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50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
Chemistry, Physics and the Born-Oppenheimer Approximation
Hendrik J. MonkhorstQuantum Theory Project
University of FloridaGainesville, FL 32611-8435
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
Outline1. Structure of Matter2. Shell Models3. Quantum Mechanics of Structure4. Born-Oppenheimer Approximation (BOA)5. Pros and Cons of BOA6. Removal of BOA7. The Coupled-Cluster Method8. The Molecular Coupled-Cluster Method9. A Solid Coupled-Cluster Method?10. Concluding Remarks
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
Preview
• This is a Tour from the Familiar to the Unfamiliar • No dazzling results with numbers, graphs• Ends with a new quantum view of structure in matter
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
• Several meanings of structure and matter. From Webster-Collegiate Dictionary,
* Structure:“…5.a the aggregate of elements of an entity in their relationship to each other….”
* Matter:”….2.a the substance of which a physical object is composed; 2.b material substance that occupies space and has weight that constitutes the observable universe, and that together with energy forms the basis of objective phenomena;…”
1. Structure of Matter
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
• In chemistry and physics we speak of * crystal structure
* molecular structure
* atomic structure * nuclear structure
• Conceptually, we imagine gradually less precise localization of particles
• Called for different concepts & methods to describe physics
• Can we bring about more unification?And should we?
( , )p n( , )A
Ze X
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
2. Shell Models
• Bohr’s(1915) atomic shells
Goeppert-Mayer (1946) nuclear shells
• Structure is shells: localization at/near radii
• Qualitatively powerful (symmetries!)• Quantitatively wanting (spectra!)
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
• Shell model forms basis for Independent Particle Models (IPMs), leading to
Aufbau PrincipPeriodic Table of Elements(Un)stable nucleiexpnts: semi-quantitative spectroscopiestheory: Hartree and Hartree-Fock orbitals
. IPM also applied to molecules and crystals (>1927)molecular orbitals, Bloch orbitals
•
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
• BUT: something had to be done about nuclei in molecules and crystals:* do they move?* if not, where are they?*expts (microwaves, X-rays, neutrons) tell where* but why?
• Are apparently not in shells, but how else to describe them?
• Nuclear positions =structure of molecules, crystals
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
3. Quantum Mechanics of Structure
• IPMs not adequate for quantitative description, and prediction
• Known as many-body or correlation problem: particles on shell interact
• Is now THE major effort to calculate corrections accurately
• Occupied sets of orbitals represent shells, unoccupied orbitals represent excitations
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
• Only Kohn-Sham (KS) implementation of Density Functional Theory (DFT) preserves this shell structure
• But KS has problems in principle & practice:* what do KS orbitals mean?* what is DFT functional?
• Other methods of correction ‘blur’ shell picture: *expression as excitations (CI, MBPT)* where are the interactions among shell particles?
• Would be desirable to stay close to shell model. How?
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
4. The Born-Oppenheimer Approximation (BOA)
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
{R}
Fix nuclei in space Calculate electron wavefunctionˆ ( , ) ( ) ( , ), : electrons, : nuclei
( ) : potential energy surface (PES){ } : multi-dimensional space min ( ) structure of molecule
e e e e
e
e
H r R E R r R r RE RR
E R
Ψ = Ψ
→
ii
1/ 4
, crystal
Calculate ( ) on PES, then total wavefunction ( , )= ( ) ( , )
Correct to second order in =
Quantum mechanical justification for structure
n
n e
e
n
X Rr R X R r R
mM
κ
Ψ Ψ
⎛ ⎞⎜ ⎟⎝ ⎠
i
i
i
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
• BOA CENTRAL to chemistry, molecular physics, solid-state physics
• Without it, where would we be?• Advantages:
* can think of structure without electrons* can ‘see’ symmetries, vibrations, rotations* internal rotations about bond* can visualize isomers, enantiomers* rather accurate for rovibrational excitations
-- just above ground states of molecules-- and lattice vibrations in crystals
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
• Disadvantages:* what is really reaction coordinate for
polyatomic molecular reactions?* difficulties for many spectroscopies
--Rydberg states--rovibronic excited states --various laser chemistries
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
,
* in principle, Born-Huang expansion can be used:( , ) ( ) ( , )
many PESs --PES picture gone (crossings, intersections) --difficult calculations* crystals: phonons
ne en e
r R X R r R↓↓
Ψ = Ψ∑
, e-phonon interaction, transport tough to deal with quantitatively
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
6. Removal of BOA What would this mean?
* treat nuclear QM at same footing as electron QM
* yet, recognize the effect of 1
Electrons are delocalized, nuclei are 'localized' within molecule and crystal W
e
n
mM
i
i
i hy this difference? Because in ground state * | | electrons delocalized * | | nuclei localized (cf. Wigner crystallization!) Highly excited states | |
loose/'vagu
e e
n n
n n
K VK V
K V
→
→
→ ≤
⇒
∼
ie' structure
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
Q: Can we bring nuclei into shell structure?A: With Coupled-Cluster (CC) method!!
* rather obvious for molecules* Wanniers for electrons & nuclei in solids* need ‘universal’ basis sets, good for eand n together
What IS the coupled-cluster method?
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
7. The Coupled-Cluster Method
3
2
Originators (for nuclei): F. Coester (1958), Coester+Kummel (1960) [exp-S method] Works for nuclei, atoms, molecules, solids,
quantum liquids & crystals ( ),...1(1 ...2
T
He
e T TΨ = Φ = + + +
i
i
i
1 2 3
1
.) .
:independent particle state (shells!) .... ,cluster operators 0 Brueckner shell orbitals
NT T T T TT
Φ
Φ= + + +
≡ →
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
; 1
, ,
( ) ( )( )
1[ , ] [ , ], ] ....2
FINITE commutator series Cluster operators correlate particles on shells in
k k k k k kk
T T
T T
T Teff
P Q P
Q Q Q T T T P T
H e E ee He E
e He H H H T H T T
−
−
= Φ Φ = −
= = =
Φ = Φ
→ Φ = Φ
= = + + +
↓
Φ
∑
i
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
Equations for ( 2,3,...)
0, 2,3,...Equation for energy
Advantages: conceptual, physical and numerical * preserves shell picture * complemented with cluster p
kT T
k
T T
T kQ e He k
EE e He
−
−
=
Φ = =
= Φ Φ
i
i
i
icture * very rapid convergence * not variational, not perturbative, no diagrams * self-consistency, like , which is coupled- method
HF orbital
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
• It has ‘conquered’ the many-body problems in* nuclear physics (just about…)* quantum chemistry (mostly…)* solid-state physics (on its way…)* quantum liquids and solids (He-4, He-3)* magnetic systems (not quite yet…)* etc…
2005: Kümmel and Bishop were awardedThe Eugene Feenberg Memorial Medal in Many-body Physics, for the CC method
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
8. The Molecular Coupled-Cluster Method
Fritz Coester (1983): "What is good for electrons [in atoms, etc.], and is good for [nucleons in] nuclei, be good for electrons and nuclei together " T
M
e n
ee ne nn nee
shoulde
T T T T T
Ψ = Φ
Φ = Φ Φ
≅ + + +
⇒
: IPM for ,Monkhorst(1987)
x x e nΦ =
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
• Electrons and nuclei on shells around a nucleus: “molecule as an atom”
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
M
Same advantages as for atoms with definite Pico/femtosecond laser chemistry with time-dependent version
for nuclei that 'carry' inner shell electronsFound basis set to perform integrals
nee
J
T
πΨ
iiiii
2 2 21, 12 12 13 13 1, 1,1312 2 .....22
12 13 1,
in 'first-quantized' (integral-equation) calculations (Harris and Monkhorst, 2005):
........20for ( ) ( )
1to 5 for ( ) ( ),(
n n n n n nm a r a r a rmmn n
ij
r r r em ij nn
ij ne ee
− − −− − − −−
=
=
∼∼ )
Angular factors can be included (Harris 2005)Just beginning......ii
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
9. A Solid Coupled-Cluster Method?
( )( )
:(traditional) correlated electronic state calculations (wannier?):nuclear vibrational calculations (Wannier?)
1 ....: ' phonon cou
ee ne nn
ne ee nn
ne
ne
TS
T T Te n
T T Te n
TSe Sn
Se
SnT
ne
e
e
e e e
e
e T e
+ +
Ψ = Φ
= Φ Φ
= Φ Φ
= Ψ Ψ
Ψ
Ψ
= + + −
i
ii
i pling'Self-consistent calculations for all (in principle)Enables structure, phonon, [e-phonon, transport,...] calculations
TooFarOutside the box? ab initio
ii
50 Years of Coupled Cluster Theory, INT, U. of WashingtonJuly 2, 2008
10. Concluding Remarks• Modern physics ‘calls for’ revisiting the notion of
structure, and how to deal with it• Accurate quantum mechanics ‘calls for’
democratic treatment of e and n• Born-Oppenheimer is an impediment now, and
Coupled-Cluster method offers a path towards that goal
• Modern computing power enables it, modern physics requires it
• Much work needs to be done, though!
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