7-33. The reduced symmetry of heteronuclear (compared to homonuclear) diatomic
Section 7-8 Beyond the Orbital Approximation
7-33. The reduced symmetry of heteronuclear (compared to homonuclear) diatomic molecules results in their having a different correlation diagram. Set up a correlation diagram for heteronuclear diatomics. Be sure to indicate that the energy levels of each type of AO are not identical for the separated atoms. Comparing correlation diagrams for homonuclear and heteronuclear molecules, does it seem reasonable that He2 is unstable, whereas the isoelectronic LiH and LiHe+ are stable molecules?
7-34. Following are some Slater orbital coefficients for some MO’s of F2 calculated by Ransil [11] (the 2pσ STOs are defined according to z axes pointing from each atom toward the other):
1σg
c1s,A = c1s,B = 0.70483
2σg
c1s,A = c1s,B = 0.17327 c2s,A = c2s,B = 0.00912 c2s,A = c2s,B = −0.67160 c2pσ ,A = c2pσ ,B = −0.00022 c2pσ ,A = c2pσ ,B = −0.08540 We see that the 1σg MO is almost entirely made from 1s AOs on A and B.
However, the 2σg MO contains what appears to be an anomalously large amount of 1s AO. This turns out to be an artifact of the fact that Slater-type 2s orbitals are not orthogonal to 1s AOs on the same center. For F2, the STO 1s, 2s overlap is 0.2377. Use this fact to construct a new orbital, 2s, that is orthogonal to 1s.
Express the 2σg MO of Ransil in terms of the basis functions 1s, 2s, and 2pσ on centers A and B. You should find the 1s coefficients much reduced.
7-35. Re for H+ equals 2.00 a.u. At this distance, E
2
elec = −1.1026 a.u. What is the value of De for H+?
2
Multiple Choice Questions (Try to answer these without referring to the text.)
1. A homonuclear diatomic MO is given by φ = 2pz,a + 2pz,b, where the z axis is the same as the internuclear axis. Which one of the following statements about φ is correct?
a) φ is an antibonding MO, symbolized σu.
b) φ is a bonding MO, symbolized πu.
c) φ is an antibonding MO, symbolized πg.
d) φ is a bonding MO, symbolized σg.
e) φ is an antibonding MO, symbolized πu.
2. For the three species N2, N+, N−, which one of the following orders for the bond
2
2
energy (i.e., bond strength) is most reasonable?
a) N2 > N+ > N−
2
2
b) N+ > N
2
2 > N−
2
c) N− > N
2
2 > N+
2
d) N− > N+ > N
2
2
2
e) Only N2 forms a bond; N+ and N− do not.
2
2
Chapter 7 The Variation Method3. According to the LCAO-MO model, which one of the following second period diatomic molecules has a double bond in the ground electronic state?
a) Li2 b) Be2 c) B2 d) C2 e) N2 4. Which one of the following statements concerning H+ is false?
2
a) The nondegenerate LCAO-MOs (without spin) must be either symmetric or anti symmetric for inversion.
b) The lowest energy MO (without spin) of the molecule is antisymmetric for inversion.
c) The MOs transform into the AOs of the helium ion as the two nuclei are fused together.
d) The ground state has a multiplicity of two.
e) The Born-Oppenheimer approximation permits finding the solution for the purely electronic wave function at each value of the internuclear distance.
5. Which one of the following statements is false for bonding MOs formed from linear combinations of AOs on atoms a and b?
a) Only AOs that have nonzero overlap can form bonding MOs.
b) Only AOs that have similar energies can form strongly bonding MOs.
c) Bonding MOs cannot have a nodal plane perpendicular to the internuclear axis and midway between a and b.
d) A p AO on b can combine with a p AO on a to form σ, π , or δ MOs.
e) A maximum of three bonding MOs can be formed from 2p AOs on a and b.
References [1] R. S. Mulliken, Rev. Mod. Phys. 2, 60, 506 (1930); 3, 90 (1931); 4, 1 (1932).
[2] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Intermolecular Forces. Wiley, New York, 1964.
[3] A. B. F. Duncan, Rydberg Series in Atoms and Molecules. Academic Press, New York, 1971.
[4] R. S. Mulliken, J. Chem Phys. 36, 3428 (1962).
[5] R. Weiss, Phys. Rev. 131, 659 (1963).
[6] T. Kinoshita, Phys. Rev. 150, 1490 (1957).
[7] C. L. Pekeris, Phys. Rev. 115, 1216 (1959).
[8] W. Kolos and L. Wolniewicz, J. Chem. Phys. 49, 404 (1968).
Section 7-8 Beyond the Orbital Approximation [9] G. W. F. Drake, M. M. Cassar, and R. A. Nistor, Phys. Rev. A 65, 54501 (2002).
[10] G. Herzberg, J. Mol. Spectrosc. 33, 147 (1970).
[11] B. J. Ransil, Rev. Mod. Phys. 32, 245 (1960).
7-33. The reduced symmetry of heteronuclear (compared to homonuclear) diatomic molecules results in their having a different correlation diagram. Set up a correlation diagram for heteronuclear diatomics. Be sure to indicate that the energy levels of each type of AO are not identical for the separated atoms. Comparing correlation diagrams for homonuclear and heteronuclear molecules, does it seem reasonable that He2 is unstable, whereas the isoelectronic LiH and LiHe+ are stable molecules?
7-34. Following are some Slater orbital coefficients for some MO’s of F2 calculated by Ransil [11] (the 2pσ STOs are defined according to z axes pointing from each atom toward the other):
1σg
c1s,A = c1s,B = 0.70483
2σg
c1s,A = c1s,B = 0.17327 c2s,A = c2s,B = 0.00912 c2s,A = c2s,B = −0.67160 c2pσ ,A = c2pσ ,B = −0.00022 c2pσ ,A = c2pσ ,B = −0.08540 We see that the 1σg MO is almost entirely made from 1s AOs on A and B.
However, the 2σg MO contains what appears to be an anomalously large amount of 1s AO. This turns out to be an artifact of the fact that Slater-type 2s orbitals are not orthogonal to 1s AOs on the same center. For F2, the STO 1s, 2s overlap is 0.2377. Use this fact to construct a new orbital, 2s, that is orthogonal to 1s.
Express the 2σg MO of Ransil in terms of the basis functions 1s, 2s, and 2pσ on centers A and B. You should find the 1s coefficients much reduced.
7-35. Re for H+ equals 2.00 a.u. At this distance, E
2
elec = −1.1026 a.u. What is the value of De for H+?
2
Multiple Choice Questions (Try to answer these without referring to the text.)
1. A homonuclear diatomic MO is given by φ = 2pz,a + 2pz,b, where the z axis is the same as the internuclear axis. Which one of the following statements about φ is correct?
a) φ is an antibonding MO, symbolized σu.
b) φ is a bonding MO, symbolized πu.
c) φ is an antibonding MO, symbolized πg.
d) φ is a bonding MO, symbolized σg.
e) φ is an antibonding MO, symbolized πu.
2. For the three species N2, N+, N−, which one of the following orders for the bond
2
2
energy (i.e., bond strength) is most reasonable?
a) N2 > N+ > N−
2
2
b) N+ > N
2
2 > N−
2
c) N− > N
2
2 > N+
2
d) N− > N+ > N
2
2
2
e) Only N2 forms a bond; N+ and N− do not.
2
2
Chapter 7 The Variation Method3. According to the LCAO-MO model, which one of the following second period diatomic molecules has a double bond in the ground electronic state?
a) Li2 b) Be2 c) B2 d) C2 e) N2 4. Which one of the following statements concerning H+ is false?
2
a) The nondegenerate LCAO-MOs (without spin) must be either symmetric or anti symmetric for inversion.
b) The lowest energy MO (without spin) of the molecule is antisymmetric for inversion.
c) The MOs transform into the AOs of the helium ion as the two nuclei are fused together.
d) The ground state has a multiplicity of two.
e) The Born-Oppenheimer approximation permits finding the solution for the purely electronic wave function at each value of the internuclear distance.
5. Which one of the following statements is false for bonding MOs formed from linear combinations of AOs on atoms a and b?
a) Only AOs that have nonzero overlap can form bonding MOs.
b) Only AOs that have similar energies can form strongly bonding MOs.
c) Bonding MOs cannot have a nodal plane perpendicular to the internuclear axis and midway between a and b.
d) A p AO on b can combine with a p AO on a to form σ, π , or δ MOs.
e) A maximum of three bonding MOs can be formed from 2p AOs on a and b.
References [1] R. S. Mulliken, Rev. Mod. Phys. 2, 60, 506 (1930); 3, 90 (1931); 4, 1 (1932).
[2] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Intermolecular Forces. Wiley, New York, 1964.
[3] A. B. F. Duncan, Rydberg Series in Atoms and Molecules. Academic Press, New York, 1971.
[4] R. S. Mulliken, J. Chem Phys. 36, 3428 (1962).
[5] R. Weiss, Phys. Rev. 131, 659 (1963).
[6] T. Kinoshita, Phys. Rev. 150, 1490 (1957).
[7] C. L. Pekeris, Phys. Rev. 115, 1216 (1959).
[8] W. Kolos and L. Wolniewicz, J. Chem. Phys. 49, 404 (1968).
Section 7-8 Beyond the Orbital Approximation [9] G. W. F. Drake, M. M. Cassar, and R. A. Nistor, Phys. Rev. A 65, 54501 (2002).
[10] G. Herzberg, J. Mol. Spectrosc. 33, 147 (1970).
[11] B. J. Ransil, Rev. Mod. Phys. 32, 245 (1960).