Accuracy of Dimer-ES Approximation on Fragment Molecular Orbital ( FMO ) Method

The fragment molecular orbital (FMO) method is recently attracting attention as a method of calculating the electronic state of macromolecular systems. To enhance the speed of the FMO method, it is necessary to apply an approximation in which SCF calculations are neglected for distant fragment (monomer) pairs (dimers) and instead the electrostatic interactions between the two monomers are calculated. This approximation is called the dimer-es approximation. The accuracy and speed brought by the dimer-es approximation depend on the minimum threshold distance between two atoms to apply the approximation. This threshold distance given in unit of van der Waals radii is named “Ldimer-es”. In this communication, we examined dependence of HF and MP2 electron correlation energy errors on “Ldimer-es”, and it is preferable to calculate FMO4-HF and FMO4-MP2 for calculation of FMO-HF and FMO-MP2 of side chain-split peptides with Ldimer-es=2.0.


Introduction
The fragment molecular orbital (FMO) method is recently attracting attention as an efficient method of calculating the electronic state of macromolecular systems [1][2][3][4].The accuracy of FMO can be improved systematically by including higher N-body terms of fragments at the expense of computational speed.Currently, it is practically possible to consider 2-body (FMO2), 3-body (FMO3), and 4-body terms (FMO4).
To enhance the speed of the FMO method, it is necessary to apply an approximation in which SCF calculations are neglected for distant fragment (monomer) pairs (dimers) and instead the electrostatic interactions between the two monomers are calculated.This approximation is called the dimer-es approximation [5,6].The accuracy and speed brought by the dimer-es approximation depend on the minimum threshold distance between two atoms to apply the approximation.This threshold distance given in unit of van der Waals radii is named "L dimer-es " in the ABINIT-MP program [7] ("RESDIM" in the GAMESS program [8]) (Fig. 1).
It was found that L dimer-es = 2.0 is appropriate for polypeptides when a side chain is united with its conjugated main chain fragment [9].When minute data on the inter fragment interaction energy (IFIE, Refs.[6,10]) are required, however, we need a more minute fragmentation scheme in which a side chain and its conjugated main chain portion are regarded as separated fragments (Fig. 2) [11].This fragmentation scheme is useful but presumably deteriorates the accuracy of the HF and MP2 electron correlation energies at the same time.Therefore, in this paper, we report accuracy of the HF and MP2 electron correlation energies (Eq. 1) in the minute fragmentation scheme defined in Fig. 2, where E total , E HF , E MP2 are FMOn total energy, FMOn HF energy, and FMOn MP2 correlation energy.

Methods
The polypeptides used for the test calculations were (Ala) 10 α-helix modelled with BIOVIA Discovery Studio 3.1 [12].Presenting the self peptide, LF9 (LSSPVTKSF), of HLA class I molecule (PDB ID: 2RFX), and super chignolin (PDB ID: 5AWL).The FMO calculations were performed with a local version of the ABINIT-MP program [7].The basis-set was fixed to 6-31G* throughout the study.HF and MP2 calculations were performed in FMO2, FMO3, and FMO4 calculations with L dimer-es = 1.0, 1.5, 2.0 and 2.5, and also in conventional HF and MP2 Molecular Orbital calculations to obtain reference values for comparison with the corresponding FMO HF and MP2 energy values.

Results and Discussion
Below molecular FMO-HF and FMO-MP2 electron correlation energies calculated with various levels (FMO2-4) and L dimer-es values (1.0-2.5) are compared with the reference MO values to examine the accuracy of FMO calculations in the minute fragmentation scheme (Fig. 2) [11].FMO2 showed extremely large calculation errors; 1.5 to 1.9 hartree for HF energy and 45 to 109 mhartree for MP2 electron correlation energy (data not shown).These errors were unacceptably large so that we do not examine FMO2 further but concentrate on FMO3 and FMO4.The behaviors of the errors in FMO3 and FMO4 were essentially similar in all the three polypeptides investigated (Figs.3-5).Hence the discussion below is expected to have a certain degree of generality in FMO calculation of arbitrary polypeptides.The errors in HF energy were negligible in FMO4 regardless of L dimer-es .Errors of HF energy and MP2 electron correlation energy by FMO3 method tend to increase as L dimer-es increases to 1.5, 2.0 and 2.5.At L dimer-es = 2.5, HF energy error is -8 mhartree,