Modeling solvent effects on electronic transitions with hybrid QM/classical

approaches

Benedetta MennucciDipartimento di Chimica e Chimica Industriale

University of PisaWeb: http://benedetta.dcci.unipi.it

Email: bene@dcci.unipi.it

Helsinki, December 12-15 2011

• 1. Methodological aspects of solvation Models◦ 1.1. Discrete vs Continuum descriptions◦ 1.2. The PCM family of methods◦ 1.3. The coupling with QM methods

Outline

• 2. Excited states & Nonequilibrium◦ 2.1. Dynamic Polarization Response◦ 2.2. Vertical Electronic Transitions◦ 2.3. Relaxation & Fluorescence

• 3. Excitation energy transfer◦ 3.1. Introduction◦ 3.2. QM description of the coupling◦ 3.3. Environment effects

Explicitly (e.g. MD trajectories or MC sampling)

How to achieve a statistically

correct description?

How to deal with the large dimension of the systems?

Molecular Mechanics

Correct description of the interactions but necessity of parameterization Generally non-polarizable

Accurate but time consumingNot straightforward to be extended to QM descriptions

All molecules (solute+solvent) are treated explicitly

Modeling solvent effects: the discrete approach

How to deal with the large dimension of the systems?

The solvent molecules disappear and they are substituted by an infinite continuum dielectric that surrounds a cavity containing the solute molecule

Approximate description of the interactions but no need of parameterizationPolarizable approach

Implicitly: use of macroscopic solvent properties (dielectric constant, refractive index, etc)

Mean field approximation but cheapStraightforward to extend to QM descriptions

How to achieve a statistically

correct description?

Solvated molecules: continuum approachSimplifying at most:

only solute is treated explicitly while solvent is replaced by a continuum dielectric

Solvated molecules: continuum approachThe cavity

Simple modelsSphere

Ellipsoid

1. van der Waals Surface (VWS): is constructed from the overlapping vdW spheres of the atoms

1

2. Solvent Accessible Surface (SAS): is the surface traced by the center of the probe molecule.

2

3. Solvent excluded Surface (SES or Connolly): is traced out by the inward-facing part of the probe sphere as it rolls on the vdW surface of the molecule. 3

Molecular models

NOT ENOUGH!

Cavitation term: work required to form the cavity

van der Waals term: dispersion and repulsion

ΔGsol = Gcav + GvdW + Gelec

As in molecular mechanics, also here we introduce a partition in terms of different interactions:

Electrostaticterm

Solvated molecules: continuum approachThe definition of the energy

ΔGsol ⇒ the free energy change to transfer a molecule from vacuum to an infinite isotropic solution.

εA charge density ρM (the solute) inside a cavity within a continuum dielectric described by its permittivity ε (the solvent)

ρMn

The electrostatic potential V has to satisfy the Poisson and Laplace equations inside and outside the cavity (together with the proper boundary conditions)

Boundary conditions

inside the cavity: ε=1

outside the cavity: ρM=0

+ −∇2V = 4πρM

−ε∇2V = 0 ∇V ⋅ n⎡⎣ ⎤⎦in

= ε∇V ⋅ n⎡⎣ ⎤⎦out

Vin =Vout

The electrostatic interactions: the Poisson equation

∇ ⋅ ε(r )

E(r )( ) = −ε

∇ ⋅∇V r( )⎡⎣ ⎤⎦ = −ε∇2V

If the dielectric is homogeneous and isotropic:

The Apparent Surface Charge

ε

ρMn

Which electrostatic potential V ?

V is the sum of the electrostatic potential VM generated by the charge distribution ρM and of the reaction potential VR generated by the polarization of the dielectric medium:

Which form for the reaction potential VR ?

V (r ) =VM (r )+VR (r )

The reaction potential is defined by introducing an apparent surface

charge (ASC) density (σ) on the cavity VR (r ) ⇒Vσ (r ) = σ (s )

r − sΓ∫ d 2s

Dielectric PCM: DPCM S. Miertuš, E. Scrocco, J. TomasiElectrostatic interaction of a solute with a continuum. A direct utilization of Ab initio molecular potentials for the prevision of solvent effects Chem. Phys. 117-129, 55 (1981)

Integral Equation Formalism: IEFPCME. Cancès, B. Mennucci and J. Tomasi A new integral equation formalism for the polarizable continuum model: Theoretical background and applications to isotropic and anisotropic dielectricsJ. Chem. Phys. 3032-3041 107 (1997)

The present

Conductor-like formulation: CPCMV. Barone, M. CossiQuantum Calculation of Molecular Energies and Energy Gradients in Solution by a Conductor Solvent Model J. Phys. Chem. A 1995-2001, 102 (1998)

A. Klamt and G. Schüürmann, COSMO: A New Approach to Dielectric Screening in Solvents with Explicit Expressions for the Screening Energy and its GradientJ. Chem. Soc. Perkin Trans. II 799-805, 2 (1993).

A PCM-like reformulation of the COSMO model

The origin (1981)

Which definition for σ?The PCM solution

The PCM family of methods

2π ε +1

ε −1⎛⎝⎜

⎞⎠⎟

I − D⎡

⎣⎢

⎤

⎦⎥Sσ IEFPCM = − 2π I − D( )VM

2π ε +1

ε −1⎛⎝⎜

⎞⎠⎟

I − D*⎡

⎣⎢

⎤

⎦⎥σ DPCM =

∂VM

∂n

Sσ CPCM = −

ε −1ε

VM

IEFPCM

DPCM

CPCM

PCM

Sσ (s) = σ (s ')s − s '

d 2s 'Γ∫

Dσ (s) = ∂∂ns '

1s − s '

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪σ (s ')d 2s '

Γ∫

Integral operators defined on the cavity surface Γ:

PCM: the numerical strategy

1. Construction of the molecular cavity in terms of interlocking spheres (centered on the solute atoms)

2. Partition of the cavity surface into N finite elements (tesserae) (Bundary Element Method)

si3. Discretization of the apparent surface charge σ

into N point-like charges q

qsi( ) = aiσ

si( )we assume that σ is constant on each element of area ai

Vector fM collects electrostatic potential (or field) produced by the solute on the surface elements:

There are basically two strategies to solve the system:

a) Inverting T by a direct method

b) By an iterative method

T(ε)q = −RfM (N × N ) linear system

fM⎡⎣ ⎤⎦ j=

EM (sj ) ⋅

n(sj ) DPCM

VM (sj ) CPCM & IEFPCM

⎧⎨⎪

⎩⎪

PCM: the discretization of the surface charge

How can we introduce a quantum mechanical description of the solute?

QM/continuumQM/MM

Continuum

modelsD

iscr

ete

mod

els

Heff Ψ = H0 + Henv( ) Ψ = E Ψ

Henv =HQM / MM + H MM QM/MM

Hcont QM/Continuum

⎧⎨⎪

⎩⎪

Effective Hamiltonian

for the solute

The effective Hamiltonian

OH

H

O H

H

OH

HHOH

HOH

HOH

HOH

HOH

HOH

HOH H

OH

HOH

HOH

HOH

HOH

HOH

HOH

HOH

HOH

HOH

HOH

HOH

HO

H

HOH

QM

MM

Three ways of treating QM/MM electrostatic interaction:

Mechanical embedding: QM calculation is performed in the gas phase: the electrostatic interaction between QM and MM regions only in the MM code, through a classical point charge model for the QM charge distribution. Not accurate model.

Electronic embedding: the classical part 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. 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.

The effective Hamiltonian: QM/MM

Heff = HQM + H MM + HQM / MM

el + HQM / MMvdW

HQM / MM

el = HQM/MMq =

m∑ qmV QM (rm )

The effective Hamiltonian: QM/MM

Polarizable embedding

HQM / MMel = HQM/MM

q + HQM/MMpol

=m∑ qmV QM (rm ) −

12 a∑ µa

ind ⋅ EQM (ra )

Electronic embedding

The solvent is described using point (atomic) charges

µaind = α a Ea

QM + EaMM{q;µ ind }( )

µ ind = B EQM + EqMM( )Induced dipoles

The solute wavefuntion depends on the solvent operator & the solvent operator depends on the wavefunction!

The solvent is described using both

charges and polarizabilities

Effective Schrodinger equation for the solute

H eff Ψ = H0 + Hcont⎡⎣ ⎤⎦ Ψ = Es Ψ

The solute wavefuntion depends on the solvent operator & the solvent operator depends on the wavefunction!

Solvent reaction potential operator

Hcont = V R = qkVkQM

k∑

The effective Hamiltonian: QM/Continuum

Solute electrostatic potential on the surface cavity:

VM (si ) = Ψ Vi Ψ +Vi

Nuclei

PCM charges:

q = −T−1RVM = QVM

V R = V Ψ Q Ψ

QM/MMpol & QM/Continuum

Solute and solvent mutually polarize

BUTan iterative procedure is necessary

It can be solved together with the standard self-consistent-field problem:

Hartree-Fock or Kohn-Sham (DFT) approach

The solute wavefuntion depends on the solvent operator & the solvent operator depends on the wavefunction!

H eff Ψ = H0 + V R⎡

⎣⎤⎦ Ψ = Es Ψ

Es = Ψ H eff ΨEigenvalue:

Internal energy

G = Ψ H0 Ψ + 1

2Ψ V R ΨElectrostatic Free

energy functional

In a thermodynamical language: we have to include the work necessary to polarize the solvent which is opposite in sign and

half in magnitude with respect to the interaction energy.

effective Hamiltonian

The variational principle can be applied but not in the standard form:here the functional to be minimized does NOT correspond to the eigenvalue

Self consistent reaction field (SCRF):a specificity of the QM/Continuum

Self consistent reaction field (SCRF)How do we introduce solvent effects?

we add a new solvent-dependent operator

EffectiveKohn-Sham operator

FKS

eff = FKS0 + X MMpol

X PCM

⎧⎨⎪

⎩⎪

X R =

12∂ q(ρ)V(ρ){ }

∂ρ

solvent operator

X MMpol =∂ qMM V(ρ){ }

∂ρ−

12∂ µ(ρ) ⋅E(ρ){ }

∂ρ

q = QV(ρ)

µ = BE(ρ) It changes at each iteration

of the SCF cycle

X MMpol (ρ) = qk

MMVkk∑ − µi (ρ)Ei

i∑

X PCM (ρ) = qi (ρ)Vi

i∑

• 1. Methodological aspects of solvation Models◦ 1.1. Discrete vs Continuum descriptions◦ 1.2. The PCM family of methods◦ 1.3. The coupling with QM methods

Outline

• 2. Excited states & Nonequilibrium◦ 2.1. Dynamic Polarization Response◦ 2.2. Vertical Electronic Transitions◦ 2.3. Relaxation & Fluorescence

• 3. Excitation energy transfer◦ 3.1. Introduction◦ 3.2. QM description of the coupling◦ 3.3. Environment effects

S1

Solvation coordinate

S0

Solvent reorganization energy

Solute in its ground State equilibrated with

the environment (equilibrium)

Solute in its excited state equilibrated

with the environment

Inertial response

NonequilibriumOnly the environment dynamic

(electronic) response readjusts: the inertial part is frozen in the initial

configuration

Excitation in solvated systemsIn a polar solvent

A Nonequilibrium model Partition of the solvent polarization into fast (electronic motions) and

slow (molecular and nuclear motions) contributions

Electronic (deformation) polarization

Orientational polarization (only

for polar solvents)

Frequency dependent permittivity

ε(ω ) = ε(∞) + ε(0) − ε(∞)1+ iωτD

Fast (electronic) response

Relaxation (orientations)

Nonequilibrium: the PCM picture

•Chromophores: QM

•Environment: continuum

•nonequilibrium:

• static (ε0) and optical dielectric constant (ε∞)

• separation of charges into dynamic and inertial components:ε

Free energy for the K excited state

Charges for the K excited state

qK (t = 0) = qKneq = qGS

in +qKdyn

qK (t →∞) = qKeq T(ε∞ )qK

dyn = −RVK

T(ε)qKeq = −RVK

qGSin = qGS

eq − qGSdyn

Within a QM/MM scheme•Chromophore: QM

•Environment: MM

•nonequilibrium:

• fixed charges which represent the inertial component

Nonequilibrium: the QM/MM picture

YES

but only if we use a polarizable force field:

the induced dipoles describe the dynamic polarization

Can we include also the dynamic part of polarization?

Equilibrium vs Nonequilibrium

Absorption energies for acrolein:

water-cyclohexane shifts (in eV)

n-π* π- π*

Exp +0.23 -0.21

neq +0.21 -0.22

eq -0.03 -0.56PCM EOM-CC/6-31+G(d)

water (polar solvent) ε=78.4, ε∞=1.8cyclohexane (apolar solvent) ε=ε∞=2.0

• State Specific (SS):• The wavefunction of the excited state is explicitly calculated together with the energy• CASSCF, CI, ….

• Linear Response (LR)• Excitation energies as poles of a linear response function of the molecule, no need of the excited state wavefunction• ZINDO, CIS, TDDFT, ...

Excitation in solvated systems:which QM approach?

For isolated systems the two approaches are “equivalent” in the limit of exact states.

Is this still valid for solvated systems?

Ground

Excited

ΔEex

Re α

ΔEex

Proper extension for MMPol or PCM but computationally expensive

Computationally efficient but approximated MMPol or PCM

responses

R. Cammi, S. Corni, B. Mennucci, J. Tomasi, J. Chem. Phys. , 122 (2005) 104513.

The LR approach is intrinsically not equivalent to SS when a non-linear effective Hamiltonian is used

• State Specific (SS):• The wavefunction of the excited state is explicitly calculated together with the energy• CASSCF, CI, ….

• Linear Response (LR)• Excitation energies as poles of a linear response function of the molecule, no need of the excited state wavefunction• ZINDO, CIS, TDDFT, EOM-CC, …

Excitation in solvated systems:which QM approach?

A BB* A*

⎛

⎝⎜⎞

⎠⎟XY

⎛⎝⎜

⎞⎠⎟=ω 1 0

0 −1⎛⎝⎜

⎞⎠⎟

XY

⎛⎝⎜

⎞⎠⎟

Aia,bj = δabδij (εa −εi )+ Kia,bj

Bia,bj = Kia, jb

Orbitals φr and orbital energies εr are obtained in the presence

of the environment

Cai,bjMMpol = − drφi

*(r )φa (r )µ

k

ind φ j*φb( ) ⋅

rk −r( )

rk −r

3k∑∫

Cai,bjPCM = drφi

*(r )φa (r ) ql φ j*φb

⎡⎣ ⎤⎦1r − sll

∑∫

Additional term induced by the environment response

+Cai,bjEnv

+Cai,bjEnv

The environment “enters” in the TDDFT equations

BUTthe response of the

solvent to the excitation is obtained using a

“transition density” and not a “state density”!

An example: the TDDFT approach

The corrected Linear response (cLR)

State specific PCM charges for the excited state

qexcPCM = qGS

PCM + qPCM (PΔ )

Unrelaxed excitation energy ω0: the response of the solvent is completely frozen in the reference ground state

Excited state densityChange in the density matrix

from the ground to the excited state: it is obtained introducing derivatives of the LR equations

Pexc = PGS + PΔ

LR

Can we recover a state-specific solvent effect still keeping the computational efficiency of LR methods?

Relaxed excitation enegy ω =ω0 +12

qexcPCM (si )

i∑ Vexc (si )

Caricato et al. J. Chem. Phys. 124, 124520 (2006)

TDB3YLP Vertical Absorption energies (in eV)

First-order correction (cLR)

3.24

3.37

3.32

3.37

3.47 3.47

3.43

Vac Diox ACN

ΔEK0

ΔEK0 ��

��

���

���

ACRO:n-π* 6.10

6.06

5.89

6.04

Vac Diox ACN

ΔEK0 ���

��

5.99

5.85

5.97 ΔEK0 ���

�� ACRO:π-π*

Caricato et al. J. Chem. Phys. 124, 124520 (2006)

However for absorption:not very large LR-cLR

differences as the two models differ ONLY for the way the

dynamic part of polarization is calculated

For fluorescence:only cLR gives a physically correct

description (excited state equilibrium)

ω0

ω0

ω0

ω0

apolar polarapolar polarGAS GAS