Fully Analytic Energy Gradient in the Fragment …...Fully Analytic Energy Gradient in the Fragment...

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Ames Laboratory Publications Ames Laboratory 2011 Fully Analytic Energy Gradient in the Fragment Molecular Orbital Method Takeshi Nagata National Institute of Advanced Industrial Science and Technology Kurt Ryan Brorsen Iowa State University Dmitri G. Fedorov National Institute of Advanced Industrial Science and Technology Kazuo Kitaura National Institute of Advanced Industrial Science and Technology Mark S. Gordon Iowa State University, [email protected] Follow this and additional works at: hp://lib.dr.iastate.edu/ameslab_pubs Part of the Chemistry Commons e complete bibliographic information for this item can be found at hp://lib.dr.iastate.edu/ ameslab_pubs/345. For information on how to cite this item, please visit hp://lib.dr.iastate.edu/ howtocite.html. is Article is brought to you for free and open access by the Ames Laboratory at Iowa State University Digital Repository. It has been accepted for inclusion in Ames Laboratory Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected].

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  • Ames Laboratory Publications Ames Laboratory

    2011

    Fully Analytic Energy Gradient in the FragmentMolecular Orbital MethodTakeshi NagataNational Institute of Advanced Industrial Science and Technology

    Kurt Ryan BrorsenIowa State University

    Dmitri G. FedorovNational Institute of Advanced Industrial Science and Technology

    Kazuo KitauraNational Institute of Advanced Industrial Science and Technology

    Mark S. GordonIowa State University, [email protected]

    Follow this and additional works at: http://lib.dr.iastate.edu/ameslab_pubs

    Part of the Chemistry Commons

    The complete bibliographic information for this item can be found at http://lib.dr.iastate.edu/ameslab_pubs/345. For information on how to cite this item, please visit http://lib.dr.iastate.edu/howtocite.html.

    This Article is brought to you for free and open access by the Ames Laboratory at Iowa State University Digital Repository. It has been accepted forinclusion in Ames Laboratory Publications by an authorized administrator of Iowa State University Digital Repository. For more information, pleasecontact [email protected]

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  • Fully Analytic Energy Gradient in the Fragment Molecular OrbitalMethod

    AbstractThe Z-vector equations are derived and implemented for solving the response term due to the externalelectrostatic potentials, and the corresponding contribution is added to the energy gradients in the frameworkof the fragment molecular orbital (FMO) method. To practically solve the equations for large molecules likeproteins, the equations are decoupled by taking advantage of the local nature of fragments in the FMOmethod and establishing the self-consistent Z-vector method. The resulting gradients are compared withnumerical gradients for the test molecular systems: (H2O)64, alanine decamer, hydrated chignolin with theprotein data bank (PDB) ID of 1UAO, and a Trp-cage miniprotein construct (PDB ID: 1L2Y). Thecomputation 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 dimerapproximation are fully analytic, which significantly reduces the computational costs. The fully analytic FMOgradient is parallelized with an efficiency of about 98% on 32 nodes.

    KeywordsPolymers, Electrostatics, Chemical bonds, Proteins, Density functional theory

    DisciplinesChemistry

    CommentsThe following article appeared in Journal of Chemical Physics 134 (2011): 124115, and may be found atdoi:10.1063/1.3568010.

    RightsCopyright 2011 American Institute of Physics. This article may be downloaded for personal use only. Anyother use requires prior permission of the author and the American Institute of Physics.

    This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/ameslab_pubs/345

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  • Fully analytic energy gradient in the fragment molecular orbital methodTakeshi Nagata, Kurt Brorsen, Dmitri G. Fedorov, Kazuo Kitaura, and Mark S. Gordon Citation: The Journal of Chemical Physics 134, 124115 (2011); doi: 10.1063/1.3568010 View online: http://dx.doi.org/10.1063/1.3568010 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/134/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonorthogonal orbital based N-body reduced density matrices and their applications to valence bond theory. II.An efficient algorithm for matrix elements and analytical energy gradients in VBSCF method J. Chem. Phys. 138, 164120 (2013); 10.1063/1.4801632 Unrestricted Hartree-Fock based on the fragment molecular orbital method: Energy and its analytic gradient J. Chem. Phys. 137, 044110 (2012); 10.1063/1.4737860 Analytic energy gradient for second-order Møller-Plesset perturbation theory based on the fragment molecularorbital method J. Chem. Phys. 135, 044110 (2011); 10.1063/1.3611020 Molecular applications of analytical gradient approach for the improved virtual orbital-complete active spaceconfiguration interaction method J. Chem. Phys. 132, 034105 (2010); 10.1063/1.3290203 Theoretical study of intramolecular interaction energies during dynamics simulations of oligopeptides by thefragment molecular orbital-Hamiltonian algorithm method J. Chem. Phys. 122, 094905 (2005); 10.1063/1.1857481

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  • THE JOURNAL OF CHEMICAL PHYSICS 134, 124115 (2011)

    Fully analytic energy gradient in the fragment molecular orbital methodTakeshi Nagata (����),1,a) Kurt Brorsen,2 Dmitri G. Fedorov,1

    Kazuo Kitaura (����),1,3 and Mark S. Gordon (�����)21NRI, National Institute of Advanced Industrial Science and Technology (AIST), 1–1-1 Umezono, Tsukuba,Ibaraki 305–8568, Japan2Ames Laboratory, US-DOE and Department of Chemistry, Iowa State University, Ames, Iowa 50011, USA3Graduate School of Pharmaceutical Sciences, Kyoto University, 46–29 Yoshidashimo Adachi, Sakyo-ku,Kyoto 606–8501, Japan

    (Received 11 November 2010; accepted 28 February 2011; published online 29 March 2011)

    The Z-vector equations are derived and implemented for solving the response term due to the exter-nal electrostatic potentials, and the corresponding contribution is added to the energy gradients in theframework of the fragment molecular orbital (FMO) method. To practically solve the equations forlarge molecules like proteins, the equations are decoupled by taking advantage of the local nature offragments in the FMO method and establishing the self-consistent Z-vector method. The resultinggradients are compared with numerical gradients for the test molecular systems: (H2O)64, alaninedecamer, hydrated chignolin with the protein data bank (PDB) ID of 1UAO, and a Trp-cage minipro-tein construct (PDB ID: 1L2Y). The computation time for calculating the response contribution iscomparable to or less than that of the FMO self-consistent charge calculation. It is also shown thatthe energy gradients for the electrostatic dimer approximation are fully analytic, which significantlyreduces the computational costs. The fully analytic FMO gradient is parallelized with an efficiencyof about 98% on 32 nodes. © 2011 American Institute of Physics. [doi:10.1063/1.3568010]

    I. INTRODUCTION

    Dynamics simulations of large molecules have becomevery important in the field of computational chemistry.1–7 Forexample, thermodynamic data such as the free energy and en-tropy can be extracted from such simulations. Since chem-ical processes occur at nonzero temperatures, an increasingnumber of molecular dynamics (MD) simulations have beenperformed using model potentials based on molecular me-chanics (MM). Another common use of computational sim-ulations is the sampling of potential energy surfaces for localand global minima using for example, Monte Carlo simula-tions. For instance, the structural data of proteins measured byX-ray or nuclear magnetic resonance (NMR) can be refinedwith the help of a computational simulation and the result-ing structure stored in the protein data bank (PDB). ClassicalMM schemes can, in principle, be replaced by more accuratemethods that consider the electronic structure explicitly.8–13

    The benefit of this replacement is that by using first princi-ples methods, more reliable and sophisticated thermodynamicproperties, dynamics, and structures can be provided.

    The cost of conventional quantum mechanical (QM)electronic structure approaches formally scales at least as N m ,where N measures the size of the molecular system and mranges from 3 for simple methods [e.g., the local densityapproximation of density functional theory (DFT)] to muchhigher values for correlated methods. Considerable efforts areinvested in developing linear scaling or O(N ) methods.14, 15

    The fragmentation methods have a fairly long history16–22 andnow it is an active area of research.13, 23–32

    a)Author to whom correspondence should be addressed. Electronic mail:[email protected] Fax: +81 29 851 5426.

    One such method is the fragment molecular orbital(FMO) method.33–36 The FMO method enables the calcula-tion of electronic states for large molecules by performingMO calculations in parallel on fragments of a large molecularsystem using external electrostatic potentials (ESPs). TheseESPs describe the electrostatic field due to charge distribu-tions of all of the other fragments. MOs of individual frag-ments are local in nature, because they are expanded in termsof basis functions on the atoms in the fragments, reducing thecomputation time significantly. Additionally, in order to re-alize linear scaling, approximations have been proposed forthe time-consuming ESP calculation in FMO34, 37 and othermethods.10 The fragment self-consistent procedure employedin FMO has been used in somewhat different forms in othermethods,18, 19,9,38, 39 while the many-body expansion of theenergy is conceptually related to the theory of intermolecu-lar interactions16 or the density expansion method.20

    Over the past decade, the FMO method has becomeone of the most extensively developed fragmentation meth-ods for the calculation of accurate chemical properties, suchas the total energy, the dipole moment, and the interac-tion energy between fragments in large systems. The FMOmethod has been interfaced with many QM methods: Sec-ond order Møller–Plesset (MP2) perturbation theory,40, 41

    coupled-cluster theory,42 DFT,43, 44 Hatree-Fock, the mul-ticonfiguration self-consistent field method,45 configurationinteraction,46, 47 time-dependent DFT,48–50 open-shells51 andnuclear-electronic orbitals.52 For these methods, the total en-ergies are in good agreement with the corresponding con-ventional QM total energies. The FMO method has been ap-plied to a number of large systems.53–66 FMO–MD has beenused67–78 to study the dynamics of various processes such as

    0021-9606/2011/134(12)/124115/13/$30.00 © 2011 American Institute of Physics134, 124115-1

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  • 124115-2 Nagata et al. J. Chem. Phys. 134, 124115 (2011)

    chemical reactions. However, the FMO–MD method suffersfrom an accumulation of errors67 with the time evolution dueto an incomplete analytic FMO energy gradient.

    The analytic FMO energy gradient reported by Kitaura etal. in 2001 (Ref. 79) was incomplete since the response termdue to the ESPs was neglected assuming that it is a small con-tribution for small basis sets. It was subsequently illustratedthat when using larger basis sets, such as 6–31G(d),80 the re-sponse contribution to the gradient is substantial. Evaluatingthe response contribution requires the solution of the coupled-perturbed Hartree–Fock (CPHF) equations. These equationsare time consuming to solve for large molecules, since theydepend not on the fragment size but on the size of the entiresystem.

    Since the original report of a partially analytic FMOgradient,79 several improvements regarding previously miss-ing terms in the FMO gradient have been implemented inthe GAMESS program package:81, 82 (a) The ESP derivativeterms including the Mulliken point charge (PTC) approxi-mation (ESP–PTC),80 (b) the derivative of the dimer energyof two distant fragments approximated as the electrostaticinteraction energy (the electrostatic dimer approximation,ES-DIM),83 and (c) the hybrid orbital projection derivatives.84

    The last and most complicated missing response term isthe subject of the present paper in which the exact responseterm for the FMO ESP gradient contribution is introduced,and the fully analytic FMO energy gradient is implemented,with an additional computation time requirement that is com-parable to the first step of the FMO calculation (the self-consistent charge, SCC, iterations). The gradient implementa-tion is also extended to the ES-DIM approximation but not yetto the ESP–PTC approximation. Consequently, the usefulnessof the FMO method is improved by establishing a completeexpression of the analytic energy gradient.

    Similarly to FMO, the response term arises in other meth-ods where the external charges are not fully self-consistent.For example, in the electronically embedded method85, 86 lay-ers are introduced in the system. This is different from frag-ments because layers are mutually inclusive, i.e., lower layersinclude higher layers, and thus the lowest layer calculationdeals with the whole system, whereas the fragment based ap-proach we take uses the locality of fragments. On the otherhand in some other fragment methods such as X-Pol87 theCPHF-related terms do not arise.

    II. FULLY ANALYTIC FMO ENERGY GRADIENT

    A. Overview of the FMO energy gradient

    In the following equations, the FMO energyexpression33–36 and the corresponding gradient equa-tions are briefly summarized. The two-body FMO (FMO2)total energy is represented as

    E =N∑I

    E ′I +N∑

    I>J

    (E ′I J − E ′I − E ′J ) +N∑

    I>J

    Tr(�DI J VI J ),

    (1)

    where E ′X is the internal fragment energy for fragment X (X= I or IJ, for monomers and dimers, respectively). VI J is the

    matrix form of the ESP for dimer IJ due to the electron den-sities and nuclei of the remaining fragments, i.e.,

    V I Jμν =∑

    K �=I J

    (uKμν + v Kμν

    ). (2)

    The one-electron and two-electron integrals in V I Jμν are,respectively,

    uKμν =∑A∈K

    〈μ| −Z A|r − RA| |ν〉, (3)

    v Kμν =∑

    λσ∈KDKλσ (μν|λσ ) , (4)

    where DKλσ is the density matrix element of fragment K and(μν|λσ ) is a two-electron integral in the atomic orbital (AO)basis. Index A runs over atoms. The dimer density matrix dif-ference �DI J in Eq. (1) is defined by

    �DI J = DI J − (DI ⊕ DJ ). (5)We note that the SCC process ensures the self-

    consistency of monomer densities with respect to each otherbut not monomer and dimer densities. The direct sum inEq. (5) means blockwise addition of two monomer matricesinto the dimer supermatrix.

    The internal fragment energies in Eq. (1) may be writtenin the form:

    E ′X =∑μν∈X

    DXμνhXμν

    + 12

    ∑μνλσ∈X

    [DXμν D

    Xλσ −

    1

    2DXμλ D

    Xνσ

    ](μν|λσ )

    +∑μν∈X

    DXμν PXμν + ENRX , (6)

    where hXμν is the X-mer one-electron Hamiltonian and the nu-clear repulsion (NR) energy is

    ENRX =∑B∈X

    ∑A(∈X )>B

    Z A Z BRAB

    . (7)

    Note that monomer and dimer densities are determinedby the MO calculations in the presence of ESPs due to theremaining monomers. For fragmentation across a covalentbond, the hybrid orbital projection (HOP) contribution

    occ∑i∈X

    2 〈i | P̂ X |i〉 =∑μν∈X

    DXμν PXμν, (8)

    must be considered in Eq. (6). The HOP operator is definedby

    P̂ X =∑k∈X

    Bk |θk〉〈θk | , (9)

    where |θk〉 is a hybrid orbital and the universal constant Bk isusually set to 106.

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  • 124115-3 Fully analytic energy gradient in FMO J. Chem. Phys. 134, 124115 (2011)

    The differentiation of E ′X with respect to nuclear coordi-nate a leads to

    ∂ E ′X∂a

    =∑μν∈X

    DXμν∂hXμν∂a

    + 12

    ∑μνλσ∈X

    [DXμν D

    Xλσ −

    1

    2DXμλ D

    Xνσ

    ]∂ (μν|λσ )

    ∂a

    +∑μν∈X

    DXμν∂ P Xμν∂a

    − 2occ∑

    i, j∈XSa,Xji F

    ′Xji

    − 4occ∑i∈X

    vir∑r∈X

    U a,Xri VX

    ri +∂ ENRX∂a

    , (10)

    where the overlap derivative matrix element is defined by

    Sa,Xi j =∑μν∈X

    C X∗μi∂SXμν∂a

    C Xν j . (11)

    The internal fragment Fock matrix element F ′Xi j is givenin terms of MOs:

    F ′Xi j = hXi j +occ∑

    k∈X[2(i j |kk) − (ik| jk)] + P Xi j , (12)

    where the MO-based projection operator matrix P Xi j is intro-duced for convenience:

    P Xi j =∑μν∈X

    C X∗μi PXμνC

    Xν j . (13)

    Throughout this study, the Roman (ijkl· · ·) and Greek(μνρσ · · ·) indices denote the molecular orbitals and AOs, re-spectively. For the response term in Eq. (10), the followingequation defined in the previous study80 is introduced:

    Ua,X,Y = 4

    occ∑i∈X

    vir∑r∈X

    U a,Xri VY

    ri . (14)

    The response terms U a,Xri associated with ESP VY

    ri in theMO basis arise from the expansion of the MO coefficientderivative in terms of the MO coefficients,88 and Eq. (14)sums the U a,Xri terms multiplied by the V

    Yri terms in order

    to simplify the gradient derivations. There are two types of

    Ua,X,Y

    terms: (a) Ua,I,I

    (i.e., X = Y) arising from the deriva-tive of the monomer terms, and (b) U

    a,X,I Jwhere X can be I,

    J, or IJ [related to the three D terms in Eq. (5)]. The deriva-tives of the MO coefficients can be written as

    ∂C Xμi∂a

    =occ+vir∑

    m∈XU a,Xmi C

    Xμm . (15)

    To obtain the occupied-virtual orbital response U a,Xri , onemust solve the CPHF equations. This will be discussed in sub-sequent subsections.

    The differentiation of the ESP energy contribution inEq. (1) with respect to nuclear coordinate a leads to

    ∂aTr(�DI J VI J )

    = ∑μν∈I J

    �DI Jμν∑

    K �=I J

    [∂uKμν∂a

    +∑

    λσ∈KDKλσ

    ∂ (μν|λσ )∂a

    ]

    − 2 ∑μν∈I J

    W I Jμν∂SI Jμν∂a

    +2∑μν∈I

    W Iμν∂SIμν∂a

    + 2∑μν∈J

    W Jμν∂S Jμν∂a

    + U a,I J,I J − U a,I,I J −U a,J,I J −2∑

    K �=I J

    ∑μν∈K

    �X K (I J )μν Sa,Kμν

    + 4∑

    K �=I J

    ∑μν∈I J

    vir∑r∈K

    occ∑i∈K

    �DI JμνUa,Kri (μν|ri) ,

    (16)where

    W Xμν =1

    4

    ∑λσ∈X

    DXμλVI Jλσ D

    Xσν (17)

    and

    �X K (I J )μν =1

    4

    ∑λσ∈K

    DKμλ

    ⎡⎣ ∑

    ζη∈I J�DI Jζη (ζη|λσ )

    ⎤⎦ DKσν.

    (18)The collection of all the U

    a,X,Yterms in Eqs. (10) and

    (16) subject to the differentiation of Eq. (1) forms the follow-ing equations (N = number of fragments):

    Ua = −

    N∑I

    Ua,I,I −

    N∑I>J

    (U

    a,I J,I J − U a,I,I − U a,J,J )

    +N∑

    I>J

    (U

    a,I J,I J − U a,I,I J − U a,J,I J ). (19)U

    ais zero when no ESP approximations are applied or

    when all the ESPs are approximated uniformly (e.g., withthe ESP–PTC approximation).80 Otherwise, U

    ais only ap-

    proximately equal to zero. Ua

    is regarded as a compensationterm arising by defining the internal fragment energies andthe ESP contribution separately in Eq. (1). If U

    a = 0, then inEq. (16) the dimer-related terms U

    a,I J,I J − U a,I,I J − U a,J,I Jneed not be evaluated, and the only terms that require the so-lution of CPHF equations come from the monomers, U a,Iri .

    B. Coupled-perturbed Hartree–Fock equations in FMO

    Calculating the fully analytic FMO energy gradients(with the aforementioned conditions on the ESP approxima-tions that lead to U

    a = 0) requires only the response termsdue to the monomers, i.e., U a,Iri . These terms can be obtainedusing the diagonal nature of the monomer Fock matrices. Thepurpose of this subsection is to derive the CPHF equations forU a,Iri in the framework of the FMO method.

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  • 124115-4 Nagata et al. J. Chem. Phys. 134, 124115 (2011)

    For monomer I, the corresponding Fock matrix rewrittenin terms of MOs is

    F Ii j = F′ Ii j + V Ii j

    = h̃ Ii j +occ∑k∈I

    [2(i j |kk) − (ik| jk)] + P Ii j ,(20)

    where the one-electron Hamiltonian in FMO is

    h̃ Ii j = hIi j + V Ii j . (21)F ′Ii j is the internal Fock matrix (i.e., the full matrix F

    Ii j with

    ESP subtracted), and P Ii j is the projection operator matrix.The differentiation of Eq. (20) with respect to nuclear co-

    ordinate a is

    ∂ F Ii j∂a

    = ∂∂a

    (h̃ Ii j +

    occ∑k∈I

    [2(i j |kk) − (ik| jk)] + P Ii j)

    .

    (22)After some algebra, Eq. (22) leads to

    ∂ F Ii j∂a

    = Fa,Ii j +occ+vir∑

    k∈I

    (U a,Iki F

    Ik j + U a,Ik j F Iik

    )

    +occ+vir∑

    k∈I

    occ∑l∈I

    U a,Ikl A′i j,kl +

    ∑K �=I

    occ+vir∑k∈K

    occ∑l∈K

    U a,Kkl 4(i j |kl).

    (23)In Eq. (23) a superscript a indicates a derivative with re-

    spect to coordinate a. The (molecular) orbital Hessian contri-bution is given by

    A′i j,kl = 4(i j |kl) − (ik| jl) − (il| jk), (24)and the Fock derivative is

    Fa,Ii j = ha,Ii j + V a,Ii j +occ∑k∈I

    [2(i j |kk)a − (ik| jk)a] + Pa,Ii j .(25)

    Equation (25) is derived from Eq. (10.7) of Ref. 88.The derivative of the one-electron Hamiltonian, ha,Ii j and thederivative two-electron integral, (i j |kl)a are defined as

    ha,Ii j =∑μν∈I

    C I∗μi CIν j

    ∂hIμν∂a

    , (26)

    (i j |kl)a =∑

    μνρσ∈IC I∗μi C

    Iν j C

    I∗ρk C

    Iσ l

    ∂(μν|ρσ )∂a

    , (27)

    (see Section 4.5 of Ref. 88 for details). The ESP derivativeV a,Ii j is defined as follows:

    V a,Ii j =∑K �=I

    (ua,Ki j +

    occ∑k∈K

    2 (i j |kk)a)

    . (28)

    F Ii j is zero by virtue of the SCF equations, if i �=j. TheHOP derivative Pa,Ii j and the one-electron contribution in the

    ESP derivative ua,Ki j are defined analogously to ha,Ii j , as MO-

    transformed derivative integrals [see Eq. (26)].

    After further algebra, Eq. (23) leads to

    ∂ F Ii j∂a

    = Fa,Ii j −(ε Ij − ε Ii

    )U a,Ii j − Sa,Ii j ε Ij

    −occ∑

    k,l∈ISa,Ikl [2(i j |kl) − (ik| jl)] +

    vir∑k∈I

    occ∑l∈I

    U a,Ikl A′i j,kl

    −∑K �=I

    occ∑k,l∈K

    2Sa,Kkl (i j |kl) +∑K �=I

    vir∑k∈K

    occ∑l∈K

    U a,Kkl 4(i j |kl),

    (29)where ε Ii is the orbital energy of MO i on fragment I and thefollowing relation88 is used

    Sa,Xi j + U a,Xji + U a,Xi j = 0. (30)Equation (29) contains two types of unknown values

    U a,Xkl to be solved: Ua,Ikl for the target fragment and U

    a,Kkl from

    the remaining fragments. Therefore, it is necessary to collect∂ F Xi j /∂a of all the monomer fragments to construct and solvethe complete CPHF equations. Consequently, a set of CPHFequations have the dimension of the whole system given inmatrix form as

    AUa = Ba0, (31)where the fragment diagonal and off-diagonal blocks of ma-trix A are given in Eqs. (32) and (33), respectively,

    AI,Ii j,kl = δikδ jl(ε Ij − ε Ii

    ) − [4(i j |kl) − (ik| jl) − (il| jk)](32)

    and

    AI,Ki j,kl = −4(i j |kl), (33)where Eq. (32) comes from the MOs of only the target frag-ment I and Eq. (33) comes from the ESP acting upon I . Theij∈I element of vector Ba0 is

    Ba,I0,i j = Fa,Ii j − Sa,Ii j ε Ij −occ∑

    kl∈ISa,Ikl [2(i j |kl) − (ik| jl)]

    −∑K �=I

    occ∑kl∈K

    2Sa,Kkl (i j |kl). (34)

    C. Z-vector method in FMO

    It is impractical to solve the CPHF equations of Eq. (31)for all nuclear coordinates of a large system. To avoid thisdifficulty, the Z-vector method is applied to the FMO CPHFequations.88

    In Eq. (16), the terms involving the unknowns U a,Kri arecollected for all of the dimer fragments IJ. The resulting con-tribution to the FMO energy gradient leads to

    a = 4N∑

    I>J

    ∑K �=I J

    ∑μν∈I J

    vir∑r∈K

    occ∑i∈K

    �DI JμνUa,Kri (μν|ri)

    =∑

    K

    vir∑r∈K

    occ∑i∈K

    U a,Kri XKri , (35)

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  • 124115-5 Fully analytic energy gradient in FMO J. Chem. Phys. 134, 124115 (2011)

    where X Kri is defined as

    X Kri = 4N∑

    (I>J )�=K

    ∑μν∈I J

    �DI Jμν (μν|ri) . (36)

    It is convenient to express Eq. (35) in vector form:

    a =∑

    K

    vir∑r∈K

    occ∑i∈K

    U a,Kri XKri = XT Ua

    = XT A−1Ba0

    = ZT Ba0. (37)This reduces the CPHF problem for solving Eq. (31) to a

    set of simultaneous equations:

    AT Z = X. (38)It is still time consuming to solve Eq. (38) directly, be-

    cause it has the dimension of the entire system. However, tak-ing advantage of the decoupled nature of the FMO method, amore clever approach can be considered by separating the di-agonal (I, I ) and off-diagonal (K , I ) blocks of A in Eq. (38):

    vir∑k∈I

    occ∑l∈I

    AI,Ikl,ri ZIkl = X Iri −

    all∑K �=I

    vir∑k∈K

    occ∑l∈K

    AK ,Ikl,ri ZKkl , (39)

    or in matrix form: (AI,I

    )TZI = X′I , (40)

    where

    X′I = XI −∑K �=I

    (AK ,I

    )TZK . (41)

    Equation (37), using definitions in Eqs. (34), (38), (40), and(41) gives the final formulation of the terms that are requiredto complete the FMO analytic gradient. The solution of theseequations is accomplished as follows, illustrated in Fig. 1.By taking the I diagonal blocks of matrix A and solving(AI,I

    )TZI = XI , one finds the initial ZI . X′I is computed

    to solve Eq. (40). The external unknowns Z Kkl in the last termof Eq. (39) are frozen, and this equation is decoupled for eachfragment I. Z is obtained by solving Eq. (40) for all fragmentsindependently, and then X′I is updated with these new valuesof ZK for the next step; the calculations are then repeated untilall elements in the Z-vector are self-consistent. This is simi-lar in both procedure and computational cost to the SCC pro-cedure in the monomer energy calculation (and it also bearsa similarity to SCF), so it will be called the self-consistentZ-vector (SCZV) method. The preconditioned conjugate gra-dient method is applied to solve Eq. (40). The convergencetest is made for all Z-vector elements. If the root-mean-squaredeviation of ZI,new−ZI,old is larger than a threshold, the pro-cedure returns to step 2 in Fig. 1.

    The SCZV method is parallelized using the generalizeddistributed data interface (GDDI).89 Because of its iterative

    FIG. 1. Schematic diagram of the SCZV procedure.

    decoupled nature, the computation time of SCZV is compara-ble to that of SCC.

    D. Application to the electrostaticdimer approximation

    The previous subsection presented a derivation in whichthe analytic FMO gradient was derived with no approxima-tions. In this subsection, the electrostatic dimer (ES-DIM) ap-proximation for the fully analytic energy gradients is intro-duced, and it is shown that the response terms arising fromthe ES-DIM approximation need not be considered, becausethey cancel out with the response term from Eq. (19).

    For separated fragments I and J, if the distance RI J (de-fined as the distance between the closest atoms in I and J di-vided by the sum of their van der Waals atomic radii) is largerthan the threshold value LES−DIM in the ES-DIM approxima-tion, the internal pair interaction energy, i.e., the correspond-ing summand in the second sum on the right-hand side of Eq.(1) can be replaced by

    E ′I J − E ′I − E ′J ≈ Tr(DI uJ ) + Tr(DJ uI )+

    ∑μν∈I

    ∑λσ∈J

    DIμν DJλσ (μν|λσ ) + �ENRI J ,

    (42)

    where the NR term �ENRI J = ENRI J − ENRI − ENRJ . In addition,the corresponding IJ term in the third sum of Eq. (1) can-cels out because �DI J = 0. The differentiation of Eq. (42)with respect to nuclear coordinate a without considering theresponse term is given elsewhere.83 Therefore, it is sufficienthere to discuss how the unknown orbital response terms in theFMO gradient are formulated. The collection of the response

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  • 124115-6 Nagata et al. J. Chem. Phys. 134, 124115 (2011)

    terms in the FMO gradient yields

    Ua + a = −

    N∑I

    Ua,I,I −

    N∑I>J

    (U

    a,I J,I J − U a,I,I − U a,J,J )

    +N∑

    I>J

    (U

    a,I J,I J − U a,I,I J − U a,J,I J )

    + 4N∑

    I>J

    ∑K �=I J

    ∑μν∈I J

    vir∑r∈K

    occ∑i∈K

    �DI JμνUa,Kri (μν|ri) .

    (43)

    As mentioned before, Ua = 0 only if there are no approx-

    imations to the ESPs. In the case of the ES-DIM approxima-tion, Eq. (43) should be reformulated.

    When the ES-DIM approximation is applied to avoidSCF calculations for dimers separated by more than LES−DIM,U

    ain Eq. (19) can be rewritten as

    Ua = −

    N∑I

    Ua,I,I

    −N∑

    I>J (RI J ≤LES−DIM)

    (U

    a,I J,I J − U a,I,I − U a,J,J )

    +N∑

    I>J (RI J ≤LES−DIM)

    (U

    a,I J,I J − U a,I,I J − U a,J,I J )

    = −N∑I

    Ua,I,I +

    N∑I>J (RI J ≤LES−DIM)

    (U

    a,I,I (J ) + U a,J,J (I ))�= 0, (44)

    where the partial terms

    Ua,X,X (Y ) = 4

    occ∑i∈X

    vir∑r∈X

    U a,Xri(uYri + vYri

    ), (45)

    describe the contribution of a single fragment denoted Y to the

    full sum over all Y in Ua,X,X

    .Also, the following relation can be used:

    Ua,I,I − U a,I,I J = U a,I,I (J ), (46)

    because [see Eq. (14)] the ESP for I J (V I J ) runs over allfragments excluding I and J, whereas the ESP for I excludesthe contribution from I. Therefore, in the difference expres-sion only J terms remain:

    V Iri − V I Jri = V I,I (J )ri ≡ u Jri + v Jri . (47)U

    ain Eq. (44) is in general nonzero and can be further

    simplified to

    Ua = −

    N∑I

    Ua,I,I +

    N∑I>J (RI J ≤LES−DIM)

    (U

    a,I,I (J ) + U a,J,J (I ))

    = −N∑

    I>J (RI J >LES−DIM)

    (U

    a,I,I (J ) + U a,J,J (I )), (48)

    where the completeness relation is used:

    N∑I

    Ua,I,I =

    N∑I

    N∑J �=I

    Ua,I,I (J ) =

    N∑I>J

    (U

    a,I,I (J ) + U a,J,J (I )).(49)

    For the derivatives of Eq. (42) the collection of responseterms for all IJ is

    N∑I>J (RI J >LES−DIM)

    [Tr

    (∂DI

    ∂auJ

    )+ Tr

    (∂DJ

    ∂auI

    )

    +∑μν∈I

    ∑λσ∈J

    ∂ DIμν∂a

    D Jλσ (μν|λσ )

    +∑μν∈I

    ∑λσ∈J

    DIμν∂ D Jλσ∂a

    (μν|λσ )]

    =N∑

    I>J (RI J >LES−DIM)

    (U

    a,I,I (J ) + U a,J,J (I ))

    −2N∑

    I �=J (RI J >LES−DIM)

    occ∑i j∈I

    Sa,Ij i(u Jji + v Jji

    ). (50)

    The first term on the right-hand side of Eq. (50) cancelsout with U

    ain Eq. (48). Since the derivative terms of Eq. (42)

    are already implemented,83 one can obtain the fully analyticenergy gradients for the ES-DIM approximation by calculat-ing the following response contribution to the gradient:

    Ua + a

    = 4N∑

    I>J (RI J ≤LES−DIM)

    ∑K �=I J

    ∑μν∈I J

    vir∑r∈K

    occ∑i∈K

    �DI JμνUa,Kri (μν|ri) .

    (51)

    E. Implementation

    To further facilitate solution of the SCZV equations,it is useful to reformulate them in the AO basis, therebyavoiding the expensive integral transformation.90 ExaminingEqs. (39), (32), and (33), the main computational effort isspent on two-electron integral terms like

    ∑kl(kl|ri)Z Kkl in

    Eq. (39). By utilizing the MO i expansion over AOs μ,

    |i〉 =∑

    μ

    Cμi |μ〉 . (52)

    The two-electron integral terms in Eq. (39) can be trans-formed according to

    vir∑k∈K

    occ∑l∈K

    (kl|ri)Z Kkl =vir∑

    k∈K

    occ∑l∈K

    C K∗μk CKνl (μν|ri)Z Kkl

    =∑μν

    (μν|ri)Z̃ Kμν, (53)

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  • 124115-7 Fully analytic energy gradient in FMO J. Chem. Phys. 134, 124115 (2011)

    where

    Z̃ Kμν =vir∑

    k∈K

    occ∑l∈K

    C K∗μk ZKkl C

    Kνl . (54)

    Using Eq. (53) avoids the full transformation of the two-electron integrals to the MO basis, which leads to a significantreduction of the computation time and memory. However, Z̃ Iμνis not symmetric; this is inconvenient and can be further im-proved by symmetrizing it:

    ZIμν =

    1

    2

    (Z̃ Iμν + Z̃ Iνμ

    ). (55)

    It is possible to rewrite the SCZV equations [Eq. (40)]

    using the symmetrized local Z-vector element ZKμν . Z

    Kμν cor-

    responds to the density matrix element Dμν in typical inte-gral programs, and therefore the SCZV method can be imple-mented using standard CPHF codes.

    The Cauchy–Schwarz inequality, which estimates thevalue of (μν|ρσ ) based on the values of a smaller set ofintegrals with repeated indices, can be used in the SCZVto obtain a further reduction of computation time. TheCauchy–Schwarz inequality screening is usually applied to

    the maximum element of the factor by which the integralsof interest are multiplied. The SCZV procedure is more ef-ficient because the Z-vector elements normally have smallervalues than the density matrix elements, and therefore, theCauchy–Schwarz integral screening can skip more terms. Inthe implementation of the response contribution, Eq. (37),the direct (AO basis) algorithm and the Cauchy–Schwarzinequality for the calculation of the two-electron integralsare included. As a result, the calculations do not need a largeamount of disk space or memory.

    III. COMPUTATIONAL DETAILS

    To verify that the response contribution that has beenderived and implemented in this study makes the FMO en-ergy gradients fully analytic, analytic gradients are comparedwith numerical gradients for molecular systems taken fromprevious studies:80, 91, 92 (H2O)64, the α-helix conformationof the alanine decamer (ALA)10 capped with –OCH3 and–NHCH3 groups, chignolin (PDB ID: 1UAO) solvated by157 water molecules, and the Trp-cage miniprotein construct(PDB ID: 1L2Y). The structures of (ALA)10 and 1L2Y were

    FIG. 2. Geometric structures of (a) (H2O)64, (b) (ALA)10 capped with CH3CO– and –NHCH3 groups, (c) chignolin solvated by 157 water molecules, and (d)1L2Y [colored by chemical elements as light grey (H), dark grey (C), blue (N), and red (O)].

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  • 124115-8 Nagata et al. J. Chem. Phys. 134, 124115 (2011)

    taken from earlier works.83, 92 Numerical gradients werecomputed with double differencing and a coordinate step of0.005 Å [except for (H2O)64 with 6–311G(d), where a 0.0005Å step was used].

    For the fragmentation of a system that does not requirefragmenting covalent bonds, such as a water cluster, the HOPoperator is not needed. For the first test calculation, a watermolecule in (H2O)64 is assigned to a fragment. The geomet-rical structure of (H2O)64 was modeled by HYPERCHEM, op-timized by AMBER94,93 and then reoptimized at the FMO-RHF/6–31G level.80 The optimized structure of (H2O)64 isdepicted in Fig. 2(a).

    For molecular systems in which fragmentation occursacross covalent bonds, the hybrid orbital operator contributesto the gradients both directly and via the response term fromthe Fock derivative, Eq. (25). The α-helix conformation of(ALA)10 with some intramolecular hydrogen bonds [shownin Fig. 2(b)] is chosen because in the case of incomplete ana-lytic gradients it is expected to have large errors in the gradi-ents compared to the β-strand or extended structures. Becausethis system is relatively small, the validity of the analytic gra-dients without approximations can also be accessed.

    The FMO method has been interfaced with the ef-fective fragment potential (EFP) method, which explicitlytreats solvent molecules by adding a one-electron potentialto the Hamiltonian.91, 94 Chignolin is immersed in 157 watermolecules described by EFPs, and its structure is optimizedusing the combined FMO/EFP method at the RHF/cc-pVDZlevel.91 The optimized structure in Fig. 2(c) reproduces thePDB NMR structure well.80

    Figure 2(d) depicts the structure of 1L2Y. Because 1L2Yis the largest system treated with the FMO method in thisstudy, it is an appropriate test case for the application of theelectrostatic dimer approximation which is necessary for effi-cient computations of large systems. In this calculation, thethreshold for the ES-DIM approximation LES−DIM was setto the default value, 2.0 and all other approximations wereturned off.34

    For systems fragmented across covalent bonds, i.e.,(ALA)10, chignolin and 1L2Y, a one residue/one fragmentpartition was adopted. The gradient calculations for (H2O)64,(ALA)10, hydrated chignolin, and 1L2Y were performed atthe RHF/6–31G(d) level and diffuse functions were added tothe carboxyl groups of 1L2Y. Additionally, RHF/6–311G(d)was also used to calculate the gradients for (H2O)64.

    The computation time of the SCZV procedure is ex-pected to be comparable to that of the SCC calculation. Thus,the timings of the calculations of fully analytic gradients inwhich both the SCC and SCZV steps were included weremeasured and compared. The timing calculations used the 32

    TABLE I. The number of GDDI groups used in dividing 31 nodes.

    Monomer step Dimer step

    (H2O)64 16 16(ALA)10 5 15Hydrated chignolin 5 151L2Y 10 15

    node SOROBAN cluster with Intel Pentium 4 central process-ing unit (CPU) 3.2 GHz nodes connected by Gigabit Ethernet.Table I lists separately for monomers and dimers the numberof GDDI groups in the gradient calculation for each system(in GDDI, in order to improve parallel efficiency, all computernodes are divided into groups and individual monomer anddimer calculations are distributed dynamically among thesegroups). Note that the number of GDDI groups in the SCZVstep is the same as in the SCC step. The parallel efficiency iscalculated for the gradient and SCZV calculations for (H2O)64using 1, 2, 4, 8, 16, and 32 nodes of the SOROBAN cluster. Forthis calculation, the number of GDDI groups is equal to thenumber of nodes.

    IV. RESULTS AND DISCUSSION

    A. Fully analytic energy gradients withoutapproximations

    In this subsection, the fully analytic gradients withoutapproximations are discussed. It is important to numericallyverify that the gradients are fully analytic using U

    a = 0[Eq. (19)].

    As mentioned previously, for (H2O)64, the energy gradi-ents do not involve the HOP term and only the response termcontributes to the analytic gradients. In Table II, the root-mean-square (RMS) value in the numerical gradients becameidentical with the new analytic gradients (0.005997 a.u.).However, the RMS value in the old (conventional) analyticgradients, in which only the response contribution is ne-glected, deviates by 0.0001 a.u. from the numerical and theanalytic ones. For the maximum absolute gradient values(MAX grad.), the new analytic gradient value is in goodagreement with the corresponding numerical value, while theconventional value differs by 0.00025 a.u. The latter is nota negligible error when aiming for fully analytic gradients.The RMS of the errors for the new analytic gradient relativeto the numerical gradient (the RMS error in the analyticgradients) is negligibly small (0.000011 a.u.), which is animprovement by a factor of 20 over the RMS error in theold analytic gradients (0.000231 a.u.). This improvementis visualized in Fig. 3(a), which plots the errors of the newanalytic gradient and the old analytic gradient relative to thenumeric gradient against the total 576 gradient elements. Thenew analytic gradient values converge to zero, while the oldanalytic gradient values have large deviations.

    For systems fragmented across covalent bonds, theHOP contribution to the response term must be considered.(ALA)10 is such a system and due to its size is a good test forcalculating numerical gradients without approximations at arelatively moderate computational cost. Note that the old an-alytic gradient method that is illustrated here did include thedirect contribution of HOP derivatives to the gradients, sincethis contribution was introduced in a previous study.84 Table IIshows that the three types of gradients are in good agreement.Even though the RMS value in the old analytic gradients isvery close to that of numerical gradients, the RMS errors andmaximum absolute gradient error for the old analytic gradientdeviates by significant amounts (0.000160 and 0.000580a.u., respectively). On the other hand, the very small errors

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  • 124115-9 Fully analytic energy gradient in FMO J. Chem. Phys. 134, 124115 (2011)

    TABLE II. The RMS and the maximum absolute values (MAX grad.) of the gradient elements for FMO2–RHF.RMS of the errors of the analytic gradients relative to the numeric gradients (RMS error) and the error in themaximum absolute gradient values (MAX error). All values are in a.u.

    Gradient RMS MAX grad. RMS error MAX error

    (H2O)64 without approximations, 6–31G(d)Numeric 0.005997 0.014090 . . . . . .Analytic 0.005997 0.014094 0.000011 0.000035Conventional 0.006132 0.014347 0.000231 0.000961

    (H2O)64 with LES−DIM = 2.0, 6–31G(d)Numeric 0.005997 0.014090 . . . . . .Analytic 0.005997 0.014094 0.000011 0.000034Conventional 0.006132 0.014347 0.000232 0.000975

    (H2O)64 with LES−DIM = 2.0, 6–311G(d)Numeric 0.007467 0.018020 . . . . . .Analytic 0.007469 0.018029 0.000003 0.000010Conventional 0.007745 0.018507 0.000433 0.002280

    (ALA)10 without approximations, 6–31G(d)Numeric 0.011658 0.043197 . . . . . .Analytic 0.011664 0.043222 0.000009 0.000039Conventional 0.011655 0.043169 0.000160 0.000580

    Hydrated (EFP) chignolin without approximations, 6–31G(d)Numeric 0.000284 0.002010 . . .Fully analytic 0.000280 0.001995 0.000017 0.000092Conventional 0.000215 0.001542 0.000191 0.001501

    1L2Y with LES−DIM = 2.0, 6–31G(d)Numeric 0.019779 0.047540 . . . . . .Analytic 0.019780 0.047543 0.000015 0.000037Conventional 0.019822 0.047418 0.000335 0.001001

    in the new analytic gradient values indicate that they arefully analytic, and that the HOP contribution to the responseterm is properly included. In Fig. 3(b), as in Fig. 3(a), onecan see that the new analytic gradients are fully analytic forevery gradient element. Comparing Fig. 3(b) ((ALA)10) withFig. 3(a) (the water cluster), the former has smaller gradienterrors for the old analytic gradients. Since the response termis related to the ESPs, the environmental potential affects thewater clusters more significantly than the polypeptide.

    Most biological processes occur in aqueous solution. Tomodel one of these processes, one may consider a protein ina water droplet. For molecular dynamics and geometry opti-mization processes, an accurate solvent model must be com-bined with the application of FMO to the solute. As the firsttest calculation toward this goal, hydrated chignolin is chosenfor the FMO/EFP framework. In order to obtain the fully ana-lytic energy gradients in this framework, it is necessary to addthe EFP derivatives to the Fock derivative term, Eq. (25) andprobably to modify the A matrix in Eq. (38). In this study,however, the EFP-related many-body polarization contribu-tion to the FMO ESP response term was neglected. There-fore, the FMO/EFP energy gradients are not fully analytic. So,the accuracy of the FMO/EFP new energy gradients is shownonly for the solute (FMO) molecule (chignolin). As seen inTable II, the accuracy of the FMO/EFP new energy gradientsis similar to that for the full treatment of FMO, but the max-imum absolute gradient error is larger (0.000092 a.u.) thanthose for the other test systems, and there are slight deviations

    in the FMO/EFP new analytic gradients in Fig. 3(c). Never-theless, the maximum absolute new analytic gradient error isimproved by a factor of 16 compared to the old analytic gradi-ent value. This encouraging result provides motivation to de-velop the fully analytic energy gradients for FMO/EFP as wellas FMO combined with the polarizable continuum model.92, 95

    For the chosen systems, the wall clock times requiredfor the SCC, the SCZV, and the total FMO calculation weremeasured. Table III shows that for the gradient calculationwithout approximations for (H2O)64, the SCC calculationtook 24.3 s, which is comparable to 29.5 s. in the correspond-ing SCZV calculation. These times are small relative to thetotal computational time of over 400 s. For (ALA)10, theSCZV calculation takes only 64% of the computation time ofthe corresponding SCC calculation. For hydrated chignolin,the SCZV calculation requires a similar percentage of thecomputation time. These results illustrate why the Z-vectorsconverge more rapidly; the calculation of two-electronintegrals is faster because the Cauchy–Schwarz inequalityis more efficient when applied to the Z-vectors rather thanto the monomer densities (used in ESPs or two-electronintegrals for monomers).

    B. Fully analytic energy gradients in theES dimer approximation

    The FMO energy gradients have been shown to be fullyanalytic by introducing the response contribution. The com-putation time in the calculation of the response term is com-

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  • 124115-10 Nagata et al. J. Chem. Phys. 134, 124115 (2011)

    FIG. 3. Errors of the analytic gradient elements relative to the numeric gradient elements for (a) (H2O)64, (b) (ALA)10 capped with CH3O– and –NHCH3groups, (c) chignolin solvated by 157 water molecules, (d) 1L2Y, all calculated at the RHF/6–31G(d) level, and (e) (H2O)64 at the RHF/6–311G(d) [all thegradient elements for (a)–(c), (e), and the representative elements for (d)]. Black diamond: fully analytic energy gradients. Yellow square: the conventionalgradients.

    parable to or less than that required for the SCC calculation.This implies that although it is practical to calculate the re-sponse term itself, it is still expensive to calculate the fullyanalytic energy gradients for a larger molecule because of thedifficulty in performing a large number of dimer SCF calcu-lations without approximations.

    As discussed above, the fully analytic energy gradientswith the ES-DIM approximation have been derived. The pur-

    pose of this subsection is to check numerically that the FMOenergy gradients are fully analytic within the ES-DIM approx-imation and to discuss the timings.

    The ES-DIM approximation was first applied to (H2O)64using both the 6–31G(d) and 6–311(d) basis sets. Table IIshows that with LES−DIM = 2.0, the RMS value, the MAX gra-dient value, and the RMS error for the new analytic gradientsare identical to those without approximations. Additionally,

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  • 124115-11 Fully analytic energy gradient in FMO J. Chem. Phys. 134, 124115 (2011)

    TABLE III. Wall clock time in the SCC, SCZV, and the total computa-tion using a 31 single 3.2 GHz Pentium 4 cluster (all in seconds), FMO2-RHF/6–31G(d).

    SCC SCZV TOTAL

    (H2O)64 without approximationsAnalytic 24.3 29.5 467.2Conventional 23.0 . . . 431.4

    (H2O)64 with LES−DIM = 2.0Analytic 29.4 28.1 261.0Conventional 24.3 . . . 227.8

    (ALA)10 without approximationsAnalytic 359.7 230.4 1240.3Conventional 358.4 . . . 1016.3

    hydrated chignolin without approximationsAnalytic 1383.6 883.2 7299.8Conventional 1382.4 . . . 6419.2

    1L2Y with LES−DIM = 2.0Analytic 3429.1 2320.7 12942.1Conventional 3380.4 . . . 10597.1

    the maximum gradient error is negligibly small. These resultsimply that the ES-DIM approximation is a suitable choice forthis system.

    The errors for the old analytic gradient increase withthe basis set, as can be seen in Table II. For example,the maximum gradient error increases from 0.000975 to0.002280 when going from 6–31G(d) to 6–311(d). Theerror in the new analytic versus numeric gradient is about15 times smaller and is probably due to the error in thenumerical gradient itself. There is also no basis set effect onthe error when the new analytic gradient is employed. Thepictorial representations of the errors in Figs. 3(a) and 3(e)demonstrate the quality of the gradients.

    For another system requiring fragmentation across cova-lent bonds, consider 1L2Y, which consists of 20 amino acidresidues. Since the calculation of numerical gradients for sucha relatively large molecule is time consuming, a subsystemis chosen, i.e., the 38 atoms defining 19 detached bonds be-tween 20 fragments (for which atoms the error in the old an-alytic gradient is the largest as found in the earlier study84).With LES−DIM = 2.0, there are 92 SCF and 98 approximated(ES-DIM) dimers. In Table II [see also Fig. 3(d)], the cor-responding RMS value and the maximum absolute gradientvalue show that the new analytic gradients are fully analytic incomparison to the numerical values. The corresponding RMSand MAX gradient errors are similar in accuracy to the fullyanalytic gradient values of the other systems. The errors aremuch improved compared to those from the old analytic gra-dient calculation, especially for the MAX gradient which ismore accurate by a factor of 27, indicating a significant con-tribution from the response term in this system.

    As shown in Table III, for (H2O)64, the total wall clocktime for the fully analytic gradient calculation with the ap-proximation (261.0 s) is less than the timing without approx-imations (467.2 s). In comparing the SCZV and the SCC cal-culation for this water cluster, it is reasonable that the formertook less computation time. When using the approximation,

    FIG. 4. The parallel efficiency using the personal computer cluster of 1, 2,4, 8, 16 and, 32 CPUs (Intel Pentium 4 3.2 GHz). Solid line: for the totalgradient calculation. Dashed line: for the SCZV calculation.

    the ratio of the computation time in SCZV to SCC is normallyless than 1. This implies that the SCZV calculation is faster;this trend is independent of the ES-DIM approximation. For1L2Y, the SCZV calculations took approximately 70% of thetime required for the SCC calculations, and it is expected thatthis ratio will decrease for larger systems.

    For (H2O)64, the parallel efficiency S(n), shown in Fig. 4,was calculated using the following expression:

    S(n) =T1Tn

    n× 100, (56)

    where T1 and Tn represent the computation time using 1node and n nodes, respectively. Hundred percent efficiencymeans that the calculation is n times faster on n nodes. Theresults show that for (H2O)64 the parallel efficiency is over97.8% for all numbers of nodes (n = 1, 2, 4, 8, 16, and 32)both in the total gradient calculation (solid line) and SCZVcalculation (dashed line). For n = 16, the parallel efficiencyin the SCZV calculation drops slightly from that for n = 8.For (H2O)64, 64 SCZV calculations should be distributedinto 16 nodes (divided into 16 GDDI groups). However, eachSCZV calculation takes a different amount of time, becausethe number of iterations and the integral screening dependupon the fragment; the GDDI calculations are dynamicallydivided over groups, and there is some granularity (i.e.,some groups finish ahead of others). This is why the parallelefficiency drops at n = 16. The superlinear scaling over 100%for a small number of nodes is also observed in FMO–RHFcalculations89 and is thought to originate from externalfactors like CPU cache efficiency.

    V. CONCLUSIONS

    The CPHF equations and the Z-vector equations in theFMO framework have been derived to compute the fully an-alytic energy gradients. One outcome of this study is thederivation and implementation of the SCZV equations, bywhich the time-consuming Z-vector equations of the wholesystem reduce to those of the monomer fragments. Addi-tionally, the SCZV procedure is parallelized by the use of

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  • 124115-12 Nagata et al. J. Chem. Phys. 134, 124115 (2011)

    GDDI.89 It was shown that the FMO energy gradients arefully analytic in the electrostatic dimer approximation. Thisleads to a significant reduction of the total computation time.Nearly fully analytic energy gradients have been successfullyimplemented for the combined FMO/EFP method as well.

    The use of the SCZV procedure is not limited to thegradient calculation. The calculation of the second derivative(Hessian) matrix is necessary to calculate important prop-erties such as the IR spectrum, Raman spectrum, and NMRchemical shifts. To calculate the fully analytic Hessian, onemust solve the CPHF equations, and the SCZV procedurewould play an essential role. The SCZV procedure could alsobe found useful in other fragmentation methods that requirethe response term (some methods87,96 do not need it withrespect to the field, although the Mulliken charge derivativesdo require it97,80).

    The analytic gradient equations have been derived atthe Hartree–Fock level of theory. For proteins, dispersionis crucial for determining the folded structure58 and forprotein-ligand binding. Therefore, the fully analytic energygradient should be extended to electron correlation methodssuch as MP2. For practical calculations, such as FMO–MDto study protein folding, it will be necessary to develop thefully analytic gradient with the point charge (ESP–PTC)approximation.

    As mentioned earlier, MD simulations are an importantapplication of FMO; however, FMO–MD has been limitedmainly to molecular clusters.70–72, 75–78 As found in this study,with the introduction of fully analytic energy gradients, re-liable MD simulation with perfect energy conservation willnow be possible.

    ACKNOWLEDGMENTS

    This work has been supported by the Next GenerationSuper Computing Project, Nanoscience Program (MEXT,Japan), and by a US National Science Foundation PetascaleApplications grant. K.B. is supported by a US Department ofEnergy Computational Science Graduate Fellowship.

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