Fragment Molecular Orbital Method

The fragment molecular orbital (FMO) method is arguably one of the most established methods in the field of fragment-based methods. The main purpose of this page is to describe the FMO method, and how the method has developed.

Origins

PIMO

The FMO method arguably got its start as the pair interaction molecular orbital (PIMO) method Kitaura et al.⁴². In the original paper, for a system split into \(N\) disjoint fragments, the Hamiltonian for the \(I\)-th fragment is written as:

\[\widehat{H}_I = \widehat{H}_I^0 + \sum_{J=1; J\neq I}^N\widehat{V}^I_{J}\]

where \(\widehat{H}_I^0\) is the Hamiltonian of the \(I\)-th fragment and \(\widehat{V}^I_{J}\) is the interaction of \(I\) with \(J\). \(\widehat{V}_{IJ}\) is given by Density Embedding. The PIMO method considers up to two-body terms, with the Hamiltonian of dimer \(IJ\) given by:

\[\widehat{H}_{IJ} = \widehat{H}_{IJ}^0 + \sum_{K=1; K\neq J\neq I}^N\widehat{V}^{IJ}_{K}\]

Analogous to the monomer Hamiltonians, \(\widehat{H}_{IJ}^0\) is the Hamiltonian of the \(IJ\)-th dimer and \(\widehat{V}^{IJ}_{K}\) is the interaction of dimer \(IJ\) with fragment \(K\). Fragment densities are converged at the one-body level and used as is in two-body terms.

Initial applications of PIMO looked at interaction energies of water trimers and tetramers. At the HF/6-31G** level of theory PIMO differed from supersystem interactions by about 0.1 kcal/mol.

FMO99

That same year, FMO proper was introduced Kitaura et al.⁴¹. Physically the equations of FMO are the same as PIMO. The difference between FMO and PIMO is conceptual and has to do with the fragment orbitals. In PIMO each fragment was a closed-shell molecule, and the intial guess orbitals are simply obtained by running the isolated closed-shell molecule. In FMO, the initial orbitals for fragment \(I\) are generated by:

• creating a reference system for fragment \(I\),

• localizing the orbitals of the reference system to the atoms in \(I\),

• discarding contributions to the orbitals from atoms outside of \(I\), and then

• renormalizing the orbitals.

How to fragment the system was briefly considered for ethanol; initial results showed the final accuarcy was quite sensitive to fragment choice (differing by up to 7 kcal/mol). The only suggestion was to not make the fragments too small. Additional calculations on propane, propanol, and methylacetamide further served as a proof-of-principles for FMO. All calculations were done using SCF/STO-3G.

FMO00

Shortly after introducing the original FMO procedure, the capping procedure was overhauled. The new FMO00 capping procedure no longer requires reference systems to obtain the initial densities. Instead, the electrons in a bond are assigned to one of the two atoms. Assuming the two bonded fragments are \(I\) and \(J\), we further assume \(I\) gets the electrons. In this case, the equations for \(J\) are solved normally, whereas the equations for \(I\) are solved with a basis set that includes part of the valence space of \(J\) (specifically the part from the bonded atom).

Also notable about this FMO incarnation is that it introduced a screening method. The FMO00 screening procedure relies on distance cutoffs (based on the shortest inter-atomic distance) to approximate the two-body interactions in some dimers. The dimer interactions for distant dimers are not neglected, rather they are approximated using the one-body densities.

All initial calculations used the SCF/STO-3G model. Application of FMO00 to propane showed that the error in the total energy (relative to the traditional calculation) decreased relative to FMO99. This study also applied FMO00 was to peptides. For the peptides the fragment size was again considered. Results were markedly better when there are two amino acids in a fragment vs. one (maximum error in the total energy decreased from 8.9 kcal/mol to around 1 kcal/mol). Relative energies of the peptides had a maximum error of about 1.5 kcal/mol.

FMO02

The next incarnation of FMO introduced approimations to the embedding method. In traditional density embedding, monomer \(I\) interacts with the density of each monomer \(J\). This amounts to a term analogous to the Coulomb build in SCF, but with the density of \(J\). Like it’s SCF brethern, the computational complexity of this terms scale as \(\mathcal{O}(N^4)\). To reduce the cost of the embedding, the FMO02 Embedding method Nakano et al.⁷⁰: was introduced. As a proof-of-concept, the study considered applications to proteins using the HF/STO-3G method. Errors, were similar to the FMO00 method, but results were obtained significantly faster.

FMO04 (Three-Body FMO)

The next development in FMO was an extension capable of handeling three-body terms Fedorov and Kitaura¹⁷. This required introducing a screening method for three body terms, i.e., FMO04 Screening, and also introduced a new capping method, FMO04 Capping, which is essentially a combination of the FMO99 Capping and FMO00 Capping methods. Three-body results focused on water clusters and peptides with the SCF method and the STO-3G, 3-21G, 6-31G*, and 6-31++G** basis sets. Notably this study was one of the first to show that FMO performs best with small basis sets, postulating that this is because of the lack of exchange and intramolecular basis set superposition error.

Summary

Name	Fragments	Embedding	Caps	Screening
PIMO	Cluster	One-body density	N/A	N/A
FMO99	N/A	One-body density	FMO99	N/A
FMO00	N/A	One-body density	FMO00	FMO00
FMO02	N/A	FMO02	FMO00	FMO00
FMO04	N/A	FMO02	FMO04	FMO04

Other FMO Development

Initial, approximate gradients for the FMO00 method were published in 2001 Kitaura et al.⁴³. This formulation neglected the coupled-perturbed Hartree-Fock contribution to the gradient, which is necessary because the densities are not obtained variationally.

Inadomi et al.³⁵ detailed a mechanism for retrieving orbital energies and densities from FMO.

Sekino et al.⁸¹ used FMO to approximate a DFT calculation, PW91/STO-3G. Also considered were the HOMO/LUMO energies, dipole moment, and quadrupole moment of the systems. Systems considered were water-ammonia clusters and DNA. Errors in energies were on the order of a millihartree, multipole moments matched to within a few tenths (units were not specified). A follow up paper Sugiki et al.⁸³ used DIIS to accelerate the convergence of the iterative charge procedure.

Komeiji et al.⁴⁴ used FMO to perform a molecular dynamics simulation of a peptide at the HF/STO-3G level of theory.

Fedorov et al.¹⁸ discussions about distributed computing of FMO with GAMESS. Presensts some SCF/6-31G* and SCF/STO-3G calculations of large water clusters.

Mochizuki et al.⁶⁸ used the SCF orbitals from an FMO calculation to do MP2. Also considered in this study were MP2 densities and partially renormalized MP2. Calculations used the 6-31G basis set and focused on a large water cluster and several proteins.