Hybrid Quantum Mechanics / Molecular Mechanics (QM/MM) Approaches -Treatment of the electrostatic QM/MM interface - Mauro Boero Institut de Physique et Chimie des Matériaux de Strasbourg University of Strasbourg - CNRS, F-67034 Strasbourg, France and @Institute of Materials and Systems for Sustainability, Nagoya University - Oshiyama Group, Nagoya Japan 1
Treatment of the electrostatic in the QM/MM interface - Errors in the QM/MM Interface (OECP) - QM/MM interface: 3 level(s) coupling Hamiltonian - MM polarization 2
From Gas-phase to complex environment Molecules in the Solids and liquids gas phase good properties reproduced using periodic boundary conditions (PBC) No environment -Structure (radial distributions) -Dynamics (diffusion) Complex disordered systems No periodicity Partitioning of the system: QM/MM 3
QM/MM Mixed Quantum-Classical QM/Interface Partitioning the system: shopping list QM/MM 1. chemical active part treated by QM methods 2. large environment that is modeled by a classical force field (MM) 3. Interface between QM and classical parts A. Laio, J. VandeVondele, and U. Rothlisberger, J. Chem. Phys . 116 , 6941 (2002); 4 A. Laio, J. VandeVondele, and U. Rothlisberger, J. Phys. Chem. B , 106 , 7300, (2002);
QM/MM Mixed Quantum-Classical Approximations in QM/MM Partitioning of a QM system into 2 parts A and B: The non-linear (NL) correction and are unknown . where we use the “nuclear density”: 5
QM/MM Mixed Quantum-Classical Approximations in QM/MM QM/MM description of the subsystem B Point charge representation of MM atoms at fix MM geometry Because of the breakdown of the point charge representation: 2-, 3- and 4-body terms are needed: 6
QM/MM Mixed Quantum-Classical Approximations in QM/MM QM description of A & MM description of B We collect all the approximations into the energy term 7
QM/MM Mixed Quantum-Classical Approximations in QM/MM This term is small when - the electronic density is well localized within QM MM ( and are small), and QM crossing - when we have a good force field for the MM par t bond ( is small) We distinguish the two cases no bonds crossing the QM/MM boundary: bonds crossing the QM/MM boundary: A. Laio, J. VandeVondele, and U. Rothlisberger, J. Chem. Phys . 116 , 6941 (2002); 8 A. Laio, J. VandeVondele, and U. Rothlisberger, J. Phys. Chem. B , 106 , 7300, (2002); review in : M. Boero, Lect. Notes Phys. 795 , pag. 81-98, Springer, Berlin Heidelberg 2010
QM/MM Mixed Quantum-Classical Approximations in QM/MM Linking E CORR ( R I ,[ ]) We parameterize the potential using atom atom centered potentials, i.e. centered on the link atoms . E CORR ( R I ,[ ]) drdr ' i OECP ( r R I , r ' R I ) i ( r ') * ( r ) V I MM where QM l * ( r ') OECP ( r , r ') Y lm ( r ) p l ( r ) 1 V I p l ( r ') Y lm m l with Gaussian-type projectors , p l ( r ) r l exp( r 2 /(2 2 2 )) The parameters are determined through a fitting procedure. 9 O.A. von Lilienfeld, I. tavernelli, U. Rothlisberger, J. Chem. Phys . 122 , 014113 (2005)
QM/MM Mixed Quantum-Classical Approximations in QM/MM According to the Hohenberg-Kohn theorem Linking “ the external potential is determined, within a trivial atom additive constant , by the electron density . This also determines the ground state wave function and all other electronic properties of the system” MM In our QM/MM scheme we optimize the parameters QM of the atom-centered potential in order to better reproduce the real quantum density in the QM volume (obtained using a full QM description of the total system). Thus we minimize the penalty function: 10 O.A. von Lilienfeld, J. Chem. Phys . 122 , 014113 (2005)
QM/MM Mixed Quantum-Classical electronic density along z-axis z Distance along the z-axis [Å] 11
QM/MM Mixed Quantum-Classical Approximations in QM/MM Example: Acetic acid (Box size: 8 Å, gas phase, 80 Ry PW cutoff) H O H O H D H O O H Dipole [D] ESP (CH 3 /D) ESP (C 1 ) 0.00 0.74 1.67 HOOC 1 -CH 3 [-0.30 + 3*0.1] HOOC 1 -D conv 2.87 0.32 0.28 (1V) HOOC 1 -D opt 1.41 - 0.03 0.54 (DCACP) 12 O.A. von Lilienfeld, I. tavernelli, U. Rothlisberger, J. Chem. Phys . 122 , 014113 (2005)
QM/MM Hamiltonian coupling additive scheme (just a reminder) i )},{R I }] H MM [{ R I }] H tot [{ i ( r i )},{ R I }] H DFT [{ i ( r i )};{ R I }] H int [{ i ( r 13
QM/MM Hamiltonian coupling: Additive scheme Scaling in a (not only) plane wave implementation: i )},{R I }] H MM [{ R I }] H tot [{ i ( r i )},{ R I }] H DFT [{ i ( r i )};{ R I }] H int [{ i ( r H DFT [{ i ( r i )};{ R I }]: QM-part: Hartree and xc interaction O ( N el N G N G ) QM/MM interface O ( N cl N G ) H int [{ i ( r i )},{R I }]: MM part: classical (ff) potential O ( N cl N cl ) H MM [{ R I }]: N is the number of plane waves or basis - set functions G N is the number of classical atoms cl 14
QM/MM Hamiltonian coupling: Electrostatics Interaction Hamiltonian N cl d 3 r ( r ) H int ( r ),{ q I , R I } q I r R I I 1 Potential acting on the QM electronic density N cl E int [ ( r ),{ q I , R I }] q I V int ( r ) ( r ) r R I I 1 Forces acting on the MM charged atoms: int E [ ( r ), { q , R }] ( r ) 3 int I I q d r r R F I I I 3 R r R I I 15
QM/MM Hamiltonian coupling: Electrostatics The nested sums (over the classical MM atoms and over the discretized QM volume) are in general computationally very expensive N cl d 3 r ( r ) H int ( r ),{ q I , R I } q I r R I I 1 N cl can be of the order of 100'000 - 500'000 N grid can be of the order of 200 200 200 16
QM/MM Hamiltonian coupling: The 3 regions scheme Region 1: NN = subset of classical atoms inside the region NN ( r ) 0 R I r d 3 r q I , r R I 1 I 1 Region 2: Classical-RESP charges interaction RESP ( , R I ) q J 1 R I r q I , r R I R J 2 I NN J QM Region 3: Multipolar expansion on MM charges [ ( r )] r R I R I r q I , 2 3 r R I I NN A. Laio, J. VandeVondele, and U. Rothlisberger, J. Chem. Phys . 116 , 6941 (2002); 17 A. Laio, J. VandeVondele, and U. Rothlisberger, J. Phys. Chem. B , 106 , 7300, (2002); review in : M. Boero, Lect. Notes Phys. 795 , pag. 81-98, Springer, Berlin Heidelberg 2010
QM/MM Hamiltonian coupling: The 3 regions scheme Divide the world in 3 domains: 1) Close to the QM region ( r < r 1 ) 2) Not too far, i.e. ESP region ( r 1 < r < r 2 ) 3) Far MM world ( r > r 2 ) 1 r 2 r Generally we test: However in all the known cases it is r 1 ~ 10-12 a.u. r 2 ~ 20-25 a.u. Only NN < MM atoms in this shell 18
QM/MM Hamiltonian coupling R1: The direct coupling R1: Direct coupling – the screened Coulomb potential ( r ) E [ ( r ),{ R I }] ff d 3 r q I r R I I NN To avoid incompatibilities due to QM (electronic) vs. classical (point charges) description of the electrostatic, we introduce the screened Coulomb potential E [ ( r ),{ R I }] ( r ) v I ( r R I ) ff d 3 r q I I NN n r n R cI where v I ( r ) n 1 r n 1 R cI n R cI is a “covalent” radius of atom I, and n is an integer (n=3) 19
QM/MM Hamiltonian coupling R2: D-RESP region R2: D-RESP: Dynamical-Restrained ElectroStatic Potential derived charges • Define atomic point charges by fitting their value to the electrostatic potential (ESP) due to the QM charge density seen by the close MM atoms • A restrain penalty function (RESP) is included, since unphysical charge fluctuations have been observed in unrestrained ESP charges during dynamics. Namely, we minimize the norm 2 D q I D q I 2 V J w q H q I R I R J J NN I QM I QM D = q I q I RESP ESP RESTRAIN 20
QM/MM Hamiltonian coupling R2: D-RESP region 2 D q I D q I 2 V J w q H q I R I R J J NN I QM I QM is minimized on the fly during the dynamics. w 0 . 10 0 . 25 w q = weight parameter to reduce charge fluctuations: q The potential V J is the Coulomb potential generated on the MM atom J by the electronic density distribution V J d 3 r ( r ) u r R J where u (| r - r J |) is a Coulomb potential modified at short range to avoid spurious over-polarization effects. 21
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