Theory and non-adiabatic effects at avoided curve crossings in

Modern understanding of chemistry is, often without that scientist are aware of it, based on the concept of the ”adiabatic potential-energy surface” (PES), ₀(R), see Fig. 2.1. The idea of a PES is that the energy of a molecular system 0(R) can be mapped out in terms of the nuclear coordinates (for example the bond lengthsR₁ and R2 in Fig. 2.1). For many systems, this representation is a reasonably well description.

In that case, trajectories calculated on this PES — which determines the forces acting on the atoms/molecules — provide a clear picture how the atoms/molecules move as a function of time. Thereby it is assumed that, unless the system is electronically ex-cited with radiation, the molecular system always stays on the PES of the electronic ground state. However, there can also be cases in which the electronic state changes non-radiatively during a dynamical event and such an event would be called an elec-tronically non–adiabatic process.(12)

The term electronically non–adiabatic is a concept of quantum mechanics. When a

2. Theory and previous results

process (e.g. a chemical reaction) is studied experimentally, it is thus far from trivial to answer the question, whether it followed an electronically adiabatic or non-adiabatic pathway. A process in which it is immediately clear, that the system must have under-gone transitions to excited electronic states is chemi–luminescence: the emission of light not resulting from heat during a chemical reaction. The emission of light comes from a radiative decay from one electronic state to another. Thus the system must have left the electronic ground state in the first place. Well–known examples for chemi–

luminescence are the reactions in glow–sticks(13) or bio–luminescence, e.g. in fireflies such as the female glowworm Lampyris noctiluca.(14)

To answer the question under what conditions non-adiabatic transitions are likely to

oc-Figure 2.1: Potential energy surface– Schematic of a two-dimensional ”potential en-ergy surface” (PES). R₁ and R₂ are bond distances. The solid curves represent contours of equal energy0(R). The PES determines the forces acting in the system and thus the trajectory (shown as thick black solid line with arrows on top). The PES is a concept based

2.1 Introdution to electronically non-adiabatic processes

for a molecular system may be written as

H(r,ˆ R) = ˆT_R+ ˆH_el(r;R), (2.1) where R and r are the vectors of the nuclear and electronic coordinates, ˆT_R is the nuclear kinetic energy operator, and ˆHel(r;R) it the electronic Hamiltonian. The semi-colon in ˆHel(r;R) is used to state, that the electronic Hamiltonian depends on r and only parametrically onR. The electronic Hamiltonian contains the entire Hamiltonian of the system except for the nuclear kinetic energy operator. It includes the electronic kinetic energy operator and the Coulomb interactions. This means, that ˆH_el(r;R) can be considered the Hamiltonian for a system with non–moving nuclei, fixed at position R.

The exact solution Ψ(r,R) diagonalizes the full Hamiltonian ˆH(r,R)

hΨ_n(r,R)|H(r,ˆ R)|Ψ_m(r,R)i=δ_nmE_n. (2.2) However, the exact solution Ψ(r,R) is almost always unknown.

Instead, computational chemists diagonalize the electronic Hamiltonian ˆH_el(r;R) and the eigenfunctions φ_m(r;R) are called the adiabatic (Born–Oppenheimer) electronic wave functions

hφ_k(r;R)|Hˆ_el(r;R)|φ_j(r;R)i=δ_kjj(R), (2.3) which are the solution of theelectronic Schr¨odinger equationwith the eigenvalues_j(R) Hˆel(r;R)|φ_j(r;R)i=j(R)|φ_j(r;R)i. (2.4) The adiabatic PES corresponding to the electronic ground state is ₀(R), the elec-tronically exited states are j(R) with j ≥ 1. The exact wave function Ψ(R,r) can now be expanded in terms of φj(r;R) with the ansatz

Ψ_n(r,R) =X

i

φ_i(r;R)Ω_i(R). (2.5)

Substituting Eq. 2.5 into the Schroedinger equation for the full Hamiltonian from Eq. 2.1, we obtain (12, 15, 16)

hTˆ_R+_j(R)−Ei

Ω_j(R) =−X

i6=j

hTˆ_ij⁽¹⁾(R) + ˆT_ij⁽²⁾(R)i

Ω_j(R), (2.6)

2. Theory and previous results

where

Tˆ_ij⁽¹⁾ = −~²

2µ hφ_i|∇_R|φ_ji · ∇_R (2.7) Tˆ_ij⁽²⁾ = −~²

2µ hφ_i|∇²_R|φ_ji (2.8)

are the first and second order non–adiabatic coupling terms, which are also called momentum and kinetic energy non–adiabatic coupling terms, respectively. In these equations, µis the reduced mass, and∇_Ris the nuclear gradient operator. If the right term of equation 2.6 is neglected — meaning no coupling between different electronic states —, we obtain

hTˆ_R+_j(R)−Ei

Ω_j(R) = 0. (2.9)

Equation 2.9 means that — in case the non–adiabatic coupling terms are neglected

— the nuclear motion is governed by a Schroedinger equation with a potential energy 0(R) that is the solution of the electronic Schroedinger equation 2.4. It is thus possible to first compute the electronic structure part 2.4 for fixed nuclei and then the nuclear dynamics part 2.9. This is known as the Born-Oppenheimer approximation.(16) When does the approximation fail? The answer can be found in the equations for the non-adiabatic coupling terms. These become large, when 1) the first or second deriva-tive of the adiabatic wave functions with the nuclear coordinates are large, meaning simply that a potential has a strong slope or curvature. In addition, the integral is large when 2) the adiabatic wave functionsφ_i and φ_j are close in energy.

There is an extensive literature on the breakdown of the Born-Oppenheimer approxima-tion in gas–phase atomic and molecular collisions (see (17, 18) and references therein).

The most common examples involve systems with multiple potential energy surfaces which classically would cross. If the symmetry of the curves is the same, the crossing is forbidden by quantum mechanics, and interactions will prevent an actual crossing. The situation is particularly simple if we are dealing with atom-atom collisions, in which

2.1 Introdution to electronically non-adiabatic processes

whereIP is the ionization potential of the alkali atom and E_A is the electron affinity of the halogen. The ionic and covalent curves which interact to form an ion-pair are both¹Σ⁺ states. Since they are of the same symmetry, there will be an avoided curve crossing at R_C = 1/∆E. As the neutral Na and I atoms approach they proceed on the neutral curve until they reach a distance of RC. At that point they can proceed adiabatically and move on to the ionic curve or non–adiabatically and stay on the neutral curve. After reaching the inner turning point of the collision, the two atoms will again cross the avoided crossing atR_C with the possibility of making and adiabatic or non–adiabtic crossing. Ion pairs result if the two crossings are adiabatic-non–adibatic or non–adiabatic–adiabatic and the probability of forming an ion pair is given by

Pion–pair= 2p(1−p) (2.10)

wherep is the probability of making a non–adiabatic transition at the crossing point.

An approximate expression for p was derived by Landau, Zener, and Stueckelberg(19, 20, 21, 22)

p=

2π

~ H₁₂²

d/dR|V_ion−V_neut|v_R (2.11) where H₁₂ is the coupling matrix element (splitting) between the neutral and ionic curves,VionandVneutare the ionic and neutral potential curves,vRis the radial velocity.

There is a large body of measurements of charge transfer collisions that are in good accord with the Landau–Zener–Stueckelberg theory.(18, 23)

Pioneering work demonstrating non–adiabatic effects at avoided curve crossings in real–

time was performed by A. Zewail and co–workers on alkali halides such as sodium iodide, see Fig. 2.2 .(24) The adiabatic electronic ground state of the molecule has ionic (Na⁺ + I⁻) character near its equilibrium bond length, and covalent (Na–I) character upon dissociation. The situation can be described with two diabatic¹ potentials (one ionic and one covalent, shown as solid lines in Fig. 2.2) that cross a bond length of 6.93 ˚A.

In the adiabatic view (dashed lines), these potentials do not cross (non–crossing rule).

In the experiment, molecules are excited to the first electronically excited adiabatic state. This is done with a femtosecond laser pulse, that excites to a superposition of different vibrational states. This superposition can be described by a wave packet,

1The termdiabatic means that nuclear kinetic energy operator ˆTR is diagonalized (instead of the electronic Hamiltonian in the adiabatic case): hφ⁰_k(r;R)|TˆR|φ⁰_j(r;R)i=δkjj(R).

2. Theory and previous results

Figure 2.2: Non–adiabatic transition at a curve crossing in the gas–phase– left:

Potential energy curves and motion of the wave packet for the NaI molecule. At the top of the figure, the different probed configurations are given. right: Temporal population of the configurations as shown of the left side. Reprinted with permission from (24). Copyright 1988, AIP Publishing LLC.

travelling back and forth in the excited state with an oscillation period of approximately 1.25 ps. Every time the wavepacket travels back and forth, some molecules undergo a non–adiabatic transition to the adiabatic ground state. This population transfer can be detected with transient femtosecond spectroscopy.

The curve crossings at alkali halides are still part of modern research, for example the photo–dissociation of alkali–halides is studied in helium nano–droplets (25) and the non–adiabatic transition probability can be tuned with Stark pulses (26). An example for a non–adiabatic transition in the gas–phase regarding the vibrational dynamics of the NO molecule has recently been found for the vibrational relaxation in the collsion system NO(X²Π(v = 1)) + Ar → NO(X²Π(v = 1)) + Ar, which was attributed to a curve crossing between the (A⁰⁰, v = 1) and (A⁰, v = 0) vibronic states of the collision system.(27)