Non-bonded interactions in |Gromacs| are pair-additive:
.. math:: V(\mathbf{r}_1,\ldots \mathbf{r}_N) = \sum_{i<j}V_{ij}(\mathbf{r}_ij);
+ :label: eqnnbinteractions1
.. math:: \mathbf{F}_i = -\sum_j \frac{dV_{ij}(r_{ij})}{dr_{ij}} \frac{\mathbf{r}_ij}{r_{ij}}
+ :label: eqnnbinteractions2
Since the potential only depends on the scalar distance, interactions
will be centro-symmetric, i.e. the vectorial partial force on particle
The Lennard-Jones potential :math:`V_{LJ}` between two atoms equals:
-.. math::
-
- V_{LJ}({r_{ij}}) = \frac{C_{ij}^{(12)}}{{r_{ij}}^{12}} -
- \frac{C_{ij}^{(6)}}{{r_{ij}}^6}
+.. math:: V_{LJ}({r_{ij}}) = \frac{C_{ij}^{(12)}}{{r_{ij}}^{12}} -
+ \frac{C_{ij}^{(6)}}{{r_{ij}}^6}
+ :label: eqnnblj
See also :numref:`Fig. %s <fig-lj>` The parameters :math:`C^{(12)}_{ij}` and
:math:`C^{(6)}_{ij}` depend on pairs of *atom types*; consequently they
.. math:: \mathbf{F}_i(\mathbf{r}_ij) = \left( 12~\frac{C_{ij}^{(12)}}{{r_{ij}}^{13}} -
6~\frac{C_{ij}^{(6)}}{{r_{ij}}^7} \right) {\frac{{\mathbf{r}_{ij}}}{{r_{ij}}}}
+ :label: eqnljforce
The LJ potential may also be written in the following form:
\sigma_{ij} &=& \left({\sigma_{ii} \, \sigma_{jj}}\right)^{1/2} \\
\epsilon_{ij} &=& \left({\epsilon_{ii} \, \epsilon_{jj}}\right)^{1/2}
\end{array}
+ :label: eqnnbgeometricaverage
This last rule is used by the OPLS force field.
term than the Lennard-Jones interaction, but is also more expensive to
compute. The potential form is:
-.. math::
-
- V_{bh}({r_{ij}}) = A_{ij} \exp(-B_{ij} {r_{ij}}) -
- \frac{C_{ij}}{{r_{ij}}^6}
+.. math:: V_{bh}({r_{ij}}) = A_{ij} \exp(-B_{ij} {r_{ij}}) -
+ \frac{C_{ij}}{{r_{ij}}^6}
+ :label: eqnnbbuckingham
.. _fig-bham:
See also :numref:`Fig. %s <fig-bham>`. The force derived from this is:
-.. math::
-
- \mathbf{F}_i({r_{ij}}) = \left[ A_{ij}B_{ij}\exp(-B_{ij} {r_{ij}}) -
- 6\frac{C_{ij}}{{r_{ij}}^7} \right] {\frac{{\mathbf{r}_{ij}}}{{r_{ij}}}}
+.. math:: \mathbf{F}_i({r_{ij}}) = \left[ A_{ij}B_{ij}\exp(-B_{ij} {r_{ij}}) -
+ 6\frac{C_{ij}}{{r_{ij}}^7} \right] {\frac{{\mathbf{r}_{ij}}}{{r_{ij}}}}
+ :label: eqnnbbuckinghamforce
.. _coul:
The force derived from this potential is:
.. math:: \mathbf{F}_i(\mathbf{r}_ij) = f \frac{q_i q_j}{{\varepsilon_r}{r_{ij}}^2}{\frac{{\mathbf{r}_{ij}}}{{r_{ij}}}}
+ :label: eqnfcoul
A plain Coulomb interaction should only be used without cut-off or when
all pairs fall within the cut-off, since there is an abrupt, large
:eq:`eqn. %s <eqnvcrf>` for simplicity as
.. math:: V_{crf} ~=~ f \frac{q_i q_j}{{\varepsilon_r}}\left[\frac{1}{{r_{ij}}} + k_{rf}~ {r_{ij}}^2 -c_{rf}\right]
+ :label: eqnvcrfrewrite
with
as:
.. math:: \mathbf{F}_i = c \, F(r_{ij}) \frac{\mathbf{r}_ij}{r_{ij}}
+ :label: eqnswitch
For pure Coulomb or Lennard-Jones interactions
:math:`F(r) = F_\alpha(r) = \alpha \, r^{-(\alpha+1)}`. The switched
force :math:`F_s(r)` can generally be written as:
-.. math::
-
- \begin{array}{rcl}
- F_s(r)~=&~F_\alpha(r) & r < r_1 \\
- F_s(r)~=&~F_\alpha(r)+S(r) & r_1 \le r < r_c \\
- F_s(r)~=&~0 & r_c \le r
- \end{array}
+.. math:: \begin{array}{rcl}
+ F_s(r)~=&~F_\alpha(r) & r < r_1 \\
+ F_s(r)~=&~F_\alpha(r)+S(r) & r_1 \le r < r_c \\
+ F_s(r)~=&~0 & r_c \le r
+ \end{array}
+ :label: eqnswitchforce
When :math:`r_1=0` this is a traditional shift function, otherwise it
acts as a switch function. The corresponding shifted potential function
then reads:
.. math:: V_s(r) = \int^{\infty}_r~F_s(x)\, dx
+ :label: eqnswitchpotential
The |Gromacs| **force switch** function :math:`S_F(r)` should be smooth at
the boundaries, therefore the following boundary conditions are imposed
on the switch function:
-.. math::
-
- \begin{array}{rcl}
- S_F(r_1) &=&0 \\
- S_F'(r_1) &=&0 \\
- S_F(r_c) &=&-F_\alpha(r_c) \\
- S_F'(r_c) &=&-F_\alpha'(r_c)
- \end{array}
+.. math:: \begin{array}{rcl}
+ S_F(r_1) &=&0 \\
+ S_F'(r_1) &=&0 \\
+ S_F(r_c) &=&-F_\alpha(r_c) \\
+ S_F'(r_c) &=&-F_\alpha'(r_c)
+ \end{array}
+ :label: eqnswitchforcefunction
A 3\ :math:`^{rd}` degree polynomial of the form
.. math:: S_F(r) = A(r-r_1)^2 + B(r-r_1)^3
+ :label: eqnswitchforcepoly
fulfills these requirements. The constants A and B are given by the
boundary condition at :math:`r_c`:
-.. math::
-
- \begin{array}{rcl}
- A &~=~& -\alpha \, \displaystyle
- \frac{(\alpha+4)r_c~-~(\alpha+1)r_1} {r_c^{\alpha+2}~(r_c-r_1)^2} \\
- B &~=~& \alpha \, \displaystyle
- \frac{(\alpha+3)r_c~-~(\alpha+1)r_1}{r_c^{\alpha+2}~(r_c-r_1)^3}
- \end{array}
+.. math:: \begin{array}{rcl}
+ A &~=~& -\alpha \, \displaystyle
+ \frac{(\alpha+4)r_c~-~(\alpha+1)r_1} {r_c^{\alpha+2}~(r_c-r_1)^2} \\
+ B &~=~& \alpha \, \displaystyle
+ \frac{(\alpha+3)r_c~-~(\alpha+1)r_1}{r_c^{\alpha+2}~(r_c-r_1)^3}
+ \end{array}
+ :label: eqnforceswitchboundary
Thus the total force function is:
.. math:: F_s(r) = \frac{\alpha}{r^{\alpha+1}} + A(r-r_1)^2 + B(r-r_1)^3
+ :label: eqnswitchfinalforce
and the potential function reads:
.. math:: V_s(r) = \frac{1}{r^\alpha} - \frac{A}{3} (r-r_1)^3 - \frac{B}{4} (r-r_1)^4 - C
+ :label: eqnswitchfinalpotential
where
.. math:: C = \frac{1}{r_c^\alpha} - \frac{A}{3} (r_c-r_1)^3 - \frac{B}{4} (r_c-r_1)^4
+ :label: eqnswitchpotentialexp
The |Gromacs| **potential-switch** function :math:`S_V(r)` scales the
potential between :math:`r_1` and :math:`r_c`, and has similar boundary
conditions, intended to produce smoothly-varying potential and forces:
-.. math::
-
- \begin{array}{rcl}
- S_V(r_1) &=&1 \\
- S_V'(r_1) &=&0 \\
- S_V''(r_1) &=&0 \\
- S_V(r_c) &=&0 \\
- S_V'(r_c) &=&0 \\
- S_V''(r_c) &=&0
- \end{array}
+.. math:: \begin{array}{rcl}
+ S_V(r_1) &=&1 \\
+ S_V'(r_1) &=&0 \\
+ S_V''(r_1) &=&0 \\
+ S_V(r_c) &=&0 \\
+ S_V'(r_c) &=&0 \\
+ S_V''(r_c) &=&0
+ \end{array}
+ :label: eqnpotentialswitch
The fifth-degree polynomial that has these properties is
.. math:: S_V(r; r_1, r_c) = \frac{1 - 10(r-r_1)^3(r_c-r_1)^2 + 15(r-r_1)^4(r_c-r_1) - 6(r-r_1)}{(r_c-r_1)^5}
+ :label: eqn5polynomal
This implementation is found in several other simulation
packages,\ :ref:`73 <refOhmine1988>`\ :ref:`75 <refGuenot1993>` but
given by:
.. math:: V(r) = f \frac{\mbox{erfc}(\beta r_{ij})}{r_{ij}} q_i q_j,
+ :label: eqnewaldsrmod
where :math:`\beta` is a parameter that determines the relative weight
between the direct space sum and the reciprocal space sum and
biod.pnpi.spb.ru / alexxy / gromacs.git / blobdiff