MO Theory H2+ and H2 solns

Solutions to Hydrogen Molecule Ion Y2, E2 = -10.16 eV (for H2 ) Y1, E1 = 1.37 eV (for H2)

Solutions to Hydrogen Molecule MOs created from combinations of p-orbitals pxA + pxB, pyA + pyB pxA - pxB, pyA - pyB pzA - pzB pzA + pzB

Solutions to Hydrogen Molecule px+ px OR py + py px - px OR py - py pz - pz pz + pz

Gerade = symmetric with inversion Parity represents center of inversion Gerade = symmetric with inversion inversion Ungerade = antisymmetric with inversion inversion

MO Energy Level Diagram for Homonuclear Diatomics lone atom s* lone atom p* 2p 2p p s s* 2s 2s s s* 1s 1s s

Molecular Term Symbols ML = S (over all e-) l l identifies “z-component” of angular momentum of an e- Symbols used to id l | l | 1 2 3 4 s p d f g

Molecular Term Symbols Angular momentum about “z-axis” for all electrons is L = |ML| Symbol used to id L L 1 2 3 4 S P D F G

Molecular Term Symbols Symbol is 2S + 1 L g/u 2S + 1 is multiplicity as already used for atomic term symbols g or u identifies overall parity To determine overall parity, make use of multiplication of symmetric and antisymmetric functions If the term is a S term, a right superscript of + or – is added to indicate whether the wavefunction is symmetric or antisymmetric with respect to reflection through a plane containing the two nuclei

Molecular Term Symbols Remember sigma orbs: From s orbs From pz orbs Remember pi orbs: From px orbs From py orbs

Molecular Term Symbols Remember sigma-star orbs: From s orbitals From p orbitals Remember pi-star orbs: From px orbitals From py orbitals

Spectroscopy – Selection Rules DL = 0, +1, -1 DS = 0 note S = Ms note W refers to spin-orbit coupling and W = L + S DW = 0, +1, -1

Molecular Term Symbols Molecular Orbitals not always so “clear-cut” Remember how orbitals change energy as go across PT Can affect MO energy pattern too

MO Energy Level Diagram for Homonuclear Diatomics Essentially LCAOs involving four orbitals are made. The sigma orbitals that we thought of as being made by the 2s orbitals are lowered in E while the sigma orbitals that we thought of as being made by the 2pz orbitals are raised in E. As you move to the right on PT, 2s and 2p energy gap increases. Early, in the period, then, this permits mixing of 2s and 2pz orbitals. Atkins, Fig 14.30

MO Energy Level Diagram for Homonuclear Diatomics (N2 and “before”) lone atom lone atom p* s 2p 2p Use this diagram for N2 and earlier in PT p s* 2s 2s s s* 1s 1s s

Taking a look at heteronuclear diatomic molecules

Taking a look at heteronuclear diatomic molecules

MOs of HF Unoccupied, E = -0.124 eV Occupied, E = -0.3523 au

MOs of HF H atom F atom H – F molecule s 1s p 2p s 2s 1s

Computational Chemistry Considering complexity of the calculations we’ve been doing, certainly, using computers to do these calcs should be useful  Computational Chemistry For polyatomic molecules can make LCAOs yMO = S ciyi - Yi constitute basis set (computational forms of atomic orbitals) Use variation theory to find ci To find structure of molecule, must move nuclei and find MOs  find structure with lowest overall energy

Computational Chemistry May “solve” for MOs using ab initio or semi-empirical methods Semi-empirical methods: empirical parameters substituted for some “integrals” to save time in calculations Ab initio methods: supposedly make no assumptions NOTE: computational chemistry may determine Energy and some other properties without using quantum chemistry Such calculations are referred to as molecular mechanics calculations

Valence Bond Theory H2 Initial approx is y = y1sA(1) y1sB(2) But, is this a symm or antisymm wavefxn? So, make LCs y+ = y1sA(1) y1sB(2) + y1sB(1) y1sA(2) y- = y1sA(1) y1sB(2) - y1sB(1) y1sA(2) In this case, turns out that y+ is lower E

ybond = [y1sA(1) y1sB(2) + y1sB(1) y1sA(2)][a(1)b(2) – a(2)b(1)] Valence Bond Theory Ground state wavefunction would be ybond = [y1sA(1) y1sB(2) + y1sB(1) y1sA(2)][a(1)b(2) – a(2)b(1)] 2 electrons in overlapping orbitals – with spins paired

Remember CH4 If try to make combinations of the valence s of C with s of H, will be different type of wavefxn, hence diff’t kind of bond than when make combination of a p of C with an s of H DON’T see any diff in bonding of 4 H’s Make LCs of valence orbitals on central atom Call these LCs hybrid orbitals Use these hybrid orbitals to make sigma bonds with H Atomic orbitals NOT used to make sigma bonds used to make pi bonds (Huckel method for conjugated)

Hybrid Orbitals Valence s and p orbitals on C  hybrids y1 = a12s + a22px + a32py + a42pz y2 = b12s + b22px + b32py + b42pz y3 = c12s + c22px + c32py + c42pz y4 = d12s + d22px + d32py + d42pz Consider ethyne Only two hybrids y1 = s + pz and y2 = s – pz Leftover px and py on one C overlap with px and py on other C

Simplification to MO Approach Huckel Approach

Symmetry of Molecules

Determining Point Groups

Special Group? Yes C∞v , D∞h , Td , Oh , Ih , Th No Cn No sh Yes S2n or S2n and i only, collinear with highest order Cn Cs Yes i No No nC2 perpendicular to Cn No C1 Yes Ci Yes Sn No sh Yes sh No n sv No n sd Yes Cnh Dnh Yes Cnv Yes No Dn Dnd Yes No Cn

C2v Character Table C2v E C2 sv(xz) sv’(yz) A1 1 A2 -1 B1 B2

Now go practice!!!

Applying Symmetry to MOs Water

MOs of Water HOMO-4 a1 Looks like s orbital on O, nbo E = -18.6035 au

MOs of Water HOMO-3 from two viewpoints a1 Looks like s orbital on O with constructive interference with c1 - bo E = -0.9127 au

MOs of Water HOMO-1 HOMO-2 b2 a1 Looks like combination of p on O (perp to C2, but in plane of molecule) with constructive interference with c2, bo Looks like combination of p on O along C2 with constructive interference with c1, bo (close to nbo) E = -0.3356 au E = -0.4778 au

MOs of Water HOMO from two viewpoints b1 Looks like p orbital on O, perpendicular to plane of molecule - nbo E = -0.2603 au

MOs of Water LUMO LUMO +1 b2 a1 Looks like combination of p on O (perp to C2, but in plane of molecule) with destructive interference with c2, abo Looks like combination of p on O along C2 with destructive interference with c1, abo E = -0.0059 au E = 0.0828 au

Filling Pattern for Water 2b2 (abo) 4a1 (abo) 1b1 (nbo) 3a1 (bo/nbo) 1b2 (bo) 2a1 (bo) 1a1 (nbo)

Molecular Spectroscopy Molecule has a number of motions Translational, vibrational, rotational, electronic Sum them to get total energy of molecule Changes may occur in any of these modes through absorption or emission of energy Vibrational: IR Rotational: Microwave Electronic: UV-Vis CHP 16, 17, 18 of text

Statistical Mechanics Quantum gives you possible energy levels (states) In a real sample, not all molecules in the same energy level With statistics and total energy, can predict (on average) how many molecules in each state Dynamic Equilibrium Role of Temperature Can predict macroscopic properties/behavior Heat capacity, pressure, etc. CHP 19, 20 of text