hybrid quantum mechanics / molecular mechanics (qm/mm ... · vj j nn wqqj h j qm note that a...

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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

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

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From Gas-phase to complex environment

Molecules in thegas phase

Solids and liquidsgood properties reproduced using periodic boundary conditions (PBC)

-Structure (radial distributions)-Dynamics (diffusion)

Complex disordered systems

No periodicityPartitioning of the system: QM/MM

No environment

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QM/MM Mixed Quantum-Classical

Partitioning the system: shopping list

1. chemical active part treated by QM methods

2. large environment that is modeledby a classical force field (MM)

3. Interface between QM and classicalparts

QM/MM

QM/Interface

A. Laio, J. VandeVondele, and U. Rothlisberger, J. Chem. Phys. 116, 6941 (2002); A. Laio, J. VandeVondele, and U. Rothlisberger, J. Phys. Chem. B, 106, 7300, (2002);

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Partitioning of a QM system into 2 parts A and B: The non-linear (NL) correction

Approximations in QM/MM


where we use the “nuclear density”:

and are unknown.

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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:



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QM description of A & MM description of B

We collect all the approximations into the energy term


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QM

MM

This term is small when

- the electronic density is well localized within QM

( and are small), and

- when we have a good force field for the MM part

no bonds crossing the QM/MM boundary:

bonds crossing the QM/MM boundary:

crossingbond


We distinguish the two cases

( is small)

A. Laio, J. VandeVondele, and U. Rothlisberger, J. Chem. Phys. 116, 6941 (2002); 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


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We parameterize the potential using atom

centered potentials, i.e. centered on the link atoms.


with Gaussian-type projectors,

QM

MMwhere

Linkingatom

The parameters are determined through a fitting procedure.

E CORR (RI ,[]) drdr'i* (r)VI

OECP (r RI ,r'RI )i (r')

VIOECP (r,r') Ylm (r) pl (r)1

m l

l

pl (r')Ylm* (r')

pl (r) r l exp(r2 /(2 22))

ECORR (RI ,[])

O.A. von Lilienfeld, I. tavernelli, U. Rothlisberger, J. Chem. Phys. 122, 014113 (2005)

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According to the Hohenberg-Kohn theorem


QM

MM

“ the external potential is determined, within a trivial additive constant , by the electron density .This also determines the ground state wave function and all other electronic properties of the system”

In our QM/MM scheme we optimize the parameters 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:

O.A. von Lilienfeld, J. Chem. Phys. 122, 014113 (2005)


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Linkingatom

electronic density along z-axis

Distance along the z-axis [Å]

z


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HOOC1-CH3

HOOC1-Dconv

(1V)

HOOC1-Dopt

(DCACP)

ESP (CH3/D) ESP (C1)Dipole [D]

1.67

2.87

1.41

0.00[-0.30 + 3*0.1]

0.32

- 0.03

0.74

0.28

0.54

Example: Acetic acid (Box size: 8 Å, gas phase, 80 Ry PW cutoff)

HO

O

H

HH

DO

O

H


O.A. von Lilienfeld, I. tavernelli, U. Rothlisberger, J. Chem. Phys. 122, 014113 (2005)


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H tot[{i(ri)},{RI }] H DFT [{i(ri)};{RI }] H int[{i(ri)},{R I}] H MM [{RI }]

QM/MM Hamiltonian coupling additive scheme(just a reminder)

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QM/MM Hamiltonian coupling: Additive scheme

Scaling in a (not only) plane wave implementation:

H tot[{i(ri)},{RI }] H DFT [{i(ri)};{RI }] H int[{i(ri)},{R I}] H MM [{RI }]

H DFT [{i(ri)};{RI }] : QM-part: Hartree and xc interaction O(Nel NG NG )

H int [{i(ri)},{R I}] : QM/MM interface O(N cl NG )

H MM [{RI }] : MM part: classical (ff) potential O(N cl N cl )

atoms classical ofnumber theis functionsset -basisor wavesplane ofnumber theis

cl

G

NN

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H int (r),{qI ,RI } qII 1

Ncl

d3r (r)r RI

Interaction Hamiltonian

Potential acting on the QM electronic density

Forces acting on the MM charged atoms:

E int[(r),{qI ,RI }](r)

qI

r RII 1

Ncl

V int (r)

int3

3int )(

}],{),([

III

II

II FRrRrrrdq

RRqrE

QM/MM Hamiltonian coupling: Electrostatics

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H int (r),{qI ,RI } qII1

Ncl

d3r (r)r RI

The nested sums (over the classical MM atoms and over the discretized QM volume) are in general computationally very expensive

Ncl can be of the order of 100'000 - 500'000Ngrid can be of the order of 200 200 200

QM/MM Hamiltonian coupling: Electrostatics

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QM/MM Hamiltonian coupling: The 3 regions scheme

Region 1: NN = subset of classical atoms inside the region

Region 2: Classical-RESP charges interaction

Region 3: Multipolar expansion on MM charges

qIqJ

RESP (,RI )RI RJJQM

I NN , r1 RI r2

qII NN [(r)] r RI

r RI3

, RI r2

qII1

NN

d 3r (r)r RI

, 0 RI r1

A. Laio, J. VandeVondele, and U. Rothlisberger, J. Chem. Phys. 116, 6941 (2002); 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

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Generally we test:

However in all the known cases it isr1 ~ 10-12 a.u. r2 ~ 20-25 a.u.

Only NN < MMatoms in this shell

Divide the world in 3 domains:1) Close to the QM region (r < r1)2) Not too far, i.e. ESP region (r1 < r < r2)3) Far MM world (r > r2)

r1 r2

QM/MM Hamiltonian coupling: The 3 regions scheme

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R1: Direct coupling – the screened Coulomb potential

QM/MM Hamiltonian couplingR1: The direct coupling

E[(r),{RI }] qIff d3r (r)

r RII NN

To avoid incompatibilities due to QM (electronic) vs. classical (point charges) description of the electrostatic, we introduce the screened Coulomb potential

E[(r ),{RI }] qIff d3r (r) v I

I NN ( r RI )

where vI (r) RcIn rn

RcIn1 rn1

is a “covalent” radius of atom I, and n is an integer (n=3)RcIn

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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 MMatoms

• 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

qI

D

RI RJ

VJI QM

J NN

2

wq qID qI

H I QM 2

ESP RESTRAINqI

D = qIRESP

QM/MM Hamiltonian couplingR2: D-RESP region

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qI

D

RI RJ

VJI QM

J NN

2

wq qID qI

H I QM 2

is minimized on the fly during the dynamics.wq = weight parameter to reduce charge fluctuations:

VJ d 3r(r) u r RJ

The potential VJ is the Coulomb potential generated on the MM atom J by the electronic density distribution

25.010.0 qw

where u(|r - rJ|) is a Coulomb potential modified at short range to avoid spurious over-polarization effects.


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Hirshfeld charges ?

wq qID qI

H I QMIn the restrain term

qIH are the so-called Hirshfeld charges. They are defined as

qIH d3r r

Iat r RI K

at r RK K

ZI

where at is the atomic (pseudo) valence charge density andZI d3rat r RI

is the bare valence charge of the I-th atom.


F. L. Hirshfeld, Theoret. Chim. Acta,44, 129-138 (1977)

AIJqI

D T J

I

J

2

AIJ

1/ RI RJ J NNwqIJ J QM

T J VJ J NNwq qJ

H J QM

Note that a short-hand notation is used in which

AIJ = high index running on

= low index running on QM

NN UQM

The Hirshfeld charges: the minimization of is a least-square procedure

where

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The Hirshfeld charges least-square procedure:

qI

D 0 AKJ qK

D T J

K

J AI

J 0

and the (analytical) solution is trivially

qID HIK

1 tKK

where andHIK AIJ AK

J

J tK AK

J T J

J


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VJRESP[(r)] qI

D[(r)]RI RJI QM

VJDFT [(r)] d3r(r) u r RJ

The D-RESP potential

vRESP (r) E RESP

(r)

E RESP

qID

I QM qI

D

(r)

while the (extra) forces on the atoms are

FJ rJE RESP

E RESP

RJ

E RESP

qID

I QM qI

D

RJ

D-RESP potential - summary

can be computed in a much more efficient way than the exact DFT potential

The corresponding potential on the electrons becomes


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= center of the multipolar expansion (geometric center of the QM system)

R3: Multipolar (MP) expansion (actually quadrupolar)

where

...))((21

)(1||

)(

5,,,

3,,

3

J

II

zyx

J

I

zyxJI

RrrRrRQ

RrrRD

RrC

Rrrrd

r

QM/MM Hamiltonian couplingR3: Multi-pole expansion region

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QM/MM Hamiltonian couplingSummary: R1 + R2 + R3

EQM / MM [(r),{RI }] qIff d 3r (r) vI (r RI )

I NN

qI

D[(r)]qJ

RI RJI QM

JMM (ESP )

C qJ

RI RJJMM(MP) D

x,y,z qJ

r RJ3

JMM(MP) (RI

r )

Q

,x,y,z qJ

r RJ5

JMM(MP) (RI

r )(RI r ) ...

R1

R2

R3

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&QMMM COORDINATESINPUTTOPOLOGYADD HYDROGENAMBERARRAYSIZES ... END ARRAYSIZESBOX TOLERANCEBOX WALLSCAPPINGCAP HYDROGENELECTROSTATIC COUPLING [LONG RANGE]ESPWEIGHTEXCLUSION fGROMOS,LISTgFLEXIBLE WATER [ALL,BONDTYPE]FORCEMATCH ... END FORCEMATCHGROMOS

HIRSHFELD [ON,OFF]MAXNNNOSPLITRCUT NNRCUT MIXRCUT ESPRESTART TRAJECTORYSAMPLE INTERACTING [OFF,DCD]SPLITTIMINGSUPDATE LISTVERBOSEWRITE LOCALTEMP [STEP fn ltg]

&END

QM/MM Hamiltonian couplingInput file

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QM/MM Hamiltonian couplingInput file

ELECTROSTATIC COUPLING [LONG RANGE]Section: &QMMM The electrostatic interaction of the quantum system with the classical system is explicitly kept

into account for all classical atoms at a distance R ≤ RCUT_NN from any quantum atom and for all the MM atoms at a distance of RCUT_NN < r ≤ RCUT_MIX and a charge larger than 0.1e (NN atoms).

MM-atoms with a charge smaller than 0.1e and a distance of RCUT_NN <r ≤ RCUT_MIX and all MM-atoms with RCUT MIX < r ≤ RCUT ESP are coupled to the QM system by a ESP coupling Hamiltonian (EC atoms).

If the additional LONG RANGE keyword is specified, the interaction of the QM-system with the rest of the classical atoms is explicitly kept into account via interacting with a multipoleexpansion for the QM-system up to quadrupolar order. A file named MULTIPOLE is produced.

If LONG RANGE is omitted the quantum system is coupled to the classical atoms not in the NN-area and in the EC-area list via the force-field charges.

If the keyword ELECTROSTATIC COUPLING is omitted, all classical atoms are coupled to the quantum system by the force-field charges (mechanical coupling). The files INTERACTING.pdb, TRAJECTORY_INTERACTING, MOVIE_INTERACTING, TRAJ_INT.dcd, and ESP (or some of them) are created. The list of NN and EC atoms is updated every 100 MD steps. This can be changed using the keyword UPDATE LIST.

The default values for the cut-offs are RCU_ NN=RCUT_MIX=RCUT_ESP=10 a.u..These values can be changed by the keywords RCUT_NN, RCUT_MIX, and RCUT_ESP withrnn ≤ rmix ≤ resp.

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The Redistributed Charge scheme (RC) is a scheme to improve the electrostatic at the linking atom.

Y. Zhang, H. Lin and D. G. Truhlar, J. Chem. Theory Comput. 3, 1378-1398 (2007)

QM/MM Hamiltonian couplingThe Redistributed Charge scheme (RC)

It tunes the electrostatic balance between quantum and classical description at the interface.

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The RC scheme is used in conjunction with hydrogen capping approach

Nomenclature for the MM atoms:

M1

M2x

M2y

M2z

M3

M3

M3

Q1

Q2x

Q2y

Q2z

Q3

Q3

Q3

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The RC scheme is used in conjunction with hydrogen capping approach

The MM partial charge of atom M1 is redistributed evenly over 3 point charges q0=qM1/n, where n is the number of M1-M2 bonds.

The capping hydrogen (or linking atom, HL) carries no charge.

Location of the charges q0:

H. Lin and D. G. Truhlar, Theor. Chem. Acc. 117, 185-199

RM 1 Rq0

RM 1 RM 2

Cq0 0.5

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QM/MM Hamiltonian couplingThe Redistributed Charge and Dipoles scheme (RCD)

The RC method introduces an error in the M1-M2 dipole.

The dipole q0R (R=|RM1-RM2|) reduces by a factor of (1 – Cq0).For Cq0 = 0.5, the contribution reduces by 50%.

In the RCD method, the values of redistributed charges q0 and of the charges on M2 atoms (labeled k = 1, 2, …) are further modified such that these contributions to the M1−M2 bond dipoles are preserved.

0

0

011

0,2,2

00

q

qkM

RCDkM

q

RCD

CCq

qqC

qq

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QM/MM Hamiltonian couplingThe Polarized-Boundary method (RCPB/RCDPB)

In the RC and RCD schemes, the QM subsystem is polarized by the MM system, but the MM subsystem is not polarized by the QM subsystem, resulting in an unbalanced treatment of the electrostatic interactions between the QM and MM subsystems.

The polarized boundary RC (PBRC) and polarized-boundary RCD (PBRCD) schemes make improvements in this regard to the RC and RCD schemes, respectively, by allowing self-consistent mutual polarizations between the QM and MM subsystems near the boundary.

In the PBRC and PBRCD schemes, the polarization of the MM subsystem due to the QM electrostatic potential is accomplished by adjusting the MMatomic partial charges in the QM/MM calculations according to the principles of electronegativity equalization and charge conservations.

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First, the MM atomic partial charges are assigned to the MM atoms, and the QM/MM calculation is performed.

The electric field generated by the QM subsystem (nucleiand electronic wavefunctions) is computed and imposed on the MM, and a new set of MM atomic partial charges is determined according to the electronegativity equalization and charge conservation.

The new set of partial charges replaces the old set of partial charges, and a new QM/MM calculation is performed with the updated partial charges (new external potential for the DFT calculation).

convergence ?

STOP

Convergence= until the variations in partial charges are smaller than preset thresholds, or until the number of iterations exceeds a preset value.

no

yes

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Charge equalization method by Rappé and Goddard.[A. K. Rappé and w. A. Goddard, W. A. (1991) J. Phys. Chem. 95, 3358-3363 (1991).]

EQ (Q1,Q2,...,QNcl) (E I 0 I

0QI 12

QI2JII

0 )I

12

QII J QJ JIJ QI d3r (r)

r RI

I

E I 0 is the unperturbed reference charge

I0 = E

Q

I0

=1/2(IP - EA) is the first order response (electronegativity)

term)Coulomb-self (repulsive responseorder second theis EA)-(IP=QE=

I02

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IIJ

JIJ0 Coulomb interaction between unit charges at I and J (at disrance R =| RI - RJ |)

Consider the classical energy expression:

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The charge equalization method is based on the minimization of the energy EQ (Q1,Q2,...,QNcl

)i.e.

under the constraints

0),...,,( 0021

IJ

JIJJIIII

NI QJQJQEQQQ

cl

1. 1 2 K Ncl

2. Qtot QII1

Ncl

This leads to a linear system of equations CD Dwhere D1 Qtot, Di i

0 10 for i 2

C1i Qi, Cij Jij J1 j for i 2

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Electronegativity equalization method by Mortier et. al.[W. J. Mortier, S. K. Ghosh, and S. Shankar, J. Am. Chem. Soc. 108, 4315- 4320 (1986) .]

The existence of a unique chemical potential everywhere in the molecule establishes the electronegativity equalization principle that Mortier write as

I (Q1,Q2,...,QNcl) ( I

0 I ) 2(I0 I )QI

QJ

RIJJI

clN 21with

and are the neutral atom electronegativity and hardness, respectively, QI, and QJ are the charges on atoms I and J, and RIJ, is the internuclear distance. The parameters and are the corrections to the neutral atom electronegativity and hardness that arise as a consequence of bonding.

I0 I

0

I I

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and are obtained by calibrating through small-molecule calculations and are transferable and consistently usable for calculating charges in any molecule.

I I

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