Energy levels of molecular orbitals

Fig. 2.1. Energy levels of molecular orbitals in formaldehyde (HOMO Highest Occupied Molecular Orbitals LUMO Lowest Unoccupied Molecular Orbitals) and possible electronic transitions.

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. 

This is a semiempirical all-valence electron quantum mechanical method, apart from the Tr-approximation and the neglect of overlap integrals, as those of Hiickel molecular orbital (HMO) theory. The method reproduces, relatively well, the shapes and the order of the energy levels of molecular orbitals. To consider the overlapping, it is possible to describe the net destabilization caused by the interaction of the two doubly occupied orbitals, the effect of which is not reproduced by HMO theory. 

Quantum mechanics provides a theoretical basis for understanding the relative energy levels of molecular orbitals and how they vary with structure. Quantum mechanics also generates a set of selection rules to predict what transitions occur in molecules. The transitions that occur in molecules are governed by quantum mechanical selection rules. Some transitions are allowed by the selection rules, while others are forbidden . The selection rules are beyond the scope of this text, but may be found in most physical chemistry texts or in the... 

Fig. 7.28 (Center) The energy level of molecular orbitals formed in the interaction of phenol and three membered ammonia cluster at the geomtry after proton-nucleus is shifted to the ammonia site. (Left) The MO levels for PhO-H at the same geometry but without the ammonia cluster. (Right) Those of the ammonia cluster without phenol. (Reprinted with permission from K. Nagashima et al., J. Phys. Chem. A 116, 11167 (2012)).

FIGURE 11.59 Energy Levels of Molecular Orbitals In Lithium Molecules... 

The determinant of matrix elements whose solution leads to the set of energy levels of molecular orbitals for a given system. 

Active Figure 14.2 Four - molecular orbitals in 1,3-butadiene. Note that the number of nodes between nuclei increases as the energy level of the orbital increases. 

We have already seen examples of semiempirical methods, in Chapter 4 the simple Hiickel method (SHM, Erich Hiickel, ca. 1931) and the extended Hiickel method (EHM, Roald Hoffmann, 1963). These are semiempirical ( semi-experimental ) because they combine physical theory with experiment. Both methods start with the Schrodinger equation (theory) and derive from this a set of secular equations which may be solved for energy levels and molecular orbital coefficients (most efficiently... 

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.

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. 

Fig. 9 presents the general qualitative energy levels and molecular orbital notations for SiH2 (D ), SiH2 (C2 ), SiX2 (C2v) and HSiX (Cs). The notation of Fig. 9 is used throughout. Contributions of X lone pairs have been neglected. 

Figure 2 Energy levels of the calculated occupied (solid line) and unoccupied (broken line) levels of molecular orbitals in 1-D silicon clusters with silicon 3d orbitals. The number of silicon atoms was varied from 2 to 20.

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... 

Figure 10.22 shows the molecular orbital energy level diagram—that is, the relative energy levels of the orbitals produced in the formation of the H2 molecule—and... 

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