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How Does a Solvent Affect Chemical Bonds? MixedQuantum/Classical Simulations with a Full CI Treatmentof the Bonding ElectronsWilliam J. Glover,§ Ross E. Larsen,† and Benjamin J. Schwartz*

Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, California 90095-1569

ABSTRACT Understanding how a solvent affects the quantum mechanics andreactivity of the chemical bonds of dissolved solutes is of fundamental importanceto chemistry. To explore condensed-phase effects on a simplemolecular solute, wehave studied the six-dimensional two-electron wave function of the bondingelectrons of the Na2 molecule in liquid argon via mixed quantum/classical simula-tion. We find that even though Ar is an apolar liquid, solvent interactions producedipole moments on Na2 that can reach magnitudes over 1.4 D. These interactionsalso change the selection rules, induce significantmotional-narrowing, and cause alarge (26 cm-1) blue shift of the dimer's vibrational spectrum relative to that in thegas phase. These effects cannot be captured via classical simulation, highlightingthe importance of quantum many-body effects.

SECTION Statistical Mechanics, Thermodynamics, Medium Effects

U nderstanding how a solvent affects the quantummechanics and reactivity of the chemical bonds ofdissolved solutes is of fundamental importance to

chemistry. The solvent's influence on chemical bonds arisesfrom alteration of a solute's electronic structure due to bothshort- and longer-ranged interactions with hundreds of near-by solvent molecules. This solvent-induced alteration of thesolute's electronic structure can alter the vibrational frequen-cies and/or dissociation energies of the solute's chemicalbonds. Traditional molecular dynamics (MD) simulationsbased on classical pairwise additive force fields are unableto explore the detailed physics underlying the solvent'sinfluence on chemical bonds because such simulations can-not account for changes in the solute's electronic structurewith the environment. Classical models can be improved byadding many-body polarization terms; it has recently beenshown that such terms are necessary to successfully repro-duce the vibrational spectrum of liquid water and ice.1,2 It isunclear, however, whether classical polarizable models canuniversally describe the changes in the solute electronicstructure that alter the nature of bond vibrations. This isparticularly true in nonpolar systems where longer-rangedelectrostatic interactions play a much smaller role than in-herently quantum mechanical interactions such as Paulirepulsion. To capture these effects, the quantum mechanicsof the solute valence electrons responsible for the chemicalbonding and the interactions of these electrons with thesurrounding solvent molecules must be accounted for at theHamiltonian level. There has been some semiempirical workdone to include solvent effects on the bonding electronicstructure in simulations examining the photodissociationdynamics of I2 in rare gas liquids/matrixes3,4 and I2

- in raregas and CO2 clusters,5-7 but to the best of our knowledge,

there has been no first-principles treatment of solvent-solutechemical bond interactions for diatomics in simple liquids.

In this paper, we go beyond a semiempirical treatment ofthe solute's electronic structure and use first-principles quan-tum mechanics to calculate the influence of a solvent on theelectrons in a chemical bond.We avoid the formidable task ofsolving Schr€odinger's equation (SE) for all of the electrons onboth the solvent and solute molecules in the simulation byadopting a mixed quantum/classical (MQC) approach, inwhich we solve the SE exactly for the valence electrons of adiatomic solute. Due to the relative simplicity of its electronicstructure, we have chosen the sodium dimer (Na2) as oursolute, andwe have investigated the chemical bond dynamicsof Na2 dissolved in liquid argon. This leads to a computationalproblem that involves finding the electronic states of the twoNa2 valence electrons in the presence of hundreds of classicalAr atoms and two Naþ core particles. We solve this problemusing our recently developed two-electron Fourier grid (2EFG)method, which is outlined for this particular application in theSupporting Information (SI). We find that even though liquidAr is an apolar solvent, its influence on the bonding electronsof Na2 is quite substantial; nonpolar interactions with Aratoms that push the valence electrons off of the nuclearcenters produce a large instantaneous dipole moment onthe dimer that can reachmagnitudes of over 0.3 e-Å (1.4 D).The presence of the solvent-induced fluctuating dipole leadsto both anew far-IRabsorption feature andanalterationof theselection rules for the dimer's infrared vibrational absorption.

Received Date: October 5, 2009Accepted Date: November 2, 2009

rXXXX American Chemical Society 166 DOI: 10.1021/jz9000938 |J. Phys. Chem. Lett. 2010, 1, 165–169

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Moreover, the solvent-induced alteration of the bondingelectron density between the Na2 nuclei leads to a large(26 cm-1) blue shift of the dimer's bond vibration relativeto that in the gas phase. Classical simulations are unable toaccount for this shift of the bond vibrational frequency becausepurely classical force fields do not capture the solvent-inducedalteration of the Na2 bonding electron density.

Our MQC approach to simulating Na2 in liquid Ar involvesfinding the ground-state solution to the SE for the two sodiumdimer valence electrons in the potential of all of the classicalsolvent atoms. The SE was solved using our recently devel-oped 2EFG method (see SI). Briefly, the valence electrons'wave function was represented on a six-dimensional realspace grid, and the SE was solved variationally at everymole-cular dynamics time step. The simulation cellwas cubicwith aside of 43.8332 Å and contained 1600 classical Ar atoms,2 classical Naþ cations, and 2 fully quantum mechanicalelectrons. Our model for the Na2 intramolecular interactionsis described in the SI and involves a Phillips-Kleinmanelectron-Naþ pseudopotential8 and classical Coulombic re-pulsion between the two Naþ nuclei. The solute-solvent coup-lingwas taken fromGervais et al.9with a slightmodification oftheir solvent polarization potential (see SI). We modeled theclassical Ar-Ar interactions with Lennard-Jones (LJ) poten-tials.10 The simulation time step was 5 fs (∼2.3�10-3 redu-ced LJ time units), and the classical particles were propagatedwith the velocity Verlet algorithm10 in the microcanonicalensemble; for our simulation cell, the Ar reduced density was0.75, and after equilibration, the Ar temperature was 120 (4 K (reduced temperature close to 1.0), placing the Ar solventin the liquid region of the LJ phase diagram.11 The electrons'wave function was expressed on a 166 cubic grid (i.e., aconfiguration interaction (CI) basis of ∼1.7� 107 functions)that spanned 14 Å in each direction and was centered in themiddle of the simulation cell where the Na2 was located. Datawere collected from a production run of 500 ps. All of thesimulation parameters and the details of the calculationsinvolved are described in the SI.

To provide a point of reference, we began by investigatingthe electronic structure of the gas-phase sodium dimer withthe same level of theory as that used for our condensed-phasesimulations. The solid black curve in Figure 1 shows theBorn-Oppenheimer potential energy curve (PEC) for thesodium dimer's bond displacement computed using our2EFG method, where the zero of energy was chosen to betwice the energy of Na0 calculated with the same grid basis.Despite our choice of a grid basis for a problem with cylind-rical symmetry, the calculated PES is smooth, exhibits a clearminimum at Re=3.27 Å with depth De=0.679 eV, and goesto zero at large displacement, indicating that the 2EFGmethodcorrectly calculates electronic structure at all distances includ-ing the dissociation limit, as expected for a CI-based method.The second derivative of the PEC at theminimumcorrespondsto a harmonic oscillator frequency of ωe= 136 cm-1. Thesevalues compare well with previous theory at a similar level toour model but using Gaussian basis functions, which foundRe=3.34 Å, De=0.458 eV, and ωe=136 cm-1.12

Having seen that the 2EFG method provides a reasonablygood description of the electronic structure of gas-phase Na2,

we now use this method to consider the properties of thedimer dissolved in liquid Ar. Perhaps the largest surprise fromthese calculations is that immersion of Na2 into simulatedliquid Ar leads to the presence of a substantial instantaneouselectric dipole moment, μB(t), on the dimer, the properties ofwhich are plotted in Figure 2. Figure 2A plots a time trace of|μB|(t) from a portion of our simulation. This figure makes itclear that there are large fluctuations in |μB|(t), reaching valuesof over 0.3 e-Å (1.4 D), with an ensemble average value of∼0.1 e-Å (∼0.5 D). The presence of such large dipolemoments is quite surprising since liquid Ar is apolar and thusexpected to be a weakly interacting solvent; this is the mainreason that it and other noble gases are used in matrixisolation studies. Since there are no charges on the solventto polarize the electrons on the dimer, the observed dipolemoment must result from short-ranged interactions with thesolvent. These “overlap-induced dipoles” result from the Paulirepulsion between the Ar and Na2 electrons,13 an effectincluded in our simulations via the e--Ar pseudopotential.Interestingly, even though classical studies have shown thatnonpolar solvation is dominated by only a few close solventatoms,14 the magnitude and direction of the induced dipoledepends on the positions of multiple Ar atoms relative to thesodiumdimer, aswe showexplicitly in the SI. Thismeans thatquantum mechanical many-body effects are important indetermining the induced dipole.

To better understand the consequences of the dipole onNa2 induced by the solvent, we investigated the time depen-dence of the instantaneous dipole moment by calculatingthe dipole time autocorrelation function, C(t)= ÆμB(t) 3 μB(0)æ,which is shown inFigure 2B. The fact that interactionswith thesolvent produce a fluctuating dipole moment with substantialmemory gives rise to an interaction-induced absorptionspectrum, which we calculated in the classical limit (viaeq 2.10 of ref 13) and plotted in Figure 2C. The spectrumshows a peak in the far-IR at ∼50 cm-1, which is wellseparated from the dimer's solution-phase oscillator fre-quency and presumably results from intramolecular rattlingmotions of the dimer in the solvent cage, similar to interpreta-tions of the far-IR spectrum of H2 in liquid Ar.15 The weakerfeature at ∼160 cm-1, which is expanded in the inset of

Figure 1. Gas-phase potential energy curve for the sodium dimer.The inset expands the region around the minimum and also plotsthe solvent-averaged intramolecular potential, ÆVintra(r)æ (purpledashed curve), described in the text.

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Figure 2C, occurs at the bond frequencyof thedimer;many-bodyinteractions with the solvent cause the normally symmetry-forbidden infrared vibrational absorption of the dimer tobecome weakly allowed in the presence of the Ar solvent.

In addition to creating a fluctuating dipole moment at thebond vibrational frequency, the change in electronic structureof the dimer induced by the solvent also alters the funda-mental nature of the bond vibration. The blue solid curve inFigure 3 shows thepower spectrum, ~C(ω), of thebondvelocityautocorrelation function for MQC-simulated Na2 in liquid Ar.The green dashed curve shows the same spectrum calculatedfrom a 1 ns all-classical simulation of this system inwhich theNa2 bond was described by a quartic fit to the gas-phase Na2PEC shown in Figure 1 and the solute-solvent interactionswere represented by a three-center fit to our calculatedNa2-Ar gas-phase potential energy surface (see SI). Forreference, the natural harmonic frequency of the gas-phasedimer in our model is shown as the black vertical dotted line.The all-classical and MQC power spectra have peaks that are

blue shifted from the gas phase by 7 and 26 cm-1, respec-tively; the solvent clearly has a substantial effect on the Na2bond vibration. The small blue shift seen in the all-classicalsimulation results from physical confinement of the vibratingsolute; thepresenceof the solventnear the outer turningpointof the vibration effectively steepens the potential holding thediatomic together, resulting in a small increase in the vibra-tional frequency. However, the fact that the all-classical simu-lation underestimates the magnitude of the vibrational blueshift seen in the MQC simulation by a factor of ∼4 indicatesthat many-body quantum effects that alter the electronicstructure of the dimer are far more important in determiningthe bond vibrational frequency.

The origin of these many-body effects on the bond vibra-tional frequency can be seenby simply considering the spatialextent of the bonding electrons' wave function; when thedimer bond length is 3.2 Å, themean radius of gyration of thequantum valence electrons' charge density for gas-phase Na2is 2.61 Å, while that for Na2 in liquid Ar is only 2.42 Å. This isconsistent with a picture in which the solvent particles com-press the electron density between the Na nuclei, leading to astiffer chemical bond. This solvent compression of the bond-ing electrons also decreases the average Na2 bond lengthfrom 3.27 ( 0.01 Å in the all-classical simulation to 3.17 (0.01 Å in the MQC simulation. Thus, the vibrational blue shiftof Na2 dissolved in liquid Ar results primarily frommany-bodyelectronic effects that cannot be adequately modeled with aclassical, gas-phase parametrization, even when all of theclassical interactions are derived from quantum mechanicalcalculations.

To further explore the origin of the vibrational blue shift ofNa2 in liquid Ar, we separated the Born-Oppenheimer poten-tial energy of the dimer into two parts, VNaNa=Vinter þ Vintra.The first term contains operators that depend only on solventdegrees of freedom, Vinter= Æψ|V̂Ar|ψæþ UNaþ-Ar, in which V̂Ar

is the argon-electron potential operator andUNaþ-Ar is the totalsodium-argon classical potential. The second term containsoperators that depend on the solute's degrees of freedom,Vintra = Æψ|V̂Na þ T̂|ψæ þ 1/R, in which V̂Na is the sodium-electron potential operator, T̂ is the kinetic energy operator forthe electrons, and R is the dimer bond length. Even though thepresence ofψ in the expectation value causesVintra to implicitly

Figure 2. Properties of the instantaneous induced dipolemomenton Na2 in liquid Ar. (a) Time trace of the induced dipole moment,μ(t), from a portion of our MQC simulation. (b) Dipole autocorre-lation function averaged over the 500 ps simulation. (c) Absorp-tion spectrum of Na2 induced from the fluctuating dipole. Theinset highlights the contribution from Na2's intramolecular vibra-tional frequency at around 160 cm-1.

Figure 3. Vibrational spectrum of Na2 calculated from the Fouriertransformof the bond velocity autocorrelation function. Solid bluecurve: from the mixed quantum/classical simulation. Dashedgreen curve: from the classical simulation, with a three-sitepotential. Dot-dashed red curve: from the classical simulationwith a modified intramolecular potential as described in the text.Dotted black line: gas-phase vibrational frequency.

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depend on the coordinates of the solvent (i.e., there is still somemany-body character in this potential that has not been com-pletely removed),wechose thisparticular separationsince in thelimit of no solvent interactions,Vintra reducesprecisely to thegas-phase Born-Oppenheimer Na2 potential (i.e., the solid curve inFigure 1). In the presence of solvent, this separation allows us togenerate an effective condensed-phase intramolecular potentialfor the dimer by calculating Vintra(r) for many solvent configura-tions by holding the solvent fixedwhilemapping out Vintra alongthe bond displacement and then ensemble averaging over thedifferent solvent configurations. This solvent-averaged intramo-lecular potential energy surface, ÆVintra(r)æ, is plottedas thepurpledashedcurve in the inset to Figure1; for reference, the gas-phasePECof thedimer is shownas the solidblackcurve.Acomparisonof the two curves shows that solvent interactions both shift theintramolecular potentialminimum to a shorter bond length andincrease the curvature at the minimum. Thus, interactions withthe solvent that compress the electronic wave function areresponsible for both the decreased bond length and increasedvibrational frequency of Na2 in solution.

Wenote that ifwe construct another all-classical simulationof this system in which we use ÆVintra(r)æ instead of the gas-phase PEC as the intramolecular dimer potential, the vibra-tional spectrum of the dimer blue shifts an additional∼9 cm-1, as shown by the red dot-dashed curve in Figure 3.This modified classical simulation, which incorporates someof the quantum effects of the solvent on the solute, clearlyprovides a better portrayal of the vibrational properties of thedimer in the condensed phase than that when the gas-phasepotential is used, but the modified classical simulation is stillunable to account for all of the shift seen in the full MQCsimulation. Thus, many-body effects in the intermolecularpart of the potential, which cannot be easily incorporated intoclassical simulations, are important in a correct description ofthe dynamics of chemical bonds in condensed phases.

Figure 3 shows that not only does interaction with thesolvent change the average vibrational frequency of thedimer, but it also changes the width of the vibrational reso-nance; the full width at half-maximum of the peak in thevibrational power spectrum increases from 4.6 cm-1 in theall-classical simulations (green dashed and red dot-dashedcurves) to 5.8 cm-1 in the MQC simulation (solid blue curve).The contributions from the physical processes underlying thewidth of a vibrational peak in a condensed-phase system canbe fairly difficult to tease apart.16 Probably the simplestcontribution to the spectral width is inhomogeneous broad-ening; different solvent configurations around the solute willresult in a static vibrational spectrum that is broadened fromthe gas phase due to a distribution of instantaneous vibra-tional frequencies. Of course, if the solvent motions that shiftthe vibrational frequency are sufficiently rapid, the spectralline width will decrease due to motional narrowing. For bothour MQC and classical systems, we calculated the inhomoge-neous contributions to the vibrational power spectrum bycalculating the distribution of frequencies associated with theinstantaneous second derivative of the PEC.We found that inboth the MQC and classical systems, the intrinsic inhomoge-neous line width of Na2 in liquid Ar was 28 cm-1. This issubstantially broader than the actual vibrational spectrum

including solvent dynamics, indicating that the vibrationalspectrum of Na2 in liquid Ar is significantly motionallynarrowed. The fact that the classical simulation predictseven narrower vibrational resonances is a sign that there iseither too much motional narrowing in the classical simu-lation or that the contribution from vibrational energy re-laxation (which is included classically in all three of oursimulations) is different between the MQC and all-classicalsimulations.

In conclusion, we have applied our two-electron Fouriergrid electronic structure method to simulate the chemicalbond dynamics of the sodium dimer in liquid argon. Byallowing the solvent to directly interact with the bondingelectrons in the solute, we have been able to investigate awhole host of many-body effects that alter the properties ofthe molecule relative to the gas phase. The most substantialchange upon solvation of Na2 in liquid Ar is the developmentof a large instantaneous dipole moment that both creates anew far-IR absorption at frequencies characteristic of thesolute rattling motions and changes the selection rules toallow (gas-phase forbidden) absorption at the solute's vibra-tional frequency. Compressionof thebonding electrons by thesurrounding solvent particles also increases the electrondensity between the solute nuclei, leading to a vibrationalfrequency that is more blue shifted than an all-classicalsimulation would predict. We also saw that even with poten-tials derived from quantum calculations and a Na2 intramo-lecular potential adjusted for a condensed-phase envi-ronment, classical simulations were still unable to reproducethe vibrational dynamics of Na2, highlighting the sensitivity ofchemical bonds to many-body interactions that only a first-principles-based quantum calculation can capture. Finally, wenote that althoughwe have focused entirely on the propertiesof the solute in its ground electronic state in this paper, our2EFG method also produces accurate electronic excitedstates, opening the exciting possibility of performing first-principles quantum simulations of the photodissociation dy-namics of chemical bonds in condensed-phase environ-ments.

SUPPORTING INFORMATION AVAILABLE Full details ofall the simulationmodels presented in this publication and explora-tion of the origin of Na2's induced dipole moment in more detail.This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Author:*To whom correspondence should be addressed. E-mail:schwartz@chem.ucla.edu.

Present Addresses:§Department of Chemistry, Stanford University.†National Renewal Energy Laboratory, Golden CO.

ACKNOWLEDGMENT This work was supported by the NationalScience Foundation under Grants CHE-0603766 and CHE-0908548and the American Chemical Society Petroleum Research Fundunder Grant 45988-AC,6. We gratefully acknowledge the California

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NanoSystems Institute and the UCLA Institute for Digital Researchand Education for use of their Beowulf clusters.

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