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The following is my notes about the GROMACS Soft-Core Potential for Free Energy Calculation


The soft-core potential between two states A and B is defined as

$\begin{split} V_{sc}(r) &=(1-\lambda)V_A(r_A) + \lambda V_B(r_B)
&=(1-\lambda)V_A(r) + \lambda V_B(r) \;\;\;\; \text{if } \alpha \sigma=0
r_A &= \left(\alpha \sigma_A^6 \lambda^p+r^6\right)^{1/6}
r_B &= \left(\alpha \sigma_B^6 (1-\lambda)^p+r^6\right)^{1/6}

When $\alpha=0$ or $\sigma=0$, the soft-core potential degenerates to the original potential. $\alpha$ is usually about 0.5 and $p=1$ or $p=2$. The recommended combination is $\alpha=0.5, p=1$.

For LJ potential,

$\alg V &= { C_{12} \over r^{12} } - {C_6 \over r^6}
V_{sc} &=C_{12}\left[{1-\lambda \over (\alpha \sigma^6 \lambda^p+r^6)^2} + {\lambda \over (\alpha \sigma^6(1-\lambda)^6+r^6)^2}\right]
&-C_6 \left[{1-\lambda \over \alpha \sigma^6 \lambda^p+r^6} + {\lambda \over \alpha \sigma^6(1-\lambda)^6+r^6}\right] \ealg$

Some special cases

$\alg V_{sc}=0 &\Rightarrow V(r_A)=0 \; V(r_B)=0
r_A=r_B &\Rightarrow \lambda=0.5 \ealg$

If A state is totally decoupled, which means $V_A=0$, in this case

$\alg V_{sc}(r) &= \lambda V_B(r_B)
&= \lambda \left\{ {C_{12} \over \left(\alpha \sigma^6 (1-\lambda)^p+r^6\right)^2} -{C_6 \over \alpha \sigma^6 (1-\lambda)^p+r^6} \right\} \ealg$


$\alg s^6 &=\alpha\sigma^6
r_{sc} &= \left(r^6+ (1-\lambda)^p s^6\right)^{1/6}
\min r_{sc} &= r &\;\; \text{when } \lambda=1
\max r_{sc} &= (r^6+s^6)^{1/6} &\;\; \text{when } \lambda=0 \ealg$

Plot of $r_{sc}$ with $\l$

\[r_{sc}/r = \left(1+(1-\lambda)^p (s/r)^6\right)^{1/6}\]

Plot of $\max r_{sc}$ with $\l$

\[\max r_{sc}/r = (1+(s/r)^6)^{1/6}\]

Plot of $V_{sc}(r)$

\[\alg V_{sc}(r, s, \l) &= \lambda \left\\{ {C_{12} \over \left(r^6+(1-\lambda)^p s^6\right)^2} -{C_6 \over r^6+(1-\lambda)^p s^6} \right\\} \\ &=4\e \lambda \left\\{ {\s^{12} \over \left(r^6+(1-\lambda)^p s^6\right)^2} -{\s^6 \over r^6+(1-\lambda)^p s^6} \right\\} \ealg\]

GMX Options

In GMX, $\alpha, \sigma, p$ is corresponding to sc_alpha, sc_sigma, sc_power, respectively. The manual said

The description of sc-sigma is not correct, actually

$\sigma= \begin{cases} & \sigma^*=(C_{12}/C_{6})^{1/6}
& \text{sc_sigma} \; \text{ if } C_6\times C_{12}=0 \end{cases}$

See the discussion





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