=== 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 energy 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:

<math display="block">E_{int} = E_{AB} - E_{A} - E_{B}
</math>

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:

<math display="block">H = F_A + F_B + W_A + W_B + V </math>

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:

<math display="block">E_{int} = \sum_{i>0, j\ge0}E^{(ij)} </math>

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''':

<math display="block">E_{SAPT} = \sum_{i>0}\sum_{j\ge0}E_{SAPT}^{(ij)}=E_{elst} + E_{ind} + E_{exch} + E_{disp} </math>

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:
[[File:SAPT.jpg|thumb|Hierrachy of SAPT theory|center]]Finally, it is worth mentioning that there are other variations of SAPT approaches. In addition, there exist other energy decomposition analysis (EDA) methods. We will revist the SAPT induction contribution in the following section, from which we know that the polarization and charge transfer (CT) are folded together in SAPT. Many other EDA approaches try to separate CT from induction but the results are usually very different from each other.

===New Potential Energy Terms===
The AMOEBA force field utilizes the MM3-style valence terms. The intermolecular terms include permanent electrostatics, polarization and van der Waals interactions.

<math display="block">E_{AMOEBA} = E_{valence} + E_{elst}^{perm} + E_{pol} + E_{vdw}</math>Although the valence interactions are important, in AMOEBA+ model we try to improve the intermolecular interactions

====Charge Penetration====

====Polarization and Charge Transfer====

===== Partition of Polarization and CT =====
Different EDA methods separate these two contributions very differently. For example, ...

Here we first tried to examine the many-body energy of AMOEBA model and compare with high-level quantum calculations.

The interaction energy of a system can be expressed as a sum of N-body interactions, where N is the number of monomers in the system. The magnitude of the interactions decrease as the order increases. The first '''many-body''' contribution is from 3-body. Practically, the contribution of 5-body and beyond is quite small comparing to lower orders. In our model, we focused on the 3-body and 4-body interaction energies. <math display="block">E_{int} = \sum_{i\ge2}^N E_{i-body}</math>

===== Atom-based polarization scaling factors =====

==== Geometry Dependent Charge Flux ====

===Other Improvements===

====Polarizability====

====van der Waals====

====Valence Parameters====

===Parametrization===

Ammm:Mm aplus

2022-07-27T15:07:44Z