Core orbital

HypcrChcrn sup )orLs MP2 (second order Mollcr-Plessct) correlation cn crgy calculation s tisin g ah initio rn cth ods with an y ava liable basis set. In order lo save mam memory and disk space, the Hyper-Chern MP2 electron correlation calculation normally uses a so called frozen -core" appro.xiniatioii, i.e. the in n er sh ell (core) orbitals are om it ted,. A sett in g in CHKM. INI allows excitation s from the core orbitals lo be included if necessary (melted core). Only the single poin t calcii lation is available for this option. ... 

Ch em uses Slater atom ic orbitals to con struct sent i-em pirical molecular orbitals. I he complete set of Slater atomic orbitals is called the basis set. Core orbitals are assumed to be chemically inactive and arc not treated explicitly. Core orbitals and the atomic nucleus form the atomic core. 

The amount of computation for MP2 is determined by the partial tran si ormatioii of the two-electron integrals, what can be done in a time proportionally to m (m is the u umber of basis functions), which IS comparable to computations involved m one step of(iID (doubly-excitcil eon figuration interaction) calculation. fo save some computer time and space, the core orbitals are frequently omitted from MP calculations. For more details on perturbation theory please see A. S/abo and N. Ostlund, Modem Quantum (. hern-isir > Macmillan, Xew York, 198.5. 

For example, the three NH bonding and three NH antibonding orbitals in NH3, when symmetry adapted within the C3V point group, cluster into ai and e mos as shown in the Figure below. The N-atom localized non-bonding lone pair orbital and the N-atom Is core orbital also belong to ai symmetry. 

The orbitals from which electrons are removed and those into which electrons are excited can be restricted to focus attention on correlations among certain orbitals. For example, if excitations out of core electrons are excluded, one computes a total energy that contains no correlation corrections for these core orbitals. Often it is possible to so limit the nature of the orbital excitations to focus on the energetic quantities of interest (e.g., the CC bond breaking in ethane requires correlation of the acc orbital but the 1 s Carbon core orbitals and the CH bond orbitals may be treated in a non-correlated manner). 

Another family of basis sets, commonly referred to as the Pople basis sets, are indicated by the notation 6—31G. This notation means that each core orbital is described by a single contraction of six GTO primitives and each valence shell orbital is described by two contractions, one with three primitives and the other with one primitive. These basis sets are very popular, particularly for organic molecules. Other Pople basis sets in this set are 3—21G, 4—31G, 4—22G, 6-21G, 6-31IG, and 7-41G. 

The simplest, and perhaps the most important, information derived from photoelectron spectra is the ionization energies for valence and core electrons. Before the development of photoelectron spectroscopy very few of these were known, especially for polyatomic molecules. For core electrons ionization energies were previously unobtainable and illustrate the extent to which core orbitals differ from the pure atomic orbitals pictured in simple valence theory. 

We have seen above how X-ray photons may eject an electron from the core orbitals of an atom, whether it is free or part of a molecule. So far, in all aspects of valence theory of molecules that we have considered, the core electrons have been assumed to be in orbitals which are unchanged from the AOs of the corresponding atoms. XPS demonstrates that this is almost, but not quite, true. 

The XPS chemical shift, for an atomic core orbital with principal and orbital... 

Figure 8.1(c) illustrates schematically the kind of process occurring in Auger electron spectroscopy (AES). The process occurs in two stages. In the first, a high-energy photon ejects an electron from a core orbital of an atom A ... 

In Figure 8.1(c) the higher-energy orbitals are indicated as being valence orbitals but, in most applications of AES, they are core orbitals. For this reason the technique is not usually concerned with atoms in the first row of the periodic table. 

Figure 8.21 shows schematically a set of lx, 2s, 2p and 3s core orbitals of an atom lower down the periodic table. The absorption of an X-ray photon produces a vacancy in, say, the lx orbital to give A and a resulting photoelectron which is of no further interest. The figure then shows that subsequent relaxation of A may be by either of two processes. X-ray fluorescence (XRF) involves an elecfron dropping down from, say, fhe 2p orbifal fo fill fhe lx... 

The lowest energy molecular orbital of singlet methylene looks like a Is atomic orbital on carbon. The electrons occupying this orbital restrict their motion to the immediate region of the carbon nucleus and do not significantly affect bonding. Because of this restriction, and because the orbital s energy is very low (-11 au), this orbital is referred to as a core orbital and its electrons are referred to as core electrons. 

Valence orbital Xij is the lowest energy solution of equation 9.23 only if there are no core orbitals with the same angular momentum quantum number. Equation 9.23 can be solved using standard atomic HF codes. Once these solutions are known, it is possible to construct a valence-only HF-like equation that uses an effective potential to ensure that the valence orbital is the lowest energy solution. The equation is written... 

The chemical bonding occurs between valence orbitals. Doubling the 1 s-functions in for example carbon allows for a better description of the 1 s-electrons. However, the Is-orbital is essentially independent of the chemical environment, being very close to the atomic case. A variation of the DZ type basis only doubles the number of valence orbitals, producing a split valence basis. In actual calculations a doubling of tire core orbitals would rarely be considered, and the term DZ basis is also used for split valence basis sets (or sometimes VDZ, for valence double zeta). 

The next step up in basis set size is a Triple Zeta (TZ). Such a basis contains three times as many functions as tire minimum basis, i.e. six s-functions and three p-functions for the first row elements. Some of the core orbitals may again be saved by only splitting the valence, producing a triple split valence basis set. Again the term TZ is used to cover both cases. The names Quadruple Zeta (QZ) and Quintuple Zeta (5Z, not QZ) for the next levels of basis sets are also used, but large sets are often given explicitly in terms of the number of basis functions of each type. 

IG This is a triple split valence basis, where die core orbitals are a contraction of six PGTOs and the valence split into three functions, represented by three, one, and one PGTOs, respectively. 

NAOs for an atomic block in the density matrix which have occupation numbers very close to 2 (say >1.999) are identified as core orbitals. Their contributions to the density matrix are removed. 

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