A Mechanistic Approach to Bond Formation in H 2

Transcription

1 A Mechanistic Apprach t Bnd Frmatin in H Frank iux Prfessr Emeritus f Chemistry Saint Jhnʹs University The centrality f the cvalent bnd t chemistry is captured in a shrt and elquent statement by Peter Atkins []. Nw we cme t the heart f chemistry. If we can understand what hlds atms tgether as mlecules we may als start t understand why, under certain cnditins, ld arrangements change in favr f new nes. We shall understand structure, and thrugh structure, the mechanism f change. The barrier t understanding is that, in spite f what ur textbks teach, ʺThe chemical bnd is a highly cmplex phenmenn which eludes all attempts at simple descriptin [].ʺ Charles Culsn, ne f the pineering therists in the applicatin f quantum mechanics t the study f chemical bnding wrte the fllwing [3]. Smetimes it seems t me that a bnd between tw atms has becme s real, s tangible, s friendly that I can almst see it. And then I awake with a little shck: fr a chemical bnd is nt a real thing: it des nt exist: n ne has ever seen it, n ne ever can. It is a figment f ur wn imaginatin... Here is a strange situatin. The tangible, the real, the slid, is explained by the intangible, the unreal, the purely mental. Many chemists and physicists have grappled with the intellectual challenge f understanding the chemical bnd. Amng thse wh have made significant cntributins are Hellmann [4], Pauling [5], Slater [6, 7], Culsn [8] and uedenberg [9-]. Amng these, uedenbergʹs wrk is the mst cmprehensive and cgent. It has received cnsiderable attentin in the pedaggical literature [3 3] as well as a number f excellent reviews in the primary literature that are accessible t nn specialists [4-6]. Unfrtunately nne f these effrts t make uedenbergʹs analysis f the cvalent bnd accessible t educatrs has had any nticeable affect n the way the chemical bnd is presented in intrductry and intermediate level undergraduate textbks [7]. Therefre, the purpse f this paper is t try again by prviding a simple tw-step mechanistic apprach t cvalent bnd frmatin in H that reveals uedenbergʹs central message - a full understanding f the nature f the chemical bnd requires cnsideratin f the rle f electrn kinetic energy, r mre apprpriately cnfinement energy [8]. The mechanism pstulates a single intermediate mlecular state fr the H bnd frmatin reactin. HH H Scaled hydrgenic wave functins will be used t calculate the initial atmic state, and the intermediate and final mlecular states. Cmputatinal details fr the intermediate and final mlecular states are available in the appendix.

2 The Initial State Schrödingerʹs equatin fr the hydrgen atm, f curse, is exactly sluble and the result in atmic units (h/ = m e = e = 4 = ; E h = x -8 J) is given belw. By cnventin the energy f the hydrgen in is zer. E T V.5E.E.5E H h h h The Hydrgen Mlecule In The H mlecular rbital is written as a linear superpsitin f scaled hydrgenic rbitals centered n the tw hydrgen nuclei, where S is the verlap integral. (a and b label the atmic rbitals, A and B the nuclear centers which are separated by a distance. where a b Ψ m = S α 3 a π exp αr α = A 3 b π exp α r = B S = ab dτ This trial wave functin and the apprpriate energy peratr lead t the fllwing variatinal energy integral, expressed in Dirac ntatin. E = a b T r A r B a b [ ( S) ] Expansin, after symmetry cnsideratins, yields the fllwing expressin, E = ( a T a) ( a T b) a r A a S b b a r A r A b which is written belw in shrt-hand ntatin. E = T aa T ab V Aaa V Abb S V Aab Minimizatin f the energy f H yields the ptimum values fr and, its grund state energy and energy cmpnents. Table summarizes the results fr the frmatin f the mlecule in frm a hydrgen atm and hydrgen in.

3 T V V ne V ne ( Aaa) V ne ( Abb) V ne ( Aab) V nn E Initial AtmicState H atm H in α = Δ Final MlecularState a b Ψ m = S α =.38 = V ne (Aaa) is the interactin f the electrn density centered n nucleus A with nucleus A. V ne (Abb) is the interactin f the electrn density centered n nucleus B with nucleus A. V ne (Aab) is the interactin f the verlap electrn density with nucleus A. These energy cntributins include the cmpanin terms invlving nucleus B as justified by mlecular symmetry. Table shws that the atmic and mlecular states individually satisfy the virial therem (E = V/ = -T), and therefre s des the bnd frmatin prcess (E = V/ = -T). At first glance Table 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 it appears that 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. That this view is an ver-simplificatin is shwn by the energy prfile (in atmic units) prvided by an ab initi calculatin f cvalent bnd frmatin in the hydrgen mlecule in using the trial mlecular rbital given abve. Figure. Ab initi energy prfile fr the frmatin the hydrgen mlecule in.

4 This energy prfile shws that cnsideratin f kinetic energy is essential in understanding bnd frmatin and mlecular stability. First, the initial drp in ttal energy as the nuclei apprach each ther is due t a decrease in kinetic energy, because the ptential energy initially increases. Secnd, the energy minimum (grund state) is achieved while ptential energy is in a steep decline. It is a sharp increase in kinetic energy that causes the energy minimum and prvides the ʺrepulsiveʺ effect necessary t cunter the (still) attractive ptential energy term. As the prfile shws nuclear repulsin desnʹt becme dminant until well after the energy minimum has been reached. Clearly a valid mdel fr the cvalent bnd requires cnsideratin f bth kinetic and ptential energy. These ab initi results are supprted by a mre empirical apprach based n the virial therem, a methd f analysis first demnstrated by Slater in 933 [5]. A serviceable representatin f mlecular energy fr diatmic mlecules (excluding nuclear kinetic energy, i.e. vibratinal degrees f freedm) is prvided by the Mrse functin. E( ) = D e exp β e The Mrse parameters (D e, e and ) can be btained frm analysis f spectrscpic data. Therefre, using the Mrse functin with the general expressin fr the virial therem (valid fr bth the classical and quantum mechanical dmains) and the expressin fr ttal energy T ( ) V ( ) D e d = d E( ) E( ) = T( ) V( ) leads t the fllwing equatins fr kinetic and ptential energy as a functin f internuclear separatin. d d T( ) = E( ) E( ) V ( ) = E( ) E( ) d d Using spectrscpic parameters frm the literature [9] fr H allws the generatin f the fllwing energy prfile in atmic units. Figure. Energy prfile fr the frmatin f the hydrgen mlecule in based n the virial therem.

5 The basic agreement between Figure (theretical, ab initi) and Figure (experimental, spectrscpic) supprts the view that bth kinetic and ptential energy cnsideratins are required fr a viable mdel f the cvalent bnd; mdels that cnsider nly electrstatic ptential energy effects are invalid and misleading. In Figures and the fact that the ptential energy increases, then decreases, and finally increases again as decreases suggests that it can be partitined int three terms. The initial decrease in kinetic energy is fllwed by an increase indicating that it cnsists f tw ppsing cntributins. Thus, uedenberg analyzed the H bnd frmatin in terms f five cntributins t the binding energy. In an effrt t make uedenbergʹs ideas accessible t undergraduates, a simple tw-step mechanistic apprach will be used t interpret cvalent bnd frmatin. As shwn belw, the mechanism pstulates an intermediate mlecular state at the equilibrium bnd length, but with the atmic value fr the rbital scale factr,. Cmputatinal details n the intermediate state are prvided in the appendix. H H H,.3 H.38,.3 T V V ne V ne ( Aaa) V ne ( Abb) V ne ( Aab) V nn E Initial AtmicState H atm H in α = Δ Intermediate MlecularState a b Ψ im = S α = = Δ Final MlecularState a b Ψ m = S α =.38 = The first step is exergic and is driven by a decrease in electrn kinetic energy. Charge delcalizatin ver the tw nuclear centers n the frmatin f the mlecular rbital brings abut a large decrease in kinetic energy (-.38 E h ) because f the larger mlecular vlume nw available t the electrn [8]. Ptential energy increases (.599 E h ) because nuclear repulsin (.4993 E h ) is larger than the decrease electrn-nucleus ptential energy ( E h ), which cnsists f the three cntributins shwn in Table. Cnstructive interference between atmic rbitals during mlecular rbital frmatin draws sme electrn density int the internuclear regin where ptential energy is higher than in the regin arund the nucleus. Thus, the significant increase in V ne (Aaa) (.3693 E h ) is the main reasn that the decrease in the attractive V ne term is less than the increase in the repulsive V nn term.

6 The secnd step is als exergic, but is driven by a decrease in ptential energy. In this step the atmic rbitals making up the mlecular rbital cntract ( increases frm t.38) t achieve the final equilibrium mlecular state. Orbital cntractin draws electrn density frm the bnd regin back tward the nuclei - it returns sme f the charge density transferred t the bnd regin in the first step back t the nuclear centers. This is evidenced by the significant decrease in V ne (Aaa) (-.54 E h ) and the change in the verlap integral frm.5856 fr = t.463 fr =.38 (see the appendix). The changes in V ne (Abb) and V ne (Aab) are relatively minr in this step. Ptential energy decreases (-.39 E h ) because the electrn is n average clser t the nuclei. Kinetic energy increases (.3 E h ) because rbital cntractin decreases the vlume ccupied by the electrn [8]. This mechanism is cnsistent with the cmputatinal results summarized in Figures and, and therefre, reiterates that a cnsideratin f bth kinetic and ptential energy effects are required t understand the nature f the cvalent bnd. Electrn-nucleus ptential energy is the nly negative (attractive) energy term and is the ʺglueʺ that hlds the mlecule tgether. Paradxically kinetic energy plays tw cntradictry rles in cvalent bnd frmatin. Its initial decrease due t charge delcalizatin funds the build up f charge in the internuclear regin and is respnsible fr the early drp in ttal energy, while its subsequent sharp increase due t rbital cntractin insures a grund state and a stable mlecular entity. T summarize further we can say that the first step is mlecular in character and is driven by a decrease in electrn kinetic energy. The secnd step, rbital cntractin, is atmic in character (drawing electrn density back t the nuclei) and driven by a decrease in electrn-nucleus ptential energy. Literature cited:. Atkins, P. W. Mlecular Quantum Mechanics, nd ed.; Oxfrd University Press, Oxfrd, UK, 983, p.5.. Kutzelnigg, W. Angew. Chem. internat. Edit. 973,, Culsn, C. A. J. Chem. Sc. 955, Hellmann, H. Z. Phys. 933, 35, Pauling, L. The Nature f the Chemical Bnd, 3rd ed.; Crnell University Press, Ithaca, NY, Slater, J. C. J. Chem. Phys. 933,, Slater, J. C. Quantum Thery f Matter, Krieger Publishing: Untingtn, NY, 977; pp Culsn, C. A. Valence, nd ed.; Oxfrd University Press, Oxfrd, UK, uedenberg, K. ev. Md. Phys. 96, 34, Feinberg, M. J.; uedenberg, K. J. Chem. Phys. 97, 54, Feinberg, M. J.; uedenberg, K. J. Chem. Phys. 97, 55, uedenberg, K. In Lcalizatin and Delcalizatin in Quantum Chemistry; Chalvet, O. et al., Eds.; eidel: Drdrecht, The Netherlands, 975; Vl. I, pp uedenberg, K.; Schmidt, M. W. J. Cmput. Chem. 7, 8, uedenberg, K.; Schmidt, M. W. J. Phys. Chem. A 9, 3, Harcurt,. D.; Slmn, H.; Beckwrth. J. Am. J. Phys. 98, 5, Baird, N. C. J. Chem. Educ. 986, 63, Harcurt,. D. Am. J. Phys. 988, 56, Nrdhlm, S. J. Chem. Educ. 988, 65, Bacskay, G. G.; eimers, J..; Nrdhlm, S. J. Chem. Educ. 997, 74, Weinhld, F. J. Chem. Educ. 999, 76(8), iux, F. Chem. Educatr 997, (6), -4.. iux, F. Chem. Educatr, 6(5), 88-9.

7 . iux, F. Chem. Educatr 3, 8(), -.. Ashkenazi, G; Kslff,. Chem. Educatr 6, (), Nrdhlm, S.; Back, A.; Bacskay, G. B. J. Chem. Educ. 7, 84, Kutzelnigg, W. Angew. Chem. Int. Ed. Eng. 973,, Melrse, M. P.; Chauhan, M.; Kahn, F. Ther. Chim. Acta 994, 88, Grdn, M. S.; Jensen, J. H. Ther. Chem. Acc., 3, An exceptin is the physical chemistry text by Jeff Davis which prvides a cncise and cherent quantum mechanically crrect interpretatin f the cvalent bnd. Unfrtunately it is lng ut f print. Davis, J. C., Advanced Physical Chemistry: Mlecules, Structure, and Spectra; The nald Press, New Yrk, 965; pp Electrns cnfined by attractive interactin with nuclei in atms and mlecules ccupy, due t their wave nature, statinary states. They are nt executing classical trajectries and cnsequently the term kinetic energy is inapprpriate. Substitutin f the de Brglie relatin (p = h/) int the classical expressin fr kinetic energy yields the electrnʹs cnfinement energy, h /m. Because the wavelength f a cnfined electrn is prprtinal t the cnfinement dimensin, it is als prprtinal t cube rt f the cnfinement vlume, V /3. Therefre the electrnʹs cnfinement energy is prprtinal t V -/3. 9. Levine, I. N. Quantum Chemistry, 6th ed., Pearsn Prentice Hall, Upper Saddle iver, NJ, 9; p 4. Appendix Overlap integral: S( α ) exp( α) α α 3 Kinetic energy integrals: Taa( α ) α S( α ) Tab( α ) α exp( α) α α 3 S( α ) Ptential energy integrals: VAaa( α ) α S( α ) VAbb( α ) α α exp( α) S( α ) α VAab( α ) αexp( α) ( α) S( α ) Intermediate Mlecular State Values f variatinal parameters: α.3 Overlap integral: S( α ).5856 Ttal Energy: Taa( α ) Tab( α ) VAaa( α ) VAbb( α ) VAab( α ).5539 Kinetic energy: Taa( α ) Tab( α ).386 Taa( α ).353 Tab( α ).79

8 Electrn-nucleus ptential energy: VAaa( α ) VAbb( α ) VAab( α ).4394 VAaa( α ).637 VAbb( α ).976 VAab( α ).5 Nuclear ptential energy:.4993 Ttal ptential energy: VAaa( α ) VAbb( α ) VAab( α ).94 Final Mlecular State Values f variatinal parameters: α.38.3 Overlap integral: S( α ).463 Ttal Energy: Taa( α ) Tab( α ) VAaa( α ) VAbb( α ) VAab( α ).5865 Kinetic energy: Taa( α ) Tab( α ).5865 Taa( α ).537 Tab( α ).67 Electrn-nucleus ptential energy: VAaa( α ) VAbb( α ) VAab( α ).67 VAaa( α ).846 VAbb( α ).339 VAab( α ).4933 Nuclear ptential energy:.4993 Ttal ptential energy: VAaa( α ) VAbb( α ) VAab( α ).73