A numerical method for the solution of one-dimensional Schrödinger-like equations with arbitrary numerical or analytical potentials is presented. The method takes advantage of matrix algebra for both obtaining several eigenvalues and eigenvectors at the same time and saving computer time. On the other hand, the method illustrates the close relationship between differential and algebraic eigenvalue problems, as well as the mathematical origin of quantization. Several examples are worked out in the text and the procedure for applying a user friendly routine to other problems is given.
Optimization algorithms and other techniques used for the crystallographic refinement of both small molecule and macromolecular X-ray crystal structures are reviewed in this work. Emphasis is made on the advantages and disadvantages of every method in its application to the refinement process of different problems, as well as on their actual implementation in current program packages. A description of the most basic (first-level) algorithms based on physical grounds, such as maximum-likelihood, least-squares, molecular dynamics, maximum-entropy, Fourier and graphical methods, and holographic methods, is followed by a second-level approach, including Newton methods, simulated annealing, and gradient methods, among others, and finally by a third-level study which includes matrix decompositions and iterative methods for the solution of the resulting linear systems ofequations. In addition, by descending to the most basicmathematical and physical level, connections with other disciplines that use almost the same tools are easily made, showing not only the interdisciplinarity of crystallography but also the flow of information between different sciences.
The experimental geometry obtained from single-crystal X-ray diffraction data for a metalladiphosphanyl carbene precursor is compared with the results of theoretical calculations made at the ab initio level by using Hartree–Fock (HF) and Density Functional Theory (DFT) methods over the carbene itself. Theoretical geometry optimizations for the singlet ground state of [ Mn(CO)4(PH2)2C: ]+ have been performed with several hybrid functionals and basis sets. Calculated geometries showed a perfect C 2v symmetry in the highest levels of calculation and were somewhat relaxed when compared with the experimental ones; for instance, with the largest basis set, the P–C–P angle found was 124.8°, whereas C–P bond distances were both 1.667 Å, compared to 103.5(3)° and 1.718(5) Å, respectively, from the experimental data. The absence of a ligand attached to the C : atom in the calculated structure, which is present in the form of iodine in the experimental complex, is probably responsible, to a certain extent, for the discrepancies. In addition to the structural computations, in order to theoretically quantify the highly electrophilic character expected for the carbene, electron affinities were calculated and found to be between 6.24 eV and 6.97 eV at different DFT levels of calculation, which confirmed the expectations. In this respect, a comparison with the analogous [Ru(CNH)4(PH2)2C:]2+ carbene is also made, showing the possibility of experimentally trapping the manganese carbene.
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