Inner-shell orbitals

Most simple empirical or semi-empirical molecular orbital methods, including all of those used in HyperChem, neglect inner shell orbitals and electrons and use a minimal basis set of valence Slater orbitals. 

There exists no uniformity as regards the relation between localized orbitals and canonical orbitals. For example, if one considers an atom with two electrons in a (Is) atomic orbital and two electrons in a (2s) atomic orbital, then one finds that the localized atomic orbitals are rather close to the canonical atomic orbitals, which indicates that the canonical orbitals themselves are already highly, though not maximally, localized.18) (In this case, localization essentially diminishes the (Is) character of the (2s) orbital.) The opposite situation is found, on the other hand, if one considers the two inner shells in a homonuclear diatomic molecule. Here, the canonical orbitals are the molecular orbitals (lo ) and (1 ou), i.e. the bonding and the antibonding combinations of the (Is) orbitals from the two atoms, which are completely delocalized. In contrast, the localization procedure yields two localized orbitals which are essentially the inner shell orbital on the first atom and that on the second atom.19 It is thus apparent that the canonical orbitals may be identical with the localized orbitals, that they may be close to the localized orbitals, that they may be identical with the completely delocalized orbitals, or that they may be intermediate in character. 

For the inner shells the outermost contour is again 0.025 Bohr 3/2. They are much steeper and, therefore, the increment is here 0.2 Bohr-3/2. Three of these inner shell contours are drawn. If the remaining inner shell contours were drawn, the inner part would be solid black. For this reason, the inner shell contours are not drawn beyond the third one and, instead, the value of the inner shell orbital at the position of the nucleus has been written into the diagram. From the figure, it is obvious that the inner shell of lithium is very similar in Li2 and LiH, and in a very practical sense transferable. However, note that the localized inner shell orbital of the lithium atom has a slight negative tail towards the other atom which yields a very small amount of antibinding. 

For the inner shell orbitals, too, one finds near-perfect transferability as was the case for lithium. 

The contribution of inner-shell correlation is taken as the difference between the CCSD(T)/MTsmall TAE with and without constraining the inner-shell orbitals to be doubly occupied. 

In addition, for thermochemical purposes we are primarily interested in the core-valence correlation, since we can reasonably expect the core-core contributions to largely cancel between the molecule and its constituent atoms. (The partitioning between core-core correlation -involving excitations only from inner-shell orbitals - and core-valence correlation - involving simultaneous excitations from valence and inner-shell orbitals - was first proposed by Bauschlicher, Langhoff, and Taylor [42]). 

A tentative explanation for the importance of connected triple excitations for the inner-shell contribution to TAE can be found in the need to account for simultaneously correlating a valence orbital and relaxing an inner-shell orbital, or conversely, requiring a double and a single excitation simultaneously. 

As seen in Table 2.1, Wlc is an acceptable fallback solution for systems for which W1 calculations are not feasible because of the number of inner-shell orbitals for heats of formation and certainly for ionization potentials, Wlch offers a significant further cost reduction over Wlh at a negligible loss in accuracy. 

Fig. 5.2 Emission of photon resulting from a vacancy (light circle) in an inner-shell orbital.

Use the data in Table 5.1 to estimate effective nuclear charges and screening constants for the inner-shell orbitals in Na+ and Cl-. Comment on the trends in screening constants. 

Which properties are least well determined by the variational method The basis functions in the LCAO expansion are either Slater orbitals with an exponential factor e r or gaussians, e ar2 r appears explicitly only as a denominator in the SCF equations thus matrix elements are of the form < fc/r 0i> these have the largest values as r->0. Thus the parts of the wavefunction closest to the nuclei are the best determined, and the largest errors are in the outer regions. This corresponds to the physical observation that the inner-shell orbitals contribute most to the molecular energy. It is unfortunate in this respect that the bonding properties depend on the outer shells. 

In a significant paper, Bauschlicher and Schaefer213 have examined the flexibility of atomic orbitals in a molecular environment, and they have shown in calculations on diatomics involving second-row atoms (among these the 32 state of PH) that only the outermost orbitals are altered during molecular formation, and hence essentially fully contracted GTO can be used for the inner-shell orbitals. We will return to this point later. For PH the contraction procedure that was used recovered 89 % of the energy obtained with an uncontracted basis set. 

The inner shell orbitals denoted by square brackets are too tightly bound to be involved in chemical reactions. The valence and outer electron determine chemical properties. 

The total electron population of each inner shell orbital will of course be two, and if the radial function were independent of the magnetic quantum number, the total contribution of such inner shell electrons to the field gradient would be zero because of the spherical symmetry of the resultant closed shell. We know from Steinheimer s work that this will not be the case and that an inner shell field gradient will indeed be present. 

The second basis set is the split-valence (or extended) 4-31G basis.m In this basis set, inner shell orbitals are written as the sum of four gaus-sian functions while valence orbitals are split into inner and outer parts consisting of three gaussians and one gaussian, respectively. Because the ratio of the inner and outer contributions is free to be determined by the SCF procedure, this basis set provides a more flexible description of the electronic distribution than ST0-3G. It has proved more reliable in energy comparisons than STO-3G. T2-i4) Wg therefore carry out for this purpose, single 4-31G calculations at the ST0-3G optimized geometries for each molecule. 

Obviously, if better flexibility is to be achieved, some decontraction of the valence shell functions must be made. In the popular 4-31G basis set this was achieved by splitting the valence shell orbitals in the ST0-4G set into two parts, the most diffuse primitive GTF being left uncontracted. In the 4-3lG basis set, each inner-shell orbital is represented by a single function containing four GTF s and... 

The deep inner shell orbitals such as Is, 2s and 2p are not very sensitive both to the scaling parameter a and to the oxidation state. On the other hand, the shallow inner shells like 3s and 3p and outer-shell orbitals 3d and 4s strongly depend upon a and the effective charge. Accordingly, the theoretical photoionization cross section computed by equation (12) is affected by the change of spatial extent of the atomic orbital. The theoretical photoionization cross sections for Fe orbitals shown in Table 2 are calculated for the photon energy of hv = 1487eV (A1 Ka) and indicated in Table 3. In the case of a=l. 0, the atomic orbitals are somewhat contracted compared with... 

It is well known that the fine structure of the K absorption edge arises fiom the photoelectric effect of a Is orbital electron of an absorption atom. It is obvious that the peaks in the rising part on XANES spectra are ascribed to the electron transition fiom an inner shell orbital to higher molecular orbitals (called as Is electron transition in this rqrort). Therefore, to estimate the probabilities of the Is electron transition by an electronic dipole theory, theoretical XANES spectrum can be caloilated using molecular orbital calculations. ... 

The other inner-shell orbitals with a low an lar momentum (s or p orbitals) are considered to be relativistically contracted with an order similar to that of the... 

The localization procedure just outlined can also be applied to the Is AO s. Their delocalized description leads naturally to a very slightly binding and antibinding MO owing to the extremely small overlap. Localizing them gives an inner shell orbital centered on each nitrogen. Due to the small overlap, these inner shell orbitals are practically pure one-center Is AO s. The one-center character of these and inner shells in this and other molecules is tacitly assumed in discussions where valence orbitals are used exclusively. 

Fig. 4. Localized (equivalent) molecular orbitals in N2 (left) and CO (right). (After K. Ruedenberg and L. S. Salmon, private communication). From top to bottom these MO s are (1) one of three equivalent components of the triple bond, (2) the left and (3) the 2 right lone pair, (4) the left and (5) the right inner shell orbitals...

The external magnetic field Bq induces currents in bonds, lone pairs and inner shell orbitals of our molecular systems, and these currents are the source of the chemical shifts. Adopting the picture of localized bond orbitals, through-space contributions are caused by charge distributions of bonds that are not directly connected to the nucleus under consideration. Provided that the charge distributions are sufficiently far away so that their electron density in the vicinity of the nucleus can be neglected, this type of interaction can be approximated by the McConnell equation ... 

---