ENERGY LEVELS OF ATOMIC ORBITALS

The value of the principal quantum number n of an orbital is a measure of the relative distance of the maximum electron density from the nucleus. An electron with a lower value of n, being on the average closer to the nucleus, is more strongly attracted to the nucleus than an electron with a higher value of n, which is at a relatively greater distance from the nucleus. 

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Figure 10.5 Energy levels of atomic orbitals, n is the principal quantum number, and the 5, p, d notation indicates the azimuthal quantum number (/). For / = 1 and above the orbital is split into multiple suborbitals (indicated by the number of lines), corresponding to the values of the magnetic quantum number m Each of these lines can hold two electrons (corresponding to spin up and spin down ), giving rise to the rules for filling up the orbitals.

According to quantum mechanics, electrons in atoms occupy the allowed energy levels of atomic orbitals that are described by four quantum numbers the principal, the azimuthal, the magnetic, and the spin quantum numbers. The orbitals are usually expressed by the principal quantum numbers 1, 2, 3, —increasing from the lowest level, and the azimuthal quantum numbers conventionally eiqiressed by s (sharp), p (principal), d (diffuse), f (fundamental), — in order. For instance, the atom of oxygen with 8 electrons is described by (Is) (2s) (2p), where the superscript indicates the munber of electrons occupying the orbitals, as shown in Fig. 2-1. 

This energy-level diagram shows the relative energy levels of atomic orbitals in a multielectron atom (in this case rubidium, Rb, atomic number 37). 

Figure 2.5 Schematic molecular orbital energy level diagram for a transition metal coordination cluster, [ML6]. (a) Energy levels of atomic orbitals of the free cation, M (b) energy levels for the six ligands, L, before bonding (c) molecular orbital energy levels for the octahedral [ML6] cluster.

Figure 2.4 Filling in the chart determines the energy levels of atomic orbitals.

Figure 3.13 Energy levels of atomic orbitals. Each orbital capable of containing 2 electrons is shown with a dash, —. The order of placing electrons in orbitals is from the lowest-lying orbitals up, as shown in the inset.

Jf(t) which we shall use in this chapter. We must mention, however, that somewhat more complicated forms have been introduced in order to consider some special aspects of the problem. Some of the most notable include that used by Bloss and Hone (which has a core level in the solid and two orbitals on the atom), those which take into account the scattering of atoms with large velocity components parallel to the surface and that recently considered by Kawai et in order to suggest a possible new experimental technique for studying energy levels of atoms in collision with surfaces. 

In contrast to channel I, which remains non-degenerate in the field, channel II splits into three dissociation branches due to the Stark effect. Atomic calculations of the H atom in a cylindrical potential oriented along the z-axis show that the energy levels of 2p-orbitals split and H(2pj.) becomes more stable relative to H(2p c) and H(2py). Because of the lateral symmetry of the potential, the degeneracy of H(2pJ and H(2py) persists. For small values of w, H(2s) is slightly less stable than H(2p ) and H(2py) while the ordering reverses when w exceeds 0.15 a.u. As a result, there exist three dissociation limits for channel II, with a nonzero cylindrical potential, which correspond to H(2s), H(2pj,), and H(2p )/H(2py). 

Which two of the four quantum numbers determine the energy level of an orbital in a multielectron atom ... 

The special stability of benzene (aromaticity) comes from the six tc electrons in three molecular orbitals made up by the overlap of the six atomic p orbitals on the carbon atoms. The energy levels of these orbitals are arranged so that there is exceptional stability in the molecule (a notional 140 kjmor1 over a molecule with three conjugated double bonds), and the shift of the six identical hydrogen atoms in the NMR spectrum (5h 7.2 p.p.m.) is evidence of a ring current in the delocalized tc system. 

Knowing these three rules—the Aufbau principle, the Pauli exclusion principle, and Hund s rule—and the energy levels of the orbitals shown in Figure 2.4, it is possible to predict the electron configurations of most atoms. And these configurations are one of the keys needed to unlock the secrets of the chemical bond. 

The electronic structure of microcrystalline silicon of one-dimensional (1-D), 2-D, and 3-D clusters were calculated using the Discrete-Variational (DV)-Xa Molecular-Orbital method. The calculated results are discussed with respect to the effect of the size and the number of dimensions on the energy levels of molecular orbitals. The energy-gap (Eg) between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO) decreases with the increase of cluster size amd the number of dimensions. It is found that including silicon 3d orbitals as basis sets decreases the Eg value. The results show that the components of silicon 3d orbitals in the unoccupied levels near LUMO are over 50 per cent. The calculated results predict that the Eg value will be close to the band gap of crystalline silicon when a 3-D cluster contadns more than 1000 silicon atoms with a diameter of 4nm. 

The ions and molecules NO, CN, CO, and N2 form an isoelectronic series. The changing nuclear charges will also change the molecular energy levels of the orbitals formed from the 2p atomic orbitals (Irr, 3ct, and Irr ). Use molecular modehng software to... 

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