Orbital Correction Method

Extension of pseudopotential theory to the transition metals preceded the use of the Orbital Correction Method discussed in Appendix E, but transition-metal pseudopotentials are a special case of it. In this method, the stales are expanded as a linear combination of plane waves (or OPW s) plus a linear combination of atomic d states. If the potential in the metal were the same as in the atom, the atomic d states would be eigenstates in the metal and there would be no matrix elements of the Hamiltonian with other slates. However, the potential ix different by an amount we might write F(r), and there arc, correspondingly, matrix elements (k 1 // 1 r/> = hybridizing the d states with the frce-eleclron states. The full analysis (Harrison, 1969) shows that the correct perturbation differs from (5K by a constant. The hybridization potential is... 

The transition-metal pseudopotentials that were discussed in Chapter 20 are also a special case of the Orbital Correction Method, in which we correct starting OPW states by admixture of other OPW s and also correct starting atomic / states by adding OPW s. 

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. 

Extended Hiickel gives a qualitative view of the valence orbitals. The formulation of extended Hiickel is such that it is only applicable to the valence orbitals. The method reproduces the correct symmetry properties for the valence orbitals. Energetics, such as band gaps, are sometimes reasonable and other times reproduce trends better than absolute values. Extended Hiickel tends to be more useful for examining orbital symmetry and energy than for predicting molecular geometries. It is the method of choice for many band structure calculations due to the very computation-intensive nature of those calculations. 

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. 

Although the Hiickel method has now been supplanted by more complete treatments for theoretical analysis of organic reactions, the pictures of the n orbitals of both linear and cyclic conjugated polyene systems that it provides are correct as to symmetry and the relative energy of the orbitals. In many reactions where the n system is the primary site of reactivity, these orbitals correctly describe the behavior of the systems. For that reason, the reader should develop a familiarity with the qualitative description of the n orbitals of typical linear polyenes and conjugated cyclic hydrocarbons. These orbitals will be the basis for further discussion in Chapters 9 and 11. 

The semiempirical molecular orbital (MO) methods of quantum chemistry [1-12] are widely used in computational studies of large molecules. A number of such methods are available for calculating thermochemical properties of ground state molecules in the gas phase, including MNDO [13], MNDOC [14], MNDO/d [15-18], AMI [19], PM3 [20], SAMI [21,22], OM1 [23], OM2 [24,25] MINDO/3 [26], SINDOl [27,28], and MSINDO [29-31]. MNDO, AMI, and PM3 are widely distributed in a number of software packages, and they are probably the most popular semiempirical methods for thermochemical calculations. We shall therefore concentrate on these methods, but shall also address other NDDO-based approaches with orthogonalization corrections [23-25],... 

On the other hand, high-level computational methods are limited, for obvious reasons, to very simple systems.122 Calculations are likely to have limited accuracy due to basis set effects, relativistic contributions, and spin orbit corrections, especially in the case of tin hydrides, but these concerns can be addressed. Given the computational economy of density functional theories and the excellent behavior of the hybrid-DFT B3LYP123 already demonstrated for calculations of radical energies,124 we anticipate good progress in the theoretical approach. We hope that this collection serves as a reference for computational work that we are certain will be forthcoming. 

The other approach, proposed slightly later by Hund[9] and further developed by Mulliken[10] is usually called the molecular orbital (MO) method. Basically, it views a molecule, particularly a diatomic molecule, in terms of its united atom limit . That is, H2 is a He atom (not a real one with neutrons in the nucleus) in which the two positive charges are moved from coinciding to the correct distance for the molecule. HF could be viewed as a Ne atom with one proton moved from the nucleus out to the molecular distance, etc. As in the VB case, further adjustments and corrections may be applied to improve accuracy. Although the imited atom limit is not often mentioned in work today, its heritage exists in that MOs are universally... 

Of the various methods of approximating the correct molecular orbitals, we shall discuss only one- the linear combination of atomic orbitals (LCAO) method. We assume that we can approximate the correct molecular orbitals by combining the atomic orbitals of the atoms that form the molecule. The rationale is that most of the time the electrons will be nearer and hence controlled by oneor the other of the two nuclei, and when this is so, the molecular orbital should be very nearly the same as the atomic orbital for that atom. The basic process is the same as the one wc employed in constructing hybrid atomic orbitals except that now we are combining orbitals on different atoms to form new orbitals that are associated with the entire molecule. We... 

A DFT-based third order perturbation theory approach includes the FC term by FPT. Based on the perturbed nonrelativistic Kohn-Sham orbitals spin polarized by the FC operator, a sum over states treatment (SOS-DFPT) calculates the spin orbit corrections (35-37). This approach, in contrast to that of Nakatsuji et al., includes both electron correlation and local origins in the calculations of spin orbit effects on chemical shifts. In contrast to these approaches that employed the finite perturbation method the SO corrections to NMR properties can be calculated analytically from... 

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. ... 

In order to make a correct analysis of such an experimental spectrum, an appropriate theoretical calculation is indispensable. For this purpose, some of calculational methods based on the molecular orbital theory and band structure theory have been applied. Usually, the calculation is performed for the ground electronic state. However, such calculation sometimes leads to an incorrect result, because the spectrum corresponds to a transition process among the electronic states, and inevitably involves the effects due to the electronic excitation and creation of electronic hole at the core or/and valence levels. Discrete variational(DV) Xa molecular orbital (MO) method which utilizes flexible numerical atomic orbitals for the basis functions has several advantages to simulate the electronic transition processes. In the present paper, some details of the computational procedure of the self-consistent-field (SCF) DV-Xa method is firstly described. Applications of the DV-Xa method to the theoretical analysises of XPS, XES, XANES and ELNES spectra are... 

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