Hydrogen atom atomic orbital representations

In either representation, (22) or (23), we see that there is residual bonding capacity remaining in the species OH. In (22) the third 2p orbital has a single electron but a capacity for two. This means more bonding can occur. In (23) a census of the electrons near the oxygen atom indicates there are only seven. The oxygen atom would be more stable if it could add one more electron. With either representation, we conclude that OH should be able to react with another hydrogen atom. See representations (24), (25). 

In NH and NFS, three p orbitals are involved in the bonding [see representation (30)]. Figure 16-10 shows the spatial arrangement implied by assuming persistence of the hydrogen atom orbitals after bonding. We expect, then, that ant-... 

The simplest possible atomic orbital representation is termed a minimal basis set. This comprises only those functions required to accommodate all of the electrons of the atom, while still maintaining its overall spherical symmetry. In practice, this involves a single (Is) function for hydrogen and helium, a set of five functions (Is, 2s, 2px, 2py, 2pz) for lithium to neon and a set of nine functions (Is, 2s, 2px,... 

In our QM systems, we have temporarily restricted ourselves to systems of one electron. If, in addition, our system were to have only one nucleus, then we would not need to guess wave functions, but instead we could solve Eq. (4.16) exactly. The eigenfunctions that are determined in that instance are the familiar hydrogenic atomic orbitals. Is, 2s, 2p, 3s, 3p, 3d, etc., whose properties and derivation are discussed in detail in standard texts on quantum mechanics. For the moment, we will not investigate the mathematical representation of these hydrogenic atomic orbitals in any detail, but we will simply posit that, as functions, they may be useful in the construction of more complicated molecular orbitals. In particular, just as in Eq. (4.10) we constructed a guess wave function as a linear combination of exact wave functions, so here we will construct a guess wave function as a linear combination of atomic wave functions (p, i.e.,... 

The chemical shifts (2.4 ppm) of acetylenic hydrogens (RCHC-H) are considerably more toward higher magnetic fields than those of alkene hydrogens (4.6 to 6.9 ppm). Show how this shielding effect might be explained in terms of the atomic orbital representation of acetylenes. 

Fig. 14.15. A schematic representation of the molecular orbitals and their energies of the donor (H, n representing the hydrogen atom Is orbital), of the acceptor (H2, bonding x antibonding x of the hydrogen molecule) as well as of the total system H3 in a linear configuration (center). The lowest-energy molecular orbital of H3 does not have any node, the higher has one, while the highest has two nodes. In all cases, we use the proximation that the molecular orbitals are built from the three Is hydrogen atomic orbitals only.

Figure 1.9. A schematic representation of the formation of bonding (o) and antibonding (a ) molecular orbitals of hydrogen (Hj) by the combination of two equivalent Is hydrogen atomic orbitals. The signs (+) and (-) do not refer to charges but rather to the sign of the wave function /, whose square (t f ) gives the probabiUty of finding the electron(s) in the volume shown.

FIGURE 12.16 Representations of molecular orbitals from linear combinations of hydrogen atomic orbitals. The MO in the middle plot is the sum of the two AOs at the top, with electron density concentrated between the nuclei. The lower MO is the difference of the two AOs, with electron density concentrated more outside the two nuclei. 

Earlier, we eonsidered in some detail how the three Ish orbitals on the hydrogen atoms transform. Repeating this analysis using the short-eut rule just deseribed, the traees (eharaeters) of the 3 x 3 representation matriees are eomputed by allowing E, 2C3, and... 

We see that each oxygen atom has residual bonding capacity. Each atom could, for example, react with a hydrogen atom to form hydrogen peroxide, as shown in electron dot representation (26). Each oxygen atom could react with a fluorine atom to form F2O2. In short, each oxygen atom is in need of another atom with mi electron in a half-filled valence orbital so that it can act as a divalent atom. 

Figure 3.10 Representations of the electron density ip2 of the Is orbital and the 2p orbital of the hydrogen atom. (b,e) Contour maps for the xe plane. (c,f) Surfaces of constant electron density. (a,d) Dot density diagrams the density of dots is proportional to the electron density. (Reproduced with permission from the Journal of Chemical Education 40, 256, 1963 and M. J. Winter, Chemical Bonding, 1994, Oxford University Press, Fig. 1.10 and Fig. 1.11.)...

Atomic orbitals are actually graphical representations for mathematical solutions to the Schrodinger wave equation. The equation provides not one, but a series of solutions termed wave functions t[ . The square of the wave function, is proportional to the electron density and thus provides us with the probability of finding an electron within a given space. Calculations have allowed us to appreciate the shape of atomic orbitals for the simplest atom, i.e. hydrogen, and we make the assumption that these shapes also apply for the heavier atoms, like carbon. 

Hyperspherical harmonics are now explicitly considered as expansion basis sets for atomic and molecular orbitals. In this treatment the key role is played by a generalization of the famous Fock projection [5] for hydrogen atom in momentum space, leading to the connection between hydrogenic orbitals and four-dimensional harmonics, as we have seen in the previous section. It is well known that the hyperspherical harmonics are a basis for the irreducible representations of the rotational group on the four-dimensional hypersphere from this viewpoint hydrogenoid orbitals can be looked at as representations of the four-dimensional hyperspherical symmetry [14]. 

As is well known, conventional hydrogenoid spherical orbitals are strictly linked to tetradimensional harmonics when the atomic orbitals for the tridimensional hydrogen atom are considered in momentum space. We have therefore studied an alternative representation, providing the Stark and Zeeman basis sets, related to the spherical one by orthogonal transformation, see eqs. (12) and (15). The latter can also be interpreted as suitable timber coefficients relating different tree structures of hyperspherical harmonics for R (Fig. 1). 

The solutions of the Schrodinger equation show how j/ is distributed in the space around the nucleus of the hydrogen atom. The solutions for v / are characterized by the values of three quantum numbers and every allowed set of values for the quantum numbers, together with the associated wave function, strictly defines that space which is termed an atomic orbital. Other representations are used for atomic orbitals, such as the boundary surface and orbital envelopes described later in the chapter. 

Neither of the Is orbitals of the hydrogen atoms of the water molecule, taken separately, transform within the group of irreducible representations deduced for that molecule. The two Is orbitals must be taken together as one or the other of two group orbitals. A more formal treatment of the group orbitals which two Is orbitals may form is dealt with in Chapter 3. 

Clearly, for a one-dimensional representation the character and the full matrix are the same thing. Hence, the incomplete projection operator is complete in these cases, and will provide the appropriate SALC unambiguously and automatically. Let us illustrate by asking what SALCs can be formed by the Is orbitals of the four hydrogen atoms in ethylene. 

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