Figure 6. 4 Two 1s Hydrogen Atomic Orbitals Combine to Form a bonding and Antibonding Molecular Orbital The bonding MO labeled o has LCAO-MO coefficients of equal sign for the two ls aOs as a result of which this mo has the same sign near the left h nucleus(A)as near the right H nucleus(B). In contrast, the antibonding MO labeled o*has LCAO-MO coefficients of different sign for the a and b is aos. as was the case in the huckel or tight-binding model outlined in the Background Material, the energy splitting between the two MOs depends on the overlap <xisaIIsB> between the two AOs An analogous pair of bonding and antibonding MOs arises when two p orbitals overlap "sideways"as in ethylene to form T and T* MOs which are illustrated in Fig. 6.5
26 Figure 6. 4 Two 1s Hydrogen Atomic Orbitals Combine to Form a Bonding and Antibonding Molecular Orbital The bonding MO labeled s has LCAO-MO coefficients of equal sign for the two 1s AOs, as a result of which this MO has the same sign near the left H nucleus (A) as near the right H nucleus (B). In contrast, the antibonding MO labeled s* has LCAO-MO coefficients of different sign for the A and B 1s AOs. As was the case in the Hückel or tight-binding model outlined in the Background Material, the energy splitting between the two MOs depends on the overlap <c1sA|c1sB> between the two AOs. An analogous pair of bonding and antibonding MOs arises when two p orbitals overlap “sideways” as in ethylene to form p and p* MOs which are illustrated in Fig. 6.5
1π Figure 6. 5 Two p, Atomic Orbitals on Carbon Atoms Combine to Form a Bonding and Antibonding Molecular Orbital The shapes of these MOs clearly are dictated by the shapes of the AOs that comprise them and the relative signs of the lcao-mo coefficients that relate the mos to AOs. For the T MO, these coefficients have the same sign on the left and right atoms; for the T* MO, they have opposite signs I should stress that the signs and magnitudes of the LCAO-MO coefficients arise as eigenvectors of the HF SCF matrix eigenvalue equation EH lhel x>Ci=E, E x>CiH
27 Figure 6. 5 Two pp Atomic Orbitals on Carbon Atoms Combine to Form a Bonding and Antibonding Molecular Orbital The shapes of these MOs clearly are dictated by the shapes of the AOs that comprise them and the relative signs of the LCAO-MO coefficients that relate the MOs to AOs. For the p MO, these coefficients have the same sign on the left and right atoms; for the p* MO, they have opposite signs. I should stress that the signs and magnitudes of the LCAO-MO coefficients arise as eigenvectors of the HF SCF matrix eigenvalue equation: Sm <cn |he | cm> Cj,m = ej Sm <cn |cm> Cj,m
It is a characteristic of such eigenvalue problems for the lower energy eigenfunctions to have fewer nodes than the higher energy solutions as we learned from several examples that we solved in the background material Another thing to note about the mOs shown above is that they will differ in their quantitative details, but not in their overall shapes, when various functional groups are attached to the ethylene molecule's C atoms. For example, if electron withdrawing groups such as Cl, Oh or Br are attached to one of the C atoms, the attractive potential experience by a T electron near that C atom will be enhanced. As a result, the bonding MO will have larger LCAO-MO coefficients Cku belonging to the tighter"basis AOs %u on this c atom. This will make the bonding t MO more radially compact in this region of space, although its nodal character and gross shape will not change. Alternatively, an electron donating group such as H,C-or t-butyl attached to one of the C centers will cause the t mo to be more diffuse(by making its LCAo-MO coefficients for more diffuse basis AOs larger) In addition to MOs formed primarily of AOs of one type(i.e, for H2 it is primarily S- type orbitals that form the o and o* MOs, for ethylene's t bond, it is primarily the c 2 AOs that contribute), there are bonding and antibonding MOs formed by combining several AOs. For example, the four equivalent C-H bonding MOs in CHa shown in Fig. 6 6 each involve C 2s and 2p as well as h Is basis aOs
28 It is a characteristic of such eigenvalue problems for the lower energy eigenfunctions to have fewer nodes than the higher energy solutions as we learned from several examples that we solved in the Background Material. Another thing to note about the MOs shown above is that they will differ in their quantitative details, but not in their overall shapes, when various functional groups are attached to the ethylene molecule’s C atoms. For example, if electron withdrawing groups such as Cl, OH or Br are attached to one of the C atoms, the attractive potential experience by a p electron near that C atom will be enhanced. As a result, the bonding MO will have larger LCAO-MO coefficients Ck,m belonging to the “tighter” basis AOs cm on this C atom. This will make the bonding p MO more radially compact in this region of space, although its nodal character and gross shape will not change. Alternatively, an electron donating group such as H3C- or t-butyl attached to one of the C centers will cause the p MO to be more diffuse (by making its LCAO-MO coefficients for more diffuse basis AOs larger). In addition to MOs formed primarily of AOs of one type (i.e., for H2 it is primarily stype orbitals that form the s and s* MOs; for ethylene’s p bond, it is primarily the C 2p AOs that contribute), there are bonding and antibonding MOs formed by combining several AOs. For example, the four equivalent C-H bonding MOs in CH4 shown in Fig. 6. 6 each involve C 2s and 2p as well as H 1s basis AOs
Figure 6. 6 The Four C-H Bonds in methane The energies of the MOs depend on two primary factors: the energies of the AOs from which the MOs are constructed and the overlap between these AOs. The pattern in energies for valence MOs formed by combining pairs of first-row atoms to form homo- nuclear diatomic molecules is shown in Fig.6.7
29 Figure 6. 6 The Four C-H Bonds in Methane The energies of the MOs depend on two primary factors: the energies of the AOs from which the MOs are constructed and the overlap between these AOs. The pattern in energies for valence MOs formed by combining pairs of first-row atoms to form homonuclear diatomic molecules is shown in Fig. 6. 7
Figure 6. 7 Energies of the Valence Molecular Orbitals in Homonuclear Diatomics Involving First-Row Atoms In this figure, the core MOs formed from the ls AOs are not shown, but only those MOs formed from 2s and 2p AOs appear. The clear trend toward lower orbital energies as one moves from left to right is due primarily to the trends in orbital energies of the constituent AOs. That is, F being more electronegative than n has a lower-energy 2p orbital than does n b Bonding, Anti-bonding, Non-bonding, and rydberg Orbitals As noted above. when valence AOs combine to form MOs. the relative signs of the combination coefficients determine, along with the Ao overlap magnitudes, the mos energy and nodal properties. In addition to the bonding and antibonding MOs discussed and illustrated earlier, two other kinds of MOs are important to know about Non-bonding MOs arise, for example, when an orbital on one atom is not directed toward and overlapping with an orbital on a neighboring atom. For example, the lone pair orbitals on H,O or on the oxygen atom of H,c=o are non-bonding orbitals. They still are described in the lcao-mo manner but their c: coefficients do not contain dominant contributions from more than one atomic center Finally, there is a type of orbital that all molecules possess but that is ignored in most elementary discussions of electronic structure. All molecules have so-called Rydberg orbitals. These orbitals can be thought of as large diffuse orbitals that describe the regions of space an electron would occupy if it were in the presence of the
30 Figure 6.7 Energies of the Valence Molecular Orbitals in Homonuclear Diatomics Involving First-Row Atoms In this figure, the core MOs formed from the 1s AOs are not shown, but only those MOs formed from 2s and 2p AOs appear. The clear trend toward lower orbital energies as one moves from left to right is due primarily to the trends in orbital energies of the constituent AOs. That is, F being more electronegative than N has a lower-energy 2p orbital than does N. b. Bonding, Anti-bonding, Non-bonding, and Rydberg Orbitals As noted above, when valence AOs combine to form MOs, the relative signs of the combination coefficients determine, along with the AO overlap magnitudes, the MO’s energy and nodal properties. In addition to the bonding and antibonding MOs discussed and illustrated earlier, two other kinds of MOs are important to know about. Non-bonding MOs arise, for example, when an orbital on one atom is not directed toward and overlapping with an orbital on a neighboring atom. For example, the lone pair orbitals on H2O or on the oxygen atom of H2C=O are non-bonding orbitals. They still are described in the LCAO-MO manner, but their Cm,i coefficients do not contain dominant contributions from more than one atomic center. Finally, there is a type of orbital that all molecules possess but that is ignored in most elementary discussions of electronic structure. All molecules have so-called Rydberg orbitals. These orbitals can be thought of as large diffuse orbitals that describe the regions of space an electron would occupy if it were in the presence of the