Separated-atom orbitals

Figure 1.10 Energy diagram for the hydrogen molecule. Combination of two atomic orbitals, is, gives two molecular orbitals, 0iec and iF moiec- The energy of Iffnoiec is lower than that of the separate atomic orbitals, and in the lowest electronic state of molecular hydrogen it contains both electrons.

It means, for example, that atomic data can only rarely be used as a substitute for molecular integrals since the atom-in-molecule orbitals are not the same as the separate atom orbitals — worse, they are no longer equivalent among themselves. An atomic self-repulsion integral (0j0, 0j0j) is different if 0j is the lone-pair hybrid of NH3 or the bond-pair hybrid as the Gillespie-Nyholm rules suggest. 

Molecular-orbital theory treats molecule formation from the separated atoms as arising from the interaction of the separate atomic orbitals to form new orbitals (molecular orbitals) which embrace the complete framework of the molecule. The ground state of the molecule is then one in which the electrons are assigned to the orbitals of lowest energy and are subject to the Pauli exclusion principle. Excited states are obtained by promoting an electron from a filled molecular orbital to an orbital which is normally empty in the ground state. The form of the molecular orbitals depends upon our model of molecule formation, but we shall describe (and use in detail in Sec. IV) only the most common, viz., the linear combination of atomic orbitals approximation. 

In any molecule, the potential close to a nucleus approximates that of the free atom (in the correct valence state) consequently, in these regions the molecular orbitals lie near the free atomic orbitals. Accordingly, it is generally assumed that the molecular orbitals may be expressed as linear combinations of the separate atomic orbitals, the LCAO approximation. The total electronic energy of the molecule may then be minimized with respect to the coefficients in this approximation and the form of the molecular orbitals thus determined. If the molecular orbitals 0, are expressed in terms of the set of atomic orbitals < b. .., N as... 

In Figures 11 and 12, are depicted some examples of molecular orbital formation from separate atomic orbitals. The illustrations are of surfaces like those of the atomic orbitals we drew in chapter 3 they are of greater physical significance than the actual orbitals themselves. Again we will stress the point that the boundary surfaces are functions of whereas the... 

The resulting probability densities are shown in Figure 8.4. The first two terms in equation (8.13) correspond to the electron density of separate atomic orbitals centred on nuclei A and B. The third term is a measure of the excess, or deficit, electron density resulting from the formation of the combined orbital. It will be important only in the region between the nuclei, where V b significant values. This term is pos-... 

The energy of electrons in a bonding molecular orbital is less than the energy of the electrons in their separate atomic orbitals. The energy of electrons in an antibonding orbital is greater than that of electrons in their separate atomic orbitals. 

Separated atoms Orbitals touch Orbitals overlap a ... 

Figure 10.31 shows the energies of the molecular orbitals cti and a relative to the atomic orbitals. Energies of the separate atomic orbitals are represented by heavy... 

In a sense, this is an application of our variational principle from Chapter 4 the lowest energy state of a particular symmetry can be found by the variational method. Say that we start with a set of separated atomic orbitals that has tTu symmetry. Then, as we change the distance between the atoms and try to adjust that wavefunction to minimize the energy, the variational method will still only give us a function of tTu symmetry, as long as we don t change the point group of the molecule. 

Figure 7-17 Correlation diagram between separated-atom orbitals and united-atom orbitals for homonuclear diatomic molecules. Energy ordinate and intemuclear separation abscissa are only suggestive. No absolute values are implied by the sketch. [Note For hJ rjg2p correlates with SdcTg, cTgSs with 3soTg. This arises because the separability of the hJ hamiltonian (in the Born-Oppenheimer approximation) leads to an additional quantum number for this molecule. In essence, the hJ wavefunction in elliptical coordinates may be written jr = L X)M (/x), The function L...

One question we can ask is this Is a minimal basis set equally appropriate for calculating an MO wavefunction for, say, B2 as F2 In each case we use 10 AOs and 2 spin functions producing a total of 20 spin MOs. With B2, however, we have 10 electrons to go into these spin MOs, and in F2 we have 18 electrons. In all but the crudest MO calculations, the total energy is minimized in a maimer that depends on the natures of only the occupied MOs. In effect, then, the calculation for B2 produces the 10 best spin MOs from a basis set of 20 spin-AOs, whereas that for F2 produces the 18 best MOs from a different basis set of 20 spin-AOs. In a sense, then, the basis for F2 is less flexible than that for B2. Of course, the use of separated atom orbitals is a conscious effort to choose that basis that best spans the same function space as the best MOs. To the extent that this strategy is successful, the above problem is obviated (i.e., if both sets are perfect, additional flexibility is useless). The strategy is not completely successful, however, and conparison of results of minimal basis set calculations down a series of molecules such as B2, C2, N2,02, and F2 may be partially hampered by this... 

Ftg. 18.4. Plots of electron charge density for an antibonding orbital solid black line) and lor separated atomic orbitals dashed line). 

Hybridization of a carbon atom means that the four electrons in the outermost energy level become unpaired, each in a separate atomic orbital. This results in carbon s ability to form four bonds with other atoms, including other carbon atoms, and creates the possibility of chains and networks of carbon atoms, which in turn is responsible for the huge number of organic compounds that exist. 

The identities of Eqs. (33) and (34) provide a more complete statement that, except for the presence of a multiplicative constant, the wave-function for two bonding electrons + one antibonding electron is equivalent to the wave-function for two electrons occnpying separate atomic orbitals with parallel spins + one electron occnpying a bonding molecnlar orbital with opposite spin. We shall make nse of this resnlt on nnmerons occasions. 

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