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1 Originally posted 2 July 2010; revised 4 March Supporting Online Material for Does the Hydrated Electron Occupy a Cavity? Ross E. Larsen,* William J. Glover, Benjamin J. Schwartz* *To whom correspondence should be addressed. Ross.Larsen@nrel.gov (R.E.L.); schwartz@chem.ucla.edu (B.J.S.) Published 2 July 2010, Science 329, 65 (2010) DOI: /science This PDF file includes: Materials and Methods SOM Text Figs. S1 to S4 Table S1 References Correction: The SOM has been revised to correct errors on two pages. In the first paragraph on page 3, there was an error in the description of the augmented basis functions used to calculate the water LUMO. Instead of an exponent for the Gaussian s-type functions of Å 2, the actual exponent used was bohr 2 ( Å 2 ). At the bottom of page 4, there was an error in the description of the values of the point charges used in the flexible simple point charge (SPC Flex) model for water. Instead of q Ox = 0.84e and q H = 0.42e, the correct values are q Ox = 0.82e and q H = 0.41e. The correct values were used in all of the calculations, so these errors did not affect any of the results or conclusions presented in either the main text or the SOM.

2 Supporting Material for Does the hydrated electron occupy a cavity? Ross E. Larsen,, William J. Glover, and Benjamin J. Schwartz Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, CA , USA Present address: National Renewable Energy Laboratory, Golden, CO , USA Present address: Department of Chemistry, Stanford University, Stanford, CA 94305, USA Corresponding authors Ross.Larsen@nrel.gov; schwartz@chem.ucla.edu. Materials and methods A new, rigorously-derived pseudopotential for the hydrated electron Formalism The pseudopotential formalism allows the complicated many-body interaction between an excess electron and a molecule to be replaced by an effective potential, which enables the properties of the excess electron to be found with a single-electron treatment. (S1) Our approach to calculating pseudopotentials is based on the orthogonalization formalism of Phillips and Kleinman, (S2) and has been described in detail in recent papers. (S3, S4) To form a pseudopotential, we begin with the Hartree-Fock (HF) molecular orbitals, including the lowest unoccupied molecular orbital (LUMO), for the molecule of interest, Ĥ ψ i = ɛ i ψ i, (1) where the Ĥ is the HF Hamiltonian and the wave functions, ψ i, are the occupied (or core) molecular orbitals with energies ɛ i. If the occupied molecular orbitals of the molecule are assumed to not change in the presence of an excess electron (the so-called frozen core approximation), (S1) the LUMO (or valence orbital) is also the wave function of an excess electron attached to the molecule; we denote the LUMO as ψ v and its energy as ɛ. To generate the pseudopotential, one forms a pseudo-orbital, φ, by adding a linear combination of the 1

3 occupied molecular orbitals to the valence orbital, φ = ψ v + a i ψ i. (2) i This prescription is not unique, so the pseudo-orbital is defined uniquely by constraining some physical property to be a minimum or maximum, in our case, by minimizing the mean kinetic energy, φ ˆT φ / φ φ. Once the pseudo-orbital has been determined, one can form the pseudopotential. (S1, S2, S3, S4) As we have discussed elsewhere, (S3, S4) however, it is of greater interest to form an effective potential, such that the one-electron Hamiltonian is the sum of the single-electron kinetic energy operator, ˆT, any one-electron potentials in the HF Hamiltonian, and the pseudopotential. When the pseudo-orbital is formed by minimizing the kinetic energy, the pseudo-orbital is nodeless, so the exact local effective potential, which is the sum of the pseudopotential and the oneelectron potentials, can be formed as: (S3) U local eff (r) = r ɛ ˆT φ. (3) r φ In the rest of this discussion, and in the body of the associated paper, we refer to the effective potential and the pseudopotential interchangeably, with the understanding that the effective potential, Eq. 3, is what is meant throughout. We note that when this effective potential is used in a one-electron Schrödinger Equation, the resulting ground-state energy matches the LUMO energy of the system exactly at the HF level, and the resulting pseudo-orbital matches the HF LUMO everywhere outside the overlap with the system valence MO s. The above pseudopotential formalism, being based on the frozen-core approximation, does not account for polarization or dispersion interactions between the molecule and excess electron. Both these types of interaction have been shown to play an important role in the binding of an excess electron to water clusters, (S5) and so might be expected to contribute to the bulk hydrated electron s energetics. Therefore, to correct for the frozen-core approximation, we add a polarization potential (given below) to the excess electron s Hamiltonian. This approach has been shown to incorporate polarization and dispersion interactions in an average fashion, and adequately predicts the binding energy of several anionic water clusters. (S6) 2

4 Numerical methods We computed the effective potential for the water-electron interaction by calculating the occupied molecular orbitals and the LUMO for a water molecule at the restricted HF level, using GAUSSIAN 98 Revision A.9. (S7) The geometry of the water molecule was fixed with OH bond lengths of Å, and an angle of 105 degrees. Because a single water molecule does not bind an excess electron, the LUMO of water is a continuum state, which is poorly represented by atom-centered gaussian functions. Thus to capture the essential features of the LUMO, and therefore the features of the pseudopotential, in the vicinity of water, we used the G basis augmented with 64 additional s- type gaussian functions placed on a cubic grid centered on the water molecule s center of mass. The grid spacing was 9 Å and the gaussian functions had exponent Å 2 ; these values were chosen after some experimentation to give good overlap between adjacent gaussians without making the basis over-complete. We verified that the shape of the LUMO in the molecular-core region was preserved in a HF calculation using the G basis without augmenting gaussians. Once the molecular orbitals were found, we formed the kinetic-energy matrix elements ψ a ˆT ψ b with a and b running over all occupied molecular orbitals and the LUMO. With these matrix elements in hand, we used the nonlinear minimization routine, Minimize, in Mathematica (S8) to find the expansion coefficients, a i in Eq. 2, that define the minimum-kinetic energy pseudo-orbital. In the molecular dynamics calculations described in this work, we solved for the hydrated electron wave function on a cubic grid. As we have noted previously, (S4, S9) the finite spacing of the grid necessitates working with a smoothed effective potential, otherwise sharp features in the effective potential could lead to unphysical artifacts, such as collapse of the wave function to a single grid point if a very attractive region of the effective potential passes over a grid point due to water motions. We smoothed the effective potential as in Ref. (S9) by convolving the pseudo-orbital with a gaussian with exponent 0.25 bohr 2 and then inserting the smoothed pseudo-orbital in Eq. 3. We chose to smooth the pseudo-orbital rather than the effective potential because this guar- 3

5 antees that the correct pseudo-orbital and energy are obtained when solving the one-electron Schrödinger equation with the smoothed effective potential. In examining the resulting effective potential, representative cuts of which are shown in the left hand panels of Figure 1 of the main paper, we noticed a significant repulsive term located between the hydrogen atoms, as well as an attractive region near the oxygen atom. These two features make the hydrogens effectively less attractive and the oxygen less repulsive than in previous bulk hydrated electron models. In combination, this means that the hydrated electron may not repel the water enough to form a cavity, or that if a cavity does form, that it may not be as strongly structured as has been seen in some models. (S10, S11, S12) To incorporate these additional features, we chose to fit the effective potential to a function of the form, U local eff (r) = 8 n=1 B n r r n i n e ρ n r r n 2 + V C, (4) where the sum runs over functions centered on the oxygen atom, the hydrogen atoms, and a midpoint site centered between the two hydrogen atoms, and V C is a sum of tapered coulomb terms from the oxygen and hydrogen atoms, q r, q r cut 2 r 2 r 3 cut r r cut, r < r cut. (5) The charges on each atom were taken to be the values of the Mulliken partial charges from the restricted HF calculation, q Ox = e and q H = e, for oxygen and hydrogen respectively. To be consistent with the electrostatics assumed in the simple point charge water model, (S13) we changed the partial charges in the fit from their HF values to q Ox = 0.82e and q H = 0.41e. The fitted parameters for the effective potential are given in Table S1. The effective potential displayed in Fig. 1 of the main text does not include the additional polarization term described above, which corrects for the frozencore approximation inherent in our pseudopotential derivation. This polarization term, V p, is identical to the damped polarization term used in Schnitker and 4

6 Rossky s e -water pseudopotential, (S10) which for each water molecule reads: V p = α 2r 4(1 exp[ (r/r c) 6 ]), (6) where α = a.u. is the isotropic polarizability of water, r c = 1.53 Å is a cutoff radius controlling the damping of the potential at the origin, and r is the radial distance from the oxygen atom. Finally, we followed Schnitker and Rossky s prescription of tapering the effective potential to zero at half the water box size over a length of 0.5 Å, using a Steinhauser fitting function. (S10) We note that the direct kinetic-energy minimization approach we used here closely resembles the method used by Turi and Borgis to calculate a waterelectron pseudopotential. (S14) Turi and Borgis formed an approximate effective potential by first choosing a simple functional form for the potential, calculating the resulting pseudo-orbital, and then varying the parameters of their potential so as to optimize the resulting pseudo-orbital. In contrast to Turi and Borgis approach, we have computed the exact (smoothed) effective potential and fit it directly. By proceeding in this fashion, we were able to identify features in the effective potential that were not represented in the fitting function assumed by Turi and Borgis, and thus we were able to include these features in our fitting function. Non-adiabatic dynamics and spectroscopy To compute the excited-state dynamics and predict the pump-probe transient absorption spectroscopy of the hydrated electron, we used the fewest-switches surface hopping algorithm of Tully. (S15) We then computed transition dipoles among the lowest eight electronic states every 10 fs in each nonadiabatic trajectory, and used our earlier formalism (S16) for inhomogeneous absorption to predict the femtosecond pump-probe transient absorption spectra. As discussed in the main text, both the ground-state equilibrium spectrum and the excited-state transient spectroscopy agree with experiment; we note that the spectroscopy computed from simulations in which the electron resides in a cavity is usually significantly ( 0.5 ev) blue-shifted from experiment. 5

7 Supporting text Orientations of transition dipole moments Figure S1 shows representative contours of the three lowest excited states of the hydrated electron. Each state has a positive and a negative lobe, with a p-like symmetry. Note that the lobes reside far from the origin, which is located at the center of mass of the ground state. Although the lowest three excited states have obvious p-like character, the transition dipoles from the quasi-spherical ground state to the p-like excited states need not be strictly orthogonal. In cavity models of the hydrated electron, the dipoles are orthogonal (with dot products between normalized transition dipoles much smaller than 0.1). (S17, S18) This near-perfect orthogonality is not found in our calculations, as shown in Fig. S2, which reveals that the transition dipoles are approximately orthogonal but that fluctuations often make the angle between them much less than 90 degrees. This is consistent with the fact that there is no polarized transient hole-burning of the hydrated electron (in contrast with the prediction of persistent polarized transient holes in the spectroscopy for electrons that reside in a cavity), as discussed in the main text. Finite size effects in hydrated electron simulations Our calculations also revealed significant changes in calculated properties of the hydrated electron as the number of water molecules was increased (keeping the grid spacing on which the electronic states are calculated the same). There was essentially no change in the electron center-of-mass to water radial distribution functions with system size, but the computed absorption spectrum has a significant red shoulder for 200 water molecules that disappears when more waters are included in the simulation cell, as shown in Fig. S3. The red shoulder disappears because adding more water molecules reduces fluctuations that allow the e aq to take on anomalously large radii of gyration, as shown in Fig. S4. The observed narrowing in the distribution of the radius of gyration is consistent with electrostriction from long-ranged Coulomb forces stabilizing the hydrated electron, as we argued in the main text. Because of these 6

8 large fluctuations, we chose only configurations in which the electron s radius of gyration was consistent with the larger 499-water molecule system when launching the excited-state nonadiabatic trajectories discussed above. Also discussed in the main text, the fact that fluctuations in the electron s properties increase dramatically as the system size is reduced has important implications for Car-Parrinello simulations of this system, which from computational constraints are limited to only a few tens of water molecules. (S19) We also observed that the approximate orthogonality between transition dipoles from the ground to the three lowest-lying excited states (see previous section) is weaker with 200 molecules than with 499 water molecules, again suggesting that several solvation shells are needed to fully determine the spectroscopic properties of a hydrated electron. 7

9 Supporting figures Figure S1: Positive (yellow) and negative (green) contours of the wavefunctions of the lowest three excited states (state 1 at top left, state 2 at top right, state 3 at bottom) of the hydrated electron calculated for a representative 499-water configuration. All waters within 7.5 A of the ground-state are shown; the point of view has been set to best show the long axis of each state. state 1 state 2 state 3 8

10 Figure S2: Distribution of dot products (ˆµ i ˆµ j = cos(θ)) between the normalized transition dipoles from the ground state to each of the three lowest excited states, ˆµ i, for i, j = 1, 2, 3 and i j. The black curve shows an exponential decay for comparison. Distribution of dot products θ = µ 1 µ 2 θ = µ 1 µ 3 θ = µ 2 µ 3 e cos( θ)/ cos(θ) 9

11 Figure S3: Calculated absorption spectra for simulation cells with 200 (red dashed curve) and 499 water molecules (green dotted curve) in the simulation cell. Normalized Absorption Spectrum waters 499 waters Energy (ev) 10

12 Figure S4: Distribution of the ground-state hydrated electron radius of gyration from mixed quantum/classical simulations with both 200 (red dashed curve) and 499 (green dotted curve) water molecules. Distribution of Rgyr waters 499 waters Radius of gyration (Å) 11

13 Supporting table Table S1: Fit parameters for the electron-water effective potential, Eq. 4. All quantities are given in atomic units. n i n ρ n B n Oxygen (r cut = ) Hydrogen 4, (r cut =2.6369) Hydrogen midpoint

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