Fundamentals of Quantum Mechanics for Chemistrygmonard.wdfiles.com › local--files › enseignement:main › chap1.pdf · 1. Fundamentals of Quantum Mechanics for Chemistry 3 Hartree-Fock

Fundamentals of Quantum Mechanics forChemistry

Gerald MONARD

Equipe de Chimie et Biochimie TheoriquesUMR 7565 CNRS - Universite Henri Poincare

Faculte des Sciences - B.P. 23954506 Vandœuvre-les-Nancy Cedex - FRANCE

http://www.monard.info/

Outline . . .1. Fundamentals of Quantum Mechanics for Chemistry

3 Hartree-Fock methods3 Density Functional Theory3 The QM scaling problem3 Semiempirical methods3 Molecular Mechanics

2. Fundamentals of QM/MM methods

3 Partionning3 QM/MM interactions3 Cutting covalent bonds3 ONIOM3 Some available software

3. Selected QM/MM applications

3 Solvent effects3 Spectroscopy3 Biochemistry

. . . Outline4. Fundamentals of Linear Scaling methods

3 QM Bottlenecks3 General ideas and solutions3 Some available software

5. Focus on some Linear Scaling methods

3 CG-DMS3 Mozyme3 Divide & Conquer

6. Selected Linear Scaling applications

3 Energy Decomposition; Charge Transfer & Polarization3 Born-Oppenheimer Molecular Dynamics

7. Parallelization of QM/MM and Linear Scaling methods

Fundamentals of Quantum Mechanics for Chemistry (1)

Some approximationsAnd there was the Schrodinger equation. . .

H0Ψ0 = E0Ψ0

where:

H0 is an Hamiltonian operator that describes a molecularsystem

Ψ0 is a wavefunction (solution of the Schrodinger equation)that describe a state of the system

E0 the energy associated to Ψ0

+ 1 equation + 2 unknowns (given H0) = +∞ solutions!

â From now on:

3 ground state3 closed shell3 non-relativistic


Born-Oppenheimer approximation

â nuclei are fixed point charges

â only electrons are represented by a wavefunction Ψ

HΨ = EelecΨ (1)

H = Te + VeN + Vee

= −1

2 ∑i

∆i︸︷︷︸kinetic energy

+ ∑i

∑K

−ZK

riK︸︷︷︸e−-nuclei inter.

+ ∑i

∑i>j

1

rij︸︷︷︸e−–e− inter.

(2)

E = Eelec + Enuclei

= 〈Ψ|H|Ψ〉+∑K

∑L>K

ZKZL

RKL


Orbital approximation

â each electron is described by a mono-electronic wavefunction: theMolecular Orbital (MO)

Ψ(1,2, . . . ,n) = ψ1(1)ψ2(2) . . .ψn(n)

â all MOs are combined in a Slater determinant (Pauli principle)

Linear Combination of Atomic Orbitals (LCAO)

â Each MO ψi is developed on a basis set of functions φµ : the AtomicOrbitals (AO)

ψi =AOs

∑µ

cµ iφµ

â the real coefficients ciµ are the unknown of the problem


Variational Principle

â The eletronic energy Eelec corresponds to a minimum with respect toeach MO ψi

∀i∂ Eelec

∂ψi= 0

+ Hartree-Fock equations

∀i Fψi = εiψi

where F is the mono-electronic Fock operator

The Hartree-Fock method (1)

The Fock operator

F(1) = Hc(1) +∑j

[Jj(1)−Kj(1)]

â Hc(1) is the one-electron core Hamiltonian

Hc(1) =−1

2∆1−∑

K

ZK

R1K

â Jj(1) is the Coulomb operator,

Jj(1) =∫

(2)ψ∗j (2)

1

r12ψj(2)dτ2

â Kj(1) is the exchange operator

Kj(1)ψi (1) = ψj(1)∫

(2)ψj(2)∗

1

r12ψi (2)dτ2


The Roothan-Hall equations

â N electrons,{

φµ

}AO basis set, closed shell, ground state

+ occupied MO = 2 electrons; virtual MO = 0 electron

ψi =AO

∑µ

cµ iφµ

â the Hartree-Fock equations can be re-written:

FC = SCε

with

3 C the matrix of cµ i coefficients3 ε the diagonal energy matrix3 S the overlap matrix:

Sµν =< φµ |φν >

The Hartree-Fock method (3)The Density MatrixFrom the MO coefficient cµ i , it is possible to build a density matrixwhose elements are:

Pµν =MO

∑j

njcµ jcν j with nj = 0 or 2 (occupation number)

The Fock matrix

Fµν = Hcµν +

AO

∑λ

AO

∑η

Pλη

[(µν |λη)− 1

2(µη |λν)

]with:

(µν |λη) =∫

(1)

∫(2)

φµ (1)φν (1)1

r12φλ (2)φη (2)dr1dr2

The Hartree-Fock energy

Eelec =1

2 ∑µ

∑ν

Pµν [Hµν + Fµν ]


The Hartree-Fock algorithm

1. Compute mono- and bielectronic integrals

2. Build core hamiltonian (invariant) Hc

3. Guess an initial density matrix

4. Build the Fock matrix F

5. Orthogonal transformation using S1/2

F′C′ = εC′

6. Diagonalization of the Fock matrix F′

The C′ coefficients are obtained

7. Inverse transformation C′→ C

8. Build the new density matrixBack to 4. unless convergence


The Hartree-Fock method is an ab initio method

â No (empirical) parameters Õ ab initio method

â Orbital approximation Õ No electronic correlation

Other ab initio methods (post-Hartree-Fock methods)

â Møller-Plesset Perturbation Theory (MP2, MP4, etc)

â Configuration Interaction (CI)

â Coupled Cluster (e.g. CCSD(T))

â MultiConfigurational Self-Consistent-Field (MCSCF)

Density Functional Theory (1)

Background

â DFT (Density Functional Theory) methods are (almost) ab initiomethods which include electronic correlation at a cost similar to aHartree-Fock calculation.In most cases, a DFT calculation is even less costly than a HFcalculation.

â DFT methods relie on the Hohenberg-Kohn theorem (1964) whichstates that the ground state energy E of a system is a functional ofthe electronic density of this system, ρ(~r). Any electronic densityρ ′(~r) other than the real electronic density will necessary lead to ahigher energy. (variational principle)


A different approachTo the opposite of ab initio methods, DFT methods try to find a simple3-dimensional ρ(~r) function and not a complex 3N-dimensional wavefunction.

ρ : R3 −→ R~r 7−→ ρ(~r)

From the Hohenberg-Kohn theorem, the energy E depends on theelectronic density.It is said that E is a functional of the electronic density:

E :(R3→ R

)−→ R

ρ 7−→ E [ρ(~r)]


The Kohn-Sham approachLet’s write:

E [ρ(~r)] = U [ρ(~r)] + T [ρ(~r)] + Exc [ρ(~r)]

with:

â U [ρ(~r)] the classical electrostatic energy

U [ρ(~r)] =nuclei

∑A

∫ −ZAρ(~r)

|~r − ~RA|d~r︸︷︷︸

electron-nuclei attraction

+1

2

∫ ∫ρ(~r)ρ(~r ′)

|~r −~r ′|d~rd~r ′︸︷︷︸

electron-electron repulsion

â T [ρ(~r)] is defined as the kinetic energy of a system with the sameelectronic density ρ(~r) but in which the electrons don’t interact

â Exc [ρ(~r)] the rest of the energy: exchange and electronic correlationcontributions to the total energy + the difference between T [ρ(~r)]and the real kinetic energy


Kohn-Sham orbitalsKohn and Sham suggest to decompose the total electronic density into asum of individual contributions for each electron:

ρ(~r) =Nα

∑i

ραi (~r) +

Nβ

∑i

ρβ

i (~r)

=Nα

∑i

|ψαi (~r)|2 +

Nβ

∑i

∣∣∣ψβ

i (~r)∣∣∣2

(α : high spin; β : low spin)

ψαi , ψ

β

i : Kohn-Sham molecular orbitals, or “auxiliary” orbitals


The Kinetic Energy operator is defined using Kohn-Sham orbitalsOne can then define T [ρ(~r)]:

T [ρ(~r)] = ∑σ=α,β

Nσ

∑i

∫ψ

σi (~r)− ∆

2ψ

σi (~r)d~r

Be careful: T [ρ(~r)] is not a real functional of the density since it is onlydefined using Kohn-Sham molecular orbitals(and not using ρ(~r)).


The Kohn-Sham EquationsBy applying the variational principle to the ground state energy:

∂ E [ρ(~r)]

∂ραi (~r)

=∂ E [ρ(~r)]

∂ρβ

i (~r)= 0

One can find the one-electron Kohn-Sham equations:

hKSψi = εiψi

with hKS: the one-electron Kohn-Sham operator

hKS =−∆

2−∑

A

ZA

|~r − ~RA|+∫

ρ(~r ′)

|~r −~r ′|d~r ′+ Vxc

and Vxc =∂ Exc [ρ(~r)]

∂ρ(~r): the exchange-correlation potential


Exchange-correlation functionalsIf the “true” exchange-correlation functional was known, the Kohn-Shamequations would give the exact electronic density of a ground statesystem.This is not the case !+ various approximations are to be made+ it exists various exchange-correlation models

First case: Exc =∫

εxc (ρ) .ρ(~r)d~r

+ local methods

Second case: Exc =∫

εxc

(ρ,~∇ρ

).ρ(~r)d~r

+

non-local methodsgradient correctedGGA

(Generalized Gradient Approximation)


Exchange-correlation potential models: Local density methodsUsually, the exchange and the correlation contributions are separated:

Exc = Ex + Ec

Ex

â LDA: Local Density Approximation

Ex =∫

ρ(~r)

(−3e2

4π

)(3π

2ρ(~r)

)1/3d~r

(exact exchange energy in ahomogeneous electron gas)

Ec

â VWN (Vosko-Wilk-Nusair)

â PZ (Perdew-Zunger)

â PW92 (Perdew-Wang, 1992)

â etc.


Exchange-correlation potential models: Non-local density methods

Ex

â PW86 (Perdew-Wang, 1986)

εPW 86x = ε

LDAx

(1 +ax2 +bx4 +cx6

)1/15

with x =|~∇ρ

ρ4/3

and a, b, and c real parameters

â B88 (Becke 1988)

εB88x = ε

LDAx −βρ

1/3 x2

1 + 6βx sinh−1 x

with β an atomic parameter

â PW91

â PBE

â etc.

Ec

â LYP (Lee-Yang-Parr)

â PW91

â PBE (Perdew-Burke-Ernzerhof)

â P86

Exc

â BP86 = B88 + P86

â BLYP = B88 + LYP

â PBE

â BPW91 = B88 + PW91


Exchange-correlation potential models: Hybrid methodsIn the hybrid methods, the exchange energy contains a part of “exact”exchange energy calculated in a similar manner as Hartree-Fock exchangeenergy (but using Kohn-Sham orbitals)Ex.: B3LYP

E B3LYPxc = E LDA

xc + a0(E HFx −E LDA

x ) + ax(E B88x −E LDA

x ) + ac(E LYPc −E LDA

c )

with a0 = 0.20, ax = 0.72, ac = 0.81

Common hybrid methods: B3LYP, PBE0, PBE1PBE

The QM scaling problem (1)

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200

wall

clo

ck C

PU

tim

e (

seconds)

number of water molecules

energy of a water cluster (3-21G basis set)

B3LYP/3-21GBLYP/3-21G

CCSD(T)/3-21GMP2/3-21G

HF/3-21G

â (H2O)n water cluster(n from 1 to 216)

â 1 energy calculations

â Gaussian G09.B01(NProcShared=4, Mem=8Gb,MaxDisk=36Gb)

â Wall clock time limit: 1 hour

â Intel(R) Xeon(R) CPU E56202.40GHz (8 cores) 32Gb RAM

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200

wall

clo

ck C

PU

tim

e (

seconds)


energy of a water cluster (6-31G* basis set)

B3LYP/6-31G*BLYP/6-31G*

CCSD(T)/6-31G*MP2/6-31G*

HF/6-31G*

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200

wall

clo

ck C

PU

tim

e (

seconds)


energy of a water cluster (6-311+G** basis set)

B3LYP/6-311+G**BLYP/6-311+G**

CCSD(T)/6-311+G**MP2/6-311+G**

HF/6-311+G**


Theoretical CPU scaling order for different QM methods

QM method Scaling

semiempirical O(N3)

DFT O(N3−N4)

Hartree-Fock O(N4)

MP2 O(N5)

CCSD(T) O(N7)

Full CI O(expN)

The (H2O)n example: n max in 1/2 hour (4 cores)

HF BLYP B3LYP MP2 CCSD(T)

3-21G 216 128 128 32 8

6-31G* 96 96 96 24 4

6-311+G** 32 32 28 16 4


How to solve the QM scaling problem?

â Moore’s Law: CPU power doubles every 18 months

+ doubling a molecular system is possible:

3 O(N3) scaling: every 18x3 months = 4.5 years3 O(N4) scaling: every 6 years3 O(N5) scaling: every 7.5 years, etc.

â Parallelism is not a valid option in the long run

3 Good speeds-up are difficult to obtain (Amdahl’s Law)3 non linear scaling of the “standard” algorithms3 standard algorithms are not parallel friendly

+ change the methods: use approximate quantum methods

3 semiempirical QM methods3 molecular mechanics (MM) force fields3 combined QM/MM methods

+ change the algorithms

3 Linear scaling algorithms

Semiempirical methods (1)

They are as old as ab initio methods

â PPP (Pariser-Parr-Pople) method + 1950s

â Extended Huckel method + 1960s

â CNDO + 1960s

â INDO + 1960s

â etc.

A shared assumption

â ab initio (HF) calculations are too time consuming

â the equations are simplified to yield accessible timings for “real”molecules

â some parameters are introduced to correct the loss of information

â these parameters are obtained from experimental data+ empirical parameters (hence the term semiempirical methods)


NDDO: Neglect of Diatomic Differential Overlap

â Most modern semiempirical methods are NDDO based:

3 MNDO (1977)3 AM1 (1985)3 PM3 (1989)3 PDDG/PM3 & PDDG/MNDO (2002)3 PM6 (2007)3 and going ...

â They are based on a simplification of the Hartree-Fock equations

Standard QM algorithm (Hartree-Fock)

Roothan Equations (closed shells)

Total energy E =1

2 ∑µ

∑ν

Pµν [Hcµν + Fµν ] +∑

A∑B>A

ZAZB

RAB

Density matrix element Pµν = 2occ

∑j

cµ jcν j (cµ j : M.O. coefficients)

Fock matrix element Fµν = Hcµν +∑

λ

∑η

Pλη

[(µν |λη)− 1

2(µη |λν)

]

bielectronic integrals (µν |λη) =∫

φµ (1)φν (1)∗1

r12φλ (2)φη (2)∗dr1dr2

The Roothan equations FC = SCε (ε : M.O. eigenvalues)

(S : overlap matrix C : M.O. coefficient matrix)

Standard QM algorithm (Hartree-Fock)

Hartree-Fock SCF algorithm

1. Compute mono- and bielectronic integrals O(N4)




5. Orthogonal transformation using S1/2 O(N3)

F′C′ = εC′

6. Diagonalization of the Fock matrix F′ O(N3)The C′ coefficients are obtained

7. Inverse transformation C′→ C

8. Build the new density matrix O(N3)Back to 4. unless convergence


NDDO approximations

â Only valence shell electrons are considered

+ core electrons are taken into account by reducing the nuclei charges(effective nuclei charge) and by introducing empirical functions tomodel the interactions between (nuclei+core electrons) and theother particles

â A minimal basis set is used.Usually: minimal Slater Type Orbital basis set

â ZDO approximation (Zero Differential Overlap):All products between basis functions corresponding to a singleelectron but centered on different atoms are neglected:

ϕAµ (i).ϕB

ν (i) = 0 if A 6= B


Consequences of the ZDO approximation

â The overlap matrix S is equal to unity: S = I

+ There is no orthogonalization step in the SCF procedure

â one-electron three-center integrals (two centers for the basisfunctions and one center for the operator) are considered to be equalto zero

â All three-center and four-center bielectronic integrals are neglected(these are the most numerous integrals)

+ The number of integrals scales as O(N2)(where N is the number of basis functions)

â ∑A

∑A>B

ZAZB

RABin HF equations is replaced by

∑A

∑A>B

fAB(RAB) a parameterized core-core repulsion function


Semiempirical SCF algorithm

1. Compute mono- and bielectronic integrals O(N2)




5. Diagonalization of the Fock matrix F O(N3)The C coefficients are obtained

6. Build the new density matrix from C O(N3)Back to 4. unless convergence


PM3 vs. ab initio

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200

wall

clo

ck C

PU

tim

e (

seconds)


energy of a water cluster (3-21G basis set vs. PM3)



HF/3-21GPM3

â (H2O)n water cluster(n from 1 to 216)

â 1 energy calculations

â Gaussian G09.B01

â Wall clock time limit: 1 hour

â Intel(R) Xeon(R) CPU E56202.40GHz (8 cores) 32Gb RAM

0

20

40

60

80

100

0 50 100 150 200

wall

clo

ck C

PU

tim

e (

seconds)


energy of a water cluster (3-21G basis set vs. PM3)



HF/3-21GPM3

â 3-21G:

3 NProcShared=43 Mem=8Gb3 MaxDisk=36Gb

â PM3:

3 NProcShared=1


An example of a NDDO method: MNDOMNDO: Modified Neglect of Differential Overlap (Dewar & Thiel: 1977)

Nomenclature:

A,B: atoms (A 6= B)

µ,ν: atomic orbitals from A

λ ,η: atomic orbitals from B

Fock matrix elements:

Fµµ = Uµµ − ∑B 6=A

Z ′B(µµ|sB sB) +A

∑ν

Pνν

[(µµ|νν)− 1

2(µν |µν)

]+∑

B

B

∑λ ,η

Pλη (µµ|λη)

Fµν = − ∑B 6=A

Z ′B(µν |sB sB) +1

2Pµν [3(µν |µν)− (µµ|νν)] +∑

B

B

∑λ ,η

Pλη (µν |λη)

Fµλ = βµλSµλ −1

2

A

∑ν

B

∑η

Pνη (µν |λη)


Parameters (1)

â Uµµ (Uss and Upp): these terms represent the one-electronone-center integrals corresponding to the sum of the kinetic energyof one electron in the atomic orbital ϕµ of A and the potentialenergy of the same electron due to its attraction by the core of A(nuclei + core electrons)

â coulombic one-center bielectronic integrals (µµ|νν) are generallynoted gµν , while exchange one-center bielectronic integrals (µν |µν)are generally noted hµν .

In the MNDO method, gµν and hµν integrals are evaluated fromexperimental spectroscopic data (oleari, 1966).There are five one-center bielectronic integrals:

gss = (ss|ss) gsp = (ss|pp) hsp = (sp|sp)

gpp = (pp|pp) gpp′ = (pp|p′p′) (pp′|pp′) = 12

[(pp|pp)− 1

2 (pp|p′p′)]


Parameters (2)

â one-electron two-center integrals βµλ are computed using theformula:

βµλ =βA

µ + βBλ

2

where βAµ and βB

λare two atomic parameters of the MNDO method.

â two-electron two-center integrals (µν |λν) are computed using amultipolar development which uses two kinds of parameters: Di andρj which are computed from the knowledge of ζ , the Slater atomicorbital coefficients

For example: for an element of the second row of the periodic table+ 5 parameters: D1,D2,ρ0,ρ1,ρ2


Parameters (3)

â core-core repulsion functions are computed using the formula:

fAB(RAB) = Z ′AZ ′B(sAsA|sBsB)[1 + e−αARAB + e−αBRAB

]where αA and αB are atomic parameters.

â In the case where (A,B) represents a hydrogen bond(A = N or O, and B = H):

fXH(RXH) = Z ′XZ ′H(sX sX |sHsH)[1 + RXHe−αXRXH + e−αHRXH

](with RXH in A)


Parameters (4)Thus, MNDO semiempirical parameters are defined by atom.

Example:C: Uss , Upp, ζ (in MNDO: ζs = ζp)

βs , βp, α, D1, D2, ρ0, ρ1, ρ2

gss , gpp, gsp, gpp′ , hsp

+ 16 parameters


Two improvements of the MNDO method: AM1 and PM3

AM1 (Austin Model 1): 1985 (Dewar et al.)PM3 (Parameter Model 3): 1989 (Stewart et al.)

+ s and p Slater atomic orbital coefficients are now different (ζs 6= ζp)

+ the core-core function is modified

fAB(RAB) = f MNDOAB (RAB)+

Z ′AZ′B

RAB

[∑k

aAk e−bAk (RAB−cAk )2

+∑k

aBk e−bBk (RAB−cBk )2

]

â aAk ,bAk ,c

Ak : atomic parameters

â AM1: k goes from 1 to 4 PM3: k goes from 1 to 2

â In the case of the PM3 method:gµν and hµν integrals are now optimized parameters

+ In the case of the carbon element:AM1: 29 parameters PM3: 23 parameters


Determination of the semiempirical parametersThe semiempirical parameters are optimized (=fitted) to reproduce agiven set of experimental data from small molecules in gas phase:

? geometrical structures ? heat of formation (∆Hf )? dipolar moments ? ionization potentials

MNDO, AM1, and PM3, etc. are different because they make use ofdifferent semiempirical equations, different number of parameters,different number of optimized parameters (experimental parameters vs.optimized parameters), and different sets of experimental data.


Advantages and disadvantages of the semiempirical methods

â A lot faster than Hartree-Fock and post-Hartree-Fock methods.

â Electronic correlation is implicitly taken into account through theuse parameters fitted from experimental data.

â Give, when properly used, better results than Hartree-Fock method.

â The quality of a semiempirical computation is dependant on the waythe semiempirical parameters have been fitted:

experimental = small gas phase + domain of validity fordata molecule semiempirical methods !



Semiempirical: what is it good for?

â , enthalpies, heats of formation

â , gas phase geometries (stable structures) of small molecules

â Y transition state geometries

â Y frequency calculations

â / intermolecular interactions (+ currently being improved)

Selected publications

MNDO Dewar, M. J. S.; Thiel, W. J. Am. Chem. Soc. 1977, 99, 4899–4907

AM1 Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. J.Am. Chem. Soc. 1985, 107, 3902–3909

PM3 Stewart, J. J. P. J. Comput. Chem. 1989, 10, 209–220

PDDG Repasky, M.; Chandrasekhar, J.; Jorgensen, W. J. Comput. Chem.2002, 23, 1601–1622

PM6 Stewart, J. J. P. J. Mol. Model. 2007, 13, 1173–1213

Molecular Mechanics (1)

How can we further speed up the calculations?

â In many problems, an accurate description of the electronicwavefunctions is not necessary

â This is true when no chemical change is performed along asimulation

+ Molecular Mechanics is a simplification of the description of amolecular system at the atomic level where no explicit electrons areconsidered

+ the energy of a system is then defined solely by the positions of thenuclei (Born-Oppenheimer approximation)


Quantum Mechanics around the equilibrium structure1 water molecule

O

H H

O

H H

O

H H

Symetric stretch

(3657 cm-1)

Asymetric stretch

(3776 cm-1)

Bend

(1595 cm-1)

Deformation around the equilibrium geometry can be modelled usingharmonic potentials.


Quantum Mechanics around the equilibrium structureMany water molecules

Water molecules in interactions:

â van der Waals contacts:

E ijvdw = εij

[(σij

Rij

)12

−2

(σij

Rij

)6]

â electrostatic dipole-dipoleinteractions

+ replaced by charge-chargeinteractions:

Eelec = ∑i

∑i>j

1

4πε0

qiqj

rij


Force Fields

â Molecular Mechanics (MM) is the application of the Newtonianmechanics (classical mechanics) to molecular systems.

â In a molecule, each atom is considered as a point charge

â The point charges interact using a parametrized force field

â A force field is an equation describing all possible interactions in amolecular system associated with pre-defined parameters:

force field = equation + parameters

â In most cases, the connectivity of the system remains constant(+ no chemical reaction)


Molecular Interactions described by a force field

��

��

��

��

��

Out−of−plane(improper torsion)

Bond rotation (torsion)

(electrostatic)Non−bonded interactions

δ+

δ−

δ+

Non−bonded interactions(van der Waals)

Angle bending

Bond stretching


Transferability / AdditivityMolecular Mechanics is based on two main assumptions:

Transferability: properties of chemical subgroups are similar either insmall molecules or large compounds (e.g.: a carbonylC=O group has very similar stretching properties in H2COor in a 10,000 atom structure)

Additivity: effective molecular energy can be expressed as a sum ofpotentials describing all interactions in the molecularsystem:

â van der Waals and electrostatic interactions(non-bonded interactions)

â bond length and angle deviations, internal torsionflexibility, etc.(bonded interactions)


Example of a force field: AMBERAMBER: general force field for the description of proteins and nucleicacids (DNA, RNA).

Epot =bonds

∑b

1

2kb(r − rb)2 +

angles

∑a

1

2ka(θ −θa)2

+dihedrals

∑d

∑n

Vn

2(1 + cos(nω− γ))

+atoms

∑i

atoms

∑j>i

{1

4πε0εr

qiqj

rij+ εij

[(σij

rij

)12

−2

(σij

rij

)6]}


An example using the AMBER force field (ff03)N-methylacetamide

Atom ResidueNumber Name Name Number

1 1HH3 ACE 12 CH3 ACE 13 2HH3 ACE 14 3HH3 ACE 15 C ACE 16 O ACE 17 N NME 28 H NME 29 CH3 NME 2

10 1HH3 NME 211 2HH3 NME 212 3HH3 NME 2


AMBER atom types and atom charges (ff03)N-methylacetamide

AtomNumber Name Type Charge

1 1HH3 HC 0.07602 CH3 CT -0.19033 2HH3 HC 0.07604 3HH3 HC 0.07605 C C 0.51246 O O -0.55027 N N -0.42398 H H 0.29019 CH3 CT -0.0543

10 1HH3 H1 0.062711 2HH3 H1 0.062712 3HH3 H1 0.0627


AMBER bond types (ff03)N-methylacetamide

Bond Number kb rbCT–HC 3 340.0 1.090CT–C 1 317.0 1.522C–O 1 570.0 1.229C–N 1 490.0 1.335N–H 1 434.0 1.010N–CT 1 337.0 1.449CT–H1 3 340.0 1.090


AMBER angle types (ff03)N-methylacetamide

Angle Number ka raHC–CT–HC 3 35.0 109.50HC–CT–C 3 50.0 109.50CT–C–O 1 80.0 120.40CT–C–N 1 70.0 116.60O–C–N 1 80.0 122.90C–N–H 1 50.0 120.00C–N–CT 1 50.0 121.90H–N–CT 1 50.0 118.04N–CT–H1 3 50.0 109.50H1–CT–H1 3 35.0 109.50


AMBER dihedral and improper types (ff03)N-methylacetamide

Dihedral Number n Vn γ

HC–CT–C–O 3 1 0.80 0.03 0.08 180.0

HC–CT–C–N 3 0 0.00 0.0CT–C–N–H 1 2 10.00 180.0CT–C–N–CT 1 2 10.00 180.0O–C–N–H 1 2 2.50 180.0

1 2.00 0.0O–C–N–CT 1 2 10.00 180.0C–N–CT–H1 3 0 0.00 0.0H–N–CT–H1 3 0 0.00 0.0

Improper Number n Vn γ

H–N–C–CT 1 2 1.1 180.0O–C–N–CT 1 2 1.1 180.0


AMBER van der Waals types (ff03)N-methylacetamide

Atom type σi εi

C 1.9080 0.0860CT 1.9080 0.1094H 0.6000 0.0157HC 1.4870 0.0157H1 1.3870 0.0157N 1.8240 0.1700O 1.6612 0.2100

E ijvdw = εij

[(σij

Rij

)12

−2

(σij

Rij

)6]

σij = σi + σj and εij =√

εi εj


Some usual force fields

AMBER Assisted Model Building and Energy Refinement (UCSF)specialized in the modelization of proteins and nucleicacids (DNA, RNA)

CHARMm Chemistry at HARvard Macromolecular Mechanics(Harvard, Strasbourg)specialized in the modelization of proteins

MM2, MM3, MM4 Allinger Molecular Mechanics (UGA)specialized in organic compounds

MMFF94 Merck Molecular Force Field (Merck Res. Lab.)specialized in organic compounds

OPLS Optimized Potentials for Liquid Simulations (Yale)

AMOEBA Polarizable force field for water, ions and proteins(WUSTL)

etc.


Simulating infinite systems

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��


Simulating infinite systems

â Periodic Boundary Conditions (PBC): a molecular system is enclosedin a box (the unit cell) and is replicated infinitely in the three spacedimensions (the images).

â Minimum Image Convention: Only the coordinates of the unit cell isrecorded. As an atom leaves the unit cell by crossing the boundary,an image enters to replace it.

+ the total number of particles is conserved.


Long-range electrostatic interactions

â The coulomb energy in periodic domains (neutral system):

Eelec =1

2

′

∑~n

∑i

∑j

qiqj

|~ri −~rj +~n|

The sum is conditionnally convergent(= slow convergence, if any)

â cut-off: if rij > rcut-off +1

rij= 0

+ non-physical but speeds up computations


The Ewald Summation

â The coulomb sum can be converted in a sum of two absolutely andrapidly convergent series in direct and reciprocal space.

â This conversion is accomplished by adding to each point charge aGaussian charge density of opposite value and same magnitude asthe point charge:

ρi (~r) =−qiα3 exp(−α

2r 2)/√

π3 (3)

where α is a positive parameter which determine the width of thegaussians

â This charge distribution screens the interaction betweenneighbouring point charges.+ fast convergence in the direct space.

â The distribution of opposite gaussian charges converges quickly inthe reciprocal space using a Fourier transform.


The Ewald SummationIt is demonstrated (by Ewald, 1921):

Eelec = U r + Um + U0

with U r the direct sum, Um the reciprocal sum, and U0 theself-interacting term (which corrects the interactions between the countercharges introduced in the system).

U r =1

2

′

∑i ,j

qiqj

4πε0∑~n

erfc(α|~ri −~rj +~n|)|~ri −~rj +~n|

(4)

Um =1

2πL3 ∑i ,j

qiqj

4πε0∑~m 6=~0

exp(−(π~m/α)2 + 2π i ~m.(~ri −~rj))

~m2(5)

U0 =−α√

π∑i

q2i (6)

~m = 2π~nL a reciprocal space vector, erfc(x) = 1− erf(x) = 1− 2√

π

∫ x0 e−u

2du


Particle Mesh Ewald (Darden et al., 1993)

â The computation time of the Ewald summations grows as O(N2)where N is the number of particles in the periodic systems.

â To speed up computations, the Particle Mesh Ewald (PME) methodhas been designed. Its computation grows as O(N log N).

â It is based on the use of a cut-off in the direct space and the use ofFast Fourier Transform (FFT) in the reciprocal space.


Molecular Dynamics (MD)

â The Molecular Dynamics (MD) is the simulation of the behavior of amolecular system along time.

â It is performed by solving the Newton’s equations of motions:

mi−→a i =

−→F i

(mi : the mass of the particle; −→a i : its acceleration;−→F i : the external forces acting on it)

â The resolution of the Newton’s equations of motions is made usingnumerical integration

â ∆t is the time incrementAt each t, the potential energy and the forces must be computed

Positions (−→xi ), velocities (−→vi ), and forces (−→Fi ) must be knowed at

each t


Molecular Dynamics IntegratorsThere are many ways of solving the Newton’s equations of motions.Verlet:

r(t + ∆t) = r(t) + ∆tv(t) +∆t2a(t)

2

a(t + ∆t) =f(t + ∆t)

m

v(t + ∆t) = r(t) +1

2∆t[a(t) +a(t + ∆t)]

Leapfrog:

v(t +1

2∆t) = v(t− 1

2∆t) + ∆ta(t)

r(t + ∆t) = r(t) + ∆tv(t +1

2∆t)

a(t + ∆t) =f(t + ∆t)

m


Conservation of the EnergyIf the system is isolated, the total energy is conserved:

mid−→v i

dt=− dE

d−→x i

with

∑i

dE

d−→x i=−→0

givesEtotal = Ekin. + Epot. = Cte


Timestep ∆t

â ∆t must be small enough to ensure the conservation of the totalenergy

â The higher the ∆t, the less energy computations are needed for agiven simulation length

â Nyquist-Shannon sampling theorem:

∆t 6 2π

√µ

k

with k the strongest force constant in the system and µ itsassociated reduced mass

â In practice, ∆t ∼ 1fs (1fs = 10−15s)


The Ergodic hypothesis

â How long a molecular dynamics simulation should be run ?

â Ergodic hypothesis: at t ∼+∞, all accessible states have beenexplored by the system.

+ it is not possible to wait t = +∞ !

â In practice: 1 year of CPU time = 31.5e6 sec.

+ typical MD length = 1 to 100 ns

+ It is difficult to ensure proper “convergence” (= that all accessiblestates have been explored)


Thermodynamical ensembles

NVE Constant number of atoms (N), constant volume (V),constant energy (E)

+ an isolated molecular system in a periodic box

NVT constant N, V, and Temperature (T)

+ the system in the periodic box is coupled to a thermostatof infinite mass (temperature coupling)

NPT constant N, T, and pressure (P)

+ the system in the periodic box is coupled to a thermostatand a barostat (temperature and pressure coupling)

+ the size of the box changes along time

µVT/µPT constant V or P, T, and chemical potential (µ)

+ multiple phases (at least two); the chemical potential isconstant for each phase.


Properties accessibles by MDNearly all non-reactive properties are accessible:

â Molecular conformations; static properties(heat of vaporization, radial distribution functions,dieletric constant, etc.)

â Dynamical properties (diffusion constant, transport, etc.)

â Phase change; state change; protein folding

â Molecular recognition; signal

â Free energy changes (solvation, alchemical transformation,thermodynamical cycle, etc.)

QM/MM Methods: Foundations (1)How to simulate a very large ”reactive” molecular system?

Quantum Mechanics

â Description of the electrons and nuclei behavior

â Allows the breaking and forming of covalent bonds

â CPU time intensive −→ limited to small systems

Molecular Mechanics

â Atoms = interacting point charges

â Bad description of chemical reaction

â Fast computations −→ suitable for large systems

QM/MM Methods: Foundations (2)

General Ideaâ Partionning of the total system

â Active part = small number of atomsDescription by Quantum Mechanics (QM) + the quantum part

â Rest of the systemDescription by Molecular Mechanics (MM) + the classical part

â The MM part acts as a perbutation to the QM part

â The coupling is called a QM/MM method


Seminal papers

â Warshel, A.; Levitt, M. J. Mol. Biol. 1976, 103, 227–249

â Singh, U. C.; Kollman, P. A. J. Comput. Chem. 1986, 7, 718–730

â Field, M.; Bash, P.; Karplus, M. J. Comput. Chem. 1990, 11,700–733

Selected reviews

â Aqvist, J.; Warshel, A. Chem. Rev. 1993, 93, 2523–2544

â Monard, G.; Jr., K. M. Acc. Chem. Res. 1999, 32(10), 904–911

â Monard, G.; Prat-Resina, X.; Gonzalez-Lafont, A.; Lluch, J. Int. J.Quant. Chem. 2003, 93(3), 229–244

â Amara, P.; Field, M. J. In Encyclopedia of ComputationalChemistry; John Wiley & Sons, Ltd, 2002

â Lin, H.; Truhlar, D. G. Theor. Chem. Acc. 2007, 117, 185–199


QM/MM Hamiltonians

H = HQM +HMM +HQM/MM

HQM/MM describes the interactions between the quantum part and theclassical part

The QM hamiltonian

HQM = −1

2

e-

∑i

∆i −e-

∑i

nuclei

∑K

ZK

riK+

e-

∑i

e-

∑i>j

1

rij+

nuclei

∑K

nuclei

∑K>L

ZKZL

RKL


The MM hamiltonian

HMM =bonds

∑b

1

2kb(r − rb)2 +

angles

∑a

1

2ka(θ −θa)2 +

dihedrals

∑d

∑n

Vn

2(1 + cos(nω− γ))

+atoms

∑i

atoms

∑j>i

{1

4πε0εr

qiqjrij

+ εij

[(σij

rij

)12

−2

(σij

rij

)6]}

The QM/MM hamiltonian

HQM/MM =−e-

∑i

classical

∑C

QC

riC︸︷︷︸e−− chargeinteractions

+nuclei

∑K

classical

∑C

ZKQC

RKC︸︷︷︸nuclei - charge

interactions

+V van der WaalsQM/MM

QM/MM Methods: Foundations (6)re-writing of the equations into electrostatic and non-electrostaticinteractions

H = Helec +Hnon-elec

Helec =−1

2

e-

∑i

∆i −e-

∑i

nuclei

∑K

ZK

riK+

e-

∑i

e-

∑i>j

1

rij︸︷︷︸standard equations

+e-

∑i

classical

∑C

−QC

riC︸︷︷︸wavefunction polarization

by external charges

Hnon-elec = HMM + V van der WaalsQM/MM +

nuclei

∑K

classical

∑C

ZKQC

RKC+

nuclei

∑K

nuclei

∑K>L

ZKZL

RKL

= HMM + V van der WaalsQM/MM + V nuclei

QM+QM/MM


QM/MM Implementations

â Helec, and V nuclei

QM+QM/MM can be computed using a standard quantummechanics code.

â The term describing the electrons-classical charge interaction isincorporated into the core Hamiltonian of the quantum subsystem(electrostatic embedded scheme).

â HMM, and V van der Waals

QM/MM are computed using standard molecularmechanics code and are relatively easy to implement.


Calibrating QM/MM interactions

â The calibration of the QM/MM interactions is the main problemfacing QM/MM methods

â The QM/MM interaction should reproduce quantitatively theinteraction between the classical and the quantum parts as if thesystem was computed fully quantum mechanically

â The quantitative reproduction of the QM/MM interactions dependson three points

1. The choice of QC or more in general the choice of the MM force field2. The choice of the van der Waals parameters to describe V van der Waals

QM/MM

3. The way the classic charges polarize the quantum subsystem


The choice of QC

â QC must be chosen to reproduce the electrostatic field due to theMM part onto the QM part

â It is a good approximation to take the charge definition from anempirical force field and incorporate those charges into Helec

â Because MM charges are designed to properly reproduceelectrostatic potentials

â However MM charges can differ greatly between force fields

â No systematic studies so far


The choice of the van der Waals components

â Specific sets of van der Waals parameters and potential energyshould be redefined to properly reproduce non-electrostatic QM/MMinteractions

+ / all these parameters are MM (QC ), QM and basis sets dependent

Selected papers

â small solute in waterFreindorf, M.; Gao, J. J. Comput. Chem. 1996, 17, 386–395Riccardi, D.; Li, G.; Cui, Q. J. Phys. Chem. B 2004, 108, 6467–6478

â protein, nucleic acidsFreindorf, M.; Shao, Y.; Furlani, T. R.; Kong, J. J. Comput. Chem. 2005,26, 1270–1278Pentikainen, U.; Shaw, K. E.; Senthilkumar, K.; Woods, C. J.;Mulholland, A. J. J. Chem. Theory Comput. 2009, 5, 396–410

â beyond Lennard-JonesGiese, T. J.; York, D. M. J. Chem. Phys. 2007, 127, 194101


Classical charge polarization

â ab initio: similar to electron-nuclei interaction

H′core

= Hcore−electrons

∑i

classical

∑C

QC

ric

E ′core

µν = < µ|H′core|ν >

= < µ|Hcore|ν >−∑i

∑C

< µ|QC

riC|ν >


Classical charge polarization: the special case for semiempirical methods

ab initio semiempirical

QM e−–nuclei⟨

µ

∣∣∣−ZKRKi

∣∣∣ν⟩ −Z ′K (µν |sK sK )

QM/MM e−–MM charge⟨

µ

∣∣∣−QCRKi

∣∣∣ν⟩ −QC (µν |sC sC )

QM nuclei–nuclei ZKZLRKL

Z ′KZ ′L(sK sK |sLsL)f (RKL)

+ Z ′KZ ′Lg(RKL)/RKL

QM/MM nuclei–MM charge ZKQCRKC

many ways. . .

â Field, M.; Bash, P.; Karplus, M. J. Comput. Chem. 1990, 11, 700–733

â Luque, F. J.; Reuter, N.; Cartier, A.; Ruiz-Lopez, M. F. J. Phys. Chem. A 2000,104, 10923–10931

â Wang, Q.; Bryce, R. A. J. Chem. Theory Comput. 2009, 5, 2206–2211

QM/MM Methods: Cutting Covalent Bonds (1)

C C

Classical Part Quantum Part

Incomplete valency

â Link Atoms

â Connection Atoms

â Local Self Consistent Field

â Generalized Hybrid Orbitals


Link atom method 1

â A monovalent atom is added along the X—Y bond= the link atom

â Usually the link atom is an hydrogen, but some implementations usea halogen-like fluorine or chlorine

â Interaction with the MM part ?It should interact with the MM part, except for the few closestatoms 2

â The link atom can be free or constrained along the X—Y bond

â Easiest implementation

â Give accurate answers as long as it is placed sufficiently far awayfrom the reactive atoms (3-4 covalent bonds)

1Field, M.; Bash, P.; Karplus, M. J. Comput. Chem. 1990, 11, 700–7332Reuter, N.; Dejaegere, A.; Maigret, B.; Karplus, M. J. Phys. Chem. A 2000, 104,

1720–1735


Connection atoms3 4

â A monovalent pseudo-atom is added at the Y position= the connection atom

â Its behavior mimics the behavior of a methyl group

â semiempirical: Antes and Thiel, 1999

â DFT (pseudo-potential): Zhang, Lee and Yang, 1999

â Pro: no supplementary atom(MM: Y atom; QM: connection atom)

â Con: Need to reparametrize each covalent bond type(C-C, C-N, etc)

3Antes, I.; Thiel, W. J. Phys. Chem. A 1999, 103(46), 9290–92954Zhang, Y.; Lee, T. S.; Yang, W. J. Chem. Phys. 1999, 110, 46

Zhang, Y. Theor. Chem. Acc. 2006, 116, 43–50


Local Self Consistent Field 5 6 7

â the two electrons of the frontier bond are described by a strictlylocalized bond orbital (SLBO)

â its electronic properties are considered as constant during thechemical reaction

â Using model systems and the MM transferability assumption ofbond properties, it is possible to determine the representation of theSLBO in the atomic orbital basis set of the quantum part

â By freezing this representation, the other QM molecular orbitals,orthogonal to the SLBOs, are generated using a local self consistentprocedure

5Thery, V.; Rinaldi, D.; Rivail, J.-L.; Maigret, B.; Ferenczy, G. J. Comput. Chem.1994, 15, 269–282

6Assfeld, X.; Rivail, J.-L. Chem. Phys. Lett. 1996, 263(1–2), 100 – 1067Monard, G.; Loos, M.; Thery, V.; Baka, K.; Rivail, J.-L. Int. J. Quant. Chem.

1996, 58(2), 153–159


Local Self Consistent FieldTo simplify:

1. The MOs describing the frontier bonds are known(transferable SLBO extracted from a model system)

⇓2. The other MOs describing the rest of the quantum fragment are

built orthogonally to the frozen orbitals with a local SCF procedure.

â LSCF is available at the semiempirical and ab initio levels

â Pro: no supplementary atom, proper chemical description of theX—Y bond

â Con: difficult to implement, especially in ab initio


Generalized Hybrid Orbitals 8

â Extension of the LSCF method

â the classical frontier atom isdescribed by a set of orbitalsdivided into two sets of auxiliaryand active orbitals

â The latter set is included in theSCF calculation, while the formergenerates an effective corepotential for the frontier atom

â Available at the semiempirical, SCC-DFTB 9 and ab initio 10 levels

â Pros and Cons similar to LSCF

8Gao, J.; Amara, P.; Alhambra, C.; Field, M. J. J. Phys. Chem. A 1998, 102,4714–4721

9Pu, J.; Gao, J.; Truhlar, D. G. J. Phys. Chem. A 2004, 108, 5454–546310Pu, J.; Gao, J.; Truhlar, D. G. J. Phys. Chem. A 2004, 108, 632–650

QM/MM Methods: the case of ONIOM (1)

Some peculiar QM/MM methods: ONIOM-like methods

What we wouldlike to model

Size of the system

Level of computationsLow Level

High Level

Large

Small 1

2(1+2)

(1)

Etotal = ELow1+2 + EHigh

1 −ELow1

QM/MM Methods: the case of ONIOM (2)

Different Approaches

â IMOMM11: QM/MM with no MM charge inclusion into the QMcore hamiltonian (no QM polarization in the original version)

â IMOMO12: QM/QM (low level QM polarization)

â ONIOM13: N-layered scheme

Etotal = ELow1+2+3 + EMedium

1+2 −ELow1+2 + EHigh

1 −EMedium1

+ Note to Gaussian Users: please use the ’EmbedCharge’ keyword ,

Cutting covalent bonds

â Link atom scheme

11Maseras, F.; Morokuma, K. J. Comput. Chem. 1995, 16, 1170–117912Humbel, S.; Sieber, S.; Morokuma, K. J. Chem. Phys. 1996, 105, 195913Svensson, M.; Humbel, S.; Froese, R. D. J.; Matsubara, T.; Sieber, S.; Morokuma,

K. J. Phys. Chem. 1996, 100, 19357–19363

Availability of QM/MM methods

Commercial and academic software (non exhaustive list)On the MM side:

â AMBER

â BOSS

â GROMACS +Gaussian/GAMESS/CPMD

On the QM side:

â CP2K

â CPMD (with GROMOS)

â Gaussian09 + ONIOMimplementation

â NWCHEM

â Qsite

Other software (non exhaustive list)

â ChemShell: a layer on top of other QM and MM software(Daresbury, UK + P. Sherwood)

â Tinker-Gaussian (Nancy, France + X. Assfeld & M. F. Ruiz-Lopez)

â Tinker-Molcas (Marseille, France + N. Ferre)

Documents

Fundamentals of Quantum Mechanics for Chemistrygmonard.wdfiles.com › local--files › enseignement:main › chap1.pdf · 1. Fundamentals of Quantum Mechanics for Chemistry 3 Hartree-Fock