Electorstatically-Embedded Many-Body Expansion (EE-MB)
The EE-MB method was proposed by Dahlke and Truhlar9. Its initial formulation amounted to applying a MBE to molecular culsters and using one of two embedding methods, EE-MB-A and EE-MB-B. The resulting fragment-based methods were termed EE-MB-A and EE-MB-B. This initial study was largely dedicated to understanding how these two embedding methods work on small water clusters (up to a pentamer). Notably errors in the absolute energy only weakly depended on the charge scheme used (Mulliken, TIP3P, and CM4 were considered), and whether EE-MB-A or EE-MB-B was used.
The next development in the EE-MB method was the development of a multi-level formulation 8. Dubbed EE-MB-CE, this new method separately applies a MBE to the SCF and correlation energies. In practice, however EE-MB-CE was always run with the full SCF energy, thus it was only the correlation energy which is truncated. Also proposed was EE-MB-CE Screening. Calculations looked at reproducing superystem MP2 energies.
In 2011, Tempkin et al.85 introduced the next version of EE-MB, sEE-MB (s stands for screened). Instead of traditional point charges, sEE-MB uses charges defined by the Screened Charge Model embedding scheme. Screened Charge Model accounts for charge penetration (when atoms are close the charge densities overlap and the electrons actually see more of the nuclear charge). The sEE-MB method was applied to water clusters where for a given MBE truncation order sEE-MB better replicated the supersystem energy than EE-MB-B.
Qi et al.72 presented the next incarnations of EE-MB termed EE-MB-NE and EE-MB-HE, where NE means nonlocal energy and HE means higher-level energy. Conceptually EE-MB-NE is the same as EE-MB-CE except that instead of performing a supersystem SCF calculation, one instead runs a local DFT calculation. EE-MB-HE is the name they give to the concept of running a multi-level calculation such that the lowest level is a supersystem calculation. Hence EE-MB-CE and EE-MB-NE are two specific cases of EE-MB-HE (specifically when the lowest level is SCF or a local DFT respectively).
Summary
It seems fair to say that the unifying theme of the various flavors of EE-MB is the use of non-iterative point charges. Unlike many of the other fragment based methods EE-MB has almost always been used to refer to a single flavor (EE-MB-B) and care has been taken to distinguish among the other flavors.
Name |
Fragments |
Embedding |
Caps |
Screening |
|
|---|---|---|---|---|---|
EE-MB-A |
N/A |
N/A |
N/A |
||
EE-MB-B |
N/A |
N/A |
N/A |
||
EE-MB-CE |
Layer 1 |
Supersystem SCF |
|||
Layer 2 |
N/A |
N/A |
|||
sEE-MB |
N/A |
N/A |
N/A |
||
EE-MB-NE |
Layer 1 |
Supersystem local DFT |
|||
Layer 2 |
N/A |
N/A |
|||
EE-MB-HE |
Layer 1 |
Low-level supersystem |
|||
Layer 2 |
N/A |
N/A |
N/A |
||
Other EE-MB Results
Dahlke and Truhlar10 considered the applicability of (presumably) EE-MB-B for performing molecular dynamics calculations on water clusters. The specific study focused on how accurately EE-MB-B replicated the supersystem gradients for a water cluster containing 64 water molecules. All calculations were done with DFT.
Dahlke et al.7 considered how well EE-MB-CE could be used to reproduce supersystem energies computed with CCSD, CCSD(T), and higher-order Moller- Plesset perturbation theory. The study focused on water hexamer isomers.
Sorkin et al.82 applies (presumably) EE-MB-B to water clusters containing a NH3 molecule. The main point was to study how well EE-MB can replicate the absolute and relative energetics of the corresponding supersystem calculations. Also considered were several methods for computing the point charges.
Leverentz and Truhlar48 revisits the discussion of EE-MB-A vs. EE-MB-B for mixed water, sulfuric acid, and ammonia clusters using DFT. The study notably considers a lot of different ways to compute the charges, but ultimately concludes the final results are pretty insensitive to the charge details.
Hua et al.29 applied EE-MB-B to Zn coordination compounds. The study specifically notes that having each ligand be one fragment, and the Zn be another led to unacceptale errors. Instead one fragment is Zn and the two closest ligands, and the remaining ligands are their own fragments. With such a fragmentation scheme EE-MB-B was able to accurately reproduce supersystem ligand disassociation energies.
Kurbanov et al.45 again considered EE-MB-B applications to Zn coordination compounds. This study considers how to fragment the compleexes in a more systematic manner. The main finding of this study is a series of guidelines for fragmenting coordination compounds. These guidelines are not a fragmentation method, since they don’t tell you how to fragment the system; rather the guidelines tell you how you can’t fragment the system. By adhereing to the aforementioned guidelines, EE-MB-B was again demonstrated to be able to predict ligand dissassociation energies.
Leverentz et al.47 uses EE-MB-B to reproduce supersystem partial charges, intermolecular charge transfer, and dipole moments of molecular clusters. Of note they were only able to obtain qualitative agreement with supersystem results for intermolecular charge transfer.
Kurbanov et al.46 applied EE-MB-B and EE-MB-CE to Zn and Cd coordination compounds. The study proposes a fragmentation method for Zn and Cd compounds that amounts to pairing the metal with the two ligands with the strongest Coulomb interaction (as deterimined by the maximum absolute point-charge-point- charge interaction). Also of interest is a discussion pertaining to how to classify the various EE-MB methods. Of particular note is the claim that there is an ambiguity in trying to distinguish between intersecting and disjoint methods.
Friedrich et al.19 applied EE-MB-B to water clusters with 26 monomers. The goal of the study was to replicate CCSD(T)/CBS results computed using the incremental method. Overall the EE-MB-B results match the incremental method results quite well.
EE-MB Reviews
Wang et al.87 reviewed the various flavors of EE-MB as well as the EE-MTA method. This review also introduces the anchor points reactive potential method which generates a semi-analytical potential energy surface by partitioning the internal coordinates of the system. They then suggest that such a method should also be considered a fragment-based method because it fragments the internal coordinates.