Lecture 4: Introduction to Quantum Dynamics

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Markus Oppel

Institute of Theoretical Chemistry Universitat¨ Wien

SHARC 2016 Lecture 4: Introduction to Quantum Dynamics

1 Introduction: General Concepts and Deﬁnitions

2 Wave packets

3 Solving the nuclear TDSE

4 Solving the nuclear TISE

5 Following wave packets experimentally: Pump-probe

Markus Oppel Quantum Dynamics October 4th, 2016 2 Introduction: General Concepts and Deﬁnitions

1 Introduction: General Concepts and Deﬁnitions

Markus Oppel Quantum Dynamics October 4th, 2016 3 Remarks

Lecture is based on: N. Balakrishnan, C. Kalyanaraman, N. Sathyamurthy, Time-dependent quantum mechanical approach to reactive scattering and related processes, Physics Reports, 280, 2, 79-144 (1997). Other suggested literature: D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective. Sausilito, CA: University Science Books (2006).

Markus Oppel Quantum Dynamics October 4th, 2016 4 The Schrodinger¨ Equation

Quantum Mechanics: (Dynamical) Behavior of a System is described by the (time dependent) Schrodinger¨ Equation: i~(∂/∂t)Ψ(t) = Hˆ(t)Ψ(t) (1)

With Hˆ(t) being the (generally time dependent) Hamiltonian of the system. A formal solution of equation (1) is: ˆ Ψ(t) = U(t, t0)Ψ(t0) (2)

ˆ U(t, t0) is the time-evolution operator (sometimes called “propagator”) ˆ U(t, t0) has the following properties Unitarity: Uˆ†Uˆ = 1 (norm/probability-density conservation) Time continuity: Uˆ(t, t) = 1 ˆ ˆ ˆ Composition property: for (t0 t1 t2): U(t2, t0) = U(t2, t1)U(t1, t0) → → ˆ 1 ˆ Time reversibility: U− (t, t0) = U(t0, t)

Markus Oppel Quantum Dynamics October 4th, 2016 5 Evaluating the time-evolution operator

In general, Uˆ can be expressed as:

" Z t # ˆ i ˆ U(t, t0) = exp − H(t 0)dt 0 . (3) ~ t0

The right hand side of equation (3) is a function of an operator. If Hˆ is explicitly time-dependent, solving equation (3) is quite complicated. For the special case of a time-independent Hamiltonian Hˆ, solving equation (3) is straightforward: ˆ i ˆ i ˆ U(t, t0) = exp − H (t t0) = exp − H∆t , (4) ~ · − ~

with ∆t = t t0. − In practice: Always work with time-independent Hamiltonians....

Markus Oppel Quantum Dynamics October 4th, 2016 6 The Hamiltonian

As seen on the first day, the full (non-relativistic) Hamiltonian describing a molecule can be separated ˆ ˆ using the Born-Oppenheimer-Approximation into an electronic Hel and a nuclear Hnuc part. ˆ Hel is usually treated by quantum chemical methods (see first day). The electronic energy of a system as a function of the coordinates of the nuclei E(R), together with the ˆ nuclear repulsion energy VN(R) enters Hnuc as part of the potential energy ˆ ˆ ˆ Hnuc = TN + E(R) + VN(R) + Vinteraction. (5) ˆ Vinteraction includes all (possibly time-dependent) interactions with the environment ˆ Interaction with an external laser field, for example Vinteraction(t)) = µE(t) Interaction with an external bath − etc, etc.

Markus Oppel Quantum Dynamics October 4th, 2016 7 Treating multiple electronic states

Treating more than one electronic state (Photochemistry!) changes the operators and wave functions to matrices and vectors. For a two state system, the Hamiltonian becomes ! H H Hˆ = 11 12 . (6) H21 H22

H11 and H22 represent the (uncoupled) electronic states

Hii = T + Vi . (7)

The off-diagonal elements H12 (usually time and/or position dependent) couple the electronic states and triggered transitions.

Markus Oppel Quantum Dynamics October 4th, 2016 8 Radiationless Transitions

Radiationless transitions are governed by the non Born-Oppenheimer terms, i.e. corrections to the simpliﬁcations made by applying the Born-Oppenheimer approximation. In the adiabatic picture, they coupling is initiated through a kinetic coupling:

non BO D a 2 a E D a a E H − = Tij = φ φ + 2 φ R φ R , (8) ij i |∇R | j i |∇ | j · ∇

while in the diabatic picture the coupling is mediated through the potential energy, i.e. we have Tij = 0 and Vij , 0 ! ! T 0 V V Hˆ = Tˆ + Vˆ = 11 + 11 12 . (9) 0 T22 V21 V22

Markus Oppel Quantum Dynamics October 4th, 2016 9 Radiative Transitions

Radiative Transitions happen within one electronic state or between different electronic states. The most common form is the interaction with an external (time-dependent) electric field within the electronic dipole approximation: field V = µ~ij E~ (t). (10) ij − · µ~ij is the (transition-)dipole moment between (i , j) or within (i = j) the electronic states, E~ (t) is the time-dependent electric laser field

E~ (t) = S(t) E~ 0 cos(ωt τ). (11) · · −

E~ 0 is the maximum ﬁeld amplitude, ω is the carrier frequency, τ a phase (usually neglected) and the shape of the laser pulse is deﬁned by choosing S(t) as either a Gaussian or sin2-type function.

Markus Oppel Quantum Dynamics October 4th, 2016 10 Wave packets

2 Wave packets

Markus Oppel Quantum Dynamics October 4th, 2016 11 Discretization of the wave function

In order to solve the Schrodinger¨ equation, the wave function and all the operators have to be represented numerically

Usual approach: Expand the wave function in a ﬁnite, orthonormal basis set: φi (x); i = 1, .., N { } “Discrete variable representation” (DVR) Also known as: Spectral projection method (“Project” the objects onto the spectrum (set of eigenvectors) of an operator) Special case of DVR: Represent the wave function on a (equally spaced) grid “Fast Fourier transform method”

Markus Oppel Quantum Dynamics October 4th, 2016 12 The fast Fourier transform method

Vˆ is diagonal in position space Tˆ is (usually) diagonal in momentum space Transformation between position and momentum space is done via the Fast Fourier Transformation (FFT) Z 1 ∞ ikx Ψ(k) = Ψ(x)e− dx = FT[Ψ(k)] (12) √2π −∞ Z 1 ∞ ikx 1 Ψ(x) = Ψ(k)e dx = FT − [Ψ(x)] (13) √2π −∞ Z dΨ(x) 1 ∞ ikx 1 = Ψ(k)(ik)e dx = FT − [(ik)Ψ(k)] (14) dx √2π −∞ 2 Z d Ψ(k) 1 ∞ = Ψ( )( )2 ikx = 1[ 2Ψ( )] 2 k ik e dx FT − k k (15) dx √2π − −∞

Markus Oppel Quantum Dynamics October 4th, 2016 13 The grid

Scaling of FFT-based method: N log(N) Criteria for N (number of grid points): Resolution of wave packet’s properties In k-space (same number of grid points N): maximum kinetic energy which can be represented k 2 Emax = max (16) kin 2m with m (effective) mass

Qndof Total number of grid points: N = i Ni , ndof = number of degree of freedoms Action of operators (scales linear with N): real*8 V(ngridmax),psi(ngridmax) ... do i=1,ngrid psi(i)=V(i)*psi(i) enddo ... Limiting factors: Memory, Computational Time, Effort of computing the PES V

Markus Oppel Quantum Dynamics October 4th, 2016 14 Solving the nuclear TDSE

3 Solving the nuclear TDSE

Markus Oppel Quantum Dynamics October 4th, 2016 15 A Closer Look at the Time Evolution Operator

Propagator for a time-independent Hamiltonian (in atomic units, ~ = 1)

ˆ ˆ ˆ iH(t t0) iH∆t U(t, t 0) = e− − = e− , (17)

with ∆t = t t 0 − How to evaluate the propagator? Different propagation schemes are in use Second-Order Differencing (SOD) Split-Operator Method (SO) Chebyshev polynomial expansion ... Most commonly used: SOD (conceptually easy) and SO (easy, numerically robust) SOD and SO require small time steps Chebyshev and other polynomial expansion schemes can be used for long time dynamics What if Hˆ is time-dependent? use small ∆t so Hˆ becomes effectively time-independent →

Markus Oppel Quantum Dynamics October 4th, 2016 16 Second Order Differencing (SOD)

Propagate one time step Ψn+1 = UˆΨn (18) Unitarity of Uˆ n 1 ˆ n Ψ − = U†Ψ (19) Substract Eq. (18) from Eq. (19) n+1 n 1 ˆ ˆ n Ψ Ψ − = (U U†)Ψ (20) − − Expand Uˆ and Uˆ † in the Taylor series and keep terms up to second order

n+1 n 1 ˆ n 3 Ψ = Ψ − (2i∆t)HΨ + O(∆t ) (21) − - Small time step ∆t needed for numerical stability - First time step (n=1) needs special treatment + Need to evaluate only HˆΨ = (Tˆ + Vˆ)Ψ = TˆΨ + VˆΨ + Suitable for complicated Hamiltonians

Markus Oppel Quantum Dynamics October 4th, 2016 17 The Split-Operator Method (SPO)

exp( iHˆ∆t) function of an operator − How to evaluate f(Oˆ )? use eigenfunctions of Oˆ as a basis → We don’t know the eigenfunctions of Hˆ, but we know eigenfunctions of Tˆ and eigenfunctions of Vˆ Ψ(k) and Ψ(x) → Unfortunately, Tˆ and Vˆ do not commute, i.e.

iHˆ∆t i(Tˆ+Vˆ)∆t iTˆ∆t iVˆ∆t e− = e− , e− e− (22)

i(Tˆ+Vˆ)∆t Expanding e− in a Tayler series, rearranging some terms and keep everything up to second order, one can show that: ˆ ˆ iHˆ∆t i T ∆t iVˆ∆t i T ∆t 3 e− = e− 2 e− e− 2 + O(∆t ) (23) + Propagator strictly unitary + Time step can be larger than for SOD + For simple Hamiltonians: Need only two FFTs per time step (one forward, one backward FFT) - Costly (and sometimes impossible) for complicated Tˆ

Markus Oppel Quantum Dynamics October 4th, 2016 18 The Propagation

Setup all necessary parameters and operators Define ndof Define kinetic energy operator T Define grid Map PES (and (transition-) dipole moments) onto grid Setup starting wave function Ψ(t = 0) Propagation: Loop over time steps ∆t

Psi=Psi0 do ISTEP=1,NTMAX t=(ISTEP-1)*dt call fft(Psi,forward) call laser(Efield,t) call spo(Psi,T,V,Efield,t,...) call fft(Psi,backward) call analyse(Psi,T,V,Efield,t,...) enddo

Markus Oppel Quantum Dynamics October 4th, 2016 19 Solving the nuclear TISE

4 Solving the nuclear TISE

Markus Oppel Quantum Dynamics October 4th, 2016 20 Calculating Ψ0

How do we get the starting wave function Ψ(t = 0) = Ψ0 - Several possibilities

1 Use analytic Ψ0, e.g. Gaussian-Type wave function

2 Ψ0 = A0 exp( α(x x0) ) (24) · − − 2 Calculate Ψ0 by solving the time independent Schrodinger¨ equation on the grid Fourier grid Hamiltonian method (FGH) → Express Hˆ = Tˆ + Vˆ in the basis of the (full) grid Diagonalize Hˆ directly.

+ Gives exact Ψ0 (variation principle) + Depending on diagonalization routine one can get several eigenfunctions at once (excited states) - Need to generate, (store), and diagonalize matrix of size N2, quickly becomes very large

3 Propagate a trial function Ψtrial in imaginary time Formally replace t by i t in Uˆ ˜ · New operator: Uˆ = exp(Hˆ ∆t) ˆ · ˆ ˆ ˆ s Propagation: Ψs = exp(H ∆t) exp(H ∆t) ... exp(H ∆t)Ψtrial = exp(H ∆t) Ψtrial · · · · · · · Apply operator s times: Ψs exp(E0 ∆t)Ψ0 ≈ · Eigenfunction Ψ0 belonging to eigenvalue E0 (cf. Power Iteration)

Markus Oppel Quantum Dynamics October 4th, 2016 21 Following wave packets experimentally: Pump-probe

5 Following wave packets experimentally: Pump-probe

Markus Oppel Quantum Dynamics October 4th, 2016 22 NaI - A Diatomic System

Simulating a pump-probe experiment on NaI Original reference: Todd S. Rose, Mark J. Rosker, and Ahmed H. Zewail, J. Chem. Phys. 91, 7415 (1989). Two consecutive laser pulses: Pump (to start the dynamics) and Probe (to stop the “reaction”) Description of the process (Prepare the initial system, Ψ0, in the electronic ground state V0) Start ﬁrst laser pulse E1 (Pump) - Excite the system onto a excited state potential V1 Stop propagation by using a second laser pulse E2 (Probe) to excite the system into the V3 potential

Pump pulse E1: 50 fs, 307 nm Two different probe pulses E2: 100 fs, 589 nm 100 fs, 620 nm

The delay time td between the pump and the probe pulse is the crucial experimental parameter!

Markus Oppel Quantum Dynamics October 4th, 2016 23 NaI - System Setup

Simulation Details: mNa mI One degree of freedom: Na-I distance r and reduced mass m˜ NaI = ∗ mNa +mI ~2 ∂2 ˆ = T 2 , (25) − 2m˜ NaI ∂r + Three electronic states: V0 (NaI ground state), V1 (Na + I−), V2 (Na∗ + I) Full Hamiltonian:     H00 H01 H02  T + V0 µ01 E(t) 0  ˆ    − ·  H = H10 H11 H12 =  µ01 E(t) T + V1 µ12 E(t) (26)   − · − ·  H20 H21 H22 0 µ12 E(t) T + V2 − · Condon approximation for the transition dipole moments: µ01 = µ12 = 1

Laser ﬁeld E(t) = E1(t) + E2(t); E1(t), E2(t): Gaussian type pulses Starting wave function Ψ   0 Ψ(t = 0) =  0  (27)   ,  0 

Ψ0: ground state eigenfunction of V0

Markus Oppel Quantum Dynamics October 4th, 2016 24