Presentation on theme: "Several tricks (Z-effective and Self Consistent Field) allow one to correct approximately for the error in using orbitals when there is electron-electron."— Presentation transcript:
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. Thomsons 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 CH 4 and NH 3 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. Chemistry 125: Lecture 11 Sept. 28, 2009 Orbital Correction and Plum-Pudding Molecules For copyright notice see final page of this file
What's Coming for Next Exam? Molecules Plum-Pudding Molecules (the "United Atom" Limit) Understanding Bonds (Pairwise LCAO) "Energy-Match & Overlap" Reality: Structure (and Dynamics) of XH 3 Molecules Atoms Orbitals for Many-Electron Atoms (Wrong!) Recovering from the Orbital Approximation Payoff for Organic Chemistry! Reactivity HOMOs and LUMOs Recognizing Functional Groups How Organic Chemistry Really Developed (Intro)
2-e Wave Function (r 1, 1, 1,r 2, 2, 2 ) a (r 1, 1, 1 ) b (r 2, 2, 2 ) = ? Multiply 1-e Wave Functions 2 22 No way can electrons be independent! They repel one another.
Tricks for Salvaging Orbitals
Pretend that the other electron(s) just reduce the nuclear charge for the orbital of interest. "Clementi-Raimondi" values for Z eff (best fit to better calculations as of 1963) Atom Z Z eff 1s He s 2p Z - effective Z eff 2s Z eff 2p C Z eff 3s Na ! ! 2s slightly less screened than 2p vice versa for Na Pretty Crude r 2Z2Z na o 1s = K e - /2 (subtle) 1s
Self-Consistent Field (SCF) 1. Find approximate orbitals for all electrons (e.g. using Z eff ) 2. Calculate potential from 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. Quit When orbital shapes stop changing Cycle back to improve 1 st orbital further, etc. etc.
Still Wrong! because real electrons are not fixed clouds. They keep out of each others way by correlating their motions. True Energy < SCF Energy What do people do about this error?
"Correlation Energy" Conceal the residual error after full SCF calculation to the Hartree-Fock limit by giving it a fancy name: Where to get correct energy (& total electron density)? by Experiment or by a Whopping Calculation: e.g. Configuration Interaction (CI) or Density Functional Theory (DFT)
If were really lucky, "Correlation Energy" might be Negligible.
"Non-bonded" Contacts (1-20) C Energy Magnitudes Should Chemists care about the error in Orbital Theory? -2 log (Energy Change kcal / mole C Core ( ) 1/2 4 Single Bonds ( ) 52Å! ( ) Changes in "correlation energy" can be ~10-15% of Bond Energy. Orbital Theory is fine for Qualitative Understanding of Bonding. C "Correlation Energy" (10 2 ) - C C 12 C Nucleus ( ) Loses 0.1 amu (E = mc 2 ) Fortunately nuclear energy is totally unchanged during chemistry! 0.001% change in nuclear energy would overwhelm all of Coulomb. correlation error bond ~ C Atom ( )
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 (x 1,y 1,z 1 ) Density of electron 2 = 2 2 (x 2,y 2,z 2 ) Total density (x,y,z) = 1 2 (x,y,z) (x,y,z) (Sum, not Product. Not a question of joint probability)
How Lumpy is the N Atom? Total = K(r 2 ) e - (2p x ) 2 = K x 2 e - (2p y ) 2 = K y 2 e - (2p z ) 2 = K z 2 e - Total = K(x 2 + y 2 + z 2 ) e - Spherical ! [from an Organic Text]
TFDCB C CC C F N is round not clover-leaf nor diamond! C N Triple Bond 2p x 2 + 2p y 2 depends on (x 2 +y 2 ) It is thus symmetrical about the z axis cross section ?
Molecules Understanding Bonds (Pairwise LCAO-MOs) Overlap & Energy-Match" Atoms 3-Dimensional Reality (H-like Atoms) Hybridization Orbitals for Many-Electron Atoms (Wrong!) Recovering from the Orbital Approximation First an aside on computer-generated MOs: Plum-Pudding MOs (the "United Atom" Limit)
What gives Atomic Orbitals their Shape? Potential Energy scales r (via ) Kinetic Energy creates nodes (Schr ö dinger) 4d 2s double the nuclear charge
Ways of Looking at an Elephant
Set of ~normal atoms Atoms with small bonding distortion (~0.05 Lewis) Single United Atom Ways of Looking at a Molecule (or a Molecular Orbital) e-density contours of H 2 Which contour should we use? Molecule from set of atoms Molecule as one atom distorted by fragmenting the nucleus Nuclei embedded in a cloud of electrons dispersed and noded by kinetic energy J. J. Thomson's Plum Pudding! (backwards) Molecule as atoms (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 H CHH H NHH H Compare MOs to AOs of Ne (4 electron pairs with n=2)
1s CH 4 NH 3 "Core" Orbitals Like 1s of C/N Tightly Held Little Distortion Contour Level e/Å 3 We'll focus on Valence Orbitals Boring!.. 8 valence e - 4 MOs 8 valence e - 4 MOs energy Three degenerate Molecular Orbitals
2s.. Spherical node
2p x.. CH 4 NH 3..
CH 4 NH 3 2p y.. CH 4 NH 3..
CH 4 NH 3 2p y.. CH 4 NH 3..
CH 4 NH 3 2p z HOMO Lewis's "unshared pair".. CH 4 NH 3.. +Unoccupied Orbitals
CH 4 NH 3 3s LUMO "HUMO?"..
2s CH 4 NH 3 3s LUMO "HUMO"..
CH 4 NH 3 3d x 2 -y 2..
CH 4 NH 3 3d x 2 -y 2..
CH 4 NH 3 3d xy..
CH 4 NH 3 3d xy..
CH 4 3d z 2..
Ethane & Methanol (Spartan 6-31G*)
7 Pairs of Valence Electrons C C HH HH H H O C H HH H Compare MOs to AOs of Ar (7 electron pairs)
2s CH 3 Orbital Energy Occupied Vacant HOMO-6 CH 3 OH Orbital Energy Occupied Vacant Rotated 90° 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, well 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-5 2p z CH 3 Orbital Energy CH 3 OH Orbital Energy
HOMO-4 2p x CH 3 Orbital Energy CH 3 OH Orbital Energy
HOMO-3 2p y CH 3 Orbital Energy CH 3 OH Orbital Energy
HOMO-2 CH 3 Orbital Energy CH 3 OH Orbital Energy 3s
HOMO-1 3d xz CH 3 Orbital Energy CH 3 OH Orbital Energy
HOMO 3d yz CH 3 Orbital Energy CH 3 OH Orbital Energy
LUMO 3d z 2 CH 3 Orbital Energy CH 3 OH Orbital Energy
LUMO+1 3p z CH 3 Orbital Energy CH 3 OH Orbital Energy
LUMO+3 LUMO+2 3p y CH 3 Orbital Energy CH 3 OH Orbital Energy
LUMO+2 LUMO+3 3p x CH 3 Orbital Energy CH 3 OH Orbital Energy
LUMO+4 3d xy CH 3 Orbital Energy
LUMO+5 3d x 2 -y 2 CH 3 Orbital Energy
LUMO+6 LUMO+4 4f CH 3 Orbital Energy CH 3 OH Orbital Energy