Physical Nature of the Covalent Bond Appendix H + H > H 2 ( ) ( )

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1 Physical Nature f the Cvalent Bn Appeni his stuy f the nature f the H cvalent bn frms a mlecular rbital as a linear cmbinatin f scale hyrgenic rbitals, LCAO-MO. he quantum mechanical integrals necessary t carry ut the require calculatins are prvie immeiately belw. Cnsult the first five sectins f Chapter 3 f ur tetbk fr an verview f the cvalent bn in the hyrgen mlecule in. he reactin fr its frmatin is, H H ----> H Fr the hyrgen atm: < E > -0.5; < > 0.5; < > -.0. he irial herem is epresse in a variety f ways: E E he evaluate integrals require fr this eercise are prvie belw: Overlap integral: Kinetic energy integrals: : ep( α) α S α, α aa α, α 3 α : ab α, : ep α α α 3 Ptential energy integrals: : α ep( α) aa α, α : α ab α, 5 3 α ep( α) Haa an Hab: tal energy integral: aa( α, ) aa( α, ) Haa α, : Hab α, Haa α, Hab( α, ) : E α, S α, : ab( α, ) ab( α, ) Bning mlecular rbital: Ψ bm α,, y, z, : ep α π y z ep S α, α y z Demnstrate wavefunctin is nrmalize: Ψ bm (,, y, z, ) y z.0000 Anti-bning mlecular rbital: Ψ am α,, y, z, : ep α π y z ep S α, α y z

2 Demnstrate wavefunctin is nrmalize: Ψ am (,, y, z, ) y z.0000 Nn-bning mlecular rbital: Ψ nm α,, y, z, : ep α π y z i ep α y z Demnstrate wavefunctin is nrmalize: ( Ψ nm (,, y, z, ) ) y z.0000 he Cvalent Bn in H ariatinal Calculatin f the Mlecular Grun State an Bn Energy he mlecular rbital fr the hyrgen mlecule in is frme as a linear cmibinatin f scale hyrgenic s rbitals centere n the nuclei, a an b. Ψ a b S Where a π ep α r α ( a) 3 b π ep α r ( b) S ab he energy peratr: H r r r r a r b he energy integral t be minimize by the variatin meth: E ( a b) H( a b) S Haa Hab S where Haa ( a a) a a a a a r a Hab ( a b) a b a b a r a r b r b a b an Haa aa aa an Hab ab ab an Hba Hab. hese integrals an the verlap integral S can be fun in the appeni abve.

3 . Plt E(α,) vs, fr α fr values f ranging frm 0. t 0 in increments f Yu can make space by pressing <enter>. Yu can eliminate space by pressing <elete>. Use the markers fr the - an y-ais t fin the ptimum internuclear separatin,, an the grun-state energy. α : :., E( α, ) Ntice hw shallw the energy minimum initially is. Yu can eagerate the minimum by pltting frm -0.6 t -0.4 n the y-ais.. Minimize the energy with respect t with α. his will require a Given/Fin slve blck an see values fr α an. eprt the ptimum value (equilibrium bn istance) an the grun-state energy. Cmpare these calculate values t the literature values f.00 a fr an E h fr the E. α : : Given E α, 0 : Fin.498 E α, % E( α, ) % E( α, ) % Demnstrate that the calculate hyrgen mlecule in energy es nt satisfy the irial herem. ab( α ) ab( α ) aa α,, : S( α, ) aa α,, : S( α, ) E( α, ).4759 E( α, ) Write the equatin fr breaking the H bn. hen use the theretical value f the grun state energy t calculate the bn energy f the hyrgen mlecule in an cmpare it t the eperimental value, 0.09 E h. Yu may recall that the hyrgen atm grun state energy is -0.5 hartree. HyrgenMleculeIn HyrgenAtm HyrgenIn BnEnergy % %.09

4 4. he first three eercises refer t a variatinal calculatin base n the hyrgen s rbital (α ). Nw treat bth α an as variatinal by minimizing the energy simultaneusly with respect t α an. Given E α, 0 α E α, 0 α : Fin( α, ) α.380 E α, Demnstrate that the calculate hyrgen mlecule in energy es satisfy the irial herem. ab( α ) ab( α ) aa α,, : S( α, ) aa α,, : S( α, ) E( α, ).0000 E( α, ) Cmpare E an with the literature values. ecalculate the H bn energy an cmpare it with the eperimental value. Cmment n the agreement between thery an eperiment as cmpare with that fun in part 3. HyrgenMleculeIn HyrgenAtm HyrgenIn BnEnergy % % E( α, ) % 6. he vibratinal frce cnstant is the secn erivative f the energy with respect t evaluate at the energy minimum. Calculate the frce cnstant an cmpare it t the eperimental value f.03 in atmic units. k : E( α, ) k k % 7. he appeni cntains the bning an antibning rbitals fr H. On ne graph plt Ψ fr the bning an antibning rbitals alng the bn ais (i.e. set y an z t zer). On anther graph plt Ψ fr the bning an antibning rbitals. Cmpare these rbitals. y : 0 z : 0 : 5, α : Ψ bm ( α,, y, z, ) Ψ am ( α,, y, z, ) 0 ( Ψ bm ( α,, y, z, ) ) ( Ψ am ( α,, y, z, ) )

5 hese figures shw that the electrn ensity is greatest at the nuclear centers, an that the bning mlecular rbital has significant electrn ensity in the internuclear regin. Just hw much charge is transfere t the bn regin will be calculate later. he antibning mlecular rbital has a ne eactly between the tw nuclei an thus remves charge frm the internuclear regin. his, f curse, is why it calle an anti-bning mlecular rbital. A Mechanism fr Cvalent Bn Frmatin in H he fllwing tw-step mechanism is prpse fr cvalent bn frmatin: he hyrgen atm an prtn frm an intermeiate, nnbning state at the equilibrium bn length (.0033) with the ptimum value f the ecay cnstant (α.38). he nn-bning state, N nm (a ib), mrphs int the final mlecular state, N bm (a b). Initial AtmicState H atm H in Intermeiate NnBningState a ib Ψ nm Final MlecularState a b Ψ bm S 8. We nw calculate the kinetic (), ptential () an ttal energy (E) fr each f the species invlve in the mechanism. In previus wrk we calculate the fllwing values fr the hyrgen atm. (he energy f the hyrgen in by cnventin is 0). E h : h : h : he energy f the nn-bning intermeiate state is calculate as fllws. E nb a a H aha bhb Haa aa aa because Hbb Haa tal: Haa α, Kinetic: aa α, Ptential: aa α,.58 Nuclear Ptential Energy: In eercise 4 we calculate the energy cntributins f the final mlecular state f H. E α, tal: Kinetic: Ptential: ab( α, ) aa α, S α, ab( α, ) aa α, S α,.730 Nuclear Ptential Energy: 0.499

6 We are nw in a psitin t cnstruct the fllwing quantitative summary f the mechanism. ne nn E Initial AtmicState H atm α H in Intermeiate NnBningState a ib Ψ nm α Final MlecularState a b Ψ bm S α Interpretatin f the Mechanism he first step is atmic an energic. It is atmic because there is n cnstructive interference between the atmic rbitals. It is energic because kinetic energy increases mre than ptential energy ecreases. he increase in kinetic energy an the ecrease in ptential energy are ue preminantly t atmic rbital cntractin frm α in the hyrgen atm t α.38 in the hyrgen mlecule in. he secn step is eergic an mlecular. he atmic rbitals frm a mlecular rbital leaing t charge reistributin relative t the nn-bning intermeiate state. his reistributin causes a ecrease in kinetic energy because the electrn is nw mre elcalize, an an increase in ptential energy because the electrn is rawn away frm the nuclear centers int the high-ptential energy internuclear regin. Overall the kinetic energy increases an ptential energy ecreases n bn frmatin, an the virial therem is satisfie. his is the surce f the cnclusin that cvalent bning is a ptential energy effect. Hwever, as this analysis shws, bth kinetic an ptential energy play imprtant rles in cvalent bn frmatin. his mechanism clarifies an imprtant issue. It is cmmnly thught, an frequently taught, that frm a ptential energy pint f view the internuclear regin is the mst favrable place fr electrn ensity because it is miway between t nuclei. Hwever, this mechanism clearly shws that the reistributin f charge int the internuclear regin ue t the interference term in the bning electrn ensity ccurs with an increase in ptential energy (0.058 E h ). Frm a purely electrstatic perspective the prefere lcatin fr an electrn is in the nucleus. he Bning Electrn Density 9. Hw much f the ttal electrn ensity, Ψ bm, actually cntributes t cvalent bning? his questin can be answere by cmparing the electrn ensity in the intermeiate nn-bning state with that f the final mlecular state. First we will emnstrate that Ψ nm (a b )/ an that this wave functin is nrmalize because a an b are nrmalize. a Ψ nm a a b a a a τ b Ψ nm shws the "atmic" electrn ensity - the electrn ensity that wul eist in the absence f cnstructive interference between the tw atmic rbitals, sa an sb.

7 he nn-bning wave functin is given in the appeni. Graph Ψ bm an Ψ nm alng the bn ais an cmpare them. ( Ψ bm ( α,, y, z, ) ) ( Ψ nm ( α,, y, z, ) ) he questin pse in sectin 9 can nw be answere. he bning electrn ensity can be efine as the charge reistributin that ccurs in step f the mechanism. Plt the ifference between the ttal electrn ensity an the nn-bning electrn ensity alng the bn ais. Interpret the plt. Fr eample, where es the electrn ensity in the internuclear regin cme frm? What is the sign n the ptential energy term fr this transfer f electrn ensity t the internuclear regin? If this is an energic prcess, hw is it fune? : Ψ bm ( α,, y, z, ) ρ b α,, y, z ( Ψ nm ( α,, y, z, ) ) ρ b ( α,, y, z) his graph shws that electrn ensity is being transferre frm the nuclear centers (lw ptential energy) t the internuclear regin (high ptential energy). his eplains why ptential energy increases uring step. he kinetic energy ecreases mre because this reistributin f charge als invlves charge elcalizatin, s it is the larger ecrease in kinetic energy that funs the transfer f charge t the internuclear regin.. he electrn ensity in the internuclear regin is equal t a plynmial in the verlap integral S an is given by 0.738S( - S)( S ). Use the value f S yu calculate earlier t evaluate the amunt f charge in the bning regin f H. Cmment n this value. Fr eample, hw es its value cmpare with a literal interpretatin f the Lewis electrn pair mel f chemical bning that says that chemical bns are frme by sharing electrns between tw nuclei. ( ) ( ) 0.738S α, S α, 0.577S α, 6. %

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