Coupling Interaction with \(\lambda\)

In this section, the scaling nonbonded and long-range correction energies with \(\lambda\) is discussed in detailed.

\[E_{\lambda} = E_{\lambda}(\texttt{VDW}) + E_{\lambda}(\texttt{Elect}) + E_{\lambda}(\texttt{LRC-VDW}) + E_{\lambda}(\texttt{LRC-Elect})\]

VDW

Soft-core

In free energy calculation, the VDW interaction between solute and solvent is scaled with \(\lambda\), non-linearly (soft-core scheme), to avoid end-point catastrophe and numerical issue

\[E_{\lambda}(\texttt{VDW}) = \lambda_{\texttt{VDW}} E_{\texttt{VDW}}(r_{sc})\]

the scaled solute-solvent distance, \(r_{sc}\) is defined as:

\[r_{sc} = \bigg[\alpha {\big(1 - \lambda_{\texttt{VDW}} \big)}^{p}{\sigma}^6 + {r}^6 \bigg]^{\frac{1}{6}}\]

where, \(\alpha\) and \(p\) are the soft-core parameters defined by user (ScaleAlpha, ScalePower) and \(\sigma\) is the diameter of atom. To improve numerical convergence of the calculation, a minimum interaction diameter \(\sigma_{min}\) should be defined by user (MinSigma) for any atom with a diameter less than \(\sigma_{min}\), e.g. hydrogen atoms attached to oxygen in water or alcohols.

To calculate the solvation free energy with thermodynamic integration (TI) method, the derivative of energy with respect to lambda (\(\frac{dE_{\lambda}(\texttt{VDW})}{d\lambda_{\texttt{VDW}}}\)) is required:

\[\frac{dE_{\lambda}(\texttt{VDW})}{d\lambda_{\texttt{VDW}}} = E_{\texttt{VDW}}(r_{sc}) + \frac{p \alpha \lambda_{\texttt{VDW}}}{6} \bigg(1 - \lambda_{\texttt{VDW}}\bigg)^{p-1} \bigg(\frac{{\sigma}^6}{{r_{sc}}^5} \bigg) F_{\texttt{VDW}}(r_{sc})\]

Electrostatic

Hard-core

In free energy calculation, the Coulombic interaction between solute and solvent can be scaled with \(\lambda\), linearly (hard-core scheme), by setting the ScaleCoulomb to false.

\[E_{\lambda}(\texttt{Elect}) = \lambda_{\texttt{Elect}} E_{\texttt{Elect}}(r)\]

where, \(r\) is the distance between solute and solvent, without any modification.

To calculate the solvation free energy with thermodynamic integration (TI) method, the derivative of energy with respect to lambda (\(\frac{dE_{\lambda}(\texttt{Elect})}{d\lambda_{\texttt{Elect}}}\)) is required:

\[\frac{dE_{\lambda}(\texttt{Elect})}{d\lambda_{\texttt{Elect}}} = E_{\texttt{Elect}}(r)\]

Warning

To avoid end-point catastrophe and numerical issue, it’s suggested to turn on the VDW interaction completely, before turning on the Coulombic interaction.

Soft-core

In free energy calculation, the Coulombic interaction between solute and solvent can be scaled with \(\lambda\), non-linearly (soft-core scheme), to avoid end-point catastrophe and numerical issue. This option can be activated by setting the ScaleCoulomb to true.

\[E_{\lambda}(\texttt{Elect}) = \lambda_{\texttt{Elect}} E_{\texttt{Elect}}(r_{sc})\]

the scaled solute-solvent distance, \(r_{sc}\) is defined as:

\[r_{sc} = \bigg[\alpha {\big(1 - \lambda_{\texttt{Elect}} \big)}^{p}{\sigma}^6 + {r}^6 \bigg]^{\frac{1}{6}}\]

where, \(\alpha\) and \(p\) are the soft-core parameters defined by user (ScaleAlpha, ScalePower) and \(\sigma\) is the diameter of atom. To improve numerical convergence of the calculation, a minimum interaction diameter \(\sigma_{min}\) should be defined by user (MinSigma) for any atom with a diameter less than \(\sigma_{min}\), e.g. hydrogen atoms attached to oxygen in water or alcohols.

To calculate the solvation free energy with thermodynamic integration (TI) method, the derivative of energy with respect to lambda (\(\frac{dE_{\lambda}(\texttt{Elect})}{d\lambda_{\texttt{Elect}}}\)) is required:

\[\frac{dE_{\lambda}(\texttt{Elect})}{d\lambda_{\texttt{Elect}}} = E_{\texttt{Elect}}(r_{sc}) + \frac{p \alpha \lambda_{\texttt{Elect}}}{6} \bigg(1 - \lambda_{\texttt{Elect}}\bigg)^{p-1} \bigg(\frac{{\sigma}^6}{{r_{sc}}^5} \bigg) F_{\texttt{Elect}}(r_{sc})\]

Warning

Using soft-core scheme to scale the coulombic interaction non-linearly, would result in inaccurate results if Ewald method is activated.

Using Ewald Summation Method, we suggest to use hard-core scheme, to scale the coulombic interaction linearly with \(\lambda\).

Long-range Correction (VDW)

The effect of long-range corrections on predicted free energies were determined for VDW interactions via a linear coupling with \(\lambda\).

\[E_{\lambda}(\texttt{LRC-VDW}) = \lambda_{\texttt{VDW}} \Delta E_{\texttt{LRC(VDW)}}\]

where, \(\Delta E_{\texttt{LRC(VDW)}}\) is the the change in the long-range correction energy, due to adding a fully interacting solute to the solvent for VDW interaction.

To calculate the solvation free energy with thermodynamic integration (TI) method, the derivative of energy with respect to lambda (\(\frac{dE_{\lambda}(\texttt{LRC-VDW})}{\lambda_{\texttt{VDW}}}\)) is required:

\[\frac{dE_{\lambda}(\texttt{LRC-VDW})}{d\lambda_{\texttt{VDW}}} = \Delta E_{\texttt{LRC(VDW)}}\]

Long-range Correction (Electrostatic)

Using Ewald Summation Method, the effect of long-range corrections on predicted free energies were determined for Coulombic interactions via a linear coupling with \(\lambda\).

\[E_{\lambda}(\texttt{LRC-Elect}) = \lambda_{\texttt{Elect}} \bigg[\Delta E_{reciprocal} + \Delta E_{self} + \Delta E_{correction} \bigg]\]

where, \(\Delta E_{reciprocal}\), \(\Delta E_{self}\), and \(\Delta E_{correction}\) are the the change in the reciprocal, self, and correction energy term in Ewald method, due to adding a fully interacting solute to the solvent.

To calculate the solvation free energy with thermodynamic integration (TI) method, the derivative of energy with respect to lambda (\(\frac{dE_{\lambda}(\texttt{LRC-Elect})}{\lambda_{\texttt{Elect}}}\)) is required:

\[\frac{dE_{\lambda}(\texttt{LRC-Elect})}{d\lambda_{\texttt{Elect}}} = \Delta E_{reciprocal} + \Delta E_{self} + \Delta E_{correction}\]