Theoretical study of the energy partition between different electronic states in the photodissociation of diatomics: from model systems to a planned experiment on NaH.

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Chapter 1

Introduction

We are perhaps not far removed from the time when we shall be able to submit the bulk of chemical phenomena to calculation.

Joseph Louie Gay-Lussac For almost a century, the power of chemistry and physics in describing processes at the atomic or molecular scale within quantum-mechanical methods has relied on the Born-Oppenheimer Approximation (BOA). Such assumption about the motion of nuclei and elec-trons which is named after Max Born and J. Robert Oppenheimer has successfully driven the development of quantum chemistry and, on the methodological level, of its computational methods.

However, the approach of BOA, also called “adiabatic approximation”, towards the effect of nuclear dynamics on electrons is not accurate enough to treat a number of physico-chemical phenomena especially when an interaction within electromagnetic radiation is taken into ac-count. Specific terms which are neglected by the theoretical treatment of BOA, called nona-diabatic couplings, are actually crucial in the correct description of photo-induced molecular dynamics.

One of the simplest examples of nonadiabatic dynamics may occur in diatomic molecules, when their photodissociation can occur in different electronic states. In that case, energy conservation is usually expressed as:

Ekin= Eini+ hν − Eel

where Ekin is the kinetic energy of the dissociated atoms, Eini is the total energy of the

initial state, and Eel is the electronic energy of the final state.

However, with ultrashort laser pulses (femto- or attosecond regime) the photon energy hν has a large indetermination. In fact, in analogy with Heisenberg’s indetermination principle, we have a similar relationship between the uncertainty on the photon energy ∆Eph= h ∆ν

and the duration of the pulse ∆t: ∆Eph ∆t ≥ ~/2

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2 CHAPTER 1. INTRODUCTION

In terms of frequency, this means ∆ν∆t ≥ 1/4π. As a consequence, a coherent pulse with ∆t ≃ 1 fs can produce a wave-packet about 5000 cm−1_{wide in the energy scale. The energy}

distribution can change substantially in the different atomic states where dissociation can occur. The same holds for different products in the photodissociation of a polyatomic.

The Landau-Zener rule [24, 25] and recent formal work [26] on nonadiabatic wave-packet dynamics indicate that the higher energy components of a wave-packet are endowed with a larger nonadiabatic transition probability. So, one may expect smaller final energies in the state where the molecule was initially excited and larger ones in the states populated by nonadiabatic transitions.

We want to understand the physical basis of this phenomenon and to determine the best conditions to observe it experimentally.

Our aim in this thesis is therefore to investigate the role that nonadiabatic couplings may play in the redistribution of total energy for nuclear wave-packets associated to interact-ing electronic states. Because the purpose of our project relies on fundamental theory, so that a reduced level of complexity was required, we considered cases where only two elec-tronic states and one nuclear coordinate were involved. Note that to determine the different components of the energy of polyatomic fragments (electronic, vibrational, rotational and translational) with sufficient accuracy is far from trivial. On the contrary, for a diatomic molecule one only needs to determine the atomic electronic state and the relative kinetic energy. Doppler spectroscopy on one of the two atoms, together with the momentum con-servation relationship, could yield both informations. So, diatomic systems seem to be the best choice also at the experimental level.

As a first instance, we conceived an ultra-simplified model within two parallel electronic potentials; this model could allow us to derive an clear mathematical characterisation about the conditions for energy and momentum conservation and their ostensible violation. Nevertheless, in order to get a more realistic picture we developed a second model regarding two electronic states in presence of an avoided-crossing. In this case, Hamiltonian matrix elements contain two diabatic potentials and an off-diagonal coupling term. Such modelling work has allowed us to assess the numerical accuracy of nonadiabatic quantum wave-packet calculations and to perform tests designed for physical considerations. Indeed, suitable val-ues for the model variables were explored to get experimentally detectable photodissociation quantum yields and energy differences between dissociating wave-packets.

As a third and final case, we studied the photodissociation of NaH, a molecule which re-sembles the previous avoided-crossing model within the two states X1_Σ+ _{and A}1_Σ+_{. In}

the literature one finds accurate potential energy curves for such electronic states, but not the nonadiabatic coupling matrix elements at the same level. Therefore, we computed the X1_Σ+ _{and A}1_Σ+ _{curves, the nonadiabatic couplings and the transition dipole by means of}

a very accurate multireference approach (CASSCF/MR-CISD). With these data we tried to simulate the photodissociation dynamics of NaH.

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Chapter 2

Quantum theory of

nonadiabatic transitions

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that the exact application of these laws leads to equations much too

complicated to be soluble.

Paul A. M. Dirac

2.1 The Born-Oppenheimer approximation.

The fundamental equation for the time evolution of quantum states is the time-dependent Schr¨odinger equation (TDSE):

i~d |Ψi

dt = ˆH |Ψi (2.1)

If the Hamiltonian itself is not time-dependent (no external fields or interacting systems that may change in time), the formal solution of this equation is:

|Ψ(t)i = ˆU (t − t0) |Ψ(t0)i = e−i ˆH(t−t0)/~|Ψ(t0)i (2.2)

where ˆU (t − t0) = exp[−i ˆH(t − t0)/~] is the time-evolution operator from time t0to time t.

With an hermitian Hamiltonian, ˆU is unitary and the normalization of the wave-function Ψ is conserved.

The eigenstates of ˆH are stationary states. Consider the time-independent Schr¨odinger equation

ˆ

H |Ψki = Ek|Ψki (2.3)

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4 CHAPTER 2. QUANTUM THEORY OF NONADIABATIC TRANSITIONS

and a state that coincides with one of the |Ψki at some time t0: |Ψ(t0)i = |Ψki. Then, eq.

(2.2) simply yields:

|Ψ(t)i = e−iEk(t−t0)/~_|Ψ

ki (2.4)

The irrelevant phase factor exp[−iEk(t − t0)/~] does not change the state properties.

However, if the initial wave-function is a linear combination of non degenerate states, we have a non-stationary state, i.e. a real time-evolution:

|Ψ(t0)i = X k Ck(t0) |Ψki (2.5) implies |Ψ(t)i =X k Ck(t0)e−iEk(t−t0)/~|Ψki (2.6)

In this expression, the time-dependence is concentrated in the ratios of the complex coeffi-cients, exp[−i(Ek− El)(t − t0)/~]: the interference between different components Ψk is the

essential feature of quantum dynamics.

The description of the molecular quantum states is usually based on two approximations: neglecting magnetic terms in the Hamiltonian and separating the nuclear and electronic motions. These go under the names of “electrostatic” or “non-relativistic” Hamiltonian approximation, and of “Born-Oppenheimer” or “adiabatic” approximation. At this level one can explain and calculate most of the molecular properties, the ground state reactivity, the structure of energetic levels that are needed for statistical thermodynamics, and many important features of the absorption and emission spectra. However, to understand the radiationless decay of electronically excited states and the interplay of excited state dynamics with optical excitation, one needs to go beyond the electrostatic and the Born-Oppenheimer approximations.

In the absence of magnetic terms, there would be no coupling between the spin of the electrons and their orbital motion (the same would hold for the nuclei, of course). In fact, no Hamiltonian term would contain the spin, but this does not mean that the spin state would have no influence on the energy. The spatial wave-functions of, say, a singlet and a triplet state, have different permutational symmetries, because of the antisymmetry principle. This has an influence on the average electron-electron repulsion and therefore on the total energy. However, within the electrostatic approximation, the three components of a spin multiplet are strictly degenerate, and the Hamiltonian matrix elements between states of different spin do vanish. This means that a pure spin state cannot evolve in time to a state of different spin: as a consequence, the lowest triplet state of a molecule would have a substantially infinite lifetime. This is not completely true, of course, because of the coupling between the electron spin and magnetic fields of various origin within the molecule. The most important term is the so called spin-orbit coupling, which originates from the motion of the electron in the presence of the charges of the nuclei and of the other electrons. When dealing with light atoms, such terms are small and the transitions between states of different spin are substantially “forbidden”.

While the electrostatic approximation is easily identified with the neglection of certain terms in the Hamiltonian, the Born-Oppenheimer approximation is usually introduced as an ansatz

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2.1. THE BORN-OPPENHEIMER APPROXIMATION. 5

concerning the form of the molecular wave-function. However, in this section we shall identify which Hamiltonian matrix elements are neglected when accepting this approximation. Let us indicate with Q the collection of the nuclear coordinates. The Q can be cartesian coordinates, referred to a laboratory frame, or internal coordinates, resulting from the sepa-ration of translational and rotational motions. However, we shall only consider the case of a “diagonal” kinetic energy operator, i.e. one that can be written as a sum of terms, each one containing the second derivative with respect to one coordinate. The electronic coordinates will be indicated with q.

The Born-Oppenheimer approximation consists in writing the molecular (“vibronic”) wave-function as a product of an electronic and a nuclear factor:

Ψ(q, Q) = ψ(q; Q) χ(Q) (2.7)

Here we mean that |χ(Q)|2 is the probability density of finding the nuclei in the positions Q1. . . Qr. . .. This implies that the electronic factor ψ(q; Q) be normalized for any choice of

the Q coordinates, i.e.:

hψ(Q) |ψ(Q) i = 1 (2.8)

Although ψ(q; Q) also depends on the nuclear coordinates Q, it does not carry any informa-tion about the nuclear probability density. |ψ(q; Q)|2 is instead the probability density to find the n electrons in q1, q2, . . . q3n (space and spin coordinates), once established that the

nuclei are in the Q1. . . Qr. . . positions. ψ(q; Q) is the electronic wave-function for “fixed

nuclei”.

All the matrix elements are evaluated by integrating first on the electronic coordinates and then on the nuclear ones.

D ψkχu Oˆ ψlχv E = hχu|Okl(Q)| χvi (2.9) where Okl(Q) = D ψk(Q) Oˆ ψl(Q) E (2.10)

only depends on the Q coordinates and, as an operator, only acts on χv(Q).

The molecular Hamiltonian, in the electrostatic approximation, has the form:

ˆ H = −X r ~2 2µr ∂2 ∂Q2 r − ~2 2m X i ∂2 ∂q2 i + V (q, Q) (2.11)

If the Q are cartesian coordinates, µr is the mass of the nucleus that is associated with the

coordinate Qr. If, on the contrary, the Qr are internal coordinates, µr is a reduced mass.

With respect to the wave-function ψ(q; Q)χ(Q), the potential energy operator V (q, Q) is a mere factor, whereas the kinetic energy terms are differential operators. Of the latter, ∂2

∂q2 i

only acts on ψ, while ∂2 ∂Q2

r acts on both ψ and χ:

∂2 ∂Q2 r ψ(q; Q)χ(Q) = ∂ 2_ψ ∂Q2 r χ + 2 ∂ψ ∂Qr ∂χ ∂Qr + ψ∂ 2_χ ∂Q2 r (2.12)

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So, by applying ˆH to the ψ(q; Q)χ(Q) wave-function, one gets: ˆ H ψ(q; Q)χ(Q) = (2.13) = χ(Q) " V (q, Q) −X i ~2 2m ∂2 ∂q2 i # ψ(q; Q) − ψ(q; Q) " X r ~2 2µr ∂2 ∂Q2 r # χ(Q)− −X r ~2 2µr 2 ∂ψ ∂Qr ∂χ ∂Qr + χ∂ 2_ψ ∂Q2 r

The request that the product ψχ be an approximation of an eigenstate of ˆH prompts us to define ψ and χ through two conditions. We first define ψ as an eigenfunction of the electronic Hamiltonian ˆHel, i.e. the molecular Hamiltonian with the exclusion of the nuclear kinetic

energy: ˆ Hel = V (q, Q) − ~2 2m X i ∂2 ∂q2 i (2.14) ˆ Hel ψk= Uk(Q) ψk (2.15)

The eigenfunctions ψk will be also called “adiabatic” wave-functions. ˆHel depends on the

nuclear coordinates, so its eigenvalues Uk and its eigenfunctions ψk also depend on the Q.

We note that ˆHel and therefore the Uk energies also contain the nucleus-nucleus Coulomb

repulsion terms. Consistently with the meaning of the electronic wave-function, eq. (2.15) can be interpreted as a Schr¨odinger equation for the electrons, at fixed nuclei. The index k has been introduced because we need to distinguish the different solutions of the equation, that correspond to the ground (k = 0) and excited electronic states.

Equation (2.13) now takes the form: ˆ H ψk(q; Q)χ(Q) = (2.16) = ψk(q; Q) " Uk(Q) − X r ~2 2µr ∂2 ∂Q2 r # χ(Q) −X r ~2 2µr 2∂ψk ∂Qr ∂χ ∂Qr + χ∂ 2_ψ k ∂Q2 r

The second condition we shall impose concerns the nuclear wave-function χ(Q). We define an Hamiltonian for the nuclear motion in the following way:

ˆ Hvib= Uk(Q) − X r ~2 2µr ∂2 ∂Q2 r (2.17)

Here Uk(Q), the “electronic energy”, plays the role of an effective potential energy that

determines the nuclear motion: the adiabatic “potential energy surface” (PES). We shall require the nuclear wave-functions to be eigenfunctions of ˆHvib:

ˆ

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2.1. THE BORN-OPPENHEIMER APPROXIMATION. 7

Since the potential energy Uk(Q) is specific of each electronic state, so are the χ

wave-functions, that will therefore carry an index k and a second index to number the solutions of the eigenvalue equation. The Hamiltonian ˆHvib and its eigenfunctions will be qualified

as “nuclear” or “vibrational”, depending on the status of the coordinates Q (cartesian or internal coordinates).

At this stage, eq. (2.13) takes an even simpler form: ˆ H ψk(q; Q)χku(Q) = (2.19) = Ekuψk(q; Q)χku(Q) − X r ~2 2µr 2∂ψk ∂Qr ∂χku ∂Qr + χku ∂2_ψ k ∂Q2 r

Till now no approximation has been introduced. However, to say that the product ψkχku

is an approximate eigenfunction of the molecular Hamiltonian, is equivalent to neglecting in the rhs of this equation the terms that imply derivatives of ψk(q; Q) with respect to the

nuclear coordinates. We are then left with the set of equations that are the basis for the understanding of molecular structure and for most of the computational work in this field: eqs. (2.15), (2.17), and

ˆ

H ψk(q; Q)χku(Q) ≃ Ekuψk(q; Q)χku(Q) (2.20)

that embodies the Born-Oppenheimer approximation. We shall here take for granted the description of the molecular levels and states, i.e. the theory and the facts of molecular spectroscopy, and we shall instead focus on the properties and consequences of the coupling terms we have neglected.

The ψkχku vibronic states, approximate eigenstates of the molecular Hamiltonian, are

cou-pled through the Hamiltonian matrix elements

Hku,lv= X r ~2 2µr Z +∞ −∞ χ∗ku t(r)_kl (Q) + 2g_kl(r)(Q) ∂ ∂Qr χlv. . . dQs. . . (2.21) where: gkl(r)= ψk ∂ ∂Qr ψl (2.22) and t(r)_kl = ψk ∂2 ∂Q2 r ψl (2.23)

The Hku,lv matrix elements are called the “nonadiabatic” (or “dynamic”, or “derivative”)

couplings. When k = l (same electronic state), g_kl(r) = 0; in fact, we can assume the eigenfunctions of ˆHel, ψk, to be real functions, and:

g_kl(r)+ g_lk(r)= ψk ∂ ∂Qr ψl + ψl ∂ ∂Qr ψk = ∂ ∂Qr hψk|ψli = 0 (2.24)

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Therefore, the g(r)_kl do not couple different vibrational states belonging to the same electronic state. As to the t(r)_kl integrals, with l = k they play the role of a small correction to the Uk

potential, with the peculiarity that the correction depends on the nuclear masses:

U_k(corrected)= Uk(Q) + X r ~2 2µr t(r)_kk(Q) (2.25)

So, the t(r)_kk corrections slightly modify the Ekulevels, but do not invalidate the approximate

separation of the electronic and nuclear motions.

We can work out a formula for the g(r)_kl matrix elements in a way similar to the demonstration of Hellmann-Feynman theorem. By differentiating eq. (2.15) we get

∂ ˆHel ∂Qr |ψli + ˆHel ∂ |ψli ∂Qr = ∂Ul ∂Qr|ψli + Ul ∂ |ψli ∂Qr (2.26)

Pre-multiplying by hψl| yields Hellmann-Feynman’s formula:

∂Ul ∂Qr = * ψl ∂ ˆHel ∂Qr ψl + (2.27)

while using hψk| (k 6= l) one gets a useful relationship involving the dynamic couplings:

ψk ∂ ∂Qr ψl = D ψk ∂ ˆHel ∂Qr ψl E Ul− Uk (2.28)

This formula shows that in general the nonadiabatic couplings between two states will be larger where their PES get closer to each other. Equations (2.27) and (2.28) are only valid for exact (or fully optimized) eigenfunctions of ˆHel, or of any other Hamiltonian with

a parametric dependence on a variable Qr. Therefore, they cannot be used as such to

compute energy gradients and couplings for approximate wave-functions. However, they may be applied when dealing with model Hamiltonians to represent analytically PES and couplings, since in that case diagonalization yields exact eigenvectors in a small basis set (see section 2.4).

To assess the Born-Oppenheimer approximation we must first of all evaluate the magnitude of the nonadiabatic couplings. Of course, the couplings do change according to the molecule and the quantum states, and in some cases they are not negligible at all. However, most of the times Hku,lv is much smaller then the nuclear kinetic energy, which is of the order of

1000 cm−1_{. In fact, the electronic wave-functions usually change much more slowly than the}

vibrational ones, as functions of the nuclear coordinates. For instance, along the stretching coordinate of a chemical bond, χkuwill oscillate several times (if it is a vibrationally excited

state) or at least will go from almost zero to a maximum and back to zero again, with a displacement of ≈ 0.1 Bohr. For the same displacement, the electronic wave-functions will just undergo a small rearrangement, for instance in the relative weights of the covalent and ionic components, in the orbital hybridization and so on. Therefore, differentiating ψk

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2.2. EXCITED STATE DYNAMICS. 9

The magnitude of Hku,lvmust be compared with the energy difference between the vibronic

states k, u and l, v. If ψk is the ground electronic state (k = 0), its vibronic states are

only coupled with others belonging to electronically excited states (l ≥ 1). For the lowest vibrational states u, the Elv− E0uenergy differences are normally larger than 10000 cm−1.

In this case, the nonadiabatic couplings will only produce small corrections in the energy levels and in the eigenfunctions: the Born-Oppenheimer approximation is well justified. For instance, accurate calculations on the BH molecule yielded energy corrections smaller than 1 cm−1_{. With heavier nuclei, the corrections will tend to be even smaller. We note that}

the orders of magnitude of the electronic energy differences, of the vibrational ones, and of the nonadiabatic couplings, depend on the inverse of the masses of electrons and nuclei, that appear in eqs. (2.14), (2.17) and (2.28): therefore, we can attribute the validity of the Born-Oppenheimer approximation to the fact that the nuclei are “heavy and slow”, while the electrons adjust almost istantaneously their motion in response to changes of the molecular geometry. By increasing the vibrational quantum number u, the Elv−E0uenergy

difference decreases and the nonadiabatic corrections become more important. Actually, large quantum numbers u are necessary to reach very distorted geometries, corresponding to higher values of the potential U0(Q). This means that, for transition states or other

distorted geometries, the approximation may be less valid than close to the equilibrium geometry.

2.2 Excited state dynamics.

For the excited electronic states we have a different situation. The energy differences Elv−

Eku, with k ≥ 1 and l > k, are usually smaller than Elv− E0u, because the electronic energy

levels become “denser” as we go up in energy. More important, each electronically excited vibronic state ψkχku is (approximately) degenerate with other states ψlχlv belonging to

lower PES. However, since the wave-function ψlχlv must be vibrationally excited (high v),

it will contain many nodes: then, the matrix element (2.28) that couples it with ψkχku

will be small. In general, we expect the couplings to be more important when the energy difference between electronic states is smaller. In any case, we want to develop a theory to understand the behaviour of an excited state that is weakly coupled with many other states, very close in energy to it. To treat this problem we shall make use of the time-dependent perturbation theory.

A common phenomenon in molecular physics is that a system, initially in a stationary state, is perturbed by an external force, such as a static electric or magnetic field, a light pulse, or the approach of another molecule. If ˆH(0)_{is the Hamiltonian of the unperturbed system,}

and ˆV is the perturbation, the full Hamiltonian is ˆ

H = ˆH(0)+ ˆV (2.29)

We have a formally identical situation even in the absence of external perturbations, when the full Hamiltonian can be partitioned in two contributions, ˆH(0)_{and ˆ}_{V , and we can assume}

that initially the system occupies one of the eigenstates of ˆH(0)_:

ˆ

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(from now on the index i takes the place of both vibronic indexes k, u or l, v, for simplicity). We shall write the time-dependent wave-function as:

Ψ(t) =X

i

Ci(t) e−iεit/~Ψi (2.31)

The square module of each coefficient in this development, |Ci(t)|2, yields the probability of

finding the system in state Ψi at time t. The time-dependent Schr¨odinger equation (TDSE)

is: ∂Ψ ∂t = − i ~( ˆH (0)_{+ ˆ}_{V )Ψ} _(2.32)

By inserting in this equation the development (2.31) we get

X i dCi dt e −iεit/~_Ψ i= −i ~ X i Cie−iεit/~V Ψˆ i (2.33)

We then multiply both members on the left by Ψ∗

j and we integrate over all coordinates.

Since hΨi|Ψji = δij, we have dCj dt = − i ~ X i Ciei(εj−εi)t/~ D Ψj Vˆ Ψi E (2.34)

If the probability |Cj(t)|2 of state Ψj increases in time, we shall say that transitions from

other states Ψi to Ψj may occur. Equation (2.34) shows that this is only possible if some

state Ψi is already populated (has a non vanishing coefficient Ci) and

Vji= D Ψj Vˆ Ψi E (2.35)

This is the basis of all selection rules, in spectroscopy or, more generally, in molecular dynamics. When the condition (2.35) is verified, we say that the perturbation ˆV couples the states Ψj and Ψi, or that these states interact through the perturber ˆV . Notice that

till now we have used the language of perturbation theory, but no approximation has been introduced: eq. (2.34) is exact.

Let us now assume that one state, Ψ0, is populated at time t = 0, i.e. that C0(0) = 1

and Ci(0) = 0 for i 6= 0 (the label 0 here does not necessarily belong to the ground state).

If we limit ourselves to sufficiently short times and not too strong perturbations, the Ci

coefficients will remain very small, with the exception of C0(t). We then approximate

dCj/dt by neglecting all the Ci with i > 0 and assuming C0(t) ≃ 1:

dCj dt = − i ~ e i(εj−ε0)t/~_V j0 (2.36)

Finally we integrate in time:

Cj(t) = − i ~ Z t 0 eiωj0t′ _V j0(t′) dt′ (2.37)

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where ~ωj0= εj−ε0= hνj0. The transition probability 0 → j, given by |Cj|2, is proportional

to the square of the coupling between the states ψ0 and ψj, |Vj0|2. The coefficients Cj

depend on time according to the initial-to-final energy difference εj− ε0, and possibly to

the time-dependence of the perturbation ˆV .

We shall consider the case of a static perturbation, i.e. Vj0 independent on time. This

is always the case when ˆV is a “small term” of the molecular Hamiltonian, such as the nonadiabatic or the spin-orbit coupling. From eq. (2.37) we have then

|Cj(t)|2= ~−2|Vj0|2 Z t 0 eiωj0t′_dt′ 2 = ~−2|Vj0|2 eiωj0t_{− 1} 2 ω2 j0 = (2.38) = 4~−2|Vj0|2 sin(ωj0t/2) ωj0 2

For t > 0 the expression (2.38) is a function of ωj0with a maximum at ωj0= 0:

lim ωj0→0 sin(ωj0t/2) ωj0 2 =t 2 4 (2.39)

Secondary maxima are found at ωj0≃ (2n + 1)π/t, n = ±1, ±2 . . ., with rapidly decreasing

values t2/(2n + 1)2π2. At ωj0= 2nπ/t we have the zeroes of the function, that correspond

to minima. The function decays to negligible values, with respect to the central maximum, for ωj0≫ 1/t, i.e. for mod εj− ε0t ≫ ~.

A weak perturbation, constant in time, cannot therefore induce transitions between levels that differ in energy by much more than ~/t: we can see this result as a manifestation of the energy conservation principle. This condition is obeyed with an accuracy that depends on the duration of the interaction. If there are just a few states Ψj with energies close to ε0,

the transition probabilities will be rather complicated functions of time and of the εj− ε0

energy differences, even within the limits of validity of perturbation theory.

In atomic and molecular physics, we frequently deal with bound states, belonging to a dis-crete spectrum, coupled with, and embedded in, a continuum of dissociative states. The example we are interested in is a bound vibrational level of an excited electronic state, in-teracting through nonadiabatic or spin-orbit couplings with almost degenerate vibrationally excited states of a lower lying electronic state. If the bound state is higher in energy than the dissociation limit for the lower electronic PES, it will be degenerate with states belonging to the dissociative continuum. The first formal treatment of this problem (although in an atomic context) is due to Fano [31] and a totally equivalent theory can be developed by Green function methods [32]. We shall use the simpler approach of perturbation theory, and introduce the approximations that yield the famous Fermi’s Golden Rule of state decay. In such cases, rather then investigating the final state of the system, we ask ourselves what is the probability that any transition occurs, i.e. the rate of decay from the initial state. To get this result, we sum up all the transition probabilities to different Ψj states. Since

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variable ε, which is the energy of the dissociative state. The matrix element Vj0 will be

replaced by V0(ε) = D Ψε Vˆ Ψ0 E (2.40)

i.e. the coupling between the discrete state Ψ0 and the dissociative state Ψε. The

normal-ization of the dissociative wave-function must be such that hΨε|Ψε′i = δ(ε−ε′). The overall

transition probability is then

P (t) = 4~−2 Z ∞ εmin |V0(ε)|2 sin(ωε0t/2) ωε0 2 dε (2.41)

where ~ωε0= ε − ε0. The final result depends on two approximations, usually well justified:

a) we consider V0(ε) substantially independent on ε in the small interval of energies within

which the integrand of eq. (2.41) is not negligible; b) we extend the lower integration limit from the dissociation energy εmin to −∞, again because the integrand becomes rapidly

negligible for |ε − ε0| ≫ ~/t. By replacing ωε0t/2 with x we have

P (t) = 2~−1|V0(ε0)|2t

Z +∞

−∞

sin2_x

x2 dx (2.42)

The integral in eq. (2.42) is a mere constant and its value is π, so:

P (t) = 2π

~ |V0(ε0)|

2

t (2.43)

This equation embodies Fermi’s Golden Rule. The total transition probability depends linearly on time, i.e. the transition rate is constant and is proportional to the square module of the coupling between the initial bound state and the dissociative ones that are close in energy to it: dP dt = 2π ~ |V0(ε0)| 2 (2.44)

We must remember that this result has been obtained within the approximation of perturba-tion theory, i.e. by assuming that the initial state only is significantly populated: therefore, in principle the rule would only be valid for sufficiently weak couplings and short times, such that P (t) ≪ 1. Ultrafast decays cannot be treated by perturbation theory and do not obey Fermi’s rule. On the other hand, in most experiments a molecule is actually isolated only in the short time scale: for longer times, collisions and other interactions with the environment will interfere with the single molecule dynamics. In other words, if the decay is slow, there is no time between two collisions to go beyond the range of applicability of perturbation theory. If we consider a sample of N (0) molecules that are in the state Ψ0 at t = 0, the

transition rate will be proportional to the number of molecules still populating the initial state at time t, N (t): dN dt = − dP dtN (t) = − 2π ~ |V0(ε0)| 2 N (t) (2.45)

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Therefore, the population will decay exponentially:

N (t) = N (0) e−t/τ (2.46)

Here τ = ~/2π |V0(ε0)|2 is called the lifetime of state Ψ0. The decay of a bound state to a

dissociative continuum is called “predissociation”.

In the previous discussion about the decay of excited states, we have assumed that the “initial state” is a vibronic product ψk(q; Q)χku(Q), i.e. that the Born-Oppenheimer approximation

somehow keeps its validity in the process of light absorption. After excitation we have a time-evolution simply because ψk(q; Q)χku(Q) is not an exact eigenstate of the molecular

Hamiltonian. Then, the question arises, why the excitation does not put the molecule into an excited eigenstate, i.e. a stationary state. In principle, this can be done, by using monochromatic light with a frequency resolution good enough as to select a single energetic level among the excited state manifold. We now examine the interaction of the molecule with a light pulse, to find out what kind of excited state is actually produced.

The radiation interacts with molecular systems mainly through the electric dipole term: ˆ

V = −~µ · ~E(t) (2.47)

Here ~µ is the molecular dipole and ~E is the electric field of the radiation, assumed approx-imately constant in space over the chromophore length scale. A pulse of linearly polarized light can be represented as

~

E(t) = ~E0A(t) cos(ωt) (2.48)

Here ~E0is a constant vector and A(t) is the “envelope” of the pulse, i.e. a function that dies

away for t → ±∞ and has its maximum value for t = 0, A(0) = 1. In this way, E0 is the

maximum amplitude of the oscillating field.

We insert the dipole perturbation in eq. (2.37), to get dCj dt = i ~ e iωj0t_~_µ j0· ~E0A(t) cos(ωt) (2.49)

Here ~µj0= hΨj|~µ| Ψ0i is the transition dipole from the ground state Ψ0to the excited state

Ψj. We then integrate from before the beginning of the light pulse (t = −∞) to after its

end (t = +∞), taking into account that 2cos(x) = eix_{+ e}−ix_:

Cj(∞) =

i

2~ ~µj0· ~E0 Z +∞

−∞

[ei(ωj0+ω)t_{+ e}i(ωj0−ω)t_{] A(t) dt} _(2.50)

i√π √

2~ ~µj0· ~E0[ ˜A(−ω − ωj0) + ˜A(ω − ωj0)] ˜

A(∆ω) is the Fourier transform of the pulse envelope A(t) and carries information about the frequency composition of the pulse. If the A(t) function is smooth enough, its transform

˜

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exp(−t2_/2τ2_{) yields ˜}_{A(∆ω) = τ exp(−τ}2_∆ω2_{/2): the bandwidth is just the inverse of the}

pulse duration. In general, a short pulse has necessarily a broad spectrum, whereas a narrow bandwidth requires a long pulse. Since we are interested in light absorption with molecular excitation, we shall only retain the contribution ˜A(ω − ωj0) in eq. (2.50), and drop the other

term, ˜A(−ω − ωj0), that is only important for stimulated emission, i.e. when ωj0< 0. So,

the excited state is

|Ψi = _~ir π₂X

i

|Ψii ~µi0· ~E0A(ω − ω˜ i0) (2.51)

We shall treat two limiting cases of great importance. The first shows why the state that is created by optical excitation can be considered a simple Born-Oppenheimer product, rather than an exact eigenstate. In the second case, we shall see how a non-stationary vibrational wave-function can be created in the excited electronic state.

We consider first a vibrational bound state χ1u belonging to the electronic excited state ψ1;

the state ψ1χ1u interacts, for instance because of nonadiabatic couplings, with a continuum

or “quasi-continuum” of vibrational states belonging to a lower lying electronic state ψ0

(not necessarily the ground state). A quasi-continuum of levels is a countable set of levels that are very dense in the energy scale, i.e. with very small energy separations between two subsequent ones: this situation is easily encountered in polyatomics. An exact eigenstate of energy Ei close to E1u is essentially a linear combination of ψ1χ1u with the states ψ0χ0v:

Ψi = aiψ1χ1u+

X

v

bviψ0χ0v (2.52)

Here we assume, for simplicity, that the ψ0χ0v states belong to a quasi-continuum rather

than to a true continuum. The χ0v are functions with many nodes, so the transition dipoles

that connect them with the ground state are very small: hψ0χ00|~µ| ψ0χ0vi ≃ 0. Therefore,

the transition dipole relative to the exact eigenstate Ψionly depends on the ai coefficient:

~

µi0= hΨi|~µ| ψ0χ00i = ai∗hψ1χ1u|~µ| ψ0χ00i (2.53)

The very existence of an absorption band in this frequency range is due to the presence of the ψ1χ1ustate, which is then called the “doorway state”. The aicoefficients depend on the

Ei− E1u energy differences and on the coupling matrix elements. In fact, the Schr¨odinger

equation for Ψiis

ˆ

H |Ψii = Ei|Ψii (2.54)

We pre-multiply by hψ1χ1u| and obtain

ai(E1u− Ei) = −

X

v

bviV1u,0v (2.55)

where V1u,0vmay be a nonadiabatic coupling such as (2.28) or a magnetic coupling. Since the

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spectrum is made of many lines (in practice, a single band) with decreasing intensity as the frequency ωi0 = (Ei−E00)/~ gets further from the central frequency ωmax= (E1u−E00)/~.

If we assume that the coupling V1u,0v changes smoothly enough as a function of the index

v, the absorption intensity will decrease approximately as (E1u− Ei)−2= ~2(ωi0− ωmax)−2

and the bandwidth will be proportional to |V1u,0v|2.

Making use of eqs. (2.51) and (2.53) we obtain, for the excited state:

|Ψi = _~ir π₂ ~µ1u,00· ~E0 X i |Ψii a∗i A(ω − ω˜ i0) = (2.56) i ~ r π 2 ~µ1u,00· ~E0 X i a∗i " ai|ψ1χ1ui + X v bvi|ψ0χ0vi # ˜ A(ω − ωi0)

If the light is highly monochromatic (long pulse), i.e. ˜A(∆ω) has a narrow peak for ∆ω = 0, the excited state will coincide with an eigenstate Ψi. This can only be done, in practice,

when the density of states is low enough (small molecules below the dissociation threshold). More often, the bandwidth of the exciting light is much larger than that of the 00 → iv transition. With ω ≃ ωmax, we can then consider ˜A(∆ω) as approximately constant in

the range of frequencies for which the transition dipoles ~µi0 and the ai coefficients are not

negligible. In this case, eq. (2.56) simplifies to

|Ψi ≃ _~ir π₂ ~µ1u,00· ~E0A(ω − ω˜ max)

" |ψ1χ1ui X i |ai|2+ X v |ψ0χ0vi X i a∗ibvi # (2.57)

Now, because of the unitarity of the matrix of the ai and bvi coefficients (eigenvectors of

the molecular Hamiltonian), we have P

i|ai|2 = 1 and Pia∗ibvi = 0. It follows that the

excited state, created by a pulse short enough, is a combination of eigenstates Ψi such

as to “reconstruct” the doorway state ψ1χ1u, without any contribution from the “dipole

forbidden” states ψ0χ0v:

|Ψi ≃ _~ir π

2 ~µ1u,00· ~E0A(ω − ω˜ max) |ψ1χ1ui (2.58) The time evolution that follows, when the light pulse has ended, can be described as the decay of the non-stationary state ψ1χ1u, “caused” by its coupling with the ψ0χ0vstates. As

we have seen, the linewidth of the transition is proportional to the square of the coupling. Remembering Fermi’s Golden Rule, we can conclude that the lifetime and the linewidth are inversely proportional to each other. These considerations rigorously apply to coherent pulses of the shape (2.48), as emitted by laser sources. The light emitted by common lamps, or the sunlight, can be described as a superposition of short pulses, with randomly variable phases and intensities. However, if we consider separately the action of each pulse, we can extend the conclusion contained in eq. (2.58) to the non monochromatic light produced by most common sources.

We now consider the case of even shorter pulses, with a larger frequency bandwidth, such as to encompass several vibrational levels E1u. On the basis of the above discussion, we

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can neglect the contribution of very excited vibrational states, belonging to lower electronic states, even if they are weakly coupled with the states ψ1χ1u. The excited wave-function

will be a combination of several vibrational states, all belonging to ψ1, i.e. a non-stationary

”wave-packets” that will move on the U1(Q) PES:

|Ψi = _~ir π₂ X

u

˜

A(ω − ω0u) ~µ1u,00· ~E0 |ψ1χ1ui (2.59)

Here ω0u= (E1u− E00)/~ is the resonance frequency for the 00 → 1u transition. If the light

pulse is ultrashort, in the femtosecond scale, we can again assume the radiation bandwidth to be so large that ˜A0(ω −ω0u) is almost independent on the vibrational quantum number u.

Moreover, we shall introduce the Franck-Condon approximation for the transition dipoles: ~

µ1u,00≃ hψ1|~µ| ψ0i hχ1u|χ00i. Then:

|Ψi = _~ir π₂ A(ω − ω˜ vert) hψ1|~µ| ψ0i · ~E0

X

u

|ψ1χ1ui hχ1u|χ00i (2.60)

Here ωvertcorresponds to the so-called “vertical excitation energy”, i.e. the value of U1− U0

at the equilibrium geometry of the ground state. Normally ωvert approximates well the

transition frequency of the strongest vibrational sub-band within the electronic absorption band. The sum in eq. (2.60) is nothing else than the development of the wave-function χ00,

in terms of the set {χ1u}. Therefore, the excited wave-function can be simply written as

|Ψi = _~ir π₂ A(ω − ω˜ vert) hψ1|~µ| ψ0i · ~E0 |ψ1χ00i (2.61)

Thus, the radiation, summing up contributions from many vibrational states belonging to the electronic excited state ψ1with “suitable” coefficients, reconstructs the starting vibrational

state χ00, only changing the electronic wave-function. Of course the wave-packets χ00,

translated into the U1potential, is not a stationary state any more, and will therefore move

in the excited state PES. Its motion, along with the radiationless transitions that may take place, is the domain of investigation of wave-packets dynamics [33] and of the experimental techniques known as “femtochemistry” [34].

We underline that this excitation scheme (translation of a vibrational wave-function, more or less unaltered, from an electronic state to another one) is a consequence of the Franck-Condon principle and can be realized only by ultrashort pulses. Moderately monochromatic light can select one vibrational level in the final electronic state, thus creating a stationary state under the aspect of the nuclear motion. The classical description of the Franck-Condon excitation, i.e. an electronic transition that does not change the nuclear coordinates and momenta, is similar to the creation of a wave-packets by an ultrashort pulse, rather than to the population of a vibrational eigenstate by (almost) monochromatic radiation.

2.3 Avoided crossings.

We have repeatedly stated that the radiationless transition rates decrease with increasing the energy gap between electronic states. Viceversa, we can suppose that the transitions

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2.3. AVOIDED CROSSINGS. 17

between almost degenerate states are particularly fast. This is the reason why, in most cases, the transitions between two excited states are faster than the final decay to the ground state: in fact, usually the ground state is well separated in energy from the rest of the electronic manifold. One important consequence is Kasha’s rule, stating that photochemical reactions and fluorescence emission have as the starting state the lowest singlet, even when the initial excitation has brought the molecule to an upper excited state. Another consequence is that even the lowest excited state may have a very short lifetime, if its PES happens to be very close to the ground state one, at accessible molecular geometries. It is therefore very interesting to understand what happens when two electronic states are (quasi-) degenerate, and to investigate the conditions to have PES degeneracies or PES crossings. The question is: can the energy difference between two electronic states vanish? In other words, can we find molecular geometries such that Uk= Ul?

Suppose we know that two electronic states ψ1 and ψ2, well identified by the properties

of their wave-functions, undergo an energy switch along a certain nuclear coordinate R. If, for any value of R, their energies get very close to each other, i.e. E2− E1 is much

smaller than the energy differences with other states, we can approximate ψ1 and ψ2 as

linear combinations of two arbitrary wave-functions η1 and η2that span the same subspace

but do not undergo considerable changes in their properties as functions of R. Because of the structure of the eigenvalue problem, this approximation tends to be exact as E2− E1

approaches zero and the interval of R to be considered is very small. The adiabatic energies will then be the eigenvalues of a 2x2 Hamiltonian matrix in the {ηi} basis, with Hij =

D ηi Hˆel ηj E : E1,2 = 1 2(H11+ H22) ± 1 2 q (H11− H22)2+ 4H122 (2.62)

(the − sign gives E1 and the + sign, E2). The two eigenvalues only coincide when two

conditions are simultaneously satisfied:

H11(R) − H22(R) = 0

H12= 0 (2.63)

Each of these two equations may have solutions for particular values of R, but in general they cannot be satisfied for the same R: so, the two potential energy curves never touch or cross (“non-crossing rule”). Important exceptions to this rule are all pairs of states with different symmetry and/or spin, since in those cases H12 identically vanishes. If H11

and H22 do cross, and H12 is not too large, we have a typical “avoided crossing”, with a

minimum energy gap E2− E1about equal to 2|H12|. By considering more than one internal

coordinate, as in all molecules with more than two atoms, we see that the two conditions can be simultaneously satisfied, therefore true surface crossings do exist and are called “conical intersections” [riferimenti]. We shall not elaborate on this topic because in this thesis we shall only deal with diatomics or single coordinate models.

In many cases we can define the wave-functions η1 and η2 so that they keep the same

distinctive features for a wide range of molecular geometries: for instance, they can be respectively ionic and covalent, or bonding and antibonding between a certain pair of atoms,

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or represent excitations localized in different parts of the molecule, and so on. The same can be done for more than two states, i.e. a transformation can be defined from the basis of the first n adiabatic states {ψ1. . . ψn} to another basis {η1. . . ηn} of states “almost”

independent on the nuclear coordinates. Such states will be called diabatic or, for reasons that will be clear later, quasi-diabatic [6, 7].

In a typical avoided crossing between two states, we can distinguish three regions along the R coordinate: a strong interaction region around R = Rxwhere H11= H22(indicated as II),

with small energy separation and large nonadiabatic couplings, and two regions (I and III) on the left and on the right of region II, where the two adiabatic energies are well separated. In region I ψ1 ≃ η1 and ψ2≃ η2, while in region III ψ1≃ η2 and ψ2≃ η1. In region II the

adiabatic states are mixtures of η1and η2with non negligible coefficients. Such a situation

was modelled by Landau and Zener [24, 25] in the following way:

H22− H11= F (R − Rx) (2.64)

H12= constant (2.65)

where F is a constant slope difference. According to formula (2.28), this model corresponds to a nonadiabatic coupling of the form:

g12(R) = ψ1 ∂ ∂R ψ2 = D ψ1 ∂ ˆHel ∂R ψ2 E U2− U1 = − H12F F2_{(R − R} x)2+ 4H122 (2.66)

The width of this Lorentzian function (FWHM = 4H12/F ) is a measure of the width of the

strong interaction region and depends on the interaction H12 and on the slope difference

F . The integral R+∞

−∞ g12(R) dR = π/2 is instead independent on the parameters F and

H12. In the Landau-Zener model the dynamics is treated semiclassically, i.e. the nuclear

motion is considered classical and the electronic wave-function evolves in time according to the electronic TDSE, where the Hamiltonian is ˆHel(R). The molecule is supposed to start

a trajectory in region I, far from the crossing point, and to end up also far but on the other side (region III). The velocity v = ˙R is considered constant.

The problem admits a mathematically exact solution, embodied in the Landau-Zener rule. If the molecule is initially in the diabatic state η1, the probability to end up in η2 after a

single passage through the avoided crossing is:

P12(dia)= 1 − e−2πH

2

12/~|vF | _(2.67)

We note that switching from a diabatic state to the other one by going through the crossing, just means to remain in the same adiabatic state, and viceversa. Therefore, in the adiabatic representation the transition probability is:

P12(adia)= e−2πH

2

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2.4. RELATIONSHIP BETWEEN THE ADIABATIC AND THE DIABATIC REPRESENTATION.19

2.4 Relationship between the adiabatic and the diabatic

representation.

The adiabatic states are almost univocally defined. Apart from a phase factor (i.e. the sign of the wave-function in the real domain) and the normalization factor, only in case of degeneracy there is some freedom in the choice of the eigenfunctions of ˆHel. On the contrary,

the diabatic states can be defined in different ways. The fact that the diabatic wave-functions are almost invariant, or very smoothly changing with the nuclear coordinates, means that the dynamic couplingsDηi

∂ ∂Qα ηj E

are small. In the one-dimensional case one can require D ηi ∂ ∂Qα ηj E

= 0 for all i and j within a given subspace of electronic states, for instance the subspace spanned by the first n adiabatic wave-functions. With more than one Qαcoordinate

this is not possible in general, because one would impose more constraints than the degrees of freedom involved in the transformation from the adiabatic to the diabatic basis. It is then common to adopt other operative definitions, whereby the couplingsDηi

∂ ∂Qα ηj E are supposedly small and possibly negligible, but not identically zero. Then, the {η1. . . ηn}

basis is called “quasi-diabatic” [riferimenti].

In this section we supply the formulas for transforming the Hamiltonian and non-adiabatic coupling matrices from an orthonormal basis of (quasi-)diabatic states |ηηηi = {|η1i . . . |ηni},

to one of adiabatic states |ϕϕϕi = {|ϕ1i . . . |ϕni}. These formulas will be used to define the

photodissociation model in section 2.5.

Definitions: |ϕϕ_{ϕi = |ηηηi C} (2.69) H=Dηηη Hˆel ηηη E (2.70) HCk = EkCk (2.71) G= ϕ ϕϕ ∂ ∂Q ϕ ϕϕ GD= ηηη ∂ ∂Q ηηη (2.72) T= ϕ ϕ ϕ ∂2 ∂Q2 ϕ ϕϕ TD= ηηη ∂2 ∂Q2 ηηη (2.73)

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Hellmann-Feynman’s type formulas:

Ct_k∂CL ∂Q = 1 (EL− EK) Ct_k∂H ∂QCL (2.74) ∂EI ∂Q = C t I ∂H ∂QCI (2.75) ∂CL ∂Q = X I CICtI ∂CL ∂Q = X I6=L 1 (EL− EI) CICtI ∂H ∂QCL (2.76) ∂CtK ∂Q = X J ∂CtK ∂Q CJC t J= − X J Ct_K∂CJ ∂Q C t J = = X J6=K 1 (EK− EJ) Ct_K∂H ∂QCJC t J (2.77)

Relationship between the first derivative coupling matrix in the adiabatic representation and in the (quasi-)diabatic one:

GKL= CtK ∂CL ∂Q + C t KGDCL= = CtK ₁ (EL− EK) ∂H ∂Q + GD CL (2.78)

Relationship between the second derivative matrix in the two representations:

TKL= ∂GKL ∂Q − ∂ϕK ∂Q ∂ϕL ∂Q = = CtK ∂2_C L ∂Q2 + 2C t KGD ∂CL ∂Q + C t KTDCL (2.79)

This formula holds both for K = L and for K 6= L, but is derived in two different ways for the two cases.

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For K 6= L the first two terms can be further elaborated: Ct_K∂ 2_C L ∂Q2 = C t K ∂ ∂Q   X I6=L 1 EL− EI CICtI ∂H ∂QCL  = = 2 EL− EK 1 EL− EK Ct_K∂H ∂QCK− C t L ∂H ∂QCL Ct_K∂H ∂QCL+ + X I6=K,6=L 1 EL− EI Ct_K∂H ∂QCIC t I ∂H ∂QCL+ 1 2CK ∂2_H ∂Q2CL   (2.80) and Ct_KGD ∂CL ∂Q = C t KGD X I6=L 1 EL− EI CICtI ∂H ∂QCL (2.81)

Therefore, with K 6= L we get:

TKL= 2 (EL− EK)2 Ct_K∂H ∂QCK− C t L ∂H ∂QCL Ct_K∂H ∂QCL+ + CtKTDCL+ 1 EL− EK Ct_K∂ 2_H ∂Q2CL+ + 2 X I6=K,6=L 1 EL− EI 1 EL− EK Ct_K∂H ∂QCIC t I ∂H ∂QCL+ C t KGDCICtI ∂H ∂QCL (2.82)

For K = L we have instead

Ct_K∂ 2_C K ∂Q2 = − X I6=K 1 (EK− EI)2 Ct_K∂H ∂QCI 2 (2.83) and therefore TKK= CtKTDCK− X I6=K 1 (EK− EI)2 Ct_K∂H ∂QCI 2 + 2X I6=K Ct_KGDCI EK− EI Ct_I∂H ∂QCK (2.84)

The matrix TD can be approximately derived from GD, by invoking the completeness of

the electronic states basis:

TD= ∂GD ∂Q − ∂ηηη ∂Q ∂ηηη ∂Q ≃ ∂GD ∂Q − ∂ηηη ∂Q|ηηη ηηη ∂ηηη ∂Q = ∂GD ∂Q + G 2 D (2.85)

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This formula can replace TD in the above equations. Note that ∂GD/∂Q is an

antisym-metric matrix while G2D is symmetric.

Let us now specialize the previous formulas for a two-state model: |ϕ1i = cos θ |η1i + sin θ |η2i

|ϕ2i = − sin θ |η1i + cos θ |η2i (2.86)

where the transformation matrix from the diabatic to the adiabatic representation is:

C=cos θ − sin θ_{sin θ} _{cos θ}

(2.87)

Adiabatic eigenvalues E1,2 can be obtained from the diabatic matrix elements HIJ by the

equation: E1,2= H11+ H22 2 ∓ " H22− H11 2 2 + H122 #1/2 (2.88)

so that the difference between adiabatic states is: ∆E = E2− E1=p∆H2+ 4H122 , where

∆H = H22− H11is the corresponding diabatic energy difference.

Applying eq. 4.15 for one adiabatic state, we can obtain a mathematical formulation for the mixing angle θ in terms of the diabatic matrix elements:

tgθ = E1− H11 H 12= ∆H 2H12 − ∆H2 4H2 12 + 1 1/2 (2.89) Ct₁C2 ∂Q= − cos 2 θ − sin2θ ∂ θ ∂Q = − ∂θ ∂Q= − 1 (1 + tg2_θ) ∂tgθ ∂Q (2.90)

The GDmatrix only has two non-zero elements, (GD)12, that we shall simply call GD, and

(GD)21= −GD. Then: C1GDC2= GD (2.91) So, eventually: G12= − ∂θ ∂Q+ GD (2.92)

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For the second derivative couplings we have:

Ct₁∂ 2_C 2 ∂Q2 = − ∂2_θ ∂Q2 (2.93) Ct₁GD ∂C2 ∂Q = 0 (2.94) Ct₁TDC2= ∂GD ∂Q (2.95)

So, the off-diagonal term is

T12= −∂ 2_θ ∂Q2 + C t 1TDC2≃ −∂ 2_θ ∂Q2 + ∂GD ∂Q (2.96) with T21= −∂ 2_θ ∂Q2−∂G_∂QD.

For the diagonal term:

Ct₁∂ 2_C 1 ∂Q2 = − ∂θ ∂Q 2 (2.97) Ct₁GD ∂C1 ∂Q = GD ∂θ ∂Q (2.98) Ct₁TDC1≃ −G2D (2.99) T11= T22= − ∂θ ∂Q 2 + Ct1TDC1≃ − ∂θ ∂Q 2 + 2GD ∂θ ∂Q− G 2 D (2.100)

In the model that will be presented in section 2.5 we shall make use of rigorously diabatic states, i.e. GD= 0 and TD= 0. So, the equations we shall need simplify to:

G12= − ∂θ ∂Q (2.101) T12= T21= − ∂2_θ ∂Q2 = ∂G12 ∂Q (2.102) and T11= T22= − ∂θ ∂Q 2 = −G2 12 (2.103)

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2.5 A two-state model of avoided crossing

The main goal of this work being to investigate how a nonadiabatic transition affects the energy distribution of a wavepacket and to determine the best conditions for the experimen-tal observation of such effects, we looked for a suitable molecular model that would allow to simulate the nonadiabatic dynamics. The ideal model would be endowed with the following properties:

1. Dynamical simplicity, for an easy interpretation of both theoretical and experimental results. A situation where a single passage through an avoided crossing occurs is a good example. The photodissociation of a diatomic molecule, producing atomic fragments, would permit an accurate experimental determination of the final energy distributions. In polyatomic systems, on the contrary, it would be hard to measure all the energy components (translational, rotational, vibrational and electronic) with the required accuracy.

2. Parameterization: the model should contain a number of parameters, concerning the potentials, the nonadiabatic coupling, the reduced mass and the light pulse features, in order to explore the dependence of the effect sought for on the system properties and on the experimental conditions.

3. Computational simplicity: unnecessary complications at the computational level should be avoided, in order to run several tests with a moderate computational burden. More-over, for testing purposes, we like to be able to compare calculations run in the diabatic and in the adiabatic representations.

On the basis of these considerations, we set up a two-state one-dimensional model, featuring an avoided crossing that can steer the photodissociation of a diatomic molecule from the excited to the ground state. The programme MODEL, written in the course of this work, computes the diabatic and adiabatic potentials and the coupling matrix elements, to be used in the calculation of stationary vibrational states and of the nonadiabatic wavepacket dynamics (see chapter VI).

Here is a short description of the model. The following matrix elements of the electronic Hamiltonian are expressed in the basis of two diabatic states η1 and η2:

H=H11 H12 H21 H22 (2.104) H11(R) = Dm h 1 − e−α(R−R0)i 2 (2.105)

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2.5. A TWO-STATE MODEL OF AVOIDED CROSSING 25 H22(R) = De+ A e−β(R−R0) (2.106) H12(R) = H21(R) = V0 e−γ(R−Rx) 2 (2.107) where:

• R is the internuclear distance.

• Dmis the well depth for the Morse potential of the η1 diabatic state.

• α is the exponential factor in the Morse potential, and is related to Dm and to the

force constant of the Morse oscillator.

• R0 is the equilibrium distance for the η1 diabatic state.

• De is the asymptotic energy of the diabatic state η2; since we choose De < Dm, De

corresponds to the dissociation energy of the molecule in its adiabatic ground state. • The parameters A and β determine the repulsive potential of the diabatic state η2.

• V0 is the maximum magnitude of the H12 coupling between the two diabatic states,

i.e. the magnitude of H12 at R = Rx.

• γ is the Gaussian exponential parameter that determines the range of distances, cen-tred at Rx, where the coupling is effective (HWHM = pln(2)/γ).

By choosing Rxapproximately coincident with the crossing point of the H11and H22diabatic

potentials, and not too large values of V0 and γ, the H12 coupling is negligible at the

R0 distance. In this case, R0 is also the equilibrium distance for the adiabatic ground

state potential energy curve, and De+ A is the vertical transition energy (see figure 2.1).

The lowest vibrational state in the ground electronic state and the excitation process are practically the same in the diabatic and adiabatic representations. Note however that this is not true with the largest V0 values we tested (0.15-0.20 a.u.).

Taking into account such mathematical settings, the adiabatic curves can be calculated as eq. 4.17. Besides the adiabatic eigenvalues E1,2, the following eigenvector matrix V can be

likewise determined: V=   1 √ 1+tg2_(θ) tg(θ) √ 1+tg2_(θ) −√tg(θ) 1+tg2_(θ) 1 √ 1+tg2_(θ)   (2.108)

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where: tg(θ) = H11−E1

H12 and θ is the mixing angle between adiabatic and diabatic states.

After the use of a specific subroutine of MODEL called EIS, which performs the diagonal-ization of the Hamiltonian matrix, the programme proceeds with the calculation of non-adiabatic coupling elements.

First of all, such calculation needs the first derivatives of the Hamiltonian matrix elements:

∂H ∂R = ∂H11 ∂R ∂H12 ∂R ∂H21 ∂R ∂H22 ∂R (2.109) ∂H11 ∂R = 2αDme −α(R−R0)h_{1 − e}−α(R−R0)i _(2.110) ∂H22 ∂R = −βAe −β(R−R0) _(2.111) ∂H12 ∂R = −2γV0(R − R0)e −γ(R−R0)2 ₌ ∂H21 ∂R (2.112)

then, the non-adiabatic coupling g12is calculated by the following equation:

g12=

V+₁ ∂H_∂RV2

(E2− E1)

(2.113)

Using the g12 terms, it is possible to determine second-derivatives-corrected adiabatic

po-tentials U1,2, if required (see sections 2.1 and 2.4):

U1,2= E1,2+

g2 12

2m (2.114)

In table 5.1 we display the values of the model parameters we have used in most calcula-tions, with the exception of V0 that was varied on a wide interval. Such parameters have

been chosen in order to obtain a crossing between the diabatic curves at about R ≃ 5.95 ˚

A. The slopes of the potential energy curves of both electronic states were set so as, during the dynamics, nuclear wave-packets on the upper state could move through the crossing with a velocity such that they would be efficiently affected by non-adiabatic couplings, and dissociation in both states would take place.

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2.5. A TWO-STATE MODEL OF AVOIDED CROSSING 27

Table 2.1: MODEL parameters and their values (all in a.u.). Parametre Values Dm 0.10 α 0.80 R0 4.00 De 0.06 A 0.06 β 1.80 Rx 5.95 γ 0.30

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28 CHAPTER 2. QUANTUM THEORY OF NONADIABATIC TRANSITIONS g12 x 0.05 E1; E2 H12 H11; H22 Dm A De _R x R0 R, a.u. en er gi es an d n on ad ia b at ic co u p li n g, a. u . 10 9 8 7 6 5 4 3 2 0.15 0.10 0.05 0 0.02 0.01

Figure 2.1: Diabatic H11, H22 and adiabatic E1, E2 curves with both diabatic H12 and

nonadiabatic g12couplings. Standard parameters (see table 5.1 with V0= max[H12] = 0.015

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Chapter 3

A very simple model of

nonadiabatic transitions

I still believe in the possibility of a model of reality, that is to say, of a theory, which represents things themselves and not merely the probability of their occurrence.

Albert Einstein

Let us now propose an extremely simple model with two diabatic states η1 and η2, one

nuclear coordinate Q and coupled by an interaction V .

Wave-packets which move over diabatic potentials are indicated as χ1(Q,t) and χ2(Q,t).

The corresponding normalized wave-function is therefore:

|Ψ(Q, t)i = χ1(Q, t) |η1i + χ2(Q, t) |η2i (3.1)

we shall indicate the matrix elements of the molecular Hamiltonian ˆH in the basis of the electronic states as Hij: H=H11 V V H22 (3.2) with H11(Q) = − 1 2µ ∂2 ∂Q2+ V1(Q) (3.3) 29

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30 CHAPTER 3. A VERY SIMPLE MODEL OF NONADIABATIC TRANSITIONS H22(Q) = − 1 2µ ∂2 ∂Q2 + V2(Q) (3.4) H12(Q) = H21(Q) = V (Q) (3.5)

The dynamics of this systems is governed by the TDSE within the following equations, where we put ~ = 1: i∂ ∂t|Ψ(Q, t)i = ˆH |Ψ(Q, t)i (3.6) i∂ ∂tχ1(Q, t) = H11χ1(Q, t) + V χ2(Q, t) (3.7) i∂ ∂tχ2(Q, t) = H22χ2(Q, t) + V χ1(Q, t) (3.8)

Representing wave-packets by means of their Fourier transformation as:

χi(Q, t) = √1

2π Z +∞

−∞

fi(k, t)eikQdk (3.9)

we can substitute equation (3.7) with

i∂ ∂tχ1(Q, t) = i Z +∞ −∞ ∂ ∂tf1(k, t)e ikQ_{dk = H} 11χ1+ V χ2 = −_2µ1 ∂ 2 ∂Q2 + V1(Q) Z +∞ −∞ f1(k, t)eikQdk + V (Q) Z +∞ −∞ f2(k, t)eikQdk = = Z +∞ −∞ k2 2µ + V1(Q) f1(k, t)eikQdk + Z +∞ −∞ f2(k, t)V (Q)eikQdk (3.10)

At this point we can multiply each member of eq. (3.10) by e−ik′_Q

and integrate over dQ; therefore, the kinetic component and the time-derivative of f , which don’t depend by nuclear coordinate, are multiplied by a Dirac delta-function δ(k − k′_{). Eventually:}

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31 i∂ ∂tf1(k ′_{, t) =} k′2 2µf1(k ′_{, t) +}Z +∞ −∞ Z V1(Q)ei(k−k ′_)Q dQ dk + Z +∞ −∞ f2(k, t) Z V (Q)ei(k−k′_)Q dQ dk = = k ′2 2µf1(k ′_{, t) +}Z +∞ −∞ f1(k, t)W1(k − k′)dk + Z +∞ −∞ f2(k, t)W (k − k′)dk (3.11) with W1(k) = 1 √ 2π Z V1(Q)eikQdQ (3.12) and W (k) = √1 2π Z V (Q)eikQdQ (3.13)

Switching k with k′_{, we finally obtain:}

i∂ ∂tf1(k, t) = k2 2µf1(k, t) + Z +∞ −∞ f1(k′, t)W1(k′− k)dk′+ Z +∞ −∞ f2(k′, t)W (k′− k)dk′ (3.14) and similarly i∂ ∂tf2(k, t) = k2 2µf2(k, t) + Z +∞ −∞ f2(k′, t)W2(k′− k)dk′+ Z +∞ −∞ f1(k′, t)W (k′− k)dk′ (3.15)

then, replacing f = g · e−ik2 t2µ , we get:

i∂ ∂tf1= i ∂ ∂t h g1(k, t)e− ik2 t 2µ i (3.16) i∂ ∂tg1= Z +∞ −∞ g1(k′, t)ei k2 −k′2 2µ tW 1(k′− k)dk′+ Z +∞ −∞ g2(k′, t)ei k2 −k′2 2µ tW (k′− k)dk′

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32 CHAPTER 3. A VERY SIMPLE MODEL OF NONADIABATIC TRANSITIONS (3.17) i∂ ∂tg2= Z +∞ −∞ g2(k′, t)ei k2 −k′2 2µ tW 2(k′− k)dk′+ Z +∞ −∞ g1(k′, t)ei k2 −k′2 2µ tW (k′− k)dk′ (3.18)

Assuming the magnitude of nonadiabatic coupling being small enough to consider the per-turbation theory appropriate for this case, we can expand the following functions, V (Q) being first-order:

gi(k, t) = g(0)i + g (1) i + O(V

2₎ _(3.19)

The zeroth-order terms are then

i∂ ∂tg (0) i (k, t) = Z +∞ −∞ g(0)i (k′, t)Wi(k′− k)e i(k2 −k′2 )t 2µ _dk′ _(3.20)

while for the first order we have

i∂ ∂tg (1) 1 (k, t) = Z +∞ −∞ g(1)1 (k′, t)W1(k′− k)ei k2 −k′2 2µ t_dk′₊ + Z +∞ −∞ g2(0)(k′, t)W (k′− k)e ik2 −k′2 2µ t_dk′ _(3.21) and i∂ ∂tg (1) 2 (k, t) = Z +∞ −∞ g(1)2 (k′, t)W2(k′− k)ei k2 −k′2 2µ t_dk′₊ + Z +∞ −∞ g1(0)(k′, t)W (k′− k)e ik2 −k′2 2µ t_dk′ _(3.22)

The total energy of the system, E =DΨ Hˆ

Ψ

E

, consists of three parts: E = E1+ E2+ E12,

where: E1= hχ1|H11| χ1i = = 1 2π Z +∞ −∞ f∗ 1(k, t)e−ikQ − 1 2µ ∂2 ∂Q2 + V1(Q) f1(k′, t)eik ′_Q dkdk′_{dQ =} = Z +∞ −∞ |f 1(k, t)|2 k2 2µdk + Z Z +∞ −∞ f∗ 1(k, t)f1(k′, t)W1(k′− k)dkdk′= = Z +∞ −∞ |g 1(k, t)|2 k2 2µdk + Z Z +∞ −∞ g∗ 1(k, t)g1(k′, t)W1(k′− k)ei k2 −k′2 2µ tdkdk′ (3.23)

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3.1. CONSTANT POTENTIALS AND GAUSSIAN WAVE-PACKETS. 33 E2= hχ2|H22| χ2i = = Z +∞ −∞ |g 2(k, t)|2 k2 2µdk + Z Z +∞ −∞ g2∗(k, t)g2(k′, t)W2(k′− k)ei k2 −k′2 2µ tdkdk′ (3.24) E12= hχ1|V | χ2i + hχ2|V | χ1i = = 2Re Z Z +∞ −∞ g∗1(k, t)g2(k′, t)W (k′− k)ei k2 −k′2 2µ t_dkdk′ _(3.25)

3.1 Constant potentials and Gaussian wave-packets.

We shall consider a very simple case with two constant diabatic potentials V1(Q) = 0 and

V2(Q) = D. In this way our equations can be simplified as:

i∂ ∂tf1(k, t) = k2 2µf1(k, t) + Z +∞ −∞ f2(k′, t)W (k′− k)dk′ (3.26) and i∂ ∂tf2(k, t) = k2 2µ+ D f2(k, t) + Z +∞ −∞ f1(k′, t)W (k′− k)dk′ (3.27)

Changing f = ge−iωt_{, with ω} 1= k 2 2µ and ω2= k2 2µ + D, we obtain: i∂ ∂tg1(k, t) = Z +∞ −∞ g2(k′, t)W (k′− k)ei(ω1−ω ′ 2)tdk′ (3.28) and i∂ ∂tg2(k, t) = Z +∞ −∞ g1(k′, t)W (k′− k)ei(ω2−ω ′ 1)t_dk′ _(3.29) where ω′

i is meant to contain k′ instead of k.

As in the general case we assume the perturbative approach as correct, so:

i∂ ∂tg (0) 1 (k, t) = ∂ ∂tg (0) 2 (k, t) = 0 (3.30) i∂ ∂tg (1) 1 (k, t) = Z +∞ −∞ g₂(0)(k′, t)W (k′− k)ei(ω1−ω′2)tdk′ (3.31)

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34 CHAPTER 3. A VERY SIMPLE MODEL OF NONADIABATIC TRANSITIONS i∂ ∂tg (1) 2 (k, t) = Z +∞ −∞ g(0)₁ (k′_{, t)W (k}′_{− k)e}i(ω2−ω1′)tdk′ (3.32)

Since the zeroth-order terms for both wave-packets are time-independent we shall write g_i(0)(k, t) = g(0)_i (k). Then: g1(1)(k, t) = Z +∞ −∞ g(0)2 (k′)W (k′− k)dk′ Z t 0 −ie i(ω1−ω2′)t′_dt′ ₌ = Z +∞ −∞ g(0)2 (k′)W (k′− k) " 1 − ei(ω1−ω′2)t (ω1− ω2′) # dk′ (3.33) g₂(1)(k, t) = Z +∞ −∞ g(0)₁ (k′_{)W (k}′_{− k)} " 1 − ei(ω2−ω′1)t (ω2− ω1′) # dk′ _(3.34)

To complete the model, we set the shape of wave-packets at t = 0. Let us suppose that initially η2 is not populated, i.e. χ2(Q, 0) = 0 and g(0)2 (k) = 0. Moreover, from eq. (3.33)

we get g1(1)(k, 0) = 0.

Let us take the initial χ1 as a Gaussian wave-packet that travels with group velocity v or

group momentum k0= µv and “momentum width” ∆k. Then

g1(0)(k) = g0exp−(k − k0)2/2∆2k (3.35) Normalization of χ1 requires g0= (2∆k)− 1 2π− 3 4

We remind that the Q-representation of this wave-packet is

χ(0)1 (Q, t) = π− 1 4∆ 1 2 kγ 1 2expγ(ik₀Q − ik2 0t/2M − ∆2kQ2/2) (3.36) with γ = (1 + i∆2kt/M )−1 (3.37)

The wave-packet spreads according to

∆Q = 1 ∆k (1 + ∆4kt2/M2) 1 2 (3.38)

while the initial distribution of momenta, which is invariant in time, is:

2π|g(0)1 (k)|2= π− 1 2∆−1 k exp −(k − k0) 2 ∆2 k (3.39)

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3.1. CONSTANT POTENTIALS AND GAUSSIAN WAVE-PACKETS. 35

In practice we have a Gaussian wave-packet with a minimum uncertainty product at t = 0. Notice however that the parameters ∆k and ∆Q are

√

2 times the uncertainties (standard deviations) on k and Q.

Considering that we get the same initial energy E = k2_{/2M with two opposite values of k,}

the distribution (3.39) translates into the initial energy distribution

ρi(E) = µ π12∆_kk_i exp −(ki− k0) 2 ∆2 k + exp −(ki+ k0) 2 ∆2 k (3.40)

where ki=√2µE is the absolute value of the initial momentum k = ±ki. In this expression,

the two Gaussian terms in square brackets represent the momentum distribution for positive k (the one with ki− k0) and for negative k (the one with ki+ k0). If k0 ≫ ∆k, only the

former counts, the latter being much smaller. Finally, we assume a simple expression for V (Q):

V (Q) = V0exp(−αQ2) (3.41)

Then

W (k) = V0

2√απexp(−k

2_/4α) _(3.42)

because of the motion of the wave-packet χ1, the interaction vanishes at t = ±∞, so we can

assume g2(1)(k, −∞) = 0 and try to compute the final wave-packet in state 2 at t = +∞.

From eq. (3.32) we get

∂g(1)₂ (k, t) ∂t = −i(8α∆k) −1 2π−54V 0· (3.43) · Z +∞ −∞ exp −(k′− k0) 2 2∆2 k −(k′− k) 2 4α + i k2_{− k}′2 2µ + D t dk′

Now, integrating first over time, we get:

g2(1)(k, +∞) = −i(8α∆k)− 1 2π− 5 4V 0 · (3.44) · Z +∞ −∞ dk′ Z +∞ −∞ dt exp −(k′− k0) 2 2∆2 k −(k′− k) 2 4α + i k2_{− k}′2 2µ + D t = = −iπ−1 4(2α∆ k)− 1 2V 0 Z +∞ −∞ dk′ _exp −(k′− k0) 2 2∆2 k −(k′− k) 2 4α δ k 2_{− k}′2 2µ + D The condition k′2 2µ = k2 2µ+ D (3.45)