Subject PHYSICAL Pper No nd Title TOPIC Sub-Topic (if ny) Module No., PHYSICAL -II QUANTUM Hückel Moleculr orbitl Theory CHE_P_M3 PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
TABLE OF CONTENTS. Lerning outcomes. Hückel Moleculr Orbitl (HMO) Theory 3. Appliction of HMO theory 3. Ethylene 4. Summry PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
. Lerning Outcomes After studying this module, you shll be ble to Apprecite the simplifiction introduced by E Hückel for studying orgnic conjugted molecules. Find the π-electron energy nd wve function for ethylene molecule Understnd the bsis of moleculr orbitl digrm for π-electron systems. Hückel Moleculr orbitl (HMO) theory HMO theory is n pproximte method which simplifies vrition method to tret plnr conjugted hydrocrbons. This theory trets the π electrons seprtely from σ electrons. Properties of the conjugted molecules re primrily determined by π-electrons. The considertion of σ-π electro seprtion in multi-electron molecule in HMO theory reduces the problem to the study of only π electrons. HMO clcultions re crried out using vrition method nd LCAO(π)-MO pproximtion. According to LCAO-MO pproximtion, the MO is written s, n i c i pzi HMO theory pproximtes the π moleculr orbitls s liner combintion of tomic orbitls. For plnr conjugted hydrocrbon, the only tomic orbitls of π symmetry re the pπ orbtils on crbon. In this module, we hve consistently ssumed the plne of the molecule s x-y plne with π orbitl in the z xis, perpendiculr to the moleculr plne. -() For two π electron system φ becomes, PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
c pz c pz -() And the pproximte energy is given by, E * * H d ^ d -(3) The Hmiltonin Ĥ incorportes the effect of the interction of π electron with the rest of the molecule (nuclei, inner electrons, σ bonds) in n verge wy In HMO method, π electrons re ssumed to be moving in potentil generted by the nuclei nd σ electrons of the molecule. The Seculr determinnt obtined for two π electron system cn be written s, H H ES ES H H ES ES c c 0 -(4) In order to solve the Seculr determinnt for n n-π electron system, Hückel treted the Hii, Hij, Sij nd Sij integrls s prmeters tht cn be evluted empiriclly by fitting the theory to experimentl results.. H H d ij ( i H ij ( i. Sij ^ * p i z j) j) p i p j z Resonnce integrl * d z Coulomb integrl p z j S ij 0 ( i ( i j) j) Overlp integrl Tking into ccount the ssumptions of HMO theory, the seculr determinnt reduces to, PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
E E 0 -(5) In this mnner, Hückel determinnt cn be generted for n n-π electron system The expnsion of n n x n Hückel determinnt yields polynomil eqution tht hs n rel roots for n-π electron system leding to n energy levels nd n moleculr orbitls. The energy of ny th moleculr orbitl (MO) is given by E x, where xis the th root of the polynomil. -(6) The vlues of the coulomb integrl α nd the resonnce integrl β re lwys negtive. If the root x is positive, then the energy level corresponds to more negtive vlue nd is more stble (Bonding moleculr orbitl) while negtive vlue of root gives ntibonding moleculr orbitl. 3. Appliction of HMO theory In this section, we shll pply HMO theory to ethylene hving π electrons with one double bond. 3. Ethylene We consider here the cse of ethylene, CH4. Ethylene is 6 electron system but HMO theory reduces this to two π electron system. PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
HMO theory trets ethylene s two electron problem, with one π electron on ech crbon tom in p-orbitl, perpendiculr to the moleculr plne. These two tomic orbitls (AOs) combine to form moleculr orbitls (MOs). Lbeling the two crbons s nd, The Seculr determinnt obtined for ethylene molecule is of the form, c H H c pz pz ES ES H H ES ES c c 0 Tking into ccount the ssumptions of HMO theory, the seculr determinnt trnsforms into Hückel determinnt s, H S H S H H S S 0 -(7) -(8) E c E c 0 PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
c c E 0 E 0 E E 0 -(9) Let, E This reduces the Hückel determinnt s, -(0) 0 -() which on expnsion gives, 0 -() So, the energies of the moleculr orbitls re, If If, E ( Bonding Moleculr Orbitl BMO), E ( Antibonding Moleculr Orbitl ABMO) The number of moleculr orbitls tht re generted using LCAO pproximtion re equl to the number of combining tomic orbitls. PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
HMO energy level digrm for ethylene Totl (π bond) energy = (α + β) -(3) [As there re two electrons in the orbitl with energy α+β] E Using λ s, the seculr equtions re obtined s c c 0 -(4) c c c c 0 0 -(5) PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
If If, c c, c c -(6) c c c -(7) Now, if we pply the normliztion condition, * d -(8) ( c c ) d pz pz c c ( pz ) d pz ( pz pz pz pz ) d If i = j, ψi = If i j, ψi = 0 c [ 0] c -(9) The sum of the squres of the coefficients is lwys unity. Moleculr λ E c c Number of orbitl nodes BMO - α+β 0 ABMO α-β c c c c PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
With this, one cn now write the two normlized wvefunctions corresponding to two Hückel moleculr orbitls for ethylene s, BMO ( pz pz ) -(0) ABMO ( pz pz ) -() The pictoril representtion of the two Hückel moleculr orbitls viz., BMO nd ABMO for ethylene is shown below. PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
The totl π electron energy (or π electron binding energy) Eπ is tken s the sum of the energies corresponding to ech π electron. For ethylene, the totl π electron energy Eπ is given by E Another relted term is π bond formtion energy which is the energy relesed when π bond is formed. Since the contribution of α is sme in the molecules s in the toms, so we cn consider the energy of two electrons, ech one in isolted nd non-intercting tomic orbitls s α, then the π bond formtion energy becomes, E ( bond formtion ) E E isolted -() -(3) In generl, E ( bond formtion) E n, where n number of C toms in the molecule For ethylene, E ( bond formtion) -(4) β is the totl π bonding energy on formtion of the ethylene molecule. PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
4. Summry HMO theory is n pproximte method which simplifies vrition method to tret plnr conjugted hydrocrbons This theory trets the π electrons seprtely from σ electrons. Properties of the conjugted molecules re primrily determined by π-electrons. HMO clcultions re crried out using vrition method nd LCAO(π)-MO pproximtion Appliction of HMO theory to ethylene molecule Ethylene is 6 electron system but HMO theory reduces this to two π electron system. ( BMO pz pz ) ABMO ( E pz pz ) Moleculr λ E c c Number of orbitl nodes BMO - α+β 0 ABMO α-β c c c c PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory
PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory