Lecture #30 page 1. Valence band approach: Molecular wavefunction described in terms of 1-electron atomic orbitals
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1 5.6 4 Lecture #3 page MOLCUL: MOLCULAR ORBITAL DCRIPTION Valence band approach: Molecular wavefunction described in terms of -ectron atomic orbitals Molecular orbital approach: Construct new set of -ectron molecular orbitals for molecules MO's extend over entire molecule Build molecular wavefunctions with those imilar to building atomic wavefunctions based on -ectron AO's. AO's derived from solution to H atom just ectron Higher atomic wavefunctions built with AO's e - MO's derived from solution to H ion just ectron Higher molecular wavefunctions built with MO's H r r R A B R r B H B R is fixed - determine ( R by fixing R at many different values H is function of just one ectron's coordinates Form MO as linear combination of AO's: ψ cs c s A B This is a -ectron MO the AO's are functions of the same ectron's coordinates. ( cs ( c s ( ψ A B Very different from valence bond theory -ectron orbitals (, N s ( s ( s ( s ( VB ψ A B A B where ectrons are centered at different atoms and Ψ is antisymmetrized.
2 5.6 4 Lecture #3 page Best values of c and c give minimum ψ not yet normalized, so can't write d ψ H r ψ drψ H ψ drψψ or explicitly ( R ( ; ( ; ψ ( ;R rψ ( r; ψ ( r; drψ r R H r R r d R R ( A B ( A B dr ( csa csb( csa csb dr c s c s H c s c s c dr s H s cc dr s H s cc dr s H s c dr s H s c drs s cc drs s cc drs s c drs s Abbreviate integrals A A A B B A B B A A A B B A B B ch cch cch ch BA BB c cc cc BA c BB ( c cc cc c ch cch cch ch BB BA BB Differentiate w.r.t. parameters c, c c cc cc c c c ch ch ch ( ( BA BB BA BA c c cc cc c c c ch ch ch ( ( BB BA BB BA BB c For best c, c, c c ( BA ( BA ( ( c H c H H c H HBA BA c HBB BB
3 5.6 4 Lecture #3 page 3 ( BA ( BA ( ( H H H c H c HBA BA HBB BB olutions given by the secular determinant ( BA ( BA ( ( H H H H HBA BA HBB BB All this follows from ψ csa csb & minimization of the energy No assumptions made about the coefficients or the integrals whenever we construct a wavefunction from two other functions, ψ cφ cφ we can immediaty solve for the energies by solving the secular determinant with the integrals as defined. We can also solve for the constants c, c for each energy. Often, and in our case, ecular determinant becomes and H H (note H is Hermitian BA BA ( ( ( ( H H H H BB BB More generally if we write wavefunction as linear combination of N functions, ψ cφ cφ... cnφn analogous treatment yids N energies solved through secular determinant H H H BA BA NA NA H H H BB BB NB NB H H H AN AN BN BN NN NN
4 5.6 4 Lecture #3 page 4 In our case, the AOs are normalized dr s s dr s s A A B B BB Note our familiar overlap integral dr s s dr s s A B BA B A and since both AO's are the same, so we have H drs H s dr s H s H A A B B BB ( ( ( ( H H c H H c H H H H ( H ( H ( ( ( H H H H ( ( ( ( H H H H H H ( R H dr s H s drs s ra rb R A A A A H dr s H s drs s ra rb R From H atom A B A B
5 5.6 4 Lecture #3 page 5 s s s s ra rb A A B B H dr sa sa J' rb R H d sa sb r K' ra R ( R J' K' J' K' nergy of H molecule-ion As before, this is total ectronic energy for protons and an ectron brought from infinite distance. H atom energy is /. The rest is the energy to form the MO. nergy of H J' K' ( R molecule-ion rative to separated H atom and proton Integrals can be evaluated analytically (McQ problems 9-3, 9-4: R R R R e R J' e K' e R 3 R R (. H molecular ion potential energy curves Molecular orbital treatment.5. (au.5. MO treatment predicts a stable molecular ground state ( R gives equilibrium bond e length, dissociation energy R/a
6 5.6 4 Lecture #3 page 6 MO xp't MO (variational s exponent R.49 au.3 A.6 A.6 A e ( R.65 au.77 ev.78 ev.5 ev e Could vary functional form of AO's to do better. DTRMIN TH MOs the values of c, c Go back to ( ( ( ( H H c H H c H H ( R ither equation for c, c can be used since all the parameters are known ( ( c H c H Using, H H H H c H c H ( ( H H H H H H c c H H H H c c c c ( ψ c s s A B Normalization ( ( drψψ c dr s i s s s c A A B B
7 5.6 4 Lecture #3 page 7 ψ ( ( s s A B Normalized ground state (bonding molecular orbital imilarly for gives antibonding MO c c ( ( s s ψ A B s A Bonding & antibonding MOs formed as linear combinations of s AOs ψ sb s A sb HB σ g symmetry HB s A sb ψ s A - sb HB σ u symmetry HB Nodal plane ψ ψ ψ symmetric about the molecular axis: "σ" symmetry symmetric upon inversion through center: gerade or "g" symmetry symmetry denoted σ g ψ ψ ψ symmetric about the molecular axis: "σ" symmetry antisymmetric upon inversion through center: ungerade or "u" symmetry symmetry denoted σ u
8 5.6 4 Lecture #3 page 8 implified nergy Lev Diagram -/ au s A σ u s B σ g hows energy at just one value of R H -ectron MOs used to build wavefunctions of larger molecules, just like H atom -ectron AOs used to build wavefunctions of higher atoms But we only have two MOs! We had lots (infinite number of AOs To get more MOs, we need to start with more AOs, e.g. ψ cs c s c s c s c p c p A B 3 A 4 B 5 za 6 zb which would yid a 6x6 secular determinant with 6 energies and 6 MOs.
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