Lectue 4. Beyond the Hückel π-electon theoy 4. Chge densities nd bond odes Chge density is n impotnt pmete tht is used widely to explin popeties of molecules. An electon in n obitl ψ = c φ hs density distibution m n m c n φm (neglecting ll ovelp m density tems such s φ nφm ). We cn thus define the π-electon chge density q m t tom m s follows: q m = ncm, (4.) whee c m is the coefficient of the bsis obitls φ m in the molecul obitl ψ ; the sum is ove ll the π obitls with n electons in ech obitl (n =0,, ). Fo the gound stte of butdiene (see Fig. 3., lectue 3) n = fo the two occupied bonding obitls nd n =0 fo the ntibonding obitls, thus the chge densities t toms nd b e q q b = c = c b + c + c b = (0.37) = (0.60) + (0.60) + (0.37) =.00 =.00 (4.) By symmety we know tht q =q d nd q b =q c. Thus the π-electon density is unity t ech tom in butdiene. Exmintion of the obitls of benzene will show tht the sme is tue fo ll the cbon toms in benzene. To intepet mny phenomen in molecules, nd it is desible to estimte the degee of double-bond chcte in the bond joining two toms. By nlogy with the chge density t n tom m, we cn define π-electon bond ode between toms m nd n s follows: p = n c c. (4.3) mn m n
Fo ethylene the two electons in the bonding π obitl, whose wvefunction gives bond ode of. Fo butdiene we hve p c c + c c 0.89, (4.4) b = b b = p c c + c c 0.45. (4.5) bc = b c b c = By symmety p b =p cd so tht ccoding to Hückel theoy the oute bonds in butdiene hve much moe double-bond chcte thn the centl bond but less thn tht in n isolted ethylene molecule. The conventionl epesenttion of butdiene s single vlence stuctue (CH =CH- CH=CH ) is in close ccod with the Hückel pictue but fils to show tht thee is some doublebond chcte in the centl bond. Fig. 4. A common epesenttion of π electon densities in ethylene, butdiene nd benzene. 4. Intoduction of othe toms into Hückel theoy In the simple Hückel π-electon theoy, it ssumes tht ll the tomic obitls in n LCAO expnsion wee simil nd hd the sme enegy, so only two pmetes, α nd β, e equied. The simple Hückel π-electon theoy povides esonble desciption of ltennt hydocbons tht
consist of only cbon nd hydogen toms. If we wish to tet molecules, such s CH =CH-CH=O, C 6 H 5 N=NC 6 H 5 then we hve to del with toms othe thn cbon toms. One wy to tet this sitution is to expess α X nd β CX of these othe toms (X) in tems of the vlues fo benzene (α C nd β CC ), α X = α C + h X β CC (4.6) nd β CX = k CX β CC, (4.7) whee h X nd k CX e empiicl pmetes. 4.3 Extended Hückel π-electon theoy The Hückel π-electon theoy includes only the π obitls nd ignoes completely the contibution due to σ obitls nd hydogen toms. Attempts hve been mde to include the σ nd hydogen obitls hve been developed fo hydocbons nd clled ll-electon theoies. The bsic fetues of the theoy e tht the tomic obitl bsis consists of ll the vlence tomic obitls nd cbon s nd p obitls. It is moe successful thn the simple Hückel π-electon theoy, but it hs mny limittions nd is not vey stisfctoy fo pol molecules. This is becuse like the Hückel π- electon theoy, the extended Hückel π-electon theoy is bsed on independent electon model, which ignoes electon-electon intections. 4.4 PPP method The Pise-P-Pople (PPP) method is simil to Hückel π-electon theoy, except tht it includes electon-electon epulsion. We cn wite H PPP =H Huckle + epulsion. (4.8) 3
4.5 Ab-initio Clcultions Ab-initio is Ltin tem nd mens fom the beginning o fom fist-pinciple. The fist b-initio clcultion ws pobbly the one on H molecule by Heitle-London in 97, but it ws not ttempted to ny lge molecules until the development of electonic computes in the 950s. The extensive poduction of b-initio clcultions begn in 960s with the widesped vilbility of pogms fo Self-Consistent Field (SCF) clcultions on polytomic molecules. Inteestingly, the eve-incesing compute powe llows us to undestnd molecul electonics tht could ultimtely poduce even moe poweful computes. Htee-Fock SCF method is one of widely used b initio methods. To undestnd the theoy, let us stt with two-electon molecule. The two electons hve obitls (wvefunctions), φ nd φ b, espectively. The wvefunction of the two electon system, Ψ(,), mybe witten s Ψ(,)= φ ()φ b (), whee numbes nd epesent the two electons. This wvefunction is used in the Htee SCF, but it hs seious poblem. Since the electon is Femion (spin is hlfintege), it hs to obey the Puli Exclusion Pinciple, which equies n ntisymmetic wvefunction when electons nd exchnge positions. Clely the wvefunction given bove is not ntisymmetic. A simple wy to constuct n ntisymmetic wvefunction fo the mny electon system is to use the Slte deteminnt, which tkes the fom of φ( ) φb( ) Ψ (,) = = [ φ( ) φb( ) φ( ) φb( ) ], (4.9) φ ( ) φ ( ) b fo the two electon system. The pefcto in the expession is equied fo nomliztion. It is esy to veify tht Ψ(,) = - Ψ(,) s expected fo ntisymmetic wvefunction. Also we note tht Ψ(,) = - Ψ(,) = 0, which mens the two electons cnnot occupied the sme stte, esult of the Puli exclusion pinciple. The bove considetions ignoe electon spin nd the molecul obitls φ nd φ b e functions of spce only. It is the stightfowd to include it in the 4
wvefunction. Electons cn hve eithe spin up(+/) o down (-/), which cn be descibed by two spin functions, α nd β, s follows α ( ) =, α( ) = 0 β ( ) = 0, β ( ) =. (4.0) The molecul obitls now hve both spce nd spin pts nd tke the foms of, αφ, βφ, αφ b nd βφ b, which e often efeed to s spin-obitls. If the two-electon system hs only one molecul obitl, φ, vilble, then the wvefunction is α () φ( ) β () φ( ) Ψ (,) =, (4.) α() φ ( ) β () φ ( ) which eflects the fct tht the fist electon tkes spin up stte, the second electon must tke spin down, vice ves. The wvefunction given by 4. is often clled esticted Htee-Fock wvefunction, which is typiclly pplied to close-shell molecules (electons e pied). Unesticted Htee-Fock wvefunction does not equie the two electons to occupy the sme obitls (spce-pt of the wvefunction). Fo exmple, α () φ( ) β () φb( ) Ψ (,) = (4.) α() φ ( ) β () φ ( ) b is n unesticted Htee-Fock wvefunction. Fo open-shell molecules, both esticted nd unesticted Htee-Fock wvefunctions e used. The Hmiltonin of the two-electon molecule is Hˆ ˆ ˆ = + V, (4.3) I + VI + = H + H + whee ˆ H = +V I nd H ˆ = +V I e the so-clled coe Hmiltonins of electons nd, espectively; ech consists of the kinetic enegy of the electon nd electosttic 5
ttction between the electon nd ll the nuclei in the molecule. The enegy of the wvefunction 4.9 o 4. is E = = * Ψ(,) Hˆ Ψ(,) dv dv [ ] ˆ ˆ ψ( )ψb( ) ψ( )ψb( ) ( H + H + )[ ψ( )ψb( ) ψ( )ψb( ) ] dvdv. (4.4) Note tht spin-obitls in Eqs. 4. nd 4. e denoted s ψ ( ) α() φ ( ), ψ b ( ) β () φ b ( ), ψ ( ) α() φ ( ) nd ψ b ( ) β () φ b ( ), espectively. Expnding the bove eqution, we cn expess the enegy s E = H + H + J K, (4.5) bb b b whee H H bb = ˆ ψ ( ) H ψ () dv, (4.6) = ˆ ψ b( ) H ψ b() dv (4.7) J b ψ ( ) ψ b() ψ () ψ b() dvdv (4.8) = clled Coulomb integl, epesenting electosttic epulsion between the two electons, nd K b ψ ( ) ψ b() ψ () ψ b() dvdv, (4.9) = clled exchnge integl, puely quntum mechnicl effect. Fo n-electon molecule, we cn genelize 4.5 s n n n E = H + ( J s K s ). (4.0) = = s= If we pply the vition theoem to the bove eqution nd vy the molecul obitls then thee is, in pinciple, minimum enegy tht cn be eched nd the wvefunction ssocited with this is clled the Htee-Fock wvefunction. 6
The mthemticl pocedue to obtin the Htee-Fock wvefunction is summized s follows. The fist step is to expess the molecul obitls s line combintions of pe-defined set of one-electon function known s bsis functions, simil to wht we hve see in LCAO ppoximtion. These bsis functions e usully centeed on the tomic nuclei nd so be some esemblnce to tomic obitls. Howeve, the mthemticl tetment is moe genel thn this nd ny set of ppopitely defined functions my be used. We define molecul obitl s N φ i = cµ iχ µ (4.) µ whee c µi e the molecul obitl expnsion coefficients. The bsis functions, χ, χ, χ N e nomlized. The omn subscipts e fo diffeent molecul obitls nd Geek subscipts e fo diffeent bsis functions. Mny b initio methods use Gussin-type function s bsis functions, which hve the genel fom, α n m l g( α, x, y, z) = cx y z e (4.) whee =x +y +z, nd n, m nd l e integes. The constnt c cn be detemined by nomlizing the function, g dv =. (4.3) The next step is to detemine the molecul obitl expnsion coefficients, c µi, which is chieved with the vition theoem in Htee-Fock theoy. The theoem sttes: If the expecttion vlue of the enegy is clculted fom n ppoximtion solution of the Schodinge eqution, then this enegy is lwys gete thn the exct gound stte enegy fom tht Hmiltonin. In othe wods, the enegy of the exct wvefunction seves s lowe bound to the enegies clculted by ny othe nomlized ppoximted wvefunction. Thus the poblem becomes one of finding the set of coefficients tht minimize the enegy of the esultnt wvefunction. The vition 7
theoem leds to the following equtions descibing the molecul obitl expnsion coefficients, c νi, deived by Roothn nd Hll: N ν = ( F ε S ) c = µ =,,... N (4.4) µν i µν νi 0 whee ε i is the enegy of n electon in molecul obitl, χ i. F µν is the clled Fock mtix nd it epesents the vege field of ll the electons on ech obitl. Fo close shell system it is given by N N coe F µν = H µν + Pλσ ( µν λσ ) ( µλνσ ) (4.5) λ= σ= whee H coe µν is mtix epesenting the enegy of single electon in the field of the be nuclei, nd P is the density mtix, given by P λσ occupied * cλ i= = icσi. (4.6) The coefficients e summed ove the occupied obitls only, nd the fcto two comes fom the fct tht ech obitl holds two electons. S µν is the ovelp mtix, s we hve seen ledy. Both the Fock nd ovelp mtices depend on the molecul obitl expnsion coefficients, so eqution 8.6 must be solved itetively. The pocedue is clled self-consistent field (SCF) method. (µν λσ) in the bove eqution is shot nottion of µν λσ ) = φµ () φν () φλ () φσ () dvdv, (4.7) ( which epesents two electon epulsion. Unde Htee-Fock tetment, ech electon sees ll of the othe electons s n vege distibution, so the instntneous electon-electon intection o coeltion is neglected. Fo exmple, the Htee-Fock wvefunction fo the gound stte of H gives pobbility density of one electon which is independent of the position of the othe 8
electon. In elity, if the fist electon is ne one nucleus then thee should be gete chnce of finding the second electon ne the othe nucleus; tht is, the positions of the two electons should be coelted. Highe level theoies, such s Density Functionl Theoy, hve been developed to include the electon-electon coeltion. 4.6 Density Functionl Theoy (DFT) The DFT ppoch is bsed on sttegy of modeling electon coeltion vi genel functionls of the electon density. A functionl is function of function. Such methods owe thei oigins to the Hohenbeg-Kohn theoem, which demonstted the existence of unique functionl which detemines the gound stte enegy nd density exctly. The theoem itself does not povide the fom of the functionl. Following the wok by Kohn nd Shm, the gound stte enegy of molecul system (excluding the nucle-nucle epulsions) cn be witten s unique functionl of the gound stte density, n(), E n( ) n( ) ' [ n( )] = T[ n( )] Ze d + dd' + E XC [ n( )], (4.8) R R In the expession, T is the kinetic enegy tem, the second tem descibes the potentil enegy of the electon-nucle ttction nd the thid tem is the epulsion between electons. The lst tem, E XC, is the so-clled the exchnge-coeltion tem, which includes: the exchnge enegy ising fom the nti-symmety of the quntum mechnicl wvefunction nd dynmic coeltion in the motions of the individul electons. Hohenbeg nd Kohn demonstted tht E XC is detemined entiely by the electon density. Pcticl implementtion of DFT equies good ppoximtion of E XC. Mny ppoximtions fo E XC hve been developed. Once the functionl of E XC is obtined, one cn detemine the gound stte enegy nd electon density by minimizing the expession 4.8. 9
Both the HF nd DFT methods hve been used to compute electon tnspot though molecules, but they suffe simil poblem the inheent inccucy in computing the LUMO enegy nd wvefunctions. The eos e, howeve, diffeent. Fo exmple, HF method oveestimtes the HOMO-LUMO gp since the LUMO enegy is too high, while DFT undeestimtes the gp. So it is cle tht both methods give bette esults if pocess is dominted by the HOMO of the molecule thn tht dominted by the LUMO. Homewok 4. Detemine the chge density nd bond ode of benzene. 4. Deive Eq. 4.5 fom Eq. 4.4. 0