Closed-shell molecular systems

The main handicap of MD is the knowledge of the function [/( ). There are some systems where reliable approximations to the true (7( r, ) are available. This is, for example, the case of ionic oxides. (7( rJ) is in such a case made of coulombic (pairwise) interactions and short-range terms. A second example is a closed-shell molecular system. In this case the interaction potentials are separated into intraatomic and interatomic parts. A third type of physical system for which suitable approaches to [/( r, ) exist are the transition metals and their alloys. To this class of models belong the glue model and the embedded atom method. Systems where chemical bonds of molecules are broken or created are much more difficult to describe, since the only way to get a proper description of a reaction all the way between reactant and products would be to solve the quantum-mechanical problem at each step of the reaction. 

A single-determinant wave-function of closed shell molecular systems is invariant against any unitary transformation of the molecular orbitals apart from a phase factor. The transformation can be chosen in order to obtain LMOs. Starting from CMOs a number of localization procedures have been proposed to get LMOs the most commonly used methods are as given by the authors of (Edmiston et ah, 1963) and (Boys, 1966), while the procedures provided by (Pipek etal, 1989) and (Saebo etal., 1993) are also of interest. It could be stated that all the methods yield comparable results. Each LMO densities are found to be relatively concentrated in some spatial region. They are, furthermore, expected to be determined mainly by that part of the molecule which occupies that given region and its nearby environment rather than by the whole system. 

The present contribution considers general electronic states of solvated molecules and is not limited to closed shell molecular compounds. For closed shell molecular systems, methods utilizing closed shell coupled-cluster electronic structure and closed shell density density functional theory for the electronic structure of the solvated system have appeared in the literature [54-67],... 

Let us study the interaction of two closed shell molecular systems andB in the ground states. First, we summarize the main ideas which we made use of in our derivation139. The interaction energy between systems A and B is defined as the... 

For the closed-shell molecular systems one thus finds... 

Of special interest to us are the properties related to the external application of uniform static electric and magnetic fields, which we will denote by F and B, respectively. To second order, the energy of a closed-shell molecular system may be written as... 

Recognition of this relationship between coupled cluster theory and MBPT has inspired research efforts to construct perturbation-based corrections to the CCSD energy to account for higher excitation contributions. Undoubtedly, the most successful and popular of these is the (T) correction first described for closed-shell molecular systems by Raghavachari et al. " In the next section, we will describe the structure of this correction using diagrammatic techniques. 

The success of this rather abrupt truncation for closed-shell molecular systems is not too surprising when one considers that the dominant terms of a perturbation expansion have been included.2 616 The next more complete approximation to attain recognition is the extended coupled-pair many-electron theory (ECPMET) of Paldus, Cizek, and Shavitt,6,17,18 which includes connected single and triple excitations,... 

Of all the methods currently used in molecular electronic structure theory, the CCSD(T) model is probably the most successful, highly accurate level, at least for closed-shell molecular systems. For many properties of interest to chemists such as molecular structure, atomization energies, and vibrational frequencies, it provides numerical data of consistently high quality, sometimes surpassing that of experiment. Nevertheless, it does fail in certain cases, in particular for systems characterized by several important Slater determinants and also for certain properties such as indirect nuclear spin-spin couplings of magnetic resonance spectroscopy. 

Coupled electron pair and cluster expansions. - The linked diagram theorem of many-body perturbation theory and the connected cluster structure of the exact wave function was first established by Hubbard211 in 1958 and exploited in the context of the nuclear correlation problem by Coester212 and by Coester and Kummel.213 Cizek214-216 described the first systematic application to molecular systems and Paldus et al.217 described the first ab initio application. The analysis of the coupled cluster equations in terms of the many-body perturbation theory for closed-shell molecular systems is well understood and has been described by a number of authors.9-11,67,69,218-221 In 1992, Paldus221 summarized the situtation for open-shell systems one must nonetheless admit... 

The sum of the total electronic energy, ee, and the energy of intemuclear repulsion, Ear- In the Hartree-Fock (SCF) method, the value of fee for a closed-shell molecular system is given by... 

To summarize, we have found that the IPs and EAs of a closed-shell molecular system may be identified with the negative orbital energies of the canonical orbitals ... 

The following presentation is limited to closed-shell molecular orbital wave-functions. The first section discusses the unique ability of molecular orbital theory to make chemical comparisons. The second section contains a discussion of the underlying basic concepts. The next two sections describe characteristics of canonical and localized orbitals. The fifth section examines illustrative examples from the field of diatomic molecules, and the last section demonstrates how the approach can be valuable even for the delocalized electrons in aromatic ir-systems. All localized orbitals considered here are based on the self-energy criterion, since only for these do the authors possess detailed information of the type illustrated. We plan to give elsewhere a survey of work involving other types of localization criteria. 

As a final example of molecular properties, consider the molecular electronic system in the presence of a static external magnetic induction B and nuclear magnetic moments M/c, corresponding to the physical situation encountered in an NMR experiment. Expanding the energy of a closed-shell electronic system in the induction and in the nuclear magnetic moments, we obtain... 

An approach for treating open-shell molecular systems by a closed-shell formalism which utilizes the similarity between the SCF equations (see Density Functional Theory (DFT), Hartree Fock (HF), and the Self-consistent Field) and those for a fictitious closed-shell system in which the odd electron is replaced by two half-electrons. 

For a closed-shell molecular electronic system, the Hartree-Fock energy (10.4.15) may, according... 

Choose LHH(spin Unrestricted Hartree-Fock) or RHF (spin Restricted Ilartree-Fock) calculations according to your molecular system. HyperChem supports UHF for both open-sh el I and closed-shell calcii lation s an d RHF for cUised-shell calculation s on ly, Th e closed-shell LHFcalculation may be useful for studyin g dissociation of m olectilar system s. ROHF(spin Restricted Open-shell Hartree-Fock) is not supported in the current version of HyperChem (for ah initio calculations). 

Huckel realized that his molecular orbital analysis of conjugated systems could be extended beyond neutral hydrocarbons He pointed out that cycloheptatrienyl cation also called tropyhum ion contained a completely conjugated closed shell six tt electron sys tern analogous to that of benzene... 

Closed-shell molecules have a multiplicity of one (a singlet). Arad-ical, with one unpaired electron, has a multiplicity of two (a doublet). A molecular system with two unpaired electrons (usually a triplet) has a multiplicity of three. In some cases, however, such as a biradical, two unpaired electrons may also be a singlet. 

Here we give the molecule specification in Cartesian coordinates. The route section specifies a single point energy calculation at the Hartree-Fock level, using the 6-31G(d) basis set. We ve specified a restricted Hartree-Fock calculation (via the R prepended to the HF procedure keyword) because this is a closed shell system. We ve also requested that information about the molecular orbitals be included in the output with Pop=Reg. 

We can now build a closed shell wavefunction by defining n/2 molecular orbitals for a system with n electrons, and then assigning electrons to these orbitals in pairs of opposite spin ... 

The Hartree-Fock equations for the /-th element of a set containing occ occupied molecular orbitals i in a closed shell system with n = 2occ electrons are [8]... 

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