Solvation Properties of Microhydrated Sulfate Anion Clusters: Insights from <i>ab</i> <i>Initio</i> Calculations

Sulfate–water clusters play an important role in environmental and industrial processes, yet open questions remain on their physical and chemical properties. We investigated the smallest hydrated sulfate anion clusters believed to have a full solvation shell, with 12 or 13 water molecules. We used <i>ab initio</i> molecular dynamics and electronic structure calculations based on density functional theory, with semilocal and hybrid functionals. At both levels of theory we found that configurations with the anion at the surface of the cluster are energetically favored compared to fully solvated ones, which are instead metastable. We show that infrared spectra of the anion with different solvation shells have similar vibrational signatures, indicating that a mixture of surface and internally solvated geometries are likely to be present in the experimental samples at low temperature. In addition, the computed electronic density of states of surface and internally solvated clusters are hardly distinguishable at finite temperature, with the highest occupied molecular orbital belonging to the anion in all cases. The equilibrium structure determined for SO₄^2–·(H₂O)₁₃ differs from that previously reported; we find that the addition of one water molecule to a 12-water cluster modifies its hydration shell and that water–water bonds are preferred over water–anion bonds

Stimulus trial-averages of EEG signal projected into UMI and PMI subspaces over a 2-delay time course (-2800ms to 3550ms relative to stimulus <i>n + 1</i> onset).

(A) Results from an example subject. Dot color indicates stimulus n’s orientation angle. Data points of adjacent stimulus orientation angles are connected by a gray line. (B) Group-average time course of scalar transform over the course of a trial (N = 42). Blue vertical line indicates the onset of stimulus n + 1. Light blue shading shows the time windows that were used to identify the dPCs. The gray shading around the curve shows standard error of the mean.</p

Fig 7 -

Angles between UMI, PMI and decision subspaces for (A) 60D RNN and (B) EEG data. Black bars indicate standard error of the mean.</p

2-back task structure in the Wan et al. [13] EEG study.

The presentation of each stimulus is followed by a 50 ms blank screen, a 200 ms radial checkerboard mask, a variable delay from 2.8 to 3.2 s, and then the next stimulus was presented, upon which the match vs. non-match response is to be made.</p

RNN model architecture.

Shown is the architecture of the 7-hidden-unit RNNs. (A) One-hot vectors corresponding to each of the 6 stimulus types are fed into the input layer, which projects to an LSTM layer with 7 hidden units. This hidden layer in turn projects to an output unit with a binary target activation (0 = non-match, 1 = match). (B) Example input and target output sequences. Two delay timesteps were installed after each stimulus presentation timestep to emulate the delay period in the 2-back EEG task. 60-hidden-unit RNNs have the same architecture except that they have 60 LSTM hidden units, and two input units that take a vector [cos 2θ, sin 2θ] (θ denoting the angle of grating orientations used in Wan et al. [13]) instead of a one-hot vector.</p

Stimulus trial-averages of RNN hidden layer activity projected into UMI and PMI subspaces over the course of a trial (stimulus <i>n</i> to <i>delay 2</i>:<i>2</i>).

(A) Results from an example 7-hidden-unit network. Dot color indicates stimulus n’s identity. (B) Time course of scalar transform over the course of a trial, averaged across 10 networks. Blue vertical line indicates the timestep when stimulus n + 1 is presented. Light blue shading shows the timesteps that were used to identify the dPCs. The gray shading around the curve indicates standard error of the mean. The gray dashed lines indicate a scalar transform of 0; the stimulus representational format is reversed after crossing this line. (C, D) Same as (A, B) but for the 60-hidden-unit RNNs. In C, data points of adjacent stimulus orientation angles are connected by a gray line.</p

IEM reconstruction of EEG recorded while subjects performed the 2-back task (<i>N</i> = 42, combining data from the pilot study and the preregistered experiment from Wan et al. [13]).

In IEM, voltage from each EEG electrode is construed as a weighted sum of responses from six orientation channels (modelled by a half-wave-rectified sinusoid raised to the 6th power), each tuned to a specific stimulus orientation, comprising the basis set. Left panel: IEM reconstruction of the stimulus during the delay in a separate one-item delayed-recognition task. This model was used to reconstruct the stimulus in the 2-back task. Right panel: Concatenation of the item n and item n + 1 stimulus events to form a trial, across which n transitions from probe to UMI to PMI in the 2-back. On the right are IEM reconstructions corresponding to the two 2 s windows centered in two 2.8 s post-mask ISIs before and after item n + 1, respectively. “*” indicates p t test), FDR-corrected for multiple comparisons. As the figure shows, IEM reconstruction of stimulus n is “flipped” relative to the training data (IEM reconstruction from delayed recognition) when it is a UMI, demonstrating priority-based remapping. (Reconstruction of the PMI was unsuccessful.) For delayed-recognition IEM reconstruction (940–1040 ms from stimulus onset), t(41) = 4.12, p n + 1 onset), t(41) = -3.02, p = 0.009; for PMI reconstruction of 2-back (1150–3150 ms from n + 1 onset), t(41) = -1.60, p = 0.116.</p

PCA visualization of LSTM hidden layer activity of an example 7-hidden-unit network (#7).

Shown is a 9-timestep time course of the 2-back task, running from stimulus n to delay 3:2. Column 1 and 2: timestep labels and example input vectors. Column 3: Time course of dimensionality-reduced LSTM hidden layer activity. Each dot in the figures indicates the unit activity from a single trial. Column 4: Same as Column 3 but now each color corresponds to one of the six stimulus types, indicating stimulus n’s identity, and the black dashed lines illustrate the “schematic” stimulus coding axis. Blue and red squares highlight the two delay timesteps used to identify UMI and PMI dPCs, respectively. Column 5: Same as Column 4 except that the colors now correspond to stimulus n’s status for the n-to-n + 2 comparison that occurs at timestep n + 2 (green: match trials, blue: nonmatch trials). Black dashed line at timestep n + 2 (yellow square) illustrates the decision-coding axis. As can be seen in Column 5, the stimulus coding axis rotates counterclockwise (in the image plane) over time such that it becomes “perpendicular” to the decision axis at timestep n + 1 and aligns with it at timestep n + 2. Percent variance explained: PC1–72.2%, PC2–15.7%.</p

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Fig A. Example 7-hidden-unit RNN trained with input following the basis function used to build IEMs in Wan et al. [1]. Shown is the 2D visualization of the LSTM hidden layer activity of this RNN. The network architecture and training procedure are identical to the 7D RNNs reported in the main text with the exception that the inputs are not one-hot vectors; instead, they are specified by the IEM basis function: R = sin6(x) (e.g., for stimulus #3, input vector is [0.0156, 0.4219, 1, 0.4219, 0.0156, 0]). Note that these results are qualitatively similar to RNNs reported in the main text (Fig 4). Fig B. Generating circular input for 60-hidden-unit RNNs. Each point on the circle can be characterized by an angle relative to the easternmost point of the circle. The coordinates of these points within the 2D space on which this circle lives are given by [cos θ, sin θ]. To construct input vectors used in our RNN model, we mapped each stimulus orientation θ to the corresponding point on the circle at 2 * θ. The multiplication by 2 is necessary to match the periodicity of the input vectors to the periodicity of the oriented grating stimuli, which have a period of 180° (i.e., the stimulus at θ is equivalent to the stimulus at θ + 180°). Fig C. Empirical test for presence of stimulus information in WM. (A) Time course of stimulus averages projected into the PMI subspace from an example 60D RNN. Data points are colored based on item n’s identity. We used the 3 timesteps prior to the presentation of n to construct a baseline distribution of dispersion values using a bootstrapping procedure. Visually, one can see stimulus information collapsing in the PMI subspace across the three timesteps that follow timestep n + 2, colored squares added to identify them for panel B. (B) The baseline distribution of dispersion values, with red dashed line indicating the 95th percentile criterion. Magenta, green and orange lines indicate the dispersion values from timesteps delay 3:1, delay 3:2, and n + 3, respectively. Table A. Cumulative percent variance explained (PEV) by top dPCs of the UMI and PMI subspaces for 7D RNN, 60D RNN and EEG data. The percentages of both stimulus and global variance explained are shown. (DOCX)</p

Electronic Structure of Aqueous Sulfuric Acid from First-Principles Simulations with Hybrid Functionals

We carried out the first ab initio molecular dynamics simulations of aqueous sulfuric acid solutions using hybrid density functionals and a concentration (∼1 mol/L) similar to that of electrolyte solutions used in photocatalytic water splitting experiments. We found that while the semilocal functional PBE greatly overestimates the degree of dissociation of the HSO₄^– ion, the hybrid functional PBE0 yields results in qualitative agreement with those of recent Raman measurements. Our findings highlight the importance of using hybrid functionals in the description of anion solvation. We further analyzed the electronic structure of the solution and found that the energy of the highest occupied molecular orbital of the anion is above that of the water valence band maximum only in the case of SO₄^2–. This indicates that SO₄^2– may be kinetically favored, instead of HSO₄^–, in scavenging photoexcited holes from photoanodes in water oxidation reactions