Going orbital

This new line of thinking about the atom (really dense nucleus with tiny particles moving around it) hit close to home (or rather, light years away from home) for many astronomers. If tiny planets in our solar system orbit around 

Remember that one of the major points of quantum mechanics is that it s nondeterministic, or rather that the future can t be determined from knowledge of the present. In our world, the classical world, when 1 throw a ball at a window, 1 can calculate exactly where it will hit, how hard it will hit, the path it will take to get to the window, and so on. Quantum mechanics is more like flipping a quarter. Sure, the last nine flips may have all landed on heads, but that new flip still has only a 50 percent chance of landing on heads or tails, regardless of what it has previously done. 

While the math underlying quantum mechanics supports the idea that an electron does not necessarily follow an orbit-looking path, there is a certain region in space where you are likely to find one if you went looking for it this region is called an orbital. Here s another way to think about it Although we can t know the exact position of an electron at all times, we can expect it to be somewhere inside our orbital a certain percentage of the time. Occasionally you encounter an orbital described as a cloud of electron density. 

Scientist Erwin Schrodinger developed a series of equations that were critical for developing quantum mechanics, a special type of physics that focuses on really small things (atoms or smaller) and how they can be both particle-like 

The solution to the Schrodinger wave-function equation for a hydrogen atom gives the orbital shape of a sphere that circulates about the nucleus of the atom. 

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Sequences such as the above allow the formulation of rate laws but do not reveal molecular details such as the nature of the transition states involved. Molecular orbital analyses can help, as in Ref. 270 it is expected, for example, that increased strength of the metal—CO bond means decreased C=0 bond strength, which should facilitate process XVIII-55. The complexity of the situation is indicated in Fig. XVIII-24, however, which shows catalytic activity to go through a maximum with increasing heat of chemisorption of CO. Temperature-programmed reaction studies show the presence of more than one kind of site [99,1(K),283], and ESDIAD data show both the location and the orientation of adsorbed CO (on Pt) to vary with coverage [284]. 

In elements of Periods 2 and 3 the four orbitals are of two kinds the first two electrons go into a spherically symmetrical orbital—an s orbital with a shape like that shown in Figure 2.7—and the next six electrons into three p orbitals each of which has a roughly doublepear shape, like those shown unshaded in each half of Figure 2.10. 

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the... 

Orbital energies and sizes go hand-in-hand small tight orbitals have large electron binding energies (i.e., low energies relative to a detached electron). For orbitals on... 

The hi orbitals would maintain their identity going to a" symmetry. Thus Bi and A2 (both A" in Cs symmetry and odd under reflection through the molecular plane) can mix. The system could thus follow the A2 component of the C( P) + H2 surface to the place... 

The advantages of INDO over CNDO involve situations where the spin state and other aspects of electron spin are particularly important. For example, in the diatomic molecule NH, the last two electrons go into a degenerate p-orbital centered solely on the Nitrogen. Two well-defined spectroscopic states, S" and D, result. Since the p-orbital is strictly one-center, CNDO results in these two states having exactly the same energy. The INDO method correctly makes the triplet state lower in energy in association with the exchange interaction included in INDO. 

The simplest example of covalent bonding is the hydrogen molecule. The proximity of the two nuclei creates a new electron orbital, shared by the two atoms, into which the two electrons go (Fig. 4.5). This sharing of electrons leads to a reduction in energy, and a stable bond, as Fig. 4.6 shows. The energy of a covalent bond is well described by the empirical equation... 

Figure 1-18. Instead of replacing the eye-bolls, new studs were welded in place of the threaded portions. They were made brittle by the heal and failed in use. Fortunately the chain prevented the lid from going into orbit.

Compute and examine the orbitals at the RHF/3-21G level in order to select the active space. We will be performing a 4-electron CAS, using 4 and 6 active orbitals. The orbitals we want are those corresponding to the rt system (where the excited electrons go) therefore, the orbitals we want will be pairs of symmetry A2 and Bp Reorder the orbitals so that six appropriate orbitals make up the active space. 

The importance of orbital overlap in determining why certain chemical reactions proceed easily while other similar reactions do not go at all was first advanced by Woodward and Hoffmann, and collectively their ideas are now known as the Woodward-Hoffmann rules. Applications of these ideas can be found in Chapter 21. 

Another useful way to think about carbon electrophilicity is to compare the properties of the carbonyls lowest-unoccupied molecular orbital (LUMO). This is the orbital into which the nucleophile s pair of electrons will go. Examine each compound s LUMO. Which is most localized on the carbonyl group Most delocalized Next, examine the LUMOs while displaying the compounds as space-filling models. This allows you to judge the extent to which the LUMO is actually accessible to an approaching nucleophile. Which LUMO is most available Least available ... 

Display the lowest-unoccupied molecular orbital (LUMO) for equatorial methylcyclohexanone. This is the orbital into which the nucleophile s pair of electrons will go. Is it larger on the axial or equatorial face A clearer picture follows from the LUMO map, which gives the value of the LUMO on the electron density surface, that is, the accessible surface of the molecule. Display the LUMO map for equatorial methylcyclohexanone. Which face of the carbonyl group is more likely to be attacked by a nucleophile Which alcohol will result ... 

The next step on the road to quality is to expand the size of the atomic orbital basis set, and I hinted in Chapters 3 and 4 how we might go about this. To start with, we double the number of basis functions and then optimize their exponents by systematically repeating atomic HF-LCAO calculation. This takes account of the so-called inner and outer regions of the wavefunction, and Clementi puts it nicely. 

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... 

Figure 11.1 shows the bond dissociation curve at the HF level with the STO-3G, 3-21G, 6-31G(d,p), cc-pVDZ and cc-pVQZ basis sets. The total energy drops considerably upon going from the STO-3G to the 3-21G and again to the 6-3IG(d,p) basis. This is primarily due to the improved description of the oxygen Is-orbital. The two different... 

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