Multi-reference methods assessment

Abstract The singlet-triplet splittings of the di-radicals methylene, trimethylene-methane, ortha-, meta- and para-benzynes, and cyclobutane-l,2,3,4-tetrone have become test systems for the applications of various multi-reference (MR) coupled-cluster methods. We report results close to the basis set limit computed with double ionization potential (DIP) and double electron attachment (DBA) equation-of-motion coupled-cluster methods. These diradicals share the characteristics of a 2-hole 2-particle MR problem and are commonly used to assess the performance of MR methods, and yet require more careful study unto themselves as benchmarks. Here, using our CCSD(T)/6-311G(2d,2p) optimized geometries, we report DIP/DEA-CC results and single-reference (SR) CCSD, CCSD(T), ACCSD(T) and CCSDT results for comparison. 

In this chapter we will provide a critical review of the use of 2- and 4-component relativistic Hamiltonians combined with all-electron methods and appropriate basis sets for the study of lanthanide and actinide chemistry. These approaches provide in principle the more rigorous treatment of the electronic structure but typically demand large computational resources due to the large basis sets that are required for accurate energetics. A complication is furthermore the open-shell nature of many systems of practical interest that make black box application of conventional methods impossible. Especially for calculations in which electron correlation is explicitly considered one needs to find a balance between the appropriate treatment of the multi-reference nature of the wave function and the practical limitations encountered in the choice of an active space. For density functional theory (DFT) calculations one needs to select the appropriate density functional approximation (DFA) on basis of assessments for lighter elements because little or no high-precision experimental information on isolated molecules is available for the f elements. This increases the demand for reliable theoretical ( benchmark ) data in which all possible errors due to the inevitable approximations are carefully checked. In order to do so we need to understand how f elements differ from the more commonly encountered main group elements and also from the d elements with which they of course share some characteristics. 

Emission spectroscopy (e.g., ICP-AES, ICP-CATS, or ICP-MS) techniques are moving to the forefront of copper determination after the instruments became available at affordable prices during the 1990s. Today, ICP-MS has reached detection limits of 5 to 50 ng Cu which renders ICP-MS compatible with ASV and CSV methods for copper determination (Lu et al. 1993). The advantages of ICP-MS include simultaneous determination of the sample multi-element profile and the assessment of isotopic species. All the measures available should be taken to prevent external contamination of the samples blanks and certified reference materials (CRMs) are the condi-... 

In order to meet the expected industrial needs for such materials, a large scale CCVD production facility has been developed in Nanocyl S.A., with a production capacity of the order of one kg of multi-walled carbon nanotubes (MWCNTs) per hour. In a typical CCVD reactor [e.g. 4], the contact between a gaseous hydrocarbon and an appropriate solid catalyst at high temperature results in the deposition of carbon nanotubes on the catalyst. The raw product that exits the reactor is therefore expected to contain carbon nanotubes, a catalyst residue and possibly amorphous carbon. The latter two products are usually referred to as impurities. A general method to assess the amoimt of impurities in a MWNT sample is not available yet. The present work reports on the use of k ton adsorption to determine the amount of nanotubes actually present in as-synthesized sample. 

Sensitivity analysis thus provides a method by which the structural stability of multi-parameter models can be assessed and described in more detail. We can say for the reference solution studied here that the Lotka-Volterra oscillator is structurally unstable to variations in I<4 and I<5 but not to variations in k, k2 and I<3. These properties of the Lotka-Volterra oscillator are, of course, well-known. The success of sensitivity analysis in unambiguously (and quantitatively) verifying these facts suggests that it will be a useful tool for the study of models which are not so well-understood. 

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