12 November 1999 Ž. Chemical Physics Letters 313 1999 701 706 www.elsevier.nlrlocatercplett Fragment molecular orbital method: an approximate computational method for large molecules Kazuo Kitaura a,), Eiji keo a, Toshio Asada a, Tatsuya Nakano b, Masami Uebayasi c a Department of Chemistry, College of ntegrated Arts and Sciences, Osaka Prefecture UniÕersity, Sakai-si, Osaka 599-8531, Japan b DiÕision of Chem-Bio nformatic, National nstitute of Health Sciences, 1-18-1 Kamiyoga, Setagaya-ku, Tokyo 158-8501, Japan c National nstitute of Bioscience and Human Technology, Tukuba-si, 305, Japan Received 5 July 1999; in final form 29 July 1999 Abstract Ž. We propose an approximate molecular orbital MO method for calculating large molecules such as proteins. Our method assigns the electrons of the molecules to fragments, and the MOs of fragments and fragment pairs are calculated to obtain the total energy of the molecule. The method avoids the MO calculation of the whole molecule and is expected to reduce the computational time drastically for large molecules. Numerical calculations were performed on propane, propanol and methylacetamide to demonstrate the accuracy of the method. The optimized geometries and the total energies were in good agreement with those from the ab initio MO calculations. q 1999 Elsevier Science B.V. All rights reserved. 1. ntroduction Large molecules such as proteins have become the targets of theoretical studies using the molecular orbital Ž MO. method as computer power has progressed. Recently, an electronic structure calculation has been performed on crambin, a protein with 46 residues, by the ab initio MO method wx 1. Computer power, however, is still insufficient to carry out the ab initio MO calculations on most proteins by the ab initio MO method. Especially geometry optimizations of the large molecules by the conventional MO method will be very hard, even in the near future. Some models, therefore, are needed to realize quantum mechanical calculations on such systems. Along ) Corresponding author. Fax: q81-0722-54-9931; e-mail: kitaura@chem.cias.osakafu-u.ac.jp these lines, mamura and co-workers have proposed the elongation method and have mainly applied the empirical and semi-empirical MO methods wx 2. Stewart and co-workers have proposed an algorithm to solve the Fock equation which avoids the diagonalization of the Fock matrix and opens the way to calculate electronic structures of large molecules at the semi-empirical MO level wx 3. Both approaches may be promising but seem to be under progress to adopt to the ab initio MO method. Fragment methods in MO theory were studied many years ago w4 6 x. The fragment MO method proposed in this work is closely related to the pair interaction molecular orbital Ž PMO. method rewx 7 and cently proposed for molecular interactions has a different background from others. Our method divides a molecule into fragments and assigns the electrons of the molecule to the fragments. The MOs 0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. Ž. P: S0009-2614 99 00874-
702 K. Kitaura et al.rchemical Physics Letters 313 1999 701 706 for the electrons in the fragment are calculated under the restriction that the MOs are localized within the fragment. After the fragment MOs are obtained for all fragments, the MO calculations for the fragment pairs are performed to obtain the total energy of the molecule. Thus our method avoids the MO calculation of the whole molecule and is expected to reduce the computational time of large molecules drastically. The numerical calculations were carried out on several molecules and the geometries and total energies were obtained in reasonable accuracy. electrons to OH. t is noted that our method does not require the assignment of nuclei to fragments. We introduce the following fragment Hamiltonian, 1 Z s H s y = y n all atoms 2 ½ i Ý 2 < r yr < i s i s N n r r J Ž. 1 ÝH 5 Ý J/ riyr i)j riyrj Ý q dr q, Ž 1. < < < < 2. Fragment MO method The molecule to be calculated is divided into fragments as illustrated in Fig. 1 for the example of propanol; the molecule is divided into the CH 3, two CH 2 and an OH fragment. The partition should be done so as not to destroy bond electron pairs. Then the electrons are assigned to fragments: 8 electrons to CH 3, 8 electrons to each CH 2 and 10 where n and N are number of electrons in the fragment and number of fragments in the molecule, respectively. Zs is the nuclear charge of atom s and r Ž r. J the electron distribution of fragment J. The electron distributions of the fragments are obtained by an iterative manner Ž vide infra.. The fragment Hamiltonian includes the electrostatic potential from the electrons in the surrounding Ž N y 1. fragments as well as the nuclear attractions from all nuclei in the molecule. Fig. 1. Schematic illustration for dividing a target molecule into fragments and transferring LMOs from reference molecules to fragments as the basis functions.
K. Kitaura et al.rchemical Physics Letters 313 1999 701 706 703 Similarly, the following fragment pair Hamiltonian is introduced, nqn J 1 all atoms Z 2 s HJs Ý ½ y = i y Ý 2 < r yr < i s i s N nqn r r J K Ž. 1 Ý H 5 Ý K/, J riyr i)j riyrj q dr q. < < < < Ž 2. The Hamiltonian also has the electrostatic potential term from the electrons in the surrounding Ž Ny2. fragments. t is noted that the electron distribution, r Ž r. K, is the one obtained from the fragment calculation and is not varied in the pair calculation. These Hamiltonians are the same as those employed in the PMO method for molecular interactions wx 7. Solving the following Schrodinger equations at an appropriate level of wavefunction such as the Hartree Fock level under the restriction on the orbitals to be localized within the appropriate fragment and the fragment pair, H C se C, H C se J J JC J, one obtains the electronic energy of the fragment, E, and the electronic energy of the fragment pair, E J. The total energy of the molecule, E, is calculated as follows; N Ý Es E y Ž Ny2. E )J J N Ý all atoms Ý s t s t s)t q ZZr< r yr <, Ž 3. where the last term of Eq. Ž. 3 is the nuclear repulsion energy. To obtain the fragment orbitals being localized within the fragment, we use the bond localized orbitals Ž LMOs. w8 10x as the basis functions for the fragment orbitals. The basis LMOs are generated beforehand from the calculations of appropriate small molecules Ž the reference molecules. which have the same group or fragment as the molecule of interest Ž the target molecule.. The LMOs of the small molecule are easily assigned to each fragment. The LMOs assigned to the fragment are transferred to the corresponding fragment in the target molecule after discarding the tails of LMOs. As for the example shown in Fig. 1, the three C H bonding LMOs, the three C H anti-bonding LMOs and the core orbital of the carbon atom of propane are used as the basis functions for the CH 3 fragment orbitals in propanol and so forth Žthe minimal basis set is assumed in the Fig. 2. The definition of the geometrical parameters of ethane and various partitions of the molecule.
704 K. Kitaura et al.rchemical Physics Letters 313 1999 701 706 statement.. The C H bond LMOs have the small coefficients of atomic basis functions which belong to the other fragments, i.e. the tails. The tails are simply discarded and the LMOs are renormalized in the procedure. Then the HF equation for the fragment and the fragment pair are solved using the LMOs as the basis functions: the Fock equation is solved in the LCMO Žlinear combination of molecular orbitals. representation w11 x. From the procedure mentioned above, one understands that the geometry of the fragments should be kept rigid in the calculation. The computational procedure of the fragment MO method is as follows: Ž. 1 Divide the target molecule into fragments and assign electrons to the fragments. Ž. 2 Prepare the LMOs for the basis functions of the fragments by performing the MO calculations of the appropriate reference molecules and calculate the initial electron density distributions of the fragments using the corresponding LMOs. Ž. 3 Construct the fragment Hamiltonians using the given electron density distributions and solve the Schrodinger equations for all fragments in the molecule to obtain the fragment energies, E and the electron density distributions. Ž. 4 Determine whether the electron density distributions of all fragments are the same as the previous ones within a specified criterion. f the density has not converged, return to the step Ž. 3 with the new density distributions. Ž. 5 Construct the fragment pair Hamiltonian using the converged density distributions, r Ž r., and solve J the Schrodinger equations for all fragment pairs to obtain E J. Ž. 6 Calculate the total energy using E and EJ ŽEq. Ž 3.. and other properties such as electron populations. The computational procedure explained above is the same as that of the PMO method wx 7 except that the fragment and fragment pair MOs are obtained under the constraint that the MOs are localized in the fragment and the fragment pair, respectively. 3. Numerical calculations All numerical calculations were carried out at the ab initio HFrSTO-3G level. The geometries of the fragments were constructed using the standard bond lengths and bond angles: R s1.54 A, CC RCH s1.09 A, R s 1.43 A, R s 0.96 A, R s 1.40 A, CO OH CN 3 RNH s 1.01 A and the sp bond angle s 109.478. The geometries of the fragments were kept rigid and the inter-fragment degrees of freedom were optimized both in the fragment MO and conventional ab initio MO calculations. The calculations were per- formed with Gaussian 94 w12 x. The accuracy of the fragment MO calculations may depend on how to divide the target molecule into fragments. The dependency was studied on ethanol by the calculations with various partitions: the cases 1 to 4 shown in Fig. 2. The LMO basis functions for the fragments were obtained from the natural localized molecular orbitals w10x of ethane. The total energies and the optimized geometrical Table 1 Total energies and optimized geometrical parameters of ethanol with various partitions a Total energy Ž a.u.. Bond length Ž A. Bond angle Ž 8. b R R u u u CO CC 1 2 3 Ab initio y152.13108 1.435 1.544 104.2 108.2 109.4 Case 1 y152.13055 Ž q0.3. 1.437 Ž q0.002. 1.541 Ž y0.003. 107.3 Ž q3.1. 107.2 Ž y1.0. 109.4 Ž q0.0. Case 2 y152.11998 Ž q7.0. 1.454 Ž q0.019. 1.582 Ž q0.038. 107.4 Ž q3.2. 108.9 Ž q0.7. 111.0 Ž q1.5. Case 3 y152.13643 Ž y3.4. 1.415 Ž y0.020. 1.564 Ž q0.020. 101.8 Ž y2.4. 106.8 Ž y1.4. 114.5 Ž q5.0. Case 4 y152.12754 Ž q2.2. 1.437 Ž q0.001. 1.553 Ž q0.008. 106.5 Ž q2.3. 107.2 Ž y1.0. 110.2 Ž q0.8. a See Fig. 2 for the geometrical parameters and the fragments used in the fragment MO calculations. n parentheses are given the error in energy, bond length and bond angle in kcalrmol, A and degrees, respectively. b The ab initio MO method.
K. Kitaura et al.rchemical Physics Letters 313 1999 701 706 705 parameters are summarized in Table 1, along with those from the ab initio MO calculations. The error in the total energy ranged from y3.4 to q7.0 kcalrmol, depending on the partitioning. The best total energy was obtained in the case 1. The geometrical parameters obtained in the case 1 agreed well Fig. 3. The total energies, optimized geometrical parameters and the fragments used in the calculations of propane, propanol and methylacetamide. The total energy, E, is in a.u. The error in the total energy, d, is in kcalrmol. The bond lengths and angles are in A and degrees, respectively. n parentheses are given the values from the ab initio MO calculations.
706 K. Kitaura et al.rchemical Physics Letters 313 1999 701 706 with those from the ab initio calculation: the bond lengths and bond angles were in error by 0.03 A and 38, respectively. The results suggest that small fragments, CH 2, in the case 2 and OH in the cases 3 and 4, rather increase the error in energy, though the data are not enough to deduce general criteria for dividing a molecule into fragments. Propane, 1-propanol and methylacetamide were tested as examples of hydrocarbons, aliphatic alcohols and molecules with peptide bonds, respectively. The reference molecules employed were propane, propanol and methylacetamide, respectively. The calculated total energies and the optimized geometrical parameters are shown in Fig. 3. For all molecules, the results are in good agreement with those from the ab initio MO calculations. The errors in the total energies are only q3.2, y2.2 and q0.7 kcalrmol for propane, propanol and methylacetamide, respectively. The errors in the geometrical parameters are also very small; the largest error in the bond length is y0.018 A in propanol and in the bond angle 48 in propanol. 4. Summary A new fragment MO method has been proposed for calculating large molecules such as proteins. The numerical calculations of several molecules reveal that the method gives the total energies and the geometries in sufficient accuracy. The error in the energy and geometry seems to become large if too small fragments are introduced in the molecule. n this aspect, the test calculations performed in this work were the most severe ones. n the practical applications of the method, one may use larger fragments and obtain better results than those shown in this work. We have carried out the fragment MO calculations on the model peptides of glycine and alanine, Ž Gly. and Ž Ala. Ž n s 3 20. n n, at the HFrSTO-3G level and obtained total energies within an error of ;1 kcalrmol, where one amino acid residue is treated as a fragment. This work will be reported elsewhere. The fragment MO method enables one to calculate the total energy of molecule without performing the calculation of whole molecule. The method is, therefore, expected to reduce the computational time of large molecule drastically. For practical applications, however, the use of the fast multipole expanw13x for Coulomb-type two-electron in- sion method tegrals involved in the electrostatic potential terms ŽEqs. Ž 2. and Ž 3.. may be critical for saving the computational time. Another advantage of the method is its ease in utilizing parallel processing, since the fragments and the fragment pairs can be calculated independently. Acknowledgements The work was supported in part by grant from the Ministry of Education Ž Grant No. 08221227.. The numerical calculations were partially carried out at the Computer Center of the nstitute for Molecular Science. References wx 1 C. Van Alsenoy, C.-H. Yu, A. Peeters, J.M.L. Martin, L. Schaefer, J. Phys. Chem. A 102 Ž 1998. 2246. wx 2 A. mamura, Y. Aoki, K. Maekawa, J. Chem. Phys. 95 Ž 1991. 5419. wx 3 J.J.P. Stewart, nt. J. Quantum Chem. 58 Ž 1996. 133. wx 4 M.D. Newton, F.P. Boer, W.N. Lipscomb, J. Am. Chem. Soc. 88 Ž 1966. 2353. wx 5 W.V. Niessen, Theor. Chim. Acta 31 Ž 1973. 111. wx 6 P. Degand, G. Leroy, D. Peeters, Theor. Chim. Acta 30 Ž 1973. 243. wx 7 K. Kitaura, T. Sawai, T. Asada, T. Nakano, M. Uebayasi, Chem. Phys. Lett. 312 Ž 1999. 319. wx 8 J.M. Foster, S.F. Boys, Rev. Mod. Phys. 32 Ž 1960. 300. wx 9 C. Edmiston, K. Ruedenberg, Rev. Mod. Phys. 34 Ž 1963. 457. w10x A.E. Reed, R.B. Weinstock, F. Weinhold, J. Chem. Phys. 83 Ž 1985. 735. w11x K. Kitaura, K. Morokuma, nt. J. Quantum Chem. 10 Ž 1976. 325. w12x M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez, J.A. Pople, Gaussian 94, Revision D.4, Gaussian, nc., Pittsburgh, PA, 1995. w13x C.A. White, B.G. Johnson, P.M.W. Gill, M. Head-Gordon, Chem. Phys. Lett. 230 Ž 1994. 8.