Exploring chemistry with the fragment molecular orbital method

The fragment molecular orbital (FMO) method makes possible nearly linear scaling calculations of large molecular systems, such as water clusters, proteins and DNA. In particular, FMO has been widely used in biochemical applications involving protein-ligand binding and drug design. The method has been efficiently parallelized suitable for petascale computing. Many commonly used wave functions and solvent models have been interfaced with FMO. We review the historical background of FMO, and summarize its method development and applications.     

Fully analytic energy gradient in the fragment molecular orbital method

The Z-vector equations are derived and implemented for solving the response term due to the external electrostatic potentials, and the corresponding contribution is added to the energy gradients in the framework of the fragment molecular orbital (FMO) method. To practically solve the equations for large molecules like proteins, the equations are decoupled by taking advantage of the local nature of fragments in the FMO method and establishing the self-consistent Z-vector method. The resulting gradients are compared with numerical gradients for the test molecular systems: (H(2)O)(64), alanine decamer, hydrated chignolin with the protein data bank (PDB) ID of 1UAO, and a Trp-cage miniprotein construct (PDB ID: 1L2Y). The computation time for calculating the response contribution is comparable to or less than that of the FMO self-consistent charge calculation. It is also shown that the energy gradients for the electrostatic dimer approximation are fully analytic, which significantly reduces the computational costs. The fully analytic FMO gradient is parallelized with an efficiency of about 98% on 32 nodes.     

The extension of the fragment molecular orbital method with the many-particle Green's function

By using the many-particle Green's function (GF) the extension of the fragment molecular orbital (FMO) method by Kitaura et al. [Chem. Phys. Lett. 313, 701 (1999)] is proposed. It is shown that the partial summation of the cluster expansion of GF reproduces the same extrapolation formula as that of FMO. Therefore we can determine the excitation energy, the transition moment, and the linear response of a molecule from GF approximated with the FMO procedure. It is also shown that no wave function exists which is consistent to the FMO results. The perturbation expansion in which the self-consistent charge approximation defines the unperturbed state is reported. By using it the three-body effects missing in the pair approximation of FMO are analyzed and the corrections to the energy and the reduced density matrices are proposed. In contrast to the previous works these new corrections are not expressed as the addition or the subtraction of the energies of fragments. They are size extensive and require only the quantities available by the FMO calculation. The accuracy of these corrections is validated with the extended Hubbard model and the several test molecules.     

Multiconfiguration self-consistent-field theory based upon the fragment molecular orbital method

The fragment molecular orbital (FMO) method was combined with the multiconfiguration self-consistent-field (MCSCF) theory. One- and two-layer approaches were developed, the former involving all dimer MCSCF calculations and the latter limiting MCSCF calculations to a small part of the system. The accuracy of the two methods was tested using the six electrons in six orbitals complete active space type of MCSCF and singlet spin state for phenol+(H(2)O)(n), n=16,32,64 (6-31G( *) and 6-311G( *) basis sets); alpha helices and beta strands of phenylalanine-(alanine)(n), n=4,8,16 (6-31G( *)). Both double-zeta and triple-zeta quality basis sets with polarization were found to have very similar accuracy. The error in the correlation energy was at most 0.000 88 a.u., the error in the gradient of the correlation energy was at most 6.x10(-5) a.u./bohr and the error in the correlation correction to the dipole moment was at most 0.018 D. In addition, vertical singlet-triplet electron excitation energies were computed for phenol+(H(2)O)(n), (n=16,32,64), 6-31G( *), and the errors were found to be at most 0.02 eV. Approximately linear scaling was observed for the FMO-based MCSCF methods. As an example, an FMO-based MCSCF calculation with 1262 basis functions took 98 min on one 3.0 GHz Pentium4 node with 1 Gbyte RAM.     

The fragment molecular orbital method for geometry optimizations of polypeptides and proteins

The fragment molecular orbital method (FMO) has been used with a large number of wave functions for single-point calculations, and its high accuracy in comparison to ab initio methods has been well established. We have developed the analytic derivative of the electrostatic interaction between far separated fragments and performed a number of restricted Hartree-Fock (RHF) geometry optimizations using FMO and ab initio methods. In particular, the alpha-helix, beta-turn, and extended conformers of a 10-residue polyalanine were studied and the good FMO accuracy was established (the rms deviations for the former two forms were about 0.2 A and for the latter structure about 0.001 A). Met-enkephalin dimer was used as a model for the polypeptide binding and computed at the 3-21G and 6-31G* levels with a similar accuracy achieved; the error in the binding energy predictions (FMO vs ab initio) was 1-3 kcal/mol. Chignolin (PDB: 1uao) and an agonist polypeptide of the erythropoietin receptor protein (emp1) were optimized at the 3-21(+)G level, with the rms deviation from ab initio of about 0.2 A, or 0.5 degrees in terms of bond angles. The effect of solvation on the structure optimization was studied in chignolin and the Trp-cage miniprotein construct (PDB:1l2y), by describing water with TIP3P. The computed structures in gas phase and solution are compared to each other and experiment.     

Geometry optimizations of open-shell systems with the fragment molecular orbital method

The ability to perform geometry optimizations on large molecular systems is desirable for both closed- and open-shell species. In this work, the restricted open-shell Hartree-Fock (ROHF) gradients for the fragment molecular orbital (FMO) method are presented. The accuracy of the gradients is tested, and the ability of the method to reproduce adiabatic excitation energies is also investigated. Timing comparisons between the FMO method and full ab initio calculations are also performed, demonstrating the efficiency of the FMO method in modeling large open-shell systems.     

Multilayer formulation of the fragment molecular orbital method (FMO)

The fragment molecular orbital method (FMO) has been generalized to allow for multilayer structure. Fragments are assigned to layers, and each layer can be described with a different basis set and/or level of electron correlation. Interlayer boundaries are treated in the general spirit of the FMO method since they also coincide with some interfragment boundaries. The question of the one- and two-layer FMO accuracy dependence upon the fragmentation scheme is also addressed. The new method has been applied to predict the reaction barrier and the reaction heat for the Diels-Alder reaction with a representative set of reactants based on dividing fragments in two layers. The 6-31G* basis set has been used for the active site and the 6-31G*, 6-31G, 3-21G, and STO-3G basis sets have been used for the substituents. Different levels of electron correlation (RHF, B3LYP, and MP2) have been applied to layers in systematic fashion. The one-layer FMO errors in the reaction barrier and the reaction heat were 2.0 kcal/mol or less for all levels applied (RHF, B3LYP, and MP2), relative to full ab initio methods. For the two-layer method the error was found to be several kcal/mol. Benchmark calculations of the activation barrier for the decarboxylation of phenylcyanoacetate by beta-cyclodextrin demonstrated that the two-layer calculations are efficient, being 36 times faster than the regular DFT, as well as accurate, with the error being 1.0 kcal/mol.     

Energy decomposition analysis in solution based on the fragment molecular orbital method

We develop the pair interaction energy decomposition analysis (PIEDA) in solution by combining the fragment molecular orbital (FMO) method with the polarizable continuum model (PCM). The solvent screening of the electrostatic interaction and the desolvation penalty in complex formation are described by this approach from ab initio calculations of fragments and their pairs. The applications to the complex of solvated sodium and chlorine ions, as well as to lysine and aspartic acid, show how the analysis helps reveal the physical picture. The PIEDA/PCM method is also applied to a small protein chignolin (PDB: 1UAO), and the solvent screening of the pair interactions is discussed.     

Time-dependent density functional theory based upon the fragment molecular orbital method

Time-dependent density functional theory (TDDFT) was combined with the two-body fragment molecular orbital method (FMO2). In this FMO2-TDDFT scheme, the system is divided into fragments, and the electron density for fragments is determined self-consistently. Consequently, only one main fragment of interest and several fragment pairs including it are calculated by TDDFT. To demonstrate the accuracy of FMO2-TDDFT, we computed several low-lying singlet and triplet excited states of solvated phenol and polyalanine using our method and the standard TDDFT for the full system. The BLYP functional with the long-range correction (LC-BLYP) was employed with the 6-31G(*) basis set (some tests were also performed with 6-311G(*), as well as with B3LYP and time-dependent Hartree-Fock). Typically, FMO2-TDDFT reproduced the full TDDFT excitation energies within 0.1 eV, and for one excited state the error was about 0.2 eV. Beside the accurate reproduction of the TDDFT excitation energies, we also automatically get an excitation energy decomposition analysis, which provides the contributions of individual fragments. Finally, the efficiency of our approach was exemplified on the LC-BLYP6-31G(*) calculation of the lowest singlet excitation of the photoactive yellow protein which consists of 1931 atoms, and the obtained value of 3.1 eV is in agreement with the experimental value of 2.8 eV.     

Fragmentation at sp2 carbon atoms in fragment molecular orbital method

In the fragment molecular orbital (FMO) method, a given molecular system is usually fragmented at sp³ carbon atoms. However, fragmentation at different sites sometimes becomes necessary. Hence, we propose fragmentation at sp² carbon atoms in the FMO method. Projection operators are constructed using sp² local orbitals. To maintain practical accuracy, it is essential to consider the three-body effect. In order to suppress the corresponding increase of computational cost, we propose approximate models considering local trimers. Numerical verification shows that the present models are as accurate as or better than the standard FMO2 method in total energy with fragmentation at sp³ carbon atoms.     

Multithreaded parallelization of the energy and analytic gradient in the fragment molecular orbital method

The fragment molecular orbital method in GAMESS is parallelized in a multithreaded OpenMP implementation combined with the MPI version of the two-level generalized distributed data interface. The energy and analytic gradient in gas phase and the polarizable continuum model of solvation are parallelized in this hybrid three-level scheme, achieving a large memory footprint reduction and a high parallel efficiency on Intel Xeon Phi processors. The parallel efficiency is demonstrated on the Stampede2 and Theta supercomputers using up to 2048 nodes (262 144 threads).     

Covalent bond fragmentation suitable to describe solids in the fragment molecular orbital method

To improve the accuracy of the fragment molecular orbital method (FMO), we introduce a new fragmentation scheme based on using frozen orbitals to describe fractioned bonds. By applying this scheme to a set of polyalanine systems of up to 40 residues for the alpha-helix and beta-strand isomers, we established its accuracy, which is considerably improved compared to the original hybrid orbital projection method used for detaching bonds in FMO. For instance, at the two-body FMO expansion with the 6-311G* basis set, the error was typically reduced 2-4 times, and for 6-31G* the accuracy increase was even larger (10 times in terms of the maximum error). For the Trp-cage protein (PDB file 1L2Y) with many charged residues, a fairly large error was observed, which was shown to become small with a larger fragment size or at the three-body level. Consequently, we applied the new scheme to the adsorption of toluene and phenol on a faujasite zeolite, and we demonstrated that good accuracy can be achieved in reproducing ab initio results.     

Theoretical study of the prion protein based on the fragment molecular orbital method

We performed fragment molecular orbital (FMO) calculations to examine the molecular interactions between the prion protein (PrP) and GN8, which is a potential curative agent for prion diseases. This study has the following novel aspects: we introduced the counterpoise method into the FMO scheme to eliminate the basis set superposition error and examined the influence of geometrical fluctuation on the interaction energies, thereby enabling rigorous analysis of the molecular interaction between PrP and GN8. This analysis could provide information on key amino acid residues of PrP as well as key units of GN8 involved in the molecular interaction between the two molecules. The present FMO calculations were performed using an original program developed in our laboratory, called “Parallelized ab initio calculation system based on FMO (PAICS)”. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009     

Analytic first and second derivatives of the energy in the fragment molecular orbital method combined with molecular mechanics

Analytic first and second derivatives of the energy are developed for the fragment molecular orbital method interfaced with molecular mechanics in the electrostatic embedding scheme at the level of Hartree-Fock and density functional theory. The importance of the orbital response terms is demonstrated. The role of electrostatic embedding upon molecular vibrations is analyzed, comparing force field and quantum mechanical treatments for an ionic liquid and a solvated protein. The method is applied for 100 protein conformations sampled in molecular dynamics (MD) to take into account the complexity of a flexible protein structure in solution, and a good agreement with experimental data is obtained: Frequencies from an experimental infrared (IR) spectrum are reproduced within 17 cm⁻¹ .     

Analytic gradient and molecular dynamics simulations using the fragment molecular orbital method combined with effective potentials

The completely analytic energy gradients are derived and implemented for the two-body fragment molecular orbital (FMO2) method combined with the model core potentials (MCP) and effective fragment potentials (EFP). The many-body terms in EFP require solving coupled-perturbed Hartree-Fock equations, which are derived and implemented. The molecular dynamics (MD) simulations are performed using the FMO2/MCP method for the capped alanine decamer and with the FMO2/EFP method for the zwitterionic conformer of glycine tetramer immersed in the water layer of 6.0 Å (135 water molecules). The results of the MD simulations using the FMO2/EFP and FMO2/MCP gradients show that the total energy is conserved at the time steps less than 1 fs.     

Extending the power of quantum chemistry to large systems with the fragment molecular orbital method

Following the brief review of the modern fragment-based methods and other approaches to perform quantum-mechanical calculations of large systems, the theoretical development of the fragment molecular orbital method (FMO) is covered in detail, with the emphasis on the physical properties, which can be computed with FMO. The FMO-based polarizable continuum model (PCM) for treating the solvent effects in large systems and the pair interaction energy decomposition analysis (PIEDA) are described in some detail, and a range of applications of FMO to biological studies is introduced. The factors determining the relative stability of polypeptide conformers (alpha-helix, beta-turn, and extended form) are elucidated using FMO/PCM and PIEDA, and the interactions in the Trp-cage miniprotein construct (PDB: 1L2Y) are analyzed using PIEDA.     

The fragment molecular orbital and systematic molecular fragmentation methods applied to water clusters

Two electronic structure methods, the fragment molecular orbital (FMO) and systematic molecular fragmentation (SMF) methods, that are based on fragmenting a large molecular system into smaller, more computationally tractable components (fragments), are presented and compared with fully ab initio results for the predicted binding energies of water clusters. It is demonstrated that, even when explicit three-body effects are included (especially necessary for water clusters due to their complex hydrogen-bonded networks) both methods present viable, computationally efficient alternatives to fully ab initio quantum chemistry.     

A study of molecular one-electron properties in terms of localized molecular orbital components

A number of molecular one-electron progperties have been analyzed by partitioning their electronic components over energy localized molecular orbitals (LMO ). The ammonia and ethane molecules, calculated in an Approximately double zeta qualtiy basis set, were considered. The partitioning of the electronic components of certain one-electron properteies over LMO allows a quantitative rationalization of the sensitivity of certain properties to basis set effects due to the differeing degree of difficulty of accurately determining different LMO as measures of the molecular electron density.     

Unrestricted Hartree-Fock based on the fragment molecular orbital method: Energy and its analytic gradient

A consideration of the surrounding environment is necessary for a meaningful analysis of the reaction activity in large molecular systems. We propose an approach to perform unrestricted Hartree-Fock (UHF) calculations within the framework of the fragment molecular orbital (FMO) method (FMO-UHF) to study large systems with unpaired electrons. Prior to an energy analysis one has to optimize geometry, which requires an accurate analytic energy gradient. We derive the FMO-UHF energy and its analytic gradient and implement them into GAMESS. The performance of FMO-UHF is evaluated for a solvated organic molecule and a solvated metal complex, as well as for the active part of a protein, in terms of energy, gradient, and geometry optimization.