- Email: [email protected]
Is solvated I3− angular?
6 July 2001
Chemical Physics Letters 342 (2001) 141±147
www.elsevier.com/locate/cplett
Is solvated I3 angular? Thorsten Koslowski a,*, Peter V ohringer b b
a Institut fur Physikalische Chemie, Universitat Freiburg, Albertstraûe 23a, D-79104 Freiburg, Germany Arbeitsgruppe fur biomolekulare und chemische Dynamik, Max-Planck-Institut fur Biophysikalische Chemie, Am Fassberg, D-37077 Gottingen, Germany
Received 20 December 2000; in ®nal form 4 May 2001
Abstract In this communication, we address the question of the geometry of the ground state of the solvated triiodide anion from a theoretical perspective. We use a simple, physically motivated and transparent model comprising a tight-binding molecular orbital scheme, point charges and an Onsager reaction ®eld to describe the anion and its liquid environment. We suggest the possibility of an angular I3 geometry originating from the coupling of a molecular dipole to the polarizable solvent once the strength of the reaction ®eld exceeds a critical value. The theoretical results are compared to recent experimental ®ndings. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The photophysics and photochemistry of the solvated triiodide anion I3 has been the subject of extensive experimental and theoretical work. In particular, the observation of IR and Raman transitions in solution that are forbidden in the D1h arrangement of atoms and the existence of I3 ions in solid inorganic compounds not exhibiting this symmetry (see e.g. [1±5]; for recent references see also [8]) has led to the question of the groundstate geometry of the triiodide anion in polar solvents like water, acetonitrile, MeOH or EtOH. Studying the transient anisotropy and the decay of the rotational excitation of solvated triiodide after excitation with 30 fs laser pulses, K uhne and V ohringer [6,7] have found strong evidence that
*
Corresponding author. Fax: +49-761-203-6189. E-mail address: [email protected] (T. Koslowski).
either the ground state or the ®rst excited state shows a nonlinear structure. The novel concept of dipole-induced symmetry breaking with reference to the triiodide anion has recently been introduced by the work of Sato et al. [8] and Lynden-Bell et al. [9]. Sato et al. [8] have computed the free energy surfaces of I3 in solution and in the gas phase using the multicon®guration self-consistent ®eld (MCSCF) method to calculate the properties of the anion and the reference interaction site model (RISM) to describe the solvent and the anion±solvent interaction. The authors come to the conclusion that the free energy surface ± studied along the bond stretching internal coordinates ± becomes virtually ¯at, and that asymmetrical structures may be thermodynamically populated due to solvent eects. Lynden-Bell et al. [9] have computed the ground-state properties and vibrational excitations of I3 on the level of quadratic con®guration interaction, QCISD(T), and have simulated the triiodide±water system using classical molecular
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 5 6 7 - X
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T. Koslowski, P. VoÈhringer / Chemical Physics Letters 342 (2001) 141±147
dynamics. Introducing a dipole moment on the I3 anion, they have found that the gas phase geometry becomes a local maximum with respect to asymmetric bond stretching once the ion is solvated. The authors express the point of view that it is likely that I3 in aqueous solution is stabilized in polar form. They clearly indicate that the perturbational nature of their approach does not permit the computation of broken symmetry equilibrium or thermodynamically averaged anionic geometries. Neither Sato et al. nor Lynden-Bell and coworkers have explicitly considered symmetry breaking with respect to the softest vibrational mode, bond bending. As there is strong experimental evidence that either the ground state or the excited state of solvated I3 does not exhibit a linear geometry [6] (see also Section 3), we will address this question in the remaining part of this work.
2. Model, theoretical methods and results In this communication, we use a simple model that contains a single relevant degree of freedom of the anion and treat the ion-solvent interaction using an Onsager reaction ®eld [16]. We do not intend to present a quantum chemical calculation on a highly sophisticated level, but want to express and clarify the physics underlying the phenomenon of molecular symmetry breaking along the bond bending coordinate in the simplest and most transparent way. The linear gas phase geometry of the triiodide anion is a consequence of both the electrostatic interaction ± partial charges of q ' e=2 reside on the terminal atoms [9] ± and the electronic structure. Of the triiodide frontier orbitals, all p MOs are populated, whereas the r LUMO is vacant. Consequently, the energy of its occupied r counterpart has the strongest contribution to the bonding energy, favouring a linear arrangement of the iodine atoms. In the simple model of the triiodide anion used here, we describe the electronic structure by a tight-binding ± or H uckel-like ± Hamiltonian,
H^
X iajb
cyia cjb tiajb ;
1
where the cyia =cia are annihilation/creation operators acting upon atomic orbitals i; j localized on atoms a; b. As in all simpli®ed treatments of late main group elements, the atomic orbital basis is restricted to three valence p functions for each atom [10±12]. Without loss of generality, the valence orbital ionization potentials of the iodine atom can be used as the zero of energy. As a consequence, no diagonal terms appear in Eq. (1). All AOs are assumed to be orthogonal. We use the potential energy curve of the diiodide molecule, I2 , to parameterize the tight-binding interactions. It can be approximated by a Morse oscillator, V r D0 f exp
2br
2 exp
brg:
2
We interpret the ®rst term within the curled brackets as a short-range repulsion induced by the core electrons and the Pauli principle, and the second term as a longer-ranged covalent attractive interaction, which has to be reproduced diagonalizing the Hamiltonian (1). Out of the six resulting molecular orbitals (r; 2 p; 2 p and r ), ®ve are occupied, and the r MO is vacant. The resulting tight-binding energy is given by VTB 0 2Vppr r0 2Vppr exp br0 . Using the 1Morse parameters D0 1:556 eV; b 1:870 A and [13], we arrive at V 0 228:9 eV and r0 2:669 A ppr VTB 3:11 eV. For the ratio of r and p matrix elements, we apply the Harrison parametrization 0 leading to Vppp 64:96 eV [14]. For nonlinear geometries, these matrix elements have to be weighted according to the Slater±Koster rules [15]. Within the computations presented here, we keep the bond lengths ®xed at the value of and use the tight-binding energy to r0 2:90 A describe the bond-bending potential. To verify our model, we have computed the potential energy curve for the symmetric bond stretching mode. In the absence and in the presence of Coulombic interactions, our model leads to reequilibrium distances of 2.853 and 2.874 A, spectively. These numbers have to be compared to the value of the harmonic ground-state potential and an average of Zanni et al. [18] r0 2:679 A observed for crystalline bond length of 2.9 A
T. Koslowski, P. VoÈhringer / Chemical Physics Letters 342 (2001) 141±147
compounds containing the triiodide anion. We have computed the force constant for symmetric 2 in the stretching, resulting in k 10:62 eV=A absence of 2 Coulomb interactions and in their presence, as compared to k 9:51 eV=A 2 . the harmonic value [17±20], k 11:36 eV= A p Consequently, the resulting frequencies x k=l exhibit an error of less than 10%, which is better than can be expected from a semiempirical potential with a parameterization based on a dierent molecule. From our perspective, strong intramolecular charge shifts should always be handled by a model that incorporates self-consistent elements, otherwise their magnitude and in¯uence is strongly overestimated. Neither the simple tight-binding approach nor the spherical reaction ®eld described below is appropriate to deal with this degree of freedom. We have thus made no attempt to model asymmetric modes and compute the associated dipole moments. In the following, we enumerate the Iodine atoms as I(1), I(2) (the central atom) and I(3). With z1 z3 1=2 and z2 0, the Coulomb interaction between the triiodide ion and a point test charge zi localized at the coordinate ~ ri is given by 2 zi e 1 0 1 V ~ ri : 3 2ri1 ri2 2ri3 4p0 It can be rewritten as 0 V ~ ri
zi e2 B B B 4p0 @
1 1 1 2ri1 2ri3 ri2 |{z} dipole
1 1 ri2 |{z}
C C C; A
point charge
4 permitting the separation of the electrostatic interactions into that between a charge-neutral dipole and the test point charge and the interaction between two point charges. Using this strategy, the dipolar contribution now originates from a charge-neutral con®guration, which is essential for the translational invariance of j~ lj. In the following, we assume that the dielectric solvent can be replaced by a polarizable medium. The interaction between the solvent and the point charge of e roughly located on the the central atom will vary
143
little with a change of the molecular geometry, provided the bond lengths r12 and r23 are ®xed. The dipole±solvent interaction can then be described by an Onsager reaction ®eld [16]: once the I3 ion is bend, the ionic dipole moment is nonzero, and the energy of the system is lowered by VORF
e2 1 r02 cos2 /=2; 4p0 2 1 rc3
5
where denotes the static dielectric constant of the solvent, r0 the equilibrium distance between two iodine atoms, rc the radius of the cavity containing the molecule, and / the bond angle. In the following, the cavity radius will be treated as an adjustable parameter. It can be estimated by assigning an eective volume to the ion that is identical to the sum of the three van der Waals spheres less half of the overlap of these spheres. [21,22], we arrive at Using rvdW 2:42 A 3 and rc 2:80 A. We note that in V 92:1 A are usually quantum chemical calculations 0:5 A added to the reaction sphere radius to account for solvent packing eects [23]. In addition to the tight-binding and the reaction ®eld energy, we consider the Coulomb repulsion between the two terminal Iodine atoms, which carry partial charges of z ' 1=2, as also attested by ab initio calculations [9]. The inclusion of van der Waals interactions or a soft-sphere repulsion between the terminal atoms does not change the picture described below even in a quantitative way. Apart from interactions stemming from the tightbinding terms, we have neglected all three-body terms like the Axilrodt±Teller triple±dipole interaction. In Fig. 1, the total energy of the system, I3 +solvent, is shown as a function of the bond angle. From top to bottom, the curves represent cavity radii of in®nity ± as appropriate to the gas We have used phase ± 3.5, 3.3, 3.2 and 3.1 A. the value of the dielectric constant of water at room temperature and at p 1 bar, 78:54, for all energy curves. In the vacuum, the potential is parabolic, with a pronounced minimum at / 1800 . With decreasing cavity radius, the potential softens considerably, until at rc 3:35 the symmetry of the system is broken: the 0:01 A, potential curve now exhibits two minima. These
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T. Koslowski, P. VoÈhringer / Chemical Physics Letters 342 (2001) 141±147
Fig. 1. Potential energy as a function of the bond angle for a reaction sphere radius of (from top to bottom) in®nity (gas phase), 3.5, 3.3, 3.2 and 3.1 A.
Fig. 2. Boltzmann-weighted bond angle probability distribu at room temperature tion for a reaction sphere radius of 3.3 A (solid line); ditto, gas phase (dotted line).
minima are more pronounced and are shifted to bond angles further away from 180° with decreasing cavity radius. For nonzero temperatures, the bond angle is subject to ensemble averaging. Within an arbitrary normalization, the corresponding Boltzmann weight is given by w / exp V / V 1800 =kB T : 6
electronic ground state (2 R u ) and iodine radicals in either the spin-orbit ground or excited state. By 2 probing the electronic 2 Pg Ru resonance of I2 with another femtosecond pulse which is temporally delayed and polarized 45° with respect to the ®rst, it is possible to reconstruct the time-dependent orientational anisotropy, r t, of the nascent diatomic fragment following bond breakage of triiodide. Provided that bond ®ssion is complete and the probe transition dipole is localized on the diatomic product, the early time decay of r t is determined by the fragment rotational kinetic energy only (inertial response). Fragment rotational excitation in photodissociation experiments can originate from (i) rotational excitation of the parent species, (ii) excited state exit channel torques, or (iii) a ground-state geometry of the parent which is bent. Hence, an assessment of the fragment rotational kinetic energy via a measurement of the inertial decay of the orientational anisotropy, provides hints as to the geometry of the potential energy surfaces relevant to bond breakage. According to Gordon, the necessary link between the early-time behavior of r t and the rotational kinetic energy can be expressed as a Taylor-series expansion for an equi-
The resulting normalized bond angle distribution at room temperature is displayed in Fig. 2 for Close to the critical reaction sphere rc 3:3 A. radius, we observe a strong population of bond angles far away from a linear arrangement. 3. Comparison with experiment Evidence for a nonlinear geometry of triiodide in either its electronic ground or excited state can be obtained from transient anisotropy experiments as described in [6]. Here, the triiodide ions are subject to optical excitation by a linearly polarized femtosecond pulse. The electronic resonances of triiodide in liquid solution are known to be dissociative leading primarily to diiodide ions in their
T. Koslowski, P. VoÈhringer / Chemical Physics Letters 342 (2001) 141±147
145
librium ensemble of diatomic rotors at canonical temperature, T [24]: " # 2 t2 2 hQ2 i 4 r t 1 2 t 7 3s2c 8I s2c Here, the free-rotor rotational correlation time is p denoted sc I=3kB T ; I is moment of inertia of the diatom (i.e. 1:1 10 44 kg m2 for diiodide), and kB is Boltzmann's constant. Deviations from a purely inertial decay at later times (i.e. t > 0) originate from damping of the free rotational ¯ow through the mean squared torques, hQ2 i, exerted onto the rotors by the surrounding solvent molecules. Note that the free-rotor correlation time sets the slowest limit for the inertial decay of r t ! 0 if the rotors were at thermal equilibrium with their bath. For diiodide ions at room temperature, this quantity is 0.94 ps. In a nonequilibrium situation, such as in photodissociations possibly yielding rotationally excited fragments, the initial fragment rotational anisotropy, r t ! 0 should nonetheless decay inertially since it is a function of the rotational kinetic energy only. Furthermore, regardless of the exact shape of the rotational distribution in which the fragments are born, their free-rotor orientational correlation time can still be determined via the inertial decay of r t. A considerably faster decay of r t as compared to the room temperature limit gives immediate evidence for either simple rotational excitation of the fragments or a pronounced evolution of the system along bending degrees of freedom which rotate the transition dipole at a rate faster than 1=sc . Fig. 3 shows a plot of an experimentally determined orientational anisotropy of the diiodide fragments following 400-nm photodissociation of triiodide ions in liquid water at room temperature. Experimental details are described in [6]. The inset displays the early-time portion of same quantity as a function of the squared time delay t2 . From the slope extrapolated to t 0, one ®nds a free-rotor time constant of 0.55 ps. This value is substantially smaller than the free-rotor time constant expected for I2 at room temperature indicating that the diatomic fragments detected at early times rotate signi®cantly faster than at thermal equilibrium with their solvent bath at 298 K. In photodisso-
Fig. 3. Orientational anisotropy of diiodide ions detected at 800 nm following 400-nm photodissociation of triiodide in liquid water at room temperature. Open circles: experimental data, curves: Gaussian approximation, r t ' exp t=sc 2 , with a free-rotor rotational correlation time of 0.55 ps (solid) and 0.94 ps (dashed). The inset displays the anisotropy as a function of the squared time delay. Open circles: experimental data, lines: inertial term of the series expansion with a free-rotor correlation time of 0.55 ps (solid) and 0.94 ps (dashed).
ciation experiments, the fragments are frequently generated in a Boltzmann distribution characterized by a rotational temperature TROT . In this case, a free-rotor time constant sc of 0.55 ps corresponds to a rotational temperature of the diiodide fragments of roughly 800 K or equivalently, a mean rotational energy of approximately 610 cm 1 . We have previously used an impulsive model for rotational excitation to estimate the groundstate structure of solvated triiodide along the bending coordinate. The model originally introduced by Levene and Valentini [25] neglects any excited-state exit channel torques and assumes an in®nitely sti diatom bond such that vibrational excitation is negligible. If the ground-state parent species is instantaneously bent at the moment of bond ®ssion, the repulsive force between the central and terminal atom leads to rotation of the diatomic fragment about its center-of-mass. For homonuclear triatomic parent species such as triiodide, energy and momentum conservation requires the rotational energy to assume the value EROT EEXC
sin2 a ; 4 cos2 a
8
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T. Koslowski, P. VoÈhringer / Chemical Physics Letters 342 (2001) 141±147
where EEXC represents the excess energy delivered by the laser pulse with respect to the asymptotic energy of the well-separated fragments. Using an excess energy of 1.2 eV provided by a 400-nm photolysis pulse, a mean rotational energy of 610 cm 1 is consistent with a bond angle of about 153°. This value is in rather good agreement with a mean bond angle of 147.5° which can be computed from the Boltzmann-weighted probability distri bution using a reaction sphere radius of 3.3 A. Here, it should be commented on the solvent dependence of this symmetry breaking phenomenon. Given the experimental uncertainties, the degree of rotational excitation and hence, the bending angle are identical to those found for liquid ethanol solutions (see [6]). Within the framework of our theoretical model, all solute± solvent interactions are phenomenologically captured in a (certainly oversimplifying) dielectric continuum description of the surrounding liquid. The instantaneous dipole±solvent interaction energy is fully described by the zero-frequency dielectric constant of the liquid and the radius of the spherically shaped cavity in which the triiodide ion is embedded. Assuming that the cavity radius does not vary signi®cantly upon changing the chemical nature of the liquid, the solvent dependence reduces to the term 1= 2 1 which is approximately 1/2 for both liquids at room temperature (i.e. 78:54 for water and 24:3 for ethanol). Yet, for nonprotic solvents of similar polarity such as acetonitrile ( 38:8), negligible excess rotational energy of the diiodide fragment has been found [6]. This discrepancy demonstrates that a more sophisticated theoretical description is necessary to take speci®c solute±solvent interactions such as hydrogen-bonding explicitly into account. However, at this stage we strongly point out that even in a structureless dielectric continuum, potential softening and ultimately symmetry breaking along the bending coordinate becomes important. 4. Conclusions In this communication, we have discussed the possibility of the existence of an angular ground-
state geometry of the triiodide anion, I3 , in polar liquids. The system has been modelled in a simple and physically transparent manner, embedding ®xed bond lengths, a tight-binding approximation to the bond-bending energy and an intraatomic Coulomb repulsion between the terminal atoms, which carry partial charges. To describe the interaction between the ion and the solvent, we have separated point charge and dipolar eects, and have treated the resulting solvent-ion interaction by applying an Onsager reaction ®eld. With increasing reaction ®eld strength ± tantamount to a decreasing reaction sphere radius rc or an increasing dielectric constant ± the potential energy curve softens, until the symmetry of the molecule is broken. We share the point of view of Sato et al. [8] that solvent eects have a strong in¯uence on the potential energy surface of the solvated triiodide anion, and the conclusions of Lynden-Bell et al. [9] that anionic symmetry breaking stems from an induced dipole±solvent interaction. From our point of view, the possibility of a symmetry broken along the bond-bending internal coordinate is worth being investigated by a more sophisticated method than the one presented here. As the energy scale associated with this type of symmetry breaking is of the order of one-tenth of an electron Volt (cf. Fig. 1), however, both the ion and the solvent would have to be treated with a considerably higher methodological and ± once simulations are applied ± statistical accuracy than described in the literature until today. Broken symmetries of solvated ions or molecules may not be restricted to the triiodide±water system: charge localization on the terminal atoms, soft modes associated with bond or torsional angle degrees of freedom and a large solvent dielectric constant are the basic ingredients to symmetry breaking that can be found in a variety of systems. We would like to note that the formation of dipolar atoms via a nonspherical deformation of the electronic charge distribution around atoms by polarization eects has been postulated by Logan [26,27] and Parrinello and coworkers [28] more than a decade ago and should not be confused with molecular symmetry breaking in liquids as discussed here.
T. Koslowski, P. VoÈhringer / Chemical Physics Letters 342 (2001) 141±147
Acknowledgements It is pleasure to thank Gundi Rink for fruitful discussions. This work was supported by the Deutsche Forschungsgemeinschaft and in parts by the Fonds der Chemischen Industrie. References [1] A.G. Maki, R. Forneris, Spectrochim. Acta A 23 (1967) 867. [2] G.C. Heyward, P.G. Hendra, Spectrochim. Acta A 23 (1967) 2309. [3] F.W. Parrett, N.J. Taylor, J. Inorg. Nucl. Chem. 32 (1970) 2485. [4] G.A. Bowmaker, S. Hacobian, Austr. J. Chem. 21 (1968) 551. [5] B.S. Ehrlich, M. Kaplan, J. Chem. Phys. 51 (1969) 603. [6] T. K uhne, P. V ohringer, J. Phys. Chem. A 102 (1998) 4177. [7] H. B ursing, J. Lindner, S. Hess, P. V ohringer, Appl. Phys. B 71 (2000) 411. [8] H. Sato, F. Hirata, A.B. Myers, J. Phys. Chem. A 102 (1998) 2065. [9] R.M. Lynden-Bell, R. Koslo, S. Ruhman, D. Danovich, J. Vala, J. Chem. Phys. 109 (1998) 9928. [10] C. Bichara, A. Pellegatti, J.-P. Gaspard, Phys. Rev. B 49 (1994) 6581.
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[11] Th. Koslowski, J. Phys.: Condens. Matter 9 (1997) 613. [12] Th. Koslowski, Zeitschrift Phys. Chemie Neue Folge 210 (1999) 45. [13] J.G. Dojahn, E.C.M. Chen, W.E. Wentworth, J. Chem. Phys. 100 (1996) 9649, and references therein. [14] W.A. Harrison, Electronic Structure and the Properties of Solids, 2nd ed., Dover, New York, 1989. [15] J.C. Slater, G.F. Koster, Phys. Rev. 94 (1954) 1498. [16] L. Onsager, J. Am. Chem. Soc. 58 (1938) 1486. [17] H. Choi, R.T. Bise, A.A. Hoops, D.M. Neumark, J. Chem. Phys. 113 (2000) 2255. [18] M.T. Zanni, J. Greenblatt, A.V. Davies, D.M. Neumark, J. Chem. Phys. 111 (1999) 2991. [19] W. Kiefer, H.J. Bernstein, Chem. Phys. Lett. 16 (1972) 5. [20] E. Johnson, A.B. Myers, J. Phys. Chem. 100 (1996) 7778. [21] L.X. Dang, B.C. Garrett, J. Chem. Phys. 99 (1993) 2972. [22] R.M. Lynden-Bell, J.C. Rasaiah, J. Chem. Phys. 107 (1997) 1981. [23] J.B. Foresman, A. Frisch, Exploring Chemistry with Electronic Structure Methods, 2nd ed., Gaussian, Pittsburgh, PA, 1996. [24] R.G. Gordon, J. Chem. Phys. 44 (1966) 1830. [25] H.B. Levene, J.J. Valentini, J. Chem. Phys. 87 (1987) 2594. [26] D.E. Logan, Phys. Rev. Lett. 57 (1986) 782. [27] D.E. Logan, J. Chem. Phys. 86 (1987) 234. [28] L.F. Xu, A. Selloni, M. Parrinello, Chem. Phys. Lett. 162 (1989) 27.
Chemical Physics Letters 342 (2001) 141±147
www.elsevier.com/locate/cplett
Is solvated I3 angular? Thorsten Koslowski a,*, Peter V ohringer b b
a Institut fur Physikalische Chemie, Universitat Freiburg, Albertstraûe 23a, D-79104 Freiburg, Germany Arbeitsgruppe fur biomolekulare und chemische Dynamik, Max-Planck-Institut fur Biophysikalische Chemie, Am Fassberg, D-37077 Gottingen, Germany
Received 20 December 2000; in ®nal form 4 May 2001
Abstract In this communication, we address the question of the geometry of the ground state of the solvated triiodide anion from a theoretical perspective. We use a simple, physically motivated and transparent model comprising a tight-binding molecular orbital scheme, point charges and an Onsager reaction ®eld to describe the anion and its liquid environment. We suggest the possibility of an angular I3 geometry originating from the coupling of a molecular dipole to the polarizable solvent once the strength of the reaction ®eld exceeds a critical value. The theoretical results are compared to recent experimental ®ndings. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The photophysics and photochemistry of the solvated triiodide anion I3 has been the subject of extensive experimental and theoretical work. In particular, the observation of IR and Raman transitions in solution that are forbidden in the D1h arrangement of atoms and the existence of I3 ions in solid inorganic compounds not exhibiting this symmetry (see e.g. [1±5]; for recent references see also [8]) has led to the question of the groundstate geometry of the triiodide anion in polar solvents like water, acetonitrile, MeOH or EtOH. Studying the transient anisotropy and the decay of the rotational excitation of solvated triiodide after excitation with 30 fs laser pulses, K uhne and V ohringer [6,7] have found strong evidence that
*
Corresponding author. Fax: +49-761-203-6189. E-mail address: [email protected] (T. Koslowski).
either the ground state or the ®rst excited state shows a nonlinear structure. The novel concept of dipole-induced symmetry breaking with reference to the triiodide anion has recently been introduced by the work of Sato et al. [8] and Lynden-Bell et al. [9]. Sato et al. [8] have computed the free energy surfaces of I3 in solution and in the gas phase using the multicon®guration self-consistent ®eld (MCSCF) method to calculate the properties of the anion and the reference interaction site model (RISM) to describe the solvent and the anion±solvent interaction. The authors come to the conclusion that the free energy surface ± studied along the bond stretching internal coordinates ± becomes virtually ¯at, and that asymmetrical structures may be thermodynamically populated due to solvent eects. Lynden-Bell et al. [9] have computed the ground-state properties and vibrational excitations of I3 on the level of quadratic con®guration interaction, QCISD(T), and have simulated the triiodide±water system using classical molecular
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 5 6 7 - X
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T. Koslowski, P. VoÈhringer / Chemical Physics Letters 342 (2001) 141±147
dynamics. Introducing a dipole moment on the I3 anion, they have found that the gas phase geometry becomes a local maximum with respect to asymmetric bond stretching once the ion is solvated. The authors express the point of view that it is likely that I3 in aqueous solution is stabilized in polar form. They clearly indicate that the perturbational nature of their approach does not permit the computation of broken symmetry equilibrium or thermodynamically averaged anionic geometries. Neither Sato et al. nor Lynden-Bell and coworkers have explicitly considered symmetry breaking with respect to the softest vibrational mode, bond bending. As there is strong experimental evidence that either the ground state or the excited state of solvated I3 does not exhibit a linear geometry [6] (see also Section 3), we will address this question in the remaining part of this work.
2. Model, theoretical methods and results In this communication, we use a simple model that contains a single relevant degree of freedom of the anion and treat the ion-solvent interaction using an Onsager reaction ®eld [16]. We do not intend to present a quantum chemical calculation on a highly sophisticated level, but want to express and clarify the physics underlying the phenomenon of molecular symmetry breaking along the bond bending coordinate in the simplest and most transparent way. The linear gas phase geometry of the triiodide anion is a consequence of both the electrostatic interaction ± partial charges of q ' e=2 reside on the terminal atoms [9] ± and the electronic structure. Of the triiodide frontier orbitals, all p MOs are populated, whereas the r LUMO is vacant. Consequently, the energy of its occupied r counterpart has the strongest contribution to the bonding energy, favouring a linear arrangement of the iodine atoms. In the simple model of the triiodide anion used here, we describe the electronic structure by a tight-binding ± or H uckel-like ± Hamiltonian,
H^
X iajb
cyia cjb tiajb ;
1
where the cyia =cia are annihilation/creation operators acting upon atomic orbitals i; j localized on atoms a; b. As in all simpli®ed treatments of late main group elements, the atomic orbital basis is restricted to three valence p functions for each atom [10±12]. Without loss of generality, the valence orbital ionization potentials of the iodine atom can be used as the zero of energy. As a consequence, no diagonal terms appear in Eq. (1). All AOs are assumed to be orthogonal. We use the potential energy curve of the diiodide molecule, I2 , to parameterize the tight-binding interactions. It can be approximated by a Morse oscillator, V r D0 f exp
2br
2 exp
brg:
2
We interpret the ®rst term within the curled brackets as a short-range repulsion induced by the core electrons and the Pauli principle, and the second term as a longer-ranged covalent attractive interaction, which has to be reproduced diagonalizing the Hamiltonian (1). Out of the six resulting molecular orbitals (r; 2 p; 2 p and r ), ®ve are occupied, and the r MO is vacant. The resulting tight-binding energy is given by VTB 0 2Vppr r0 2Vppr exp br0 . Using the 1Morse parameters D0 1:556 eV; b 1:870 A and [13], we arrive at V 0 228:9 eV and r0 2:669 A ppr VTB 3:11 eV. For the ratio of r and p matrix elements, we apply the Harrison parametrization 0 leading to Vppp 64:96 eV [14]. For nonlinear geometries, these matrix elements have to be weighted according to the Slater±Koster rules [15]. Within the computations presented here, we keep the bond lengths ®xed at the value of and use the tight-binding energy to r0 2:90 A describe the bond-bending potential. To verify our model, we have computed the potential energy curve for the symmetric bond stretching mode. In the absence and in the presence of Coulombic interactions, our model leads to reequilibrium distances of 2.853 and 2.874 A, spectively. These numbers have to be compared to the value of the harmonic ground-state potential and an average of Zanni et al. [18] r0 2:679 A observed for crystalline bond length of 2.9 A
T. Koslowski, P. VoÈhringer / Chemical Physics Letters 342 (2001) 141±147
compounds containing the triiodide anion. We have computed the force constant for symmetric 2 in the stretching, resulting in k 10:62 eV=A absence of 2 Coulomb interactions and in their presence, as compared to k 9:51 eV=A 2 . the harmonic value [17±20], k 11:36 eV= A p Consequently, the resulting frequencies x k=l exhibit an error of less than 10%, which is better than can be expected from a semiempirical potential with a parameterization based on a dierent molecule. From our perspective, strong intramolecular charge shifts should always be handled by a model that incorporates self-consistent elements, otherwise their magnitude and in¯uence is strongly overestimated. Neither the simple tight-binding approach nor the spherical reaction ®eld described below is appropriate to deal with this degree of freedom. We have thus made no attempt to model asymmetric modes and compute the associated dipole moments. In the following, we enumerate the Iodine atoms as I(1), I(2) (the central atom) and I(3). With z1 z3 1=2 and z2 0, the Coulomb interaction between the triiodide ion and a point test charge zi localized at the coordinate ~ ri is given by 2 zi e 1 0 1 V ~ ri : 3 2ri1 ri2 2ri3 4p0 It can be rewritten as 0 V ~ ri
zi e2 B B B 4p0 @
1 1 1 2ri1 2ri3 ri2 |{z} dipole
1 1 ri2 |{z}
C C C; A
point charge
4 permitting the separation of the electrostatic interactions into that between a charge-neutral dipole and the test point charge and the interaction between two point charges. Using this strategy, the dipolar contribution now originates from a charge-neutral con®guration, which is essential for the translational invariance of j~ lj. In the following, we assume that the dielectric solvent can be replaced by a polarizable medium. The interaction between the solvent and the point charge of e roughly located on the the central atom will vary
143
little with a change of the molecular geometry, provided the bond lengths r12 and r23 are ®xed. The dipole±solvent interaction can then be described by an Onsager reaction ®eld [16]: once the I3 ion is bend, the ionic dipole moment is nonzero, and the energy of the system is lowered by VORF
e2 1 r02 cos2 /=2; 4p0 2 1 rc3
5
where denotes the static dielectric constant of the solvent, r0 the equilibrium distance between two iodine atoms, rc the radius of the cavity containing the molecule, and / the bond angle. In the following, the cavity radius will be treated as an adjustable parameter. It can be estimated by assigning an eective volume to the ion that is identical to the sum of the three van der Waals spheres less half of the overlap of these spheres. [21,22], we arrive at Using rvdW 2:42 A 3 and rc 2:80 A. We note that in V 92:1 A are usually quantum chemical calculations 0:5 A added to the reaction sphere radius to account for solvent packing eects [23]. In addition to the tight-binding and the reaction ®eld energy, we consider the Coulomb repulsion between the two terminal Iodine atoms, which carry partial charges of z ' 1=2, as also attested by ab initio calculations [9]. The inclusion of van der Waals interactions or a soft-sphere repulsion between the terminal atoms does not change the picture described below even in a quantitative way. Apart from interactions stemming from the tightbinding terms, we have neglected all three-body terms like the Axilrodt±Teller triple±dipole interaction. In Fig. 1, the total energy of the system, I3 +solvent, is shown as a function of the bond angle. From top to bottom, the curves represent cavity radii of in®nity ± as appropriate to the gas We have used phase ± 3.5, 3.3, 3.2 and 3.1 A. the value of the dielectric constant of water at room temperature and at p 1 bar, 78:54, for all energy curves. In the vacuum, the potential is parabolic, with a pronounced minimum at / 1800 . With decreasing cavity radius, the potential softens considerably, until at rc 3:35 the symmetry of the system is broken: the 0:01 A, potential curve now exhibits two minima. These
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Fig. 1. Potential energy as a function of the bond angle for a reaction sphere radius of (from top to bottom) in®nity (gas phase), 3.5, 3.3, 3.2 and 3.1 A.
Fig. 2. Boltzmann-weighted bond angle probability distribu at room temperature tion for a reaction sphere radius of 3.3 A (solid line); ditto, gas phase (dotted line).
minima are more pronounced and are shifted to bond angles further away from 180° with decreasing cavity radius. For nonzero temperatures, the bond angle is subject to ensemble averaging. Within an arbitrary normalization, the corresponding Boltzmann weight is given by w / exp V / V 1800 =kB T : 6
electronic ground state (2 R u ) and iodine radicals in either the spin-orbit ground or excited state. By 2 probing the electronic 2 Pg Ru resonance of I2 with another femtosecond pulse which is temporally delayed and polarized 45° with respect to the ®rst, it is possible to reconstruct the time-dependent orientational anisotropy, r t, of the nascent diatomic fragment following bond breakage of triiodide. Provided that bond ®ssion is complete and the probe transition dipole is localized on the diatomic product, the early time decay of r t is determined by the fragment rotational kinetic energy only (inertial response). Fragment rotational excitation in photodissociation experiments can originate from (i) rotational excitation of the parent species, (ii) excited state exit channel torques, or (iii) a ground-state geometry of the parent which is bent. Hence, an assessment of the fragment rotational kinetic energy via a measurement of the inertial decay of the orientational anisotropy, provides hints as to the geometry of the potential energy surfaces relevant to bond breakage. According to Gordon, the necessary link between the early-time behavior of r t and the rotational kinetic energy can be expressed as a Taylor-series expansion for an equi-
The resulting normalized bond angle distribution at room temperature is displayed in Fig. 2 for Close to the critical reaction sphere rc 3:3 A. radius, we observe a strong population of bond angles far away from a linear arrangement. 3. Comparison with experiment Evidence for a nonlinear geometry of triiodide in either its electronic ground or excited state can be obtained from transient anisotropy experiments as described in [6]. Here, the triiodide ions are subject to optical excitation by a linearly polarized femtosecond pulse. The electronic resonances of triiodide in liquid solution are known to be dissociative leading primarily to diiodide ions in their
T. Koslowski, P. VoÈhringer / Chemical Physics Letters 342 (2001) 141±147
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librium ensemble of diatomic rotors at canonical temperature, T [24]: " # 2 t2 2 hQ2 i 4 r t 1 2 t 7 3s2c 8I s2c Here, the free-rotor rotational correlation time is p denoted sc I=3kB T ; I is moment of inertia of the diatom (i.e. 1:1 10 44 kg m2 for diiodide), and kB is Boltzmann's constant. Deviations from a purely inertial decay at later times (i.e. t > 0) originate from damping of the free rotational ¯ow through the mean squared torques, hQ2 i, exerted onto the rotors by the surrounding solvent molecules. Note that the free-rotor correlation time sets the slowest limit for the inertial decay of r t ! 0 if the rotors were at thermal equilibrium with their bath. For diiodide ions at room temperature, this quantity is 0.94 ps. In a nonequilibrium situation, such as in photodissociations possibly yielding rotationally excited fragments, the initial fragment rotational anisotropy, r t ! 0 should nonetheless decay inertially since it is a function of the rotational kinetic energy only. Furthermore, regardless of the exact shape of the rotational distribution in which the fragments are born, their free-rotor orientational correlation time can still be determined via the inertial decay of r t. A considerably faster decay of r t as compared to the room temperature limit gives immediate evidence for either simple rotational excitation of the fragments or a pronounced evolution of the system along bending degrees of freedom which rotate the transition dipole at a rate faster than 1=sc . Fig. 3 shows a plot of an experimentally determined orientational anisotropy of the diiodide fragments following 400-nm photodissociation of triiodide ions in liquid water at room temperature. Experimental details are described in [6]. The inset displays the early-time portion of same quantity as a function of the squared time delay t2 . From the slope extrapolated to t 0, one ®nds a free-rotor time constant of 0.55 ps. This value is substantially smaller than the free-rotor time constant expected for I2 at room temperature indicating that the diatomic fragments detected at early times rotate signi®cantly faster than at thermal equilibrium with their solvent bath at 298 K. In photodisso-
Fig. 3. Orientational anisotropy of diiodide ions detected at 800 nm following 400-nm photodissociation of triiodide in liquid water at room temperature. Open circles: experimental data, curves: Gaussian approximation, r t ' exp t=sc 2 , with a free-rotor rotational correlation time of 0.55 ps (solid) and 0.94 ps (dashed). The inset displays the anisotropy as a function of the squared time delay. Open circles: experimental data, lines: inertial term of the series expansion with a free-rotor correlation time of 0.55 ps (solid) and 0.94 ps (dashed).
ciation experiments, the fragments are frequently generated in a Boltzmann distribution characterized by a rotational temperature TROT . In this case, a free-rotor time constant sc of 0.55 ps corresponds to a rotational temperature of the diiodide fragments of roughly 800 K or equivalently, a mean rotational energy of approximately 610 cm 1 . We have previously used an impulsive model for rotational excitation to estimate the groundstate structure of solvated triiodide along the bending coordinate. The model originally introduced by Levene and Valentini [25] neglects any excited-state exit channel torques and assumes an in®nitely sti diatom bond such that vibrational excitation is negligible. If the ground-state parent species is instantaneously bent at the moment of bond ®ssion, the repulsive force between the central and terminal atom leads to rotation of the diatomic fragment about its center-of-mass. For homonuclear triatomic parent species such as triiodide, energy and momentum conservation requires the rotational energy to assume the value EROT EEXC
sin2 a ; 4 cos2 a
8
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T. Koslowski, P. VoÈhringer / Chemical Physics Letters 342 (2001) 141±147
where EEXC represents the excess energy delivered by the laser pulse with respect to the asymptotic energy of the well-separated fragments. Using an excess energy of 1.2 eV provided by a 400-nm photolysis pulse, a mean rotational energy of 610 cm 1 is consistent with a bond angle of about 153°. This value is in rather good agreement with a mean bond angle of 147.5° which can be computed from the Boltzmann-weighted probability distri bution using a reaction sphere radius of 3.3 A. Here, it should be commented on the solvent dependence of this symmetry breaking phenomenon. Given the experimental uncertainties, the degree of rotational excitation and hence, the bending angle are identical to those found for liquid ethanol solutions (see [6]). Within the framework of our theoretical model, all solute± solvent interactions are phenomenologically captured in a (certainly oversimplifying) dielectric continuum description of the surrounding liquid. The instantaneous dipole±solvent interaction energy is fully described by the zero-frequency dielectric constant of the liquid and the radius of the spherically shaped cavity in which the triiodide ion is embedded. Assuming that the cavity radius does not vary signi®cantly upon changing the chemical nature of the liquid, the solvent dependence reduces to the term 1= 2 1 which is approximately 1/2 for both liquids at room temperature (i.e. 78:54 for water and 24:3 for ethanol). Yet, for nonprotic solvents of similar polarity such as acetonitrile ( 38:8), negligible excess rotational energy of the diiodide fragment has been found [6]. This discrepancy demonstrates that a more sophisticated theoretical description is necessary to take speci®c solute±solvent interactions such as hydrogen-bonding explicitly into account. However, at this stage we strongly point out that even in a structureless dielectric continuum, potential softening and ultimately symmetry breaking along the bending coordinate becomes important. 4. Conclusions In this communication, we have discussed the possibility of the existence of an angular ground-
state geometry of the triiodide anion, I3 , in polar liquids. The system has been modelled in a simple and physically transparent manner, embedding ®xed bond lengths, a tight-binding approximation to the bond-bending energy and an intraatomic Coulomb repulsion between the terminal atoms, which carry partial charges. To describe the interaction between the ion and the solvent, we have separated point charge and dipolar eects, and have treated the resulting solvent-ion interaction by applying an Onsager reaction ®eld. With increasing reaction ®eld strength ± tantamount to a decreasing reaction sphere radius rc or an increasing dielectric constant ± the potential energy curve softens, until the symmetry of the molecule is broken. We share the point of view of Sato et al. [8] that solvent eects have a strong in¯uence on the potential energy surface of the solvated triiodide anion, and the conclusions of Lynden-Bell et al. [9] that anionic symmetry breaking stems from an induced dipole±solvent interaction. From our point of view, the possibility of a symmetry broken along the bond-bending internal coordinate is worth being investigated by a more sophisticated method than the one presented here. As the energy scale associated with this type of symmetry breaking is of the order of one-tenth of an electron Volt (cf. Fig. 1), however, both the ion and the solvent would have to be treated with a considerably higher methodological and ± once simulations are applied ± statistical accuracy than described in the literature until today. Broken symmetries of solvated ions or molecules may not be restricted to the triiodide±water system: charge localization on the terminal atoms, soft modes associated with bond or torsional angle degrees of freedom and a large solvent dielectric constant are the basic ingredients to symmetry breaking that can be found in a variety of systems. We would like to note that the formation of dipolar atoms via a nonspherical deformation of the electronic charge distribution around atoms by polarization eects has been postulated by Logan [26,27] and Parrinello and coworkers [28] more than a decade ago and should not be confused with molecular symmetry breaking in liquids as discussed here.
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Acknowledgements It is pleasure to thank Gundi Rink for fruitful discussions. This work was supported by the Deutsche Forschungsgemeinschaft and in parts by the Fonds der Chemischen Industrie. References [1] A.G. Maki, R. Forneris, Spectrochim. Acta A 23 (1967) 867. [2] G.C. Heyward, P.G. Hendra, Spectrochim. Acta A 23 (1967) 2309. [3] F.W. Parrett, N.J. Taylor, J. Inorg. Nucl. Chem. 32 (1970) 2485. [4] G.A. Bowmaker, S. Hacobian, Austr. J. Chem. 21 (1968) 551. [5] B.S. Ehrlich, M. Kaplan, J. Chem. Phys. 51 (1969) 603. [6] T. K uhne, P. V ohringer, J. Phys. Chem. A 102 (1998) 4177. [7] H. B ursing, J. Lindner, S. Hess, P. V ohringer, Appl. Phys. B 71 (2000) 411. [8] H. Sato, F. Hirata, A.B. Myers, J. Phys. Chem. A 102 (1998) 2065. [9] R.M. Lynden-Bell, R. Koslo, S. Ruhman, D. Danovich, J. Vala, J. Chem. Phys. 109 (1998) 9928. [10] C. Bichara, A. Pellegatti, J.-P. Gaspard, Phys. Rev. B 49 (1994) 6581.
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