Mirror plane (of molecule) 2. the Coulomb integrals for all the carbon atoms are assumed to be identical. . = 0 : if atoms i and j are non-bonded.

Transcription

1 6 Hükel Theory This theory was originally introdued to permit qualitative study of the π-eletron systems in planar, onjugated hydroarbon moleules (i.e. in "flat" hydroarbon moleules whih possess a mirror plane of symmetry ontaining all the arbon atoms, and in whih the atoms of the arbon skeleton are linked by alternating double and single arbon-arbon bonds when the bonding is represented in a loalised fashion). It is thus most appropriate for moleules suh as benzene or butadiene, but the approah and onepts have wider appliability. Basi Assumptions. the atomi orbitals ontributing to the π-bonding in a planar moleule (e.g. the so-alled p π orbitals in a moleule suh as benzene) are antisymmetri with respet to refletion in the moleular plane; they are therefore of a different symmetry to the atomi orbitals ontributing to the σ-bonding and may be treated independently. Mirror plane (of moleule) s or p σ orbital p π orbital. the Coulomb integrals for all the arbon atoms are assumed to be idential. i.e. small differenes in α-values due to the different hemial environment of C atoms in a moleule suh as are negleted.. all resonane integrals between diretly-bonded atoms are assumed to be the same; whilst those between atoms that are not diretly bonded are negleted. i.e. φ ˆ φ dτ : if atoms i and j are diretly σ-bonded. i H j. : if atoms i and j are non-bonded. 4. all overlap integrals representing the overlap of atomi orbitals entred on different atoms are negleted. i.e. φi φ j. dτ : if i j (note - if i j then φi φ j. dτ sine it is assumed that the atomi orbitals are normalized) 6.

2 A Closer Look at the Seular Determinant The basi form of the seular determinant for the bonding arising from the overlap of two orbitals (from 4.9) is reprodued below. For three overlapping orbitals the approah outlined in Chapter 4 leads to a seular determinant of the form: From a omparison of the two seular determinants given above, it is beoming lear that all suh seular determinants have a harateristi struture:. eah row and olumn may be assoiated with one of the atomi orbitals; thus the first row and first olumn ontain information about the nature of orbital and its interations with the other orbitals, the seond row and seond olumn ontain information about the nature of orbital and its interations with the other orbitals.. The diagonal set of elements (omprised of those elements where row intersets olumn, row intersets olumn,.. and so on) inlude the values of the relevant Coulomb integrals (α, α et.).. The off-diagonal elements (omprised of those elements having different row numbers and olumn numbers) are equal to the relevant resonane integrals (e.g. at the intersetion of row and olumn ) This struture is summarised below, where the rows and olumns have been labelled with numbers identifying the assoiated atomi orbital: Orbital labels Off-diagonal elements onsist of the resonane integrals; in this instane that orresponding to the interation between orbitals on atoms & On-diagonal elements inorporate the Coulomb integrals for eah atom; in this ase that for atom. 6.

3 Linear Conjugated Hydroarbons C Moleules (-atom hain) Seular Determinant / Equation α E α E α E x x x where () x x( x. x.) (. x.) + (. x.) x x x x x x( x ) x or x ± i.e. x +,, Now E α x So the energies of the moleular orbitals are: E α. E α E α +. - Highest Energy - Lowest Energy 6.

4 The seular equations are: (α E) (α E) (α E) or, in terms of x, x + + [] + x + [] + + x [] For x (i.e. E α ) [] [] + i If we now apply the normalisation ondition ( - see Appendix ) i.e. For x (i.e. E α +. ) [] [] []. +. i If we now apply the normalisation ondition ( - see Appendix ) i ( ) (.5). (.77) 6.4

5 i.e. For x + (i.e. E α. ) [] [] []. +. i If we now apply the normalisation ondition ( - see Appendix ) (.5) (.77) i.e. In summary, Nodes Eletron Energy α.44 ( ANTI-BONDING ) of p z α ( NON-BONDING ) α +.44 ( BONDING ) < Letures - examples of bond order and harge distribution alulations for allyli speies > 6.5

6 General Solution (n-atom hain ; e.g. C n H (n+) onjugated polyenes) 4 (n ) n The seular determinant has the same basi form, whatever the hain length, as illustrated below: n x x x x n x Consequently the solutions also have the same basi form, whatever the hain length, and it an be demonstrated that: Orbital oeffiients: are given by s sin π ks n + [4] where n - total number of atoms in the onjugated hain s - atom number ( i.e.,,., n ) k - quantum number, identifying the MO (,,., n ) and the onstant of proportionality an be determined by applying the normalisation ondition. Example: onsider the highest energy MO ( k ) of the three arbon hain ( n ). i.e. πs s sin 4 π 6π sin, sin, 4 4 9π sin 4 These oeffiients are not normalized - to ahieve this we need to multiply them all by the same onstant value, hosen to ensure that the new values of the oeffiients satisfy the ondition that. i 6.6

7 From the general expression for the oeffiients given above it an be seen that the relative signs and sizes an also be visualised using the following trigonometri onstrution. Proedure :. Draw the n-regularly spaed atoms (,,., n ) in a straight line and then add two imaginary atoms (labelled and (n + ) ) at either end of the hain.. Sketh portions of sine waves between these two imaginary end atoms, ensuring that the imaginary end atoms orrespond to nodes of the sine wave.. The lowest energy MO has no other nodes and orresponds to half a sine wave; the next MO has one additional node in the middle and orresponds to a omplete sine wave; the next MO has two additional nodes. and so on. e.g. for the third MO ( k ) of the three arbon hain ( n ). Sinusoidal variation of amplitude of orbital oeffiients Value of s 4 Real atoms Orbital energies: are given by πk E α + os [5] n + where n - total number of atoms in the onjugated hain k - quantum number, identifying the MO (,,., n ) Note that the osine funtion varies only between the limiting values of and +. πk i.e. os + n + hene ( α + ) E ( α ) Consequently all moleular orbital energies must lie within an energy range of 4, ± of the original atomi orbital energy. Reall also that for an n-atom hain, in whih eah atom ontributes one atomi orbital to the onjugated π-system, there will be n overlapping atomi orbitals giving rise to n moleular orbitals. 6.7

8 Sine all these MOs are onfined to a fixed energy range, it follows that the average energy separation must derease as n inreases. This is illustrated below in an eletron energy diagram whih also shows the eletron oupany (for the neutral moleule) for the first three members of the series : Eletron energy (orbital energy) α α LUMO HOMO α + n 4 large Note:. As n inreases: the HOMO-LUMO separation dereases - onsequently the photon energy required to exite an eletron from the HOMO to LUMO also dereases. i.e. hν E E HOMO E LUMO dereases as n inreases. For onjugated arbon atom hains the photon energy hanges from the UV (for small n ) to the visible (large n ). Consequently, moleules with extended onjugated systems are oloured.. As n : the separation between any two energy levels dereases towards zero and the energy levels (although still disrete in priniple) effetively merge to give a ontinuous band of energy levels. The width of this band is 4 and is therefore determined by the effetiveness of overlap of the individual atomi orbitals and hene the strength of interation and the magnitude of the resonane integral. The system approahes the metalli state - that is to say that eletrial ondution an readily our (sine the HOMO-LUMO separation is essentially zero) and all wavelengths of visible light are readily absorbed. 6.8

9 Cyli Conjugated Hydroarbons General Solution (n-atom ring ; C n H n, yli onjugated hydroarbons) n n The seular determinant has the same basi form, whatever the ring size, as illustrated below: n x x x x n x Consequently the solutions also have the same basi form, whatever the ring size, and it an be demonstrated that the: Orbital energies: are given by πk E α + os [6] n where n - total number of atoms in the onjugated ring system k - quantum number, identifying the MO (,,,., n ) but, given the periodi and "even" (symmetrial about θ ) nature of the osine funtion, this permitted range of k-values may also be written in the form k, ±, ±,., k, ±, ±,., n n ± for even n. for odd n. Note that the osine funtion varies only between the limiting values of and +, and beause the osine funtion is an even funtion, the energy depends only on k. 6.9

10 πk i.e. os + n hene ( α + ) E ( α ) It is possible to represent equation [6] as a geometrial onstrution, and this is illustrated below. Consider n 6 ( e.g. C 6 H 6 - benzene) ; then the expression for E is where πk E α + os 6 k, ±, ±, (sine n is even). E Cirle of radius + α π 6 4π 6 + os π 6 Whilst the above diagram highlights the relationship between the geometrial onstrution and the equation for E, it also suggests that the energies of the orbitals may also be obtained using a simpler onstrution - namely by drawing the orresponding regular polyhedron (i.e. a hexagon for n 6) insribed inside the irle with one apex oinident with the bottom of the irle. E α + α α + α + α + < Leture - other examples, e.g. n, n 5 > 6.

11 Note : for these yli onjugated systems. The lowest-energy MO is always non-degenerate. The highest-energy MO may be non-degenerate (if n is even) or degenerate (if n is odd).. All the remaining solutions form pairs of degenerate MOs One onsequene of this arrangement of the MOs is that only speifi numbers of eletrons an be aommodated if a stable moleule is to result - this is embodied in the. Hükel Rule A stable, losed-shell onjugated yli struture is obtained for (4N + ) eletrons. i.e. stable strutures are obtained for, 6,,. eletrons. < Leture - examples of stable yli systems > What are the Wavefuntions? The wavefuntions are generally omplex (i.e. ontain imaginary parts), the exeptions being the non-degenerate solution(s). The MO of lowest energy ( k ) is always non-degenerate ( irrespetive of whether n is even or odd) and has the same oeffiient for eah and every ontributing atomi orbital - there are therefore no angular nodes in the wavefuntion. e.g. for n 6 Top view Side view The moleular plane is a nodal plane for all the π-mos, but for this partiular MO there are no angular nodal planes in the wavefuntion and all the interations between orbitals on adjaent atoms are of a bonding nature. 6.

12 The highest-energy MO of even n systems is also non-degenerate. The magnitude of the oeffiient is the same for eah and every ontributing atomi orbital but the sign hanges between adjaent atoms. e.g. for n 6 In this ase there are three angular nodal planes as marked (---) on the diagram - the wavefuntion thus hanges sign between eah pair of atoms and all the interations between orbitals on adjaent atoms are of an anti-bonding nature. The remaining degenerate pairs of solutions are omplex wavefuntions but it is possible to generate ompletely real linear ombinations of these whih are still solutions of the Shrödinger equation with the same energies (in the same way that the p x and p y atomi orbitals may be onstruted from the p + and p atomi orbitals). The oeffiients for the real funtions are: e.g. for n 6 πks ψ k + os. φs s n πks ψ k sin. φ s s n k ± s k ± s 6.