Lecture 4: Introduction to Quantum Dynamics 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) = H^(t)Ψ(t) (1) With H^(t) being the (generally time dependent) Hamiltonian of the system. A formal solution of equation (1) is: ^ Ψ(t) = U(t; t0)Ψ(t0) (2) ^ U(t; t0) is the time-evolution operator (sometimes called “propagator”) ^ U(t; t0) has the following properties Unitarity: U^yU^ = 1 (norm/probability-density conservation) Time continuity: U^(t; t) = 1 ^ ^ ^ Composition property: for (t0 t1 t2): U(t2; t0) = U(t2; t1)U(t1; t0) ! ! ^ 1 ^ Time reversibility: U− (t; t0) = U(t0; t) Markus Oppel Quantum Dynamics October 4th, 2016 5 Evaluating the time-evolution operator In general, U^ can be expressed as: " Z t # ^ i ^ 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 H^ is explicitly time-dependent, solving equation (3) is quite complicated. For the special case of a time-independent Hamiltonian H^, solving equation (3) is straightforward: ^ i ^ i ^ 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 ^ ^ using the Born-Oppenheimer-Approximation into an electronic Hel and a nuclear Hnuc part. ^ 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 ^ nuclear repulsion energy VN(R) enters Hnuc as part of the potential energy ^ ^ ^ Hnuc = TN + E(R) + VN(R) + Vinteraction: (5) ^ Vinteraction includes all (possibly time-dependent) interactions with the environment ^ 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 H^ = 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 jrR j j i jr j j · r 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 H^ = T^ + V^ = 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(!t τ): (11) · · − E~ 0 is the maximum field amplitude, ! 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 f g “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 V^ is diagonal in position space T^ is (usually) diagonal in momentum space Transformation between position and momentum space is done via the Fast Fourier Transformation (FFT) Z 1 1 ikx Ψ(k) = Ψ(x)e− dx = FT[Ψ(k)] (12) p2π −∞ Z 1 1 ikx 1 Ψ(x) = Ψ(k)e dx = FT − [Ψ(x)] (13) p2π −∞ Z dΨ(x) 1 1 ikx 1 = Ψ(k)(ik)e dx = FT − [(ik)Ψ(k)] (14) dx p2π −∞ 2 Z d Ψ(k) 1 1 = Ψ( )( )2 ikx = 1[ 2Ψ( )] 2 k ik e dx FT − k k (15) dx p2π − −∞ 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) ^ ^ ^ 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 H^ is time-dependent? use small ∆t so H^ becomes effectively time-independent ! Markus Oppel Quantum Dynamics October 4th, 2016 16 Second Order Differencing (SOD) Propagate one time step Ψn+1 = U^Ψn (18) Unitarity of U^ n 1 ^ n Ψ − = UyΨ (19) Substract Eq. (18) from Eq. (19) n+1 n 1 ^ ^ n Ψ Ψ − = (U Uy)Ψ (20) − − Expand U^ and U^ y in the Taylor series and keep terms up to second order n+1 n 1 ^ 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 H^Ψ = (T^ + V^)Ψ = T^Ψ + V^Ψ + Suitable for complicated Hamiltonians Markus Oppel Quantum Dynamics October 4th, 2016 17 The Split-Operator Method (SPO) exp( iH^∆t) function of an operator − How to evaluate f(O^ )? use eigenfunctions of O^ as a basis ! We don’t know the eigenfunctions of H^, but we know eigenfunctions of T^ and eigenfunctions of V^ Ψ(k) and Ψ(x) ! Unfortunately, T^ and V^ do not commute, i.e. iH^∆t i(T^+V^)∆t iT^∆t iV^∆t e− = e− , e− e− (22) i(T^+V^)∆t Expanding e− in a Tayler series, rearranging some terms and keep everything up to second order, one can show that: ^ ^ iH^∆t i T ∆t iV^∆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 T^ 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.