1 Introduction and overview

1.1 Goals

1. Link between Chemical Kinetics (PC II), Molecular Quantum Mechanics (PC III) and Spectroscopy (PC IV)

2. Show how reversible quantum mechanics relates to macroscopic irreversible reac- tion kinetics, introduction to quantum dynamics, molecular quantum dynamics, quantum scattering and modern experimental methods.

1.2 Overview

Microscopic quantum systems obey the time-dependent Schr¨odingerequation (TDSE) which is a linear differential equation.

d i Ψ(r, t) = Hˆ Ψ(r, t) where Hˆ is the Hamiltonian of the the system ~dt (1.1) 2 d2 Hˆ = − ~ + V (r, t) 2m dr2 Equation (1.1) is time reversible, its solutions are oscillatory or simple exponential decays

⇓

Mesoscopic systems obey master equations which are linear differential equations (e.g.Pauli equations)

d~p = K~p ~p : population vector (1.2) dt Equation (1.2) is not time-reversible and its solutions relax towards equilibrium of maxi- mal entropy (2nd law of thermodynamics).

⇓

dck Macroscopic systems obey dt = f[cj, ck, cl, ...... ka(τ), kb(τ)...]. In general non-linear differential equations for macroscopic concentration variables. The solutions describe relaxation of the system into equilibrium.

1 1 Introduction and overview

1.3 Classical mechanics

• Dynamics of the particles are described as trajectories in phase space with coordi- nates ~qk and momenta ~pk. • State of system is a point in 6N -dimensional phase space (3N coordinate and 3N momenta)

• Hamiltonian formulation: introduced by William Rowan Hamilton (1805-1865), Irish physicist who also introduced Hamilton’s principle, Hamiltonian mechanics, Hamilton-Jacobi equation, Quaternions, Nabla symbol, tensors etc.) dq ∂H k ≡ q˙ = dt k ∂p k (1.3) dpk ∂H ≡ p˙k = dt ∂qk

• Alternatives: Newton mechanics, Lagrangian mechanics

• Many calculations rely on classical mechanics over potential-energy hyper- surfaces (PES) calculated within the Born-Oppenheimer Approximation (BOA). ”Classical trajectory” methods.

• In many cases, calculation of complete PES is too demanding

• use of empirical force field = Molecular dynamics (MD)

• use of density functional theory (DFT) to calculate forces → Car-Parrinello molec- ular dynamics

• Nobel prize in chemistry 2013:Martin Karplus: 1964/1965 treated H + H2 reac- tion classically. Kuppermann and Wyatt in 1975 reported full quantum treatment that agrees with classical calculation. Michael Levitt, Ariel Warshell extended classical-trajectory methods to treat complex systems, including molecules of bio- logical interest.

1.4 Quantum mechanics

• Observable physical variables correspond to operators

• Classical Hamilton’s equations → quantum mechanical Heisenberg equations.

dˆqk ˙ 1 ≡ qˆk = [ˆqk, Hˆ ] dt i~ dˆpk ˙ 1 (1.4) ≡ pˆk = [ˆpk, Hˆ ] dt i~ Hˆ = Tˆ + Vˆ

2 1 Introduction and overview

N 2 2 2 ! 1 X pˆx,k pˆy,k pˆz,k For N massive point-like particles: Tˆ = + + 2 mk mk mk k=1 ∂ (1.5) wherep ˆ = −i , x ~∂x ∂2 pˆ2 = − 2 x ~ ∂x2 If the interaction between particles can be expressed as a potential, the corre- sponding operator Vˆ is multiplicative.

• The Heisenberg equations of motion can be written in matrix form amenable to direct numerical implementation.

• The Schr¨odinger picture is more common: state of system specified by wave func- tion ψ = function of all coordinates and time. Probability density:

2 p(x, y, z, ....., xn, yn, zn, t) = |ψ(x, ....t)| (1.6) = ψ.ψ∗

Temporal evolution of ψ(x, t) given by TDSE

∂ψ(x, t) i = Hψˆ (x, t) (1.7) ~ ∂t Describes evolution of ”pure” or ”dynamical” state. A statistical mixture of many quantum states is described by density operatorρ ˆ ⇒ quantum analogue of classical probability density in phase space.

1.5 Quantum-classical correspondence: the Ehrenfest Theorem

The Ehrenfest theorem relates the time derivative of the expectation value of the momen- ∂Vˆ (q) tum (ˆp) and position (ˆq) operators to the expectation value of the force F = h− ∂t i. The time-dependent expectation value of an operator Aˆ, is defined as follows Z hAˆi = Ψ∗AˆΨdτ (1.8)

Differentiating with respect to time one gets

d d d ∂Aˆ hAˆi = h Ψ|AˆΨi + hΨ|Aˆ Ψi + hΨ| Ψi (1.9) dt dt dt ∂t Using TDSE one can write

∂ i Ψ(r, t) = Hˆ Ψ(r, t) (1.10) ~∂t

3 1 Introduction and overview

∂ i Ψ∗(r, t) = −Hˆ Ψ∗(r, t) (1.11) ~∂t Equation (1.9) can be modified using equations (1.9), (1.10), (1.11)

d 1 1 ∂ hAˆi = h−Hˆ Ψ|AˆΨi + hΨAˆ|Hˆ Ψi + hΨ| AˆΨi dt i i ∂t ~ ~ (1.12) 1 ∂Aˆ = hΨ|[A,ˆ Hˆ ]|Ψi + h i i~ ∂t If the operator Aˆ is not explicitly time-dependent then the equation becomes

d i hAˆi = hΨ|[A,ˆ Hˆ ]|Ψi (1.13) ~dt For the positionq ˆ and momentump ˆ operators one can write, d i~ hqˆi = hΨ|[ˆq, Hˆ ]|Ψi dt (1.14) d i hpˆi = hΨ|[ˆp, Hˆ ]|Ψi ~dt ˆ pˆ2 ˆ Using the Hamiltonian H = 2m + V (q) one can obtain the Ehrenfest theorem d hpˆi hqˆi = (1.15) dt m

d ∂Vˆ hpˆi = h− i (1.16) dt ∂q These equations are not a closed set of ordinary differential equations (ODE), as a ∂Vˆ result it is necessary to know the wave function Ψ in order to calculate h− ∂t i. If equation 1.16 has been more ”classical,”

d ∂V (hqi) hpi = − , (1.17) dt ∂ hqi d ˆ ˆ 0 We expand the derivative of the potential- energy operator writing dq V (q) = V (q) 1 Vˆ 0(q) = Vˆ 0(hqi) + (q − hqi)Vˆ 00(hqi) + (q − hqi)2Vˆ 000(hqi) + ...... (1.18) 2 The expectation value of Vˆ0(q) can be approximated by 1 hVˆ (q)i ' Vˆ 0(hqi) + χVˆ 000(hqi) (1.19) 2 where χ ≡ q2 − hqi2. First term is the classical term and second term is the ”error” made when using the classical expression. This error depends on the width of the wavepacket and the third

4 1 Introduction and overview derivative of the potential. For up to harmonic potentials, the ”error” term vanishes and the classical equation of Ehrenfest can apply. For more general potentials, the ex- pectation values of position and momentum follow the classical trajectories for compact (small width) wavepackets, for short times (little wavepacket dispersion) or for nearly harmonic potentials.

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