Same idea for polyatomics, keep track of identical atom e.g. NH 3 consider only valence electrons F(2s,2p) H(1s)

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1 XIII 63 Polyatomic bonding -09 -mod, Notes (13) Engel Balance: nuclear repulsion, positive e-n attraction, neg. united atom AO ε i applies to all bonding, just more nuclei repulsion biggest at low separation, attraction at high - minimum Extend LCAO-MO for Simple Polyatomics, 1 st - Hetero atom diatomic bonding - Engel Ch.16.11, requires adjustment to homonuclear diatomic picture Consider H-F - F more electronegative, higher IP than H If ionize, electron goes to vacuum, free to fly away Free - both same, bigger IP means more negative energy Ex. HF bonding Result: bonding MO is closer in energy to F 2p than H 1s, σ b will look more like F-2p, and with 2 electrons, it will have more electron density close to F than to H create a dipole moment by displace e -

2 XIII 64 Same idea for polyatomics, keep track of identical atom e.g. NH 3 consider only valence electrons F(2s,2p) H(1s) split of bonding from non bond is large, bonds are strong Note all σ, no π - separation of p x, p y not preserved, easiest way for chemist hybrid idea bonding interaction bigger energy than s-p split promote s equivalent to p treat s, p as virtually degenerate can mix sp 3 4 orbitals point at apex tetrahedral sp 2 3 orbitals 120 apart(leaves p z π) double bond sp 2 orbitals 180 apart (p x,y orbital 2π) triple bond sp 3 1 = s + p z + p x + p y sp 3 2 = s + p z - p x - p y sp 3 3 = s - p z + p x - p y sp 3 4 = s - p z - p x + p y sp 2 1 = s + p x + p y sp 2 2 = 2s - p x - p y sp 2 3 = p x - p y sp 1 = s + p z sp 1 = s - p z

3 XIII 65 All of these take some chemical intuition balance electronegativity and how molecules form C 2 H 2 linear, C 2 H 4 planar, {NH 3 non planar, BH 3 planar} A little more depth We took simplistic separation approach How do we evaluate energies / realistic wave/fct imagine AO s as a basis set of orbitals guesses H = φ i Hφ i / φ i φ i = H ii /S ii expectation value (i,i)-diagonal H = φ i Hφ j / φ i φ j = H ij /S ij interaction value (i,j) off-diagonal These are called Coulomb integrals (H ij ) energy operator which are normalized to overlap integrals (S ij ) So if ψ = c i φ i i.e. if MO made of AOs, then want to optimize c i -- use variation principle: H / c i = 0 e.g. let: ψ = c A φ A + c B B φb E = [c 2 A H AA + 2c A c B AB HB (as in σ b ) then -- demonstrate + c B 2 H BB ]/[c A 2 + 2c A c BBS AB + c B 2 ] set E/ c A = 0, E/ c B = 0 2 equations, 2 unknowns B easiest (Levine) differentiate both sides secular equations, {E[c 2 A +2c A c B AB+c 2 B ]}/ c A = {c 2 A H AA +2c A c BBH AB +c 2 B H BB }/ c A SB E/ c A [c 2 A +2c A c B AB+c 2 B ] + E[2c A +2c BBS AB ] = [2c A H AA +2c B AB] SB HB First term = 0 (min), rest: 0 = c A (H AA - E)+ c B (H AB - ES AB ) same for H BB

4 Solution: energies from secular determinant: XIII 66 (polynomial, quadratic H AA E H AB ES AB 2 solution E 1,E 2 ) 0 = HAB ES AB H BB E general wave function (LCAO-MO): ψ = c i φ i Secular Determinant n-aos give n-mos n - E K values H 11 E H 21 ES 21 H n1 ES n1 H 12 ES 12 H 22 E : 0 = H 13 ES 13 : : : : H 1n ES 1n H nn E Then insert E K s into secular equations (n - equations) Σ ij (H ij - E k S ij )c k j = 0 n-e k makes n separate sol n solve set of simultaneous equations for coefficients get a set of n - c i K for each (n-values) E K Delocalization major aspect of MO model, see above, all MOs are solutions to secular determinant (energies) and associated set of secular equations (wave functions), so each has contributions from all the AOs - delocalized Valence Bond method, localized view, chemical appeal Idea, AOs on one atom overlap those on another and share and electron pair bond

5 XIII 67 H 2 : ψ VB (1,2) = φ A (1)φ B (2) + φ A (2)φ B (1) electrons identical, need both forms, bonding - increase e - density between Contrast this with the MO picture ψ MO (1,2) = σ b (1) σ b (2) = [φ A +φ B ](1) [φ A +φ B ](2) = φ A (1)φ B (2)+φ A (2)φ B (1)+φ A (1)φ A (2)+φ B (2)φ B (1) VB 2 e - state ionic states (excited) Difference, MO too much emphasis on ionic states, VB too much bond strength (e - all between nuclei) Bring them together VB add excited state (ionic) contribution (e - promoted) ψ VB (1,2) = φ A (1)φ B (2)+φ A (2)φ B (1) + λ[φ A (1)φ A (2)+φ B (2)φ B (1)] MO add excited configuration, σ*(1) σ*(2) σ b (1) σ b (2) λ'[σ*(1) σ*(2)] = φ A (1)φ B (2)+φ A (2)φ B (1) + (1-λ')[φ A (1)φ A (2)+φ B (2)φ B (1)] Adjusting λ, λ' will bring these to the same functional form Can use variation method to optimize λ, λ' Poly atomic VB select AOs with valence e - to mix, share e -, form 2e - bond, promote e - to get hybrids C (2s) 2 (2p) 2 C (2s) 1 (2p)3 C (sp 3 ) 4 Then can form local bonds: C-H: φ = C (sp 3 ) 1 +H(1s) 1 And C-C: φ = C a (sp 3 ) 1 + C b (sp 3 ) 1 etc.

6 XIII 68 Problem: VB completely local, so misses out on delocalized systems like π and interactions between bond networks VSEPR model describe molecular geometry based on e - pair repulsion, in VB bonds, lone pairs Minimize energy by distance, spread, open angles Diatomic linear, trivial, one bond Triatomic linear or bent, depend on lone pairs O=C=O - two bonds around C, both double, linear H-O-H - two bonds and two lone pairs around O Approx tetrahedral, pairs more space Rules: 1. bonds to ligands and lone pairs repel 2. lone pairs require more angular space than bonds 3. More electronegative ligand increase space, more electroneg. central atom, decrease space 4. multiple bond needs more space than single bond Ex. CH 4 tetrahedral, angle between C-H bonds, 109º NH 3 pyramidal, lone pair 4 th position, angle ~107º H 2 O bent, two lone pairs other positions, angle ~105º CO 2 linear, SO 2 bent (lone pair) C 2 H 2 linear, C 2 H 4 planar, C 2 H 6 each ~ tetrahedral

7 XIII 69 Solving MO problem for real systems: Clearly all of this gets complex for large molecules most quantum chemistry problems solved using computers AO ψ ~ Σ i c i φ i φ AO i ~ Σ i a k f k f k functions, e.g. STO e -αr, Gaussian e -α r 2 sometimes fix a k s by some optimization (e.g. solving atoms) always optimize c i by minimize energy can also vary geometry to optimize energy structure determination Hückel Model of π-systems -- a little different π-bonds weaker so usually highest E bond (Aufbau last to fill) symmetry out of the plane not mix with σ-orbital Hückel Theory empirical approximation solve π-system separately, idea is π not mix with σ ψ = i c i φ i φ i = 2p z i just sum over p z orbital on atoms i this ignores the σ orbitals - lower energy Ex. Butadiene ψ k = c k 1φ 1 + c k 2φ 2 + c k 3φ 3 + c k 4φ 4 normal secular determinant: H 11 ε H 12 εs 12 H 13 εs 13 H 14 εs 14 H 21 εs 21 H 22 ε H 23 εs 23 H 24 εs 24 0 = H 31 εs 31 H 32 εs 32 H 33 ε H 34 εs 34 H 41 εs 41 H 42 εs 42 H 43 εs 43 H 44 ε

8 XIII 70 Rules: 1. Let S ij = 0 i j -- no overlap between atoms 2. let H ii = α (i = j) -- All treated the same, diagonal 3. let H ij = β i = j ± 1, near neighbor interact 4. let H ij = 0 non-near neighbor do not interact Hückel secular determinant: α ε β = β α ε β 0 0 β α ε β 0 0 β α ε solution divide entire matrix (determinant) by β then let x = (α ε)/β 0 = x x x x det - expand by minors (show): x 4 3x = 0 solution (just quadratic eq.): x = ±0.618, ±1.618 Plug into: x = (α ε)/β to get expressions for ε ε 1 ± = α ± β ε 2 ± = α ± β Butadiene has 4 π-electrons, could have 2 in each orbital: E g = 2ε ε 2 + = 4α β - ground state: (1π) 2 (2π) 2 E ex = 2ε ε ε 2 - = 4α β excited state: (1π) 2 (2π) 1 (3π) 1

9 ΔE = β transition in uv: XIII 71 4 Plug ε k into: (Hij ε k δ ij )c k j = 0 for j = 1 4 (δ ij -no S ij ) i= 1 Ex: (α ε k )c 1 + βc 2 = 0 βc 1 + (α ε k )c 2 + βc 3 = 0 βc 2 + (α ε k )c 3 + βc 4 = 0 βc 3 + (α ε k )c 4 = 0 4 equations (j) solve simultaneously for c j, repeat each ε k ψ 1π = 0.37 p p p p 4 ψ 2π = 0.60 p p p p 4 ψ 3π = 0.60 p p p p 4 ψ 4π = 0.37 p p p p 4 (0 nodes) (1 node) (2 nodes) (3 nodes) Hückel MO s (phasing, shapes - side view) for butadiene: # nodes E

10 XIII 72 Hückel MO s for benzene very similar, but since cyclic get an extra term in determinant, i.e. H 16 and H 61 0 try it! What if system not π-system? QM calculational Complexity builds-up semi empirical extend the idea above Extended Hückel let atoms differ, calculate S ij but valence electrons interaction empirical parameter (i.e. include σ + π but parameterize H ij ) CNDO MNDO INDO basically variations of neglecting overlap (NDO) and including parameters for H ij ab initio Compute solution to Schrödinger Equation no parameters but many approximations SCF {typically Hartree-Fock set up w/f and use to calculate potential (V(r i )) and re-solve -- keep cycling until no change in V(r i )} Being a little more precise add spin: Multielectron wave function - obey Pauli (anti-symmetric w/r/t exchange of electrons) -- use determinant form: ψ (r 1,r 2,r 3,,r N ) = φ A (r 1 ) φ B (r 1 ) φ C (r 1 ) φ N φ A (r 2 ) φ B (r 2 ) : φ A (r 4 ) φ N (r n ) This will be anti-symmetric w/r/t exchange of electron i.e. ψ (r 1, r 2, r 3, ) = -ψ (r 1, r 3, r 2, ) etc. since exchange electron --> exchange rows

11 XIII 73 Still product of one-electron functions (MO) average V(r ij ) Solving ab-initio problem involves center integrals Put up example of what it means big problem with center often neglect some Gaussian orbitals g nmk (r) = (x n + y m + z k ) e -αr 2 make integrals simpler, rep. AOs as: STO = c n g n (r) Calculations get very big tend to scale like n 4 n 5 n = number basis functions (the g n (r) above) so small molecules very quick (do on PC) big molecules become impossible Methods work well for molecules up to ~100 atoms (or ~1000 basis functions represent AO s) multi center integrals need large memory / disk takes time (now dedicated PC s or clusters) Alternate approach: Density Functional Theory (DFT) Hohenberg-Kohn ρ(r) electron density for grd state in principle can determine w/f: ψ (r 1,, r n ) Advantage calculations become easier / bigger system Disadvantage involves parameters / approximations (variation works differently) [now part of standard Quantum Chemistry programs] Solution density orbital density etc. (self-consist)

12 Goals: 1. molecular properties structure, bonding, charge distribution, dipole moments spectral transitions IR(vibration) -grd st property UV (electronic) - more difficult, excited state 2. Structure optimize geometry minimum E 3. Kinetics reaction surface E as fct geometry Very large molecules e.g. proteins / nucleic Use results of Quantum Mechanics create U el (R) an energy function that varies with nuclear position minimize geometries: molecular mechanic model XIII 74 dynamics / trajectories: molecular dynamics