Atomic orbitals of hydrogen

Now we can calculate the ground-state energy of H2. Here, we only use one basis function, the Is atomic orbital of hydrogen. By symmetry consideration, we know that the wave function of the H2 ground state is... 

Fig. 2-34. Electron energy bands of liquid water formed from atomic orbitals of hydrogen and oxygen atoms.

The tt bond of the ethene molecule can be formulated very much like the bond in.the hydrogen molecule (Section 21-2A), with the difference that the bonding is achieved by the overlap of two 2p atomic orbitals of carbon rather than two Is atomic orbitals of hydrogen. 

The decrease in ionic character from hydrogen fluoride to hydrogen iodide is marked and due to increasing symmetry in the distribution of the bonding electrons. At first sight, the bond is a simple one, compounded of the Is and the atomic orbitals of hydrogen and halogen respectively. 

These Hartree orbitals resemble the atomic orbitals of hydrogen in many ways. Their angular dependence is identical to that of the hydrogen orbitals, so quantum numbers and m are associated with each atomic orbital. The radial dependence of the orbitals in many-electron atoms differs from that of one-electron orbitals because the effective field differs from the Coulomb potential, but a principal quantum number n can still be defined. The lowest energy orbital is a Is orbital and has no radial nodes, the next lowest s orbital is a 2s orbital and has one radial node, and so forth. Each electron in an atom has associated with it a set of four quantum numbers (n, , m, mfj. The first three quantum numbers describe its spatial distribution and the fourth specifies its spin state. The allowed quantum numbers follow the same pattern as those for the hydrogen atom. However, the number of states associated with each combination of (n, , m) is twice as large because of the two values for m. ... 

FIGURE 6.21 Correlation diagram for HF. The 2s, 2p and 2py atomic orbitals of fluorine do not mix with the Is atomic orbital of hydrogen, and therefore remain nonbonding. 

Figure 7.1 Overlap of the atomic orbitals of hydrogen. Positive phase for wave functions (orbitals) shown red, negative as white...

The atomic orbitals of hydrogen are labeled by quantum numbers. Three integers are required for a complete specification. 

FIGURE 22.1 Electron energy bands of liquid water made of atomic orbitals of hydrogen and oxygen [5] CB = conduction band, VB = valence band, sc > —1.2 eV, gv < —9.3 eV, gg > 8 eV. 

The second situation referred to above, viz systems containing conjugated double bonds, is perhaps more important to the present discussion. The classical example of such a system is benzene. The molecular orbital treatment regards the six G—G bonds and the six G—H bonds as completely localized molecular orbitals compounded out of carbon sp2 hybrid atomic orbitals and the hydrogen s orbital. So far the treatment is identical with the electron pair theory, discussed in Chapter 4. The G—G bonds will be or bonds formed by the overlap of two sp2 hybrid atomic orbitals, one from each carbon atom and the C—H bonds will also be a bonds formed by the overlap of one sp2 hybrid atomic orbital of carbon with the s atomic orbital of hydrogen. The six carbon 2p atomic orbitals that remain will form completely non-localized molecular orbitals. Thus each 2pt electron will be regarded as existing in the field of six nuclei and will possess a wave function of the form ... 

Using equation 1.16, determine the energies of atomic orbitals of hydrogen with = 1, 2, 3, 4 and 5. What can you say about the relative spacings of the energy levels ... 

We begin our consideration of chemical bonding by looking at the simplest possible molecule, H2. The molecular orbitals derived for this system form the basis of the molecular orbitals for all other diatomic molecules, in much the same way that the atomic orbitals of hydrogen form the basis for all atomic orbitals. 

Fig. 23 Molecular orbital of the hydrogen molecule, , composed of the atomic orbitals of hydrogen atoms,...

According to the orbital overlap model (Section 1.6F), a triple bond is described in terms of the overlap of sp hybrid orhitals of adjacent carbons to form a sigma bond, the overlap of parallel 2py orbitals to form one pi bond, and the overlap of parallel 2p orbitals to form the second pi bond. In ethyne, each carbon forms a bond to a hydrogen by the overlap of an sp hybrid orbital of carbon with a 15 atomic orbital of hydrogen. 

It has been proved that each of the Hartree-Fock orbitals has the same asymptotic dependence on the distance from the molecule (N.C. Handy, M.T. Matron, HJ. Silverstone,PM x. Rev. 180 (1969) 45), i.e. const - exp( —V—2Ema. r), where Cma. is the orbital energy of HOMO. Earlier, people thought the orbitals decay as exp(—2c,-r), where c,- is the orbital energy expressed in atomic units. The last formula, as is easy to prove, holds for the atomic orbitals of hydrogen atoms (see p. 178). R. Ahlrichs,... 

A plot of RDF,j against r is shown in Figure 2.4, and indicates that the maximum probability of finding the electron is at a distance (= 52.917726 pm) from the nucleus. One common pictorial representation of atomic orbitals is the solid figure, or boundary surface, in which there is a 95% chance (a probability of 0.95) of finding the electron. The 95% boundary surface for the Is atomic orbital of hydrogen is shown in Figure 2.5. It has a radius of 160 pm. 

Figure 2.5 The 95% boundary surface for the 1s atomic orbital of hydrogen...

The wave function for an atom simultaneously depends on (describes) all the electrons in the atom. The Schrodinger equation is much more complicated for atoms with more than one electron than for a one-electron species such as a hydrogen atom, and an explicit solution to this equation is not possible even for helium, let alone for more complicated atoms. We must therefore rely on approximations to solutions of the many-electron Schrodinger equation. One of the most common and useful of these is the orbital approximation. In this approximation, the electron cloud of an atom is assumed to be the superposition of charge clouds, or orbitals, arising from the individual electrons these orbitals resemble the atomic orbitals of hydrogen (for which exact solutions are known), which we described in some detail in the previous section. Each electron is described by the same allowed combinations of quantum numbers (w, m(, and /,)... 

Fig. 1.11 Cross-sections through the (a) Is (no radial nodes), (b) 2s (one radial node), (c) 3s (two radial nodes), (d) 2p (no radial nodes) and (e) 3p (one radial node) atomic orbitals of hydrogen. The rabitals have been generated using the program Orbital Viewer (David Manthey, www.OTbitals.com/orb/index.html).

The last statement of the above example mentioned a set of orbitals. A similar procedure can be applied to a combination of atomic orbitals, just as the four sp hybrid orbitals above were treated in combination. For example, in methane the four Is atomic orbitals of hydrogen, which the symmetry operations in move onto each other, represent such a set of orbitals. Perhaps not so surprisingly, the set of four Is atomic orbitals can also be labeled as A 0 T2 in the point group. 

The Atomic Orbitals of Hydrogen 367 The expectation for the average electron position in this orbital is then... 

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