QMMM Approaches for Solvatochromic Shifts Assessing the Quality of QMMM Approaches to Describe Vacuo-to-water Solvatochromic Shifts

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QM/MM Approaches for Solvatochromic Shifts

Assessing the Quality of QM/MM Approaches to Describe

Vacuo-to-water Solvatochromic Shifts

Luca Nicoli,1Tommaso Giovannini,1and Chiara Cappelli1

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy.

(*Electronic mail: chiara.cappelli@sns.it)

(*Electronic mail: tommaso.giovannini@sns.it)

(Dated: 28 October 2022)

The performance of different Quantum Mechanics/Molecular Mechanics embedding models to compute vacuo-to-water

solvatochromic shifts are investigated. In particular, both non-polarizable and polarizable approaches are analyzed and

computed results as compared to reference experimental data. We show that none of the approaches outperforms the

others and that errors strongly depend on the nature of the molecular transition. Thus, we prove that the best choice of

embedding model highly depends on the molecular system, and that the use a speciﬁc approach as a black-box can lead

to signiﬁcant errors and sometimes totally wrong predictions.

I. INTRODUCTION

Focused models have a long standing tradition in com-

putational chemistry for the simulation of spectral proper-

ties of complex systems.1–4 Among them, quantum mechan-

ics/molecular mechanics (QM/MM) approaches have become

very popular,1,2,5–7 due to their strengths in dealing with many

diverse external environments, ranging from strongly interact-

ing solvents3to biomolecular environments.8–10 Indeed, the

increasing popularity of QM/MM is linked to their ability

to describe target/environment interactions with an atomistic

detail.2,11

When applied to solvated systems, the most common

QM/MM partition consists of treating the solute at the QM

level, and the solvent in terms of classical MM force ﬁelds.

For a given QM level, the quality of QM/MM results strongly

depends on the physics lying behind the speciﬁc approach

which is exploited to model the interaction between the QM

and MM layers.7The latter is generally limited to electrostatic

terms, being non-electrostatic contributions only rarely taken

into account.12–15

The MM layer can be modeled in terms of a set of ﬁxed

multipoles placed at atomic sites, thus yielding the so-called

Electrostatic Embedding (EE) approach.2As a consequence,

the MM layer polarizes the QM density, but not vicev-

ersa. However, a correct physical description of an inter-

acting solute-solvent systems requires mutual solute-solvent

polarization effects to be considered.16–18 Thus, many dif-

ferent polarizable embedding have been proposed and amply

tested.3,8,16–23

In polarizable QM/MM approaches, MM fragments are en-

dowed with polarizable multipolar charge distributions which

are modiﬁed as a result of the interaction with the QM density,

and viceversa.7The physically consistent description which is

then obtained, permits to compute remarkably accurate val-

ues of many spectroscopic signals, especially when polariz-

able QM/MM approaches are coupled to accurate procedures

to sample the conﬁgurational phase-space.24–28 The various

QM/MM approaches differ from the speciﬁc way the electro-

static and polarization terms are modeled. The latter not only

modiﬁes the solute’s ground state density, but also its response

properties.

Despite the increasing interest in exploiting QM/MM ap-

proaches to describe spectral properties, the performance

of the different QM/MM approaches has only rarely been

investigated.29,30 Therefore, the ideal model to be employed

for a given application has not been clearly deﬁned yet.

In this work, we present extensive comparison of the re-

sults obtained by applying a selection of QM/MM embedding

models to the calculation of vacuo-to-water solvatochromic

shifts. The approaches are chosen because they conceptually

span diverse classes of models that are employed in the liter-

ature. In particular, we employ the EE (as speciﬁed by means

of the TIP3P parametrization),31 where MM atoms are de-

scribed in terms of ﬁxed charges, the polarizable Fluctuating

Charges (FQ),3,32–34 where polarization effects are described

in terms of a set of charges that vary as a response to the ex-

ternal electric potential.4Discrete Reaction Field (DRF)16,35

is an example of amply used approaches which model polar-

ization effects in terms of a set of induced dipoles assigned

to MM atoms.16,20,35–37 More sophisticated models are used

to reﬁne DRF electrostatic description in terms of ﬁxed mul-

tipolar expansions.19,38,39 The last approach is the Fluctuat-

ing Charges and Fluctuating Dipoles (FQFµ) model,21 where

each MM atom is assigned a charge and dipole which can

vary as a result of polarization effects. While EE and DRF

directly follow from an electrostatic multipolar expansion of

the energy,40 FQ is grounded in conceptual DFT,41 and FQFµ

can be seen as a pragmatical extension of FQ.21

Each embedding approach models QM/MM interactions

according to the order of the multipolar expansion of the MM

variables (ﬁxed and/or polarizable). From the numerical point

of view, such an interaction also depends on the parameters

deﬁning the speciﬁc model: ﬁxed atomic charge q(for EE and

DRF), atomic electronegativity χand chemical hardness η

(for FQ and FQFµ), and atomic polarizability α(for DRF and

FQFµ). The numerical values of such parameters clearly de-

termine the QM/MM interaction, and in turn computed spec-

troscopic signals.42 Thus, in this paper a total of eight differ-

ent parameter sets, which are speciﬁcally developed for the

aqueous environment, are compared.13,16,21,31,32,43

The manuscript is organized as follows. In the next sec-

arXiv:2210.15412v1 [physics.chem-ph] 27 Oct 2022

QM/MM Approaches for Solvatochromic Shifts 2

tion, we brieﬂy recap the theoretical foundations of the

QM/MM embedding approaches which are exploited in this

work. Then, their performance are tested to describe vacuo-

to-water solvatochromic shift of a set of 11 medium-to-large

molecules, for which experimental data are available in the

literature. Results are also discussed in terms of the physico-

chemical description of the QM/MM interaction and the na-

ture of the solute’s transition.

II. THEORETICAL MODELLING

The total energy of a QM/MM system reads:3

Etot =EQM +EMM +Eint

QM/MM (1)

where EQM and EMM are the energies of the QM and MM por-

tions, respectively. By neglecting non-electrostatic (disper-

sion/repulsion) interactions, the QM-MM interaction energy

Eint

QM/MM can be expressed as:

Eint

QM/MM =Eele

QM/MM +Epol

QM/MM (2)

where the electrostatic Eele

QM/MM and possibly polarization

Epol

QM/MM energy terms are highlighted. In a generic deﬁnition

of a force ﬁeld, MM atoms can be endowed with a ﬁxed mul-

tipolar distribution M(charges, dipoles, quadrupoles, ...) and

additional quantities D, accounting for polarization effects.

By this, the various polarizable or non-polarizable QM/MM

approaches differ in the way they deﬁne Mand D, and be-

cause they account or neglect polarization terms (i.e. D). By

assuming a classical electrostatic interaction between the QM

and MM portions, the total energy in Eq. 1 can be rewritten

as:

Etot [ρ,D] =EQM[ρ(r)] + M†ZTM(r)ρ(r)dr

2D†AD +D†ZTD(r)ρ(r)dr+D†TM

(3)

where the Amatrix describes the self interaction of the polar-

ization sources; Tis a block matrix, which takes into account

the interaction between the ﬁxed and polarizable MM distribu-

tions. Tξ(r)(ξ=M,D) collects QM/MM electrostatic inter-

action kernels40 (see Sec. S1.1 in the Supplementary Material

– SM for more details).

Within a Kohn–Sham (KS) density functional theory (DFT)

formulation, by differentiating Eq. 3 with respect to ρ, the

QM/MM Fock Matrix ˜

Fis recovered. By minimizing Eq. 3

with respect to D, the equations which describe the polariza-

tion of the MM portion are obtained. This allows us to deﬁne

the coupled QM/MM equations:

δEtot [ρ,D]

δρ(r)=h0

KS[ρ(r)] + ˆvemb(r) = ˜

F(4)

δEtot [ρ,D]

δD=Θ[ρ,D] = 0 (5)

where h0

KS is the common KS operator, given by:

KS =−1

2∇2−∑

|r−Rm|+Zρ(r0)

|r−r0|dr0+δEXC

δρ(r)(6)

where EXC is the exchange-correlation energy functional.

In Eqs. 4 and 5, ˆvemb(r)and Θ[ρ,D]are deﬁned as:

ˆvemb(r) = M†TM(r) + D†TD(r)(7)

Θ[ρ,D] = AD +ZTD(r)ρ(r)dr+TM (8)

The solutions of Eqs. 4 and 5 deﬁne the ground state (GS)

QM density and the polarization vector D.

Vertical excitation energies can be computed by resorting

to the linear response (LR) formulation of the time-dependent

DFT (TDDFT) formalism (see Sec. S1 in the SM for more

details).3,20,35,44. In the case of QM/MM approaches, LR-

TDDFT equations45 are modiﬁed to account for the presence

of the MM layer.44 In particular, the MM environment mod-

iﬁes excitation energies through two mechanisms: (i) mod-

iﬁcation of energy and spatial distribution of GS molecular

orbitals (MOs), usually referred to as indirect effect; (ii) inclu-

sion of additional terms in LR-TDDFT equations, which ac-

count for the mutual interaction between the MM layer and the

transition QM density. The latter contribution is usually called

“direct effect”, and is only present in case of polarizable em-

bedding approaches. Note that state-speciﬁc formulations of

polarizable embedding models have also been proposed. They

speciﬁcally account for the relaxation of the solute density in

the excited state of interest, while discarding the dynamical

aspects of solute-solvent interactions, which are instead con-

sidered in the LR formalism.44,46 It is ﬁnally worth noting that

local ﬁeld effects induced on the QM moiety due to the polar-

ization of the MM portion to the external radiation ﬁeld, are

not taken into account in this work, although they may affect

computed oscillator strengths.47

A. Embedding Models

The equations reported in the previous section are general

enough to constitute a uniﬁed framework, which can be spec-

iﬁed for the various embedding approaches that are exploited

in the present work. The latter differ in the way D,Mare

deﬁned.

1. Electrostatic embedding (EE): each atom in the MM re-

gion is endowed with a ﬁxed charge i.e. M= [qM]and

D= [0]. Therefore, the MM layer polarizes the QM

density but not viceversa, thus it only indirectly affects

the QM solute’s response properties.

2. Fluctuating Charges (FQ) approach: each MM atom is

endowed with a charge, whose value is not ﬁxed, but

varies as a result polarization effects.3,4,48,49 Thus, M=

[0]and D= [q], with qbeing the polarizable charges.

The parameters entering the FQ models, thus determin-

ing the qcharges, are the atomic electronegativity χ

QM/MM Approaches for Solvatochromic Shifts 3

and chemical hardness η, which are theoretically de-

ﬁned in conceptual DFT.41 Polarization follows from

the electronegativity equalization principle,50,51 which

allows to deﬁne atomic partial charges in terms of the

constrained minimum of a suitable energy functional.3

More details on the FQ model can be found in section

S2 in the SM.

3. Discrete Reaction Field (DRF): each MM atom is

endowed with a ﬁxed charge qand a polarizable

dipole µ,16,35 This approach to model polarization ef-

fects is exploited also by other polarizable QM/MM

approaches.7,17,20,39 Thus, in this case, M=qMand

D= [0,µ]. Additional details about DRF can be found

in section S3 in the SM.

4. Fluctuating Charges and Fluctuating Dipoles (FQFµ):

each MM atom is endowed both with a polarizable

charge qand a polarizable dipole µ.14,21,44,52,53 FQFµ

is a pragmatic extension of the FQ model, where D=

[q,µ]. The parameters that need to be set are the atomic

electronegativity χ, chemical hardness ηand atomic

polarizability α. Additional information on FQFµis

reported in section S4 in the SM.

To better understand analogies and differences between the

aforementioned approaches, they are schematically speciﬁed

for the water molecule in Fig. 1.

FIG. 1. Graphical explanation of the variables associated to EE, FQ,

DRF and FQFµFFs for the water molecule.

III. COMPUTATIONAL DETAILS

We apply the aforementioned QM/MM approaches to the

calculation of vacuo-to-water solvatochromic shifts. To this

end, we select eleven molecules (see Fig. 2) for which ex-

perimental UV-Vis absorption spectra in aqueous solution are

available in the literature.54–66 The variety of molecular size,

together with the different sign of experimentally measured

solvatochromic shifts, makes this set an ideal test-bed for em-

bedding models.

In order to sample the solute-solvent phase-space, molec-

ular dynamics (MD) simulations are performed. In particu-

lar, we run MD simulations of I,II,IV,V,VI,VII,IX, and

XI in aqueous solution without imposing any constraints on

the solute’s geometry. On the other hand, the solute geom-

etry is kept frozen at the PCM67 optimized structure during

MD runs of III,VIII, and X, because only minor geometrical

FIG. 2. Molecules studied in this work. Iacrolein, II

para-nitroaniline, III 1-methyl-8-oxyquinolinium betaine,

IV 4-aminophtalimide, Vsyn-CH3-CH3bimane, VI 4-2-

[4-(dimethylamino)phenyl]ethenyl-1-methylpyridinium, VII

doxorubicin, VIII Reichardt’s betaine, IX 4,4-diﬂuoro-

4-bora-3a,4a-diaza-s-indacene (BODIPY), X1-methyl-4-

[(oxocyclohexadienylidene)ethylidene]-1,4-dihydropyridine or

Brooker’s merocyanine (MB), XI 5-methylcytidine.

distortions are expected, due to their limited ﬂexibility. All

MDs are performed according to the protocols previously re-

ported by some of the present authors (see Refs. 14, 26, 42,

44). For each molecule, a series of uncorrelated snapshots are

extracted from MD runs, and for each snapshot, a spherical

droplet with a variable radius depending on the solute intrin-

sic size is cut. The droplet radius ranges from 15 Å (molecule

I) to 25 Å (molecule VIII), and it is set to retain all solute-

solvent interactions (see Tab. I for the average number of wa-

ter molecules included for each system). For each snapshot,

the absorption spectrum is calculated and convoluted with a

Gaussian function of FWHM of 0.3 eV. The ﬁnal UV/Vis ab-

sorption spectrum, is then obtained as the average over the

set of uncorrelated snapshots extracted from MD runs. Note

that the convergence of the computed spectra as a function of

the number of snapshots (see Tab. I) has been checked. Simi-

larly to previous studies,14,44,54 for each investigated molecule

we perform an extra set of calculations in which all water

molecules that are placed at a distance lower than 3.5 Å from

each solute atom are described at the QM level, whereas the

remaining ones are treated by using the FQFµforce ﬁeld. The

resulting approach is called QM/QMw/FQFµ(QMw). Note

that, within this approach, a proper QM description of hy-

drogen bonding interactions is introduced. QM/QMw/FQFµ

results are obtained as an average on the minimal number of

geometries (70 structures for I, 60 structures for II, and 20

structures for the remaining molecules) which guarantee the

convergence of the spectra (see Section S5.1 in the SM).

For each system, the solvatochromic shift (∆E) is calculated

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QM/MMApproachesforSolvatochromicShiftsAssessingtheQualityofQM/MMApproachestoDescribeVacuo-to-waterSolvatochromicShiftsLucaNicoli,1TommasoGiovannini,1andChiaraCappelli1ScuolaNormaleSuperiore,PiazzadeiCavalieri7,56126Pisa,Italy.(*Electronicmail:chiara.cappelli@sns.it)(*Electronicmail:tommaso.giovannin...

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QMMM Approaches for Solvatochromic Shifts Assessing the Quality of QMMM Approaches to Describe Vacuo-to-water Solvatochromic Shifts

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