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Use this search facility to find out more about the profile of our HPC-Europa2 visitors, the type of work they have been doing, and their project achievements.
Combining organic species with gold, or transition metals in general, usually has a substantial impact on the optical, electronic, and magnetic properties of the compound. Trinuclear complexes of coinage metals have been found to posses interesting photochemical properties. Cyclic trinuclear Au(I) complexes are especially intriguing, as they exhibit bright phosphorescence at room temperature, tuneable across the whole visible spectrum, charge-transfer properties, and are sensitive to multiple stimuli. They are the main focus of this project. The specific systems of this project will be the stacked, cyclic trinucelar Au(I) carbeniate complexes, recently synthesised and experimentally characterised at the University of North Texas. The compounds have been found to potentially provide the broad emission bands necessary for the production of white light. For this highly coveted feature, a careful tuning of the organic ligands is necessary. Further, the emission and excitation spectra of the complexes have been found to crucially depend on the packing arrangements of the species. The photophysical properties of these species depend heavily on intermolecular interactions, the most important of which is the aurophilic attraction. In order to first conclusively establish the correct structure of these complexes, a combination of experiment and high-level modelling is necessary. Establishing a reliable method for modelling this class of compounds, would enable in silico tuning of the properties. Thus, a main objective of this project is to uncover the theoretical methods capable of delivering the required accuracy. Studying gold and gold-containing complexes computationally is very challenging. Gold has many electrons, increasing the computational effort of simulating their correlated behaviour. The problem is further complicated by the significant effects arising from relativity. Density functional theory has thus been extensively used in the study of gold-containing species, due to the usually high quality:computational cost ratio. Standard DFT methods do not provide a sufficiently accurate description of the aurophilic attraction bonding gold atoms together, however. As shown by Johansson, Furche, and co-workers, already structural properties require functionals that include true non-local effects, that is, meta-generalised gradient approximations. Electronic excitations computed at DFT level also suffer from problems. These difficulties are reasonably well-known in the case of normal organic compounds, but very little is known of the performance of DFT when it comes to accurately describing excitations for complexes incorporating several heavy atoms. From a practical point-of-view, the most problematic short-coming of contemporary DFT is, arguably, the inability to describe charge-transfer excitations correctly. These types of excitations can be expected to be important in the trinuclear Au-complexes of the project. Thus, wave-function based methods need to be employed. The complexes to be studied are large, and even with supercomputing power, the choice of reliable quantum chemical methods that can be employed is limited. The approximate second-order coupled cluster method CC2 has recently been very cost-efficiently implemented, and will be one of the methods used in the study. The main work-horse of the study will, however, be the second-order algebraic diagram construction ADC(2) approximation. This method was recently thoroughly benchmarked and applied in the host group, and found to be a cost-effective alternative..
However efficiently implemented, computing excitations at correlated wave-function level is still restricted to a scalar relativistic approximation. Spin-orbit (SO) effects can however be large when the systems contain heavy metals. Therefore, excitation energies including spin-orbit coupling will be performed at DFT level.
The project got well underway, although due to exceptionally long queueing times, only a minor part of the planned calculations could be finished during the visit. The initial benchmark-type studies on smaller systems were succesfully completed, and a good understanding of the required level of theory for studying the gold complexes was obtained, however. Importantly, the collaboration between host and guest was crucial for obtaining the insight required for completing the project. The collaboration continues, as access to the supercomputing facilities is available also beyond the limits of the physical visit.
As a side-product of the visit, other projects where the interests of the host and guest coincide were uncovered, which lead to further collaborative projects, where the strengths of both groups combined are expected to lead to highly interesting science.
In this Project we aim to analyze the aromaticity and the different distortive nature of s and p electrons employing the energy decomposition analysis (EDA) technique as used in previous works by Prof. Bickelhaupt’s method for the study of the aromaticity of classical organic molecules. This is a very promising molecular orbital (MO) model of aromaticity that allows understanding the reasons behind the fact that aromatic molecules such as benzene have a regular structure with delocalized bonds, while antiaromatic molecules like cyclobutadiene prefer distorted geometries with localized bonds. In none of the systems analyzed by Prof. Bickelhaupt and coworkers does the p-electron system favour a symmetric, delocalized ring. It is the sigma-electron system that forces the system to acquire its regular structure.
This Project is carried out in cooperation with Prof. Dr. F. Matthias Bickelhaupt of the Vrije Universiteit Amsterdam. During the realization of the project the computational resources of SARA were extensively used. All calculations were performed on Huygens installation.
In this Project we were interested in the analysis of s and p electron distortive character in the case of all-metal aromatic clusters, and, in particular, in the quintessential Al42- aromatic species. Just for comparison we added the isoelectronic B42- and Ga42- species. In addition, we compared the results obtained with those derived for the C4H4, C4H42+, and C4H42- species. Finally, the Al44- species having ambiguous aromatic character was also included in the study. The ADF software package based on Density Functional Theory (DFT) has been used for calculations. Calculations were performed using BP86 functional in conjunction with a triple-zeta quality Slater-type basis set augmented with two sets of polarization functions (TZ2P). For the EDA analysis we have considered two diagonal X2 fragments of the X4 clusters analyzed and we have changed the angle between the two fragments from 90 to 100ş. In this way we change from D4h to D2h molecular symmetries and we can discuss which energy terms increase or decrease during this process. Preliminary results of our calculations for Al42- show that the orbital interaction term of the EDA is clearly distortive while the Pauli repulsion prefers the symmetric structure. Since there is only Pauli repulsion among the s electrons it is possible to sum the s orbital interaction term and the Pauli repulsion term to arrive at the conclusion that the s electrons favour the symmetric structure in Al42- while the p electrons are slightly distortive but much less than in C4H4 for instance. Similar results were obtained for the rest of the systems.
The Finite Element Tearing and Interconnecting method Dual-Primal (FETI-DP) method is one of the non-overlapping domain decomposition methods, which was published by Farhat and his co-workers in the article [FarhatEtAll-01] in 2001. The domain decomposition methods divide the original domain into several smaller subdomains. The FETI-DP method was developed due to problems with singular subdomain matrices in the original FETI method. The FETI-DP method is the combination of the FETI method and Schur complement method.
The FETI-DP method divide unknowns into two categories – corner unknowns and remaining unknowns. The remaining unknowns are further split into remaining interface unknowns and internal unknowns. The continuity conditions are enforced by Langrange multipliers which are defined on the remaining interface unknowns and also by corner nodes. The corner unknowns ensure the non-singularity of the subdomain matrices. The remaining unknowns are eliminated and the coarse problem is after obtained. The matrix of the coarse problem is symmetric and positive-definitive. Therefore the coarse problem can be solved by the conjugate gradient method. More information about FETI-DP method can be found in the article [FarhatEtAll-01] or in the book [Kruis06] by Kruis.
The selection of the the corner nodes where are corner unknowns defined deserves special attention. A definition of the corner nodes was published in the original article [FarhatEtAll-01] by Farhat. The corner nodes are there defined as
D1: Cross-points - It means the nodes which belonged to more than two subdomains
D2: The set of nodes located at the beginning and end of each edge of each subdomain.
But the definition is not suitable for all possible meshes. If the original domain is divided into two subdomains and the first subdomain is surrounded by the second domain, then there is no cross-points and there is not beginning and the end of any edge. Recently, strong influence of the definition of the corner unknowns on the condition number of the subdomain matrix has been observed in the work [KabelikovaEtAll-09] by Kabelíková et all. The large condition numbers of subdomain matrices significantly deteriorate the convergence of the iterative methods used for the solution of the coarse problem. There is a minimal needed number of corner nodes. In the case of two-dimensional meshes, plane strain and plane stress problems require two different nodes, three nodes are better for plate problems. Therefore, the minimum number of needed nodes is three in the case of two-dimensional meshes. In the case of three-dimensional meshes must be selected three non-collinear nodes. Theoretically can be selected all nodes on the subdomain boundaries, but then the FETI-DP method transforms itself to the Schur complement method.
There is no software known to author which can be use for the selection of corner nodes. This was a motivation for a development of an algorithm for the selection such nodes. The algorithm will be developed with help of the graph theory and several heuristic rules.
[FarhatEtAll-01] Farhat, C. and Lesoinne, M. and LeTallec, P. and Pierson, K. and Rixen, D.: “FETI-DP: A Dual-Primal Unified FETI Method-Part I: Faster Alternative to the Two-Level FETI Method” , International Journal for Numerical Methods in Engineering, vol. 50, pages = 1523 – 1544.
[KabelikovaEtAll-09] Kabelíková, P. and Dostál, Z. and Kozubek, T. and Markopoulos, A.: “Generalized inverse matrix evaluation using graph theory”, In Proceedings of the Modelling 2009, Blaheta, R., Starý J. (ed.),Institute of Geonics AS CR, Ostrava, Czech Republic, 2009.
[Kruis-06] Kruis, J.: “Domain Decomposition Methods for Distributed Computing”, Saxe-Coburg Publications, edition 1st, 2006.
There were developed two independent algorithm for selection of the corner nodes. The first algorithm is used for two-dimensional meshes and the second algorithm for three-dimensional meshes. Both algorithm are based on the Graph theory. Therefore there will be defined several necessary terms from the Graph theory. A graph G(V,E) consist of a finite set V of elements called vertices and a finite set E of elements called edges. Nodes of a finite element mesh can be mapped to the set V of graph vertices. Vertices vi and vj are connected by an edge if the appropriate nodes belong to a edge of the finite element. Otherwise, there is no edge between vi and vj. This graph is called nodal graph. The degree dG(v) of vertex v in graph G is the number of edge of G incident with v. A walk in a graph G = (V, E) is a ﬁnite sequence of vertices v0 , v1 , . . . , vk such that (vi−1, vi ), 1 ≤ i ≤ k is an edge in the graph G. The nodes v0 and vk are called end vertices of the walk. A walk is a trail if all its edges are distinct.
In the case of the two-dimensional algorithm, the graph B(V,E) is established form boundary nodes between subdomains. The vertex degree is established for each vertex in the graph. The corner nodes is then defined as
vertices which vertex degree is equal to one
vertices which vertex degree is more than two
The algorithm, which selects corner nodes, based on the vertex degree is called minimal number algorithm. A control of chosen nodes must be done after the selection procedure. The number of corner nodes is controlled. The minimum number of needed nodes is three. If there are no enough nodes from the minimal number algorithm then the another nodes must be selected.
The next control is aimed on the geometrical relation between corner nodes. If the selected nodes are too close each other, the subdomain matrix has usually very large condition number. A distance between corner nodes are controlled . The next condition is a non-collinearity condition. This is necessary in order to avoid several nodes in one line which lead also to large condition number of subdomain matrix.
It is possible to add further corner nodes by an extended algorithm for adding corner nodes. The extended algorithm is based on the restriction of the graph B(V,E) into several subgraphs. The subgraphs Sj (V,E) are defined as the open trail between two corner nodes in the graph B(V,E). The further corner nodes can be added as the centre of such subgraph or the walk can be split into k part and the corner node can be added at the end of such part of the walk.
The several numerical test was done with help of SMP computer Ness and supercomputer HECToR.
The following behaviour of the algorithm was observed. Higher number of corner nodes reduces the number of iterations in the conjugate gradient method and therefore also the computational time is reduced. If an optimal number is reached, the number of iterations still decreases but the total computational time starts to grow because time of condensation of the matrix contains entries related to the corner nodes prevails over time saved by the reduced number of iterations. The tests also showed the ability of the proposed algorithm to select the minimum number of corner nodes in the case of very general domains. The FETI-DP method can be therefore used without manual selection of corner nodes.