Representation of MO

TABLE 2.1. SCF Total Energies (liartrees) of CH4, NH3, H2O, anti HE as a Function of Basis Set  

Hartree-Fock theory is a rigorous ab initio theory of electronic structure and has a vast array of successes to its credit. Equilibrium structures of most molecules are calculated almost to experimental accuracy, and reasonably accurate properties (e.g., dipole moments and IR and Raman intensities) can be calculated from HF wave functions. Rela- 

Another way that additional configurations can be added to the the ground-state wave function is by the use of Moller-Plesset perturbation theory (MPPT). As it happens, a Hamiltonian operator constructed from a sum of Fock operators has as its set of solutions the HF single determinantal wave function and all other determinantal wave 

If carried out with a good basis set [6-31G(d) or better], the benefits of MPPT, carried out to second order (MP2), include moderate improvements in structures and relative energies and often significant improvement in the values of secondary properties such as dipole moments, vibrational frequencies, infrared and Raman absorption intensities, and NMR chemical shifts. Modem quantum chemistry codes such as the GAUSSIAN package incorporate analytical calculation of MP2 forces and force constants. Although it adds substantially to the time required to carry out the calculations, the results of MPPT usually make the extra effort worthwhile. 

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Many other methods, of course, take advantage of invariances with respect to orbital transformations to obtain alternative representations of wavefunctions, such as for example the commonly employed localization procedures for the doubly-occupied MOs from Hartree-Fock calculations [11-15]. We give here a brief account of procedures that particularly seek a valence bond representation of MO wavefunctions. 

Figure 2.5. Three different representations of MO 5 of NH3. (a) one-dimensional plot of orbital value along the symmetry axis. Points at which the value is +0.1 are marked, (b) Three-dimensional surface connecting all points at which the orbital value is +0.1 (unshaded) or -0.1 (shaded), (c) Two-dimensional cross section of (b) in a plane containing the symmetry axis.

Note that Eq. (4.17) does not specify the locations of the basis functions. Our intuition suggests that they should be centered on the atoms of the molecule, but this is certainly not a requirement. If this comment seems odd, it is worth emphasizing at this point that we should not let our chemical intuition limit our mathematical flexibility. As chemists, we choose to use atomic orbitals (AOs) because we anticipate that they will be efficient functions for the representation of MOs. However, as mathematicians, we should immediately stop thinking about our choices as orbitals, and instead consider them only to be functions, so that we avoid being conceptually influenced about how and where to use them. 

The idea to employ a finite basis set of AOs to represent the MOs as linear combinations of the former apparently belongs to Lennard-Jones [68] and had been employed by Hiickel [37] and had been systematically explored by Roothaan [38]. That is why the combination of the Hartree-Fock approximation with the LCAO representation of MOs is called the Hartree-Fock-Roothaan method. 

The existence of a charge density presupposes a concomitant set of MOs and Sect 2.3 describes how chemical reactivity can be based on the notion of Frontier Molecular Orbital (FMO) control i.e. the most important orbital interactions are between the HOMO on one species and the LUMO on the other or vice versa. FMO control, together with electrostatic charge control, provides a powerful qualitative basis for interpreting reactivity. The relative energies and compositions of MOs are vital and many computer programs now provide 3-dimensional representations of MOs to facilitate analysis. 

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