Orbitals construction

Use the four-step procedure described earlier. Begin with the Lewis structure for the molecule, and then identify the appropriate hybrid orbitals. Construct a a bond framework, and complete the bonding picture by assembling the bonds from the unhybridized p orbitals. 

These compounds are often referred to as hypervalent. The apical bonds in a trigonal bipyramid are described by molecular orbitals constructed from a p-orbital on the central atom and o-bonding orbitals (p- or sp hybrid) on the apical ligands. The molecular orbitals can be drawn as ... 

If an excited state is concerned, this is done under the restriction that the function should be orthogonal to all of the lower-energy states. We may specify these as the uni-configurational Hartree-Fock wave functions . The "best orbitals constructing the determinants in these wave functions are in general not orthogonal to each other. 

Figure A.14 Energy diagram for the adsorption of an atom on a d-metal. Chemisorption is described with molecular orbitals constructed from the d-band of the metal and atomic orbitals of the adatom. The chemisorption bond in b) is weaker, because the antibonding chemisorption orbital is partially filled (compare Fig. A.5).

The Tt orbitals constructed through combination of individual C - H bonds can interact with the unpaired spin in much the same way as seen before for... 

Evaluation of these integrals and expansion of the above determinant leads to a polynomial of degree N in , the roots of which give the energies of the molecular orbitals constructed in the above manner. The coefficients Cfj may be determined by substitution of the particular E into the simultaneous linear equations covered by the above determinant together with the normalization equation... 

Therefore, in order for boron to form a 77-bond perpendicular to the molecular plane to each of the chlorine atoms, it must use three hybrid orbitals constructed from one AO which belongs to V and a pair of AOs which belong to r Table 11-2.2 shows that only the following orbitals meet these requirements ... 

It is instructive to examine some diagrams showing how the signs of the w s vary around the ring. For benzene, using the orbitals constructed above, we can make the following drawings ... 

Symbols A and B cepresem otoms j, y, anti z represent orkntaoori ef lhe p crtilals. Note that the "atomic orbital labels (e.g.. Is) can apply only to molecular orbitals constructed by mi King two orbitals of lhe same type. We shall see later (page (64) that this is an overamplificaiion. 

Another array of orbitals with lhe same sort of linear combinations is that of the rr system of cydopolyencs (benzene, naphthalene) with which you are probably already Familiar The molecular Orbitals constructed from sets of parallel p orbitals are both bonding and antibonding at various energies. The system b particularly stable if Huckd s rule 2m + 2 electrons, see also Chapter 15) e obeyed In the case of an mb rate array of hythogen atoms (Is1), the situation is unstable (as you should have questioned immcdiniclvl it reverts to an array of H2 mdeoics ... 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

There are two parameters in the atomic coulomb functions, the effective nuclear charge and the quantum defect. The values of these were taken by Johnson and Rice from available spectral data. The effective atomic charge was adjusted to give the correct ionization potential of the molecule, 9.25 eV, requiring thereby z = 0.8243. The quantum defects of carbon were taken from the appropriate atomic series and were 1.04 for the 5-state and 0.73 for the p-states. It is interesting to compare the calculated molecular quantum defects (i.e., those corresponding to the Johnson-Rice LCAO function) with those which can be obtained from the various benzene Rydberg series.218 The asymptotic form of the elu orbital constructed from s atomic functions is... 

This function is zero for all points equidistant front the two nuclei (that is, it has a nodal plane). The orbital y>j is large in the region between the nuclei (where lsA and 1% overlap and the electrostatic potential is low) and is generally referred to as a bonding orbital. Similarly, y>t, which keeps its electron away from the internuclear region, is antibonding. The two functions and yj2 are analogous to the symmetric and antisymmetric orbitals for the one-dimensional model. A similar transformation can be applied and two equivalent orbitals constructed. These are... 

Ground state 02 and the chosen 3drr Rydberg level have the same 3Zg" symmetry but differ in their molecular orbital construction ... 

It is evident from a comparison of 5 and 4 that the reflection transforms >pi into i/j2. The reader should verify that the a reflection and the C2 rotation also transform symmetry correct. The result we have found will generally hold when molecular orbitals constructed by the LCAO method from hybrid atomic orbitals are subjected to symmetry operations. Each of those orbitals in the set of MO s that is not already symmetry correct will be transformed by a symmetry operation into another orbital of the set. 

FIGURE 8.2 Both Is and 2p orbitals give symmetric probability distributions. Combining the two into a superposition state breaks this symmetry, because the 2p orbital constructively interferes with Is in one lobe and destructively interferes in the other. 

Orbital construction and diagonalization of the Cl matrix are alternated until the calculations converge. 

Then the same procedure as in the construction of the natural orbital basis for the HF calculations is used, and the truncated set of natural orbitals that is obtained is used to expand the transition density matrix, and hence the index-pair (k, /) is contracted. In practice this means that instead of making one large transformation with the transition density matrix in Eq. (9-19), two smaller ones are done first one from the ACBS to truncated orbital basis, then to state basis. For reasonable thresholds in the orbital construction, td, this will lead to a speed-up of the QMSTAT calculations. Details of the two procedures described above can be found in reference [139],... 

Many of the principles and techniques for calculations on atoms, described in section 6.2 of this chapter, can be applied to molecules. In atoms the electronic wave function was written as a determinant of one-electron atomic orbitals which contain the electrons these atomic orbitals could be represented by a range of different analytical expressions. We showed how the Hartree-Fock self-consistent-field methods could be applied to calculate the single determinantal best energy, and how configuration interaction calculations of the mixing of different determinantal wave functions could be performed to calculate the correlation energy. We will now see that these technques can be applied to the calculation of molecular wave functions, the atomic orbitals of section 6.2 being replaced by one-electron molecular orbitals, constructed as linear combinations of atomic orbitals (l.c.a.o. method). 

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