Generalized Energy Based Fragmentation (GEBF)
GEBF is the culmanation of the Divide and Conquer Local Correlation (DCLC) methods.
Development of GEBF was motivated by a need to treat large macromolecules/clusters that are charged or highly polar 59. The original GEBF method, which we term the GEBF07 method, consists of the GEBF07 Fragmentation Method, with Standard Distance capping, a slight variation of SMF06 Screening, and an Iterative Point Charge Embedding. GEBF’s original presentation also includes an energy equation which is simply a one- body GMBE corrected for the self-interaction of the point charges (i.e., it is assumed that the energies of the fragments include not only the interaction of the fragment with the point charges, but also the interactions among the point charges). Initial applications of GEBF focused on peptides and water clusters with the HF and MP2 methods. In addition to absolute energies this study also considered dipole moments and static polarizabilities.
Hua et al.33 introduced the GEBF08 method which differs from the GEBF07 method in nearly all regards. In particular GEBF08 uses the GEBF08 Fragmentation Method fragmentation method, Iterative Point Charge Embedding (I assume since the GEBF07 method is cited, but the article is not clear on this point), Standard Distance capping, and no (?) screening. GEBF08 was applied to geometry optimizations and vibrational frequency calculations. The GEBF08 energies, optimized geometries, and vibrational frequencies were compared to the conventional results. Small basis SCF (both HF and DFT) results, for several types of systems (clusters, peptides, and hydrogelators) are presented.
The next methodologic development came when Hua et al.30 updated the fragmentation method of the GEBF08 method. The resulting GEBF10 method uses the GEBF10 Fragmentation Method fragmentation method which is similar to the original GEBF07 Fragmentation Method method with a series of “extension rules”. GEBF10 was applied to a series of large organic molecules. Computed properties included absolute energies, relative energies, and optimized geometries all with small basis HF or DFT. Compared to the supersystem results GEB10 did well for geometries, but the errors in the energetics were often off by more than 1 kcal/mol.
Hua et al.31 noted that the GEBF10 method performed poorly for systems containing important non-covalent interactions. To combat this, Hua et al.31 introduced a new fragmentation method GEBF12 Fragmentation Method. GEBF12 Fragmentation Method extends GEBF10 Fragmentation Method by augmenting the original set of fragments with additional fragments formed from pairs of pseudoatoms (plus the pseudoatoms near that pair). The resulting GEBF12 method was then applied to several peptides and large molecules. GEBF12 performed better than GEBF10 (for example for binding energies of the large molecules GEBF10 erred from supersystem results by about 3.8 kcal/mol whereas GEBF10 only erred by about 0.5 kcal/mol). Of note the role of BSSE corrections (as computed using the Distance-Based CP correction) was also considered.
In looking at explicitly correlated CCSD(T) energies of water clusters, Wang et al.89 introduced the GEBF14 method. GEBF14 differs from GEBF10 in the fragmentation method, more specifically GEBF14 uses GEBF14 Fragmentation Method which is essentially GEBF08 Fragmentation Method combined with GEBF10 Fragmentation Method. By only applying GEBF to the correlation energy, Wang et al.89 were able to replicate supersystem explicitly correlated MP2/CBS results to within about 0.3 kcal/mol (errors when GEBF was also applied to the SCF component of the energy were about 1 kcal/mol). The authors attributed the better performance of only applying GEBF to the correlation energy to a better treatment of BSSE.
In looking at host/guest complexes, Li et al.56 realized that GEBF14 often resulted in one fragment which contained all of, or most of, the pseudoatoms (presumably because of ties in pseudoatom distances?). Regardless, to avoid this, Li et al.56 introduced GEBF19 Fragmentation Method. GEBF19 was applied to 10 systems from the S30L database (TODO: cite). At the M06-2X/def2-TZVP level of theory, GEBF19 was able to reproduce supersystem binding energies to better than 1 kcal/mol; however, more careful analysis showed that this was a result of error cancellation.
Cheng et al.5 wanted to combine GEBF with machine-learning. To that end they introduced an approximation to the GEBF14 method by not embedding the fragments. The resulting GEBF20 method was then used to combine the energies, gradients, and dipole moments computed by a machine-learning model. GEBF20 was then used to perform MD runs for a series of alkanes.
GEBF Summary
Note
We need to verify that all of the embedding methods are truly the same. In particular I think some of the methods only do one iteration where others iterate to consistency every time. Another variation is in how the charges are computed, namely are they taken from a specific fragment, supersystem calculation, or some other way.
Name |
Fragments |
Embedding |
Caps |
Screening |
|---|---|---|---|---|
GEBF07 |
||||
GEBF08 |
N/A |
|||
GEBF10 |
N/A |
|||
GEBF11 |
Manual |
N/A |
||
GEBF12 |
N/A |
|||
GEBF14 |
N/A |
|||
GEBF19 |
N/A |
|||
GEBF20 |
N/A |
N/A |
As the above table shows, there are a number of different methods which have been termed “GEBF”. We suspect that the authors of the GEBF method originally felt that it was the energy equations which defined GEBF. This is supported by the follow up studies which changed the fragmentation method still calling the method GEBF. However, the GEBF energy equation is nothing more than the inclusion-exclusion principle (accounting for the self-interaction of the point charges), so such a definition would mean CG-MTA, or any other overlapping fragment-based method, should also be labeled as GEBF. The GEBF authors eventually realized:cite:Yuan_2016,Li_2019 the IEP equivelence. At this point, it is reasonable to assume that it is the closed form which results from the IEP that determines whether a method is GEBF or not, particularly whether there is a self-energy correction. However, this hypothesis is also disproved when Cheng et al.5 debuts the GEBF20 method, which does not include any electronic embedding. Ultimately, it appears that it is simply the authors of the study which determine whether or not the method is GEBF or not.
Other GEBF Results
Li et al.49 used GEBF07 with manually specified fragments (it’s not immediately clear if the embedding was iterative) as the QM part of a QM/MM calcualtion. This method was applied to solvated polyethylene.
Dong et al.12 uses a GEBF08 approximation to B3LYP (with van der Waals correctiosn) to model aromatic oglioamides. Of note this study included the Distance-Based CP BSSE corrections which were found to be a good approximation to full counterpoise corrections.
Yang et al.90 applied GEBF08 to water clusters containing 20 to 30 water molecules in order to search for low energy geometries. Results using B3LYP and MP2 were optimized and corrected for ZPVE. Some comparison with supersystem results was presented (errors were around 1 kcal/mol), but the majority of conclusions relied on fragment-based methods alone.
Jiang et al.38 applied GEBF11 (GEBF10 with user-defined fragments) to computing vibrational circular dichromism spectra at the DFT level of theory. The authors report that the VCD spectra were in good agreement with supersystem spectra as well as experimental.
Hua et al.32 applied GEB07 (presumably; the paper is not clear on the exact details and instead presents references) to poly-alanine systems containing up to 40 residues. Optimized structures, energies, and enthalpies were computed using DFT (B3LYP and M06-2X). Errors in GEBF approximated energies, relative to supersystem results, were about 2 kcal/mol; errors were only computed for systems with 10 alanines.
Yang et al.91 looked at water clusters trapped in MOF pores using QM/MM. Two types of clusters were considered: neutral and protonated. The QM region of the neutral cluster was too big for a conentional B3LYP calculation so the authors relied on GEBF (presumably either GEBF07 or GEBF10). The geometries were optimized using QM/MM and the cluster geometries agreed well with experiment. No comparison of GEBF to supersystem calculations was presented.
Li53 applied GEBF10 to methanol disolved in water. Methanol dissolved in clusters containing up to 1115 water molecules were considered. To reach the large cluster size an ONIOM was utilized where the highest level of theory was GEBF approximated MP2-F12/aDZ, the middle layer was GEBF approximated MP2/aDZ, and the lowest level was DFTB (presumably with no GEBF approximation). As part of the calibration supersystem MP2-F12/aDZ results for a cluster with 12 water molecules were computed and compared to the GEBF approximated results; errors were less than 1 kcal/mol. Of note this study also used the Distance-Based CP method.
Fang et al.15 applied GEBF14 to periodic systems. In their approach, the lattice energy of the unit cell is estimated by performing GEBF on all fragments in a super cell that have at least one pseudoatom within the unit cell. the supercell is then embedded in an infinite field of point charges which gives rise to an Ewald term. There is then discussion of a compensation field to correct for truncating the infinite field into a finite field; I admittedly am unclear if this is in addition to the Ewald term or a replacement for it. Either way the resulting method is calibrated for phase II ice against periodic BL3LYp/6-31G*. With the compensation field, and 5 waters per fragment, GEBF matches the supersystem result to within 1 mH. Fang et al.15 then go on to compute the lattice constants for 10 molecular crystals using either CCSD(T)-F12/aTZ or MP2-F12/aTZ plus a CCSD(T)/aDZ correction (depending on the size of the molecules). The GEBF-approximated lattice energies agree with experiment to within 6 kJ/mol.
Li et al.54 describes the Linear Scaling Quantum Chemistry (LSQC) program. LSQC is a program capable of running GEBF calculations and CIM calculations by leveraging existing quantum chemistry packages. For GEBF calculations, LSQC can leverage Gaussian and Molpro. For CIM, LSQC relies on a locally modified version of GAMESS. Also presented were GEBF-M06-2X/6-31G* optimized structures and IR spectra, GEBF-MP2/6-31G* optimized structures, and a scaling plot of GEBF-MP2/6-31G*.
Liu et al.66 modeled the conformation dynamics of three-station molecular shuttles with DFT. To determine which DFT was most suitable, reference MP2 binding energies were computed using the GEBF method.
Fang et al.14 extended periodic GEBF14 calculations to include IR and Raman spectra. Comparisons to conventional periodic DFT (PBE0-D2/6-311G(d,p)) on imidazole and CO2 demonstrated that PBC-GEBF could faithfully approximate conventional supersystem calculations. For urea and ammonia borane crystals, PBC-GEBF calculations at the MP2/6-311++G(d,p) level of theory were compared to experiment. Discrepency between PBC-GEBF and experiment was worse, but the results still overall match well.
Zhang et al.100 presented AIMD simulations of polypetides using GEBF11. Simulations were performed at several levels of theory. Initial Comparisons of energy differences for the polypetides showed a fairly large descrepancy across the methods considered, notably GEBF-M06-2X/6-31G differed from conventional M06-2X by about 10 kcal/mol. DFTB differed by almost 100 kcal/mol. It was noted that GEBF approximated SCF/STO-3G and M06-2X/STO-3G simulations had large energy drift (presumably because the gradients were too inaccurate from neglegting response terms).
Li et al.60 extended GEBF to excited states (specifically the GEBF14 flavor). In these calculations, the excitations were assumed localized to a set of pseudoatoms. Fragments containing the active pseudoatoms were treated with an excited state method. Fragments without active pseudoatoms are treated with the ground state method. For a conjugated aldehyde, comparison of GEBF-\(\omega\)B97XD to conventional TD-DFT results (for the 6-31G, 6-311G(d,p), 6-311++G(d,p), and cc-pVTZ bases) showed errors less than 0.1 eV. Solvatochromatic shifts of acetone in solvent (solvents of water, methanol, acetonitrile, and carbon tetrachloride) computed with GEBF-\(\omega\)B97XD match experiment to about 0.1 eV. A similar analysis of pyridine and uracil in water again showed GEBF-\(\omega\)B97XD could match supersystem results well.
Yuan et al.96 compares GEBF14 to the EE-MB method. The discussion focuses on absolute energies of water clusters using M06-2X, HF, and MP2 (and several basis sets). Consistent with Bettens (TODO: cite Bettens BSSE), the study finds that the EE-MB method converges poorly (likely because of BSSE). One notable example is M06-2X/6-311++G** for clusters with 20 waters; here the maximum unsigned error exceeds 11 mH and does not seem to have converged by an 8-body MBE. EE-MB is also shown to have a fairly large sensitivity to the level of theory. Interestingly, using loose n-mer screening rectifies these problems somewhat.The study goes on to demonstrate that GEBF14 does not suffer from these problems, and that using supersystem basis sets with GEBF14 does not appreciably change the result. The authors attribute this to a BSSE cancellation effect stemming from the IEP nature of GEBF.
Tao et al.84 used the GEBF14 approximated \(\omega\)B97XD/6-311++G(d,p) model to analyze the properties of water clusters containing 50 molecules. The study was primarily interested in characterizing the different types of hydrogen bonds which are present and explaining how the types of hydrogen bonds affect the freezing of water.
Liu et al.65 uses GEBF approximated M06-2X/6-31G(d) for computing the energies and gradients needed for AIMD. The study looked at pseudorotaxane and compared AIMD results to classical MD results.
Li et al.62 used PBC-GEBF to optimize molecular crystal geometries for a QM/MM calculation. Once the geometries were optimized QM/MM calculations were run to compute excited states. The QM region contained dimers or trimers of and MM charges were taken from the PBC-GEBF calculation.
Wang et al.88 undertook a joint experimental/theory study on crystals of DNA bases. PBC-GEBF-PBE(D3B3)/6-311+G(d,p) was used to optimized the crystal structures and to compute the IR vibrational spectra. Geometries and spectra were compared to experiment and matched well.
Zhang et al.101 used GEBF14 and PBC-GBEF14 to establish benchmark energetics for water clusters containing 144 and 64 waters, respectively. Benchmarks relied on the CCSD(T)-F12b/cc-pVDZ-F12 model. Relative energies and binding energies are computed. The remainder of the study compares the performance of several flavors of DFT against the benchmarks.
Yuan et al.95 used GEBF14 to compute benchmark energetics for water clusters involving 32 and 64 water molecules. The benchmarks presente are relative energies of different isomers, computed with the GEBF-CCSD(T)/CBS model. Calibration was done by comparing to supersystem MP2/cc-pVTZ results. For water 32, restricting fragments to at most 6 waters resulted in unsigned errors of 1.2 kcal/mol; this decreases to 0.2 kcal/mol if fragments may contain 8 waters. For water 64, restricting to 8 waters resulted in mean unsigned errors of 1.4 kcal/mol; this reduces to 0.4 kcal/mol if the maximum is raised to 10 waters. The remainder of the study considered how various density functional theory approaches performed compared to the GEBF-CCSD(T)/CBS results.
Zhao et al.103 applied GEBF14 to NMR shifts. Considered a number of systems including a Trp-cage mini protein, CH3CN solvated in CHCl3, a oligoamide beta sheet, and a supramolecular aggregate. The GEBF approximated NMR shifts are compared with both supersystem results and experiment. Overall, the predicted NMR shifts match well, with errors typically less than 1 ppm.
Li et al.63 applied GEBF14 to ionic liquids. Of particular note the authors found that choosing neutral pseudoatoms (one cation and one anion per pseudoatom) led to better results than simply letting each cation/anion be a pseudoatom. GEBF based on one ion per pseudoatom fragments sometimes erred by more than 20 kcal/mol! Also computed relative energies, optimized geometries, IR spectra, and thermodynamic values. The authors considered the effect of ensuring the same fragments throughout the optimization (as opposed to picking the closest ion pairs in each step) and somewhat surprisingly found that this had little affect on the geometry and spectra, but could lead to large errors in the thermodynamic values.
Yuan et al.94 applied GEBF14 to supramolecular coordination complexes. In doing so they stipulated that pseudoatom creation should avoid breaking metal- ligand bonds. GEBF approximated DFT energies, relative energies, optimized geometries, IR spectra, and NMR chermical shifts are compared to supersystem calculations and found to match well. On this basis, GEBF approximated IR and NMR spectra are then used to help assign experimental peaks.
Fu et al.20 applied GEBF14 and PBC-GEBF14 to studying the excited state spectra of uracil. To this end GEBF14 was combined with QM/MM. The study looked aqueous, amorphous, and crystalline uracil. GEBF approximated excited state energies were computed using TD-\(\omega\)B97X-D/6-311++G(d,p) and compared to experiment. For aqueous and crystalline uracil peak maximums were off by 10 nm.
Zhao et al.102 introduced a variant of the PBC-GEBF method. More specifically, up to this point PBC-GEBF had been done on crystals formed from small molecules. As such pseudoatoms were taken to be entire molecules. The new “fragment-based” variant allowed the molecules in the crystal to be broken into multiple pseudoatoms. The new method was applied to a series of systems and with various flavors of DFT. Compared to experimental NMR parameters, both flavors performed well.
Li et al.61 applied PBC-GEBF14 to ionic crystals. The study proposed two mechanisms for forming pseudoatoms: one ion per pseudoatom or pairs of ions (one cation and one anion). The results establish that defining pseudoatoms using pairs of ions leads to much better resutls. The resulting methodology is then used to predict optimized geometries, IR spectra, Raman spectra, and NMR shifts. PBC-GEBF results are compared to experiment and conventional periodic DFT. Generally speaking good agreement between PBC-GEBF and the benchmarks is seen.
Hong et al.28 updated the equations used for the gradients of PBC-GEBF to use fractional coordinates instead of combinatorial coordinates. The revamped gradients are compared directly to analytic gradients from conventional periodic DFT calculations. The gradients are then used for geometry optimization (results are compared to both conventional calculations and experiment). Finally IR spectra are computed.
GEBF Reviews
Li and Li50 provides a brief introduction to fragment-based methods primarily focusing on GEBF (including its earlier incarnations).
Li et al.52 reviewed the progress of GEBF to date. New results presented here included an analysis of the gradients for a hydrogelator with and without neglect of the response terms (results were at the HF/6-31G* level of theory) and ab initio molecular dynamic simulations of alanine peptides (using the M06-2X/STO-3G level of theory).
Fang et al.16 reviewed the progress of PBC-GEBF and presented new comparisons to both periodic supersystem calculations as well as experiment. Results provided included optimized geometries, lattice energies, vibrational frequency shifts, and Raman spectra. Most results were obtained with the M06-2X functional and/or MP2. Generally speaking agreement between PBC-GEBF and the supersystem or experiment were good.
Li et al.55 was a follow up review to Li et al.52. In this review advancements since the 2014 review were summarized. In particular, the PBC variant of GEBF and properties thereof. The majority of the review focuses on select calculations from various types of systems including: ionic liquids, molecular crystals, coordination compounds, and host-guest systems.