A General Sparse Tensor Framework for Electronic Structure Theory.

Linear-scaling algorithms must be developed in order to extend the domain of applicability of electronic structure theory to molecules of any desired size. However, the increasing complexity of modern linear-scaling methods makes code development and maintenance a significant challenge. A major contributor to this difficulty is the lack of robust software abstractions for handling block-sparse tensor operations. We therefore report the development of a highly efficient symbolic block-sparse tensor library in order to provide access to high-level software constructs to treat such problems. Our implementation supports arbitrary multi-dimensional sparsity in all input and output tensors. We avoid cumbersome machine-generated code by implementing all functionality as a high-level symbolic C++ language library and demonstrate that our implementation attains very high performance for linear-scaling sparse tensor contractions.

Fast, accurate evaluation of exact exchange: The occ-RI-K algorithm.

Construction of the exact exchange matrix, K, is typically the rate-determining step in hybrid density functional theory, and therefore, new approaches with increased efficiency are highly desirable. We present a framework with potential for greatly improved efficiency by computing a compressed exchange matrix that yields the exact exchange energy, gradient, and direct inversion of the iterative subspace (DIIS) error vector. The compressed exchange matrix is constructed with one index in the compact molecular orbital basis and the other index in the full atomic orbital basis. To illustrate the advantages, we present a practical algorithm that uses this framework in conjunction with the resolution of the identity (RI) approximation. We demonstrate that convergence using this method, referred to hereafter as occupied orbital RI-K (occ-RI-K), in combination with the DIIS algorithm is well-behaved, that the accuracy of computed energetics is excellent (identical to conventional RI-K), and that significant speedups can be obtained over existing integral-direct and RI-K methods. For a 4400 basis function C68H22 hydrogen-terminated graphene fragment, our algorithm yields a 14× speedup over the conventional algorithm and a speedup of 3.3× over RI-K.

Efficient implementation of the pair atomic resolution of the identity approximation for exact exchange for hybrid and range- separated density functionals.

An efficient new molecular orbital (MO) basis algorithm is reported implementing the pair atomic resolution of the identity approximation (PARI) to evaluate the exact exchange contribution (K) to self-consistent field methods, such as hybrid and range-separated hybrid density functionals. The PARI approximation, in which atomic orbital (AO) basis function pairs are expanded using auxiliary basis functions centered only on their two respective atoms, was recently investigated by Merlot et al. [J. Comput. Chem. 2013, 34, 1486]. Our algorithm is significantly faster than quartic scaling RI-K, with an asymptotic exchange speedup for hybrid functionals of (1 + X/N), where N and X are the AO and auxiliary basis dimensions. The asymptotic speedup is 2 + 2X/N for range separated hybrids such as CAM-B3LYP, ωB97X-D, and ωB97X-V which include short- and long-range exact exchange. The observed speedup for exchange in ωB97X-V for a C68 graphene fragment in the cc-pVTZ basis is 3.4 relative to RI-K. Like conventional RI-K, our method greatly outperforms conventional integral evaluation in large basis sets; a speedup of 19 is obtained in the cc-pVQZ basis on a C54 graphene fragment. Negligible loss of accuracy relative to exact integral evaluation is demonstrated on databases of bonded and nonbonded interactions. We also demonstrate both analytically and numerically that the PARI-K approximation is variationally stable.

Electrode-assisted catalytic water oxidation by a flavin derivative.

The success of solar fuel technology relies on the development of efficient catalysts that can oxidize or reduce water. All molecular water-oxidation catalysts reported thus far are transition-metal complexes, however, here we report catalytic water oxidation to give oxygen by a fully organic compound, the N(5)-ethylflavinium ion, Et-Fl(+). Evolution of oxygen was detected during bulk electrolysis of aqueous Et-Fl(+) solutions at several potentials above +1.9 V versus normal hydrogen electrode. The catalysis was found to occur on glassy carbon and platinum working electrodes, but no catalysis was observed on fluoride-doped tin-oxide electrodes. Based on spectroelectrochemical results and preliminary calculations with density functional theory, one possible mechanistic route is proposed in which the oxygen evolution occurs from a peroxide intermediate formed between the oxidized flavin pseudobase and the oxidized carbon electrode. These findings offer an organic alternative to the traditional water-oxidation catalysts based on transition metals.

Electronic properties of N(5)-ethyl flavinium ion.

We investigated the electronic properties of N(5)-ethyl flavinium perchlorate (Et-Fl(+)) and compared them to those of its parent compound, 3-methyllumiflavin (Fl). Absorption and fluorescence spectra of Fl and Et-Fl(+) exhibit similar spectral features, but the absorption energy of Et-Fl(+) is substantially lower than that of Fl. We calculated the absorption signatures of Fl and Et-Fl(+) using time-dependent density functional theory (TD-DFT) methods and found that the main absorption bands of Fl and Et-Fl(+) are (π,π*) transitions for the S(1) and S(3) excited states. Furthermore, calculations predict that the S(2) state has (n,π*) character. Using cyclic voltammetry and a simplistic consideration of the orbital energies, we compared the HOMO/LUMO energies of Fl and Et-Fl(+). We found that both HOMO and LUMO orbitals of Et-Fl(+) are stabilized relative to those in Fl, although the stabilization of the LUMO level was more pronounced. Visible and mid-IR pump-probe experiments demonstrate that Et-Fl(+) exhibits a shorter excited-state lifetime (590 ps) relative to that of Fl (several nanoseconds), possibly due to faster thermal deactivation in Et-Fl(+), as dictated by the energy gap law. Furthermore, we observed a fast (23-30 ps) S(2) → S(0) internal conversion in transient absorption spectra of both Fl and Et-Fl(+) in experiments that utilized pump excitations with higher energy.

Mechanism of N(5)-ethyl-flavinium cation formation upon electrochemical oxidation of N(5)-ethyl-4a-hydroxyflavin pseudobase.

We investigated the oxidation behavior of 5-ethyl-4a-hydroxy-3-methyl-4a,5-dihydrolumiflavin (pseudobase Et-FlOH) in acetonitrile with the aim of determining if the two-electron oxidized Et-FlOH(2+) undergoes a release of hydroxyl cation and the production of 5-ethyl-3methyllumiflavinium cation (Et-Fl(+)). The focus of this work is to investigate the possibility of using Et-FlOH as a catalyst for water oxidation. The cyclic voltammetry demonstrates that Et-FlOH exhibits two one-electron oxidation potentials at +0.95 and +1.4 V versus normal hydrogen electrode (NHE), with the second oxidation potential being irreversible. The production of Et-Fl(+) is observed in the cyclic voltammetry of Et-FlOH and has been previously assigned to the release of OH(+) from the two-electron oxidized Et-FlOH(2+). The results of our study show that this is not the case: (i) we performed bulk electrolysis of the electrolyte solution at +2 V and then added Et-FlOH to the electrolyzed solution. We found that Et-Fl(+) is produced from this solution, even though Et-FlOH itself was not oxidized; (ii) reactions of Et-FlOH with chemical oxidants (ceric ammonium nitrate, nitrosyl tetrafluoroborate, and tetrabutylammonium persulfate) demonstrate that Et-Fl(+) production occurs only in the presence of strong Lewis acids, such as Ce(4+) and NO(+) ions. On the basis of these results, we propose that the production of Et-Fl(+) in the electrochemistry of Et-FlOH occurs because of the shift in the Et-FlOH/Et-Fl(+) acid-base equilibrium in the presence of protons released during anodic oxidation. We identified two sources of protons: (i) oxidation of traces of water present in the acetonitrile releases oxygen and protons and (ii) two-electron oxidized Et-FlOH(2+) releases protons located on the N(5)-alkyl chain. The release of protons from Et-FlOH(2+) was confirmed by cyclic voltammetry of Et-FlOH in the presence of pyridine as a base. The first oxidation peak of Et-FlOH at +0.95 V is reversible in the absence of pyridine. The addition of pyridine leads to the shift of the oxidation potential to a less positive value, which is consistent with a proton-coupled electron transfer (PCET). Furthermore, the anodic current increases, and the cathodic peak becomes irreversible, giving rise to two additional reduction peaks at -0.2 and -1 V. The same reduction peaks were observed in the high scan rate cyclic voltammogram of Et-FlOH in the absence of pyridine, implying that the release of protons indeed occurs from Et-FlOH(2+). To determine which functional group of Et-FlOH(.+) is the source of protons, we performed DFT calculations at the B3LYP/6-311++G** level of theory for a reaction of Et-FlOH(.+) with pyridine and identified two proton sources: (i) the >N-CH(2)- group of the N(5) alkyl chain and (ii) the -OH group in the 4a-position of the radical cation. Because the appearance of new reduction peaks at -0.2 and -1.0 V occurs in the model compound that lacks -OH protons (Et-FlOMe), we conclude that the proton removal occurs predominantly from the >N-CH(2)- moiety.