# pbam-auto

PB-AM is an analytical solution to the linearized Poisson-Boltzmann equation for multiple spherical objects of arbitrary charge distribution in an ionic solution. More details on the method are available in Lotan, Head-Gordon (2006). The physical calculations are uses to perform various actions on a system of molecules such as calculation of energies, forces, torques, electrostatic potentials, and Brownian dynamics schemes. This fast method coarse-grains all molecules of the system into single spheres large enough to contain all molecule atoms.

Todo

If there’s only one mode to PBAM, let’s call it `pbam` instead of `pbam-auto`. Documented in https://github.com/Electrostatics/apbs-pdb2pqr/issues/498

The current implementation of PB-AM in APBS includes:

- Calculation of energies, forces and torques
- Calculation of electrostatic potentials
- Brownian dynamics simulations

Keywords for this calculation type include:

ELEC pbam-auto keywords:

- 3dmap
- diff
- dx
- grid2d
- gridpts
- mol
- ntraj
- pbc
- pdie
- randorient
- runname
- runtype
- salt
- sdie
- temp
- term
- termcombine
- units
- xyz

## Background information

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. \(A^{(i)}\) is solved for through an iterative SCF method.

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

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

where \(\langle M, N \rangle\) denotes the inner product. Forces can be obtained from

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