Basis orbitals

E.( ), possesses a complete set of eigenfiinctions, the matrix F whose dimension Mis equal to the number of atomic basis orbitals, has M eigenvalues e. and M eigenvectors whose elements are the. . Thus, there are... 

The basis orbitals coimnonly used in the LCAO-MO process fall into two primary classes ... 

For both types of orbitals, the coordinates r, 0 and cji refer to the position of the electron relative to a set of axes attached to the centre on which the basis orbital is located. Although STOs have the proper cusp behaviour near the nuclei, they are used primarily for atomic- and linear-molecule calculations because the multi-centre integrals which arise in polyatomic-molecule calculations caimot efficiently be perfonned when STOs are employed. In contrast, such integrals can routinely be done when GTOs are used. This fiindamental advantage of GTOs has led to the dominance of these fimetions in molecular quantum chemistry. 

Much effort has been devoted to developing sets of STO or GTO basis orbitals for main-group elements and the lighter transition metals. This ongoing effort is aimed at providing standard basis set libraries which ... 

Jordan oompared the use of plane wave and oonventional Gaussian basis orbitals within density funotional oaloulations in ... 

Because symmetry operators eommute with the eleetronie Hamiltonian, the wavefunetions that are eigenstates of H ean be labeled by the symmetry of the point group of the moleeule (i.e., those operators that leave H invariant). It is for this reason that one eonstruets symmetry-adapted atomie basis orbitals to use in forming moleeular orbitals. 

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution the Fj y matrix elements depend on the Cy i LCAO-MO eoeffieients whieh are, in turn, solutions of the so-ealled Roothaan matrix Hartree-Foek equations- Zy Fj y Cy j = 8i Zy Sj y Cy j. One should also note that, just as F ( )i = 8i (l)j possesses a eomplete set of eigenfunetions, the matrix Fj y, whose dimension M is equal to the number of atomie basis orbitals used in the LCAO-MO expansion, has M eigenvalues 8i and M eigenveetors whose elements are the Cy j. Thus, there are oeeupied and virtual moleeular orbitals (mos) eaeh of whieh is deseribed in the LCAO-MO form with Cy j eoeffieients obtained via solution of... 

The basis orbitals eommonly used in the LCAO-MO-SCF proeess fall into two... 

However, beeause the atomie basis orbitals are attaehed to the eenters, and beeause these eenters are displaeed in forming V, it is no longer true that = 0 the variation in... 

Each of these factors can be viewed as combinations of CSFs with the same Cj and Cyj coefficients as in F but with the spin-orbital involving basis functions that have been differentiated with respect to displacement of center-a. It turns out that such derivatives of Gaussian basis orbitals can be carried out analytically (giving rise to new Gaussians with one higher and one lower 1-quantum number). 

The total molecular energy is invariant to all transformations involving basis orbitals, just as any physical event is invariant under any transformation of coordinates. But just as the proper choice of coordinates helps in visualizing physical events, so the choice of the proper orbital basis is helpful in visualizing molecular properties. 

The three highest occupied orbitals of sulfoxides are the lone pairs ns and n0, as well as the 7iso bond210. The 1,3-dithietane 1-oxide adds a lone-pair ionization and destabilizes the n0 and nso radical-cation states compared with thietane oxide. According to a hyperconjugative MO model, the ns+ combination in 1,3-dithietane is destabilized by about leV relative to the basis orbital energy a(ns) due to the combination with the... 

Basicities 565-567 Basis orbitals 2-5 Benzisothiazolinone dioxides 163 Benzofurans 638 Benzoic acids, pKa values of 586 Benzoxathiazine 2-oxides 71 Benzyne intermediates 306 Beta scales 559... 

Figure 1.5. Localized molecular orbitals formed from the atomic basis orbitals and electronic transitions for the carbonyl group.

The Hiickel energy levels and p basis orbital coefficients for formaldehyde and ethylene are shown in Figure 4.9. If we examine the orbital interactions for case 1, we can see that in the orientation such that the lone pair on oxygen is oriented in a way that a bond can begin to form between the oxygen lone pair and the carbon ir orbital, there are two possible orbital interactions. 

In most of the acyclic examples of the di-7r-methane rearrangement studied, there has been methyl substitution on the central (C-3) atom. We should expect that electron-withdrawing substituents (relative to —CH3) on C-3 would lower the energy of the basis orbitals on C-3 and slow the transformation shown in the diagrams for schemes (3) and (4). 

The determinantal wave function in Eq. (21) is built [23] from complex dynamical spin orbitals Even when the basis orbitals ut in Eq. (22) are orthogonal these dynamical orbitals are nonorthogonal, and for a basis of nonorthogonal atomic orbitals based on Gaussians as those in Eq. (24) the metric of the basis becomes involved in all formulas and the END theory as implemented in the ENDyne code works directly in the atomic basis without invoking transformations to system orbitals. 

This shortcoming could be avoided and the treatment improved by assigning different basis energies Aex(, and Aen,i to exocyclic and endocyclic basis orbitals jr/L and by adjusting the cross terms, but this is not really worthwhile. (For a more detailed discussion of the model see elsewhere173.)... 

The parameters are defined in equations 45-48. The parameters A, B and T are negative quantities if the orientation of the basis orbitals is defined as shown in diagram 43, and the same is true for AA if we assume that the energy of the cr-orbital lies below that of the two jr-orbitals as indicated in presentation 44. The orbital energies j are obtained by solving the secular determinant given by equation 49, which yields the solutions given in equations 50. 

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