Many Electron Theory: Particles and holes. Unitary transformations.

Transcription

1 Many Electon Theoy: Paticles and holes. Unitay tansfomations. Continued notes fo a wokgoup Septembe-Octobe 00. Notes pepaed by Jan Lindebeg, Septembe 00

2 Heny Eying Cente fo Theoetical Chemisty Equivalencies in the Paise-Pa-Pople model. Hückel established a emakably useful model fo the chaacteization of conjugated systems, paticulaly the catacondensed aomatic hydocabons. Chales Coulson and his students, especially H. C. Longuet-Higgins gave this model a athe unique status as both a tool fo the applied chemist and a fomalism that seved to develop concepts such as chages and bond odes. The one-electon pictue in the Hückel model cannot account fo specta such that tiplets and singlets ae diffeent. Thus it came to Paise and Pa and to Pople to offe the genealization which caies thei name. Basic to the PPP-model is the appeciation of the smallness of Coulomb inteaction integals that involve poducts of atomic obitals that ae { } othogonal. The spin obital basis, B: us s = 1,, M, is now the set of so called π-obitals, two of opposite spin fo each atom. We extend the { } notation so that the basis is B: us, u s s = 1,, M. Negative subscipts efe to β-type spin obitals, while positive ones concen α-type. Standad atomic obitals ae nonothogonal with ovelap integals between neighboing atoms in the ange Löwdin othogonalization offes an othonomal set, as simila as possible to the oiginal one. An illustation fom the ethylene molecule seves as an indicato fo the easons behind the concept of Zeo Diffeential Ovelap, ZDO. Two simple Slate type atomic obitals ua(), ub() ae centeed on the two cabon nuclei and give contou diagams in a plane though the nuclei pependicula to the molecula plane such as in Fig. 1: Jan Lindebeg

3 Heny Eying Cente fo Theoetical Chemisty u a u b Thei ovelap integal is S, the symmetically othonomalized obitals ae uˆ = u κ u λ; uˆ = u κ u λ; a a b b b a cos( ω) sin( ω) κ = ; = ;sin= S. cos λ cos ω ω ω They have diagams such as in Fig. : û = u κ u λ û = u κ u λ û = u κ u λ a a b a a b a a b 3 Jan Lindebeg

4 Heny Eying Cente fo Theoetical Chemisty A slight distotion is pesent. The poduct density of the othonomal obitals gives a density as given in Fig. 3: ˆ ˆ uu a b Regions of positive and negative density altenate and have small amplitudes. Calculations have shown that Coulomb integals involving this density give small values. Neglect of such integals constitutes ZDO. Hückel neglected matix elements between obitals that ae fathe away than neighboing atoms. This featue is nomally used in the PPP model as well. The model hamiltonian is then H( PPP)= α( aa + a a )+ s βs aas + a a s ( + ) + γa aa a + 1 sγs a a + a a asas a sa s ( pq s) = δpqδsγ p Altenant hydocabons ae those whee the cabon atoms can be sepaated into two disjoint subsets so that an atom in one set only has neighbos in the othe set. Numbeing can then be aanged so that one set is labeled by Jan Lindebeg 4

5 Heny Eying Cente fo Theoetical Chemisty even numbes, o staed as used conventionally, and the othe then has odd labels and ae unstaed. Matix elements β s equal zeo when both indices ae eithe even o odd. It was shown in Hückel theoy that thee is a paiing between occupied and unoccupied molecula obitals fo altenant hydocabons and it was late demonstated that the PPP model peseves this featue. It is a consequence of a symmety opeation that is epesented by a unitay opeato. Hee we conside the fom U = exp ( Π) ; Π = π ( aa a a)= Π The elementay annihilation and ceation opeatos tansfom as follows and thus a, Π π [ ]= [ a, a a ] = π a ; a, Π π [ ]= [ a, a a ] = π a ; [ ] = a, Π π [ a, a a] = π a ; [ a, Π] = π [, ] = π a a a a ; Ua U = a ; Ua U = a ; Ua U = a ; Ua U = a ; The tansfomation of the hamiltonian can then be woked out to be s UH( PPP) U = α( a a + aa)+ s βs a a s + aas + γa a aa + 1 sγs a a + aa a sa s asas = ( α + γ + sγ s )( 1 aa a a )+ H( PPP) ( + ) and with cetain conditions thee is an invaiance unde this tansfomation. 5 Jan Lindebeg

6 Heny Eying Cente fo Theoetical Chemisty The dipole moment opeato is epesented, in the PPP model, as ( ) D= er a a + a a and is "odd" unde the tansfomation. So is the dipole velocity opeato, + ( ) D= 1 [ D, H( PPP) e ]= R R β a a a a ih ih s s s s s The detemination of optical otatoy powe equies a epesentation of the magnetic moment opeato. London consideed the diamagnetic popeties of conjugated systems in the Hückel model and concluded that a pope fom fo the intoduction of a magnetic field in the hamiltonian was to let the obitals be field dependent and to obtain field dependent β-paametes: βs( B)= β ie s( 0) exp[ h c B ( R Rs) ] It follows that H( PPP; B)= H( PPP; 0) B M+ H PPP ie = ( ; 0) B s h c ( R Rs) βs( aas + a a s)+ and the esult is that the magnetic moment opeato changes sign unde the tansfomation. NOTES: J. L. & Y. Ö.: Deivation and Analysis of the Paise-Pa-Pople Model J. Chem. Phys. 49(1968)716. J. L. Consistency Requiement in the Paise-Pa-Pople Model. Chem. Phys. Lett. 1(1967)39. J. Michl: Seveal papes in JACS on magneto-optical popeties in (?) J. L. & J. Michl: On the Inheent Optical Activity of Oganic Disulfides J. Am. Chem. Soc. 9(1970)619. Jan Lindebeg 6

7 Heny Eying Cente fo Theoetical Chemisty Petubation expansion, van Vleck foms. Conflicting views on the bonding in conjugated systems, the valence bond o esonance theoy as opposed to molecula obital desciptions, ae illuminated by a study of the PPP-model in the limit of sepaated atoms. This is the situation when the β-paametes ae consideed small and the tem whee they occu in the hamiltonian is taken as a petubation. The othe tems commute with the atomic numbe opeatos,, and zeoth ode states ae detemined only by the n = a a + a a eigenvalues of these. Nomally we expect that all atoms have one electon and thee will then be a manifold of M degeneate states to be dealt with by degeneate petubation theoy. Van Vleck consideed such a poblem by means of unitay tansfomations. A geneal opeato appoach to petubation theoy consides the fom [ + ] ( ) = + + [ ] exp iθ H V exp iθ H { V i Θ, H }+ i[ Θ, V] 1 { [ Θ, [ Θ, H] ]}+ and the idea is to eliminate the fist ode tem, so that V + i[ Θ, H]= 0 [ + ] ( )= + [ ]+ exp iθ H V exp iθ H Θ, V The analysis is limited hee to fist ode tems in Θ. i A solution to the fist ode equation is obtained by obseving that and that d dt [ ] = [ ] = iht iht iht iht iht iht e Θe ie H, Θ e e Ve 7 Jan Lindebeg

8 Heny Eying Cente fo Theoetical Chemisty iht iht e a e = a exp itα + itγa a + it s γ sns = aa a exp itα + it s γ sns + aa a exp itα + itγ + it s γ sns The exponential opeatos commute with the othes. Futhe algebaic manipulations leads to an expession fo the second ode petubation opeato that is expessed in tems of atomic spin opeatos: S ex iey ez = ( aa + a a)+ ( aa + a a)+ a a a a ( ) The esult is a Heisenbeg spin hamiltonian of the fom J s H eff 1 s s s = J + S S s βsβ = s β sβs β s = α αs + γ γs + p, s γp γ sp np s This hamiltonian acts in the space whee thee is pecisely one electon on evey atomic site. Negative values fo the effective exchange integals J indicates a pefeence fo antipaallel aangements of spins on neighboing atoms and a possible paiing of obitals into spin singlets, the valence bond type functions. Eigenstates of the effective hamiltonian can be woked out fom the ules of valence bond theoy as given by Eying-Walte-Kimball. Josef Paldus has studied the PPP-model extensively and with egad to petubation theoy and Al Matsen has given an extensive account of the use of the symmetic goup fo hamiltonians such as the PPP one. The Hubbad model obtains when only one-cente γ's ae kept. Jan Lindebeg 8