1 The chemical bond •Why do some combinations of atoms form stable molecules? •Why other combinations of atoms do not form stable molecules? •What determines the bond strength? •What determines the molecular geometry? •What determines if a chemical reaction will take place? •What determines the properties of molecules (optical, electrical, magnetic, etc). Lewis model, Octet rule, VSEPR (Valence Shell Electron Pair Repulsion) Quantum Chemistry Fundamental, Quantitative, Demanding (But – commercial software packages are available and commonly used nowadays) Phenomenological, Qualitative, Easy to apply 2 The electrons are much lighter than the nuclei: 1836 1. The electrons move much faster than the nuclei! p e m m = ⇒ The Born-Oppenheimer approximation (1927) 1. Solve the for different frozen configuraions of the nuclei. 2. Electronic Schrodinger equation electronic potential surfac "Connect-the-dot" to form . 3. Sol es Nuclear Schrodinger eqve the uation for each of the electronic surfaces. 4. The corresponds to the Minimum of the electronic po molecular geometry tential surface. Strategy E(R) R R eq R 3 + 2 The H molecule The molecular orbitals (MO's) at R= . eq R E n e r g y g "Gerade" (Symmetrical with respect to inversion through Angular momentum component along bond axis = the center of the molecule): 1 1s The atomic Ground s (- ,- ,- ) ( tat , e , or . 0. ). g xyz xyz sσ σ ψψ ⇒ = ⇒ ⇒ 2 + R++ no "*" Bonding: energy lower than H(1s)+H , bital in the dissociated molecule (when R ): high electronic density in H ( 1 ) the region H(1s)+H . between nuclei, well g sσ →∞ →∞ ⇒ ⎯⎯⎯ → -shaped potential. * u "Ungerade" (anti-symmetrical with respect to in Angular momentum com version through First excited the center ponent alon of the mole state cule): g bon (- 1 1 d axis = 0. ,- ,- ) ( , , :. ). s u s xyz xyz σ ψ σ ψ=− ⇒ ⇒ + + + * R 2 * Anti-bonding The atomic orbital i : energy higher than n the dissociated molecule (when R ): H(1s)+H , low electronic den H ( 1 ) si H(1s)+H . ty in the region between nucle u sσ →∞ →∞ ⎯ ⇒ ⇒ ⎯⎯ → i, repulsive potential surface. 4 g "Gerade" (Symmetrical with respect to inversion thro Angular momentum component ugh the center of 2 2s The ato the mo along lecul 2-nd excited e): (- ,- bond axis = , s - tate . ) 0 ,) . (,. g xyz xy s zψ σ σ ψ ⇒ ⇒ = ⇒ + 2 + R+ mic orbital in the dissociated molecule (when R ): H ( 2 no "*" Bonding: energy lo )H(2 wer than H(2s)+H . s)+H . g sσ →∞ →∞ ⎯⎯⎯ → ⇒ * u "Ungerade" (anti-symmetrical with respect to inve Angular momentum compon rsion through the 3-rd cent ent along bond axi er of the molecule excited s ): ( s = 0 ta 2t -,-,-) (, ,) . . s e: . 2 u xyz xyz sσ σ ψψ ⇒ ⇒ =− ⇒ * R+ + + 2 The atomic orbital in the dissociated molecule (when R ): H ( * Anti-bonding: energy 2) H(2s)+H higher than H(2s)+ . H u sσ →∞ ⇒ →∞ ⎯⎯⎯ → g "Gerade" (Symmetrical with respect to inversion th Angular momentum compone rough the cente 5-t r o nt along bond axis = f the molecule): (- h e , xc - ited s 0 2 2s The a ta ,- ) ( , , . ) t te . . g z xy p zxyzψψ σ σ ⇒ ⇒ ⇒ = R++ 2 + z z omic orbital in the dissociated molecule (when R ): H ( no "*" Bonding: energy lower than H 2) H( ( 2p )+ 2p ) H H +. . zg pσ →∞ ⎯ ⇒ →∞ ⎯⎯ → * u "Ungerade" (anti-symmetrical with respect to in Angular momentum com version through 7-th excited the center ponent alon of the mole state: cule): g bon (- d axis = 0 ,- ,- ) ( , , . 2 2s . ) . zu xyz p xyzψ σ σ ψ ⇒ ⇒ =− R+ 2 z z + +* The atomic orbital in the dissociated molecule (when R ): * Anti-bonding: energy higher than H(2p )+ H( 2 ) H(2p)+H. H zu pσ →∞ ⎯⎯ → ⇒ ⇒→∞ ⎯ 5 ** g "Gerade" (Symmetrical with r 22 es 6-th excited st pect t Angular momentum com o inversion through ponent along the ce ate: nter of the molecule): (- , ,. bo -,- nd ax )(,, . ) 2 . is x g yg h pp xyz xyz ππ π ψ π ψ = ⇒ ⇒ =± ** ++ xy xy RR+ + + + 2x2y 2p ,2p The atomic orbital in the dissociated molecule (when R ): H ( 2 ) H(2 * Anti-bonding: ene p)+H, H( 2 ) rgy higher H than H(2p )+H or H(2p )+H (2p )+H . xggy ppππ →∞ →∞ ⇒→ ⎯⎯⎯ →⎯⎯ ⇒ ⎯ u "Gerade" (anti-symmetrical with respect to inversion through the center of the molecule): Angular momentum component along bond 4-th exci (- ,- , axis ted state 2 -) . 2 , : , 2 . ( xyuu h xy pp zxy π π ψ π ψ π ⇒=± ⇒ =− ++ x xy RR+ + + + 2x2y y 2p ,2p The atomic orbital in the dissociated molecule (when R ): H ( 2 ) H(2p )+H , H ( 2 ) H(2 * Bonding: energy lower than H(2p )+H or H(2p )+ ,) p )+H . . H yuxu pp z ππ →∞ →∞ ⇒→∞ ⎯⎯ ⇒ ⎯ →⎯⎯→ + 2 H is the only molecule for which the Schrodinger equation has an analytical solution Treatment of more complex molecules has to be based on approximations •Ideally, the approximations employed should allow for systematic improvement until the result is indistinguishable from the exact result. •The actual calculation of the approximations is usually performed of computers. 6 The variational principle 00 01 0 0 0 If the energy levels of a system are given by E ... (E is the ground state energy), then the expectation value of the energy (the average energy) is never greater than E () (:.) N NN HPEE E PEE E E=++ < ≥ < < " (The average over a set of values cannot be smaller than the smallest value in the set). 1 1 1 1. Introduce an for the wavefunction that depends on the adjustable parameters ,, : . 2. Evaluate the expectation value of "ed the ucat ene ed guess rgy as a " function o (; , , f ) ,, : guP P ess P C H CC xC CC ψ… … … {} 1 111 ** 111 0 (,,) (;,,) (;,,). (3. Minimize ( , , ) with respect to , , : 4. And get an approximation for the g ,, ) (,, ) . round state energy and wave function: P P guess P guess P PPP HC C CC xCCH xCCdx Min H C C H C C ECC ψψ…= … … …= …… ≥… ∫ ** ** 01 0 1 (1) The approximation can only be improved by adding more parameters. (2) The extremum points (local minim (,, ); () (;,, a, maxima Rema and r suddle points) provide ks: ppr . a ) P guess P EHC C x xC Cψψ≈… ≈ … oximate excited energy levels and wave functions. The LCAO (Linear Combination of Atomic Orbitals) approximation + 2 * 1 1 1 2 1 1 1 2 Consider the following educated guess for a MO of H : () (, , ) 0 1 1 (, , ) 0 ( Symme or t ) ry () : . A AS A S A S AA B BS B S A MO AB A AB B S g BB B AB u AB B R HRCC C CC s CC s HRCC C CR CC C C C R CC ψ ψ ψ ψ ψ ψσψ ψ ≈+ ∂ ⎫ = ⎪ ∂ =⇒∝+≈ ⎪ ⎬ ∂ =− ⇒ ∝ − ≈ ⎪ = ⎪ ∂ ⎭ =⇒= =− 1 A S ψ 1 B S ψ 7 ( ) ( ) ( ) ()()() 2 2 111 2 2 11 2 1 2 11 2 2 BAA A SS S AA A S B B B BB SSS ψψ ψ ψψ ψ ψ ψψ ψ ψψ =+− +++ = − ( ) ( ) 2 1 2 1S A S B ψ ψ+ *+ gu 2 Electronic potential surfaces for 1s and 1s states of H from the LCAO approximation. σσ Correlation diagram and electronic configuration 8 Hartree method for many-electron molecules 1 11 1 2 111 111 12 12 2 22 2 222 2222 2 222 1 11111 22222 1 12 2 2 11 1 11 1 (, 11 ,) 11 (, ,) ( 1 (, ,) 11 (, ,) (, , 1 ) 1 ,,) 1 1 A ABB T Rr r J dx dy dz x y z TJxyzE T Rr r dx dy dz x y z J TJxyz xyz Rr Exyz R r r r R r H ψ ψψ ψ ψψ +− − ⎛⎞ −− +− − ⎛⎞ −−+ = ⎜⎟ + =++− = ⎜⎟ ⎝⎠ = = ⎫ ⎪ ⎪ ⎬ ⎪ ⎝ ⎭ ⎠ ⎪ ∫∫∫ ∫∫∫ 222 2222 2111 1111 1 , Solved by the ,,) (,,); LCAO approximation ( ,, (,,)xyz xyzxy Ezxyz Eψψ ψ×= =+ A B R r 12 r 1A r 1B 1 2 r 2B r 2A 2 The H molecule First period homonuclear diatomics 1 2g -1 =255 kJmol , 1/ 2, 1.06(1s): eq BEBORAH σ + == 2 2 -1 g =431 kJmol , 1, 0.7(1s): 4 eq BEBORAH σ == 2 1 2 -*1 gu =251 kJmol ,(1s) 1/2, 1.() 8s 01: eq BE BH ORe Aσσ + == 22 2g g (1s)(1s): =0, , 0 eq BE BO RHe σσ ==∞ () 1 Bond energy; Bond order= # bonding electrons-# antibonding electron 2 BE BO== 9 Second period homonuclear diatomics = The consists of the occupied AO's in the atoms. It will lead to enough MO's for hosting all the electron minim s um basis o (no more # o th ( a # n t f AO's wo pai of AO's red el f M ect 1) r s O' ons ⇒ * per MO). Two AO's contribute significantly to the bond formation, only if their atomic energy levels are very close. Two atomic orbitals on different atoms contribute significantly ( t 2) (3 o b) ond * Stated without a proof. formation only if they have substantial overlap (=constructive interferenc ) * e. 2 12 2 2 2 2 2 *** 2 * 2 2 * 2 2 2 2 12 2 2 2 22 z z z z y x x y x yxy A AA p Ss A pss A A g z gg uzuu uy ux B BB p B p B B p p BB p p AA gx gyp ppp p ss p p p pp ψ ψψ ψψ ψ ψ ψ ψ ψψ σ σσ σ π π ππ ψ ψ ψ ⎧ +⇒ ⎧⎧+⇒ +⇒ ⎪⎪⎪ ⎨⎨⎨ −⇒−⇒ −⇒ ⎪⎩⎩ ⎩ ⎧ +⇒ ⎧ +⇒ ⎪⎪ ⎨⎨ −⇒ −⇒ ⎩ ⎩ The z axis coincides with the bond axis. 2 * 2 22 2 2 z z z z g z B p B p p A p z A u p p ψ ψ ψ σ ψ σ ⎧ +⇒ ⎪ ⎨ −⇒ ⎪ ⎩ 2 2 2 2 * 2 2 x x x x ux B p B A p g p A p x p p ψ ψ ψψ π π ⎧ +⇒ ⎪ ⎨ −⇒ ⎪ ⎩ * *0 2 and 2 can be obtained by rotating 2 and 2 around the z axis by 90 . uy gy ux gx pp pp ππ ππ z y (perpendicular to plane of the page) x 10 Energy ordering of MO’s (second period diatoms) g uu The ordering of the 2 and the (2,2) is reversed! z xy p pp σ ππ 7Z ≤ 8Z ≥ * Not important for bonding since: (1) Same occupation in bonding and anti-bonding MO's. (2 "Chemis 1, try is dominated by the valence electrons" ) inner shell electrons = small overlap ( ) 1 . gu ssσσ⇒ Overlap: Electron-electron repu 2,2 lsion: 2 :-) 2 :-( :-( 2,2 gz g uxux uxz ux p p p p p pσ σ π π ππ > > 22 2 22222 The overlap dominates at (the electron repulsion is compensated for by the greater attraction to the n , u , c " " (and their ions) Z8 ," ", , , (and t :-) (2),(2 (2)leus) ) g xuxz u E OF Ne Li Be B C N pp pEE πσ π⇒ ≥ < The repulsion dominates a (2 t (2) heir ions ,( 2 ) ) Z7 ) ux uxgz EpEpEpππσ⇒> ≤ 11 2s 2s 2 g sσ * 2 u sσ 2 g z pσ * 2 uz pσ 2,2,2 x yz ppp 2,2,2 x yz ppp * 2 g x pπ * 2 g y pπ 2 ux pπ 2 uy pπ 1 1 2.67 105 2 eq BO RA BE kJ mol Li − = = = Diamagnetic Li Li 2s 2s 2 g sσ * 2 u sσ 2 g z pσ * 2 uz pσ 2,2,2 x yz ppp 2,2,2 x yz ppp * 2 g x pπ * 2 g y pπ 2 ux pπ 2 uy pπ eq 1 0 R2.45 9 2 BO A BE kJmol Be − = = = Diamagnetic Be Be 12 2s 2s 2 g sσ * 2 u sσ 2 g z pσ * 2 uz pσ 2,2,2 x yz ppp 2,2,2 x yz ppp * 2 g x pπ * 2 g y pπ 2 ux pπ 2 uy pπ 1 1 1.59 289 2 eq BO RA BEkJmol B − = = = Paramagnetic B B 2s 2s 2 g sσ * 2 u sσ 2 g z pσ * 2 uz pσ 2,2,2 x yz ppp 2,2,2 x yz ppp * 2 g x pπ * 2 g y pπ 2 ux pπ 2 uy pπ 1 2 1.24 599 2 eq BO RA BE kJ mol C − = = = Diamagnetic C C 13 2s 2s 2 g sσ * 2 u sσ 2 g z pσ * 2 uz pσ 2,2,2 x yz ppp 2,2,2 x yz ppp * 2 g x pπ * 2 g y pπ 2 ux pπ 2 uy pπ 1 3 1.10 942 2 eq BO RA BE kJ mol N − = = = Diamagnetic N N 2s 2s 2 g sσ * 2 u sσ 2 g z pσ * 2 uz pσ 2,2,2 x yz ppp 2,2,2 x yz ppp * 2 g x pπ * 2 g y pπ 2 ux pπ 2 uy pπ 1 2 1.21 494 2 eq BO RA BEkJmol O − = = = Paramagnetic 2 Why O is chemically active O O 14 2s 2s 2 g sσ * 2 u sσ 2 g z pσ * 2 uz pσ 2,2,2 x yz ppp 2,2,2 x yz ppp * 2 g x pπ * 2 g y pπ 2 ux pπ 2 uy pπ 1 1 1.41 154 2 eq BO RA BE kJ mol F − = = = Diamagnetic F F 2s 2s 2 g sσ * 2 u sσ 2 g z pσ * 2 uz pσ 2,2,2 x yz ppp 2,2,2 x yz ppp * 2 g x pπ * 2 g y pπ 2 ux pπ 2 uy pπ 0 (except at v Unstab ery lo le w temp.) 2 BO Ne = Ne Ne 15 Heteronuclear diatomic molecules •No inversion symmetry (u and g labels are meaningless). •MO’s may consist of linear combinations of different AO’s. •Bonding MO’s = Higher probability for the electron to be in the vicinity of the more electronegative atom; Anti-bonding MO’s = Higher probability for the electron to be in the vicinity of the less electronegative atom. •The molecule has a dipole moment. δ+ δ− More electronegativeLess electronegative AO’s of B AO’s of O 2 ** * 2 2 * 2 Anti-bondin Bonding MO 2.5 Paramagn g MO i et c B Bs B B Bs O Os O O Os OB BO C C C C C C C C ψ ψ ψ ψ =+ < =− > = BO B O B O 16 AO’s of C AO’s of O 22 * 2 * * 2 * Anti-bondi Bonding MO 3 Diamagneti ng MO c B Cs C B Cs O Os O C O O s O B C C C C C C O C C ψ ψ ψ ψ =+ < =− > = CO C O AO’s of N AO’s of O 2 ** * 2 2 * 2 Anti-bondin Bonding MO 2.5 Paramagn g MO i et c B Ns N B Ns O Os O O Os ON BO C C C C C C C C ψ ψ ψ ψ =+ < =− > = NO N O NO 17 H F HF +- HF Hybridization and molecular geometry sp hybridization (linear geometry) The energy increase due to mixing of excited AO's may be compensated for if it leads to more overlap. MO's that consist of linear combinations of hybrid AO's (linear combinations Ratio of th O nal: e A ⇒ 's of the isolated atom) 18 sp 2 hybridization (triangular geometry) BH 3 B HH H sp 3 hybridization (tetrahedral geometry) H C H H H H N H H H O H 19 as in diatoms (sp hy1. bridization) bonds: .COσ− 1x 2x 3 1 x * y2y3y * C2p( ) C C2p( ) 2p C2p( ) C2p () C2p () ) , ,, ( , 2. bonds: delocalized MO's. AB AB n nb xx b yyy x OC OCO ππ πππ π π ++ + ⇒ ⇒ + A O B OC 2 CO COO 2S 2S(2 ) z pspσ + Delocalization is Missing in the Lewis diagram .. 2BO= 20 NOO [ ] 2 NO − 2s 2s 2 y p 2 y p 2 sp 2 (2) x sp pσ + (2 ) nb z pπ (2 ) z pπ NOO .. 1.5BO= 2-butene H CCC C HH HH H H H 3 sp 2 sp 21 Structural isomerization: ()() * 1 1 2 *h The barrier may be removed by photochemically exciting an electron (Double bond) from the MO to the MO (single bond: ) ν ππ πππ ⎯⎯→ ()( ) 1 1 * π π 2 π Delocalized orbitals in cojugated organic molecules:π CCCC 1,3 butadiene 2 sp Stronger bonds Similar to Particle in 1D box 22 Benzene All C-C bonds are identical, And intermediate in strength between a single and a double bond. Fullerenes, nanotubes and all that… Buckministerfullerene eitan Microsoft PowerPoint - Chap12.ppt