Orbital-based methods

Orbital-based methods can be used to compute transition structures. When a negative frequency is computed, it indicates that the geometry of the molecule corresponds to a maximum of potential energy with respect to the positions of the nuclei. The transition state of a reaction is characterized by having one negative frequency. Structures with two negative frequencies are called second-order saddle points. These structures have little relevance to chemistry since it is extremely unlikely that the molecule will be found with that structure. 

This technique has been applied occasionally to orbital-based methods, where it is called seam searching. The rest of the techniques mentioned in this chapter are applicable to semiempirical, density functional theory (DFT), and ah initio techniques. 

Many semiempirical methods have been created for modeling organic compounds. These methods correctly predict many aspects of electronic structure, such as aromaticity. Furthermore, these orbital-based methods give additional information about the compounds, such as population analysis. There are also good techniques for including solvation elfects in some semiempirical calculations. Semiempirical methods are discussed further in Chapter 4. 

Nearly all liquid simulations have been done using molecular mechanics force fields to describe the interactions between molecules. A few rare simulations have been completed with orbital-based methods. It is expected that it will still be a long time before orbital-based simulations represent a majority of the studies done due to the incredibly large amount of computational resources necessary for these methods. 

Computed optical properties tend not to be extremely accurate for polymers. The optical absorption spectra (UV/VIS) must be computed from semiempiri-cal or ah initio calculations. Vibrational spectra (IR) can be computed with some molecular mechanics or orbital-based methods. The refractive index is most often calculated from a group additivity technique, with a correction for density. 

Broadly speaking it is still an orbital based method of course but not one that corresponds to the elementary concept of a particular number of electrons in the shells of an atom. 

Garmer DR, Stevens WJ (1989) Transferability of molecular distributed polarizabilities from a simple localized orbital based method. J Phys Chem 93 8263... 

Empirical Valence Bond (EVB) [163,164] simulations as an alternative to Molecular Orbital based methods. 

Of the orbital-based methods, the earliest remains the most widely used method that developed by Mulliken and called the Mulliken population. The total number of electrons in a molecule N must equal the integral of p(r) over all spaces. For simplicity, we will examine the case of HF wavefunction then this integral can be... 

The NBO method can be nsed for ab initio calcnlations at the HF and any correlated levels as well as for DFT methods. The results do not strongly depend on the size of the basis set, which is one reason that the NBO method has replaced the ontdated Mnlliken popnlation analysis, for example, for the calculation of atomic partial charges. But there is a price that has to be paid for the advantages. Like any orbital-based method for partitioning the electronic charge into atomic and bonding domains, the choice of the selection procedure has some arbitrary character that needs to be known in order to jndge the qnality of the results. 

At the same time that band-theoretical methods have been rapidly developing in accuracy and breadth of applicability, there has been substantial progress in qualitative or semiquantitative atomic-orbital-based methods, whether based upon accurate atomic potentials or upon para-... 

In 1958, Nesbet extended Brueckner s theory for infinite nuclear mat-ter to nonuniform systems of atoms and molecules. By consideration of the CISD problem in which the electronic Hamiltonian is diagonalized within the basis of the reference and all singly and doubly excited determinants, Nesbet explained that Brueckner theory allows one to construct a set of orthonormal molecular orbitals for which the correlated wavefunction coefficients for all singly excited determinants vanish. Unfortunately, the construction of the set of orbitals that fulfill this Brueckner condition can be determined only a posteriori from the single excitation coefficients computed in a given orbital basis. As a result, the practical implementation of Brueckner-orbital-based methods has... 

Perhaps the greatest need for Brueckner-orbital-based methods arises in systems suffering from artifactual symmetry-breaking orbital instabili-ties, " ° where the approximate wavefunction fails to maintain the selected spin and/or spatial symmetry characteristics of the exact wavefunction. Such instabilities arise in SCF-like wavefunctions as a result of a competition between valence-bond-like solutions to the Hartree-Fock equations these solutions typically allow for localization of an unpaired electron onto one of two or more symmetry-equivalent atoms in the molecule. In the ground Ilg state of O2, for example, a pair of symmetry-broken Hartree-Fock wavefunctions may be constructed with the unpaired electron localized onto one oxygen atom or the other. Though symmetry-broken wavefunctions have sometimes been exploited to produce providentially correct results in a few systems, they are often not beneficial or even acceptable, and the question of whether to relax constraints in the presence of an instability was originally described by Lowdin as the symmetry dilemma. ... 

One aspect of the approximations to the density functional theory is that some of the integrals required to solve the above equation can not be evaluated in closed form even for the gaussian-orbital basis functions used in this work. Therefore some numerical work is required to solve the above equations. An extensive literature on our gaussian-orbital-based methods exists and the interested readers are referred to Refs. [48-56] for a more detailed discussion of the numerical methods used here. Here we provide a brief discussion on some of these algorithms. 

The computational approaches to interpreting the wavefunction have split along two basic approaches. The earliest procedures were based on distributing the elearons between the atoms in a molecule based on the occupancy of the atomic orbitals. Thus, the Hilbert space is subdivided on the basis of where the orbitals are centered. We term this approach the orbital-based method, and discuss three major procedures for dissecting the orbitals. 

A number of alternative orbital-based methods have been proposed. We discuss a few of them only briefly because none are currently widely used. The... 

The second term of (11) describes the fictitious kinetic energy of the electrons. The term contains an arbitrary parameter (fictitious mass parameter) with appropriate units of energy times a squared time. The explicit form used in (11) is for orbital based methods. 

The early popularity of valence bond methods declined, because the molecular orbital calculations were more amenable to writing efficient computer algorithms. Recently the programming of valence bond methods has improved because it also benefitted from the rapid development of computer technology and programs have been developed which are competitive in accuracy and economy with programs for the Hartree-Fock method and other molecular orbital-based methods. These developments are due to and have been described by Gerratt, Cooper, Karadakov and Raimondi and summarised by Li and McWeeny in 2002, van Lenthe and co-workers in 2005 and Shaik and Hiberty s reviews in 2008 [155-158]. 

D. R. Garmer and W. J. Stevens,/. Phys. Chem., 93, 8263 (1989). Transferability of Molecular Distributed Polarizabilities from a Simple Localized Orbital Based Method. 

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