Open-shell computational methods

With radicals there is no convenient method like the Hartree-Fock-Roothaan procedure commonly used for closed-shell systems. In contrast, the open-shell theory is typical of a number of methods suggested which differ in accuracy from the viewpoint of true SCF theory, in range of applicability, complexity, and computing feasibility. A critical survey of open-shell SCF methods reported by Berthier D covers the literature up to 1962. We shall not duplicate that review here we propose rather to note some features of open-shell methods relevant to their computation feasibility and to mention procedures published after 1962. The unrestricted treatments that assume different space orbitals for different spins will be disregarded here because the restricted wave functions... 

Dr. Gey (Berlin) found our classification of open shell SCF methodes somewhat inaccurate in distinguishing between the methodes using coupling operators and the density matrix method inasmuch as the latter only represents a more elegant formulation of the former. This is certainly true, but from the practical point of view the two computational techniques are different so that our classification is perhaps still reasonable. 

Although LHF is often a better theoretical treatment of open-shell systems than the RHF (half-electron) methods, it takes longer to compute. Separate matrices for electrons of each spin roughly double the length of the calculation. ... 

Nearly every technical difficulty known is routinely encountered in transition metal calculations. Calculations on open-shell compounds encounter problems due to spin contamination and experience more problems with SCF convergence. For the heavier transition metals, relativistic effects are significant. Many transition metals compounds require correlation even to obtain results that are qualitatively correct. Compounds with low-lying excited states are difficult to converge and require additional work to ensure that the desired states are being computed. Metals also present additional problems in parameterizing semi-empirical and molecular mechanics methods. 

LORG (localized orbital-local origin) technique for removing dependence on the coordinate system when computing NMR chemical shifts LSDA (local spin-density approximation) approximation used in more approximate DFT methods for open-shell systems LSER (linear solvent energy relationships) method for computing solvation energy... 

Next, you choose a UHF or RHF calculation. HyperChem can compute open-shell (non-singlet) systems with either the half-electron RHF or the UHF method (see Hartree-Fock Wave Functions on page 37). 

From the above it should be clear that UHF wave functions which are spin contaminated (more than a few percent deviation of (S ) from the theoretical value of S S + 1)) have disadvantages. For closed-shell systems an RHF procedure is therefore normally preferred. For open-shell systems, however, the UHF method has been heavily used. It is possible to use an ROHF type wave function for open-shell systems, but this leads to computational procedures which are somewhat more complicated than for the UHF case when electron correlation is introduced. 

From the point of view of the present state of computational possibilities, an extension of open-shell methods to a and a + rr electronic systems is rather tempting. This extension is easy for the method of Longuet-Higgins and Pople e.g., the CNDO/2 method is amenable to radicals having one nondegenerate open shell if in the original terms for F matrix elements (66),... 

The determination of the electronic structure of lanthanide-doped materials and the prediction of the optical properties are not trivial tasks. The standard ligand field models lack predictive power and undergoes parametric uncertainty at low symmetry, while customary computation methods, such as DFT, cannot be used in a routine manner for ligand field on lanthanide accounts. The ligand field density functional theory (LFDFT) algorithm23-30 consists of a customized conduct of nonempirical DFT calculations, extracting reliable parameters that can be used in further numeric experiments, relevant for the prediction in luminescent materials science.31 These series of parameters, which have to be determined in order to analyze the problem of two-open-shell 4f and 5d electrons in lanthanide materials, are as follows. 

Because of the inherent limitations of such semiempirical procedures, they can only be relied upon for yielding predictions for a limited set of data, the range of which includes the set of experimental data used for their parametrization. As such data are less abundant for open-shell species, such as radical ions, it is not surprising that there are examples of dramatic failures of semiempirical methods in predicting their electronic spectra, some of which will be discussed later. Ab initio methods are not burdened by these limitations but, as mentioned above, they require additional computations to account for dynamic electron correlation. 

The identification of unknown chemical compounds isolated in inert gas matrices is nowadays facilitated by comparison of the measured IR spectra with those computed at reliable levels of ab initio or density functional theory (DFT). Furthermore, the observed reactivity of matrix isolated species can in some instances be explained with the help of computed reaction energies and barriers for intramolecular rearrangements. Hence, electronic structure methods developed into a useful tool for the matrix isolation community. In this chapter, we will give an overview of the various theoretical methods and their limitations when employed in carbene chemistry. For a more detailed qualitative description of the merits and drawbacks of commonly used electronic structure methods, especially for open-shell systems, the reader is referred to the introductory guide of Bally and Borden.29... 

Quantum chemical calculations have become a valuable tool in the research of reactive intermediates. Unfortunately, there is no unique computational method that can be uniformly applied in all cases and give an accurate answer at a practical cost. A variety of computational methods are available, each with its own weaknesses and advantages. The species that are of interest in this chapter and which often have unpaired electrons, pose specific problems in calculating their properties, and some care in choosing appropriate methods is necessary for obtaining meaningful results. In this respect, an excellent guide for calculations on open-shell molecules has been recently published [51]. 

Also the variational 2-RDM method for computing the 2-RDM of a singlet state may be applied directly to open-shell molecules without significant modification. 

Computational quantum chemistry has emerged in recent years as a viable tool for the elucidation of molecular structure and molecular properties, especially for the prediction of geometrical parameters, kinetics and thermodynamics of highly labile compounds such as nitrosomethanides. However, they are difficult objects for both experimental (high toxicity, redox lability, high reactivity, explosive character etc.) and computational studies, even with today s sophisticated techniques (e.g. NO compounds are often species with open-shell biradical character which requires the application of multi-configuration methods). 

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