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, SHIFT, and SWITCH functions.

Potential Calculation

Coulomb 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 Ewald separation parameter according to

\[\alpha = \frac {\sqrt{-\log (Tolerance)}}{r_{cut}}\]

, where Tolerance is 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 Calculation

Virial 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.

SHIFT

This option forces the electrostatic energy to be zero at Rcut distance.

Potential Calculation

Coulomb interactions between atoms can be modeled as

\[E(\texttt{SHIFT}) = \frac{q_i q_j}{4\pi \epsilon_0} \Big( \frac{1}{r_{ij}} - \frac{1}{r_{cut}} \Big)\]
Virial Calculation

Virial is basically the negative derivative of energy with respect to distance, multiplied by distance, Eq. 4. Coulomb force between atoms can be modeled as,

\[W(\texttt{SHIFT}) = \frac{q_i q_j}{4\pi \epsilon_0} \Big( \frac{1}{r_{ij}} \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}^2} \Big)\]

SWITCH

This option in CHARMM or EXOTIC force field forces the electrostatic energy to be zero at Rcut distance.

Potential Calculation

Coulomb interactions between atoms can be modeled as,

\[E(\texttt{SWITCH}) = \frac{q_i q_j}{4\pi \epsilon_0} \bigg( \Big(\frac{r_{ij}}{r_{cut}} \Big)^2 - 1.0\bigg)^2 \frac{1}{r_{ij}}\]
Virial Calculation

Virial is basically the negative derivative of energy with respect to distance, multiplied by distance, Eq. 4. Coulomb force between atoms can be modeled as,

\[W(\texttt{SWITCH}) = \frac{q_i q_j}{4\pi \epsilon_0} \Bigg[ \bigg( \Big(\frac{r_{ij}}{r_{cut}} \Big)^2 - 1.0\bigg)^2 \frac{1}{{r_{ij}}^2} - \bigg( \frac{4}{{r_{cut}}^2} \bigg) \bigg( \Big(\frac{r_{ij}}{r_{cut}} \Big)^2 - 1.0\bigg) \Bigg] \times \frac{\overrightarrow{r_{ij}}}{r_{ij}}\]

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 Calculation

Coulomb interactions between atoms can be modeled as,

\[E(\texttt{SWITCH})=\frac{q_i q_j}{4\pi\epsilon_0\epsilon_1}\bigg(\frac{1}{r_{ij}}+\varphi_{1}(r_{ij})\bigg)\]

, where \(\epsilon_1\) is the dielectric constant, which in MARTINI force field is equal to 15.0 and \(\varphi_{\alpha}(r_{ij})\) is defined in Eq. 13-16.

Virial Calculation

Virial is basically the negative derivative of energy with respect to distance, multiplied by distance, Eq. 4. Coulomb force between atoms can be modeled as,

\[W(\texttt{SWITCH})=\frac{q_iq_j}{4\pi\epsilon_0\epsilon_1}\bigg(\frac{1}{{r_{ij}}^2}+d\varphi_1(r_{ij})\bigg)\times\frac{\overrightarrow{r_{ij}}}{r_{ij}}\]

, where \(d\varphi_1 (r_{ij})\) is defined in Eq. 18.