Chemistry, Physics and the Born-Oppenheimer Approximation

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


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


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


• 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


• 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


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!)


• 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

•


• 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


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


• 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?


4. The Born-Oppenheimer Approximation (BOA)


{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


• 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


• Disadvantages:* what is really reaction coordinate for

polyatomic molecular reactions?* difficulties for many spectroscopies

--Rydberg states--rovibronic excited states --various laser chemistries


,

* 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


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


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?


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= + + +

≡ →


; 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


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


• 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


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 =


• Electrons and nuclei on shells around a nucleus: “molecule as an atom”


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


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


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!

Documents

Chemistry, Physics and the Born-Oppenheimer Approximation