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Vibrational Symmetry Breaking of NO3 in Aqueous Solution: NO Asymmetric Stretch Frequency Distribution and Mean Splitting† Sai G. Ramesh,§ Suyong Re,§,¶ Jean Boisson,§ and James T. Hynes*,§,‡ Chemistry Department, E´cole Normale Supe´rieure, and Centre National de la Recherche Scientifique, Unite´ Mixte de Recherche 8640, 24 rue Lhomond, 75005 Paris, France, and Department of Chemistry and Biochemistry, UniVersity of Colorado, Boulder, Colorado 80309-0215 ReceiVed: April 20, 2009; ReVised Manuscript ReceiVed: July 13, 2009
We apply a solute-solvent approach to a theoretical study of vibrational symmetry breaking in aqueous NO3 solution. Experimental infrared and Raman spectra have shown that the NO asymmetric stretches, which are degenerate for the isolated anion, are split by 35-60 cm-1 in dilute solution. As an initial step to calculating the spectra, we have computed the distribution of energies, or the “static spectrum”, and the resulting mean splitting of the two NO asymmetric stretch eigenstates in an aqueous milieu. These have been obtained in a two-mode treatment that considers only the NO asymmetric stretch mode pair as well as a full six-mode treatment. In both sets of calculations, six eigenstates, namely, the ground state, the two NO asymmetric stretch fundamentals, and its three overtones, were determined to suffice for converged energy distributions and mean splittings. The couplings between these six states are driven by the solvent forces on the anion’s modes, which were extracted from molecular dynamics simulations. The solvent forces on the two central modes were found to give rise to a majority of the computed mean splitting of 21.7 cm-1. The distribution of NO asymmmetric stretch excitation energies with these two modes alone was found to have a Maxwell-Boltzmann shape. The solvent forces on the in-plane bends were found to modestly reduce the splitting size and slightly alter the width of the parent distribution. The symmetric stretch force was found to have no effect on the splitting but instead resulted in a widening on the distribution shape. The force gradients were found to have a weak effect on both the eigenvalue distribution and the mean splitting. 1. Introduction The nitrate anion, NO3-, is a key species in the atmospheric sciences. Its mineral salts are very hygroscopic, so that aerosols containing them are able to seed cloud formation.1 Important heterogeneous chemistry, including stratospheric ozone depletion,2 occurs on nitrate aerosols. NO3- is relevant in various other atmospheric contexts,3-5 often in connection with the issue of its production via acid dissociation of nitric acid HNO3 at aqueous aerosol surfaces, a topic of much recent experimental7 and theoretical6 research; for an extensive discussion, see ref 6a. The anion is also important in its own right in the solution chemistry context. NO3- can function as a probe of its local environment via, say, the spectroscopy of its six vibrational modes. Conversely, the solvent effect on the intramolecular dynamics has attracted considerable interest. For example, Waterland et al. have considered the photodynamics of the NO3 in hydrogen bonding and polar solvents using resonance Raman spectroscopy.8 In connection with the anion as a probe, many infrared (IR) and Raman spectral investigations were published in the late 1960s and early 1970s for several nitrate salts and a range of concentrations, tending toward the high end.9 These works drew inferences on the nature of the anion hydration and specific cation effects based upon the changes in the positions †
Part of the “W. Carl Lineberger Festschrift”. * To whom correspondence should be addressed. E-mail: hynes@ spot.colorado.edu. § ´ Ecole Normale Supe´rieure, and Centre National de la Recherche Scientifique. ‡ University of Colorado. ¶ Present address: Theoretical Biochemistry Laboratory, RIKEN Advanced Science Institute 2-1, Hirosawa, Wako, Saitama 351-0198, Japan.
and widths of NO3-’s vibrational bands. More recent results along these lines are due to Zhang et al.,10 Grassian and coworkers,1 and Liu et al.11 Of the vibrational bands observed in the above studies, the NO asymmetric stretch band (ν3, ∼1370 cm-1) features prominently. It is the most intense infrared band. The underlying pair of NO asymmetric stretch vibrations is degenerate for the D3h-symmetric isolated anion. The absence of spatial symmetry in solution lifts the degeneracy of the NO asymmetric stretches, resulting in a doublet in both IR and Raman spectra. This effect has been referred to as “symmetry breaking” and is the focus of this article.12 While this band splitting increasing with concentration, a splitting of 35-60 cm-1 has been reported in dilute solutions.1,8-11 On this basis, it has been suggested that the symmetry breaking can arise from the nitrate-water interaction alone; it does not require a cation’s presence, though the effect is clearly enhanced by it. Keeping to this dilute limit, this article presents an initial theoretical study on the ν3 symmetry breaking in aqueous NO3-. Theoretical studies of nitrate symmetry breaking are few in number, including ab initio computations on nitrate-water clusters1,8,11 as well as mixed quantum NO-3 -classical H2O (QM/ MM) simulations.13 The former have been typically coupled with experimental work. Among them, Waterland et al. have combined ab initio data with neutron scattering and Raman spectral data on several crystalline nitrates to relate the rootmean-square NO bond length distortion to the NO asymmetric stretch frequency splitting.8 QM/MM simulations by Lebrero et al. showed periods of time, lasting about 100-200 fs, where a NO bond length and its oxygen’s negative charge are both larger or smaller than those for the other two NO bonds.13 More
10.1021/jp903626t 2010 American Chemical Society Published on Web 08/18/2009
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TABLE 1: Vibrational Modes and Harmonic Frequencies (in cm-1) of D3h-Symmetric NO3label
mode
symm.
ωia
1 2 (3x, 3y) (4x, 4y)
NO sym. stretch out-of-plane (OOP) bend NO asym. stretch ONO in-plane bends
A1′ A2′′ E′ E′
1056.7 830.8 1407.0 734.0
a
The ωi have been adjusted for a better match with available aqueous-phase fundamental energies. See Appendix A.2.
recently, Boisson et al. have shown through QM/MM simulations that two interleaved sets of such bond length asymmetric structures can be defined.14,15 One set is only slightly more populated than the other, and the anion switches between them with a fast, 100 fs time scale. However, these valuable previous efforts have not provided a theoretical framework for the interpretation of the spectrum and the splitting; we take a first step in that provision here. The present work, as part of our ongoing studies on nitrate hydration,14-16 attempts a more complete quantum treatment of NO3-’s vibrations, and thereby the symmetry breaking, by employing the perturbative solute-solvent coupling method. We focus upon the manner in which water solvent forces on the anion, derived from molecular dynamics simulations, influence the distribution of excitation energies and the resulting mean splitting of the NO asymmetry stretch (ν3) band. The question of whether the forces on the two central NO asymmetric stretch modes alone suffice to explain the observed effect is addressed. Further, the effects of including the solvent forces on the remaining four modes are also analyzed. We shall be primarily concerned with average measures in this article, postponing to the future a study of the dynamics in the system, including the aqueous-phase infrared spectrum of NO3-, and the origin of the solvent forces on its vibrations. The remainder of this article is organized as follows. We start in section 2 with definitions of a few essential quantities and outline the scheme of our calculations using them. A simplification of the symmetry breaking problem to the two core NO asymmetric stretch modes is attempted in section 3, where we compare classical and quantum treatments. Section 4 describes a progressive advance from the two-mode to the full, six-mode treatment of the problem. We summarize our findings and conclude in section 5. 2. Preliminaries Table 1 shows the vibrational modes of nitrate, numbered as per spectroscopic convention. The x, y tags for the doubly degenerate E′ modes come from the axis along which the N atom motion points. The relative atomic displacements for our target modes, the NO asymmetric stretches, are depicted in Figure 1. The Hamiltonian for the reference D3h-symmetric anion is written as
Ho )
∑ 21 ωi(Pi2 + Qi2) + Vanh(Q)
Figure 1. The degenerate NO asymmetric stretch normal modes.
vibrational data in the aqueous milieu; these numbers are shown in Table 1. The anharmonic vibrational potential is developed as a Taylor series to sixth order. The details of the computation of the ωi and Vanh are presented in Appendix A.2. The solvent influence on these vibrations is contained in the interaction potential
δV(t) ) -
∑ FiQi - ∑ GijQiQj i
(2)
iej
As its notation signifies, δV(t) is the fluctuating part of the solute-solvent interaction, 〈δV(t)〉 ) 〈Fi〉 ) 〈Gij〉 ) 0. The average, constant part that we have subtracted usually provides the gas-to-solution-phase frequency shifts.17 We have effectively absorbed it into Ho by the adjustment of the ωi. The time evolution of the forces (Fi) and force gradients (Gij) on the NO3 ’s modes, which are evaluated at the equilibrium geometry of the anion, is governed by the solvent dynamics. For the present calculations, we carry out molecular dynamics simulations with the NO3--H2O force field developed by Thomas et al.;18 Appendix B gives the specifics of the simulations. An important result is that the simulation forces and force gradients have Gaussian distributions, with σ(Fi) ≈ 100 cm-1 (∼180 cm-1 for F2) and σ(Gij) ≈ 1-10 cm-1. We analyze symmetry breaking in aqueous nitrate through the solution of
H(t) ) Ho + δV(t)
(3)
Its quantum treatment, the solute-solvent coupling approach, shall occupy most of this article, and it involves the following scheme. Ho is cast as a matrix in a basis of multimode harmonic product functions. Diagonalization of this matrix gives the eigenstates ψi, and eigenvalues Eo,i. These are then used to construct matrix elements for eq 3
Hij(t) ) Eo,iδij + δVij(t) δVij(t) ) 〈ψi |δV(t)|ψj〉
(4)
(1)
i
This form implies that we use a single reference NO3- structure to treat symmetry breaking, as opposed to (multiple) in-plane distorted ones. Since it is not immediately obvious that our choice suffices, we give our reasoning in Appendix A.1. As the harmonic part of Ho shows, the Q are rectilinear normal coordinates in unitless form. The harmonic frequencies, ωi, that we use in this article have been adjusted to better suit the mean
We identify the eigenfunctions with sizable NO asymmetric stretch character in the solution of H(t). The description of these eigenfunctions and their eigenvalues are the pieces of information that we shall need in this partial study of the symmetry breaking in aqueous NO3-. The quantum treatment will be interspersed with a simpler, classical approach as needed. The role of the solvent forces, Fi, in the title phenomenon is the primary focus of this article. We eschew a cumbersome notational distinction between forms of δV(t) with and without
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the force gradients, Gij. Unless explicitly indicated otherwise, δV(t) henceforth means only the linear part of eq 2. 3. Two-Mode Analysis Instead of directly plunging into a full six-mode treatment of symmetry breaking, we first examine only the two NO asymmetric stretches, Q3x and Q3y. This two-mode simplification is of intrinsic value as it is apt to be transparent, and its results can serve as a reference for the more complex six-mode problem. For this section alone, the 3 in the mode subscripts is dropped. The Hamiltonian for this two-mode (2m) subproblem is 2m H2m(t) ) H2m o + δV (t) 1 1 2 2 2 2 3 2 H2m o ) ω(Px + Py ) + ω(Qx + Qy ) + f(Qx - 3QxQy ) 2 2 δV2m(t) ) -FxQx - FyQy (5)
where we have retained only the cubic anharmonicity. Its particular form reflects the three-fold symmetry of the anion. The anharmonicity constant, f, is a negative number, and our ab initio computations, described in Appendix A.2, yield a value of -49.4 cm-1. δV(t) has been limited to just the solvent forces, assuming their dominance in the solvent influence; these will be checked by including the force gradients at the end of the analyses. 3.1. Classical Treatment. With only a few terms present, H2m(t) lends itself to a simple classical treatment. The starting point is the observation that Qx ) Qy ) 0 is not the equilibrium position for the two-mode potential. A completion-of-squares step
1 ω(Q2x + Q2y ) - FxQx - FyQy ) 2 Fx 2 Fy 1 1 + ω Qy ω Qx 2 ω 2 ω
[
]
[
Fx ω F y Qy ) Qy ω
3f ω
(7)
This coefficient will be important throughout this paper. With the values of f and ω quoted above, R ≈ 0.1. The ellipsis in eq 6 represents dropped constant and linear terms. The prefactors for the latter contain Rn, n g 2, that lead to weak secondary effects. Our focus is the modified quadratic portion of eq 6, from which we wish to extract new frequencies. To this end, we define the polar form of the forces F ) √Fx2 + Fy2
[
θ(t) ) tan-1(Fy /Fx)
1 1 cos θ sin θ 2 2 R(t) ) 1 1 -sin θ cos θ 2 2
]
(8)
Applying a coordinate rotation Q′ ) R(t)Q′′, the 2 × 2 matrix of force constants is transformed as
[
R(t)T
] [
ω - 2RFx 2RFy R(t) 2RFy ω + 2RFx ω - 2RF 0 ) 0 ω + 2RF
]
(9)
The quadratic part of two-mode Hamiltonian simplifies to
1 2 [ωP2 x + (ω - 2RF)Qx ] + 2 1 + (ω + 2RF)Q2 [ωP2 y y ] 2 for which
]
2
F2x + F2y 2ω
eliminates the linear terms and yields (Fx/ω, Fy/ω) as the instantaneous potential minimum. The above equation signals for a change of variables
Qx ) Qx -
R)-
ω′′ ) √ω(ω ( 2RF) ≈ ω ( RF
(10)
are the new frequencies. The two variable changes above express the modification of the character of the degenerate pair of NO asymmetric modes due to the solvent. It is particularly clear that Fx and Fy have no effect without the intramolecular anharmonicity. The most conspicuous change is, of course, that the harmonic
Px ) Px Py ) Py
Upon re-expressing the cubic anharmonicity in the new variables, additional quadratic terms are generated. The new Hamiltonian now reads
1 2 H2m(t) ) ω(P2 x + Py ) + 2 ω - 2RFx 2RFy 2RFy ω + 2RFx
[
1 [Q Q ] × 2 x y Qx 2 + f(Q3 x - 3QxQy ) + ... Qy (6)
][
]
where the coefficient gauging the nitrate-solvent coupling is
Figure 2. Distribution of split harmonic frequencies of the NO asymmetric stretches based on the distribution of the solvent force magnitude, F(F), on them. See text around eq 11.
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frequencies of the new modes are split by 2RF () -6fF/ω). To leading order, this is the extent of symmetry breaking within the two-mode framework. For any given distribution of Fx and Fy, 〈F〉 > 0, resulting in some nonzero mean splitting of harmonic frequencies. The simulations described in Appendix B yield Fx and Fy as uncorrelated quantities, both with a Gaussian spread of the same width σ. The distribution for F is therefore the product of these Gaussians with the angle integrated out, F(F) ) F/σ2 exp(-F2/2σ2). With this, and the estimates R ≈ 0.1 and σ(F) ≈ 95 cm-1, we depict the distribution of the split frequencies ω ( RF in Figure 2. The mean splitting for this distribution will be
2R〈F〉 ) √2πRσ
(11)
whose numerical value is 23.8 cm-1. This compares reasonably well to the 35 cm-1 experimental estimate by Hudson et al.; see Figure 3 of ref 1b. We now introduce the force gradients from eq 2
-GxxQ2x - GxyQxQy - GyyQ2y into δV2m(t). Clearly, these directly affect the quadratic potential. Working through the coordinate transformations, we obtain the modified frequencies as
ω′′ ≈ ω (
1 [2RFx + (Gxx - Gyy)]2 + [2RFy - Gxy]2 2√
(12)
This equation and eq 10 are the two-mode analogues of the solvent-induced frequency change formula for a single oscillator that has been severally employed in the literature.19 To estimate the size of the splitting with the force gradients, we employ the simulation results (Appendix B) that the Gij’s also have a Gaussian distribution. The ones we need have the widths σ(Gxx - Gyy) ) σ(Gxy) ≈ 3.3 cm-1. Taking four terms under the square root in eq 12 to have mutually uncorrelated Gaussian distributions, we randomly sample values for them. This procedure yields a splitting of 24.1 cm-1, a value which hardly differs from the estimate with Fx and Fy alone.20 3.2. Two-Mode Quantum Treatment: Identification of a Core-State Sextet. Turning now to a quantum analysis, our first task is to obtain eigenstates and eigenvalues of Ho2m in eq 5. We choose a rather small basis, consisting of only the harmonic product functions with a total of 0, 1, and 2 quanta of vibrational excitation in the NO asymmetric stretch modes, Qx and Qy. This makes for a basis size of 6, and we shall demonstrate its sufficiency a few lines below. The notation 3mx3ny refers to a product basis function with m and n vibrational quanta in Qx and Qy, respectively. (The 3 comes, of course, from the spectroscopic numbering scheme in Table 1.) 1/2 x x y When symmetrized, the basis is partitioned into three pairs, namely, A 1′: 00 and [1/21/2](32x + 32y ); E′: x 3 1 and [1/2 ](3 2 - 3 2); y x y x y 2m and E′: 3 and -3 3 . (The minus sign on -3 3 is required for correct “orientation”.) Being of A ′ symmetry, H only y 1 1 1 1 1 1 o permits intrapair couplings. The Ho2m matrix is therefore composed of three 2 × 2 matrices, the eigenvectors for which are trivially obtained as
0.0
νgs )
2m Eo,x ) 1399.2
νx )
2m Eo,y ) 1399.2
νy )
2m Eo,gs )
2m Eo,2x ) 2821.7 2νx ) 2m Eo,2y ) 2821.7 2νy ) 2m Eo,2A ) 2813.9 2νA )
c1 ) 0.9973,
00 1 x [32 - 3y2] √2 c1 3y1 -c23x13y1 1 x y -c2 3x1 +c1 [32 - 32] √2 -c2 3y1 -c13x13y1 1 x [32 + 3y2] √2 c2 ) 0.0739 c1 3x1 +c2
(13)
We use ν’s to refer to eigenstates henceforth; 2νx refers to an overtone state of E′x symmetry. The expansion coefficients, c1 and c2, are the same for the E′x and E′y states, as per symmetry requirements. Note that c2 depends on f, the cubic anharmonicity constant
c2 ≈ -
3f √2ω
f
