Beyond the Orbital Approximation

TABLE 7-3 Energies and Dipole Moments for Hydrogen Fluoride Calculated by the Variation Method Using Different Basis Sets  

These problems with basis set adequacy are difficult to overcome conpletely. Fortunately, with a certain amount of experience, insight, and caution, it is nevertheless possible to carry out variational calculations and interpret their results to obtain reliable and useful information. 

Most of our discussion of the variation method has been restricted to calculations within the orbital approximation. To avoid leaving an inaccurate impression of the capabilities of the variation method, we shall briefly describe some calculations on some small (two-electron) systems where the method can be employed to its fullest capabilities. These calculations are listed in Table 7-4. 

The calculation on He by Kinoshita expresses the spatial part of the wavefunction as 

TABLE 7-4 Results of Some Very Accurate Variational Calculations on Two-Electron Systems 

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Whereas the one-electron exponential form Eq. (5.5) is easily implemented for orbital-based wavefunctions, the explicit inclusion in the wavefunction of the interelectronic distance Eq. (5.6) goes beyond the orbital approximation (the determinant expansion) of standard quantum chemistry since ri2 does not factorize into one-electron functions. Still, the inclusion of a term in the wavefunction containing ri2 linearly has a dramatic impact on the ability of the wavefunction to model the electronic structure as two electrons approach each other closely. 

Of course, there are methods used in ab initio work that even go beyond the orbital approximation altogether, but this is discussed further below. 

To improve upon die mean-field picture of electronic structure, one must move beyond the singleconfiguration approximation. It is essential to do so to achieve higher accuracy, but it is also important to do so to achieve a conceptually correct view of the chemical electronic structure. Although the picture of configurations in which A electrons occupy A spin orbitals may be familiar and usefiil for systematizing the electronic states of atoms and molecules, these constructs are approximations to the true states of the system. They were introduced when the mean-field approximation was made, and neither orbitals nor configurations can be claimed to describe the proper eigenstates T, . It is thus inconsistent to insist that the carbon atom... 

The necessity for going beyond the HF approximation is the fact that electrons are further apart than described by the product of their orbital densities, i.e. their motions are con-elated. This arises from the electron-electron repulsion operator, which is a sum of ten-ns of the type... 

The set of atomic orbitals Xk is called a basis set, and the quality of the basis set will usually dictate the accuracy of the calculations. For example, the interaction energy between an active site and an adsorbate molecule might be seriously overestimated because of excessive basis set superposition error (BSSE) if the number of atomic orbitals taken in Eq. [4] is too small. Note that Hartree-Fock theory does not describe correlated electron motion. Models that go beyond the FiF approximation and take electron correlation into account are termed post-Flartree-Fock models. Extensive reviews of post-HF models based on configurational interaction (Cl) theory, Moller-Plesset (MP) perturbation theory, and coupled-cluster theory can be found in other chapters of this series. ... 

Because of its severe approximations, in using the Huckel method (1932) one ignores most of the real problems of molecular orbital theory. This is not because Huckel, a first-rate mathematician, did not see them clearly they were simply beyond the power of primitive mechanical calculators of his day. Huckel theory provided the foundation and stimulus for a generation s research, most notably in organic chemistry. Then, about 1960, digital computers became widely available to the scientific community. 

In addition, if one goes beyond the Hartree-Fock approximation to something like the configuration interaction approach there is an important sense in which one has gone beyond the picture of a certain number of electrons into a set of orbitals.10 If one insists on picturing this, then rather than just every electron being in eveiy possible orbital... 

After planetary accretion was complete there remained two groups of surviving planetesimals, the comets and asteroids. These populations still exist and play an important role in the Earth s history. Asteroids from the belt between Mars and Jupiter and comets from reservoirs beyond the outer planets are stochastically perturbed into Earth-crossing orbits and they have collided with Earth throughout its entire history. The impact rate for 1 km diameter bodies is approximately three per million years and impacts of 10 km size bodies occur on a... 

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