Atomic orbitals approximate relative energies

FIGURE 4.5 Approximate relative energies of the subshells in the many-electron atom, reflecting the shifts due to nuclear shielding. The Is and 2s orbital energies are too low to appear in this plot. Actual relative energies will vary with number and excitation of electrons. 

Approximate relative energies of the atomic orbitals up to the 5s level are shown in Figure 1-8. With its help, we can give an electronic configuration to every atom in the periodic table. To do so, we follow three rules for assigning electrons to atomic orbitals ... 

Figure 1-8 Approximate relative energies of atomic orbitals, corresponding roughly to the order in which they are filled in atoms. Orbitals of lowest energy are filled first degenerate orbitals are filled according to Hund s rule.

The Schrodinger equation can be solved approximately for atoms with two or more electrons. There are many solutions for the wave function, ij/, each associated with a set of numbers called quantum numbers. Three such numbers are given the symbols n, , and mi. A wave function corresponding to a particular set of three quantum numbers (e.g., n = 2, = 1, mi = 0) is associated with an electron occupying an atomic orbital. From the expression for ij/y we can deduce the relative energy of that orbital, its shape, and its orientation in space. 

In case of multielectron atoms the energies of various orbitals depend not only upon the nuclear charge but also upon the other electrons present in the atom. It is impossible to calculate the exact energies of various orbitals in a multielectron atom. However, approximate values of their energies can be obtained from the special data. Relative order of energies of various orbitals in all multielectron atoms is same and is illustrated in following figure. 

Molecular orbitals are characterized by energies and amplitudes expressing the distribution of electron density over the nuclear framework (1-3). In the linear combination of atomic orbital (LCAO) approximation, the latter are expressed in terms of AO coefficients which in turn can be processed using the Mulliken approach into atomic and overlap populations. These in turn are related to relative charge distribution and atom-atom bonding interactions. Although in principle all occupied MOs are required to describe an observable molecular property, in fact certain aspects of structure and reactivity correlate rather well with the nature of selected filled and unfilled MOs. In particular, the properties of the highest occupied MO (HOMO) and lowest unoccupied MO (LUMO) permit the rationalization of trends in structural and reaction properties (28). A qualitative predictor of stability or, alternatively, a predictor of electron... 

Having recognized that the muffin-tin approximation, widely used in molecular calculations, is rather severe, they use the overlapping atomic spheres concept (60). This concept has been considered to lead to an improved description of ionisation potentials of molecules where a substantial fraction of the charge due to the valence electrons is distributed over the interatomic region of constant potential (60). The SCF Xa calculations on Nis—CO and Ni4—CO clusters yield three main peaks, one due to the 5-band and two which can be related to the n and 5relative energies of the peaks agree well with the experimental values. In contradiction to the initial tentative assumption of Eastman and Cushion and to the CNDO results of Blyholder, the first peak of the adsorbed CO is found to be due to the 1 n orbital and the second due to the 5a orbital. 

Since accurate calculations are not usually feasible, it is necessary to find means of securing fair approximations to molecular orbitals and of estimating their relative energies. The simplest way is to build them up out of atomic orbitals, taking... 

The essential difference between the nature of the m.o.s in BeH2 and in CO2 is that the peripheral atoms in carbon dioxide (O atoms) contribute with four valence atomic orbitals and not just one as in BeH2. However, the 2s a.o. of the oxygen atom is of sufficiently low energy relative to the 2p a.o.s (as well as relative to the carbon valence a.o.s) in order to be considered - to an acceptable approximation for many purposes - as a core orbital (besides the Is orbitals of C and O). We thus have three (from O) -I- four (from C) +three (from 0)= 10 a.o.s to define 10 valence m.o.s. Orthogonality relations associated with the symmetry of the various orbitals enable those 10 a.o.s to be divided into three sets as shown in Fig. 9.1. 

The presence of these three electronic states, as well as the sizable spin-orbit (SO) splitting in the F and Cl atoms (1.15 kcaEmol for F and 2.52 kcaEmol for Cl [25]) raises two important questions (1) what is the reactivity of the excited spin-orbit state relative to that of the ground state and (2) how well is the dynamics of the reaction described by calculations on a single, electronically adiabatic potential energy surface (PES). If the reaction were to proceed adiabatically on a single PES, as would be predicted by the Bom-Oppenheimer (BO) approximation, then the excited SO state should not react.[26, 27]... 

FIGURE 6.9 Correlation diagram for H J in the linear combination of atomic orbitals (LCAO) approximation. The bonding orbital is stabilized relative to the noninteracting system by the energy difference Af. 

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