William A. Goddard, III, wag@wag.caltech.edu 316 Beckman Institute, x3093

1. Ch120a-Goddard-L01 Lecture 13 February 5, 2014 Homonucleardiatomics Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy Course number: Ch120a Hours: 2-3pm Monday, Wednesday, Friday William A. Goddard, III, wag@wag.caltech.edu 316 Beckman Institute, x3093 Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Teaching Assistants:Sijia Dong <sdong@caltech.edu> Samantha Johnson <sjohnson@wag.caltech.edu>

2. Homonuclear Diatomics

3. Homonuclear Diatomics Molecules – the valence bond view Consider bonding two Ne atoms together Clearly there will be repulsive interactions as the doubly occupied orbitals on the left and right overlap, leading to repulsive interactions and no bonding. In fact as we will consider later, there is a weak attractive interaction scaling as -C/R6, that leads to a bond of 0.05 kcal/mol, but we ignore such weak interactions here The symmetry of this state is 1Sg+

4. Halogen dimers Next consider bonding of two F atoms. Each F has 3 possible configurations (It is a 2P state) leading to 9 possible configurations for F2. Of these only one leads to strong chemical binding This also leads to a 1Sg+ state. Spectroscopic properties are listed below . Note that the bond energy decreases for Cl2 to Br2 to I2, but increases from F2 to Cl2. we will get back to this later.

5. Di-oxygen or O2 molecule Next consider bonding of two O atoms. Each O has 3 possible configurations (It is a 3P state) leading to 9 possible configurations for O2. Of these one leads to directly to a double bond This suggests that the ground state of O2 is a singlet state. At first this seemed plausible, but by the late 1920’s Mulliken established experimentally that the ground state of O2 is actually a triplet state, which he had predicted on the basis of molecular orbital (MO) theory. This was a fatal blow to VB theory, bringing MO theory to the fore, so we will consider next how Mulliken was able to figure this out in the 1920’s without the aid of computers.

6. The homonuclear diatomic correlation diagram Mulliken knew the ordering of the atomic orbitals and considered how combinations of the atomic orbitals would change as the nuclei were pushed together to eventually form a united atom. First consider the separate atoms limit where there is a large but finite distance R separating the atoms. The next slide shows the combinations formed from 1s, 2s, and 2p orbitals.

7. Separated atoms limit Note that in each case we get one bonding combination (no new nodal plane) and one antibonding combination (new nodal plane, red lines)

8. Splitting of levels General nodal arguments allow us to predict that But which is lower of and which is lower of Here the nodal plane arguments do not help

9. At large R 2ps better bonding than 2pp In earlier lectures we considered the strength of one-electron bonds where we found that Since the overlap of ps orbitals is obviously higher than pp We expect that bonding antibonding

10. Separated atom limit MO notation Separated atoms notation

11. MO Homonuclear Diatomics Molecules – Ne2compare the VB and MO views 2 3 anti bonds 4 Consider bonding two Ne atoms together 4 VB 2 3 bonds 2 1s22s22p6=10 e 1s22s22p6=10 e 20 e 2 MO view 3 antibonds cancel 3 bonds, thus no net bonding Generally bond ~ 1/(1+S) Antibond ~ 1/(1-S) Where S= overlap. Thus net is antibonding left and right overlap, leading to repulsive interactions and no bonding. 2 2

12. MO Homonuclear Diatomics Molecules – F2compare the VB and MO views 0 2 anti bonds 4 Consider bonding two Ne atoms together 4 VB 2 3 bonds 2 1s22s22p5=9 e 1s22s22p5=9 e 20 e 2 MO view 2 antibonds cancel 2 bonds, but still net single bond 2 2

13. MO Homonuclear Diatomics Molecules – O2compare the VB and MO views 0 1 anti bonds 2 Consider bonding two Ne atoms together 4 VB 2 3 bonds 2 1s22s22p5=9 e 1s22s22p5=9 e 20 e 2 MO view 1 antibonds cancel 1 bonds, but still net double bond But now we have 2 electrons to put into the two pgx and pgy orbitals, can get either singlet or triplet states Lets examine this in more detail 2 2

14. States based on (p)2 D- S- S+ D+ Have 4 spatial combinations Which we combine as where x and y denote px and py φ1, φ2 denote the angle about the axis and F is independent of φ1, φ2 Rotating about the axis by an angle g, these states transform as

15. States arising from (p)2 Adding spin we get xy-yx better than xy+yx Because no chance for both electrons at same spot (like Hund’s rule on spin of atoms) Triplet lower by Kxy, singlet higher by Kxy O2 Energy (eV) xx-yy better than xx+yy Because decrease chance for both electrons at same spot (like Hund’s L rule atoms) Triplet lower by Kxy, singlet higher by Kxy MO theory explains the triplet ground state and low lying singlets Ground state

16. States arising from (p)2 Adding spin we get MO theory explains the triplet ground state and low lying singlets exact MO 1.636 O2 Energy (eV) (p)2 0.982 0.0 Ground state

17. States arising from (p)2 exact MO Spin and parity forbidden 1.636 O2 Energy (eV) (p)2 0.982 Spin and angular momentum forbidden 0.0 Ground state Mulliken knew that these splittings would be ~ 1 eV in infrared region, so he looks near dusk to get the longest path of the sunlight through the atmosphere and saw the transition Immediately MO big winner and VB big loser

18. MO First excited configuration of O2 (pu)3(pg)3 (1pu)3 (1pu)3 (1pg)3 (1pg)3 0 2 (1pg)2 Ground 4 excited 2 1Su+ 2 1Du 3Su- 2 3Su+ 1Su- Only dipole allowed transition from 3Sg- 3Du 2 Strong transitions (dipole allowed) DS=0 (spin) Sg Su or Pu but S-  S- 2 18

19. The states of O2 molecule Moss and Goddard JCP 63, 3623 (1975) (pu)3(pg)3 (pu)4(pg)2

20. MO Homonuclear Diatomics Molecules – N2compare the VB and MO views 0 0anti bonds 0 Consider bonding two Ne atoms together 4 VB 2 3 bonds 2 1s22s22p3=7 e 1s22s22p3=7 e 20 e 2 MO view 0 antibonds get triple bond Get singlet state, 1Sg+ 2 2

21. More on N2 The elements N, P, As, Sb, and Bi all have an (ns)2(np)3 configuration, leading to a triple bond Adding in the (ns) pairs, we show the wavefunction as This is the VB description of N2, P2, etc. The optimum orbitals of N2 are shown on the next slide. The MO description of N2 is Which we can draw as

22. GVB orbitals of N2 Re=1.10A R=1.50A R=2.10A

23. Hartree Fock Orbitals N2

25. MO Homonuclear Diatomics Molecules – N2+compare the VB and MO views 0 0anti bonds 0 VB 3 2 2.5bonds 2 1s22s22p3=7 e 1s22s22p3=6 e 20 e 2 This suggests that N2+ 2Pu ground state 3 2Sg+1st exc. state 1 2Su+2nd exc. state 2 1 But really ground state is 2Sg+, 1st exc. State is 2Pu 2 Something wrong with order of orbitals

26. United atom limit Next consider the limit in which the two nuclei are fused together to form a united atom For N2 this would lead to a Si atom. Here we get just the normal atomic aufbau states 1s < 2s < 2p < 3s < 3p < 4s,3d < 4p etc But now we consider an itty bity elongation of the Si nucleus toward two N nuclei and how the atomic states get perturbed For the 1s orbital all that happens is that the energy goes up (less electron density on the nuclei) and the symmetry becomes sg

27. 2s and 2p united atom orbitals Similarly 2s just goes to 2sg (and a lower binding) But the 2p case is more interesting For the 2ps state the splitting of the nuclei lead to increased density on the nuclei and hence increased binding while for 2pp there is no change in density Thus 2psu < 2ppu

28. Summarizing united atom limit Note for 3d, the splitting is 3ds < 3dp < 3dd Same argument as for 2p

29. Summary more united atom levels

30. Correlation diagram for Carbon row homonuclear diatomics C2 N2 O2 F2 United atom limit O2+ separated atom limit N2+

31. Using the correleation diagram Choices for N2 In order to use the correlation diagram to predict the states of diatomic molecules, we need to have some idea of what effective R to use (actually it is the effective overlap with large R small S and small R large S). Mulliken’s original analysis [Rev. Mod. Phys. 4, 48 (1932)] was roughly as follows. 1. N2 was known to be nondegenerate and very strongly bound with no low-lying excited states 2 4 2 2 4 2 4 2 2 2 2

32. N2 MO configurations This is compatible with several orderings of the MOs Largest R 2 4 2 4 2 2 2 4 2 2 Smallest R 2

33. N2+ But the 13 electron molecules BeF, BO, CO+, CN, N2+ Have a ground state with 2S symmetry and a low lying 2S sate. In between these two 2S states is a 2P state with spin orbital splitting that implies a p3 configuration This implies that Is the ground configuration for N2 and that the low lying states of N2+ are This agrees with the observed spectra

34. Correlation diagram for Carbon row homonuclear diatomics C2 N2 O2 F2 United atom limit O2+ separated atom limit N2+

35. O2 MO configuration (1pg)2 2 For O2 the ordering of the MOs Is unambiguous 4 2 2 2 Next consider states of (1pg)2 2 2

36. The configuration for C2 2 1 1 2 1 2 4 3 4 4 2 2 2 2

37. The configuration for C2 2 1 1 2 1 2 4 3 4 4 2 2 From 1930-1962 the 3Pu was thought to be the ground state. Now exper. Ground state is 1Sg+ is (0.09ev lower) 2 2

38. Ground state of C2 MO configuration Have two strong p bonds, but sigma system looks just like Be2 which leads to a bond of ~ 1 kcal/mol The lobe pair on each Be is activated to form the sigma bond. The net result is no net contribution to bond from sigma electrons. It is as if we started with HCCH and cut off the Hs

39. C2, Si2,

41. Low-lying states of C2

44. The configuration for Si2 Si2 has this configuration C2has this configuration 2 1 1 2 1 2 4 3 4 4 2 2 Effectively Si2 has lower overlap than C2 (toward Sep. atom limit) 2 2

45. Do B2, Be2, Li2, Li2+

46. MO and VB view of He dimer, He2 MO view ΨMO(He2) = A[(sga)(sgb)(sua)(sub)]= (sg)2(su)2 VB view ΨVB(He2) = A[(La)(Lb)(Ra)(Rb)]= (L)2(R)2 Net BO=0 Pauli  orthog of R to L  repulsive Substitute sg = R +Land sg = R -L Get ΨMO(He2) ≡ ΨMO(He2)

47. Van der Waals interactions For an ideal gas the equation of state is given by pV =nRT where p = pressure; V = volume of the container n = number of moles; R = gas constant = NAkB NA = Avogadro constant; kB = Boltzmann constant Van der Waals equation of state (1873) [p + n2a/V2)[V - nb] = nRT Where a is related to attractions between the particles, (reducing the pressure) And b is related to a reduced available volume (due to finite size of particles)

48. Noble gas dimers No bonding at the VB or MO level Only simultaneous electron correlation (London attraction) or van der Waals attraction, -C/R6 • LJ 12-6 Force Field • E=A/R12 –B/R6 • = De[r-12 – 2r-6] • = 4 De[t-12 – t-6] • = R/Re • = R/s where s = Re(1/2)1/6 =0.89 Re Ar2 s Re De

49. London Dispersion The weak binding in He2 and other noble gas dimers was explained in terms of QM by Fritz London in 1930 The idea is that even for a spherically symmetric atoms such as He the QM description will have instantaneous fluctuations in the electron positions that will lead to fluctuating dipole moments that average out to zero. The field due to a dipole falls off as 1/R3 , but since the average dipole is zero the first nonzero contribution is from 2nd order perturbation theory, which scales like -C/R6 (with higher order terms like 1/R8 and 1/R10)

50. London Dispersion The weak binding in He2 and other nobel gas dimers was explained in terms of QM by Fritz London in 1930 The idea is that even for a spherically symmetric atoms such as He the QM description will have instantaneous fluctuations in the electron positions that will lead to fluctuating dipole moments that average out to zero. The field due to a dipole falls off as 1/R3 , but since the average dipole is zero the first nonzero contribution is from 2nd order perturbation theory, which scales like -C/R6 (with higher order terms like 1/R8 and 1/R10) Consequently it is common to fit the interaction potentials to functional forms with a long range 1/R6 attraction to account for London dispersion (usually referred to as van der Waals attraction) plus a short range repulsive term to account for short Range Pauli Repulsion)