lecture 6. many-electron atoms. pt.4. physical significance of hartree-fock solutions: electron...
Post on 30-Dec-2015
221 Views
Preview:
TRANSCRIPT
Lecture 6. Many-Electron Atoms. Pt.4.Physical significance of Hartree-Fock
solutions:Electron correlation, Aufbau principle, Koopmans’ theorem & Periodic trends
References
• Ratner Ch. 9.5-, Engel Ch. 10.5-, Pilar Ch. 10• Modern Quantum Chemistry, Ostlund & Szabo (1982) Ch. 3.3 • Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.7• Computational Chemistry, Lewars (2003), Ch. 5
• A Brief Review of Elementary Quantum Chemistryhttp://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.htmlhttp://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html
• Electron-electron repulsion• Indistinguishability
Helium Atom First (1 nucleus + 2 electrons) (Review)
We cannot solve this Schrödinger equation analytically. (Two electrons are not separable nor independent any more.)
A series of approximations will be introduced.
1. Electron-electron repulsion (correlation)
The 1/r12 term removes the spherical symmetry in He.
~H atom electron at r1
~H atom electron at r2
newly introduced
: Correlated, coupled
Hartree-Fock equation (One-electron equation)
spherically symmetric
&
- Two-electron repulsion operator (1/rij) is replaced by one-electron operator VHF(i), which takes it into account in an “average” way.
- Any one electron sees only the spatially averaged position of all other electrons.
- VHF(i) is spherically symmetric.
- (Instantaneous, dynamic) electron correlation is ignored.
- Spherical harmonics (s, p, d, …) are validangular-part eigenfunctions (as for H-like atoms).
- Radial-part eigenfunctions of H-like atoms are not valid any more. optimized
Veff includes
• A single Slater determinant never corresponds to the exact wave function.
EHF > E0 (the exact ground state energy)
• Correlation energy: a measure of error introduced through the HF scheme
EC = E0 EHF (< 0)
– Dynamical correlation
– Non-dynamical (static) correlation
• Post-Hartree-Fock method (We’ll see later.)– Møller-Plesset perturbation: MP2, MP4, …
– Configuration interaction: CISD, QCISD, CCSD, QCISD(T), …
– Multi-configuration self-consistent-field method: MCSCF, CAFSCF, …
Electron Correlation (P.-O. Löwdin, 1955)Ref) F. Jensen, Introduction to Computational Chemistry, 2nd ed., Ch. 4
Solution of HF-SCF equation:Z- (measure of shielding)0 0.31
1.72 2.09
8.49 8.69
2.422.58
2.782.86
3.153.17
3.513.55
3.873.90
4.244.24
8.888.93
9.109.71
9.3610.11
9.7310.52
9.9310.88
10.2411.24
more shieldedless shielded
Solution of HF-SCF equation:Effective nuclear charge
(Z- is a measure of shielding.)
higher energy, bigger radius lower energy, smaller radius
www.periodictable.com/Properties/A/AtomicRadius.v.wt.html
Source: www.chemix-chemistry-software.com/school/periodic_table/atomic-radius-elements.html
larger smaller
As well as the total energy, one also obtains a set of orbital energies.
Remove an electron from occupied orbital a.
Orbital energy = Approximate ionization energy
Physical significance of orbital energies (i):Koopmans’ theorem (T. C. Koopmans, 1934)
Physica,1, 104
Ostlund/SzaboCh.3.3
length
energy
Atomic orbital energy levels & Ionization energyof H-like atoms
2
20
0
4
ema
e
Total energy eigenvalues are negative by convention. (Bound states)
...3,2,1 with 32 222
02
42
nne
eZEn
depend only on the principal quantum
number.
1Ry
Minimum energy required to remove an electron from the ground state
IE (1 Ry for H)
atomic units
Hartree-Fock orbital energies i & Aufbau principle
degenerate
For H-like atoms
”
”
Hartree-Fock orbital energies i depend on
both the principal quantum number (n) and the angular quantum number (l).
Within a shell of principal quantum number n,
ns np nd nf …
Electronegativity (~ IE + EA)
~Lowest Unoccupied
AO/MO (LUMO)
~Highest Occupied
AO/MO (HOMO)
small
small
high
high
low or deep
low or deep
small
large
large
large
Na + Cl+ NaCl Na+ + Cl
Periodic trends of many-electron atoms: Electronegativity
http://www.periodictable.com/Properties/A/Electronegativity.bt.wt.html
Periodic trends of many-electron atoms: 1st ionization energy
http://www.periodictable.com/Properties/A/IonizationEnergies.bt.wt.html
Periodic trends of many-electron atoms: Electron affinity
http://www.periodictable.com/Properties/A/ElectronAffinity.bt.wt.html
top related