Hybridized orbital method

Pu JZ, JL Gao, DG Truhlar (2005) Generalized hybrid-orbital method for combining density functional theory with molecular mechanicals. ChemPhysChem 6 (9) 1853—1865... 

Let us start by using the hybrid orbital method to predict the structure of methane. Methane, CH4, is composed of a carbon atom and four hydrogen atoms. The carbon atom has an electron configuration of Is2 2s2 2p2. Each hydrogen atom has an electron configuration of Is1. Experiments showed that the geometry of the... 

The local self-consistent field (LSCF) or fragment SCF method has been developed for treating large systems [105,134-139], in which the bonds at the QM/MM junction ( frontier bonds ) are described by strictly localized bond orbitals. These frozen localized bond orbitals are taken from calculations on small models, and remain unchanged in the QM/MM calculation. The LSCF method has been applied at the semiempirical level [134-137], and some developments for ab initio calculations have been made [139]. Gao et al. have developed a similar Generalized Hybrid Orbital method for semiempirical QM/MM calculations, in which the semiempirical parameters of atoms at the junction are modified to enhance the transferability of the localized bond orbitals [140]. Recent developments for ab initio QM/MM calculations include the method of Phillip and Friesner [141], who use Boys-localized orbitals in ab initio Hartree-Fock QM/MM calculations. These orbitals are again taken from calculations on small model systems, and kept frozen in QM/MM calculations. 

Covalent crystals are held together by strong, highly directional bonds usually described by the valence bond hybrid orbital method. Each atom is part of a large extended single molecule that is the crystal itself. Because of the nature of their bonds, covalent crystals have very high melting points and are hard and brittle. 

P. Amara, M. J. Field, C. Alhambra and J. Gao The generalized hybrid orbital method for combined quantum mechanical/molecular mechanical calculations formulation and tests of the analytic derivatives, Theoret. Chem. Acc. 104, 336-343 (2000). 

Hybrid orbital methods differ from the link atom models in that no extra atoms are introduced into the system. Instead, an atom (usually a tetravalent sp carbon) at the covalent boundary is designed to have both a QM and a MM character. This is typically done by defining a set of hybrid sp orbitals on the atom, some of which are fixed and not included in the QM calculation and the remainder of which are allowed to participate. 

Both the above schemes were not entirely satisfactory and several modifications have been suggested. Rivail and coworkers have developed a hybrid orbital method, similar in spirit to that of Warshel and Levitt, in which the QM atom at the junction is taken to have a normal complement of orbitals but the hybrid orbital which points towards the MM atom is kept frozen. The form of this orbital is not optimized in the QM calculation although it is counted in the evaluation of the QM energy terms and its interaction with the MM atoms is carefully parametrized. The hybrid orbital method is relatively easy to implement for semiempirical QM methods (as done by Warshel and Rivail et al.) but becomes more complicated for ab initio HF wavefunctions. An equivalent approach is... 

Generalized Hybrid Orbital Method for Combined Quantum Mechanical/Molecular Mechanical Calculations Formulation and Tests of the Analytical Derivitives. 

The total electron density contributed by all the electrons in any molecule is a property that can be visualized and it is possible to imagine an experiment in which it could be observed. It is when we try to break down this electron density into a contribution from each electron that problems arise. The methods employing hybrid orbitals or equivalent orbitals are useful in certain circumsfances such as in rationalizing properties of a localized part of fhe molecule. Flowever, fhe promotion of an electron from one orbifal fo anofher, in an electronic transition, or the complete removal of it, in an ionization process, both obey symmetry selection mles. For this reason the orbitals used to describe the difference befween eifher fwo electronic states of the molecule or an electronic state of the molecule and an electronic state of the positive ion must be MOs which belong to symmetry species of the point group to which the molecule belongs. Such orbitals are called symmetry orbitals and are the only type we shall consider here. 

In this case, there are three equivalent hybrid orbitals, each called sp (trigonal hybridization). This method of designating hybrid orbitals is perhaps unfortunate since nonhybrid orbitals are designated by single letters, but it must be kept in mind that each of the three orbitals is called sp. These orbitals are shown in Figure 1.4. The three axes are all in one plane and point to the comers of an equilateral triangle. This accords with the known structure of boron trifluoride (BF3), a planar molecule with angles of 120°. 

When combining QM with MM methods, the partitioning of the system will often intersect a chemical bond. This bond is usually chosen to be a carbon-carbon single bond (whenever possible) and three major coupling methods have been developed, which are referred to as the link-atom [54] , pseudo-atom/bond [55] and hybrid-orbital [56] approach, respectively. In the link atom approach the open valency at the border is capped by a hydrogen atom, and most DFTB QM/MM implementations are based on this simple scheme [49, 50] or related variations [57], Recently,... 

A method which is similar to the Pariser-Parr-Pople method for the n electron system and is applicable to common, saturated molecules has been proposed by Pople 28>. This method is called the CNDO complete neglect of differential overlap) SCF calculation. Katagiri and Sandorfy 29> and Imamura et al. °) have used hybridized orbitals as basis of the Pariser-Parr-Pople type semiempirical SCF calculation. 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

Hybrid orbitals may be considered as perfectioned AOs, adopted in the calculation of localized MOs in polyatomic molecules, with the LCAO method (cf. section 1.17.1). In the case of hybrid orbitals sp, the four linear combinations of s and p orbitals (Tbi, Tc2, Te, Te ) that lead to tetrahedral symmetry are... 

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