# Methods¶

## Exclusions and 1-4 Interactions.¶

Atoms within a molecule that are close by in the chain, *i.e.* atoms
that are covalently bonded, or linked by one or two atoms are called
*first neighbors, second neighbors* and *third neighbors*, respectively
(see Fig. 33). Since the interactions of atom **i** with atoms
**i+1** and **i+2** are mainly quantum mechanical, they can not be
modeled by a Lennard-Jones potential. Instead it is assumed that these
interactions are adequately modeled by a harmonic bond term or
constraint (**i, i+1**) and a harmonic angle term (**i, i+2**). The
first and second neighbors (atoms **i+1** and **i+2**) are therefore
*excluded* from the Lennard-Jones interaction list of atom **i**; atoms
**i+1** and **i+2** are called *exclusions* of atom **i**.

For third neighbors, the normal Lennard-Jones repulsion is sometimes
still too strong, which means that when applied to a molecule, the
molecule would deform or break due to the internal strain. This is
especially the case for carbon-carbon interactions in a
*cis*-conformation (*e.g.* *cis*-butane). Therefore, for some of these
interactions, the Lennard-Jones repulsion has been reduced in the GROMOS
force field, which is implemented by keeping a separate list of 1-4 and
normal Lennard-Jones parameters. In other force fields, such as
OPLS 103, the standard Lennard-Jones
parameters are reduced by a factor of two, but in that case also the
dispersion (r\(^{-6}\)) and the Coulomb interaction are scaled.
GROMACS can use either of these methods.

## Charge Groups¶

In principle, the force calculation in MD is an \(O(N^2)\) problem. Therefore, we apply a cut-off for non-bonded force (NBF) calculations; only the particles within a certain distance of each other are interacting. This reduces the cost to \(O(N)\) (typically \(100N\) to \(200N\)) of the NBF. It also introduces an error, which is, in most cases, acceptable, except when applying the cut-off implies the creation of charges, in which case you should consider using the lattice sum methods provided by GROMACS.

Consider a water molecule interacting with another atom. If we would
apply a plain cut-off on an atom-atom basis we might include the
atom-oxygen interaction (with a charge of \(-0.82\)) without the
compensating charge of the protons, and as a result, induce a large
dipole moment over the system. Therefore, we have to keep groups of
atoms with total charge 0 together. These groups are called *charge
groups*. Note that with a proper treatment of long-range electrostatics
(e.g. particle-mesh Ewald (sec. PME), keeping charge groups
together is not required.

## Treatment of Cut-offs in the group scheme¶

GROMACS is quite flexible in treating cut-offs, which implies there can be quite a number of parameters to set. These parameters are set in the input file for grompp. There are two sort of parameters that affect the cut-off interactions; you can select which type of interaction to use in each case, and which cut-offs should be used in the neighbor searching.

For both Coulomb and van der Waals interactions there are interaction type selectors (termed vdwtype and coulombtype) and two parameters, for a total of six non-bonded interaction parameters. See the User Guide for a complete description of these parameters.

In the group cut-off scheme, all of the interaction functions in Table 9 require that neighbor searching be done with a radius at least as large as the \(r_c\) specified for the functional form, because of the use of charge groups. The extra radius is typically of the order of 0.25 nm (roughly the largest distance between two atoms in a charge group plus the distance a charge group can diffuse within neighbor list updates).

Type | Parameters | |
---|---|---|

Coulomb | Plain cut-off | \(r_c\), \({\varepsilon}_{r}\) |

Reaction field | \(r_c\), \({\varepsilon}_{rf}\) | |

Shift function | \(r_1\), \(r_c\), \({\varepsilon}_{r}\) | |

Switch function | \(r_1\), \(r_c\), \({\varepsilon}_{r}\) | |

VdW | Plain cut-off | \(r_c\) |

Shift function | \(r_1\), \(r_c\) | |

Switch function | \(r_1\), \(r_c\) |

# Virtual interaction sites¶

Virtual interaction sites (called dummy atoms in GROMACS versions before 3.3) can be used in GROMACS in a number of ways. We write the position of the virtual site \(\mathbf{r}_s\) as a function of the positions of other particles \(\mathbf{r}\)\(_i\): \(\mathbf{r}_s = f(\mathbf{r}_1..\mathbf{r}_n)\). The virtual site, which may carry charge or be involved in other interactions, can now be used in the force calculation. The force acting on the virtual site must be redistributed over the particles with mass in a consistent way. A good way to do this can be found in ref. 104. We can write the potential energy as:

The force on the particle \(i\) is then:

The first term is the normal force. The second term is the force on particle \(i\) due to the virtual site, which can be written in tensor notation:

where \(\mathbf{F}_{s}\) is the force on the virtual site and \(x_s\), \(y_s\) and \(z_s\) are the coordinates of the virtual site. In this way, the total force and the total torque are conserved 104.

The computation of the virial ((13)) is non-trivial when
virtual sites are used. Since the virial involves a summation over all
the atoms (rather than virtual sites), the forces must be redistributed
from the virtual sites to the atoms (using (3)) *before*
computation of the virial. In some special cases where the forces on the
atoms can be written as a linear combination of the forces on the
virtual sites (types 2 and 3 below) there is no difference between
computing the virial before and after the redistribution of forces.
However, in the general case redistribution should be done first.

There are six ways to construct virtual sites from surrounding atoms in
GROMACS, which we classify by the number of constructing atoms. **Note**
that all site types mentioned can be constructed from types 3fd
(normalized, in-plane) and 3out (non-normalized, out of plane). However,
the amount of computation involved increases sharply along this list, so
we strongly recommended using the first adequate virtual site type that
will be sufficient for a certain purpose. Fig. 34 depicts 6 of
the available virtual site constructions. The conceptually simplest
construction types are linear combinations:

The force is then redistributed using the same weights:

The types of virtual sites supported in GROMACS are given in the list below. Constructing atoms in virtual sites can be virtual sites themselves, but only if they are higher in the list, i.e. virtual sites can be constructed from “particles” that are simpler virtual sites.

On top of an atom. This allows giving an atom multiple atom types and with that also assigned multiple, different bonded interactions. This can espically be of use in free-energy calculations.

The coordinates of the virtual site equal that of the constructing atom:

(6)¶\[\mathbf{r}_s ~=~ \mathbf{r}_i\]The force is moved to the constructing atom:

(7)¶\[\mathbf{F}_i ~=~ \mathbf{F}_{s}\]As a linear combination of two atoms (Fig. 34 2):

(8)¶\[w_i = 1 - a ~,~~ w_j = a\]In this case the virtual site is on the line through atoms \(i\) and \(j\).

On the line through two atoms, with a fixed distance (Fig. 34 2fd):

(9)¶\[\mathbf{r}_s ~=~ \mathbf{r}_i + a \frac{ \mathbf{r}_{ij} } { | \mathbf{r}_{ij} | }\]In this case the virtual site is on the line through the other two particles at a distance of \(|a|\) from \(i\). The force on particles \(i\) and \(j\) due to the force on the virtual site can be computed as:

(10)¶\[\begin{split}\begin{array}{lcr} \mathbf{F}_i &=& \displaystyle \mathbf{F}_{s} - \gamma ( \mathbf{F}_{is} - \mathbf{p} ) \\[1ex] \mathbf{F}_j &=& \displaystyle \gamma (\mathbf{F}_{s} - \mathbf{p}) \\[1ex] \end{array} ~\mbox{ where }~ \begin{array}{c} \displaystyle \gamma = \frac{a}{ | \mathbf{r}_{ij} | } \\[2ex] \displaystyle \mathbf{p} = \frac{ \mathbf{r}_{is} \cdot \mathbf{F}_{s} } { \mathbf{r}_{is} \cdot \mathbf{r}_{is} } \mathbf{r}_{is} \end{array}\end{split}\]As a linear combination of three atoms (Fig. 34 3):

(11)¶\[w_i = 1 - a - b ~,~~ w_j = a ~,~~ w_k = b\]In this case the virtual site is in the plane of the other three particles.

In the plane of three atoms, with a fixed distance (Fig. 34 3fd):

(12)¶\[\mathbf{r}_s ~=~ \mathbf{r}_i + b \frac{ (1 - a) \mathbf{r}_{ij} + a \mathbf{r}_{jk} } { | (1 - a) \mathbf{r}_{ij} + a \mathbf{r}_{jk} | }\]In this case the virtual site is in the plane of the other three particles at a distance of \(|b|\) from \(i\). The force on particles \(i\), \(j\) and \(k\) due to the force on the virtual site can be computed as:

(13)¶\[\begin{split}\begin{array}{lcr} \mathbf{F}_i &=& \displaystyle \mathbf{F}_{s} - \gamma ( \mathbf{F}_{is} - \mathbf{p} ) \\[1ex] \mathbf{F}_j &=& \displaystyle (1-a)\gamma (\mathbf{F}_{s} - \mathbf{p}) \\[1ex] \mathbf{F}_k &=& \displaystyle a \gamma (\mathbf{F}_{s} - \mathbf{p}) \\ \end{array} ~\mbox{ where }~ \begin{array}{c} \displaystyle \gamma = \frac{b}{ | \mathbf{r}_{ij} + a \mathbf{r}_{jk} | } \\[2ex] \displaystyle \mathbf{p} = \frac{ \mathbf{r}_{is} \cdot \mathbf{F}_{s} } { \mathbf{r}_{is} \cdot \mathbf{r}_{is} } \mathbf{r}_{is} \end{array}\end{split}\]In the plane of three atoms, with a fixed angle and distance (Fig. 34 3fad):

(14)¶\[\mathbf{r}_s ~=~ \mathbf{r}_i + d \cos \theta \frac{\mathbf{r}_{ij}}{ | \mathbf{r}_{ij} | } + d \sin \theta \frac{\mathbf{r}_\perp}{ | \mathbf{r}_\perp | } ~\mbox{ where }~ \mathbf{r}_\perp ~=~ \mathbf{r}_{jk} - \frac{ \mathbf{r}_{ij} \cdot \mathbf{r}_{jk} } { \mathbf{r}_{ij} \cdot \mathbf{r}_{ij} } \mathbf{r}_{ij}\]In this case the virtual site is in the plane of the other three particles at a distance of \(|d|\) from \(i\) at an angle of \(\alpha\) with \(\mathbf{r}_{ij}\). Atom \(k\) defines the plane and the direction of the angle.

**Note**that in this case \(b\) and \(\alpha\) must be specified, instead of \(a\) and \(b\) (see also sec. Virtual sites). The force on particles \(i\), \(j\) and \(k\) due to the force on the virtual site can be computed as (with \(\mathbf{r}_\perp\) as defined in (14)):(15)¶\[\begin{split}\begin{array}{c} \begin{array}{lclllll} \mathbf{F}_i &=& \mathbf{F}_{s} &-& \dfrac{d \cos \theta}{ | \mathbf{r}_{ij} | } \mathbf{F}_1 &+& \dfrac{d \sin \theta}{ | \mathbf{r}_\perp | } \left( \dfrac{ \mathbf{r}_{ij} \cdot \mathbf{r}_{jk} } { \mathbf{r}_{ij} \cdot \mathbf{r}_{ij} } \mathbf{F}_2 + \mathbf{F}_3 \right) \\[3ex] \mathbf{F}_j &=& && \dfrac{d \cos \theta}{ | \mathbf{r}_{ij} | } \mathbf{F}_1 &-& \dfrac{d \sin \theta}{ | \mathbf{r}_\perp | } \left( \mathbf{F}_2 + \dfrac{ \mathbf{r}_{ij} \cdot \mathbf{r}_{jk} } { \mathbf{r}_{ij} \cdot \mathbf{r}_{ij} } \mathbf{F}_2 + \mathbf{F}_3 \right) \\[3ex] \mathbf{F}_k &=& && && \dfrac{d \sin \theta}{ | \mathbf{r}_\perp | } \mathbf{F}_2 \\[3ex] \end{array} \\[5ex] ~\mbox{where }~ \mathbf{F}_1 = \mathbf{F}_{s} - \dfrac{ \mathbf{r}_{ij} \cdot \mathbf{F}_{s} } { \mathbf{r}_{ij} \cdot \mathbf{r}_{ij} } \mathbf{r}_{ij} ~\mbox{, }~ \mathbf{F}_2 = \mathbf{F}_1 - \dfrac{ \mathbf{r}_\perp \cdot \mathbf{F}_{s} } { \mathbf{r}_\perp \cdot \mathbf{r}_\perp } \mathbf{r}_\perp ~\mbox{and }~ \mathbf{F}_3 = \dfrac{ \mathbf{r}_{ij} \cdot \mathbf{F}_{s} } { \mathbf{r}_{ij} \cdot \mathbf{r}_{ij} } \mathbf{r}_\perp \end{array}\end{split}\]As a non-linear combination of three atoms, out of plane (Fig. 34 3out):

(16)¶\[\mathbf{r}_s ~=~ \mathbf{r}_i + a \mathbf{r}_{ij} + b \mathbf{r}_{ik} + c (\mathbf{r}_{ij} \times \mathbf{r}_{ik})\]This enables the construction of virtual sites out of the plane of the other atoms. The force on particles \(i,j\) and \(k\) due to the force on the virtual site can be computed as:

(17)¶\[\begin{split}\begin{array}{lcl} \mathbf{F}_j &=& \left[\begin{array}{ccc} a & -c\,z_{ik} & c\,y_{ik} \\[0.5ex] c\,z_{ik} & a & -c\,x_{ik} \\[0.5ex] -c\,y_{ik} & c\,x_{ik} & a \end{array}\right]\mathbf{F}_{s} \\ \mathbf{F}_k &=& \left[\begin{array}{ccc} b & c\,z_{ij} & -c\,y_{ij} \\[0.5ex] -c\,z_{ij} & b & c\,x_{ij} \\[0.5ex] c\,y_{ij} & -c\,x_{ij} & b \end{array}\right]\mathbf{F}_{s} \\ \mathbf{F}_i &=& \mathbf{F}_{s} - \mathbf{F}_j - \mathbf{F}_k \end{array}\end{split}\]From four atoms, with a fixed distance, see separate Fig. 35. This construction is a bit complex, in particular since the previous type (4fd) could be unstable which forced us to introduce a more elaborate construction:

- (18)¶\[\begin{split}\begin{aligned} \mathbf{r}_{ja} &=& a\, \mathbf{r}_{ik} - \mathbf{r}_{ij} = a\, (\mathbf{x}_k - \mathbf{x}_i) - (\mathbf{x}_j - \mathbf{x}_i) \nonumber \\ \mathbf{r}_{jb} &=& b\, \mathbf{r}_{il} - \mathbf{r}_{ij} = b\, (\mathbf{x}_l - \mathbf{x}_i) - (\mathbf{x}_j - \mathbf{x}_i) \nonumber \\ \mathbf{r}_m &=& \mathbf{r}_{ja} \times \mathbf{r}_{jb} \nonumber \\ \mathbf{x}_s &=& \mathbf{x}_i + c \frac{\mathbf{r}_m}{ | \mathbf{r}_m | } \end{aligned}\end{split}\]
In this case the virtual site is at a distance of \(|c|\) from \(i\), while \(a\) and \(b\) are parameters.

**Note**that the vectors \(\mathbf{r}_{ik}\) and \(\mathbf{r}_{ij}\) are not normalized to save floating-point operations. The force on particles \(i\), \(j\), \(k\) and \(l\) due to the force on the virtual site are computed through chain rule derivatives of the construction expression. This is exact and conserves energy, but it does lead to relatively lengthy expressions that we do not include here (over 200 floating-point operations). The interested reader can look at the source code in`vsite.c`

. Fortunately, this vsite type is normally only used for chiral centers such as \(C_{\alpha}\) atoms in proteins.The new 4fdn construct is identified with a ‘type’ value of 2 in the topology. The earlier 4fd type is still supported internally (‘type’ value 1), but it should not be used for new simulations. All current GROMACS tools will automatically generate type 4fdn instead.

A linear combination of \(N\) atoms with relative weights \(a_i\). The weight for atom \(i\) is:

(19)¶\[w_i = a_i \left(\sum_{j=1}^N a_j \right)^{-1}\]There are three options for setting the weights:

- center of geometry: equal weights
- center of mass: \(a_i\) is the mass of atom \(i\); when in free-energy simulations the mass of the atom is changed, only the mass of the A-state is used for the weight
- center of weights: \(a_i\) is defined by the user