS-96)

gromos_torsion Cosine based torsion
k Force constant (kJ/mol)
aeq Equilibrium angle $\alpha_{{\mathrm{eq}}}$ (deg)
index Array with exactly three indices (relative to molecule)

$$ u(\alpha) = \frac{1}{2}k(\cos(\alpha) - \cos(\alpha_{{\mathrm{eq}}}))^2 $$ where $\alpha$ is the angle between vector 1→0 and 1→2 (numbers refer to the position in index).

Proper periodic dihedral

periodic_dihedral Proper periodic dihedral
k Force constant (kJ/mol)
n Periodicity (multiplicity) of the dihedral (integer)
phi Phase angle $\phi_{\mathrm{syn}}$ (deg)
index Array with exactly four indices (relative to molecule)

$$ u(r) = k(1 + \cos(n\phi - \phi_{\mathrm{syn}})) $$ where $\phi$ is the angle between the planes constructed from the points 0,1,2 and 1,2,3 (numbers refer to the position in index).

Improper harmonic dihedral

harmonic_dihedral Improper harmonic dihedral
k Harmonic spring constant (kJ/mol/rad²)
deq Equilibrium angle $\alpha_{\mathrm{eq}}$ (deg)
index Array with exactly four indices (relative to molecule)

$$ u(\phi) = \frac{1}{2}k(\phi - \phi_{\mathrm{eq}})^2 $$ where $\phi$ is the angle between the planes constructed from the points 0,1,2 and 1,2,3 (numbers refer to the position in index).

Geometrical Confinement

confine Confine molecules to a sub-region
type Confinement geometry: sphere, cylinder, or cuboid
molecules List of molecules to confine (names)
com=false Apply to molecular mass center
k Harmonic spring constant in kJ/mol or inf for infinity

Confines molecules in a given region of the simulation container by applying a harmonic potential on exterior atom positions, $\mathbf{r}_i$:

$$ U = \frac{1}{2} k \sum_{i}^{\text{exterior}} f_i $$

where $f_i$ is a function that depends on the confinement type, and $k$ is a spring constant. The latter may be infinite which renders the exterior region strictly inaccessible. During equilibration it is advised to use a finite spring constant to drive exterior particles inside the region. Should you insist on equilibrating with $k=\infty$, ensure that displacement parameters are large enough to transport molecules inside the allowed region, or all moves may be rejected. Further, some analysis routines have undefined behavior for configurations with infinite energies.

Available values for type and their additional keywords:

sphere Confine in sphere
radius Radius ($a$)
origo=[0,0,0] Center of sphere ($\mathbf{O}$)
scale=false Scale radius with volume change, $a^{\prime} = a\sqrt[3]{V^{\prime}/V}$
$f_i$ $\vert\mathbf{r}_i-\mathbf{O}\vert^2-a^2$

The scale option will ensure that the confining radius is scaled whenever the simulation volume is scaled. This could for example be during a virtual volume move (analysis) or a volume move in the $NPT$ ensemble.

cylinder Confine in cylinder along $z$-axis
radius Radius ($a$)
origo=[0,0,*] Center of cylinder ($\mathbf{O}$, $z$-value ignored)
$f_i$ $\vert (\mathbf{r}_i-\mathbf{O})\circ \mathbf{d}\vert^2 - a^2$

where $\mathbf{d}=(1,1,0)$ and $\circ$ is the entrywise (Hadamard) product.

cuboid Confine in cuboid
low Lower corner $[x,y,z]$
high Higher corner $[x,y,z]$
$f_i$ $\sum_{\alpha\in {x,y,z} } (\delta r_{i,\alpha})^2$

where $\delta r$ are distances to the confining, cuboidal faces. Note that the elements of low must be smaller than or equal to the corresponding elements of high.

Solvent Accessible Surface Area

sasa SASA Transfer Free Energy
radius=1.4 Probe radius for SASA calculation (Å)
molarity Molar concentration of co-solute
dense=true Flag specifying if a dense or a sparse version of a cell list container is used
slices=25 Number of slices per particle when calculating SASA (the more, the more precise)

Calculates the free energy contribution due to

  1. atomic surface tension
  2. co-solute concentration (typically electrolytes)

via a SASA calculation for each particle. The energy term is:

$$ U = \sum_i^N A_{\text{sasa},i} \left ( \gamma_i + c_s \varepsilon_{\text{tfe},i} \right ) $$

where $c_s$ is the molar concentration of the co-solute; $\gamma_i$ is the atomic surface tension; and $\varepsilon_{\text{tfe},i}$ the atomic transfer free energy, both specified in the atom topology with tension and tfe, respectively. Will use cell lists if a geometry is either cuboid or sphere. The dense option specifies if a dense implementation (memory heavy but faster) or a sparse one (slightly slower but light) of a cell list container will be used.

Alternative schemes

Scheme PBC Cell list Note
sasa Default
sasa_reference Use for debugging
freesasa Uses FreeSASA

Penalty Function

This is a version of the flat histogram or Wang-Landau sampling method where an automatically generated bias or penalty function, $f(\mathcal{X}^d)$, is applied to the system along a one dimensional ($d=1$) or two dimensional ($d=2$) reaction coordinate, $\mathcal{X}^d$, so that the configurational integral reads,

$$ Z(\mathcal{X}^d) = e^{-\beta f(\mathcal{X}^d)} \int e^{-\beta \mathcal{H}(\mathcal{R}, \mathcal{X}^d)} d \mathcal{R}. $$

where $\mathcal{R}$ denotes configurational space at a given $\mathcal{X}$. For every visit to a state along the coordinate, a small penalty energy, $f_0$, is added to $f(\mathcal{X}^d)$ until $Z$ is equal for all $\mathcal{X}$. Thus, during simulation the free energy landscape is flattened, while the true free energy is simply the negative of the generated bias function,

$$ \beta A(\mathcal{X}^d) = -\beta f(\mathcal{X}^d) = -\ln\int e^{-\beta \mathcal{H}(\mathcal{R}, \mathcal{X}^d)} d \mathcal{R}. $$

Flat histogram methods are often attributed to Wang and Landau (2001) but the idea appears in earlier works, for example by Hunter and Reinhardt (1995) and Engkvist and Karlström (1996).

To reduce fluctuations, $f_0$ can be periodically reduced (update, scale) as $f$ converges. At the end of simulation, the penalty function is saved to disk as an array ($d=1$) or matrix ($d=2$). Should the penalty function file be available when starting a new simulation, it is automatically loaded and used as an initial guess. This can also be used to run simulations with a constant bias by setting $f_0=0$.

Example setup where the $x$ and $y$ positions of atom 0 are penalized to achieve uniform sampling:

energy:
- penalty:
    f0: 0.5
    scale: 0.9
    update: 1000
    file: penalty.dat
    coords:
    - atom: {index: 0, property: "x", range: [-2.0,2.0], resolution: 0.1}
    - atom: {index: 0, property: "y", range: [-2.0,2.0], resolution: 0.1}

Options:

penalty Description
f0 Penalty energy increment (kT)
update Interval between scaling of f0
scale Scaling factor for f0
nodrift=true Suppress energy drift
quiet=false Set to true to get verbose output
file Name of saved/loaded penalty function
overwrite=true If false, don't save final penalty function
histogram Name of saved histogram (optional)
coords Array of one or two coordinates

The coordinate, $\mathcal{X}$, can be freely composed by one or two of the types listed in the next section (via coords).

Reaction Coordinates

The following reaction coordinates can be used for penalising the energy and can further be used when analysing the system (see Analysis). Please notice that atom id’s are determined by the order of appearance in the atomlist whereas molecular id’s follow the order of insertion specified in insertmolecules.

General keywords Description
index Atom index, atom id or group index
indexes Array of atomic indexes ([a,b] or [a,b,c,d])
range Array w. [min:max] value
resolution Resolution along the coordinate (Å)
dir Axes of the reaction coordinate, $e.g.$, [1,1,0] for the $xy$-plane

Atom Properties

coords=[atom] Property
x, y or z $x$-, $y$- or $z$-coordinate of the $i$th particle, $i$=index
q Charge of the $i$th particle, $i$=index
R Distance of the $i$th particle from the center of the simulation cell, $i$=index
N Number of atoms of id=index

Molecule Properties

coords=[molecule] Property
active If molecule is active (1) or inactive (0); for GCMC ensembles
angle Angle between instantaneous principal axis and given dir vector
com_x, com_y or com_z Mass-center coordinates
confid Conformation id corresponding to frame in traj (see molecular topology).
end2end Distance between first and last atom
Rg Radius of gyration
mu_x, mu_y or mu_z Molecular dipole moment components
mu Molecular dipole moment scalar ($e$Å/charge)
muangle Angle between dipole moment and given dir vector
N Number of atoms in group
Q Monopole moment (net charge)
atomatom Distance along dir between 2 atoms specified by the indexes array
cmcm Absolute mass-center separation between group indexes a and b or atomic indexes ab and cd
cmcm_z $z$-component of cmcm
mindist Minimum distance between particles of id indexes[0] and indexes[1]
L/R Ratio between height and radius of a cylindrical vesicle
Rinner Average $d$ of id=indexes[0] for particles having a smaller $d$ than id=indexes[1]

Notes:

  • the molecular dipole moment is defined with respect to the mass-center
  • for angle, the principal axis is the eigenvector corresponding to the smallest eigenvalue of the gyration tensor
  • Rinner can be used to calculate the inner radius of cylindrical or spherical vesicles. $d^2=\bf{r} \cdot$dir where $\bf{r}$ is the position vector
  • L/R can be used to calculate the bending modulus of a cylindrical lipid vesicle
  • Rg is calculated as the square-root of the sum of the eigenvalues of the gyration tensor, $S$. $$ S = \frac{1}{\sum_{i=1}^{N} m_{i}} \sum_{i=1}^{N} m_{i} \bf{t_i} \bf{t_i^T} $$ where $\bf{t_i} = \bf{r_i} - \bf{cm}$, $\bf{r_i}$ is the coordinate of the $i$th atom, $m_i$ is the mass of the $i$th atom, $\bf{cm}$ is the mass center of the group and $N$ is the number of atoms in the molecule.

System Properties

coords=[system] Property
V System volume
Q System net-charge
Lx, Ly or Lz Side length of the cuboidal simulation cell
height Alias for Lz
radius Radius of spherical or cylindrical geometries
N Number of active particles
mu System dipole moment scalar (𝑒Å)
mu_x, mu_y or mu_z System dipole moment components (𝑒Å)

The enclosing cuboid is the smallest cuboid that can contain the geometry. For example, for a cylindrical simulation container, Lz is the height and Lx=Ly is the diameter.

Multiple Walkers with MPI

If compiled with MPI, the master process collects the bias function from all nodes upon penalty function update. The average is then re-distributed, offering linear parallelization of the free energy sampling. It is crucial that the walk in coordinate space differs in the different processes, e.g., by specifying a different random number seed; start configuration; or displacement parameter. File output and input are prefixed with mpi{rank}.

The following starts all MPI processes with the same input file, and the MPI prefix is automatically appended to all other input and output:

yason.py input.yml | mpirun --np 6 --stdin all faunus -s state.json

Here, each process automatically looks for mpi{nproc}.state.json.

Constraining the system

Reaction coordinates can be used to constrain the system within a range using the constrain energy term. Stepping outside the range results in an inifinite energy, forcing rejection. For example,

energy:
    - constrain: {type: molecule, index: 0, property: end2end, range: [0,200]}

Tip: placing constrain at the top of the energy list is more efficient as the remaining energy terms are skipped should an infinite energy arise.