Molecular Tailoring Approach (MTA)
MTA Method Development
In the original MTA approach, Gadre et al.25 sought a means of reducing the cost of constructing the SCF density matrix. To do this the system of interest was fragmented into overlapping fragments and the density matrix of each fragment is computed. Pieces of the fragment density matrices are then combined to form an approximation to the supersystem’s density matrix. How one should choose the pieces for use in the supersystem density matrix is not well defined at this point in the development of MTA. Initial results focused on zeolite and peptides using small basis sets. Interestingly because the supersystem density is not converged, comparisons are made against the supersystem’s density matrix and electrostatic potential, as opposed to the usual absolute energy.
In a follow-up study Babu and Gadre1 proposed the MTA03 Fragmentation method and with it the MTA 03 variant of MTA. Like in the original MTA study:cite:Gadre1994, the density matrix of the supersystem is assembled from the density matrices of the fragments. Unlike, in the original manuscript an algorithm for doing so is provided. Applications focused on three systems: 5,7,8-trimethyltocol, zeolite, and a peptide using small basis SCF. Results primarily focused on comparing the electorstatic potential computed by MTA to that of the supersystem.
The next development 27 in MTA, came from realizing the IEP could be used to compute the energy, and energy derivatives, of a fragmented system. Initial applications focused on replicating the energy of medium-sized organic molecules using small-basis HF, MP2, and B3LYP. Also considered were geometry optimizations, and vibrational frequencies. This study is also one of the earliest applications of MTA06 Fragmentation and with it the MTA 06 variant of MTA.
For benzene clusters, Mahadevi et al.67 note that a 1-body MTA expansion still neglects some many-body interactions. They suggested a screening method which amounts to manually adding back in two-, three-, etc. body interactions which do not appear in the 1-body MTA expansion. For larger benzene clusters (hexamers, for example) these higher-body corrections can be more than a kcal/mol. Conceptually this correction is very similar to SMF Screening 05. The accompanying numerical results demonstrated that BSSE was quite prevelant in the benzene tetramers. Despite this fact, results for larger clusters did not correct for BSSE.
Furtado et al.21 noted that up to this point in MTA development, errors (relative to the supersystem calcualtion) resulting from MTA tend to be basis set independent. They also note that when applying MTA to a correlated method the majority of the error comes from the SCF part of the energy. On the basis of these observations, Furtado et al.21 propsed the “grafting” correction to MTA. Grafting involves:
Compute the SCF energy of the supersystem with a large basis set, \(E_{SCF/High}\).
Using MTA compute the MP2 correlation energy with the same large basis set, \(E_{MTA-MP2/High}\).
Compute the MP2 energy of the supersystem with a small basis set, \(E_{MP2/Low|}\).
Using MTA compute the MP2 correlation energy with the same small basis set, \(E_{MP2/Low|}\).
The final energy is then estimated as \(E_{SCF/High}\) plus \(E_{MTA-MP2/High}\) plus a correction (the difference between \(E_{MP2/Low|}\) and \(E_{MP2/Low|}\)) which accounts for the error in the fragmentation. Alternatively, one may view the energy as \(E_{SCF/High}\) plus \(E_{MP2/Low|}\) plus a basis set correction (the difference between \(E_{MTA-MP2/High}\) and \(E_{MP2/Low|}\)). Either way grafting amounts to a multi-level fragment-based method. To demonstrate grafting Furtado et al.21 considered MP2 energies of water clusters; compared to supersystem MP2/aug-cc-pVDZ energies the grafting procedure resulted in errors on the order of 0.3 mH.
Isegawa et al.36 combined the electrostatic embedding of the EE-MB method with the MTA. It is perhaps worth noting this MTA development is somewhat unconventional in that it has not been made by the Gadre group. The study considers how four parameters: size of fragments, fragment boundaries, type of background charge, and type of cap atom affect the approximation. Admittedly the description of the method is very hard to follow and it is not exactly clear what the final method is. It is perhaps worth noting that this study proposes that one can simply truncate the IEP early (they specificlly state they will only consider at most intersections involving two ragments). In genernal this leads to unacceptable errors. For example, consider a system of four non-interacting hydrogen atoms. Arbitrarily labeling the atoms 0, 1, 2, and 3 we choose to create three fragments: 012, 013, and 123. Applying the IEP:
The first three terms are the energies of the fragments, the next three are the pair-wise intersections, and the last term is the one ternary intersection. Even with non-interacting systems we can’t neglect the ternary interaction as it is an error of half a hartree (i.e., the energy of a hydrogen atom). In the case of Isegawa et al.36’s study, the approximation works because it’s not actually an approximation. More specifically their fragmentation scheme is such that the higher-order intersections always cancel out.
MTA Summary
Like many fragment-based methods, there is no “the MTA method” as the definition of the method has changed over the years without modifying the name. The following table summarizes the variants of MTA, which have appeared. Note that the names of the variants are our names, and were not present in the original manuscripts.
Name |
Fragments |
Embedding |
Caps |
Screening |
|---|---|---|---|---|
MTA 03 |
N/A |
N/A |
||
MTA 06 |
N/A |
N/A |
||
MTA 10 |
Manual |
N/A |
N/A |
In this table “manual” fragmentation refers to the fact that the user specified the fragments. Based on the presentation of MTA in the literature it is reasonable to assume that the authors of MTA really consider the IEP to be the key ingrediant of what distinguishes MTA from other fragment methods.
Other MTA Results
Babu et al.2 used the MTA 03 variant of MTA (and possibly an early version of the MTA 06 variant, but the description in the paper is too vague for me to be certain) to approximate the density matrix of an ibuprofen crystal with the HF/STO-3G and HF/6-31G(d,p) levels of theory. Comparisons between the supersystem density matrices were made at the HF/STO-3G level of theory.
Gadre et al.24 provided additional details about the MTA06 Fragmentation procedure and saw the MTA 06 variant of MTA be applied to additional studies of organic molecules. Calulations used the HF and B3LYP methods with the 6-31G(d) and 6-31G(d, p) basis sets. Absolute energies, gradients, optimized geometries, and molecular electorstatic potential surfaces were compared to the supersystem results.
Elango et al.13 applied the MTA 06 variant of MTA to boric acid nanotubes and nanorings. Comparisons to B3LYP/3-21G supersystem energies and (partial) geometry optimizations. Additional results with CG-MTA approximated B3LYP and MP2 are reported (6-31+G(d,p) basis set). Of note BSSE calculations are foregone due to cost and an expectation that they will not change the results.
Jose and Gadre39 considered optimized structures of lithium clusters using small basis DFT. Results for larger clusters relied on the MTA 06 variant of MTA (disclaimer it is possible that the fragments were manually defined, but it is not clear from the presentation). Several properties were computed for each cluster including: the adiabatic ionization potential (which is notable as it requires computing the energy of the cationic cluster) and the polarizability.
Rahalkar et al.74 used the MTA 06 variant of MTA to computatethe Hessian matrix at the HF, B3LYP, and MP2 levels of theory (with small basis sets). Errors in computed vibrational frequencies were less than a wavenumber.
Jose and Gadre40 used the MTA 06 variant of MTA to study CO2 clusters with DFT. Optimized geometries and vibrational frequencies were considered. For vibrational frequencies, CG-MTA matched supersystem calculations to within about a wavenumber.
Yeole and Gadre92 applied MTA to conjugated \(\pi\) systems with small basis DFT and MP2. Single point energies, gradients, and optimized geometries were computed and compared to the results of the respective full calculation. Overall the results were promising as long as relatively large fragments were used (fragments had radii of 6+ Angstroms).
Rahalkar et al.75 paired the MTA 06 variant of MTA with small basis IMS-MP2 and IMS-RI-MP2 (AFAIK IMS is a disk-based MP2 algoritm in the GAMESS package). Comparisons of MTA energies to FMO energies were also presented. Overall the comparison shows that three-body FMO performs about as well as MTA, although it is noted that the error in the FMO results tends to increase with basis set size.
Yeole et al.93 applied MTA (exactly which variant is unclear, possibly MTA 10, without screening) to CO2 clusters with the focus of finding local minima. MTA was used to approximate single point energies and gradients at the MP2/CBS level of theory (extrapolations of MP2/aDZ and MP2/aTZ). No comparisons to full system calculations were presented, instead comparisons were made to previously published results that leveraged a Lennard-Jones potential.
Rahalkar et al.76 is conceptually similar to 93 except that instead of CO2 clusters the focus was on acetylene clusters.
Rahalkar and Gadre73 used the density matrix from the MTA 06 variant of MTA to build a Fock matrix. Subsequent diagonalization of the Fock matrix yielded MOs. The study uses relatively small fragments and in turn the results are a mixed bag. Some HOMO-LUMO gaps are modeled well, others aren’t. Of note they consider several conjugated systems.
Sahu et al.80 applied the grafted MTA procedure to additional clusters. Of note they also introduced the idea of an R-goodness for two-, three-, body interactions. The manuscript suggests that these higher-body R-goodness parameters could be used to influence the fragmentation procedure; however, a clear description of how to do this is not provided.
MTA Reviews
Gadre et al.23 reviewed applications of MTA geometry optimization, frequency calculations, as well as property computations to clusters.
Rahalkar et al.77 reviewed the MTA method up to that point. The presentation primarily distinguishes MTA from other methods by pointing out that MTA is capable of geometry optimizations and that MTA has been applied to large systems. Also of note this review gives a nice explanation of the MTA 06 fragmentation method.
Gadre et al.26 is a review primarily focusing on understanding the structure of molecular clusters (from both the experimental and theoeritcal perspective); however, there is a bit of disucssion of how fragment-based methods, including MTA, have been used to study the problem.
Sahu and Gadre79 review MTA primarily in light of its applications to predicting minima of molecular clusters.