When are electrons fast and are Born and Oppenheimer correct?

Transcription

1 When are electrons fast and are Born and Oppenheimer correct? Wim J. van der Zande Department of Molecular and Laser Physics University of Nijmegen Han-Sur-Lesse, Winter 2003

2 Contents of the Lectures 0. Introduction: Some history 1a. The Schrödinger Equation 1b. Potential Energy Surfaces: reality or tool? 1c. Does nature want to minimize energy? 1d. Spectral structure and dynamic time scales? 1d. What is the limit of spectral structure: intra molecular energy relaxation? 2a. Schrödinger Equation and the Born-Oppenheimer Separation 2b. Potential energy surfaces in diatomics 2c. Non-adiabatic interactions 2d. Diabatic and adiabatic states 2e. Progress related to multi-channel quantum defect theory

3 Four hours of Lectures 3a. Collisions with 0 ev electrons: fast or slow 3b. Continua in molecular physics 3c. Dissociative recombination: radiation, dissociation and autoionization 3d. A mystery for theoretical chemists

4 Lecture A short history: Robert Oppenheimer Julius Robert Oppenheimer was born in on 22nd April, He studied at Harvard University before working with Ernest Rutherford at Cambridge University and later with Max Born in Gottingen, Germany.

5 Lecture A short history: Max Born was born in Breslau, Germany, on 11th December, He studied physics at the University of Gottingen and obtained his doctorate in Grandfather of Olivia Newton-John

6 Lecture One-particle Schrödinger Wave- Equation: Kinetic energy Potential energy Wave function Erwin Schrödinger Born: 12 Aug 1887 in Erdberg, Vienna, Austria Died: 4 Jan 1961 in Vienna, Austria

7 Lecture One-particle Schrödinger Wave- Equation: time dependent Potential energy: if time independent, then phase factor: the magic of QM The observable!! Another is transition strength One-particle Schrödinger Wave-Equation: time independent

8 Lecture The Molecular Schrödinger Wave- Equation: all interactions of known electrostatic nature Total kinetic energy Total potential energy, i, j nuclei/electrons The Hamiltonian in the case of a molecule, A,B: nuclei; i electrons (in atomic units)

9 Lecture b Atomic Units: m e (mass electron)= 1 a.u. ( kg) r e (radius first Bohr orbit)) = 1 a.u. ( meter) t = 1 a.u. (time of 1 rad of the round trip of 13.6 ev electron) ( sec) q e (charge electron) = 1 a.u. ( C) Derived: v = 1 a.u ( c/α = /137 m/s) E= 1.a.u. (27.21 ev, J = 1 Rydberg) h/(2π) = 1 (a.u.of energy)*1 a.u. of time) =1 ( Js) c= 137 a.u. of velocity

10 Lecture The Equation Providing: Potential Energy Surface E e (R): multidimensional potential energy surface: Electrons Calculated Away: Is this an observable?

11 Lecture This figure shows O 3, where the bond angle of ozone is constrained at 117, and the two bond lengths form the x- and y-axes of the graph:

12 Lecture Complex PES and PES complexities

13 Lecture Complex PES and PES complexities Next lecture! Eugene Paul Wigner Born: 17 Nov 1902 in Budapest, Hungary Died: 1 Jan 1995 in Princeton, New Jersey, USA Radiationless transition

14 Lecture Discussion Points: (1) Does nature want to go to minimum in PES? (2) How does nature go from reactants to products? (3) Single versus very many collisions (4) Internal degrees of freedom as local energy bath (entropy)

15 Lecture b Discussion Points: (1) Does nature want to go to a minimum in PES? Answer: NO, no such a law exists. Only conservation of energy in a closed system. Hence preference is observed in case of dissipation possibilities (multiple collisions, radiation) or the practical situation that the density of states in degrees of freedom not shown in the PES effectively generate continuum allowing dissipation.

16 Lecture c Discussion Points: (1) How does nature go from reactants to products? Answer: - Start at proper energy (total energy above TS) - Embed reactants in collisions such that thermal fluctuations help the way up and over the TS - Remove products

17 Lecture Eigenvalue Problems and Time Scales of Associated Classical Motion. The solution of the SE: eigen-energies excitation strengths E. τ > h Energy spacing Classical period E elec E vib E rot E IVR

18 Lecture tau (sec) E (ev) E(cm-1) Label 1.52E electronic 3.04E E #DIV/0! E vibration 4.13E #DIV/0! E rotation 2.07E H2 E elec #DIV/0! 4.13E rotation E vib E rot E IVR 2.07E heavy #DIV/0! E pi-sigma 2.07E symmetry E. τ > h

19 Lecture E. τ > h: energy spacing and classical periods 12 Particle in box E n = n 2 h 2 /(8mL 2 ) E n+1.n =(2n+1)h 2 /(8mL 2 ) v= (2E/m) = nh/(2ml) t=(2l/v) = 4mL 2 /(nh) Dus: t = h/ E n+1.n

20 Lecture E. τ > h Harmonic Oscillator E = hf t = f -1 = h/ E

21 Lecture Intra-molecular processes and discrete spectral structures. H 3 Discussions. IVR ISC When does spectral structure disappear in IVR/ISC Where does the energy go in IVR/ISC What determines the visibility of the structure State 1 State 2 excitatie State 0

22 Lecture b Discussions. Where does the energy go in IVR/ISC Intra-molecular processes and discrete spectral structures. IVR ISC Answer: nowhere, total energy is a conserved quantity, IVR requires such a high density of state that even a (super) high resolution laser excites a coherent (initially) wavepacket (bright state) State 1 excitatie State 0 State

23 Lecture c Discussions. When does spectral structure disappear in IVR/ISC Answer: (a) this question depends on the resolution of the person who asks the question (b) when radiation causes line broadening more than level spacing (the dark states become a statistical bath) (c) then the line becomes homogeneously broadened Intra-molecular processes and discrete spectral structures. IVR State 1 State 2 excitatie State 0 ISC

24 Lecture Born-Oppenheimer Separation To be solved q 1 nuclei, q 2 electrons Ansatz/ Wisdom and Requirement: can we separate H?

25 Lecture Consequence: = = =

26 Lecture The Hamiltonian The term giving problems for separation: Step 1 write: And substitute

27 Lecture Remove kinetic operator of the nuclei: define electronic Hamiltonian Because: Substitution gives: If the following would hold Then: =

28 Lecture b

29 Lecture In reality we get: = Next step: multiply with φ e (r;r) and integrate over electronic coordinates: <φ e (r;r) φ e (r;r)>=0. This gives: T N φ N (R) + (E e +V NN E tot ) φ N (R) = Σ(1/(2M A ){<φ e (r;r) φ e (r;r)> φ N (R) + A <φ e (r;r) 2 φ e (r;r)> φ N (R) } Nuclear SE Non- BO terms

30 Lecture h 2 These terms cause non-adiabatic interaction terms or: when non-zero the concept of potential curves breaks down Gives rise to Adiabatic Correction as it behaves as a potential (Born-Huang) <φ e (r;r) 2 φ e (r;r)> E e (R)+V NN +<φ e (r;r) 2 φ e (r;r)> The Adiabatic potential

31 Lecture Now we have a more complete description but a hard to solve problem. If {φ e,i (r;r)} forms a complete set of Eigenfunctions of H el then: The solution of the any problem can be written as: Ψ(r; R) = Σ φ e,i (r;r) φ N,i (R) and the Equation to solve becomes: T N φ Ni (R) + (E ei +V NN E tot ) φ Ni (R) = ( 1 / 2M ){<φ ei (r;r) φ ei (r;r)> φ Ni (R) + <φ ei (r;r) 2 φ ei (r;r)> φ Ni (R)} =0 + Σ( j 1 / 2M ){<φ ei (r;r) φ ej (r;r)> φ Nj (R)+ <φ ei (r;r) 2 φ ej (r;r)> φ Nj (R) } Coupling terms (coupled equations needed for solution)

32 Lecture The equation below implies that the physics of a system may take place on multiple potential energy surfaces simultaneously (hence the coupled equations): T N φ Ni (R) + (E ei +V NN E tot ) φ Ni (R) = ( 1 / 2M ){<φ ei (r;r) 2 φ ei (r;r)> φ Ni (R)} + Σ( 1 / 2M ){<φ ei (r;r) φ ej (r;r)> φ Nj (R)+ <φ ei (r;r) 2 φ ej (r;r)> φ Nj (R) } j Note: couplings are in some sense always the result of a primitively chosen basis set. For all problems a basis of eigenfunctions exist. Couplings is often not a physical interaction between physical entities.

33 Lecture Non-adiabatic interaction terms, some properties? Complete set: <φ i φ j > = 0, i j (I) <φ i d/dr φ j > 0, i j (note: = d/dr in one dimension) (II) <φ i d/dr φ i > = 0, because: d<φ i φ i >/dr =0 = 2 <φ i d/dr φ i > d<φ i φ j >/dr =0 = <φ i d/dr φ j > + <φ j d/dr φ i > (II) <φ i d 2 /dr 2 φ j > 0, all i,j (operator antisymmetric) hence: d 2 /dr 2 term gives both diagonal adiabatic correction, and interaction between states

34 Lecture Character change Two state diatomic system. Diabatic States? 1/R Na + Cl - attraction Adiabatic potentials Suspicion: <φ 1 d/dr φ 2 > 0 The pragmatic solution: Diabatic states: recipe in general subjective sacrifice: <φ 1 H el φ 2 > = 0 demand: <φ 1 d/dr φ 2 > = 0

35 Lecture crossing H el -coupling Small Large Diabatic states crossing avoided crossing Adiabatic states d/dr-coupling Sharp, high Broad, low avoided crossing

36 Lecture Consequences and some aspects in words: - for adiabatic states a procedure exists! - for diabatic states no universal procedure exist! - possibilities: retain all CI coefficients determined for R e for all R... use derived quantities as a dipole moment to keep character constant use adiabatic states and generate locally diabatic states - Couplings due d 2 /dr 2 not taken into account.

37 Lecture φ d1 (r; R)= cos(θ(r)) φ a1 (r ; R) + sin(θ(r)) φ a2 (r ; R) φ d2 = -sin(θ) φ a1 + cos(θ) φ a2 < φ d1 φ d2 > =0 define two state rotation, θ= θ(r) such that: < φ d1 d/dr φ d2 > = 0 Two state diatomic system: Consequences (1): < φ a1 d/dr φ a2 > = ½ (V ad1 -V ad2 ) at distance of closest approach (2): < φ a1 d/dr φ a2 > dr= π/2 (!) (3): V d1 = cos 2 (θ) V ad1 +sin 2 (θ) V ad2 V d2 = cos 2 (θ) V ad2 +sin 2 (θ) V ad1 (3): V a1,2 = 0.5{(V d1 +V d2 ) 2 ± ((V d1 -V d2 ) 2 + 4H 122 ) 1/2 }

38 Lecture b crossing H el -coupling Small Large Diabatic states crossing avoided crossing Adiabatic states d/dr-coupling Sharp, high Broad, low avoided crossing

39 Lecture c crossing H el -coupling Small Large Diabatic states crossing avoided crossing Adiabatic states d/dr-coupling Sharp, high Broad, low avoided crossing

40 Diabatic to Adiabatic T N φ Nd1 (R) + (E d1 E tot ) φ Nd1 (R) = H el φ Nd2 (R) T N φ Nd2 (R) + (E d2 E tot ) φ Nd1 (R) = H el φ Nd2 (R) (numerical more simple, no derivative) Should be equivalent to: T N φ Na1 (R) + (E a1 E tot ) φ Na1 (R) = < φ ea1 d/dr φ ea2 > dφ Nd2 (R)/dR T N φ Na2 (R) + (E a2 E tot ) φ Na2 (R) = < φ ea2 d/dr φ ea1 >dφ Nd1 (R)/dR Special: if H el 0 than nature is perfectly diabatic and < φ ea1 d/dr φ ea2 > π/2.δ(r-r c ) (θ(r)= π/2 Heavyside(R c ))

41 Lecture The world of large series of electronically highly excited states: Rydberg States. n,l( 1, <<n) + e - n,l( n-1) + + Molecular structure: probed at small r (small l)

42 Lecture The world of large series of electronically highly excited states: Rydberg States. Outside area: analytical (H-like) n,l( 1, <<n) + Electron core scattering: deceleration/ acceleration (phase shift quantum defect energy shift: All Rydberg states get their particular character at the core only! For molecules: quantum defect = QD(R)

43 Lecture Difference between the Ion and the Rydberg States is: scattering of the electrons on the core, which affects the description of other electrons etc. Potential curve of Ion State is slightly different: the quantum defect: δ(r) constant but a function of R. <d/dr> coupling and <d 2 /dr 2 > adiabatic correction follow directly from δ(r) and do no longer require a separate non-bo calculation (Child, Stolyarov: Phys. RevA,2001) +

44 O+O+O O 2 + O( 1 S) WHY?? Aurorae, O( 1 S)

45 Dissociative Recombination The reaction in the case of: O (v=0) + e - (at 0 ev) O( 33 P) P) + O( O( 33 P) P) ev ev O( 11 D) D) + O( O( 33 P) P) ev ev O( 11 D) D) + O( O( 11 D) D) ev ev O( O( 33 P) P) + O( O( 11 S) S) ev ev O( O( 11 D) D) +O( 11 S) S) +0.8 ev ev nm nm

46 When electrons meet an ion.... When electrons meet an ion.... E near 0 ev e- E col (1 Å)=13.6 ev Autoionization O + O σ 1/E, P(per hit)= fs Dissociation (10 fs) Radiation (ns)

47 Dissociative Recombination a simplified potential energy diagram Doubly excited repulsive neutral state Ionic state AB + + (e - at rest at ) Energy (Rydberg state) Kinetic energy Fragment A,B formation Internuclear separation

48 Dissociative Recombination what determines its efficiency? O 2+ + (e - at )?? Energy *Interaction strength 2?? Internuclear separation

49 14 12 O 2 + A 2 Π u O + ( 4 S)+O( 3 P) 10 O 2 + X 2 Π g energy (ev) E 3 Σ u 1 1 u + f ' 1 Σ u O( 1 D)+O( 1 S) O( 1 D)+O( 1 D) O( 3 P)+O( 1 D) 2 - B 3 Σ 1 3 u Π u O( 3 P)+O( 3 P) internuclear separation (Ångström)

50 NO + : confrontation with statistical model with spinconservation 120% 100% 80% 60% 40% X, 0 ev a) 100% 90% 80% 70% 60% 50% 40% 30% X, 1.25 ev b) 20% 20% 10% 0% Statistical Model Spin-Selection Experimental Results 0% Statistical Model Spin-Selection Experimental Results 35% 30% 35% c) d) 30% 25% 25% 20% 20% 15% 15% 10% 10% 5% 5% 0% Statistical Model Spin Selection Experimental Results 0% Statistical Model Spin-Selection Experimental Results X, 5 a 3 Σ+, 0 ev =

51 From 2 to 3 atoms: also 3 fragments? The neutral product branching ratios in DR of H 3 +, NH 2 +, H 2 O +, D 3 O +, CH 2 +, and CH 5 +

52 Mysteries in DR The quantum chemistry suggests great specificity: however rates/cross sections are nearly system independent The quantum chemistry suggests selective dissociations: however a statistical model provides predictive power, WHY? (note: nature does not goes to lowest state!) Would quantum chemistry predict three body fragmentation in DR of polyatomic ions Is the essential difference that the world of many excited states is very different from ground state intuition?

53 Dissociative recombination of HD + Q 1 H 2 ** ( 1 Σ g+ ) 1/E and Rydberg resonance s

54 Dissociative recombination of HD + (J=0,1,,4) Cross section times energy: close to 1/E 3 mev x 0.1