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We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The component-wise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. At the time step n , the orthogonality of the wave function being updated to other wave functions is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to all other wave functions known at the current time, while the normalization of this wave function is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to this wave function at t n + 1 / 2. The unconditional energy stability is nontrivial, and it comes from a subtle treatment of the two-electron integral as well as a consistent treatment of the two projections. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature. • This paper proposes a novel and efficient numerical scheme for the Kohn-Sham gradient flow based model. • The scheme is an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme. • The scheme does not modify the original energy, allows large time step sizes, and solves small systems at each time step. • The rigorous proof is presented in the paper. • Several numerical examples are illustrated to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

[Display omitted] • The porous few-layer bimetallic co-doped g-C 3 N 4 is prepared via a facile bottom-up method. • The bimetallic synergetic regulating effect on the electronic structure can be produced in Co/Ni-g-C 3 N 4. • The superior performance results from the improved charge separation efficiency and visible light harvestability. • This work provides meritorious guidance for new bimetallic co-doped g-C 3 N 4 photocatalysts. The low specific surface area and low charge transfer efficiency of conventional graphite carbon nitride (g-C 3 N 4) are the main obstacles to its application in photocatalytic CO 2 reduction. In this paper, graphite carbon nitride was protonated by phosphoric acid (H 3 PO 4), and a new few-layer porous carbon nitride was prepared by intercalation polymerization with doping bimetal in the cavity of g-C 3 N 4. Under visible light irradiation, the CO formation rate of Co/Ni co-doped g-C 3 N 4 can reach 13.55 μmol g−1 h−1, which was 3.9 times higher than that of g-C 3 N 4 (3.49 μmol g−1 h−1). The density functional theory (DFT) calculations showed that the addition of Co and Ni in the cavity of g-C 3 N 4 can induce bimetallic synergistic regulation of the electronic structure, thus improving the separation efficiency of charges and visible light capture ability of g-C 3 N 4. Our work has great reference value for designing and synthesizing novel bimetallic co-doped g-C 3 N 4 photocatalytic materials. [ABSTRACT FROM AUTHOR]

In this paper, we propose a parallel orbital-updating based optimization method for electronic structure calculations. With our method, the solution of the minimization problem for the Kohn-Sham energy functional with respect to N orbitals is replaced by the solution of N independent minimization problems for the energy functional with respect to one orbital and the orthogonalization of the N updated orbitals. This new method allows a two-level parallelization. This feature makes our approach has a great advantage in large scale parallel computing. The numerical experiments show that our new method is reliable and efficient. Hence, our new method has a great potential for large scale electronic structure calculations on modern supercomputers. • Propose a parallel orbital-updating based optimization approach for KS model. • The new approach has a feature of two-level parallelization. • Provide two practical parallel orbital-updating based optimization algorithms. • Numerical experiments show that these new algorithms are reliable and efficient. [ABSTRACT FROM AUTHOR]