Molecular Orbital. Reading: DG , 3.1-5, 4; MT 3.1. Lewis Structure. G. N. Lewis (UC, Berkeley, 1915)
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1 Molecular Orbital Reading: DG.-4, 3.-5, 4; MT 3. Lewis Structure G. N. Lewis (UC, erkeley, 95) Octet Rule: Closed shell configuration of 8 surrounding e
2 Resonance The best structure has the fewest formal charges and has the negative charge on the highest electronegativity atom. Valence ond theory eitler, London (97) L. Pauling (orbital hybridization) Localized orbital approach Investigating ground state molecules, molecular geometry ond dissociation energy
3 Molecular Orbital theory Mulliken Delocalized orbital approach Unoccupied orbital Spectroscopic properties (ionization, excited states) Valence ond Theory Valence bond theory (VT) is a localized quantum mechanical approach to describe the bonding in molecules. VT provides a mathematical justification for the Lewis interpretation of electron pairs making bonds between atoms. VT 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 VT, 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: symmetric about the internuclear axis bonds: have a node on the inter-nuclear axis and the sign of the lobes changes across the axis. 3
4 Valence ond Theory Valence bond theory treatment of bonding in and F F s p A s s A F s p z axis p z p z This gives a s-s bond between the two atoms. This gives a p-p bond between the two F atoms. For the VT treatment of bonding, we ignore the anti-bonding. Valence bond theory treatment of bonding in O z axis O p z p z s p This gives a p-p bond between the two O atoms. O s p p y p y z axis (the choice of p y is arbitrary) This gives a p-p bond between the two O atoms. In VT, bonds are predicted to be weaker than bonds because there is less overlap. O O Lewis structure Double bond: bond + bond Triple bond: bond + bonds The Lewis approach and VT predict that O is diamagnetic this is wrong! 4
5 Valence ond theory A 0.74 A ond dissociation energy: 03 kcal/mol Our Goal Use atomic orbitals to compose a wavefunction that accurately describe these observations. 5
6 Energy 3/ s ( a 0 ) e r / a 0 0 a s A() s () - -4 ev f Experimental curve Internuclear distance Energy s ( ) a 0 3/ e r / a 0 g s A() s () s A() s () 0 b a - -4 ev f s A() s () s A() s () eitler-london Wavefunction Experimental curve Internuclear distance 6
7 Variation Principle: If an arbitrary wavefunction is used to calculate the energy, then the value obtained is never less than the true value. * ( x) ( x) dx *( x) ( x) 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
8 s ( ) a 0 3/ e r / a s Z a 3/ 00 ( ) 0 e Zr / a 0 Energy g Z 3/ s ( a 0 ) e Zr / a c b a s A() s () s A() s () Zeff=.7-4 ev f Experimental curve Internuclear distance 8
9 Promoting electron Orbital ybridization s A p za A N(s N(s A p p za z ) ) A s p z s contribution p contribution A N( s A pza) A sa pza + + sb - pzb N( s pz ) =0. 9
10 Energy g N(s A A N(s p p za z ) ) 0 c b a - -4 ev d f ) () () () A( A Experimental curve Internuclear distance Energy g 0 c b a - -4 ev d e f () () () () [ () () () ()] A A A Experimental curve A =6% Internuclear distance 0
11 Energy Valence bond theory is all about promoting e and hybridizing orbitals into the shape required by experiments.! g 0 c b a - -4 ev d e f Internuclear distance F N( pzf sf ) + Covalent ond + s sf pzf lp N( pzf sf ) + + sf pzf =0.5 Lone pair
12 promotion s s px py pz s s px py pz E 500kcal / mol enefit of hybridization. Increase in e density between nuclei, enhanced bonding. Decrease in e-e repulsion by formation of a lone-pair orbital, LP Orbital more directed away from P than the Original s LP Ionic Contribution: 50% s () () s () ev=96.4 kj/mol=3. kcal/mol F F F F () () ()
13 [ s a () pyo() s a () pyo()] [s b(3) pzo(4) s b(4) pzo(3)] Valence bond theory is all about promoting electron and hybridizing orbitals into the shape required by experiments! y a p y - O z s p z b 04.5 a b dv 0 a dv 3
14 y p y - O s + - a + p z b z a N[cos p b N[cos p zo zo sin p sin p yo yo s s o o ] ] Orthogonality and Normalization 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.: n m 0 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, i.e.: n n 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. 4
15 a N[cos p b N[cos p zo zo sin p sin p yo yo s ] o s ] 04.5 a b dv 0 a o dv Orthogonality Normalization 0.5 N / p a 0.55 p b zo zo 0.7 p yo 0.7 p 0.45 s yo o 0.45 s o [ s a () pyo() s a () pyo()] [s b(3) pzo(4) s b(4) pzo(3)] New Wavefunction: [ s a () a () s a () a ()] [s b(3) b(4) s b(4) b(3)] Unit Orbital Contribution: The contribution of any atomic orbital to all hybrid Orbitals must add up to.0. 5
16 Orbital contribution 0.55 p a 0.55 p b Chem zo 04A, UC, erkeley yo zo 0.7 p 0.7 p 0.45 s yo o 0.45 s o p zo p yo s o a b 30% 30% 60% % 50% 50% 00% % 0% 0% 40% % onding: Average 80% p character, 0% s character What s left? 40% p +60% s 0.63 p 0.77 s lp zo o y p y - O s + - lp 0.63 p 0.77 s zo o a + p z b z 0.55 p a 0.55 p b zo zo 0.7 p yo 0.7 p 0.45 s yo o 0.45 s o 6
17 lp lp ( p ( p x x ) lp ) lp onding orbital p character s character 80% 0% lp orbital 70% 30% Directionality 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: N s p 3 s N s s The p orbitals on N are oriented along the X, Y, and Z axes so we would predict that the angles between the p-s bonds in N 3 would be 90. We know that this is not the case. 7
18 onding orbital lp orbital N p character 07.3 s character 77% 3% 69% 3% p 0 s s Promotion s s p
19 (s a p x ) b (s p x ) p character s character onding orbital 50% 50%
20 ybridization The problem of accounting for the true geometry of molecules and the directionality of orbitals is handled using the concept of hybrid orbitals. ybrid orbitals are mixtures of atomic orbitals and are treated mathematically as linear combinations of the appropriate s, p and d atomic orbitals. Linear sp hybrid orbitals A s orbital superimposed on a p x orbital s s p The / are normalization coefficients. p The two resultant sp hybrid orbitals that are directed along the x-axis (in this case) 3
21 Example of the orthogonality of and s p s p s p s p s s s p s p p p Thus our hybrid sp orbitals are orthogonal to each other, as required. ybridization Valence bond theory treatment of a linear molecule: the bonding in e e e* s e p The promotion energy can be considered a part of the energy required to form hybrid orbitals. e* (sp) sp p s s e The overlap of the hybrid orbitals on e with the s orbitals on the atoms gives two e- (sp)-s bonds oriented 80 from each other. This agrees with the VSEPR theory prediction. 4
22 onding orbital p character s character 67% 33% Valence bond theory treatment of a trigonal planar molecule: the bonding in 3 * s p * (sp ) sp p y x This gives three sp orbitals that are oriented 0 apart in the xy plane be careful: the choice of axes in this example determines the set of coefficients. x 3 6 s p py x s 3 s p py 6 p x 5
23 The coefficients in front of each atomic wavefunction indicate the amount of each atomic orbital that is used in the hybrid orbital. The sign indicates the orientation (direction) of the atomic orbitals. Remember that you have to use each atomic orbital completely (columns) and that each hybrid must be normal (rows). Check this by summing the squares of the coefficients. x 3 6 s p py x s p py s 6 p x /3 + /6 + / = So this hybrid is normal /3 + /6 + / = So this hybrid is normal /3 + 4/6 = So this hybrid is normal The signs in front of the coefficients indicate the direction of the hybrid: : -x, +y : -x, -y 3 : +x, 0y y /3 + /3 + /3 = So the entire s orbital has been used /6 + /6 + 4/6 = So the entire p x orbital has been used / + / = So the entire p y orbital has been used x Valence bond theory treatment of a trigonal planar molecule: the bonding in 3 * sp p 3 s s s The overlap of the sp hybrid orbitals on with the s orbitals on the atoms gives three - (sp )-s bonds oriented 0 from each other. This agrees with the VSEPR theory prediction. 6
24 onding orbital p character s character 75% 5% Valence bond theory treatment of a tetrahedral molecule: the bonding in C 4 C C* s p C* (sp 3 ) sp 3 C This gives four sp 3 orbitals that are oriented in a tetrahedral fashion s px py pz s px py pz s px py pz s px py pz
25 Valence bond theory treatment of a tetrahedral molecule: the bonding in C 4 C C* s p C* (sp 3 ) sp 3 4 s s s s C The overlap of the sp 3 hybrid orbitals on C with the s orbitals on the atoms gives four C- (sp 3 )-s bonds oriented from each other. This provides the tetrahedral geometry predicted by VSEPR theory.
26
27 Trigonal bipyramidal
28 Valence bond theory treatment of a trigonal bipyramidal molecule: the bonding in PF 5 3s 3p PF 5 has a VSEPR theory AX 5 P geometry so we need hybrid 3d orbitals suitable for bonds to P* 5 atoms. ns and np combinations can only P* (sp 3 d) 3d provide four, so we need to use nd orbitals (if they are available). 3s 3p z 3p y 3p x 3d z sp 3 d z The appropriate mixture to form a trigonal bipyramidal arrangement of hybrids involves all the ns and np orbitals as well as the nd z orbital. Valence bond theory treatment of a trigonal bipyramidal molecule The orbitals are treated in two different sets. x 3 6 s p py x s 3 s p py 6 p x These coefficients are exactly the same as the result for the trigonal planar molecules because they are derived from the same orbitals (sp ) 4 pz 5 pz d z d z These coefficients are similar to those for the sp hybrids because they are formed from a combination of two orbitals (pd). Remember that d orbitals are more diffuse than s or p orbitals so VT predicts that the bonds formed by hybrids involving d orbitals will be longer than those formed by s and p hybrids. 3
29 Valence bond theory treatment of a trigonal bipyramidal molecule: the bonding in PF 5 P* (sp 3 d) 3d F s p F F s p s p F s p F s p The overlap of the sp 3 d hybrid orbitals on P with the p orbitals on the F atoms gives five P-F (sp 3 d)-p bonds in two sets: the two axial bonds along the z-axis (80 from each other) and three equatorial bonds in the xy plane (0 from each other and 90 from each axial bond). This means that the 5 bonds are not equivalent! octahedral 4
30 S S* 3s Valence bond theory treatment of an octahedral molecule: the bonding in SF 6 3p 3d S* (sp 3 d ) 3d F F F F F F 3s 3p z 3p y 3p x 3d z 3d sp 3 d x -y The overlap of the sp 3 d hybrid orbitals on S with the p orbitals on the F atoms gives six S-F (sp 3 d )-p bonds 90 from each other that are equivalent. You can figure out the normalization coefficients. 5
31 Valence bond theory treatment of bonding: a hypervalent molecule, ClF 3 3s 3p Cl 3d Cl* F F Cl F Cl* (sp 3 d) F F F 3d There are five objects around Cl so the geometry is trigonal bipyramidal and the shape is given by AX 3 E (T-shaped). The overlap of the sp 3 d hybrid orbitals on Cl with the p orbitals on the F atoms gives three P-F (sp 3 d)-p bonds in two sets: the two axial bonds along the z-axis (less than 80 from each other because of the repulsion from the lone pairs) and the one equatorial bond halfway between the other Cl bonds. Again, the bond lengths will not be the same because there is more d contribution to the axial hybrid orbitals.
32 VT does not adequately explain:. Excited state properties O O. Paramagnetism of oxygen: presence of unpaired electrons 3. ond in hypervalent molecules, energy cost too high Lewis structure
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