After applying the united-atom “plum-pudding” view of molecular orbitals, introduced in the previous lecture, to a more complex molecule, this lecture introduces the more utilitarian concept of localized pairwise bonding between atoms. Formulating an atom-pair orbital as a sum of atomic orbitals creates an electron difference density by means of the cross product that enters upon squaring a sum. This “overlap” term is the key to bonding. The hydrogen molecule is used to illustrate how close a simple sum of atomic orbitals comes to matching reality, especially when the atomic orbitals are allowed to hybridize. The overlap integral is related to the hybridization of carbon-atom orbitals. Chemistry 125: Lecture 12 Sept. 30, 2009 Overlap and Atom-Pair Bonds For copyright notice see final page of this file

1-Fluoroethanol

Wire

1s (F) Core 1

1s(O) Core 2

1s(C 1 ) Core 3

1s(C 2 ) Core 4

1s(valence)

2p x

2p y rotate

2p y rotate

2sp z (up)

2sp z (down)

3d xy

e-density contours of H 2 Single “United Atom” distorted by a fragmented nucleus Which contour should we use? The Plum-Pudding View of Molecular Orbitals Shows Generality of Kinetic-Energy-Based Clouds Atoms with weak bonding But One Must Probe Harder to Gain a Useful Understanding of Chemical Bonds

Pairwise LCAO MOs 1 √2 ( AO a + AO b )  (x 1,y 1,z 1 ) = SUM (L inear C ombination ) of AOs (like hybridization, but with two atoms) Why is this form sensible? “True” molecular orbitals extend over entire molecules, but we want to understand local bonds as

H 2 at Great Distance 1 2 ( AO A 2 + AO B 2 )    (x 1,y 1,z 1 ) = H 2 at Bonding Distance? 1 √2 ( AO A + AO B )  (x 1,y 1,z 1 ) = + AO A  AO B error?negligible!  +

Overlap (A  B) Creates Bonding If we approximate a molecular orbital as a sum of atomic orbitals: and square to find electron density: then subtract the average of the atom electron densities: we find bonding, the difference electron density due to overlap: Looks very good near nuclei (A near A, B near B) “By-product” of squaring a sum. A completely different instance of multiplying! (NOT two electrons) “By-product” of squaring a sum.  < (normalization) < Shifts e-density from atoms _ to overlap region.

in Ain B Wells far apart Wells far apart Total Energy of Particle "Mixing" localized   s for double minimum Wells close together in AB Antibonding Holds A & B together Black line is energy Blue line is  Bonding! Stabilization of Particle e-Density Grows e-Density Shrinks

AA 2 BB 2 Where is  A  B significant? no yesa littleno!  b small yes! Where is  A 2 significant? At the center 2  A  B is as large as  A 2 +  B 2 Electron Density nearly Doubled! “Overlap Integral” (   A  B ) measures net change from atoms. Region of Significant Overlap 

92.9% of Total Electronic Energy (almost all of which was already present in the atoms) High accuracy is required to calculate correct value of the Bond Energy, the difference between atoms and molecule. (Cf. X-ray difference density) Total e-Density Difference Density 1s (atomic) 0.02 e/a o 3 Coutoured at e/a o 3 Coutoured at e/a o 3 State-of-the-art Comparison 40 years ago Laws & Lipscomb, Isr. J. Chem. 10, 77 (1970)  A =  B = e -  /2 52% of Bond Energy H H

1s (atomic) 1s (optimized exponent)  A =  B = e -  /2.4 Total e-Density Difference Density 52% % 0.04 of Bond Energy Very crudest model shows most of bond. General spread increases bonding density/stabilization. shift from atom to bond Adjust molecular orbital to lower the average “total energy”. This makes  more realistic, because the true energy is the lowest possible (according to the “variational principle”).  A =  B = e -  /2 of Bond Energy H H 2  larger shift from atom to bond

1s (optimized exponent) 0.04 General spread increases bonding density/stabilization. 73% of Bond Energy Hybridization’s directed spread improves bonding density. 2  larger shift from atom to bond Total e-Density Difference Density Hybridized (+ some SCF) (96.7% 1s; 0.6% 2s; 2.7% 2p) % of Bond Energy 100% 1sHybrid: 96.7% 1s 0.6% 2s 2.7%2p Helps A  B overlap (but at the cost of 3% n=2 character) 3  Larger shift from beyond nucleus to bond A more realistic way to deform the atomic orbital:

+ some correlation (CI) 0.11 Total e-Density Difference Density Hybridized (+ some SCF) (96.7% 1s; 0.6% 2s; 2.7% 2p) % of Bond Energy Density ~unchanged >2  better Directed spread improves bonding density. (How so?) 76% of Bond Energy

Pairwise LCAO-MO Looks like atoms (especially near nuclei) (the Main Event for electron enery; ~ 6x larger than bond) <1 √2 ( AO A + AO B )  (x 1,y 1,z 1 ) = Virtues: Builds up e-density between nuclei (through Overlap - the source of Bonding) Hybridizing AOs provides flexibility (unlimited if you use all H-like AOs in hybrid) Easy to formulate and understand (but keep it simple - valence shell is fairly good) Smooths  to lower kinetic energy [though ultimate contraction toward nuclei raises it again]

Pairwise LCAO-MO <1 √2 ( AO A + AO B )  (x 1,y 1,z 1 ) = <1 2 (AO A 2 + AO B AO A AO B ) == Atoms Bond (overlap / product) >1>1 >1>1 Anti

Overlap & Energy-Match Bonding depends on

Consider how the Overlap Integral (the “sum” of A x B over all space) depends on the Distance between two Carbon Atoms and on Hybridization of their 2s/2p Atomic Orbitals. (energetically cheaper than hybridizing 1s/2p of H)

2s C Overlap Scale Diameter of node for 2s C is 0.7 Å Sliding together to 1.4Å (~CC bond distance) superimposes the two 'X's xx

2s x C Overlap Scale 2s x x x x x x x Sliding together to 1.4Å superimposes the two 'X's Overlap Integral = 0.41 ! Guess the overlap integral,  A  B (remember that  A  A = 1)

C Overlap Overlap Integral Å s-p  2s2p  2s2p    + x - + x + 2p  xx s-s p-p  CCCCCC  and  are “orthogonal” (net overlap = 0) to -1 at D = 0 to 0 at D = 0 to 1 at D = 0 p-p   (sigma) is Greek “s” MO analogue of s AO. (no node through nuclei)  (pi) is Greek “p” MO analogue of p AO. (nodal plane through nuclei)

End of Lecture 12 Sept. 30, 2009 Copyright © J. M. McBride Some rights reserved. Except for cited third-party materials, and those used by visiting speakers, all content is licensed under a Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0).Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0) Use of this content constitutes your acceptance of the noted license and the terms and conditions of use. Materials from Wikimedia Commons are denoted by the symbol. Third party materials may be subject to additional intellectual property notices, information, or restrictions. The following attribution may be used when reusing material that is not identified as third-party content: J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0