benedetta mennucci

Benedetta MennucciDipartimento di Chimica e Chimica Industriale

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

Applications of quantum mechanical continuum solvationmodels to the study of molecular properties and

spectroscopic features of molecular solutes

Applications of quantum mechanical continuum solvationmodels to the study of molecular properties and

spectroscopic features of molecular solutes

IMA Thematic Year on Mathematics and Chemistry:

SolvationTutorial:

Theories of Solvation Within Quantum

Chemistry

December 7, 2008

Quantum-Mechanical issue: the QM model has to properly describe the effects of solute-solvent mutual polarization at both electronic and structural level

Physical issue:the solvation model has to properly includethe main physical interactions giving rise to solvent-induced change on the property/spectroscopic signal

The QM description of solvent effects on molecular properties and

spectroscopies

The QM description of solvent effects on molecular properties and

spectroscopies

the Energy

The Quantum Mechanical issueThe Quantum Mechanical issue

We need a proper definition of molecular properties

We have to start from a quantity which is the basic element of any QM description:

The molecular property can be defined in terms of the change of the energy of the system with respect to a perturbation (an external

electric, or magnetic field, a geometrical deformation …): thisdefinition is still valid for a solvated system

The energy is expanded in a Taylor series in the perturbation strength λ

The nth-order property is the nth-order derivative of the energy

For example, by considering fourtypes of perturbations: external

electric (F) or magnetic field (B), nuclear magnetic moment (nuclearspin, I) and a change in the nuclear

geometry (R).

PropertyF B I R

F B I R

n n n n

n n n n

EF B I R

+ + +∂∝∂ ∂ ∂ ∂

The Quantum Mechanical issueThe Quantum Mechanical issue

IR spectra

NMR spectra

MO-LCAOFinite basis approximation:

Orthonormality † =C SC 1

=FC SCε

= χCφ

† †K K

K

= =∑D CC c cDensity matrix:

( ) ( )1 ( )2 NNE tr tr V= + +Dh DG D

( )1 2| | .. | | ...K=C c c c

( ) ( ) ( )= + − = +F h J D K D h G D

| |h hαβ α βχ χ= ( )| | | |G D g gαβ σδ α σ β δ α σ δ βσδ

χ χ χ χ χ χ χ χ= −∑

|Sαβ α βχ χ=

Fock matrix

An example: the Hartree-Fock description

Energy

( ) ( )( ) ( ) ( ) ( )1 ( )2 NN

E tr tr tr Vλ λ λ λ

λ∂

= + − +∂

Dh DG D S DFD

HF: First derivative

Easy to calculate: only derivatives of integrals on the atomic basis

Example:

First-order algorithms for geometry optimizations

( ) 0stationary

E xx

∂=

∂xx0 x0+δx

First derivative: geometry optimization in solution

Derivatives of the geometrical parameters which define the

cavity surface

Differentiation of the reaction terms = derivatives of ASC charges

Ri i

i

vac vac q= + = +∑F F FV V

( ) 0stationary

E xx

∂=

∂xx0 x0+δx

( ) ( ) ( )( ) ( ) (( ) ( )) ( )1 1(12 2

( ))2

Rx x xx RxN

xN

E tr tr tr Vtr Ux

∂= + + − + +

∂Dh DG D S D DDV D F

Effective Fock :

( ) ( )( ) ( )

2( , ) ( , ) ( , ) ( , )

( ) ( ) ( ) ( ) ( ) ( )

1 ( )2

( )

NNE tr tr V tr

tr tr tr

λ η λ η λ η λ η

η λ η λ λ ηη λ∂

= + + −∂ ∂

+ + −

Dh DG D S W

D h D G D S W( ) ( ) [ ] ( ) ( )( )η η η η η= + + +W D FD DF D DG D D DFD

HF: Second derivative

( )How to get ?ηD

More difficult: we need to calculatederivatives of the density matrix

Example:

IR Vibrational frequencies and intensities and NMR parameters

− =FD DF 0

( )( )† † †= = =DD CC CC C1C D

(0) (1) (1) (0) (1) (0) (0) (1) 0− + − =F D D F F D D F(0) (1) (1) (0) (1)+ =D D D D D

Switching on the pertubation and expanding with respect to it:

Coupled perturbed Coupled perturbed HFHF

(1) (1) (1) (0) (0) (1)( ) ( )= + +F h G D G D

First-order Coupled Perturbed Hartree-Fock (CPHF):

Hartree Fock stationary conditions

First-order changein the Fock matrix

Coupled perturbed equations in the presence of a polarizable dielectric

And in solution?And in solution?

Effective Fock :

ASC reaction charges q:

As a result the final density matrix (and the corresponding response properties) will be changed by the presence of the polarizable

environment

( )Rk k

kq=∑V D V

va Rc= +FF V

(1) (1) (1) (0) (0) ( (1) (0) (0) () 1)1 ( ) ( )( ) ( ) R R= + + + +F h G D VD V DG D

(0) (1) (1) (0) (1) (0) (0) (1) 0− + − =F D D F F D D F

Direct effects: solvent induced changes in the solute electronic charge

The physical issue: a tentative schematization of solvent

effects


effects

Indirect effects:solvent induced changes in the solute geometry and/or in the relative energies of different conformers

Direct vs Indirect effects: an exampleDirect vs Indirect effects: an example

N-Methyl Acetylproline Amide (NAP) in water:

a simple model for peptides

The conformation of NAP in polar, protic solvents, such as water or alcohols, is still controversial.

In particular, it has been recently demonstrated that the results obtained by using Amber MD simulations strongly depend on the force field exploited (ff99 vs. ff03).

H2C

CH2

CH2

CHNH2

O

N

O

H3C

φψ

Oh, K.-I.; Han, J.; Lee, K.-K.; Hahn, S.; Han, H.; Cho, M. J. Phys. Chem. B 2006, 110, 13335.

PII310 Helix I

Boltzmann Populations (%)

H2C

CH2

CH2

CHNH2

O

N

O

H3Cφ

ψ

C7Conformational analysis: three stable conformers

Conformational analysis: Conformational analysis: three stable conformersthree stable conformers

-PII

1310 Helix I

99C7

In gas-phaseB3LYP/ 6-311++G**

C. Cappelli and B. Mennucci J. Phys. Chem. B 2008, 112, 3441.

The C7 conformer dominates in gas-phase due to the

presence of a stabilizing intra-

molecular H-bond.

PII310 Helix I

68-PII

281310 Helix I

499C7

In waterPCM-B3LYP/ 6-311++G**

In gas-phaseB3LYP/ 6-311++G**

Boltzmann Populations (%)

C7

C. Cappelli and B. Mennucci J. Phys. Chem. B 2008, 112, 3441.

Conformational analysis: three stable conformers

Conformational analysis: Conformational analysis: three stable conformersthree stable conformers

H2C

CH2

CH2

CHNH2

O

N

O

H3Cφ

ψ

In water the other two conformers become

important: they present a better interaction with the solvent (polar groups are

more exposed to the solvent)

1500 1550 1600 1650 1700 1750 1800 18500

5

10

15

I (ar

b. u

nits

)

ν (cm-1)

1500 1550 1600 1650 1700 1750 1800 1850

-300

-200

-100

0

100

200

300

400

500

R (a

rb. u

nits

)

ν (cm-1)

1500 1550 1600 1650 1700 1750 1800 1850

-1500

-1000

-500

0

500

1000

R (a

rb. u

nits

)

ν (cm-1)

1500 1550 1600 1650 1700 1750 1800 18500

10

20

30

40

I (ar

b. u

nits

)

ν (cm-1)

C7

310HelixI

IR

VCD

In water

Infrared and vibrational circular dichroism spectra of single conformers: direct effects

Infrared and Infrared and vibrationalvibrational circular circular dichroismdichroism spectra of spectra of single conformers: single conformers: direct effectsdirect effects

PII

Bothposition and intensity of

peakschange

passing fromgas-phase to

water

In gas-phase

B3LYP/6-311++G**

Amide IAmide II

1500 1550 1600 1650 1700 1750 1800 1850

-1000

-500

0

500

1000

R (a

rb. u

nits

)

ν (cm-1)

1500 1550 1600 1650 1700 1750 1800 18500

5

10

15

20

25

30 gas water water@gas_population

I (ar

b. u

nits

)

ν (cm-1)

Exp taken from: Oh, K.-I.; Han, J.; Lee, K.-K.; Hahn, S.; Han, H.; Cho, M. J. Phys. Chem. B 2006, 110, 13335–13365.

Averaged Infrared and vibrational circular dichroismspectra: direct vs indirect effects

Averaged Infrared and Averaged Infrared and vibrationalvibrational circular circular dichroismdichroismspectra: direct spectra: direct vsvs indirect effectsindirect effects

EXP

B3LYP/ 6-311++G**

Bulk versus

specificeffectsDirect effects:

solvent induced changes in the solute electronic charge

Indirect effects:solvent induced changes in the solute geometry and/or in the relative energies of different conformers


effects


effects

Bulk versus specific solvationBulk versus specific solvation

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

Macro- and micro-solvation

Bulk (averaged) effects Local (specific) effects

To understand solvent effects at the molecular scale, some questions need to be addressed:

How does solvation at the solute surface differ from that of bulk?Are there local rigid structures of solvent at the solute surface?What are the time scales for solvent dynamics at the solute surface?

No significant differences between surface and bulk: continuum models

Rigid and/or persistent solvent structures at the surface: ?

Bulk versus specific solvationBulk versus specific solvation

Solvent structures at the surfaceSolvent structures at the surfaceSupermolecule = solute surrounded by some explicit solvent molecules

It properly accounts for short-range effects.

The different components of the supermolecule can be describedat the same level (QM) or using an hybrid approach: a better level

for the solute and a more approximated level for the solventmolecules (semiempirical or MM)

The SupermoleculeThe SupermoleculeHow many explicit solvent molecules are needed?

From the chemicalanalysis of the system

3 possible H-bonds: 2 on the O(C) and 1 on the H(N)

Radial distributionfunctions for the O-H pairs

From MD simulations

---- CO-H

---- NH-O

O-Hw RDF: On average, 2 hydrogenatoms of 2 water molecules hydrate the carbonyl group by means of hydrogenbonding.

(N)H-OW: The integration number up to the first minimum gives 1 water molecule H-bonded to the H(N).

An example: N-methyl acetamide in water

The SupermoleculeThe Supermolecule

Which configuration?

◊ From QM geometry optimizations: the most stable configuration

◊ From MD simulations:

Proper description for strongly interacting solute-solvent systems giving rise to stable clusters.The structure of strong H-bonded solute-solvent can be properly described with QM geometry optimizations

Better for weaker solute-solvent interactions described by a more dynamic situation.

From MD simulations: how does it work?

First we perform an analysis of the radial and spatial distribution functions so to obtain information on how solvent molecules pack

in “shells” around some specific atoms in the solute

Then we select solute-solvent clusters on the basis of a cutoff distance (rcut): each cluster includes all solvent molecules inside

the sphere of radius= rcut

The value used for rcut has to be chosen so the represent the first solvation shell.

Finally, in order to achieve a statistically meaningful description of the liquid, we introduce averages on many

different clusters obtained with the same cutoff

Clusters from MD simulationsClusters from MD simulations

Calculations of the property of interest have to be repeated for all the clusters so to obtain a correct average value

Advantage:

Proper description of weak solute-solvent specific interactions which cannot be represented by a single

configuration obtained from a QM geometry optimization

Disadvantage:

Quite demanding from a computational point of view

How can we include long-range effects?

Enlarging the dimension of the supermolecule:


OH

H

O H

H

O H

HHOH

HO

H

HO

H

HOH

HO

H

H O 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

Problems:

1. By increasing the dimensions, the accuracy of the QM level has to be largelyreduced, or an hybrid QM/MM has to be introduced

2. The problem of statistical representativity becomes very difficult: increasingthe dimension of the cluster we need more and more clusters so to get a correct picture of the liquid

How can we include long-range effects?


Adding an “external” continuum:

the solvated supermolecule

The main problems disappear:

1. The dimensions do not increase, we do not need to reduce the accuracy of the QM level, or to shift to an hybrid QM/MM method

2. The statistical representativity is automatically satisfied by using the continuum description in terms of the solvent bulk properties.

An example of bulk versus specific effects: Nuclear Magnetic Resonance (NMR) spectraAn example of bulk versus specific effects:

Nuclear Magnetic Resonance (NMR) spectra

( ( )( ) )local neigh solventbourσ σ σ σ= + +

Electron density around the nucleus studied

Neighboring chemical groups in the molecule studied

Solvent molecules around the nucleus studied

νL= γBloc/2π = (1- σ) γB0/2π

The resonace frequency for a particular nucleus depends on its molecular environment. This is why NMR is such a useful tool

for structure determination… and solvent effects.

Shieldingconstant

Pyridazine Pyrimidine

Pyrazine

15N nuclear shieldings of diazines in various solvents

An example of bulk versus specific effects: Nuclear Magnetic Resonance (NMR) spectraAn example of bulk versus specific effects:

Nuclear Magnetic Resonance (NMR) spectra

Different electronicdistributions:

different interactionswith a solvent

5 10 15 20 25 30 35 40 45

5

10

15

20

25

30

35

40

45

In acetone In DMSO In water

PCM

(pp

m)

Exp (ppm)

N

N

N

N

NN

B3LYP(GIAO)/6-31+G(d,p)

In water the calculated values are smaller than what observed:

a part of the solvent effect is missing

15N nuclear shieldings of diazines in solution

10 15 20 25 30 35 40 45

10

15

20

25

30

35

40

45

Calc

(pp

m)

Exp (ppm)

In gas With PCM

QM optimizedsupermolecule

A continuum-only approach

Specific effects are now correctlyaccounted for but only with inclusion of bulk effects (PCM clusters) the observed

solvent effect is reproduced

-5 0 5 10 15 20 25 30 35 40 45

A more difficult case: N-Methyl-acetamide in acetone


B3LYP/6-311+G(d,p)

Error (ppm) with respect to exp

B. Mennucci, J.M. Martinez, J. Phys. Chem. B, 109, 9830 (2005)

<MD>+PCM

Nuclear shieldings: 17O (red) and 15N (blue)

Overestimation for O

Why a so different accuracy forthe two nuclei?

Very good for N

PCM : Only continuum

PCM

HC

HH

HC

HH

O

Weak but persistent interactionsWeak but persistent interactions


From MD simulation:

Spatial Distribution Function (SDF)

Methyl groups create a cage which prevents the polar part of the acetone molecules to be in close contact with NMA oxygen.

Bulk effects introduced by the PCM leads to an excessive polarization on NMA oxygen.

An homogeneous sphere-like distribution of

methyl groups is found around the NMA oxygen

-5 0 5 10 15 20 25 30 35 40 45



B3LYP/6-311+G(d,p)

Error (ppm) with respect to exp


PCM

<MD>+PCM

Nuclear shieldings: 17O (red) and 15N (blue)

Overestimation for O

Very good for N

Only averages on clusters extractedfrom MD snapshots correctly

reproduce the solvation on oxygen

<MD>+PCM: averages on MD clusters+continuum

PCM : Only continuum

Some conclusive remarks

Solvent effects are difficult to define: they act differently on different quantities

Solvation free energies are generally “globally sensitive”:

they are properly described by a continuum model even when specific effects are present

Molecular response properties are generally “locally sensitive”:

they cannot be properly described by a continuum model when specific effects are present

Solvent effects on molecular properties: Global versus local

Solvent effects on molecular properties: Global versus local

Persistent versus LabilePersistent versus LabileA useful simplification for “locally sensitive” properties:distinction between persistent and labile interactions, based on their characteristic residence time.

► persistent interactions:

► labile interactions:

QM optimized supermolecules can be used and the effects of outer-shell molecules can be included with the continuum model.

QM optimized supermolecules are not representative of the real situation; structures extracted from MD simulations are to be preferred.

Necessity of a double average on the solvent molecules:

for the first shell: average on different MD clusters

for the outer shells: an implicit average through the continuum

Which Theoretical Model?

Not a unique answer!

The choice has to be preceded by an analysis on:

(i) the chemical system,

(ii) the properties of interest,

(iii) the required accuracy,

(iv) the computational cost

Solvation is an intrinsically dynamic and long-range phenomenon: statistical treatments involving averaging and

fluctuations from averages are important.

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