Method overview

PB-AM is an analytical solution to the linearized Poisson-Boltzmann equation for multiple spherical objects of arbitrary charge distribution in an ionic solution. The solution can be reduced to a simple system of equations as follows:

\[ A = \Gamma \cdot (\Delta \cdot T \cdot A + E) \]

Where \(A^{(i)}\) represents the effective multipole expansion of the charge distributions of molecule \(i\). \(E^{(i)}\) is the free charge distribution of molecule \(i\). \(\Gamma\) is a dielectric boundary-crossing operator, \(\Delta\) is a cavity polarization operator, \(T\) an operator that transforms the multipole expansion to a local coordinate frame. More details on the method are available in [LoHe06]. Once \(A^{(i)}\) has been solved, through an iterative SCF method, physical properties of the system can be computed, as detailed in the next section.

Physical calculations

Interaction energies

From the above formulation, computation of the interaction energy (\(\Omega^{(i)}\)) for molecule i, is given as follows:

\[\Omega^{(i)}=\frac{1}{\epsilon_s} \sum_{j \ne i}^N \left \langle T \cdot A^{(j)} , A^{(i)} \right \rangle \]

Where \(\langle . . . \rangle\) denotes an inner product.

Forces and Torques

When energy is computed, forces follow as:

\[ \textbf{F}^{(i)} = \nabla_i \Omega^{(i)}=\frac{1}{\epsilon_s} [ \langle \nabla_i \,T \cdot A^{(j)} , A^{(i)} \rangle + \langle T \cdot A^{(j)} , \nabla_i \, A^{(i)} \rangle ]\]

The method to calculate the torque \(\boldsymbol{\tau}^{(i)}\) on molecule is outside the scope of this manual, but is discussed extensively in [LoHe06]