Chemistry, Physics and the Born- Oppenheimer Approximation

Transcription

1 Chemistry, Physics and the Born- Oppenheimer Approximation Hendrik J. Monkhorst Quantum Theory Project University of Florida Gainesville, FL

2 Outline 1. Structure of Matter 2. Shell Models 3. Quantum Mechanics of Structure 4. Born-Oppenheimer Approximation (BOA) 5. Pros and Cons of BOA 6. Removal of BOA 7. The Coupled-Cluster Method 8. The Molecular Coupled-Cluster Method 9. A Solid Coupled-Cluster Method? 10. Concluding Remarks

3 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

4 1. Structure of 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;

5 In chemistry and physics we speak of * crystal structure * molecular structure * atomic structure * nuclear structure A (, e X) Z ( p, n) 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?

6 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!)

7 Shell model forms basis for Independent Particle Models (IPMs), leading to Aufbau Princip Periodic Table of Elements (Un)stable nuclei expnts: semi-quantitative spectroscopies theory: Hartree and Hartree-Fock orbitals. IPM also applied to molecules and crystals (>1927) molecular orbitals, Bloch orbitals

8 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

9 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

10 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?

11 4. The Born-Oppenheimer Approximation (BOA)

12 i Fix nuclei in space i Calculate electron wavefunction Hˆ Ψ ( r, R) = E ( R) Ψ ( r, R), r : electrons, R: nuclei E e e e e e {R} ( R) : potential energy surface (PES) { R} : multi-dimensional space min E ( R) structure of molecule, crystal i i i e Calculate X ( R) on PES, then total wavefunction Ψ( rr, )= X( R) Ψ ( rr, ) n n e m e Correct to second order in κ = Mn Quantum mechanical justification for structure 1/ 4

13 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

14 Disadvantages: * what is really reaction coordinate for polyatomic molecular reactions? * difficulties for many spectroscopies --Rydberg states --rovibronic excited states --various laser chemistries

15 * in principle, Born-Huang expansion can be used: Ψ = ne Ψe ne, (, rr) X ( R) (, rr) many PESs --PES picture gone (crossings, intersections) --difficult calculations * crystals: phonons, e-phonon interaction, transport tough to deal with quantitatively

16 i i i i 6. Removal of BOA What would this mean? * treat nuclear QM at same footing as electron QM me * yet, recognize the effect of 1 Electrons are delocalized, nuclei are 'localized' within molecule and crystal Why this difference? Because in ground state * K V electrons delocalized e e * K V nuclei localized (cf. Wigner crystallization!) n n Highly excited states K V loose/'vague' structure n M n n

17 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 e and n together What IS the coupled-cluster method?

18 7. The Coupled-Cluster Method i Originators (for nuclei): F. Coester (1958), Coester+Kummel (1960) [exp-s method] i Works for nuclei, atoms, molecules, solids, He 3 quantum liquids & crystals ( ),... T 1 2 i Ψ= e Φ= (1 + T + T +...) Φ. 2 Φ :independent particle state (shells!) T = T1+ T2 + T3 + T N T 1...,cluster operators 0 Brueckner shell orbitals

19 P =Φ Φ ; Q = 1 P Q = Q, Q T = T, T P = T k k k k k k k T T He ( Φ ) = Ee ( Φ) T T ( e He ) Φ = EΦ T T 1 e He = Heff = H + [ H, T] + [ H, T], T] FINITE commutator series i Cluster operators correlate particles on shells in Φ

20 i Equations for T ( k = 2,3,...) T T Q e He Φ= 0, k = 2,3,... k iequation for energy T T E e He k i Advantages: conceptual, physical and numerical * preserves shell picture * complemented with cluster picture E = Φ Φ * very rapid convergence * not variational, not perturbative, no diagrams * self-consistency, like HF, which is coupled- orbital method

21 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 awarded The Eugene Feenberg Memorial Medal in Many-body Physics, for the CC method

22 8. The Molecular Coupled-Cluster Method Fritz Coester (1983): "What is good for electrons [in atoms, etc.], and is good for [nucleons in] nuclei, should be good for electrons and nuclei together " T Ψ M = e Φ Φ=Φ Φ e n T T + T + T + T ee ne nn nee Φ x : IPM for x = e, n Monkhorst(1987)

23 Electrons and nuclei on shells around a nucleus: molecule as an atom

24 isame advantages as for atoms i Ψ M with definite i Pico/femtosecond laser chemistry with time-dependent version i T nee J π for nuclei that 'carry' inner shell electrons ifound basis set to perform integrals in 'first-quantized' (integral-equation) calculations (Harris and Monkhorst, 2005): m13 2 mn 1, n a12r12 a13 r13... an 1, nrn 1, n r r... r e 2m n 1, n m 20for ( ij) = ( nn) ij 1to 5 for ( ij) = ( ne),( ee) i Angular factors can be included (Harris 2005) ijust beginning...

25 9. A Solid Coupled-Cluster Method? iψ = e iψ iψ S Se Sn T Φ Tee+ Tne + Tnn = e ΦΦ ( )( ) e n Tne Tee Tnn = e e Φ e Φ Tne = e Ψ Ψ Se Sn e n :(traditional) correlated electronic state calculations (wannier?) :nuclear vibrational calculations (Wannier?) Tne ie = 1 + T +...: ' e phonon coupling' ne iself-consistent calculations for all (in principle) ienables ab initio structure, phonon, [e-phonon, transport,...] calculations TooFarOutside the box?

26 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!