A Mechanistic Approach to Bond Formation in H 2

Transcription

1 A Mechanistic Apprach t Bnd Frmatin in H Frank Riux Department f Chemistry Cllege f Saint Benedict Saint Jhnʹs University St. Jseph, MN 5674 Intrductin Ruedenbergʹs innvative analysis f the cvalent bnd ( ) has received cnsiderable attentin in the pedaggical literature ( ) as well as a number f excellent reviews in the primary literature that are accessible t nn specialists (4 6). he purpse f this study is analyze three mechanisms fr cvalent bnd frmatin in H that are suitable fr an undergraduate curse in quantum chemistry and reveal Ruedenbergʹs central message a full understanding f the nature f the chemical bnd requires cnsideratin f the rle f electrn kinetic energy. All three mechanisms pstulate a single intermediate state fr the reactin fr H bnd frmatin. HH H Scaled hydrgenic wave functins will be used t calculate the initial atmic state, the final mlecular state and the intermediate states. Mathematical details can be fund in the Appendix. he Hydrgen Atm Using the fllwing s rbital s(, r) exp r in a variatinal calculatin n the hydrgen atm yields the fllwing result fr the energy in atmic units. H he first term is electrn kinetic energy and the secnd term is electrn nucleus ptential energy. Minimizatin f H with respect t yields =, and the fllwing result fr the grund state energy. H.5..5 he Hydrgen Mlecule In he H mlecular rbital is written as a linear superpsitin f scaled hydrgenic rbitals centered n the tw hydrgen nuclei, a and b. S ab is the verlap integral, bm sa(, ra) sb(, rb) S (, R ) R Sab(, Rab ) exprab Rab ab ab A variatinal calculatin using bm yields the results in the fllwing table.

2 ne AtmicState H atm H in = MlecularState a b m = S =.8 R = able shws that the atmic and mlecular states individually satisfy the virial therem ( = / = ), as des the bnd frmatin prcess ( = / = ). At first glance able and the virial therem suggest that chemical bnding is gverned slely by electrstatics. In bnd frmatin, in the transitin frm atms t a mlecule, kinetic energy increases, ptential energy decreases, and ttal energy decreases. Frm this perspective, clearly ptential energy must be the key factr in the frmatin f a stable mlecule, because it has the same sign as the change in ttal energy, and a decrease in energy is the signature f stability. hat this srt f reasning leads t an incrrect interpretatin f the cvalent bnd will be illustrated by the mechanistic analyses that fllw. Mechanism I he first mechanism is ne that might appear familiar t general chemistry students. Fr example, when we describe the bnding in methane t undergraduates we generally invke a mechanism that uses the cncepts f atmic prmtin [s p > s p ], hybridizatin [s p > (sp hybrids) 4 ], and bnd frmatin thrugh the verlap f atmic rbitals. he H bnd frmatin mechanism described here cnsist f just tw steps: atmic prmtin and cvalent bnd frmatin (H H > H* H > H ). his simple mechanism has previusly been discussed in the review literature (6) and has been invked in a study f the cvalent bnd in H (). * H H H.8 H H.8,R. ne AtmicState H atm H in = Intermediate xcitedatmicstate H atm H in = MlecularState a b m = S =.8 R =

3 In the first step the hydrgen atm rbitals prepare fr bnding by cntracting frm = t =.8, the ptimum value f the final mlecular wave functin. his step is atmic and endergic (.8 h ), increasing the kinetic energy mre (.66 h ) than it decreases the ptential energy (.8 h ). he ptential energy decreases because the electrn is drawn clser t the nucleus. he kinetic energy increases because f the increased cnfinement f the electrn in the cntracted atmic rbital kinetic energy is inversely prprtinal t the square f the average distance f the electrn frm the nucleus, r inversely prprtinal t vlume f the electrn distributin raised t the / pwer, /. he secnd step cnsists f the frmatin f a mlecular wave functin by the superpsitin (linear cmbinatin) f the prmted atmic rbitals. he electrn density f this mlecular rbital is, bm = a b S = a ab b S Mlecular rbital frmatin distributes the electrn density ver the entire mlecule and this charge delcalizatin brings abut a significant decrease in kinetic energy (.798 h ). Ptential energy increases slightly (.65 h ) because nuclear repulsin ( h ) mre than ffsets the increase in electrn nuclear attractin (.44 h ) due t the fact that the electrn nw interacts with bth nuclei. his step is exergic (.48 h ) because kinetic energy decreases mre than ptential energy increases. In ther wrds, in this mechanism charge delcalizatin funds (drives) the frmatin f the chemical bnd. Mechanism II Mechanism II pstulates an intermediate nn bnding mlecular state that has the equilibrium values f and R. he nn bnding character this state is revealed by examining its electrn density. nm = a ib = a b he electrn density has n interference term and simply places half the electrn density in each atmic rbital. he energy f the nn bnding state as a functin f and R and is prvided in the Appendix. his intermediate state subsequently relaxes t the final mlecular bnding state. he calculatins fr this mechanism are summarized in able III. ne AtmicState H atm H in = Intermediate NnBndingState a ib nm = =.8 R = MlecularState a b bm = S =.8 R =

4 Frmatin f the nn bnding mlecular state is endergic (.47 h ) because rbital cntractin increases kinetic energy (.664 h ) mre than it decreases ptential energy (.58 h ). he kinetic energy increase fr this intermediate state is identical t that in Mechanism I. he ptential energy decrease is less than that fr Mechanism I because nuclear nuclear repulsin ( h ) in the intermediate state is greater than ʺelectrn ther nucleusʺ attractin [.487 h =.75 h (.8 h )] he subsequent relaxatin t the bnding mlecular state is exergic (.7 h ) because charge delcalizatin decreases kinetic energy (.799 h ) mre than charge redistributin t the bnd regin increases electrn nucleus ptential energy (.58 h ). As in Mechanism I we have an endergic atmic step fllwed by a exergic mlecular step. his mechanism clarifies an imprtant issue. It is cmmnly thught, and frequently taught, that frm a ptential energy pint f view the internuclear regin is the mst favrable place fr electrn density because it is midway between t nuclei. Hwever, this mechanism clearly shws that the redistributin f charge int the internuclear regin due t the interference term in the bnding electrn density ccurs with an increase in ptential energy. hat this is nt sme arcane quantum mechanical phenmenn is revealed by a simple classical electrstatic calculatin []. Frm a purely electrstatic perspective the prefered lcatin fr an electrn is in the nucleus. If a stable chemical bnd is due t the build up f charge density in the internuclear regin, then this mechanism shws that kinetic energy plays a crucial rle in bringing abut this effect. Mechanism III he third mechanism pstulates an initial mlecular state, rather than a prmted atmic state r a nn bnding mlecular state fr the intermediate. he initial mlecular state has the equilibrium bnd length, but the atmic value fr the rbital scale factr,. he calculatins based n this mechanism are shwn belw. H H H,R. H.8,R. ne AtmicState H atm H in = Intermediate BndingState a b im = S = R = MlecularState a b m = S =.8 R =

5 he intermediate mlecular state is calculated at the equilibrium bnd length (R =. a ) fr a mlecular rbital with =.. Charge delcalizatin ver the tw nuclear centers n the frmatin f the mlecular rbital brings abut a large decrease in kinetic energy (.8 h ). Ptential energy (.599 h ) increases because nuclear repulsin ( h ) is larger, as in Mechanism II, than electrn ther nucleus attractin (.494 h ). his step is exergic and we see again that it is kinetic energy drives cvalent bnd frmatin. In the secnd step f this mechanism the atmic rbitals making up the mlecular rbital cntract ( increases frm t.8) t achieve the final equilibrium mlecular state. his rbital cntractin decreases electrn nucleus ptential energy (.9 h ) mre than it increases kinetic energy (. h ), s this step is als exergic. Hwever, as Ruedenberg pinted ut, this step is essentially atmic in character. Orbital cntractin draws electrn density ut f the bnd regin back tward the nuclei. In ther wrds it returns sme f the charge density transferred t the bnd regin in the first step back t the nuclear centers. Summary he three mechanisms examined pstulate three different intermediate states: () prmted atmic; () nn bnding mlecular; () intermediate mlecular. Hwever, they all tell the same stry; kinetic energy plays a crucial rle in chemical bnd frmatin, and therefre viable mdels fr the cvalent bnd require a cnsideratin f bth kinetic and ptential energy. he virial therem ntwithstanding (as Ruedenberg has said), the frmatin f a chemical bnd is nt simply an electrstatic phenmena. Appendix: Cmputatinal Details Atmic energy cntributins: H H H Bnding mlecular energy cntributins: expr R S R R bm R R exp R S R R bm R R R R expr S R R exp R R bm R bm R bm R Nn bnding mlecular energy cntributins: nbm R nbm R R exp R nbm R nbm R nbm R

6 Atmic State H ( ).5 H ( ). H ( ).5 Mlecular State: bm (.8.).5865 bm (.8.).7 bm (.8.).5865 Mechanism I Intermediate xcited Atmic State H (.8).766 H (.8).8 H (.8).477 Mechanism II Intermediate Nnbnding Mlecular State nbm (.8.).4595 nbm (.8.).766 nbm (.8.).58 Mechanism III Intermediate Bnding Mlecular State bm (.).86 bm (.).94 bm (.).559 Literature cited:. Ruedenberg, K. Rev. Md. Phys. 96, 4, 6 5. Feinberg, M. J.; Ruedenberg, K. J. Chem. Phys. 97, 54, Feinberg, M. J.; Ruedenberg, K. J. Chem. Phys. 97, 55, Ruedenberg, K. In Lcalizatin and Delcalizatin in Quantum Chemistry; Chalvet, O. et al., ds.; Reidel: Drdrecht, he Netherlands, 975; l. I, pp 45.. Harcurt, R. D.; Slmn, H.; Beckwrth. J. Am. J. Phys. 98, 5, Baird, N. C. J. Chem. duc. 986, 6, Harcurt, R. D. Am. J. Phys. 988, 56, Nrdhlm, S. J. Chem. duc. 988, 65, Bacskay, G. G.; Reimers, J. R.; Nrdhlm, S. J. Chem. duc. 997, 74, Weinhld, F. J. Chem. duc. 999, 76(8), Riux, F. Chem. ducatr 997, (6), 4.. Riux, F. Chem. ducatr, 6(6), Riux, F. Chem. ducatr, 8(),.. Ashkenazai, G; Kslff, R. Chem. ducatr 6, (),.. Nrdhlm, S.; Back, A.; Bacskay, G. B. J. Chem. duc. 7, 84(7),. 4. Kutzelnigg, W. Angew. Chem. Int. d. ng. 97,, Melrse, M. P.; Chauhan, M.; Kahn, F. her. Chim. Acta 994, 88, Grdn, M. S.; Jensen, J. H. her. Chem. Acc.,, Added references: Recently Klaus Ruedenberg and frmer student (nw clleague) Mike Schmidt published tw exhaustive studies n the quantum mechanical principles f the cvalent bnd. ʺWhy Des lectrn Sharing Lead t Cvalent Bnding? A ariatinal Analysisʺ J. Cmput. Chem. 7, 8, 9 4. ʺPhysical Understanding thrugh ariatinal Reasning: lectrn Sharing and Cvalent Bndingʺ J. Phys. Chem. A, 9. (),