Intermolecular Energy and Virial Function (Electrostatic)
In this section, the virial and energy equation of electrostatic interaction for different potential function are discussed in details.
Ewald
This option calculate electrostatic energy using standard Ewald Summation Method.
Note
Once this option is activated, it would override the the electrostatic calculation using VDW, EXP6, SHIFT, and SWITCH functions.
Potential CalculationCoulomb interactions between atoms can be modeled as
\[E(\texttt{Ewald}) = E_{real} + E_{reciprocal} + E_{self} + E_{correction}\]\(E_{real}\): Defines the short range electrostatic energy according to
\[E_{real} = \frac{1}{4\pi \epsilon_0} \frac{1}{2} \sum_{i =1}^{N} \sum_{j = 1}^{N} q_i q_j \frac{erfc(\alpha r_{ij})}{r_{ij}}\], where \(\alpha\) is
Ewaldseparation parameter according to\[\alpha = \frac {\sqrt{-\log (Tolerance)}}{r_{cut}}\], where
Toleranceis a parameter, controlling the desired accuracy.\(E_{reciprocal}\): Defines the long range electrostatic energy according to,
\[E_{reciprocal} = \frac{1}{\epsilon_0 V} \frac {1}{2} \sum_{\overrightarrow{k} \ne 0}^{} \frac {1}{\overrightarrow{k}^2}\exp\bigg(\frac {-\overrightarrow{k}^2}{4 \alpha^2}\bigg) \Bigg[ {\Big| R_{sum} \Big|}^2 + {\Big| I_{sum} \Big|}^2 \bigg]\], where \(\overrightarrow{k}\) is reciprocal vector, \(R_{sum}\) and \(I_{sum}\) are,
\[R_{sum} = \sum_{i=1}^{N} q_i \cos \big(\overrightarrow{k}.\overrightarrow{x_i}\big)\]\[I_{sum} = \sum_{i=1}^{N} q_i \sin \big(\overrightarrow{k}.\overrightarrow{x_i}\big)\]\(E_{self}\): Defines the self energy according to,
\[E_{self} = -\frac{\alpha}{4\pi \epsilon_0 \sqrt{\pi}} \sum_{i=1}^{N} {q_i}^2\]\(E_{correction}\): Defines intra-molecule nonbonded energy,
\[E_{correction} = -\frac{1}{4\pi \epsilon_0} \frac{1}{2} \sum_{j=1}^{N }\sum_{l =1}^{N_j} \sum_{m = 1}^{N_j} q_{j_l} q_{j_m} \frac{erf(\alpha r_{j_l j_m})}{r_{j_l j_m}}\]Virial CalculationVirial is basically the negative derivative of energy with respect to distance, multiplied by distance, Eq. 4. Coulomb force between atoms can be modeled as,
\[W_{Ewald} = W_{real} + W_{reciprocal}\]\(W_{real}\) defines the short range electrostatic and \(W_{reciprocal}\) defines the long range electrostatic force according to,
\[W_{real} = \frac{1}{4\pi \epsilon_0} \frac{1}{2} \sum_{i =1}^{N} \sum_{j = 1}^{N} q_i q_j \bigg[ \frac{erfc(\alpha r_{ij})}{r_{ij}} + \frac{2\alpha}{ \sqrt{\pi}} \exp(-\alpha^2 {r_{ij}}^2) \bigg] \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}^2}\]\[\begin{split}\begin{split} W_{reciprocal} = \frac{1}{\epsilon_0 V} \frac {1}{2} \sum_{\overrightarrow{k} \ne 0}^{} \Bigg[\frac {1}{\overrightarrow{k}^2}\exp\bigg(\frac {-\overrightarrow{k}^2}{4 \alpha^2}\bigg) \bigg( {\Big| R_{sum} \Big|}^2 + {\Big| I_{sum} \Big|}^2 \bigg) \bigg( 1 - \frac{\overrightarrow{k}^2}{2\alpha^2} \bigg) \Bigg] +\\ \sum_{i=1}^{N} \frac{1}{\epsilon_0 V} \sum_{\overrightarrow{k} \ne 0}^{} \Bigg[ \frac {q_i}{\overrightarrow{k}^2}\exp\bigg(\frac {-\overrightarrow{k}^2}{4 \alpha^2}\bigg) \bigg[ I_{sum} \times\cos(\overrightarrow{k}.\overrightarrow{x_i}) - R_{sum} \times \sin(\overrightarrow{k}.\overrightarrow{x_i}) \bigg] \Bigg] \times \big( \overrightarrow{k}.\overrightarrow{r_{ic}} \big) \end{split}\end{split}\], where \(\overrightarrow{r_{ic}}\) is the vector between atom and the center of the mass of the molecule.
VDW
Using VDW potential type without Ewald method, simply uses coulomb energy to calculate the electrostatic potential.
Potential CalculationCoulomb interactions between atoms can be modeled as
\[E_{\texttt{Elect}}(r_{ij}) = \frac{q_i q_j}{4\pi \epsilon_0 r_{ij}}\]Virial CalculationVirial is basically the negative derivative of energy with respect to distance, multiplied by distance.
\[W_{\texttt{Elect}}(r_{ij}) = -\frac{dE_{\texttt{Elect}}(r_{ij})}{r_{ij}}\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}} = F_{\texttt{Elect}}(r_{ij}) \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}}\]\[F_{\texttt{Elect}}(r_{ij}) = \frac{q_i q_j}{4\pi \epsilon_0} \Big( \frac{1}{{r_{ij}}^2} \Big)\]
EXP6
Using EXP6 potential type without Ewald method, simply uses coulomb energy to calculate the electrostatic potential.
Potential CalculationCoulomb interactions between atoms can be modeled as
\[E_{\texttt{Elect}}(r_{ij}) = \frac{q_i q_j}{4\pi \epsilon_0 r_{ij}}\]Virial CalculationVirial is basically the negative derivative of energy with respect to distance, multiplied by distance.
\[W_{\texttt{Elect}}(r_{ij}) = -\frac{dE_{\texttt{Elect}}(r_{ij})}{r_{ij}}\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}} = F_{\texttt{Elect}}(r_{ij}) \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}}\]\[F_{\texttt{Elect}}(r_{ij}) = \frac{q_i q_j}{4\pi \epsilon_0} \Big( \frac{1}{{r_{ij}}^2} \Big)\]
SHIFT
This option forces the electrostatic energy to be zero at Rcut distance.
Potential CalculationCoulomb interactions between atoms can be modeled as
\[E_{\texttt{Elect}}(r_{ij}) = \frac{q_i q_j}{4\pi \epsilon_0} \Big( \frac{1}{r_{ij}} - \frac{1}{r_{cut}} \Big)\]Virial CalculationVirial is basically the negative derivative of energy with respect to distance, multiplied by distance.
\[W_{\texttt{Elect}}(r_{ij}) = -\frac{dE_{\texttt{Elect}}(r_{ij})}{r_{ij}}\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}} = F_{\texttt{Elect}}(r_{ij}) \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}}\]\[F_{\texttt{Elect}}(r_{ij}) = \frac{q_i q_j}{4\pi \epsilon_0} \Big( \frac{1}{{r_{ij}}^2} \Big)\]
SWITCH
This option in CHARMM or EXOTIC force field forces the electrostatic energy to be zero at Rcut distance.
Potential CalculationCoulomb interactions between atoms can be modeled as,
\[E_{\texttt{Elect}}(r_{ij}) = \frac{q_i q_j}{4\pi \epsilon_0} \bigg( \Big(\frac{r_{ij}}{r_{cut}} \Big)^2 - 1.0\bigg)^2 \frac{1}{r_{ij}}\]Virial CalculationVirial is basically the negative derivative of energy with respect to distance, multiplied by distance.
\[W_{\texttt{Elect}}(r_{ij}) = -\frac{dE_{\texttt{Elect}}(r_{ij})}{r_{ij}}\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}} = F_{\texttt{Elect}}(r_{ij}) \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}}\]\[F_{\texttt{Elect}}(r_{ij}) = \frac{q_i q_j}{4\pi \epsilon_0} \Bigg[ \bigg( \Big(\frac{r_{ij}}{r_{cut}} \Big)^2 - 1.0\bigg)^2 \frac{1}{{r_{ij}}^2} - \bigg( \frac{4}{{r_{cut}}^2} \bigg) \bigg( \Big(\frac{r_{ij}}{r_{cut}} \Big)^2 - 1.0\bigg) \Bigg]\]
SWITCH (MARTINI)
This option in MARTINI force field smoothly forces the potential energy to be zero at Rcut distance and starts modifying the potential at Rswitch = 0.0 distance.
Potential CalculationCoulomb interactions between atoms can be modeled as,
\[E_{\texttt{Elect}}(r_{ij})=\frac{q_i q_j}{4\pi\epsilon_0\epsilon_1}\bigg(\frac{1}{r_{ij}}+\varphi_{E, 1}(r_{ij})\bigg)\], where \(\epsilon_1\) is the dielectric constant, which in
MARTINIforce field is equal to 15.0 and \(\varphi_{E, \alpha = 1}(r_{ij})\) is defined as:\[\begin{split}\varphi_{E, \alpha}(r_{ij}) = \begin{cases} -C_{\alpha} & r_{ij} \leq r_{switch} \\ -\frac{A_{\alpha}}{3} (r_{ij} - r_{switch})^3 -\frac{B_{\alpha}}{4} (r_{ij} - r_{switch})^4 - C_{\alpha} & r_{switch} < r_{ij} < r_{cut} \\ 0 & r_{ij} \geq r_{cut} \end{cases}\end{split}\]\[A_{\alpha} = \alpha \frac{(\alpha + 1) r_{switch} - (\alpha +4) r_{cut}} {{r_{cut}}^{(\alpha + 2)} {(r_{cut} - r_{switch})}^2}\]\[B_{\alpha} = \alpha \frac{(\alpha + 1) r_{switch} - (\alpha +3) r_{cut}} {{r_{cut}}^{(\alpha + 2)} {(r_{cut} - r_{switch})}^3}\]\[C_{\alpha} = \frac{1}{{r_{cut}}^{\alpha}} -\frac{A_{\alpha}}{3} (r_{cut} - r_{switch})^3 -\frac{B_{\alpha}}{4} (r_{cut} - r_{switch})^4\]Virial CalculationVirial is basically the negative derivative of energy with respect to distance, multiplied by distance.
\[W_{\texttt{Elect}}(r_{ij}) = -\frac{dE_{\texttt{Elect}}(r_{ij})}{r_{ij}}\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}} = F_{\texttt{Elect}}(r_{ij}) \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}}\]\[F_{\texttt{Elect}}(r_{ij})=\frac{q_iq_j}{4\pi\epsilon_0\epsilon_1}\bigg(\frac{1}{{r_{ij}}^2}+\varphi_{F, 1}(r_{ij})\bigg)\], where \(\varphi_{F, \alpha = 1} (r_{ij})\) is defined as:
\[\begin{split}\varphi_{F, \alpha}(r_{ij}) = \begin{cases} 0 & r_{ij} \leq r_{switch} \\ A_{\alpha} (r_{ij} - r_{switch})^2 + B_{\alpha} (r_{ij} - r_{switch})^3 & r_{switch} < r_{ij} < r_{cut} \\ 0 & r_{ij} \geq r_{cut} \end{cases}\end{split}\]