Molecular Sciences. Introduction of concepts and the basics of molecular physics

Transcription

1 Molecular Sciences Introduction of concepts and the basics of molecular physics Jochen Küpper (Andrea Trabattoni) Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron DESY, Hamburg Department of Physics & Department of Chemistry, Universität Hamburg The Hamburg Center for Ultrafast Imaging, Universität Hamburg 3. & 7. August 2017 based on the UHH master module The Basis of Modern Molecular Physics CFEL SCIENCE Controlled Molecule Imaging

2 Molecules in fields Manipulation of translational and rotational motion 2 cos θz cos θx,y cos θ Filsinger, Erlekam, von Helden, JK, Meijer, Phys. Rev. Lett. 100, (2008) Holmegaard, Nielsen, Nevo, Stapelfeldt, Filsinger, JK, Meijer, Phys. Rev. Lett. 102, (2009) Nevo, Holmegaard, Nielsen, Hansen, Stapelfeldt, Filsinger, Meijer, JK, Phys. Chem. Chem. Phys. 11, 9912 (2009)

3 Rigid diatomic rotor Rotation of Diatomic Molecules classical energy: r m 1 m 2 E r 1 2 I!2 with I m 1 r m 2r 2 2 (8) with the reduced mass µ m 1m 2 m 1 +m 2 and bond length r r 1 + r 2 one obtains E r 1 2 µr2! 2 (9) with L I! this yields quantum mechanical energy: E r L2 2I (10) E r h2 8 2 I J(J + 1) (11) E r is inversely proportional to I E r scales with rotational quantum number as J(J + 1) Jochen Küpper (CFEL, DESY, UHH) Molecular Sciences 16. & 18. August / 126

4 Pendular States Orientation of a Permanent Dipole Polar Molecules in Static Electric Fields µε µε V µ (θ) = µε cosθ 0o 90o 180o θ Hamiltonian Dimensionless Hamiltonian H B~J 2 + V µ (45) H B ~J 2 + µ B cos ~J 2 +! cos (46) - - Dimensionless interaction strenght parameter! µ B (47) Mixing operator couples states with equal M but J s differing by ±1 cos (48) Cylindrical symmetry around, therefore uniform in. c.f. dimensional calculation in Y.-P. Chang et al, Comp. Phys. Comm. 185, (2014) Jochen Küpper (CFEL, DESY, UHH) Molecular Sciences 16. & 18. August / 179

5 Schrödinger Equation Pendular States Polar Molecules in Static Electric Fields Schrödinger Equation Representation in the free rotor basis JKi H B ~J 2! cos E B (49) JM! E 1X J M a JM J (!) JMi (50) Matrix elements D JM ~J JM E J(J + 1) (51) hjm cos J + 1Mi r (J + M + 1)(J M + 1) (2J + 1)(2J + 3) (52) hjm cos JMi 0 (53) r (J + M)(J M) hjm cos J 1Mi (54) (2J 1)(2J + 1) Jochen Küpper (CFEL, DESY, UHH) Molecular Sciences 16. & 18. August / 179

6 Energies and Wavefunctions Pendular States Energies and Wavefunctions Eigenfunctions are coherent linear superpositions of the field-free rotor basis functions: hybridization. Wavefunctions are of mixed parity (i. e., parity is different from ±1). With equation (50) and parity of basis functions p J,M ( 1) J p J, the parity of the mixed wavefunction is mixed as well: X p a J p J (55) J Parity conservation does not apply to state of indefinite parity. Such a state can be oriented and have a space-fixed electric dipole moment. In a state of definite parity, the space-fixed electric dipole moment is zero. Jochen Küpper (CFEL, DESY, UHH) Molecular Sciences 16. & 18. August / 179

7 Example calculation Pendular States Energies and Wavefunctions Group work pendular states (teams of 2, 15 min) set up the Hamiltonian matrix for fixed M and! ensure convergence by using an appropriately sized Hamiltonian matrix okay, let s use M 0 and! 1, dimension (3 3) and the known matrix elements r (J + M + 1)(J M + 1) H J,J+1 hjm cos J + 1Mi (2J + 1)(2J + 3) hjm cos JMi 0 r (J + M)(J M) H J,J 1 hjm cos J 1Mi (2J 1)(2J + 1) H J,J D JM ~J JM E J(J + 1) What can you say about the eigenstates under these conditions (numbers)? What is the energy of the ground state? What is the wavefunction of the ground state? What is the parity of the ground state? Jochen Küpper (CFEL, DESY, UHH) Molecular Sciences 16. & 18. August / 179

8 Example Calculation I Solution Pendular States Energies and Wavefunctions set up a matrix for fixed M and! and a dimension that ensures convergence of eigenproperties obtained by diagonalization Ĥ *.,! q J 1 (J 1 + 1) (J1 +M+1)(J! 1 q (J2 +M)(J 2 M) M+1) (2J 1 +1)(2J 1 +3) 0 q (J2 +M+1)(J 2 (2J 2 1)(2J 2 +1) J 2 (J 2 + 1)! 0! M+1) (2J 2 +1)(2J 2 +3) q (J3 +M)(J 3 M) (2J 3 1)(2J 3 +1) J 3 (J 3 + 1) + / - for instance, for M 0 and! 1 get Ĥ *., 1 q 0(0 + 1) 1 (0+0+1)(0 0+1) q (1+0)(1 0) (2 0+1)(2 0+3) 0 q (1+0)(1 0) q (2 1+1)(2 1+3) (2+0)(2 0) (2 1 1)(2 1+1) 1(1 + 1) (2 2 1)(2 2+1) 2(2 + 1) + / - Jochen Küpper (CFEL, DESY, UHH) Molecular Sciences 16. & 18. August / 179

9 Example Calculation II Solution Pendular States Energies and Wavefunctions this yields the (numerical) Hamiltonian matrix Ĥ *., Diagonalization yields the eigenvalues and eigenvectors + / C 1 ( , , ) (56) C 2 ( 0.264, , ) (57) C 3 ( , , ) (58) This calculation defines the ground state with sufficient accuracy: E( J 0, M 0)! B (59) J 0, M 0 E! J 0, M 0i J 1, M 0i + (60) J 1, M 0i The eigenfunctions of the Hamiltonian are the coherent linear superpositions of the field-free-rotor basis functions Jochen Küpper (CFEL, DESY, UHH) Molecular Sciences 16. & 18. August / 179

10 Example Calculation III Solution Pendular States Energies and Wavefunctions we speak of hybridization of the wavefunctions (basis functions) For our ground state we get the parity X p a j p j ( 1) ( 1) ( 1) 2 (61) J 0.712, ±1 Since the parity is not defined, such a state can be (is) oriented. The system in this state has a space-fixed dipole moment! Jochen Küpper (CFEL, DESY, UHH) Molecular Sciences 16. & 18. August / 179

11 E ective Dipole Moment Pendular States Energies and Wavefunctions The (negative) derivative of the energy with respect to the field strength is the effective dipole moment of the quantum E/B hcos i max arccos hcos i (63) low field seeker E increases with increasing. high field seeker E decreases with increasing. θ Jochen Küpper (CFEL, DESY, UHH) Molecular Sciences 16. & 18. August / 179

12 Molecules in fields Selection of quantum states and species U Stark = 1 2 E2 ( cos 2 +? ) µecos U Stark = ~µ ~E ~F Stark = ~ ru Stark f(µ, ~ re)

13 Horke, Chang, Długołęcki, JK, Angew. Chem. Int. Ed. 53, (2014, VIP) Understanding and controlling the dynamics of highly excited molecular systems Separating para and ortho water a ~F Stark =0 b ~F Stark1 c ~F Stark2

14 Trippel, Chang, Stern, Mullins, Holmegaard, JK, Phys. Rev. A (2012) Spatial separation of neutral clusters using the m/µ deflector pure samples of indole-water (indole-(h 2 O) 1 ) energy (GHz) a A = GHz B = GHz C = GHz μ = D E (kv/cm) 0 energy (GHz) 100 c 0 A = MHz 200 B = MHz C = MHz μ = 4.4 D E (kv/cm) 0 b d 0 energy (GHz) 100 A = MHz 200 B = MHz C = MHz μ = D E (kv/cm) energy (GHz) 100 A = MHz 200 B = MHz C = MHz μ = 2.63 D E (kv/cm)

15 Trippel, Chang, Stern, Mullins, Holmegaard, JK, Phys. Rev. A 86, (2012) Chang, Horke, Trippel, JK, Int. Rev. Phys. Chem. 34, (2015) Spatial separation of neutral clusters using the m/µ deflector pure samples of indole-water (indole-(h 2 O) 1 ) 20 x20 15 I (arb. unit) Y (mm)

16 energy (cm -1 ) Filsinger, Erlekam, von Helden, JK, Meijer, Phys. Rev. Lett. 100, (2008) Filsinger, JK, Meijer, Hansen, Maurer, Nielsen, Holmegaard, Stapelfeldt, Angew. Chem. Int. Ed. 48, 6900 (2009) Conformer selection with the m/µ deflector y (mm) valve 10 kv ground 0 1 x (mm) skimmers E (kv/cm) detection laser deflector time-of-flight mass spectrometer 0.41 m 0.15 m 0.22 m (0.40 m) MCP trans-3-aminophenol μ = 0.77 D -3-4 μ = 2.3 D cis-3-aminophenol electric field strength (kv/cm)

17 Molecules in fields Alignment and orientation with electric fields

18 3D alignment: Larsen, Hald, Bjerre, Stapelfeldt, Seideman, Phys. Rev. Lett. 85, 2470 (2000) 3D orientation: Nevo, Holmegaard, Nielsen, Hansen, Stapelfeldt, Filsinger, Meijer, JK, Phys. Chem. Chem. Phys. 11, 9912 (2009) Molecular alignment and orientation Connecting molecular and laboratory frame Z θ z cos cos 2 orientation alignment isotropic aligned oriented cos =0 cos 2 =1/3 cos =0 cos 2 > 1/3 cos > 0 cos 2 > 1/3

19 Molecular alignment and orientation Connecting molecular and laboratory frame 2 cos θz cos θx,y cos θ Filsinger, Erlekam, von Helden, JK, Meijer, Phys. Rev. Lett. 100, (2008) Holmegaard, Nielsen, Nevo, Stapelfeldt, Filsinger, JK, Meijer, Phys. Rev. Lett. 102, (2009) Nevo, Holmegaard, Nielsen, Hansen, Stapelfeldt, Filsinger, Meijer, JK, Phys. Chem. Chem. Phys. 11, 9912 (2009)

20 Toward time-resolved imaging of chemical dynamics khz-rate manipulation experiments 1 mj, 30 fs 10 mj, 40 fs khz (>70 nm bandwidth) recently added Coherent Astrella (6 mj 35 fs) for UV, mid-ir, etc. Trippel, Mullins, Müller, Kienitz, Długołęcki, JK, Mol. Phys. 111, (2013, Bretislav Friedrich Festschrift)

21 Scenarios of rotational dynamics in OCS (X, v=0, J=0) Adiabatic alignment with a 485 ps pulse ) 2 W/cm 12 Intensity ( Time (ps) D θ 2 cos Energy Intensity J=2 J=1 Temporal laser profile Time J=2 J=1 J=0 J=0 Time Trippel, Mullins, Müller, Kienitz, Omiste, Stapelfeldt, González Férez, JK, Phys. Rev. A 89, (R) (2014)

22 Scenarios of rotational dynamics in OCS (X, v=0, J=0) Intermediate-case alignment with a 50 ps pulse experiment 2 W/cm 12 in θ 2D 2 cos cos 2 θ2d b I max = W/cm 2 I max = W/cm 2 I max = W/cm 2 I max time in ps time (ps) 2 W/cm 12 in 10 I max theory time (ps) cos 2 θ2d A simple two state wave packet, a working coherent-control experiment and a strongly-driven quantum pendulum Achievable degree of alignment is comparable to adiabatic case! Trippel, Mullins, Müller, Kienitz, Omiste, Stapelfeldt, González Férez, JK, Phys. Rev. A 89, (R) (2014)

23 Scenarios of rotational dynamics in OCS (X, v=0, J=0) Intermediate-case alignment with a 50 ps pulse A molecular movie at slowdown (80 ps in 80 s) Trippel, Mullins, Müller, Kienitz, Omiste, Stapelfeldt, González Férez, JK, Phys. Rev. A 89, (R) (2014)

24 Scenarios of rotational dynamics in OCS (X, v=0, J=0) Impulsive alignment with a 50 fs pulse ) 2 W/cm 12 Intensity ( Time (ps) D θ 2 cos Energy Intensity J=2 J=1 Temporal laser profile Time J=2n J=2 J=1 J=0 J=0 Time Trippel, Mullins, Müller, Kienitz, Omiste, Stapelfeldt, González Férez, JK, Phys. Rev. A 89, (R) (2014)

25 Scenarios of rotational dynamics in OCS (X, v=0, J=0) Very strong alignment using two short pulses cos 2 θ P momentum (arb. u.) Exp t (ps) <cos 2 θ> Populations 1 cos 2 θ P momentum (arb. u.) 1 P momentum (arb. u.) HWHM 4.4 º Time in ps t in (ps) ps P momentum (arb. u.) cos 2 3D J=0 80 % I=8 TW/cm 2 J=0 95 % I=8 TW/cm 2 J=0 80 % I=7 TW/cm 2 J=0 95 % I=7 TW/cm 2 signal (arb. unit) ) max < 4.5 Exp. Th θ (º) with Arnaud Rouzée, MBI Berlin

26 Scenarios of rotational dynamics in OCS (X, v=0, J=0) Imaging quantum-revivals with Arnaud Rouzée, MBI Berlin

27 Mixed-field orientation: The flea on Schrödinger s cat or how tiny perturbations localize the wave function I ac =5 10 I 11 W/cm 2 ac = W/cm 2

28 Building a double-minimum potential with pertinent asymmetry I dc = 1 kv/cm

29 Imaging molecular dynamics with controlled molecules JK, et al (53 authors), Phys. Rev. Lett., 112, (2014)

30 Coherent diffractive imaging of isolated molecules 100 pm precision from 0.5 nm -1 diffraction (λ = 620 pm) I (normalized) Detector E TSL Detector E YAG E FEL E YAG photons/pixel photons/pixel a c Simulation Experiment 0.01 simultaneous diffractive-imaging 1 determination of p (arb. u.) p (arb. u.) x x pm 700 pm atomic resolution distance and angle 800 pm pm Experiment electron diffraction: Hensley, Yang, Centurion, Phys. Rev. Lett. 109, (2012) a c NoYAG 0.4 a b d s (nm ) YAG Iodine-Iodine (pm) p (arb. u.) z p (arb. u.) z 0.7 y O O ( ) ( ) diffraction data yields <cos 2 θ>2d=0.8 (vs. 0.84) r(i I) 800 pm JK, et al (53 authors), Phys. Rev. Lett., 112, (2014) b d x (vs. 700 pm)

31 Imaging of Isolated Molecules with Ultrafast Electron Pulses Hensley, Yang, Centurion, Phys. Rev. Lett. 109, (2012) Centurion, J. Phys. B 49, (2016)

32 Diffractive imaging of (rotational) dynamics 100 fs temporal resolution! Gühr, Centurion, Wang, et al (23 authors), Nat. Comms. 7, (2016)

33 Molecular frame photoelectron angular distributions of 3D-oriented benzonitrile molecules Direct imaging of electronic molecular structure E probe e- y x E stat z detector Estat = 467 V/cm Econtrol = W/cm 2 Eprobe = W/cm 2 Holmegaard, Hansen, Kalhøj, Kragh, Stapelfeldt, Filsinger, JK, Meijer, Dimitrovski, Abu-samha, Martiny, Madsen, Nature Phys. 6, 428 (2010)

34 Laser induced electron diffraction e - rescattered Strong laser field Direct imaging of nuclear molecular structure

35 game : electron trajectories in an external field