Ammm:Mm aplus

Introduction

One is recommended to read through and understand the background knowledge of polarizable force field AMOEBA, in order to have a better learning experience of the next generation AMOEBA+ force field. As indicated by the name, AMOEBA+ is based on the AMOEBA framework with incoporation of essential physical effects that are missing in AMOEBA model. With the short-range charge penetration and charge transfer effects being included, AMOEBA+ potential is able to follow the energy components by the Symmetry Adapted Perturbation Theory (SAPT).

Below brief description of the SAPT theory is introduced, followed by the specific interactions/physical effects that are incoporated in AMOEBA+ potential. In the last part, brief summary of the current status of the parametrization of the AMOEBA+ force field is given.

SAPT Method

Intermolecular interactions are small (comparing to the intramolecular interactions). For example, helium dimers at equalibrium only has interaction atoms of -0.02 kcal/mol. However, the intermolecular interactions are important for describing a lot of phenomina in chemistry, physics and biology.

Starting from a dimer AB composed of two monomers, A and B, the interaction energy is computed as:

$E_{int}=E_{AB}-E_{A}-E_{B}$

This is often called supermolecular approach to calculate intermolecuar energies. The total energy of each fragment is usually at least 4 orders of magnitude larger than the interaction energy. So this calculation is largely relies on error cancellation. In addition, this approach only gives one number of that interaction, which limits its interpretation ability. From a perturbative approach, SAPT is able to calculate the interaction energy accurately, meanwhile provides energy terms with physical interpretation.

The Hamiltonian of SAPT is constructed as follows:

$H=F_{A}+F_{B}+W_{A}+W_{B}+V$

where F is the Fock operator, W is the intranonomer correlation operator and V is the intermonomer interaction operator. In this approach, the intramonomer correlation is included via the second perturbation expansion involving the Møller-Plesset (MP) fluctuation potential. Interaction actions are then expressed as:

$E_{int}=\sum _{i>0,j\geq 0}E^{(ij)}$

where i(j) is the orders of V(W). By grouping together different orders of the corrections above, SAPT decompose the interaction energy into four energy terms, namely, electrostatics, induction, exchange and dispersion:

$E_{SAPT}=\sum _{i>0}\sum _{j\geq 0}E_{SAPT}^{(ij)}=E_{elst}+E_{ind}+E_{exch}+E_{disp}$

The simplest SAPT approximation that gives reasonable total interaction energy is SAPT0, in which the intramonomer correlation is completely neglected. SAPT0 is sometimes useful for understanding the nature of interactions between relatively big fragments (hundreds of atom). The highest order of SAPT is obtained when i=3 and j=2, which corresponds to SAPT2+3. Alternatively, there are truncated levels of SAPT, including SAPT0, SAPT2, SAPT2+ and SAPT2+(3). The detailed hierachy grouping of SAPT theory can be seen below:

Hierrachy of SAPT theory

New Potential Energy Terms

These energy terms are newly added, which means they are not in the AMOEBA model.

Polarization and Charge Transfer

Atom-based polarization scaling factors