Chem 104A, UC, Berkeley Chem 104A, UC, Berkeley Molecular Orbital Reading: DG 2.11-14, 3.1-5, 4; MT 3.1 Lewis Structure G. N. Lewis (UC, Berkeley, 1915) Octet Rule: Closed shell configuration of 8 surrounding e 1 Chem 104A, UC, Berkeley Resonance The best structure has the fewest formal charges and has the negative charge on the highest electronegativity atom. Chem 104A, UC, Berkeley Valence Bond theory Heitler, London (1927) L. Pauling (orbital hybridization) Localized orbital approach Investigating ground state molecules, molecular geometry Bond dissociation energy 2 Chem 104A, UC, Berkeley Molecular Orbital theory Mulliken Delocalized orbital approach Unoccupied orbital Spectroscopic properties (ionization, excited states) Chem 104A, UC, Berkeley Valence Bond Theory Valence bond theory (VBT) is a localized quantum mechanical approach to describe the bonding in molecules. VBT provides a mathematical justification for the Lewis interpretation of electron pairs making bonds between atoms. VBT asserts that electron pairs occupy directed orbitals localized on a particular atom. The directionality of the orbitals is determined by the geometry around the atom which is obtained from the predictions of VSEPR theory. In VBT, a bond will be formed if there is overlap of appropriate orbitals on two atoms and these orbitals are populated by a maximum of two electrons. bonds: have bonds: a node on the symmetric about inter-nuclear axis the internuclear and the sign of axis the lobes changes across the axis. 3 Valence Bond TheoryChem 104A, UC, Berkeley Valence bond theory treatment of bonding in H2 and F2 F 2s 2p H 1s1 H 1s1 A B F A B 2s 2p z axis 2pz 2pz This gives a 1s-1s bond This gives a 2p-2p bond between between the two H atoms. the two F atoms. For the VBT treatment of bonding, we ignore the anti-bonding. Chem 104A, UC, Berkeley Valence bond theory treatment of bonding in O2 z axis O 2s 2p 2pz 2pz This gives a 2p-2p bond O between the two O atoms. 2s 2p OO z axis Lewis structure 2p 2p (the choice of 2py is arbitrary) y y Double bond: bond + bond Triple bond: bond + 2 bonds This gives a 2p-2p bond between the two O atoms. In VBT, bonds The Lewis approach and VBT are predicted to be weaker than predict that O is diamagnetic – bonds because there is less overlap. 2 this is wrong! 4 Chem 104A, UC, Berkeley Valence Bond theory A B 0.74 A Bond dissociation energy: 103 kcal/mol Chem 104A, UC, Berkeley Our Goal Use atomic orbitals to compose a wavefunction that accurately describe these observations. 5 Chem 104A, UC, Berkeley 1 1 Energy 1s ( )3/ 2 er / a0 a0 1s (1)1s (2) a A B 0 -2 f Experimental curve -4 eV 0.5 1.0 1.5 Internuclear distance 1 1 3/ 2 r / a0 Chem 104A, UC, Berkeley Energy 1s ( ) e a0 g 1sA (1)1sB (2) 1sA (2)1sB (1) a 0 b 1sA (1)1sB (2) 1sA (2)1sB (1) -2 Heitler-London Wavefunction f Experimental curve -4 eV 0.5 1.0 1.5 Internuclear distance 6 Chem 104A, UC, Berkeley Variation Principle: If an arbitrary wavefunction is used to calculate the energy, then the value obtained is never less than the true value. * (x)H(x)dx H *(x)(x) Chem 104A, UC, Berkeley Variation Principle: If an arbitrary wavefunction is used to calculate the energy, then the value obtained is never less than the true value. (x) Can always minimize E (x) (x) 7 Chem 104A, UC, Berkeley 1 1 1s ( )3/ 2 er / a0 a0 + + 1 Z 3/ 2 Zr / a0 1s 100 ( ) e a0 Chem 104A, UC, Berkeley 1 Z Energy 1s ( )3/ 2 eZr / a0 a0 g a b 0 c 1sA (1)1sB (2) 1sA (2)1sB (1) -2 Zeff=1.17 f Experimental curve -4 eV 0.5 1.0 1.5 Internuclear distance 8 Chem 104A, UC, Berkeley Promoting electron Orbital Hybridization - + + A 2pzA A N(1sA 2 pzA ) 1sA N(1s 2 p ) + + - B B zB B 2pzB 1sB 1 s contribution 1 2 2 p contribution 1 2 Chem 104A, UC, Berkeley - + + A N(1sA 2 pzA ) A za 1sa 2p + + - B B N(1sB 2 pzB ) zb 1sb 2p =0.1 9 Chem 104A, UC, Berkeley Energy N(1s 2 p ) g A A zA B N(1sB 2 pzB ) a b 0 c -2 d A (1) B (2) A (2) B (1) f Experimental curve -4 eV 0.5 1.0 1.5 Internuclear distance Chem 104A, UC, Berkeley Energy g a b 0 c A (1) B (2) A (2) B (1) -2 d [ (1) (2) (2) (1)] e A A B B f Experimental curve -4 eV =6% 0.5 1.0 1.5 Internuclear distance 10 Chem 104A, UC, Berkeley Energy Valence bond theory is all about promoting e and hybridizing orbitals into the shape required by experiments.! g a b 0 c -2 d e f -4 eV 0.5 1.0 1.5 Internuclear distance Chem 104A, UC, Berkeley Covalent Bond - + + F N(2 pzF 2sF ) + + H1s zF 2sF 2p N(2 p 2s ) - + + lp zF F + + H zF 2sF 2p =0.5 Lone pair 11 Chem 104A, UC, Berkeley promotion 2 2 2 2 1 2 1 2 2 2 1s 2s 2 px 2 py 2 pz 1s 2s 2 px 2 py 2 pz E 500kcal / mol 1 eV=96.4 kJ/mol=23.1 kCal/mol Benefit of hybridization 1. Increase in e density between nuclei, enhanced bonding 2. Decrease in e-e repulsion by formation of a lone-pair orbital, LP Orbital more directed away from BP than the Original s LP Ionic Contribution: 50% 1sH (1)F (2) 1sH (2)F (1) F (2)F (1) Chem 104A, UC, Berkeley 12 Chem 104A, UC, Berkeley [1sHa (1)2 pyo (2) 1sHa (2)2 pyo (1)][1sHb (3)2 pzo (4) 1sHb (4)2 pzo (3)] Valence bond theory is all about promotingChem 104A, electron UC, Berkeley and hybridizing orbitals into the shape required by experiments! - y py s + O - 104.5 + pz HH dv 0 a b a b 2 a dv 1 z 13 Chem 104A, UC, Berkeley - y py s + O - + pz HH a b z N[cos 2 p sin 2 p 2s ] a 2 zo 2 yo o N[cos 2 p sin 2 p 2s ] b 2 zo 2 yo o Orthogonality andChem Normalization 104A, UC, Berkeley Two properties of acceptable orbitals (wavefunctions) that we have not yet considered are that they must be orthogonal to every other orbital and they must be normalized. These conditions are related to the probability of finding an electron in a given space. Orthogonal means that the integral of the product of an orbital with any other orbital is equal to 0, i.e.: 0 nm where n m and means that the integral is taken over “all of space” (everywhere). Normal means that the integral of the product of an orbital with itself is equal to 1, i.e.: 1 nn This means that we must find normalization coefficients that satisfy these conditions. Note that the atomic orbitals () we use can be considered to be both orthogonal and normal or “orthonormal”. 14 N[cos 2 p sin 2 p 2s ] Chem 104A, UC, Berkeley a 2 zo 2 yo o N[cos 2 p sin 2 p 2s ] b 2 zo 2 yo o 104.5 dv 0 Orthogonality a b 2 0.5 dv 1 Normalization a N 1/ 1.25 a 0.552 pzo 0.712 pyo 0.4522so b 0.552 pzo 0.7122 pyo 0.4522so [1sHa (1)2 pyo (2) 1sHa (2)2 pyo (1)][1sHb (3)2 pzo (4) 1sHb (4)2 pzo (3)] New Wavefunction: [1sHa (1)a (2) 1sHa (2)a (1)][1sHb (3)b (4) 1sHb (4)b (3)] Chem 104A, UC, Berkeley Unit Orbital Contribution: The contribution of any atomic orbital to all hybrid Orbitals must add up to 1.0. 15 a 0.55Chem2 pzo 104A, 0.71 UC,2 Berkeleypyo 0.4522so Orbital contribution b 0.552 pzo 0.7122 pyo 0.4522so a b 30% 30% 60% 2 pzo 0.552 30% 2 p 50% 50% 100% yo 0.712 50% 20% 20% 40% 2so 0.4522 20% Bonding: Average 80% p character, 20% s character What’s left? 40% p +60% s lp 0.632 pzo 0.772so Chem 104A, UC, Berkeley - y py s - + O lp 0.632 pzo 0.772so + pz HH a b z a 0.552 pzo 0.712 pyo 0.4522so b 0.552 pzo 0.7122 pyo 0.4522so 16 Chem 104A, UC, Berkeley p s 1 (2 p ) lp1 2 x lp character character 1 Bonding 80% 20% lp1 (2 px lp ) 2 orbital lp 70% 30% orbital DirectionalityChem 104A, UC, Berkeley The bonding in diatomic molecules is adequately described by combinations of “pure” atomic orbitals on each atom. The only direction that exists in such molecules is the inter-nuclear axis and the geometry of each atom is undefined in terms of VSEPR theory (both atoms are terminal). This is not the case with polyatomic molecules and the orientation of orbitals is important for an accurate description of the bonding and the molecular geometry. Examine the predicted bonding in ammonia using “pure” atomic orbitals: 2s 2p N The 2p orbitals on N are oriented along the X, Y, 3 H and Z axes so we would predict that the angles 1s 1s 1s between the 2p-1s bonds in NH3 would be 90°.