For copyright notice see final page of this file Chemistry 125: Lecture 11 Sept. 27, 2010 Orbital Correction and Plum-Pudding Molecules Several tricks (“Z-effective” and “Self Consistent Field”) allow one to correct approximately for the error in using orbitals when there is electron-electron repulsion. Residual error is hidden by naming it “Correlation energy.” J.J. Thomson’s Plum-Pudding model of the atom can be modified to visualize the form of molecular orbitals. There is a close analogy in form between the molecular orbitals of CH4 and NH3 and the atomic orbitals of neon, which has the same number of protons and electrons. The underlying form, dictated by kinetic energy, is distorted by pulling protons out of the Ne nucleus to play the role of H atoms. The same is true for more complicated molecules, because Schrödinger was right. For copyright notice see final page of this file

What's Coming for Next Exam? Atoms Orbitals for Many-Electron Atoms (Wrong!) Recovering from the Orbital Approximation Molecules Plum-Pudding Molecules (the "United Atom" Limit) Understanding Bonds (Pairwise LCAO) "Energy-Match & Overlap" Reality: Structure (and Dynamics) of XH3 Molecules Reactivity HOMOs and LUMOs Recognizing Functional Groups Payoff for Organic Chemistry! How Organic Chemistry Really Developed (Intro)

Multiply 1-e Wave Functions 2 a(r1,q1,f1)  b(r2,q2,f2) = Multiply 1-e Wave Functions Y(r1,q1,f1,r2,q2,f2) attr to R. Feynman: “Thereﾕs a reason physicists are so successful with what they do, and that is they study the hydrogen atom and the helium ion and then they stop.” Orbitals can’t work!

Tricks for Salvaging Orbitals

r 1s = K e-r/2 2Z r  r nao Z - effective ! ! Pretty Crude Pretend that the other electron(s) just reduce the nuclear charge for the orbital of interest. "Clementi-Raimondi" values for Zeff (best fit to better calculations as of 1963) Atom Z Zeff 1s He 2 1.69 Zeff 3s Na 11 10.63 6.57 6.80 2.51 Zeff 2s Zeff 2p C 6 5.67 3.22 3.14 1s = K e-r/2 2s slightly “less screened” than 2p r  r 2Z nao r Z ! 2p 2s 1s ! but vice versa for Na (subtle)

Self-Consistent Field (SCF) 1. Find approximate orbitals for all electrons (e.g. using Zeff) 2. Calculate potential energy due to fixed, point protons and fixed clouds for all electrons but one. 3. Use this new potential to calculate an. an..improved orbital for that one electron. 4. Repeat steps 2 and 3 to improve the orbital for another electron. . . . Improve all orbitals one by one. Cycle back to improve 1st orbital further, etc. etc. Quit ? when orbital shapes (and energy) stop changing

True Energy < SCF Energy Still Wrong! because real electrons are not fixed clouds. They keep out of each other’s way by correlating their motions. True Energy < SCF Energy What do people do about this error?

Conceal the residual error after full SCF calculation to the “Hartree-Fock” limit by giving it a fancy name: "Correlation Energy" Where to get correct energy (& total electron density)? by Experiment or by a Much More Complex Calculation: e.g. “Configuration Interaction” (CI) or “Density Functional Theory” (DFT)

If we’re really lucky, "Correlation Energy" might be Negligible.

Energy Magnitudes Should Chemists care about the error in Orbital Theory? 8 6 2 4 ~ 12C Nucleus (2  109) Fortunately nuclear energy is totally unchanged during chemistry! Loses 0.1 amu (E = mc2) 0.001% change in nuclear energy would overwhelm all of Coulomb. - C • C C+6 + + - + - - - + + + - Coulombville C Core (2  104) correlation error ≈ bond C Atom (3  103) Changes in "correlation energy" can be ~10-15% of Bond Energy. log (Energy Change kcal / mole) 1/2  4 Single Bonds (2  102) C "Correlation Energy" (102) "Non-bonded" Contacts (1-20)  Correlation Orbital Theory is fine for Qualitative Understanding of Bonding. -2 He•He @ 52Å! (2  10-6)

Orbitals can't be “true” for >1 electron, because of e-e repulsion but we'll use them to understand bonding, structure, energy, and reactivity * * Resort to experiments or fancy calculation for precise numbers.

If we use orbitals, how should we reckon total electron density? Density of electron 1 = 1 2(x1,y1,z1) Density of electron 2 = 2 2(x2,y2,z2) Total density (x,y,z) = 1 2(x,y,z) + 2 2(x,y,z) (Sum, not Product. Not a question of joint probability)

How Lumpy is the N Atom? Spherical ! (2px)2 = K x2 e- (2py)2 = K y2 e- (2pz)2 = K z2 e- Total = K(x2 + y2 + z2) e- Total = K(r2) e- Spherical ! [from an Organic Text]

2px2 + 2py2 depends on (x2+y2) It is thus symmetrical F N TFDCB ? C N Triple Bond cross section is round not clover-leaf nor diamond! 2px2 + 2py2 depends on (x2+y2) It is thus symmetrical about the z axis

Atoms Molecules 3-Dimensional Reality (H-like Atoms) Hybridization Orbitals for Many-Electron Atoms (Wrong!) Recovering from the Orbital Approximation Molecules Understanding Bonds (Pairwise LCAO-MOs) “Overlap & Energy-Match" First an aside on computer-generated MOs: Plum-Pudding MOs (the "United Atom" Limit)

What Spreads and Shapes Atomic Orbitals? Potential Energy scales r (via ) double the nuclear charge Kinetic Energy creates nodes (Schrödinger) 2s 4d

Ways of Looking at an Elephant http://en.wikipedia.org/wiki/Image:Blind_monks_examining_an_elephant.jpg "Blind monks examining an elephant" by Itcho Hanabusa. 1888 Ukiyo-e print illustration from Buddhist parable

Ways of Looking at a Molecule (or a Molecular Orbital) from set of atoms Molecule as one atom Molecule as atoms Set of ~normal atoms Single “United Atom” Atoms with small bonding distortion (~0.05 Lewis) distorted by fragmenting the nucleus J. J. Thomson's Plum Pudding! (backwards) Nuclei embedded in a cloud of electrons dispersed and “noded” by kinetic energy Which contour should we use? e-density contours of H2 (worth a quick look)

How the Plums Distort Hydrogen-Like Kinetic-Energy Puddings

Methane & Ammonia Spartan 6-31G. calculates good SCF MOs (on my laptop We want to understand them visually.

4 Pairs of Valence Electrons Compare MOs to AOs of Ne (4 electron pairs with n=2) H C N H 4 Pairs of Valence Electrons

Contour Level 0.001 e/Å3 CH4 NH3 Boring! 1s 8 valence e-  4 MOs .. .. energy Three “degenerate” Molecular Orbitals Two “degenerate” Molecular Orbitals CH4 NH3 1s "Core" Orbitals Like 1s of C/N Tightly Held Little Distortion Boring! We'll focus on Valence Orbitals Contour Level 0.001 e/Å3

.. 2s Contour Level 0.03 e/Å3

.. 2s “Spherical” node

.. CH4 NH3 2px

.. CH4 NH3 CH4 NH3 2py

.. CH4 NH3 CH4 NH3 2py

CH4 NH3 CH4 NH3 2pz HOMO .. Lewis's "unshared pair" +Unoccupied Orbitals HOMO Highest Occupied Molecular Orbital .. CH4 NH3 CH4 NH3 2pz Lewis's "unshared pair" No proton stabilizes this lobe.

Lowest Unocupied Molecular Orbital "HUMO?" .. LUMO Lowest Unocupied Molecular Orbital .. CH4 NH3 3s

.. 2s CH4 NH3 3s LUMO

.. .. .. .. .. CH4 NH3 3dx2-y2

.. CH4 NH3 3dx2-y2

.. .. .. .. .. CH4 NH3 3dxy

.. CH4 NH3 3dxy

.. .. .. .. 3dz2 3dz2 CH4

Ethane & Methanol (Spartan 6-31G*)

7 Pairs of Valence Electrons H 7 Pairs of Valence Electrons Compare MOs to AOs of Ar (7 electron pairs) O C H

Rotated 90° CH3CH3 Orbital Energy Occupied Vacant 2s Pedantic Note: with two “heavy” atoms there are two boring “core” orbitals. For the purpose of making atomic analogies to study valence-level molecular orbitals, we’ll use the atomic 1s orbital to stand for the set of molecular core orbitals. Thus we start with 2s rather than 1s for valence-level MOs, which will in truth include tiny nodes around the heavy nuclei. HOMO-6 CH3OH Orbital Energy Occupied Vacant

CH3CH3 Orbital Energy CH3OH 2pz HOMO-5

CH3CH3 Orbital Energy CH3OH 2px HOMO-4

CH3CH3 Orbital Energy CH3OH 2py HOMO-3

CH3CH3 Orbital Energy CH3OH 3s HOMO-2

CH3CH3 Orbital Energy CH3OH 3dxz HOMO-1

CH3CH3 Orbital Energy CH3OH 3dyz HOMO

CH3CH3 Orbital Energy CH3OH LUMO 3dz2 LUMO

CH3CH3 Orbital Energy CH3OH LUMO+1 3pz LUMO+1

CH3CH3 Orbital Energy CH3OH LUMO+2 3py LUMO+3

CH3CH3 Orbital Energy CH3OH LUMO+3 3px LUMO+2

CH3CH3 Orbital Energy LUMO+4 3dxy

CH3CH3 Orbital Energy LUMO+5 3dx2-y2

CH3CH3 Orbital Energy CH3OH LUMO+6 4f LUMO+4

1-Fluoroethanol

Wire

Core 1 1s (F)

Core 2 1s(O)

Core 3 1s(C1)

Core 4 1s(C2)

1s(valence)

2px

2py rotate

2py rotate

2spz (up)

2spz(down)

3dxy

Single “United Atom” distorted by a fragmented nucleus Atoms with The Plum-Pudding View of Molecular Orbitals Shows Generality of Kinetic-Energy-Based Clouds e-density contours of H2 But One Must Probe Harder to Gain a Useful Understanding of Chemical Bonds Single “United Atom” distorted by a fragmented nucleus Atoms with weak bonding Which contour should we use?

End of Lecture 11 Sept. 27, 2010 Copyright © J. M. McBride 2009,2010. 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). 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