Hydrogen 2s orbital

In this table, eis is the energy of a hydrogenic Is orbital, fi2s the energy of a hydrogenic 2s orbital. Before we worry about comparison with experiment, there are a couple of loose ends that have to be tidied up. 

Figure 1.21 Attempt to restore a match to the hydrogen 2s orbital using the Slater and sto-3g) approximations of Figure 1.20. Only the general feature of the valence region behaviour of the exact hydrogen function is reproduced.

Figure 4.21a The results for the assumption that the Dunning split can be scaled to return a good representation of the hydrogen 2s orbital.

The wavefunction described with the optimum a,y corresponds well with a somewhat perturbed lslj 2s configuration. Two of the electrons depend on the electron-nuclear distance with a values (screening parameters) that correspond to a partially screened interaction with the - -3-charged Li nucleus in a split-shell electron distribution. The third electron (that with pre-multiplying r) has an a value somewhat larger than for a hydrogenic 2s orbital, indicative of the fact that the inner-shell electrons do not completely shield the Li nucleus. The electron-electron a values all reflect the existence of electron-electron repulsion, with the effect most pronounced for the Is-ls interaction. All these observations are consistent with the notion that the exponentially correlated wavefunction gives an excellent zero-order description of the electronic structure of Li. 

At what distance from the nucleus is an electron in a hydrogen 2s orbital most likely to be found Hint there are two maxima in... 

FIGURE 1.8 Probability of the electron position for the hydrogen 2s orbital. (Thanks to Prof Peter Knowles for this figure.)... 

Sketch the probability of finding an electron in the 2s orbital of hydrogen at distance r from a hydrogen nucleus as a function of r as a contour map with heavy lines at high probability and light lines at low probability. How does this distribution differ from the Is orbital ... 

Section 2 6 Bonding m methane is most often described by an orbital hybridization model which is a modified form of valence bond theory Four equiva lent sp hybrid orbitals of carbon are generated by mixing the 2s 2p 2py and 2p orbitals Overlap of each half filled sp hybrid orbital with a half filled hydrogen Is orbital gives a ct bond... 

Why stop with a single hydrogenic Is orbital on either centre A little thought shows that the 2s and 2po- orbitals might also make a contribution to the bonding, and so we could write... 

And what if the basis functions are centred on different atoms The CNDO solution to the problem is to take all possible integrals such as those above to be equal, and to assume that they depend only on the atoms A and B on which the basis functions are centred. This satisfies the rotational invariance requirement. In CNDO theory, we write the two-electron integrals as pab and they are taken to have the same value irrespective of the basis functions on atom A and/or atom B. They are usually calculated exactly, but assuming that the orbital in question is a Is orbital (for hydrogen) or a 2s orbital (for a first row atom). 

The number of electrons in an atom affects the properties of the atom. The hydrogen atom, with one electron, has no electron-electron repulsions therefore, all the orbitals of a given shell in the hydrogen atom are degenerate. For instance, the 2s-orbital and all three 2p-orbitaIs have the same energy. In many-electron atoms, however, the results of spectroscopic experiments and calculations show... 

At this stage, it looks as though electron promotion should result in two different types of bonds in methane, one bond from the overlap of a hydrogen ls-orbital and a carbon 2s-orbital, and three more bonds from the overlap of hydrogen Is-orbitals with each of the three carbon 2/ -orbitals. The overlap with the 2p-orbitals should result in three cr-bonds at 90° to one another. However, this arrangement is inconsistent with the known tetrahedral structure of methane with four equivalent bonds. 

The wave functions nlm) for the hydrogen-like atom are often called atomic orbitals. It is customary to indicate the values 0, 1, 2, 3, 4, 5, 6, 7,. .. of the azimuthal quantum number / by the letters s, p, d, f, g, h, i, k,. .., respectively. Thus, the ground-state wave function 100) is called the Is atomic orbital, 200) is called the 2s orbital, 210), 211), and 21 —1) are called 2p orbitals, and so forth. The first four letters, standing for sharp, principal, diffuse, and... 

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