Semi-local

Mueller, C. R., and Eyring, H., J. Chan. Phys. 19, 1495, Semi-localized orbitals. I. The hydrogen molecule. Combination of Inui (1941) and Coulson-Fischer (1949). 

Higuchi, J., J. Chem. Phys. 27, 825, (ii) Semi-localized bond orbital treatment of the allyl radical. Extension of VB. 

Kristyan, S., Pulay, P., 1994, Can (Semi)Local Density Functional Theory Account for the London Dispersion Forces , Chem. Phys. Lett., 229, 175. 

However, it is more convenient to determine the NLMOs directly by a numerical procedure56 that incorporates higher perturbation corrections of all orders. As mentioned in Section 3.2.4, the Slater determinant of semi-localized NLMOs... 

The nA— obc and nc— oab delocalizations lead to semi-localized (NLMO) orbitals cuab c and cuA Bc, which can be written as... 

Figure 4.45 A metal-ligand m,—orbital splitting diagram depicting interaction of the metal-atom d NAO and ligand nL NBO to form semi-localized NLMOs of the coordination complex, with splitting energy Aed. = < d/NLMO — fd> (NAO).

The pseudo-potentials used here are of the 1-dependent semi-local type, according to the expression of Barthelat and Durand The single valence electron pseudo-potential for the [Na ] core has been widely used in accurate standard valence calculations The argon atoms are represented via [Ar]... 

II est naturel de se demander si, sous les hypotheses du th orkme, X est (globalement) quasi-projectif sur S. Nous montrons qu il en est bien ainsi si S est localement noeth rien, r gulier, de dimension 1 dans ce cas, tout fais-ceau inversible sur X, qui est ample sur les fibres maximales, est S-ample (chap. VIII). Par contre, il existe un torseur sous un schema ab lien, sur une base semi-locale normale, de dimension 2, qui n est pas projectif (chap. XIII). 

Comme X a un nombre f ini de points, X est mdme slffine et par suite est le spectre d un anneau semi-local. 

Bemarque XI 2.3. II se peut que si S est normal et si G est un S sch ma en groupes lisse sur S, k fibres connexes, alors tout espace homog ne sous 9 qui est trivial (i.e. de la forme G/H, ok H est un sous-sch ma en groupes de G) soit semi-localement quasi-projectif sur S. Par contre, nous verrons au chapitre XIII un exemple d espace principal homogene X sous un schema ab lien G, sur une base affine normale, qui a est pas semi-localement projectif. Pour un tel X, il existe done un ensemble fini de points de X qui n est pas contenu dans un ouvert affine. 

Il existe un morphisme S —— S, entier surjectif dont les fibres n ont qu un nombre fini de points, tel que X. soit semi-localement quasi-projectif sur S. ... 

S, F 1 image reciproque de F dans S qui, avec les hypotheses faites, est un ensemble fini. Comme X est semi—localement quasi-projectif sur S , il existe un ouvert U de S , contenant F , tel que X t soit... 

Remarques XIII 3.1. a) Dans l exemple precedent, puisque X est d ordre infini, X n est pas projectif sur S (XIII 2.3 ii)), done n est pas semi-localement projectif (XI 2.1). II revient- au meme (XI 2.2) de dire qu il n e-xiste pas d ouvert affine de X, qui contient les points maximaux des deux fibres speciales (cf. XI 2.3). 

Lemme XIV 1.6. Soient S un schema, G un S-schema abeiien, P un S-torseur sous G, repr6sentable et semi-localement projectif sur S. Alors P verifie... 

Cela rksulte du fait classique qu un schema lisse et projectif sur un schema affine semi-local, possede une quasi-section finie ktale. 

EGA IV 17.1.1) k fortiori, P est trivial. Nous sommes done ramen s au cas oil X est normal. Par passage k la limite sur les voisinages ouverts contenant P, on se ramene au cas ou X est affine semi-local. Quitte alors a... 

The description of bonding at transition metal surfaces presented here has been based on a combination of detailed experiments and quantitative theoretical treatments. Adsorption of simple molecules on transition metal surfaces has been extremely well characterized experimentally both in terms of geometrical structure, vibrational properties, electronic structure, kinetics, and thermo-chemistry [1-3]. The wealth of high-quality experimental data forms a unique basis for the testing of theoretical methods, and it has become clear that density functional theory calculations, using a semi-local description of exchange and correlation effects, can provide a semi-quantitative description of surface adsorption phenomena [4-6]. Given that the DFT calculations describe reality semi-quantitatively, we can use them as a basis for the analysis of catalytic processes at surfaces. 

In the last three cases above the authors have made the radial form of the potential dependent on the angular part of the wavefunction on which it operates, recognizing that the potential experienced by an electron in, for example, the 3p orbital of chlorine is different from that in the 3s. Such potentials are termed semi-local. This dependence is particularly important when there are valence orbitals in an atom which have angular momenta which are not present in the core, e.g. the 3d orbital of the first row of transition metals. 

The complexity of the functional form for Cc0Te increases from equation (67) to equation (71). To remove the necessity of accepting any particular predetermined functional form Kahn and Goddard29 evaluated a semi-local potential by making use of... 

Ab initio effective hamiltonians (i) Semi-local forms... 

Table 3 A comparison of semi-local and non-local pseudopotential calculations of ionization energies o/Fe (a.u.)...

Each calculation is for a valence state, not a spectroscopic state. The energies are relative to Fe 4s23d9, and are the differences between two separate SCF calculations. 6 Ref. 24. This reference includes comparison of many other excitation energies using AE and semi-local VE calculations. e Refs. 4, 31. 

The semi-local pseudopotential24 used for the calculations of Table 3 was based on a parameterization for the neutral atom. Melius, Olafson, and Goddard also included in their paper some calculations based on the single valence electron ion Fe7+. As expected, this parameterization leads to far worse results, which differ on average from the all-electron results by 0.05 a.u., thus emphasizing the importance of the contribution of valence-valence interaction to the effective potential [equation (51)]. 

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