Applications of Quantum Dynamics: Using the Time-Dependent Schrödinger Equation. Graham Worth

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1 Applications of Quantum Dynamics: Using the Time-Dependent Schrödinger Equation Graham Worth Dept. of Chemistry, University College London, U.K. 1 / 34 II. Solving the TDSE To solve the TDSE need The Hamiltonian in a set of suitable coordinates Potential surfaces and non-adiabatic couplings Kinetic energy operator The form of the initial wavepacket An algorithm to integrate the TDSE and evolve wavepacket in time A method of extracting the information of interest 2 / 34

2 The Standard Method Have chosen suitable coordinates and obtained W, now solve i Ψ(q, t) = HΨ(q, t) (1) t Expand the wavefunction in a direct-product basis set N 1 Ψ(q 1,... q f, t) = j 1 =1 N f j p =1 A j1...j f (t) χ (1) j 1 (q 1 ) χ (f ) j f (q f ) (2) and substituting this into TDSE (or using a variational principle), i Ȧ j1,...j f = j 1 χ (f ) j f H χ (1) l 1 χ (f ) l f A l1,...l f. (3) l 1,...l f χ (1) or simply iȧ = HA (4) 3 / 34 The Hamiltonian matrix elements Need to evaluate matrix elements (integrals) H JL = j 1 χ (f ) j f H χ (1) l 1 χ (f ) l 1,...l f χ (1) l 1,...l f χ (1) l f = j 1 χ (f ) j f T + V χ (1) l 1 χ (f ) l f (5) As written an N f N f matrix of multi-dimensional integrals! 1. Usually T = κ T κ T JL = κ χ (κ) j κ T κ χ (κ) l κ δ Jκ L κ (6) N N matrices, which by a suitable choice of basis functions can be solved analytically 4 / 34

3 2. The potential is a local operator V (q). Thus if basis functions are localised, χ (κ) j δ(q q j ) then V JL = χ (1) j 1 χ (f ) j f V χ (1) l 1 χ (f ) l f = V (q j1... q jf )δ JL (7) To enable conditions 1 and 2 use a DVR basis set in which X ij = φ i ˆx φ j UXU T = x (8) where φ i are an analytically known FBR. And so χ α = i U αi φ i (9) are the DVR, χ α δ(x x α ). Various DVRs possible. Beck et al Phys. Rep. (2) 324: 1 5 / 34 Problem 3: Expandind TDSE in a basis set results in an N f N f matrix of multi-dimensional integrals! Solution: To solve this use a DVR basis set in which KE integrals known analytically and PE integrals diagonal. Result 3: Problem is discretised on a grid Various DVRs possible. Beck et al Phys. Rep. (2) 324: 1 6 / 34

4 Integrating the TDSE Full solution is Split-operator method. Ψ(t) = e i Ĥt Ψ() (1) e i Ĥt e i Tt e i Vt (11) so divide propagation into short steps and approximate Ψ(t + δt) = e i 2 V δt e i T δt e i 2 ˆV δt Ψ(t) (12) Chebyshev Propagation Represent propagator by polynomial expansion: e i Ĥt Ψ = n a n P n (H)Ψ (13) where P n (H) are generated by a recurrence relationship 7 / 34 Larger systems: The MCTDH Method Standard method nuclear wavefunction expanded in a basis set: N 1 N f Ψ(Q 1,..., Q f, t) =... A j1...j f (t) j 1 =1 j f =1 f κ=1 χ (κ) j κ (Q κ ) (14) Variational equations of motion for A. iȧj = L Φ J H Φ L A L (15) Exponential increase in computer resources N f 8 / 34

5 Larger systems: The MCTDH Method The Multiconfiguration Time-Dependent Hartree Method n 1 n f Ψ(Q 1,..., Q f, t) =... A j1...j f (t) j 1 =1 j f =1 f κ=1 ϕ (κ) j κ (Q κ, t) (16) Variational equations of motion for A and ϕ. iȧj = L Φ J H Φ L A L (17) i ϕ (κ) = ϕ (κ) j (q κ ) = ( 1 P (κ)) ( ρ (κ)) 1 H (κ) ϕ (κ) (18) N κ k=1 a (κ) kj (t)χ (κ) k (q κ ) (19) Reduced computer resources n f 9 / 34 N 5. n 1 f N f standard (MB) n f MCTDH (MB) 2 2, , ,25, ,5, Can also combine modes to reduce effective dimensionality and thus treat 2 DOFs MCTDH also starting point for more powerful methods and approximations: ML-MCTDH, GMCTDH, vmcg, DD-vMCG Reviews: Beck et al Phys. Rep. () 324:1 Meyer and Worth TCA (3) 19:251 Meyer, Gatti, andworth Multi-Dimensional Quantum Dynamics: MCTDH Theory and Applications (29) Wiley-VCH 1 / 34

6 Summary To solve the time-dependent by propagating a wavefunction Select appropriate coordinates Obtain potential energy surface(s) (and couplings) Choose a primitive basis set Choose integration scheme with controllable error Result 4: The TDSE can be solved to provide complete information on the propagating system. Problem 4a: Resources grow exponetially. Limits to a few atoms. Problem 4b: Production of potential surfaces is huge amount of work. Limits applicability. Problem 4c: Potential surfaces need to be in diabatic picture Next step is to set up, run and analyse simulation 11 / 34 Initial Wavepacket The time-dependent Schrödinger Equation is an initial value problem. Start in a single eigenstate then initiate process. 1. Multiply by (transition) dipole operator to model absorption of a photon Ψ(t = ) = ˆµψ i (2) 2. For a reaction AB + C Ψ(t = ) = ψ i,ab χ C (R) (21) where χ C (R) = Ne α (R R ) 2 e i p(r R ) (22) localises incoming atom with an incoming momentum p 12 / 34

7 Finding Eigenstate An elegant way to find the ground-state is to use energy relaxation. In this wavepacket propagated in imaginary time. As Ψ(q, t) = i c i ψ i e i E i t (23) if it τ then Ψ(q, τ) = i c i ψ i e 1 E i τ (24) and all c i except c. Can also use this to obtain excited states. 13 / 34 Expectation Values Typical observables of interest to the system dynamics are coordinates, momentum and their spread < q κ >, < q 2 κ > and < p κ >, < p 2 κ >. (25) Energy flow into modes can be obtained using a zero-order Hamiltonian, e.g. if H = 2 i=1 ω i 2 ( 2 q 2 i + q 2 i ) + q 1 q 2 (26) then evaluate < E i >= ω i 2 ( 2 q 2 i + q 2 i ) (27) 14 / 34

8 State Populations Finally, diabatic state populations are obtained from P α = Ψ(t) α α Ψ(t) (28) where α is a diabatic electronic state. Adiabatic state populations require transformation. If γ (ad) = α S αγ α (29) then P (ad) γ = αβ Ψ(t) α S αγ S γβ β Ψ(t) (3) 15 / 34 If a reaction occurs into different channels, e.g. AB + CD AC + BD AD + BC Then probability of a state-to-state reaction is given by S-Matrix Ψ βm (E) = S βm,αn (E)Ψ αn (E) (31) where α, β are channels; m, n internal (vibrational) quantum numbers and E total energy. Sum over S-matrix elements at relevent energy to get the cumulative reaction probability N(E) = S βm,αn (E) 2 (32) βm,αn which in turn are related to the thermal rate constant k(t ) = 1 hq where Q is the partition function N(E)e E kt de (33) 16 / 34

9 Branching ratios and reactivity Want to now how much goes into different channels. Divide wavefunction Ψ(t) = Ψ (t) + Ψ γ (t) (34) γ where Ψ γ (t) = Θ γ (R R c )Ψ(t) (35) Amount in a particular channel γ is Ψ γ (t) Ψ γ (t) Change is the flux t Ψ γ(t) Ψ γ (t) = i Ψ(t) [H, Θ γ ] Ψ(t) = Ψ(t) ˆF Ψ(t) (36) And total amount of system that has flowed into γ is σ γ = dt ˆF(t) (37) 17 / 34 Flux analysis When a dissociative channel is present, must add a CAP where W γ = ax b Θ γ. Can now write σ γ = H = H sys iw γ (38) dt Ψ(t) W γ Ψ(t)(t) (39) Can use same formalism to get scattering matrix elements ν S γν,αν(e) 2 = with g(τ) = g w (τ) + g θ (τ) and g w (τ) = T τ 2 π (E) 2 Re T dτ g(τ)e ieτ (4) dt Ψ(t) W γ Ψ(t + τ) (41) g θ (τ) = 1 2 Ψ(T τ) Θ γ Ψ(T ), (42) 18 / 34

10 Spectral Transition Rate For weak light fields, Fermi s Golden Rule states that the probability for a transition is related to square of the Transition dipole moment µ fi = dτ Ψ f µψ i = Ψ f µ Ψ i by rate µ 2 fi Using the Born-Oppenheimer Approximation, Ψ(R, r) = ψ(r)χ(r) we find that and µ fi = ψ f ˆµ ψ i χ v χ v I fi µ el fi 2 χ v χ v 2 19 / 34 The Franck-Condon Principle For a molecule, the intensity of a transition from state Ψ i to Ψ f in a light field is proportional to F.C. factors I fi χ v χ v 2 where χ v χ v lower vibrational level upper vibrational level 2 / 34

11 Types of transitions: Bound-bound ν ν 21 / 34 Types of transitions: Bound-unbound Water band à 22 / 34

12 Predissociation S 2 absorption (top) and emission (bottom) Ricks and Barrow Canad. J. Phys. 47: 2423 (1969) 23 / 34 Golden Rule Spectrum In the time-dependent picture, the FC absorption spectrum is the Fourier Transform of the autocorrelation function σ(ω) ω dt C(t)e i(e +ω)t where E denotes the ground-state energy and where the autocorrelation function is defined as, (43) C(t) = Ψ() Ψ(t) = Ψ ( t 2 ) Ψ( t 2 ) (44) Using eigenvalue representation, C(t) = i c i c i e iω i t (45) and dt C(t)e i(e +ω)t = dt i c i c i e i(e i E ) e iω i t = i c i c i δ(e i E ) (46) 24 / 34

13 Absorption spectrum I I = e σcl (47) S 2 S 1 Abs[C(t)] Energy hν Time [fs] S Intensity Q Energy [ev] 25 / 34 Golden Rule derived from first-order perturbation theory. Spectrum valid for: weak fields direct processes Due to finite propagation multiply C(t) by 1. f (t) = exp( t/τ). (48) and g k (t) = cos k ( πt 2T ) ( Θ k = 1 usually best choice 1 t T ) (49) Energy [ev] T =1 fs. 26 / 34

14 Effect of damping H = i=1,2 ω i 2 ( 2 Q 2 i + Q 2 i ) ( E1 + κ (1) Q + 1 λq 2 λq 2 E 2 + κ (2) Q 1 ) Eigenvalue spectrum /home/graham/graphics/pyr_spec/pyr2_diag Spectrum; cos filter /home/graham/graphics/pyr_spec/pyr2_qd Intensity Eigenvalues Intensity Dynamics τ = 1 fs Energy [ev] Energy [ev] Spectrum; cos filter /home/graham/graphics/pyr_spec/pyr2_qd Spectrum; cos filter /home/graham/graphics/pyr_spec/pyr2_qd Intensity Dynamics Intensity Dynamics 2 1 τ = 4 fs τ = 1 fs Energy [ev] Energy [ev] 27 / 34 Add light field to Hamiltonian Time-resolved spectra Ĥ = ĤM + ĤML where Time-resolved photo-electron spectrum: Need to describe free-electron continuum, e.g. Ψ(R, t) = X + Ã + ψ i µ ij.e(t) ψ j (5) ij i=1,2 Spectrum is long-time population of continuum states I(ω) i=1,2 de X I i (E) (51) de Ψ(t ) X I i (E) 2 (52) Domcke and Stock Adv. Chem. Phys. (1997) 1: 1 Seel and Domcke J. Chem. Phys. (1991) 95: / 34

15 Time-resolved Photo-electron Spectrum: Toluene I (ω) = lim C X + i (ω, t) 2 t i X + Create Resonance ν 6a / ν 1b16b Arbitrary Units Probe Time/ps Ion energy/cm-1 3ps 6ps Arbitrary units Davies et al PCCP (1) 12: 9872 Computationally expensive... Richings and Worth JCP (14) 141: / 34 Summary Grid-based quantum dynamics can provide a numerically exact solution to the TDSE Require a primitive basis MCTDH a general algorithm for polyatomic systems Aim is to simulate a signal that can be related to an experiment Initial wavepacket maps onto experimental conditions Signal obtained from propagated wavefunction Branching ratios Spectrum Can then analyse simualation to obtain molecular picture and detailed information: energy flow between modes, state populations, etc. from expectation values. 3 / 34

16 Input File Quantics program driven by an ascii input file. ections reflect the choices to be made: PRIMITIVE-BASIS-SECTION: OPERATOR-SECTION: INIT_WF-BASIS-SECTION: SPF-BASIS-SECTION: INTEGRATOR-SECTION: RUN-SECTION: the coordinates, no. grid points etc. file containing the operator the initial wavefuncion the no. of MCTDH basis functions details of the integration scheme type of calculation to be made 31 / 34 RUN-SECTION name = hh2 propagation tfinal = 12.d tout = 1.d tpsi = 1.d psi gridpop steps end-run-section OPERATOR-SECTION opname = h3j alter-labels CAP_rd = CAP [ ] # starting point, strength, order end-alter-labels end-operator-section SPF-BASIS-SECTION rd = 14 rv = 1 theta = 1 end-spf-basis-section PRIMITIVE-BASIS-SECTION #Label DVR N Parameter rd sin 68 1.d 9.4d # xi, xf rv sin 48.6d 6.24d # xi, xf theta leg 31 even # l_z, sym (all/even/odd) end-primitive-basis-section INTEGRATOR-SECTION CMF/var =.1, 1.d-5 BS/spf = 7, 1.d-6 SIL/A = 3, 1.d-5 end-integrator-section INIT_WF-SECTION build rd gauss 4.5d -8.d.25d # r,p, sigma_r rv eigenf H2 pop=1 # pop=1 -> ground state theta leg sym # l_z, l, sym/no-sym end-build end-init_wf-section end-input 32 / 34

17 HAMILTONIAN-SECTION mode rd rv theta Operator File Defines the operators for the coordinates rd, rv etc. The coordinate labels are free to be chosen. The Hamiltonian has the form H = s H s = s c s h (1) s (x1)h (2) s (x2)... (53) operators h (κ) s are defined in table. E.g..5/mass_rv 1 q^-2 j^2 is H 2 = 1 m rv 1 (rv) 2 ĵ 2 (54) ĵ 2 is the angular momentum squared operator for theta. Note that the potential in this example is a special case, V. This is defined separately in the LABELS-SECTION. Any parameters can be given in the PARAMETERS-SECTION 33 / 34 OP_DEFINE-SECTION title H+H2 Reactive Scattering in Jacobian Coordinates. J=. Minimum of H_2 curve: ev. Zero point energy of vib.:.27 ev end-title end-op_define-section PARAMETER-SECTION mass_rd = , H-mass # Reduced mass of H--H2 system mass_rv =.5, H-mass # Reduced mass of H2 molecule jtot = # Total angular momentum jbf = # Projection on BF axis (K, or Omega). end-parameter-section HAMILTONIAN-SECTION modes rd rv theta /mass_rd q^-2 1 j^2.5/mass_rv 1 q^-2 j^2 1. KE KE 1 1. V end-hamiltonian-section LABELS-SECTION V = lsth {jacobian} end-labels-section # The following one-dimensional hamiltonian is used to determine the # eigenstates of H_2. These are then used as initial rv-spf s. HAMILTONIAN-SECTION_H modes rd rv theta KE v:h end-hamiltonian-section end-operator 34 / 34