The effect of polarizable environment on two-photon absorption cross sectionscharacterized by the equation-of-motion coupled-cluster singles and doublesmethod combined with the effective fragment potential approachKaushik D. Nanda, and Anna I. Krylov

Citation: The Journal of Chemical Physics 149, 164109 (2018); doi: 10.1063/1.5048627View online: https://doi.org/10.1063/1.5048627View Table of Contents: http://aip.scitation.org/toc/jcp/149/16Published by the American Institute of Physics

THE JOURNAL OF CHEMICAL PHYSICS 149, 164109 (2018)

The effect of polarizable environment on two-photon absorption crosssections characterized by the equation-of-motion coupled-clustersingles and doubles method combined with the effective fragmentpotential approach

Kaushik D. Nanda and Anna I. KrylovDepartment of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA

(Received 16 July 2018; accepted 1 October 2018; published online 29 October 2018)

We report an extension of a hybrid polarizable embedding method incorporating solvent effects inthe calculations of two-photon absorption (2PA) cross sections. We employ the equation-of-motioncoupled-cluster singles and doubles method for excitation energies (EOM-EE-CCSD) for the quantumregion and the effective fragment potential (EFP) method for the classical region. We also introducea rigorous metric based on 2PA transition densities for assessing the domain of applicability ofQM/MM (quantum mechanics/molecular mechanics) schemes for calculating 2PA cross sections.We investigate the impact of the environment on the 2PA cross sections of low-lying transitions inmicrohydrated clusters of para-nitroaniline, thymine, and the deprotonated anionic chromophore ofphotoactive yellow protein (PYPb). We assess the performance of EOM-EE-CCSD/EFP by comparingthe 2PA cross sections against full QM calculations as well as against the non-polarizable QM/MMelectrostatic embedding approach. We demonstrate that the performance of QM/EFP improves whenfew explicit solvent molecules are included in the QM subsystem. We correlate the errors in the 2PAcross sections with the errors in the key electronic properties—identified by the analysis of 2PA naturaltransition orbitals and 2PA transition densities—such as excitation energies, transition moments,and dipole-moment differences between the initial and final states. Finally, using aqueous PYPb,we investigate the convergence of 2PA cross sections to bulk values. Published by AIP Publishing.https://doi.org/10.1063/1.5048627

I. INTRODUCTION

Electronic structure methods enable highly accurate cal-culations of molecular structures, energetics, and properties;however, the domain of applicability of ab initio methods interms of the size of a system is limited by their steep scaling andlarge computational costs. Despite significant progress in com-puter hardware and algorithms, full quantum-chemical mod-eling of condensed-phase systems remains elusive. This lim-itation can be circumvented by using multi-scale approachescombining a high-level quantum mechanical (QM) descrip-tion of a system (i.e., a solvated chromophore) with a moreapproximate treatment, such as classical molecular mechan-ics (MM), of the environment. Among many different fla-vors, explicit solvent approaches are particularly attractivebecause, in contrast to implicit solvent models (e.g., polar-izable continuum), they are capable of describing specificinteractions such as solvent-solute hydrogen bonding and saltbridges.

Numerous hybrid multi-scale methods1 differ in theirdescription of the interactions between the QM subsystem(treated at a high level of theory) and the environment. Inthe most commonly used QM/MM approach, classical forcefields are used to describe the environment (hence, MM) andthe interaction between the QM and MM parts is describedby electrostatic embedding.2 This basic QM/MM strategy has

been successfully used for describing both the ground- andexcited-state energetics and properties of large systems.3–6 TheQM/MM approach can be improved by using polarizable forcefields in which the charge distributions in the MM part canrespond to the changes in the electronic distribution in the QMpart.

The quality of force fields (whether polarizable or not)is essential for obtaining reliable electronic properties of theQM subsystem. Force fields are commonly derived usingempirical parameterization aiming at reproducing particularobservables. A more rigorous description can be achieved byusing force fields derived from first principles, such as in theeffective fragment potential (EFP) approach.7–11 In the EFPmethod, the MM subsystem is fragmented into smaller sub-units (fragments), and the force-field parameters of each ofthese fragments are obtained from ab initio calculations. Thefragments are frozen (i.e., their structures do not change),which enables using a fixed set of parameters computedonce. In principle, these parameters can be computed “onthe fly,” but this, of course, leads to increased computationalcosts. The QM/EFP scheme has been shown to yield accu-rate solvatochromic shifts, ionization and detachment ener-gies, electron-attachment energies, and redox potentials ina variety of solvated systems including protein-bound chro-mophores.12–19 QM/EFP is similar to the polarizable embed-ding (PE) approach.20–23 The description of electrostatics and

0021-9606/2018/149(16)/164109/14/$30.00 149, 164109-1 Published by AIP Publishing.

164109-2 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

polarization is nearly identical in QM/EFP and PE; QM/EFPalso includes (non-empirical) terms describing dispersion andexchange-repulsion interactions.

Higher-order response properties such as two-photonabsorption (2PA) cross sections are particularly sensitive tothe environment.24 For example, the 2PA cross sections of redfluorescent proteins with identical DsRed-like chromophores(DsRed, mCherry, and mStrawberry) differ by a factor offive.25 As pointed out in a recent validation study, the inclu-sion of solvent effects on 2PA cross sections is criticallyimportant for unambiguous comparison between theory andexperiment.26,27

Nonlinear response properties are formally given by sum-over-states expressions, which depend on the wave functionsand excitation energies of all electronic states of the sys-tem. Thus, in contrast to the one-photon transition momentsexpressed as the matrix elements between the initial and thefinal states only, the quality of the description of nonlinearproperties, such as 2PA cross sections, depends on the qualityof the representation of the entire spectrum by a model Hamil-tonian. This is the primary reason why modeling 2PA crosssections is challenging.

The QM/MM approach has a number of limitations thatmake reliable calculations of 2PA cross sections difficult. First,QM/MM schemes with only one chromophore in the QM partcannot describe delocalized states, such as those in molecularaggregates28 or charge-transfer (CT) states delocalized overseveral solvent molecules.29 This issue can be partially reme-died by a careful choice of the QM subsystem, for example,by including some solvent molecules. Second, small errors inthe QM/MM excitation energies (present in the denominatorof the formal expression) and transition dipole moments canaccumulate leading to large errors in the calculated 2PA crosssections. Third, the quality of computed 2PA cross sections issensitive to the degree of electronic correlation included in themodel Hamiltonian.

In the context of 2PA calculations, the EFP method hasbeen employed in combination with time-dependent densityfunctional theory (TDDFT)30 but not with high-level wave-function methods such as the equation-of-motion coupled-cluster singles and doubles method for excitation energies(EOM-EE-CCSD).31–33 Coupled-cluster methods have beenemployed within the polarizable embedding scheme in thecalculations of 2PA cross sections.22 In this article, weextend our EOM-CCSD implementation of the 2PA cross sec-tions34–36 to include the effect of solvent environment viaQM/EFP embedding. We assess the quality of the EOM-EE-CCSD/EFP 2PA cross sections (i.e., computed with theEOM-EE-CCSD wave functions embedded in the polariz-able environment of EFP fragments) relative to the fullEOM-EE-CCSD calculations and identify the main sourcesof errors. We consider low-lying transitions in small micro-solvated clusters of three chromophores: para-nitroaniline(pNA), thymine, and the deprotonated anionic chromophoreof photoactive yellow protein (PYPb). To analyze the role ofsolvent polarization, we compare the EOM-EE-CCSD/EFP2PA cross sections for the microhydrated clusters withthe results from two non-polarizable QM/MM schemes:(a) EOM-EE-CCSD/(EFP without polarization effects) and

(b) EOM-EE-CCSD/MM, wherein the solvent molecules arerepresented by CHARMM22 point charges37,38 (we denotethese calculations as QM/CHARMM). To assess the effect ofbulk solvation, we calculate the 2PA cross sections of the PYPbchromophore in water and investigate the convergence of theresults with respect to the QM size by including an increasednumber of water molecules in the QM subsystem.

II. THEORYA. The QM/EFP scheme

In the QM/EFP embedding scheme, the Hamiltonian ofthe whole QM/MM system is written as

H full = HQM + HMM + HQM/MM , (1)

where HQM and HMM describe the QM and MM parts, respec-tively, and HQM /MM describes the interaction between thesetwo subsystems. In the QM/EFP approach, the MM systemis broken into effective fragments and its energy, EMM , iscomputed as the sum of many-body electrostatic, polariza-tion, dispersion, and exchange-repulsion interactions betweenthe fragments

EMM = Eelec + Epol + Edisp + Eexch. (2)

In calculations of solvated species, each solvent molecule istreated as an EFP fragment. The EFP method can also beapplied to macromolecules via a fragmentation scheme thatcan deal with breaking covalent bonds.17

In polar and hydrogen-bonded solvents such as the onesstudied here, electrostatics and polarization are the leadingterms for inter-fragment interactions. The electrostatic inter-action energy, Eelec, is computed using Stone’s distributedmultipole analysis39–41 by placing permanent multipoles (upto octopoles) at the atoms and bond midpoints. Only the inter-actions of the induced dipoles with the permanent multipolesand the induced dipoles on other fragments are included in thecalculation of Epol. A detailed description of the dispersionand exchange-repulsion terms as well as the damping func-tions needed to correct the artificial charge-penetration effectsand to prevent “polarization catastrophe” at close distancesbetween the permanent and induced multipoles can be foundelsewhere.9,10,42–44

The QM system is embedded in the polarizable environ-ment such that the electrostatic and polarization interactionsbetween the QM and MM subsystems are included explicitlyin the one-electron part of HQM ,

HQMpq = 〈φp |H0 + V elec + Vpol |φq〉, (3)

where φs are the molecular orbitals. V elec is the Coulombpotential due to the nuclear charges and permanent multi-poles of the fragments, whereas Vpol is the potential due tothe induced multipoles. The induced dipoles on each frag-ment are calculated self-consistently with each other and withthe Hartree-Fock (HF) wave function using a two-level itera-tive scheme. The converged induced dipoles of the referenceHF determinant are fixed during subsequent calculations ofthe ground- and excited-state wave-function amplitudes (e.g.,T amplitudes in CC and excitation amplitudes in EOM-CC).

164109-3 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

The contribution to the polarization energy due to the interac-tions of the induced dipoles with the QM electronic density,Epol ,QM /MM , is included in the state energies. Since the inducedmultipoles calculated at the HF level are kept fixed in the calcu-lations of the excited-state wave functions, their contributionto the excitation energies is the same for all excited states.In other words, beyond the HF level, the polarizable environ-ment is not allowed to respond to the change in the electronicwave function. State-specific energy corrections due to theresponse of the polarizable environment to different electrondistributions of each state can be computed perturbatively12

and were shown to improve the quality of computed energydifferences.13,14,16–19 Here, we omit these corrections becausethis state-specific treatment cannot be easily incorporated intothe response equations solved in 2PA calculations. In the cur-rent implementation,12 the dispersion and exchange-repulsioninteractions between the QM and MM systems (Edisp,QM /MM

and Eexch ,QM /MM ) are treated approximately, as interactionsbetween the EFP fragments, and are added to the state ener-gies; thus, they do not contribute to wave functions, excitationenergies, and other transition properties. The total energies ofthe ground and excited states and the excitation energy can bewritten as

EQM/MMgr = 〈Ψgr |H0 + V elec + Vpol

HF |Ψgr〉 + EMM + Epol,QM/MMHF

+ Edisp,QM/MM + Eexch,QM/MM , (4)

EQM/MMex = 〈Ψex |H0 + V elec + Vpol

HF |Ψex〉 + EMM + Epol,QM/MMHF

+ Edisp,QM/MM + Eexch,QM/MM , (5)

and

ΩQM/MMex,gr = 〈Ψex |H0 + V elec + Vpol

HF |Ψex〉

− 〈Ψgr |H0 + V elec + VpolHF |Ψgr〉, (6)

where Ψgr and Ψex are the ground- and excited-state wavefunctions. Note that in CIS (configuration interaction singles),TDDFT, and EOM-CC calculations, excitation energies arecomputed directly and not as energy differences from state-specific calculations.

B. The EOM-EE-CCSD/EFP method

Our choice of the electronic-structure method for the QMsystem is EOM-EE-CCSD31,45,46 in which the excited-statewave functions are given by

|Ψ〉 = ReT |Φ0〉. (7)

Here, Φ0 is the reference determinant and eT |Φ0〉 is theCCSD wave function. The CCSD cluster operator T and theEOM-CCSD operator R comprise single and double excitationoperators

T = T1 + T2 and R = R0 + R1 + R2. (8)

The amplitudes of the operator T satisfy the following equa-tions:

〈Φµ |H |Φ0〉 = 0, (9)

where H = e−T HeT is the similarity-transformed Hamiltonianand Φµ denote singly and doubly excited determinants. In theEOM/EFP scheme, the one-particle component of the Hamil-tonian H is given by Eq. (3). The EOM amplitudes Rk and the

corresponding energies Ek for EOM-CCSD states are obtainedby diagonalizing H in the space of the reference, singly, anddoubly excited determinants

HRk = EkRk . (10)

Since H is non-Hermitian, its left (〈Ψk | = 〈Φ0Lk |) and right(|Ψk〉 = |RkΦ0〉) eigenstates are not Hermitian conjugates, butform a biorthonormal set

〈Φ0LmRnΦ0〉 = δmn. (11)

An important feature of the EOM-CCSD/EFP scheme is thatit preserves the biorthonormality of the target EOM statesbecause the induced multipoles are computed self-consistentlyat the HF step and kept fixed during the calculations of theCCSD and EOM amplitudes. This biorthogonality is impor-tant for computing transition properties such as 1PA and 2PAtransition moments.

C. 2PA within the EOM-EE-CCSD/EFP scheme

We compute the 2PA transition moments and cross sec-tions for EOM-CCSD wave functions following the approachdescribed in Refs. 34 and 36. Formally, the 2PA transi-tion moments are given by the following sum-over-statesexpressions:

Mxyf←i(ω1,ω2) = −

∑n

(〈Ψf |µ

y |Ψn〉〈Ψn |µx |Ψi〉

Ωni − ω1

+〈Ψf |µ

x |Ψn〉〈Ψn |µy |Ψi〉

Ωni − ω2

)(12)

and

Mxyi←f (−ω1,−ω2) = −

∑n

(〈Ψi |µ

x |Ψn〉〈Ψn |µy |Ψf〉

Ωni − ω1

+〈Ψi |µ

y |Ψn〉〈Ψn |µx |Ψf〉

Ωni − ω2

). (13)

Here, ω1 and ω2 are the frequencies of the two absorbedphotons satisfying the 2PA resonance condition ω1 + ω2

= Ωfi with Ωni = En − Ei. To derive expressions for theEOM-CCSD 2PA transition moments, we replace the dipole-moment operator, µa, with the similarity-transformed dipole-moment operator, µa = e−T µaeT , and the wave functions,〈Ψk |s and |Ψk〉s, with the EOM-CCSD target states, 〈Φ0Lk |sand |RkΦ0〉s, in the above expressions. Then, we recast theseexpressions into the numerically and formally equivalentform

Mxyf←i(ω1,ω2) = −

(〈Φ0Lf | µ

y |Xω1,xi 〉 + 〈Xω1,x

f | µy |RiΦ0〉)(14)

and

Mxyi←f (−ω1,−ω2) = −

(〈Φ0Li | µ

x |X−ω2,yf 〉 + 〈X−ω2,y

i | µx |RfΦ0〉),

(15)

where X and X are the first-order response wave functionscomputed by solving auxiliary response equations. This refor-mulation of the sum-over-states expressions leads to practicalexpressions of the 2PA moments in terms of just the zeroth-and first-order wave functions of the initial and the final states.The response equations are

Xω1,xk (H − Ek + ω1) = 〈Φ0Lk | µ

x |ΦI〉 (16)

164109-4 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

and

(H − Ek − ω1)Xω1,xk = 〈ΦI | µ

x |RkΦ0〉, (17)

where ΦI s denote the Slater determinants from the target-state manifold (e.g., the reference, singly excited, and doublyexcited determinants for EOM-EE-CCSD).

D. Summary of the algorithm

The EOM-EE-CCSD/EFP method is implemented inthe Q-Chem quantum-chemistry package.47,48 The procedurefor calculating 2PA cross sections involves the followingsteps:

1. First, using a standard procedure,9,10 all MM solventmolecules are replaced by rigid effective fragmentsplaced at the same position and orientation as the originalsolvent molecules. The QM subsystem is now in the fieldof these effective fragments.

2. An iterative self-consistent field procedure is thenemployed such that the induced dipoles on each EFPfragments are self-consistently iterated with the HF wavefunction and with induced dipoles of other EFP frag-ments. The converged induced dipoles are then kept fixedin the subsequent steps of the calculation.

3. The CCSD T - and Λ-amplitudes and EOM-EE-CCSDR- and L-amplitudes and energies are then computedusing the HF wave function and orbitals calculated inthe second step. Using these amplitudes, the CCSDand EOM-EE-CCSD state and transition densities areformed for calculating the state and transition dipolemoments.

4. In the 2PA step, the excitation energies without the state-specific corrections (due to the polarization response ofthe environment to the wave functions of the groundand excited states) are used to determine the photonfrequencies according to the 2PA resonance condition.The procedure for calculating the 2PA cross sections isdescribed in Refs. 34 and 36.

E. Characterization of 1PA and 2PA transitionsin terms of natural transition orbitals

Valuable insight into solvent effects can be gained byunderstanding electronic transitions (1PA, 2PA, etc.) in termsof molecular orbitals. This entails mapping the changesin electronic density induced by excitation onto pairs ofmolecular orbitals. For 1PA transitions, such maps are pro-vided by the reduced transition one-particle density matrices(1PDMs), γ, [

γf←i

]pq= 〈Ψf |p

†q|Ψi〉, (18)

where p† and q create and annihilate electrons in molecularorbitals φp and φq, respectively. Importantly, γ is related tothe experimental observables, e.g., the 1PA transition momentis given by

µaf←i = 〈Ψf |µ

a |Ψi〉 =∑pq

[γf←i

]pq

[µa]

pq. (19)

The elements of transition 1PDMs can be interpreted asthe amplitudes of an excitation operator that generates the final

state from the initial correlated state

Ψf =∑pq

[γf←i

]pq

p†qΨi + higher excitations. (20)

The squared norm of γ,

| |γ | |2 =∑pq

γ2pq, (21)

gives the weight of one-electron character of the transition andprovides a bound to the respective expectation values by virtueof the Cauchy-Schwarz inequality.49,50

Transition 1PDM can be used to define an exciton’s wavefunction (Ψexc), which is a two-particle quantity that containsessential information about the changes in electron densityupon the transition51–54

Ψexc(rH , rP) =∑pq

[γf←i

]pqφp(rH )φq(rP), (22)

where rH and rP are the hole and particle (electron) coordi-nates (using rH = rP = r, Ψexc is reduced to the transitiondensity).

Equations (20) and (22) allow one to interpret the indi-vidual elements of γ as weights of particular configura-tions. For example, using localized orbitals, one can expressthe probability of finding the exciton in a spatial domainD as

1

| |γf←i | |2

∑p∈D,q∈D

[γf←i

]2

pq. (23)

As discussed in Subsection II F, this reasoning can be extendedto quantify the extent of charge-transfer character.55,56

Further simplification in exciton’s representation can beachieved by applying singular-value decomposition (SVD) toγ. The SVD procedure reduces the description of electronicexcitation to one-electron transitions between pairs of orbitalscalled natural transition orbitals54,56–59 (NTOs). Using NTOs,the exciton’s wave function can be written as

Ψexc(rP, rH ) =∑

K

σK φK ,P(rP)φK ,H (rH ), (24)

where φK ,P and φK ,H are the particle and hole orbitals obtainedby SVD of γ and σK are the corresponding singular values.Usually, only a few σK s are significant, giving rise to a simpleinterpretation of excited-state characters in terms of excitationsbetween hole and particle orbitals. By using NTOs, one canexpress the inter-state matrix elements between many-bodywave functions in terms of matrix elements between orbitals,facilitating the connection between molecular orbital theoryand experimental observables

µaf←i = 〈Ψf |µ

a |Ψi〉 =∑

K

σK 〈φK ,P |µa |φK ,H〉. (25)

A detailed description of the NTO analysis and its implemen-tation in Q-Chem is given in Refs. 54 and 55.

We recently extended the concepts of transition 1PDMsand NTOs to 2PA transitions.36 In addition to enablingvisualization of a 2PA transition in terms of molecularorbitals, the 2PA NTO analysis provides insight into thecharacter of the virtual state involved and the type of 2PAtransition.

164109-5 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

F. Conditions necessary for using fragment approachfor describing 2PA transitions in the condensed phase

Let us now discuss the domain of applicability of the frag-ment approaches for modeling 2PA. With an aim to providea metric quantifying whether the conditions needed for theQM/EFP treatment are met by a particular 2PA transition, weinvoke the NTO analysis of the respective perturbed 1PDM.As per Eq. (25), the 2PA moments can be expressed in termsof matrix elements between the respective hole and particleorbitals

Mabf←i(ω1,ω2) =

∑K

σaK 〈φ

aK ,P |µ

b |φaK ,H〉, (26)

where φaK ,P and φa

K ,H are the K th particle and hole NTOs forthe 2PA a-component (a, b = x, y, z) transition 1PDM with asingular value of σa

K .Since the NTOs are linear combinations of atomic orbitals

(χs), i.e.,φK =

∑ν

dKν χν , (27)

Eq. (26) can be expressed in the basis of χs as

Mabf←i(ω1,ω2) =

∑K ,ν,ρ

σaK dK ,P

ν dK ,Hρ 〈 χν |µ

b | χρ〉. (28)

The corresponding 2PA a-component transition 1PDM in theatomic orbitals basis is

[γa

f←i

]νρ=

∑K

σaK dK ,P

ν dK ,Hρ . (29)

By partitioning the QM system into the chromophore (A) andsolvent (B), we can partition the above 1PDM into the fourparts, A←Aγa

f←i,A←Bγa

f←i,B←Aγa

f←i, and B←Bγaf←i, where C←Dγ

represents the contribution of the hole residing on fragment Dand a particle residing on fragment C

C←Dγaf←i =

∑ν∈C,ρ∈D

[γa

f←i

]νρ

. (30)

The probability of finding the hole on fragment D and theparticle on fragment C is now given by

C←DQaf←i =

1

| |γaf←i | |

2

∑ν∈C,ρ∈D

[C←Dγa

f←i

]2

νρ

=| |C←Dγa

f←i | |2

| |γaf←i | |

2. (31)

The four probabilities corresponding to the four fragment1PDMs give a measure of the charge-transfer contributionswithin and across fragments into the 2PA transition moment.This is similar to the charge-transfer (CT) numbers used toquantify the extent of charge transfer in electronic transi-tions.54,55 Breaking the system into a QM solute and an MMsolvent does not introduce large errors only when the proba-bility of finding both the hole and particle is predominantly onthe solute, i.e., when

A←AQaf←i ≈ 1. (32)

In other words, the performance of fragment approaches suchas QM/EFP deteriorates with increasing contributions of thesolvent to the 2PA transition 1PDMs.

We computed these CT numbers for the 2PA transitionsin microhydrated clusters studied here as well as for the rep-resentative snapshots from solvated PYP simulations. Theanalysis reveals that the charge-transfer probabilities betweenthe chromophore and the solvent are low (A←BQa

f←i +B←AQa

f←i< 0.01). The probability of finding both the hole and particleNTOs on the solvent molecules is even lower. This indicatesthat the QM/MM fragmenting of these systems should notintroduce large errors into the 2PA cross sections because the2PA transition is predominantly localized on the chromophore.This encouraging finding is far from obvious as in contrast to1PA transitions for which only the initial and final states needto be localized on the solute, one could expect a noticeablecharge-transfer character in the virtual states.

III. COMPUTATIONAL DETAILS

All calculations were carried out using the Q-Chemquantum-chemistry package.47,48 We used the B3LYP/6-31+G∗ optimized geometry for pNA, the RI-MP2/cc-pVTZgeometry for thymine, and the CCSD/6-31+G∗ geometry forPYPb in the calculations of bare chromophores. The struc-tures used in the QM/EFP calculations for the microsol-vated clusters of pNA, thymine, and PYPb were taken fromRefs. 12, 60, and 61, respectively; they are shown in Fig. 1.We used the EOM-EE-CCSD/6-31+G∗ level of theory withinthree different QM/MM schemes in which the MM subsys-tem was represented by the (1) EFP fragments, (2) EFPfragments without polarization, and (3) CHARMM22 pointcharges. Core electrons were kept frozen in all calculations.In each of these systems, we characterized the lowest sym-metric 2PA transition considering degenerate photons. ForpNA, we also considered non-degenerate photons, the firstphoton having a frequency of 25 400 cm−1 (3.1492 eV).We used the TheoDORE package53 for computing the

FIG. 1. The studied microhydrated clusters of pNA, thymine, and PYPb.

164109-6 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

charge-transfer probabilities following the 2PA NTO calcu-lations with Q-Chem.

In the QM/EFP calculations, all solvent water moleculeswere replaced by the standard effective fragments placedat the same positions and orientations as the initial solventmolecules.9,10 This means that the Cartesian geometries of theinitial structures and the QM/EFP model clusters are slightlydifferent. To eliminate the ambiguity due to slight geometrychange, we used these QM/EFP geometries in all calculationsof the microsolvated clusters, meaning that the structures of thewater molecules in extended QM subsystems are exactly thesame as in the effective fragments. All Cartesian geometriesare provided in the supplementary material.

To model bulk solvation, we performed molecular dynam-ics (MD) simulations with a PYPb chromophore and onesodium cation in a periodic box of 1717 water molecules usingthe NAMD software.62 We used the CHARMM22 parametersfor water and the sodium cation; the parameters for PYPb aregiven in the supplementary material. After the initial minimiza-tion for 20 ps (with 2 fs time step), we equilibrated the systemfor 2 ns (with 1 fs time step) in an NPT ensemble at 298 K and1 atm and followed up with a production run of 1 ns. From thistrajectory, we picked 21 snapshots separated by 50 ps for the2PA calculations using EOM-EE-CCSD/6-31+G∗//EFP (thepdb files for the snapshots are given in the supplementarymaterial). Along the equilibrium trajectory, the sodium cationis more than 10 Å from the PYPb chromophore. In the 2PAcalculations, we included only the chromophore and watermolecules.

To investigate the effect of the QM size on the computed2PA cross section, we systematically extended our QM subsys-tem by including explicit water molecules at the phenolate andcarboxylic acid ends of PYPb. In these calculations, the struc-tures of QM water molecules were identical to the structures inthe MD snapshots. Using these snapshots, we also performedCIS/6-31+G∗//EFP calculations to investigate the convergenceof the key components—excitation energies, transition dipolemoments, and dipole-moment differences between the groundand excited states. In these CIS/EFP calculations, the structuresof the model systems were obtained by using the standard EFPprocedure9,10 on the MD snapshots, meaning that the structuresof water molecules in the QM system are slightly different fromthe structures in the MD snapshots.

IV. RESULTS AND DISCUSSIONA. Molecular orbital framework for 2PA transitionsin pNA, thymine, and PYP chromophore

To understand the effect of microsolvation on the 2PAtransitions of the three benchmark molecules, we begin bycharacterizing these transitions in the bare chromophores. Weaim to understand the nature of solute-solvent interactions byanalyzing the orbital character of the transitions. We calculatedthe 2PA NTOs using our wave-function analysis toolkit.36 Theleading NTO pairs and the molecular orientations are shownin Fig. 2.

In our calculations, the pNA molecule is in the xz plane,with the z axis along the dipole moment. The lowest symmet-ric 2PA transition has a large microscopic cross section (see

FIG. 2. Dominant NTOs (isovalue = 0.05) for the (a) z-component transition1PDM of the lowest symmetric 2PA transition in pNA with degenerate andnon-degenerate photons, (b) x- and y-component transition 1PDMs of the low-est 2PA transition in thymine with degenerate photons, and (c) x-component1PDM of the lowest 2PA transition in the PYPb chromophore with degeneratephotons.

Table I) and is dominated by the zz component of the 2PA tran-sition moment. The NTO analysis of the z component 1PDMshows that this transition is well described by just one NTO pairwith a large singular value. This NTO pair (shown in Fig. 2)reveals the intramolecular charge-transfer character: upon thistransition, the dipole moment increases due to the shift inthe electronic density towards the electron-withdrawing nitrogroup. A more detailed analysis of this transition can be foundin Ref. 36. The charge-transfer character is further supportedby analyzing the participation ratios (PRNTO) computed for thez-component 1PDM and its componentωDMs (see Ref. 36 fordefinitions). Whereas the PRNTO for the z-component 1PDMis ∼1-2 for both degenerate and non-degenerate cases, thePRNTOs for its component ωDMs are >20. This is a signatureof intramolecular charge-transfer transitions. Intramolecularcharge-transfer 2PA transitions are driven by the change in thedipole moment upon excitation (〈Ψf |µa|Ψf 〉 − 〈Ψi |µa|Ψi〉), asgiven by an approximate expression for the 2PA transitionmoment

Mabf←i(ω1,ω2) ≈ −

〈Ψf |µb |Ψi〉

ω1

(〈Ψf |µ

a |Ψf〉 − 〈Ψi |µa |Ψi〉

)−〈Ψf |µ

a |Ψi〉

ω2

(〈Ψf |µ

b |Ψf〉 − 〈Ψi |µb |Ψi〉

),

(33)

which reduces to

Maaf←i(ω) ≈ −2

〈Ψf |µa |Ψi〉

ω

(〈Ψf |µ

a |Ψf〉 − 〈Ψi |µa |Ψi〉

)(34)

for degenerate photons and diagonal elements of the 2PAtransition-moment tensor.36,63,64 Equation (34) also shows that

164109-7 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

TABLE I. 2PA cross sections with degenerate and nondegenerate photons computed with EOM-EE-CCSD/6-31+G∗//MM for the microsolvated clusters of para-nitroaniline. % errors relative to the full QM approachin parentheses. (Ω in eVs, all other quantities in a.u.)

System QM MM Ω f δ2PA δ2PA

ω1 = ω2 ω1 , ω2

pNA pNA 4.55 0.47 4543 6296

pNA + 2H2O pNA + 2H2O 4.49 0.44 4848 7012pNA + H2O EFP 4.49 0.44 4961(2.3) 7182(2)pNA EFP 4.48 0.45 5059(4.4) 7339(5)pNA EFP (no pol) 4.50 0.45 4990(2.9) 7181(2)pNA CHARMM 4.49 0.45 5018(3.5) 7236(3)

pNA + 4H2O pNA + 4H2O 4.46 0.43 4920 7237pNA + H2O EFP 4.46 0.45 5161(4.9) 7590(5)pNA EFP 4.45 0.45 5286(7.4) 7843(8)pNA EFP (no pol) 4.48 0.45 5143(4.5) 7506(5)pNA CHARMM 4.50 0.45 4994(1.5) 7166(1)

pNA + 6H2O pNA + 6H2O 4.46 0.43 4884 7159pNA EFP 4.48 0.45 5206(6.6) 7575(6)pNA EFP (no pol) 4.51 0.45 5025(2.9) 7168(0)pNA CHARMM 4.51 0.45 5051(3.4) 7230(1)

in this case, the 2PA transition moment is proportional tothe transition dipole moment (〈Ψf |µa|Ψi〉) and inversely pro-portional to the excitation energy (Ωfi = ω1 + ω2). Thus,understanding the difference in the permanent dipole momentsof the ground and excited states, transition dipole moment,and excitation energy in the microsolvated clusters is the keyto understanding how solvent impacts 2PA cross sections.For example, bulk solvation stabilizes the excited state morethan the ground state because of a larger excited-state dipolemoment, resulting in the lowering of the excitation energy rel-ative to the gas phase. However, in microsolvated clusters, thechange in the excitation energy depends on the relative ori-entation of the solute molecules and the extent to which theystabilize or destabilize the hole and particle orbitals.

In our calculations, thymine (Cs) is in the xy plane. Thelowest A′ transition with degenerate photons has a smallmicroscopic cross section dominated by the Mxx and Myy

components. The NTO analyses of the x- and y-componentsof the respective 1PDMs show that each of these compo-nent transitions can be described by a single pair of NTOs,which indicates that the transition is accompanied by someintramolecular charge transfer, away from the oxygen of thecarbonyl group across the methyl group. This is further val-idated by the PRNTOs for these 1PDMs (∼1-2) and theircomponent ωDMs (>20). So, this 2PA transition can also beexplained by Eq. (34) and the changes in its 2PA cross sectionin microhydrated clusters can be explained by understandingthe corresponding changes in the transition moments, differ-ences in permanent dipoles, and excitation energy. The shift inelectronic density increases the dipole moment of the moleculein the excited state, which suggests red shift in aqueoussolution.

PYPb also has Cs symmetry and is in the xy plane in ourcalculations. For this chromophore, the lowest symmetric 2PAtransition with degenerate photons has a large microscopic

2PA cross section with a dominant Mxx component. The 2PANTO analysis for the x-component 2PA 1PDM gives two dom-inant NTO pairs, which represent the opposing channels ofelectronic density flow upon the 2PA transition. The domi-nant channel reveals charge transfer from the carboxylic acidto the phenolate and the second channel reveals the chargetransfer from the phenolate to the carboxylic acid. The latterchannel corresponds to the orbital character of the 1PA transi-tion between these states shown in the supplementary material(Fig. S1). Thus, the dominant NTO pair indicates that the vir-tual state is dominated by the initial state, which has a largerelectronic dipole than the excited state.65 By contrast, in thestudied transition in pNA, the character of the virtual stateis dominated by the final state with a larger dipole momentthan the ground state and only one NTO pair. Quantitativeanalysis of the PRNTOs for this 1PDM (∼1-2) and its compo-nentωDMs (>30) also shows that this 2PA transition in PYPbhas some intramolecular charge-transfer character. Moreover,the respective transition dipole moment is also large. Equa-tion (34), therefore, explains the large 2PA cross section forthis transition.

B. Microsolvated p-nitroaniline clusters

In all microhydrated clusters of pNA, the excitation energycalculated with the full QM approach is slightly lower thanthat of the bare chromophore. This indicates that the watermolecules, which are clustered around the nitro group, stabi-lize the particle orbital of the lowest symmetric 2PA transitionmore than the hole orbital. Moreover, the character of thedominant 2PA NTOs for this transition does not change uponmicrosolvation. This is expected for weakly interacting solute-solvent systems. The change in the oscillator strength is alsosmall. Table S1 in the supplementary material indicates thatthe difference in permanent dipole moments of the ground and

164109-8 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

excited states increases, which along with the slight decreasein the excitation energy increases the 2PA cross section byup to 8% for degenerate photons and 15% for non-degeneratephotons in accordance with Eq. (33). Across the three clus-ters, the full QM cross section increases from dihydratesto tetrahydrates and decreases slightly from tetrahydrates tohexahydrates.

The QM/EFP scheme with just the pNA molecule in QMreproduces this trend but slightly exaggerates the changesacross these clusters with errors of about 2%-9% relativeto the respective full QM cross sections (Table I). Omittingthe polarization contributions of the EFP fragments in theQM/EFP calculations results in smaller errors (2%-5%) rel-ative to the full QM cross sections. This approach, however,yields slightly smaller cross sections for the 2PA transitionwith non-degenerate photons for the hexahydrate than thedihydrate. Similarly, the QM/CHARMM approach also yieldssmaller errors relative to the full QM cross sections across thethree clusters but fails to reproduce the trend obtained with thefull QM calculation. In fact, QM/CHARMM shows a decreasein the cross sections for both sets of photons from dihydrate tothe tetrahydrate and an increase from tetrahydrate to hexahy-drate. Yet, the small differences between QM/EFP, QM/EFPwith polarization turned off, and QM/CHARMM preclude usfrom arguing in favor of any of these approaches. Such smalldifferences also indicate that for these systems the impactof intermolecular polarization on 2PA cross sections is notlarge.

We also report the QM/EFP results for the dihydrate andtetrahydrate clusters with the QM subsystem comprising thepNA and its nearest hydrogen-bonded water. These calcula-tions result in smaller errors of 2%-5% relative to the fullQM cross sections. Similar reduction of the QM/EFP errorsupon inclusion of the hydrogen-bonded waters in the QM sys-tem was reported for the core-ionization energies of solvatedglycine.19

The errors in the 2PA cross sections for the three QM/MMschemes relative to the full QM results can be traced back tothe slightly smaller errors in the excitation energies, transitionmoments, and dipole-moment differences with the QM/EFPscheme as tabulated in the supplementary material (Table S1).We note that the errors in the excitation energies for these threeQM/MM approaches relative to the full QM results are verysmall (<1%), but the errors in the transition dipole momentsand dipole-moment differences are slightly larger (up to 2%-3% for both).

C. Microsolvated thymine clusters

The orbital character of the lowest A′ 2PA transition inthymine does not change upon microsolvation. In the full QMcalculation, the microsolvation changes the excitation energyby up to 3%, the oscillator strength by up to 14%, and the 2PAcross sections by up to 20% (we note that the magnitude ofthe microscopic cross section is small). In the full QM cal-culation, the excitation energy of this transition decreases inT1–T2 relative to the bare thymine (see Table II), but in the T3cluster, the excitation energy is higher than in bare thymine.The difference in trends can be attributed to the followingstructural feature: only in the T3 cluster, the water molecule is

TABLE II. 2PA cross sections with degenerate photons computed with EOM-EE-CCSD/6-31+G∗//MM for the microsolvated clusters of thymine. % errorsrelative to the full QM approach in parentheses. (Ω in eVs, all other quantitiesin a.u.)

System QM MM Ω f δ2PA

T T 5.64 0.24 104

T1 T1 5.57 0.26 118T EFP 5.58 0.25 104(12)T EFP (no pol) 5.58 0.25 105(11)T CHARMM 5.57 0.25 106(10)

T2 T2 5.51 0.21 114T EFP 5.50 0.21 109(4)T EFP (no pol) 5.52 0.22 109(4)T CHARMM 5.53 0.22 108(5)

T3 T3 5.68 0.23 89T EFP 5.69 0.23 91(2)T EFP (no pol) 5.67 0.23 94(6)T CHARMM 5.67 0.23 95(7)

T11 T11 5.54 0.27 122T EFP 5.55 0.25 102(16)T EFP (no pol) 5.57 0.25 102(16)T CHARMM 5.56 0.25 104(15)

T12 T12 5.49 0.24 120T1 EFP 5.48 0.24 117(3)T2 EFP 5.50 0.23 106(12)T EFP 5.49 0.23 103(14)T EFP (no pol) 5.51 0.23 103(14)T CHARMM 5.50 0.23 103(14)

T112 T112 5.46 0.25 125T11 EFP 5.45 0.25 121(3)T2 EFP 5.47 0.23 104(17)T EFP 5.47 0.24 101(19)T EFP (no pol) 5.49 0.24 101(19)T CHARMM 5.48 0.24 101(19)

near the carbonyl group that is across from the methyl group,thereby stabilizing the hole orbital shown in Fig. 2(b). Allthree QM/MM schemes reproduce this trend in the excitationenergy across the clusters with negligible errors (<1%) relativeto the full QM calculation. The 2PA cross section decreasesfrom T1 to T2 to gas-phase thymine to T3. All three QM/MMschemes, however, yield smaller 2PA cross sections for the T1cluster than for the T2 cluster, with comparable errors. Forthe T3 cluster, QM/EFP (2% error) does slightly better thanthe other two schemes (6%-7% error). The errors in the 2PAcross sections relative to the full QM calculations can be tracedback to even smaller errors in the transition moments anddipole-moment differences between the ground and excitedstates (Table S2). For thymine clusters, these differences acrossthe three QM/MM schemes are small. The larger excitationenergies and smaller dipole-moment differences between theground and excited states of thymine compared to pNA andPYPb explains why these thymine systems have smaller 2PAcross sections.

The results for the thymine dihydrates and trihy-drates computed with the three QM/MM schemes are also

164109-9 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

comparable, with errors ranging between 11% and 20% rela-tive to the full QM calculation. This indicates that polarizationhas a negligible effect on the 2PA cross sections in thymineclusters. Similar to our observation for the pNA clusters, inclu-sion of explicit water molecules in the QM subsystem forthymine polyhydrates reduces the errors of the QM/EFP cal-culation. In terms of the structure, the T12 dihydrate can beviewed as a superposition of T1 and T2 monohydrate clusters.Considering the large QM/EFP errors in the 2PA cross sectionfor the T1 cluster, the inclusion of the T1 fragment (the tildeindicates that the T1 fragment and T1 cluster are structurallysimilar) in the QM subsystem reduces the error in the 2PAcross sections from 14% to about 3% for the T12 cluster. Sim-ilarly, the structure of the T112 trihydrate can be described asa superposition of the T11 dihydrate and the T2 monohydrate.The QM/EFP errors in the 2PA cross section relative to the fullQM calculation are reduced from 19% to 3% when the T11fragment (structurally similar to the T11 cluster) of T112 isincluded in QM.

D. Microsolvated clusters of the anionicPYP chromophore

The lowest excited state of the phenolate chromophoreof PYP (PYPb) has a large 2PA cross section (Table III).In the microsolvated clusters, the 2PA cross section for thistransition increases significantly (up to 110% in the PYPb-WPWP1 cluster). We also note that the variations in the exci-tation energy for this transition are large across the threeclusters. In the PYPb-WP1 and PYPb-WPWP1 clusters, thewater molecules are hydrogen-bonded with the phenolate oxy-gen. This interaction stabilizes the hole NTO of the transi-tion (see Fig. S1 of the supplementary material) more thanthe particle NTO and results in an increase in the excita-tion energy. In the PYPb-WCWP1 cluster, water moleculesmake hydrogen bonds with both the phenolate and the car-boxylic acid group. This stabilizes both the hole and particleNTOs to a similar extent, and the excitation energy does notchange significantly in this cluster relative to the bare PYPbchromophore.

The three QM/MM schemes show small errors (<2%)in the excitation energy relative to the full QM calcula-tion. The QM/EFP errors are slightly smaller than thoseof the other two non-polarizable QM/MM schemes. This isexpected for a charged chromophore for which mutual polar-ization is more pronounced. Interestingly, QM/EFP overes-timates the excitation energies for all clusters, whereas theother two QM/MM schemes underestimate the excitationenergies.

In the PYPb-WP1 and PYPb-WCWP1 clusters, theQM/EFP errors in the 2PA cross sections relative to the full QMcalculation are smaller than the errors of the other two QM/MMschemes, which again signifies that solvent polarization playsan important role in these clusters. The errors for the PYPb-WPWP1 cluster are comparable across all three schemes. Theerrors in the 2PA cross sections can be traced back to smallererrors in the transition moments and electronic dipole differ-ences between the ground and excited states given in Table S3.Interestingly, QM/EFP yields more accurate dipole-momentdifferences than the other two schemes but performs poorlyfor the transition moments.

In Sec. IV E, we investigate the effect of bulk solvationon PYPb and show that the accuracy of the 2PA cross sec-tions can be significantly improved by including nearest watermolecules in the QM subsystem.

E. Anionic PYP chromophore in aqueous solution

Table S4 in the supplementary material and Fig. 3 showthe excitation energies and microscopic 2PA cross sectionswith degenerate photons for the 21 MD snapshots of PYPb insolution computed with QM/EFP. We use the following nota-tions for calculations with different QM subsystems: (a) PYPbmeans that only PYPb is included in QM, (b) PYPb+ meansthat PYPb plus all water molecules within 2.5 Å of the phe-nolate oxygen are included, (c) +PYPb includes PYPb plus allwater molecules within 2.5 Å of the carboxylic acid group,(d) +PYPb+ includes PYPb plus all water molecules within2.5 Å of the phenolate oxygen and the carboxylic acid group,

TABLE III. 2PA cross sections with degenerate photons computed with EOM-EE-CCSD/6-31+G∗//MM for themicrosolvated clusters of PYPb. % errors relative to the full QM calculation in parentheses.

System QM MM Ω f δ2PA

PYPb PYPb 3.21 1.06 9578

PYPb-WP1 PYPb-WP1 3.25 1.01 16862PYPb EFP 3.26 0.99 15961(5)PYPb EFP (no pol) 3.23 1.00 15353(9)PYPb CHARMM 3.22 1.00 15291(9)

PYPb-WCWP1 PYPb-WCWP1 3.21 1.07 16878PYPb EFP 3.22 1.02 15177(10)PYPb EFP (no pol) 3.20 1.02 14469(14)PYPb CHARMM 3.20 1.01 14500(14)

PYPb-WPWP1 PYPb-WPWP1 3.37 0.98 20095PYPb EFP 3.41 0.94 18155(10)PYPb EFP (no pol) 3.32 0.96 18105(10)PYPb CHARMM 3.31 0.97 18212(9)

164109-10 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

FIG. 3. (a) Excitation energies, (b) 1PA transition moments, (c) dipole-moment differences between the ground and excited states, and (d) microscopic 2PAcross sections for the 21 snapshots with varying size of the QM subsystem. The horizontal lines indicate average properties for PYPb (blue), PYPb+ (red), and+PYPb+ (black) calculations across the 21 snapshots. The shaded area indicates the standard deviation for the +PYPb+ calculations across the 21 snapshots.

(e) PYPb++ includes PYPb plus all water molecules within3.5 Å of the phenolate oxygen, and (f) PYPb+++ includesPYPb plus all water molecules within 4.0 Å of the phenolateoxygen.

With just the PYPb chromophore in QM, the average2PA cross section across the snapshots shows about 28%increase relative to the bare chromophore (Table III) witha large standard deviation. With an exception of snapshot#15, the 2PA cross sections are higher than the referencegas-phase value. As expected, the excitation energies acrossall snapshots increase relative to the bare chromophore. Thetransition moments and dipole-moment differences betweenthe initial and the final states are given in Table S4 (sup-plementary material). We note that the transition moments,in general, decrease, while the dipole-moment differencesincrease, the latter overcompensating the former resultingin higher 2PA cross sections in solution relative to the gasphase.

The QM/EFP results for the microsolvated clusters showthat 2PA cross sections are sensitive to the choice of the QMsubsystem and that the inclusion of solvent molecules in QMcauses a noticeable change in the cross section. We carriedout a similar analysis for the PYPb chromophore in solution,wherein we extend the QM subsystem by including more watermolecules in the QM part. First, we included the first solva-tion shell (defined by the cutoff distance of 2.5 Å) aroundthe phenolate oxygen (denoted by PYPb+). We see that thePYPb+ 2PA cross sections increase significantly (increase inthe average is ∼15%) and excitation energies decrease (byup to 0.11 eV) across the snapshots relative to the QM/EFP

results with just the chromophore in QM. On the other hand, theinclusion of the first solvation shell of water molecules aroundthe carboxylic acid group (denoted by +PYPb) increases the2PA cross sections to a smaller extent than the PYPb+ calcu-lations in five of the six snapshots for which we conductedsuch calculations. In general, the excitation energies increaseslightly (up to 0.02 eV) in the +PYPb calculations rela-tive to the QM/EFP calculations with just the chromophorein QM.

We note that the 2PA cross sections for the +PYPb+QM/EFP calculations, wherein the QM subsystem includesthe first solvation shells around both the phenolate oxy-gen and the carboxylic acid group, can be approximatedby adding the differences between the PYPb+ and PYPbresults to the differences between the +PYPb and PYPbresults

+PYPb+δ2PA =PYPbδ2PA +

(PYPb+δ2PA −

PYPbδ2PA

)+

(+PYPbδ2PA −

PYPbδ2PA

). (35)

The increase in the average 2PA cross section across snapshotsfor the +PYPb+ calculations is ∼22%. These changes in the2PA cross sections between QM/EFP calculations with differ-ent QM subsystems can be traced back to the correspondingchanges in the transition moments and dipole-moment differ-ences between the ground and excited states given in Table S4(supplementary material).

Since the changes due to the inclusion of water moleculesin QM at the two ends of the PYPb chromophore are approx-imately additive, we test if the inclusion of the first solvation

164109-11 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

shell in QM gives sufficiently converged results by includ-ing water molecules beyond the first solvation shell only atthe phenolate end. For snapshots #13 and #17, we include allwater molecules within 4.0 Å from the phenolate oxygen. Forsnapshot #13, the excitation energy and 2PA cross section forthe PYPb+++ calculations show less than a 1% change rela-tive to the results for the PYPb+ calculation with just the firstsolvation shell in QM. The change in the transition momentsfrom PYPb+ to PYPb+++ for this snapshot is positive, whichcancels the negative change in the dipole-moment differenceleading to an overall smaller change in the 2PA cross section.For snapshot #17, the change in the 2PA cross section fromPYPb+ to PYPb++ (water molecules within 3.5 Å) is about5%. However, the 2PA cross section change by <1% fromPYPb++ to PYPb+++ calculations. The transition momentsand dipole-moment differences also show negligible changesfrom PYPb++ to PYPb+++. These calculations clearlysuggest that the convergence of the 2PA cross sections withthe size of QM subsystem depends on the convergence of theexcitation energies, transition moments, and dipole-momentdifferences.

Given the above results, a crucial question is then: Howfast do these properties converge with the size of the QM sub-system? In other words, how large the QM solvation shellaround the chromophore should be for the converged bulkresults? While we could not carry out such convergence stud-ies with EOM-EE-CCSD, we can investigate the convergenceof the key electronic properties contributing to the 2PA crosssections using a lower-level method. Towards this goal, weinvestigated the convergence of the excitation energies, tran-sition moments, and dipole-moment differences by comput-ing them with the CIS/EFP scheme for all snapshots (theresults for snapshots #13 and #17 are presented in Fig. 4).For this analysis, we gradually extend the QM subsystem byincluding more and more water molecules in QM using theEFP-modified geometries (as in the case of the microhydratedclusters, this strategy allows us to compare the properties of dif-ferent embedded QM subsystems using identical structures).In all model structures, the chromophore is oriented in the sameway such that the x-component of the 1PA transition momentand dipole-moment difference always dominates. We approxi-mate the corresponding Mxx moment according to Eq. (34) andthen estimate the error in the 2PA cross section in terms of theerror in this 2PA transition moment as a function of the QMsize. The use of the EFP-modified extended QM subsystemdoes not introduce significant differences in the behavior ofthe excitation energies, total transition moments, and dipole-moment differences in our analysis. The differences in theseproperties for these two EFP-modified snapshots across dif-ferent embedded QM subsystems (given in Table S5 of thesupplementary material) are small when compared to the cor-responding values for embedded QM subsystems constructedwith the geometries taken from the MD snapshots (given inTable S6 of the supplementary material for snapshots #13 and#17).

We expand the QM subsystem by adding waters withina cutoff distance from either the phenolate oxygen or thecarboxylic acid group of the chromophore. We vary thecutoff distance from 0 Å (chromophore only) to 5 Å.

FIG. 4. CIS/6-31+G∗//EFP calculations for snapshots #13 and #17 showingthe behavior of key electronic properties as a function of the size of the QM sub-system. (a) The number of water molecules as a function of the cutoff distancefrom the phenolate oxygen or the carboxylic acid group, (b) excitation ener-gies, (c) x-component transition moments, (d) x-component dipole-momentdifferences between the ground and excited states, and (e) xx-component 2PAtransition moments computed using Eq. (34) as functions of the number ofQM water molecules.

For snapshots #13 and #17, the maximum number ofwater molecules included in QM was 54 and 43, respec-tively [Fig. 4(a)]. The cutoff distance of 2.5 Å gives the

164109-12 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

structure corresponding to the +PYPb+ calculations discussedabove.

Figure 4(b) shows the behavior of the excitation energyas a function of the QM water molecules. In both snapshots,we observe an initial steep drop from the calculation withthe bare chromophore in QM to the +PYPb+ calculation.The excitation-energy curve for snapshot #17 plateaus afterabout 20 water molecules (∼4 Å cutoff) are included in QM.By contrast, this curve for snapshot #13 shows a slow grad-ual decrease and plateaus when about 42 water molecules(∼4.6 Å cutoff) are included in the QM. Including the firstsolvation shell (+PYPb+ calculations) around the two endsof PYPb reduces the error in the excitation energy from3.5% to 2.6% and 1.6% to 0.8% for snapshots #13 and #17,respectively, relative to the calculations with the largest QMsubsystems.

Figure 4(c) shows the behavior of the x-component tran-sition moment as a function of QM water molecules. After aninitial steep increase upon addition of the first solvation shell,the changes are much smaller for both snapshots. The errorsin the +PYPb+ calculations are 1.8% and 0.4% for snapshots#13 and #17, respectively, relative to the calculations with thelargest QM subsystems.

Figure 4(d) shows the behavior of the x-componentdipole-moment difference as a function of QM watermolecules. As for the excitation energies, the dipole-momentdifference for snapshot #17 is well converged after the inclu-sion of 20 water molecules in the QM subsystem. However,the convergence is slower for snapshot #13 for which conver-gence is reached only after∼45 molecules are included in QM.Whereas the absolute errors relative to the calculation with thelargest QM subsystem are small for snapshot #17 (<2.5%),the corresponding errors are slightly larger for snapshot #13(up to 6.2%). Including the first solvation shell decreases theabsolute errors for snapshot #17 from 0.9% to 0.1%, while theabsolute errors increase for snapshot #13 from 4.3% to 5.5%.

Figure 4(e) shows the approximate Mxx transitionmoment, calculated using Eq. (34), as a function of the numberof QM water molecules. Similar to the behavior of the excita-tion energy and dipole-moment difference, the xx-component2PA transition moment for snapshot #17 changes by no morethan 3.3% after the inclusion of ∼20 water molecules. On theother hand, the change is larger and convergence is slower(after inclusion of ∼45 water molecules in QM) for snapshot#13. This is highlighted in that the absolute errors in +PYPb+estimates for this 2PA transition moment are 6.2% for snapshot#13 as against 0.3% for snapshot #17. Since the approximateerror in the 2PA cross sections is twice the error in the dom-inant Mxx component, for snapshot #13, we estimate that ourbest EOM-EE-CCSD 2PA cross section (+PYPb+ calculation)is within ∼12.5% from the converged bulk value.

Figure S2, Table S5, and the discussion in the supple-mentary material provide the CIS/EFP error analysis for all21 snapshots as a function of the cutoff distance. The errorsin the average microscopic 2PA cross sections for PYP and+PYP+ calculations are expressed in terms of the errors inthe Mxx transition moments relative to the converged values[Eq. (S1)]. These errors are −12.6% and −3.3%, respectively.This clearly shows that the errors in the average 2PA cross

sections relative to the converged values drop significantlywhen the water molecules in the first solvation shell are treatedat the QM level. Thus, we estimate that our best bulk-averagedvalue of the EOM-CCSD 2PA cross section (+PYPb+ calcu-lation) is about 3-4% below the converged result. The analysisof the convergence of 2PA cross sections with respect to thesize of the QM subsystem shows that the first solvation shellprovides a good balance between computational cost and con-verged results for the final averaged spectra, despite slowerconvergence for individual snapshots. A similar conclusionfor another polarizable embedding QM/MM scheme has beenreported in Ref. 66 for converged X-ray absorption spectraaveraged over multiple snapshots.

V. CONCLUSIONS

We extended the EOM-CCSD method of calculating 2PAcross sections to condensed-phase calculations, wherein thequantum system is embedded in the polarizable environmentrepresented by EFP fragments. We also presented a met-ric based on transition 1PDMs that helps us to assess theapplicability of a particular QM/MM embedding scheme forstudying 2PA transitions.

We analyzed the solvent effects on 2PA cross sections inmicrohydrated clusters of pNA, thymine, and the PYPb chro-mophore. We benchmarked the EOM-EE-CCSD/EFP resultsagainst full EOM-EE-CCSD calculations for these systems.When only the chromophore is included in the QM subsys-tem, the errors in the EOM-EE-CCSD/EFP 2PA cross sectionsrelative to the full EOM-EE-CCSD treatment are <20%. TheEOM-EE-CCSD/EFP method captures the main trends in the2PA cross sections as the full QM calculations. A system-atic inclusion of explicit solvent molecules in QM revealsthat the calculated 2PA cross sections are sensitive to theQM size. When nearest water molecule(s) are added to theQM subsystem, the EOM-EE-CCSD/EFP errors drop down to∼5%.

In the EOM-EE-CCSD/EFP method, we turned off thecontribution of the polarization effects and analyzed itsimpact on the 2PA cross sections. We find that polariza-tion has a large impact on the EOM-EE-CCSD/EFP 2PAcross sections of (anionic) PYPb clusters but not for clus-ters of thymine and pNA. We also compared our resultswith the EOM-EE-CCSD/MM approach, wherein the sol-vent molecules are represented by CHARMM charges. Whilewe do not see significant differences in the results withthese three QM/MM schemes for thymine and pNA clus-ters, EOM-EE-CCSD/EFP performs better for the PYPbclusters.

We also performed NTO analyses of the 2PA transitionsin order to understand their orbital character and to obtaininsight into the impact of the solvent. The analyses revealedthe intramolecular charge-transfer character of these 2PA tran-sitions, which then allowed us to evaluate the 2PA transitionmoments from three key components: excitation energies, 1PAtransition moments, and dipole-moment differences betweenthe ground and excited states. The errors in the EOM-EE-CCSD/EFP 2PA cross sections relative to the full EOM-EE-CCSD results were then traced back to the errors in the

164109-13 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)

excitation energies, 1PA transition moments, and dipole-moment differences.

We have also reported the EOM-EE-CCSD/EFP 2PAcross section of the lowest symmetric transition in PYPbin solution. Our analysis shows that inclusion of few watermolecules in the QM subsystem reduces the errors in the exci-tation energies, 1PA transition moments, and dipole-momentdifferences significantly relative to the converged values withextended QM subsystems. Consequently, the inclusion ofwater molecules in the QM improves the calculated 2PA crosssection significantly.

SUPPLEMENTARY MATERIAL

See supplementary material for (1) 1PA NTOs for thestudied transitions in pNA, thymine, and PYPb in the gasphase; (2) key quantities such as excitation energies, transitionmoments, dipole-moment differences, and 2PA cross sectionsfor the microhydrated clusters and aqueous PYPb calculatedwith EOM-EE-CCSD/EFP and CIS/EFP methods; (3) force-field parameters for PYPb; (4) Cartesian coordinates; and (5)a separate .txt file containing pdbs for the 21 MD snapshots ofPYPb in solution.

ACKNOWLEDGMENTS

A.I.K. acknowledges support by the U.S. National Sci-ence Foundation (Grant No. CHE-1566428). K.D.N. acknowl-edges helpful discussions with Dr. Ilya Kaliman on EFPand Dr. Atanu Acharya and Tirthendu Sen on NAMDsimulations.

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