theoretical models for the solvent effect · theoretical models for the solvent effect. models for...

Benedetta MennucciDipartimento di Chimica e Chimica Industriale

Web: http://benedetta.dcci.unipi.it Email: [email protected]

Theoretical models for the solvent effect

Theoretical models for the solvent effect

Models for the solvent effectsModels for the solvent effects

1. Introduction:Main aspects of the modelling of molecular systems in condensed phase.

2. Discrete Methods: Main elements of Force Fields.Elements of Monte Carlo (MC) and Molecular Dynamics (MD) simulations.Some extensions.Potentialities and limits.

3. Continuum Methods: The Onsager MethodThe molecular cavityThe electrostatic problem.Apparent surface charge (ASC) methods

4. The Polarizable Continuum Model (PCM)Discretization of the cavity surfaceAlternative PCM formulationsExtension to interphases liquid/liquid and liquid/gas and to liquid crystals

5. PCM: the quantistic problem.The “reaction field” operator and the Self-Consistent Reaction Field (SCRF) method.The problem of the outlying charge.

6. The calculation of solvation free energyThe definition of the free energy functional Nonelectrostatic contributionsVibrational and rotational contributions

Models for the solvent effectsModels for the solvent effects7. Solvent effects on molecular properties and spectroscopiesElectric properties (polarizability and hyperpolarizability)Molecular Vibrations: IR and Raman spectroscopiesMagnetic properties: nuclear shielding nuclear and spin-spin coupling constant

10. PCM in Gaussian 03The construction of the inputThe analysis of the output

8. Excited electronic states in solutionThe dynamics of the solvent responseAbsorptions: the Franck-Condon approximationRelaxation and Emissions

9. Solute-solvent specific effectsThe “solvated supermolecule” approachThe statistical problem

1. Introduction

Main aspects for the modelling of molecular systems in condensed phase.

Atomic scale Nanoscale(0.1-10 nm)

Nanoscale(10-15-10-9 s)

Atomic scale

Mesoscale(10-1000 nm)

Mesoscale(10-8-10-2 s)

Macroscale (> 1 s)

Macroscale (> 1 μ)

Time and length Scale in Simulations

Solvent effects on molecularproperties and processes

Membrane

InteractionsInteractions

As first step, any theoretical model requires the definition of the interactions acting among particles.

Isolated moleculeelectrons and nuclei to form the molecule

Liquidsmolecules to form the solution

– Bulk polarization– Site binding or specific solvation– Preferential hydration– Acid/base chemistry– …

ze: chargeα: polarizabilityμ: dipole moment I: ionizationpotential

Liquids: Non-Covalent InteractionsLiquids: Non-Covalent InteractionsTypical energies

(kcal/mol)

60

4

dip-dip: < 1

HB: 3-5

< 1

< 2

Empirical (molecular mechanics, MM)

Classical description (no electrons). Very fast, but limited in scope. Need of introducing parameters. Feasible on large systems

Modeling of Intramolecular Interactions. Modeling of Intramolecular Interactions.

Quantum-mechanical (electronic description)

1. Semi-empirical

Involving some empirical parameters. Feasible for large systems - Generallyinadequate for weak interactions and/or accurate studies

2. Ab initio

From first principles. Computationally expensive, but of more general scope.Feasible on “small-medium” systems.

Modeling of Intermolecular Interactions. Modeling of Intermolecular Interactions.

Discrete (or miscroscopic description)

1. Statistical-mechanics (Monte Carlo, Molecular Dynamics)

Classical MM (force fields) for all molecules.

2. QM/MM

QM for a small part of the system and MM for all the rest.

Continuum

QM molecular system (the solute) placed in a cavity within a continuous medium (the solvent) characterized by macroscopic properties

2. Discrete Methods

Main elements of Molecular Mechanics: the Force Field.

How does Molecular Mechanics (MM) work?

How does Molecular Mechanics (MM) work?

• Two main aspects:

– Functions that describe the forces acting on the atoms» Force Field

– Numerical integration methods, to calculate the motion of the atoms due to the forces acting on them

» Molecular simulations (MD, MC, etc)

Functions that describe the forces acting on the atoms: the Force Fields

Functions that describe the forces acting on the atoms: the Force Fields

• a force field is comprised of – functional forms for the interactions– numerical parameters to define functional forms– atom types to define parameters

• each atomic number is divided into atom types, based on bonding and environment (e.g. carbon: sp3, sp2, sp, aromatic, carbonyl, etc.)

• parameters are assigned based on the atom types involved (e.g. different C-C bond length and force constant for sp3-sp3 vs sp2-sp2)

• parameters chosen to fit structures (in some cases also vibrationalspectra, steric energies)

• individual terms are approximately transferable (e.g. CH stretch in ethane almost the same as in octane)

• many examples: MM2, MM3, Amber, Sybyl, Dreiding, UFF, MMFF, etc.

• differ by the functional forms and parameters• not mix and match - each developed to be internally

self consistent• some force field use united atoms (i.e. H's condensed

into the heavy atoms) to reduce the total number of atoms (but with a reduction in accuracy)

Force FieldsForce Fields

Atoms as spheres and bonds as springs, interactions between particles in terms of analytical functions derived from classical

mechanics.

Energy = Stretching Energy + Bending Energy + Torsion Energy + Non-Bonded Interaction Energy

The Anatomy of a Force-FieldThe Anatomy of a Force-Field

• Many force fields use just a quadratic term:Estr = Σi ki (ΔRi)2 (harmonic oscillator)(ki= force constant for bond i)but the obtained bond energy for stretched bonds is too large

• Morse potential is more accurate, but is usually not used because of expense :

Estr = Σi Di[1-exp(-βΔRi)]2

Estr(R) = E(R0) + dE/dR (R – R0) + ½ d2E/dR2 (R – R0)2 + …..R0 = equilibrium value (not-stretched bond)

D

Bond Stretch TermBond Stretch Term

• usually a quadratic polynomial is sufficient :

Ebend = Σi ki (θi – θ0)2

• for very strained systems (e.g. cyclopropane)

a higher polynomial is better

Ebend = Σ ki (Δθi)2 + k’i (Δθi)3 + k”i (Δθi)4 + . . .

θ0=109º θ0=60º

Angle Bend TermAngle Bend Term

Torsional TermTorsional Term

• most force fields use a single cosine with appropriate barrier multiplicity, n

Etors = Σ A[1+cos(nτ – φ)]

A controls the amplitude of the curve

n controls its periodicity

φ shifts the entire curve along the rotation angle axis (τ).

Non-bonded termsNon-bonded terms

• electrostatic• van der Waals

– Repulsive part– Attractive part

• Hydrogen bonds

Enon-bond = Ees + EvdW + EHbond

Electrostatic species• “Monopole” (point charges)

– Ions – Acid groups –(COO)-

– Basic groups –(NH3)+

• Dipole: +ne & -ne separated charges– Usually partial charges

• δ+C=Oδ- δ+H—Nδ-

Electrostatic Interactions• Monopole-Monopole

• Dipole-Dipole

Coulomb i jes

i j ij

q qE

R>

= ∑

( )3 cos - 3 cos cosA Bes A B

µ µER

χ α α=

• Overlap of charge densities• It becomes important when atoms are close• It rapidly increases with distance

Van der Waals interactionsVan der Waals interactions

Energy

r/Å01 2 3 4 5

• Dispersive interactions (London): attractive forces between momentary induced dipoles (they do not require permanent dipoles)

– They are always present– They are of the order of 0.01 to 2 kcal/mol (weak forces)– They increase with the number of electrons

Repulsive component

Attractive component

Attractive “London Dispersion”– Negative– Less steep than

repulsionTotal

Ener

gyr/Å0

1 2 3 4 5

Repulsion- positive- Very steep

Most favorable separationMost favorable separationSum of van der Waals radii of the two atoms

Atoms cannot go too close: they approximated as rigid spheres


• Lennard-Jones potential

R

EvdW

Rij

ε

21/6σij

Rij

dispersionrepulsion

12 6

vdWij ij

E 4R R

ij ijij

i j

σ σε

<

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟= −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠∑i j


• σ and ε are tabulated for each atom

ij terms are obtained as arithmetical and geometrical averages:

σij = (σi + σj)/2; εij = (εi εj)1/2

Hydrogen Bonding InteractionsHydrogen Bonding Interactions

• In some force fields they are represented with a specific van der Waalsterm in which ε is large and σ small.

• Alternatively an extra term is introduced:EHbond = Σ A rij

-12 - C rij-10

• In many force fields one implicitly takes into account the H-bond in the balance between electrostatic and non-bonded terms

They have energies of the order of 2–5 kcal mol–1 (covalent single bonds are of the order of 60–110 kcal mol–1).

δ− δ+

Classically Electronically

2. Discrete Methods

Elements of Molecular Dynamics (MD) and Monte Carlo (MC) simulations

Some extensions

Potentialities and limits of discrete methods

Molecular Dynamics (MD)Molecular Dynamics (MD)

Molecular dynamics is an extension of the molecularmechanics approach

It is based on the idea that the atoms of the molecule feel forcesand want to move.

Each atom is treated as a particle responding to Newton’s equations of motion.

Integration of these equations with successive time steps lead tothe trajectory of the atom over time in the form of a list of

positions & velocities

It is a deterministic technique: given an initial set of positions and velocities, the subsequent time evolution is in principle completely determined

It is a statistical mechanics method: it is a way to obtain a set of configurationsdistributed according to some statistical distribution function, or statisticalensemble.

The time evolution of a set of interacting atoms is followed by integrating the classical (Newton) equations of motion :

2

2

dtdm

ddV r

r=−

V = potential energy at position r defined in terms of the force field (generally limited to a two-body term V(i,j))

m=F a


Simulation box

A fixed (thousands) number of solvent molecules is explicitly taken into account: this forms the simulation box

A realistic model of a solution requires a very large number of solvent molecules to be included along with the solute.

However the computational times should remain feasible

Simply placing the solute in a box of solvent is not sufficient:a large number of solvent molecules will be at the edge of the

solvent and the surrounding vacuum. This is obviously not a realistic picture of a bulk fluid.

The dimension of the box and the number of molecules contained are related: they have to represent the correct density of the liquid

Molecular Dynamics (MD): liquid solutionsMolecular Dynamics (MD): liquid solutions

Periodic boundaryconditions

As a particle moves out of the simulation box, animage particle moves in toreplace it.

To prevent the outer solvent molecules from ‘boiling’ off into space periodic boundary conditions are employed

The particles being simulated are enclosed in a box which is then replicated in all three dimensions to give a periodic array:

Computationally intensive: it requires the evaluation of an infinite number of interacting pairs.

A non-bonded cutoff distance is defined such that each atom ‘sees’ onlyone image of all of the other atoms: the complexity is greatly reduced.


Systems containing thousands of atoms, simulation times ranging from a few picoseconds (10-12 s) to hundreds of nanoseconds (10-9 s).

The simulation is “safe” when the simulation time is >> than the relaxation time of the quantities we are interested in.

A time step of 0.5-2.0 fs ascertains that all aspects of molecular motion are sampled correctly.Large molecules have a broad range of vibrational frequencies. To make sure that at least one full period of low-frequency oscillations is sampled, the simulation time should be at least 10 ps.

How many molecules, Which times?


Time average of the function B[R(t)] that describes the instantaneous value of the property B as a function of the position R in the phase-space:

A set of time-correlated points (a trajectory) in the phase-space are generated starting from a suitable set of coordinates and velocities, computing for each

time step the property B[R(t)]:

[ ] [ ]0

1( ) lim ( )t

B B R t B R t dtτ

τ τ→∞= = ∫

[ ]1

1 ( )t

B B R tτ

τ =

= ∑

Calculation of a physical quantity with MD


Stochastic techniques: use random number generation.

The physical system is described in terms of probability distribution functions, PDFs.

These describe the evolution of the system, whether in space, or energy, or time, or even some higher dimensional phase space.

MONTE CARLO (MC)MONTE CARLO (MC)

The goal: to simulate the physical system by random sampling from PDFs and by performing the supplementary

computations needed to describe the system evolution.

Stochastic techniques: use random number generation. MONTE CARLOMONTE CARLO

ξ =random number between 0 and 1V(n) < V(o) yes (potential energy decreases)

V(n) > V(o) yes only if ξ ≤ exp(- β ΔV)

The trial move rn=ro+kδrmax is accepted?

The recipe is:Prepare the system in a configuration o (old), find a new configuration n byrandom displacements.

Many simulations are performed (multiple “trials” or “histories”) and the desired result is taken as an average over the number of observations :

Physical property B

( ) exp ( ) /( )

exp ( ) /j jj

j jj jj

X V kTB X P

V kT

⎡ ⎤−⎣ ⎦= =⎡ ⎤−⎣ ⎦

∑∑ ∑r r

rr

1. MC requires to calculate only energy, while MD requires both energies and forces

2. In MC the outcome of each trial move only depends upon its immediate predecessor (no ‘memory’), in MD it is possible to predict the configuration of the system at any time

3. MD has a kinetic energy contribution to the total energy whereasin a MC simulation the total energy is determined directly from the potential energy function.

4. MD provides information about the time dependence of the properties of the system

Monte Carlo (MC) / Molecular Dynamics (MD)Monte Carlo (MC) / Molecular Dynamics (MD)

In the “ergodic” hypothesis:

time average = ensemble average

MD simulations = MC simulations

Advantages1. Very effective to study the structure of the liquid2. Thermodynamic quantities can be obtained3. MD: dynamical properties and time/dependent

phenomena can be studied

Technical Problems1. Size of the sample2. Boundary conditions3. Quality of the force field4. Cut off for the interactions


• Radial Distribution Function (RDF), g(r):– Measures the number of atoms at distance r from a given

atom compared with the number at the same distance in an ideal homogeneous gas at the same density

– Gives information on how molecules pack in ‘shells’ of neighbors, as well as average structure


Important outputs of molecular simulations

MD:• Mean residence time (MRT) :

– Measures the length of the mean permanence of an atom within a given distance r from another given atom

– Gives information on the dynamics of the solvent structure

- O(NMA) H(water)

- H(NMA) O(water)

The 3D equivalentSpatial Distribution Function (SDF)

Radial Distribution Function (RDF)Radial Distribution Function (RDF)N-methyl-acetamide (NMA) in water

O-Hw RDF: a very well-defined first peak centered at 1.83 Å that integrates to 2.3. On average, two hydrogen atomsof two water molecules hydrate the carbonyl group bymeans of hydrogen bonding.

Approach unity for large distances: the localdensity converges to the ideal reference.

(N)H-OW: a much less structured solvent. The first peak, centered at 2.05 Å not so sharply defined and the integration number up to the first minimum is 1.1.

M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids(Clarendon Press, Oxford, 1987)

A. R. Leach, Molecular Modelling, 2nd edition(Prentice Hall, Harlow, 2001)

D. Frenkel and B. Smit, Understanding Molecular Simulation(Academic Press, San Diego, 2002)


Literature

And the electrons ?

• The system is divided in two subsystems: an inner region (QM) where quantum-mechanics is used and an outer region (MM) where a classical field is used, interacting with each other

The hybrid QM/MM MethodThe hybrid QM/MM Method


The van der Waals term (through a Lennard-Jones potential) also ensures that the QM and MM systems will not get too close because of a lack of electronic structural description of the solvent MM system

/ /elec vdW

eff QM MM M QM MMQ MMH H HHH= + + +

Effective Hamiltonian

Hamiltonian forthe QM solute

molecule

OH

H

O H

H

O H

HHOH

HO

H

HO

H

HOH

HO

H

HO

H

HOH H

OH

H OH

HOH

HO

HH

OH

HOH

HO

H

HOH

HOH

H O H

HO H

H O H

HO

H

HOH

QM

MM

Solvent-solventMM interaction

(force field)

12 6

/1 1

4S M

vdW ms msQM MM ms

s m ms ms

HR Rσ σε

= =

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∑∑

/1 1 1

S N Melec s m sQM MM

s i mis ms

eq Z qHr R= = =

⎡ ⎤= − +⎢ ⎥

⎣ ⎦∑ ∑ ∑Electrostatic QM-MM interaction

Van der Waals QM-MM interaction


Mechanical embedding: QM calculation is performed in the gas phase, without electronic coupling to the environment. The electrostatic interaction between QM and MM regions only in the MM code, through a classical point charge model for the QM charge distribution (e.g. a potential derived charge model). Not accurate model.

Three ways of treating QM/MM electrostatic interaction:

Electronic embedding: the classical partition appears as an external charge distribution (e.g. a set of point charges) in the QM Hamiltonian: polarization of the QM region by the MM charge distribution in the QM electronic structure calculation. The partial charges used to describe the MM distribution are usually derived from force fields. More accurate model.

Polarized embedding: the polarization of the MM region in response to the the QM charge distribution is also included. The most accurate model but also complex and expensive.

/1 1 1

S N Mele s m sQM MM

s i mis ms

eq Z qHr R= = =

⎡ ⎤= − +⎢ ⎥

⎣ ⎦∑ ∑ ∑


The total energy of the fluid system at an instantaneous configuration during an MC or MD simulation is:

However

Ab initio methods are too slow to be practical in MC and MD simulations because millions of electronic structure calculations are needed.Consequently, much less accurate semiempirical methods have been primarily utilized in QM/MM simulations and thus the method is not more accurate than standard MD or MC.

The attractive feature of the method is that it synthesizes the accuracy/generality offered by QM calculations and the computational

efficiency of MM representations.

/ /elec vdW

eff QM MM QMQ MMMMMEE EEH E= Ψ Ψ = + + +

Car and Parrinello (1985): efficient scheme to perform molecular dynamics on a quantum mechanical potential surface (ab initio molecular dynamics AIMD)

Ab Initio Molecular DynamicsAb Initio Molecular Dynamics

Ab initio molecular dynamicsForces on nuclei derived from

instantaneous electronic structure

NucleiClassical mechanics

ElectronsQM theory

[ ]

[ ]

{ },{ }

{ },{ }constraints

QMI i

J JJ

QMI i

ii

E RM R

R

E Rmφ

φ

φφ

φ

∂= −

∂

∂= − +

∂

Dynamics of nuclei on the QM potential surface

Fictitious dynamics of the electronic orbitals

Ab Initio Molecular DynamicsAb Initio Molecular Dynamics

Originally formulated for solids, it combines the Density Functional Theory (DFT)approach to electronic structure calculation with simulation of finite temperature dynamics:

it avoids parameterized force fields!

The technique is still limited in the number of molecules and in the times which can be explored

Not yet competitive with classical MD simulations

QM/MMJ. Gao, in: K.B. Lipkowitz, D.B. Boyd (Eds.), Reviews in Computational Chemistry, vol. 7, VCH, New York, 1996.

AIMDR. Car, M. Parrinello, Phys. Rev. Lett. 55 (1985) 2471

D. Marx, J. HutterModern Methods and Algorithms of Quantum Chemistry,Proceedings, Second Edition, J. Grotendorst (Ed.),NIC Series, Vol. 3, pp. 329, 2000.

LiteratureLiterature

Condensed phase

Change in the point of view

A solvated molecule

From condensed phases to “solvated” molecules:a step back in the level of complexity

From condensed phases to “solvated” molecules:a step back in the level of complexity

Explicit consideration of a large number of trajectories (MD) or of MC steps

Time consumingVery difficult to extend to QM descriptions

How to achieve a statistically

correct description?

Discrete models

How to deal with the large dimension of the systems?

Approximated descriptions: Force Fields

Necessity of large parameterization; Generally non-polarizable potentials; Necessity of unphysical periodic boundaries; Problems with long-range interactions (cut-off)

Solvated molecules: which approach?Solvated molecules: which approach?

Continuum modelsDiscrete models

Solute placed into a cavity within a continuousmedium with macroscopic properties

Solvated molecules: which approach?Solvated molecules: which approach?

Continuum models

How to deal with the large dimension of the systems?

The solvent molecules disappear and they are substituted by a continuous dielectric of infinite extent that surrounds a cavity containing the solute molecule

No need to introduce force fieldsPolarizable dielectricComplete inclusion of long-range interactions

Implicitly accounted for using the macroscopic properties of the continuum (the dielectric

permittivity, the refractive index, etc)

Computationally cheapStraightforward to extend to QM descriptions

How to achieve a statistically

correct description?

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