Theoretical Studies of Charge Recombination Reactions

Theoretical studies of charge recombination reactions

Siﬁso Musa Nkambule

Akademisk avhandling f¨oravl¨aggandeav licentiatexamen vid Stockholms universitet, Fysikum. January, 2015.

Abstract

This thesis is based on theoretical studies that have been done for low-energy reactions involving small molecular systems. It is mainly focusing on mutual neutralization of oppositely charged ions and dissociative recombination of molecular ions. Both reactions involve highly excited electronic states that are coupled to each other. Employing electronic structure methods, the electronic states relevant to the reactions are computed. It is necessary to go beyond the Born-Oppenheimer approximation and include non-adiabatic effects. This non-adiabatic behaviour plays a crucial role in driving the reactions. Schemes on how to go about and include these coupling elements are discussed in the thesis, which include computations of non-adiabatic and electronic couplings of the resonant states to the ionisation continuum. The nuclear dynamics are studied either semi-classically, using the Landau-Zener method or quantum mechanically, employing the time-independent and time-dependant Schrödinger + equations. Reactions studied here are mutual neutralization in collision of H + H−, where both the semi-classical and quantum mechanical methods are employed. Total and differential cross sections are computed for all hydrogen isotopes. We also perform quantum mechanical + studies of mutual neutralization in the collision of He + H−. For this system not only the non-adiabatic couplings among the neutral states have to be considered, but also the electronic coupling to the ionization continuum. Worth noting in the two systems is that reactants and the final products belong to different electronic states, which a coupled together, hence for the reaction to take place there must be non-adiabatic transition. + Lastly, the direct mechanism of dissociative recombination of H2O is studied. Here time- dependant quantum mechanical methods are employed and the dissociation cross section is computed. Acknowledgements

I am most grateful to my supervisor Asa˚ Larson who has supported and encouraged me in both scientiﬁc and less scientiﬁc issues since I began my doctoral studies. I am also sincerely grateful, for the same reason to Nils Elander. I am also very thankful to Ann Orel for the valuable discussion we have had and her patience when I commit the many errors in handling the data. I wish to thank my friends and colleagues at the Stockholm University, Department of Physics. List of papers

This work is based on the following papers;

I Diﬀerential and total cross sections of mutual neutralization in low-energy + collisions isotopes of H + H−. Siﬁso M. Nkambule, Nils Elander and Asa˚ Larson, in manuscript.

+ II Landau-Zener studies of mutual neutralization in collisions of H + H− and + Be + H−. Hanae Miyano Hedberg, Siﬁso Nkambule and Asa˚ Larson, Journal of Physics B: Atomic, Molecular and Optical Physics, 47 (22), 2014

III Studies of HeH: DR, RIP, VE, DE, PI, MN,.... Asa˚ Larson, Sifiso Nkambule, Ermelie Ertan, Josefin Söderand Ann E. Orel, EPJ web of Conferences accepted for publication, 2015.

+ IV Theoretical study of the mechanism of H2O dissociative recombination. Siﬁso M. Nkambule, Asa˚ Larson, S. Fonseca dos Santos and Ann E. Orel, in manuscript.

Reprints made with permission from publishers.

i Contributions from author

Paper I. I participated in deriving the formulas for calculating the cross section, carried out the nuclear scattering calculations and analysed the results. I also wrote the ﬁrst draft of the article.

Paper II. I calculated the electronic couplings that enter the Landau-Zener formula from previous structure calculations. I also performed the two-by-two adiabatic-diabatic transformation transformation and I also participated in the discussions about the computations of the other set of electronic couplings and on the discussions about the paper.

Papers III. I performed the diabatization of HeH resonant states. I also participated in deriving the formulas relevant for the adiabatic-diabatic states transformation and I carried out the nuclear scattering calculations and analysis of results.

Paper IV. I participated in carrying out the quasidiabatization scheme, extracted some resonance energies and autoinization widths from the eigenphase sum data and performed the one- dimensional wave packet calculations and analysis. I also participated on the discussions about the paper.

ii Contents

1 Introduction 1 1.1 Mutual Neutralization ...... 2 1.2 Dissociative Recombination ...... 3 1.3 Overview of computation scheme ...... 5

2 Computational details 6 2.1 Molecular Schr¨odingerequation ...... 6 2.1.1 Born-Oppenheimer approximation and potential energy surfaces . . . . . 7 2.2 Quantum chemistry calculations ...... 9 2.2.1 The Hatree-Fock method ...... 9 2.2.2 Conﬁguration interaction method ...... 11 2.3 Electronic scattering calculations ...... 12 2.3.1 Basic scattering theory and resonance ...... 12 2.3.2 Complex Kohn variational method ...... 14 2.4 Adiabatic to diabatic transformation ...... 19

3 Nuclear Dynamics 23 3.1 Time-dependant wave propagation ...... 23 3.1.1 1D wave dynamics ...... 23 3.1.2 The MCTDH method ...... 25 3.2 Time-independent nuclear scattering ...... 26 3.3 Semi-classical Landau-Zener studies ...... 27

4 Discussion of attached papers 29 + 4.1 Paper I: Diﬀerential cross section for mutual neutralization of H + H− ..... 29 4.1.1 Scattering of nuclei species ...... 29 + 4.2 Paper II: Landau-Zener studies of mutual neutralization in collisions of H + H− + and Be + H−...... 34 4.3 Studies of HeH: DR, RIP, VE, DE, PI, MN,...... 36 + 4.4 Paper IV: Theoretical study of the mechanism of H2O dissociative recombination 38

5 Conclusion and outlook 42

A Deriving spin multiplicity factors for bosons and fermions i

iii List of Figures

1.1 The Mutual neutralization process ...... 3 1.2 Direct dissociative recombination ...... 4 1.3 Indirect dissociative recombination ...... 5

2.1 Resonance features ...... 14 2.2 Elastic electron scattering ...... 18 2.3 Adiabatic and Diabatic potential energy curves ...... 19 2.4 Quasidibatization of the water molecule ...... 20

3.1 1D Wave Propagation ...... 25

4.1 Differential cross section for all isotopes ...... 31 4.2 Differential cross section for hydrogen at different energies ...... 31 4.3 Total cross section ...... 32 4.4 Landau-Zener cross section ...... 33 4.5 Branching ratios ...... 33 4.6 Radial couplings ...... 34 4.7 Rotational angle ...... 35 4.8 Adiabatic and Diabatic states ...... 35 4.9 Electronic couplings ...... 35 4.10 HeH MN cross section ...... 37 4.11 HeH MN branching ratios ...... 38 4.12 Asymetric streth Widths ...... 39 4.13 Symetric strecth Potential energy curves ...... 40 4.14 Symetric streth Cross sections ...... 40 4.15 1D total cross section ...... 41

iv List of Tables

4.1 Cross section ratios ...... 32

v Chapter 1

Introduction

This thesis is based on a theoretical study of low energy (< 10 eV) molecular collision processes, occurring in low temperature ionized environments, namely; mutual neutralization (MN) and dissociative recombination (DR). Such processes are of significance in many fields of research, such as primordial and pre galactic gas reactions, interstellar gas clouds chemistry, planetary ionosphere chemistry and plasma physics. In the divetor section of a fusion plasma facility, like the ITER [1] project, undesirable rest products that can be harmful to the environment are created and it is important to understand the processes taking place in this region of the plasma [2–4]. Molecular H2 gas is also believed to play a crucial role in the formation of the first stars and galaxies, and as such the chemical evolution of gases at the early epochs are crucial if one needs to understand the competing process in the formation of H2 [5–12]. A strong motivation for this research has been to get a better understanding of how molecular ions involved in these reactions are destroyed and the kind of new fragments that are formed. The theoretical framework for studying microscopic phenomena involving molecules is a quantum mechanical picture, using the Schrödingerequation. The fundamental property of molecules is the large ratio of the mass of nuclei to electron mass, which is about 3680 even + for the lightest molecule of H2 . This property thus implies that the electronic motion and nuclear motions are viewed as uncoupled from each other, or the electrons can instantaneously adjust to the slower motion of the nuclei. This enables a description based on the so-called Born-Oppenheimer approximation [13], which takes advantage of this mass difference between the particles and makes great ease in describing and studying the molecular reaction based on a quantum mechanical description. It enables us to view the system in the adiabatic picture, (to be further described in Chapter 2) and it means the electrons stay in the same quantum state while the nuclei position are changing. In this model, the wavefunction of the molecule is divided into electronic and nuclear components and in solving the Schrödingerequation involving only the latter, at a fixed nuclear positions, the eigenvalues are obtained, and if obtained for a range of internuclear positions, the so-called potential energy surface (potential energy curve in the case of diatomic molecules) can be computed. This model forms the basis of most chemistry models aimed at describing chemical phenomena at microscopic model [14]. However, this approach of the molecular phenomena is applicable, to a good approximation, for processes that involve electronic states well separated in energy. This usually includes reactions occurring in the ground electronic state [15, 16]. On the other hand, reactions involving excited electronic states do not fall in this category. For processes driven by the dynamics on highly excited electronic states, the Born-Oppenheimer approximation can often not be applied. These electronic states can lie close to each other, and as a result, non-adiabatic transitions from one electronic state to another may occur [17]. When studying non-adiabatic processes, a change of electronic state basis is usually advan- tageous [18]. For computational efficiency, it is usually beneficial to transform the adiabatic

1 states (coupled by non-adiabatic couplings) to diabatic states (coupled by electronic couplings). This is further discussed in Section 2.4. This thesis reports on two types of reactions that are characteristically driven by non- adiabatic effects, namely; mutual neutralization (MN) and dissociative recombination(DR). In dissociative recombination, a molecular cation captures an electron forming an excited neutral molecule that dissociates into neutral fragments. The electron capture is here driven by a coupling between a neutral state and the ionisation continuum. Mutual neutralization is a process where oppositely charged molecular or atomic ions collide and charge transfer results in formation of neutral fragments. Here the system starts in one electronic state (ion-pair state) and driven by the non-adiabatic couplings, it ends up in another state (covalent state). Central to both these reactions are the facts that they involve highly excited states and the processes depend on how the various electronic states couple to each other. In these studies, electronic structure calculations are first carried out to obtain the adiabatic potential energy surfaces and the relevant couplings. Above the ground state of the ion, electron scattering calculations [19] are performed to obtain accurate energy positions of resonant states and the couplings to the ionisation continuum. These couplings are related to the lifetimes of the resonant states. Once the potential energy surfaces and couplings are obtained, nuclear dynamics relevant to each of the described processes are studied quantum mechanically using either time-dependent wave packet or time-independent nuclear scattering methods. The goal is to compute cross section and final state distributions. The idea is to determine what fragments are formed or what bonds are broken. The differential cross section, defined as the ratio of the scattered flux in a given solid angle and the incoming flux, can then be computed. The total cross section is obtained by integrating the differential cross section over all solid angles. In the wave packet studies, the cross section is computed by analysing the dissociating flux. When time-independent methods are used the cross section is obtained from the asymptotic from of the radial wave function. Section 1.1 and 1.2 outlines a general description of the reactions. In Chapter 2, brief descriptions of the methods used when calculating potential energy surfaces and couplings are given. The theoretical approaches to model the nuclear dynamics of the processes are described in Chapter 3 and the results of the attached papers I-IV are discussed in Chapter 4.

1.1 Mutual Neutralization

A mutual neutralization process involves two oppositely charged atomic or molecular ions, that collide and exchange an electron to form neutral separated fragments. Schematically, for collisions of atomic ions, the process can be written as

+ ( ) ( ) A + B− A ∗ + B ∗ . (1.1) → Here the asterisk (*) denotes the fact that A and/or B may be electronically excited. Fig. 1.1 illustrates possible pathways in which the reaction can take place. The reaction starts in the ion-pair state, which is a highly excited electronic state. The system moves towards smaller internuclear distances, as illustrated in Fig 1.1. The blue and red arrows shows different pathways for products in different final states. The process is non-adiabatic since to pass from the reactants to the product side, two or more electronic states have to be crossed. Only two covalent states are shown in the figure, but in reality there can be many states lying close to each other. At specific internuclear distances the curves can get close to each other and as illustrated by the arrows, there are a number of different pathways that can lead to the MN process. The potential energy curve of the ion-pair state, having an attractive Coulomb potential, crosses the potentials of low lying covalent states at large internuclear separations. Due to the

2 tt n ti o tbei rdtoa es ic h lcrncnatinz n detach autoinization and state. an resonant autoionize by the can given of electron is lifetime detachment the the electron since to the proportional sense inversely for traditional resonant is probability a a which The the called width, in has to to is stable molecule. which attach corresponds formed not the molecule, time-scale temporarily state from is collision intermediate can a it electron neutral electron cause and the The the state can if However, time-scale. electron dissociation most vibrational the The energy dissociate. molecule’s drive low to fragments. and comparison, a separated molecule in to the how mass dissociates is huge then ask energy a that low to a molecule captures question neutral cation molecular fascinating a a form which in to process electron a is (DR) recombination Dissociative Recombination Dissociative 1.2 and 33] 32, [27, experimentally both studied previously 35]. been and [34, has accelerated theoretically it then and meV) are system merged. beams few small ion are relatively a two rings to The two (down K). the few energies where a interact center-of-mass to to controlled (down allowed rings well temperatures are storage and charged low ion low electrostatic very oppositely facility, doubly at very the this and at in In separately kept stored 31]. are be [30, which will reactions is ions Stockholm molecular molecular in and energy commissioning atomic Moseley low under by studying currently one is the at is that aimed collision eV DESIREE, higher 3 named facility, at below beam result mainly experimental merged done, only been The have studies a have [27–29]. approaches using experimental energies theoretical [21] also refined Lewis and more and [24–26] then Bates employed Since by been out model. carried 23] is was [22, study section Landau-Zener theoretical semi-classical cross first The the involved. energy, electrons collision of low At ( cross . energy the collision large [20]. the energies, law be to collision threshold to proportional inversely low expected be at is to found ions, reaction charged the for oppositely section the between arrows attraction blue and Coulomb red the by A( shown states as quantum neutralization different to in lead products can forming that pathways possible five are 1.1: Figure hr r w ieetmcaim nwihteD rcs a aepae ietand direct a place; take can process DR the which in mechanisms different two are There He involving reaction MN the is here studied reaction, Another H of reaction The uulnurlzto rcs nteaibtcpcue ntepeetsse there system present the In picture. adiabatic the in process neutralization Mutual

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. + B - + H + − hsi loa also is This . tal. et 2] The [29]. an indirect mechanism. In the direct mechanism, the cation AB+ captures the electron to a resonant state AB∗∗, as illustrated in Eq. (1.2) for a diatomic molecule;

+ AB + e− AB∗∗ A + B. (1.2) → → The resonant state has a repulsive potential energy curve and the system dissociates into neutral fragments. The relevant potential energy curves are schematically shown for a diatomic molecule in Fig. 1.2.

+ AB Energy

AB** Internuclear distance Figure 1.2: Schematic illustration of the direct dissociative recombination process. The ion + AB captures an electron to a doubly excited neutral state, AB∗∗, followed by dissociation.

Before the potentials of the resonant state and the ion cross, there is a possibility of re- emitting the captured electron, a process known as autoionization. Once the ion potential has been crossed, autoionization is no longer possible and the resonant state continues to dissociate into neutral fragments, A and B. These fragments might be excited. Due to the Coulomb attraction between the molecular cation and the electron, the low energy DR cross section follows the same Wigner’s threshold law [20] as in low energy MN reactions. In the indirect mechanism, the ion captures an electron to a ro-vibrationally excited Rydberg state, AB∗, which then couples to a resonant state and the system dissociate into fragments as described by Eq. (1.3) + AB + e− AB∗ AB∗∗ A + B. (1.3) → → → Fig 1.3 shows the pathway for the indirect DR mechanism.

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3]aentdie ycpueit eoatsae Instead state. resonant a into capture by driven not are [37] * + ** Internuclear distance oepeieeeg oiin ftersnn ttsrel- states resonant the of positions energy precise More : h lcrnceeg uvs(ufcs r bandus- obtained are (surfaces) curves energy electronic The : 5 AB ∗ hc hnculst doubly a to couples then which , Chapter 2

Computational details

This chapter discusses the theoretical background that underlines the processes studied in papers I-IV. The non-relativistic time-independent Schr¨odingerequation for an isolated molecular system is described. Throughout the thesis atomic units are used, i.e., ℏ = ke = me = e = 1, h where ℏ = 2π , h is the Planck constant, ke is the Coulomb constant, while me and e are the electron mass and the elementary charge, respectively.

2.1 Molecular Schr¨odingerequation

The non-relativistic Hamiltonian, Hmol, that describes an isolated molecular system is of the form; Hmol = Tn + Te + Vnn + Vne + Vee (2.1) where Tn is the kinetic energy of all N nuclei (speciﬁed by a vector R), each of mass Mi (i=1, 2 ,..., N); N 1 T = 2 , (2.2) n − 2M ∇Ri i=1 i ∑ and Te is the kinetic energy of all n electrons (speciﬁed by a vector rj); n 1 T = 2 . (2.3) e − 2∇rj j=1 ∑ Vnn is the nuclear repulsion term between two nuclei of charge Zi and Zj respectively, expressed as N N Z Z V = i j , (2.4) nn R R i=1 j>i i j ∑ ∑ | − | while Ven, represents the interaction between electrons and nuclei and is of the form

n N Z V = j . (2.5) en − r R i=1 j=1 i j ∑ ∑ | − | The electron-electron repulsion energy, Vee, is n n 1 V = . (2.6) ee r r i=1 j>i i j ∑ ∑ | − | The complete many body Schr¨odingerequation has the form

HmolΨ(R, r) = EΨ(R, r). (2.7)

6 Here E is the total energy, R is the set of nuclear coordinates and r is the set of electronic coordinates.

2.1.1 Born-Oppenheimer approximation and potential energy surfaces In solving Eq. (2.7) for a molecular system, it is necessary to make approximations. One of the most common approximation is the Born-Oppenheimer approximation [13], which separates the electronic and nuclear motions. The approximation is based on the notable mass diﬀerences between electrons and even the lightest nuclei, which in turn implies that the electrons move faster relative to the nuclei. The non-relativistic Hamiltonian can be written as

Hmol = Tn + Hel. (2.8)

Here Hel is the electronic Hamiltonian. In the Born-Oppenheimer approximation, the total wavefunction separated into a nuclei term and electronic term. This leads us to approximate the total molecular wavefunction Ψ(R, r) as a product of the electronic and nuclear wavefunctions, Ψ(R, r) = ϕ(r; R)χN (R). (2.9) The electronic Schr¨odingerequation can be expressed in the form

a a a Helϕi (r; R) = ϵi (R)ϕi (r; R) (2.10)

a By solving Eq. (2.10), the electronic energies, ϵi (R), which depend parametrically on the a nuclear separation are obtained. The electronic wave function, ϕi (r; R) now depends parametrically on the nuclear coordinates. If the electronic energies are obtained for various R, a potential energy surface (or a potential energy curve in a case of diatomic molecule) is obtained. These potential energy surfaces (curves) are called adiabatic, hence the subscript “a”, since it is assumed that electrons respond instantly to the nuclear motion. For diatomic systems, adiabatic potential energy curves of electronic states with the same symmetry are not allowed to cross [39]. Once the electronic problem has been solved, and a complete set of orthonormal electronic a states, ϕi , has been obtained for all possible nuclei conﬁgurations, the full molecular wave function, Ψ(R, r), can be expanded as

∞ a a Ψ(R, r) = χi (R)ϕi (r; R). (2.11) i=1 ∑ If we insert Eq. (2.11) in Eq. (2.7), we obtain

∞ ∞ a a a a (Tn + Hel) χi (R)ϕi (r; R) = E χi (R)ϕi (r; R). (2.12) i=1 i=1 ∑ ∑

For simplicity, we now consider a diatomic case of masses M1 and M2, with reduced mass µ = M1M2 1 2 , the nuclear vector R becomes R for the nuclei degree of freedom, and TN = 2µ R, M1 + M2 ∇ with the centre of mass translation and rotational motions transformed away [40]. If Eq. (2.12) a is multiplied on the left by ϕj ∗(r,R) , then integrate over the electronic co-ordinates and use the othornormality of the electronic states, a set of coupled time- independent diﬀerential equations for the nuclear motion is obtained;

7 1 ∂2 ∞ 1 ∂ 1 χa(R) + [ϵa(R) E] χa(R) F a + Ga χa(R) = 0, (2.13) −2µ ∂R2 i i − i − µ ij · ∂R 2µ ij j j=1 ∑ [ ] a where Fij is a matrix that contains the first order non-adiabatic coupling elements, defined as ∂ F a (R) = ϕa ϕa , (2.14) ij ⟨ i |∂R| j ⟩ a and Gij is the matrix with second order non-adiabatic coupling elements, defined as

∂2 Ga (R) = ϕa ϕa . (2.15) ij ⟨ i |∂R2 | j ⟩ a a Fij contains only oﬀ-diagonal elements, while Gij also has diagonal terms, which are referred a to as adiabatic corrections. If the electronic wave functions are real, Fij is an anti-symmetric matrix. Eq. (2.13) can be compactly written in matrix form as

1 ∂ 1 ∂ χa + [ϵa E] χa 2Fa + Ga χa = 0, (2.16) −2µ ∂R − − 2µ ∂R [ ] a a where all the χi are collected in column vector χ . a In practice, the off-diagonal contributions from Gij are smaller than the contribution from a Fij and hence they are usually neglected [41]. Thus, we focus our attention on the first order elements. Starting from Eq. (2.10), it is possible to derive from the off-diagonal Hellmann- Feynman theorem [42], the following expression

∂H ∂ ϕ el ϕ = (ϵ ϵ ) ϕ ϕ . (2.17) ⟨ i| ∂R | j⟩ j − i ⟨ i|∂R| j⟩ for the non-adiabatic coupling element

a ∂Hel a ϕi ϕj F a (R) = ⟨ | ∂R | ⟩. (2.18) ij − ϵa ϵa j − i Unless it vanishes by symmetry, the right hand side numerator of Eq. (2.18) is finite whilst the denominator will depend on the energy separation of the i-th and the j -th adiabatic electronic a states. It is thus evident that Fij is very significant in the regions where those electronic states lie very close in energy. Neumann and Wigner [43] proved that for diatomic molecules, the potential energy curves corresponding to the same electronic symmetry will not cross, and the adiabatic potential energy curves will have avoided crossings when the coupling elements are large. Polyatomic molecules do not posses the non-crossing rule. Instead potential energy surfaces may become degenerate and at the point (or seam) of degeneracy a conical intersection exist [44]. The non-adiabatic coupling between these states may then become infinite. Non-adiabatic effects are driving processes such as dissociative recombination, mutual neutralization, and photochemical reactions, therefore it is imperative to go beyond the Born- Oppenheimer approximation when studying such reactions, which is further discussed in Sec- tion 2.4. In order to obtain the potential energy surfaces (curves), electronic structure calculations have been carried out and brief descriptions of the various methods are given below.

8 2.2 Quantum chemistry calculations

In carrying out the electronic structure calculations, quantum chemistry techniques have been performed and in this section they will be briefly described. For a system with few electrons, like H2 or HeH, the Full Configuration Interaction (FCI) method has been used. The Multireference Configuration Interaction (MRCI) method has been used to perform calculations for systems + with a larger number of electrons, like in the study of DR of H2O .

2.2.1 The Hatree-Fock method The standard approach for describing electrons in a molecule is by using a set of one particle functions. These one particle functions, called molecular orbitals, are constructed in such a way that they take care of the electronic spatial and spin distribution (so-called spin orbitals). The total wave function of a molecule is expressed as a product of the spin orbitals. For electrons, which are fermions, the total wave function must be antisymmetric with respect to the interchange of two electronic coordinates, as a consequence of the Pauli exclusion principle. This is achieved by writing the electronic wave function in a form of a Slater determinants,

ζ1(1) ζ2(1) ζn(1) ζ (2) ζ (2) ··· ζ (2) 1 1 2 n ϕSD = . . ···. . , (2.19) √n! . . .. .

ζ1(n) ζ2(n) ζn(n) ··· where ζ (j) denotes the molecular spin orbital i for electron j in an n electron system. The i ζi functions are constructed in such a way that they are orthonormal to one another. This trial wave function will now consist of the single Slater determinant. By making such an approximation, where the electron-electron repulsion is only added as an average effect, the implications are that electron correlation is not well taken care of. If we now define a one-electron operator, hî, as the sum of the kinetic energy and the electron-nuclei interaction of electron i;

1 N Z hˆ = 2 j , (2.20) i −2∇ri − r R j=1 i j ∑ | − | and a two-electron operator,g ˆij, for the electron-electron repulsion as 1 ˆg = . (2.21) ij r r | i − j| Thus the electronic Hamiltonian can now be simply expressed as n n n

Hel = hî + ˆgij + Vnn. (2.22) i i=1 j>i ∑ ∑ ∑ The energy of this Slate-type wavefunction is given by [45] n n n E = ϕ H ϕ = h + (ȷ κ ) + V , (2.23) SD ⟨ SD| el| SD⟩ i ij − ij nn i=1 i=1 j>i ∑ ∑ ∑ where h = ζ (1) hˆ ζ (1) , (2.24) i ⟨ i | 1| i ⟩ ȷ = ζ (1)ζ (2) ˆg ζ (1)ζ (2) , ij ⟨ i i | 12| j j ⟩ κ = ζ (1)ζ (2) ˆg ζ (1)ζ (2) . ij ⟨ i i | 12| j i ⟩ 9 The term hi is called a one electron integral and it represents the energy of an electron occupying orbital i. The terms ȷij and κij are the two-electron integrals representing repulsion energy between two electrons (coulomb integral) and the exchange term, respectively. In the Hatree-Fock (HF) method [45, 46] (usually referred to as self-consistent field method), the molecular orbitals are determined by applying the variational principle, which minimizes the energy. If we define a coulomb (ˆȷi) and an exchange (ˆκi) operator, defined by their action on a spin orbital, such that

ȷˆ ζ (2) = ζ (1) ˆg ζ [1) ζ (2) , (2.25) i| i ⟩ ⟨ i | 12| i ⟩| i ⟩ κˆ ζ (2) = ζ (1) ˆg ζ [1) ζ (2) , i| i ⟩ ⟨ i | 12| j ⟩| i ⟩ The energy can now be expresses as

n n n E = ζ hˆ ζ + ( ζ ȷˆ ζ ζ κˆ ζ ) + V . (2.26) SD ⟨ i| i| i⟩ ⟨ j| i| j⟩ − ⟨ j| i| j⟩ nn i=1 i=1 j>i ∑ ∑ ∑

We now have deﬁne a one electron operator, Fˆi, that describes the electron kinetic energy, attraction to all nuclei and electron repulsion to all other electron as the Fock operator,

n Fˆ = hˆ + (ˆȷ κˆ ) . (2.27) i i j − j ∑[j=1 The Fock operator can be inserted in Eq. (2.26) and by applying the Lagrange energy minimi- sation technique [45], we arrive at the Hatree-Fock equation;

Fˆiζi = ϵiζi, (2.28) where ϵi has the interpretation of orbital energy. Eq. (2.28) is a pseudo-eigenvalue equation, since the Fock operator depends on all occupied molecular orbitals, as can be clearly seen from the dependence of the operator on the exchange and Coulomb operators. In practice, Eq. (2.28) is solved by expanding each of the molecular orbitals in some prede- ﬁned M basis set of atomic orbitals,

ζi = Cıiξi (2.29) ı=1 ∑ where Cı are the expansion coeﬃcients. This expansion can now be inserted into the Hatree- Fock equation and after multiplying on the left by ξν and integrating it results to the Roothaan- Hall equations [45] for a closed shell system,

M M

FνıCıi = ϵi SνıCıi, (2.30) ı=1 ı=1 ∑ ∑ where

S = ξ ξ , and (2.31) νı ⟨ ν| ı⟩ F = ξ Fˆ ξ . νı ⟨ ν| i| ı⟩ Eq. (2.70) can be written compactly in matrix form as

FC = SCϵ. (2.32)

10 Eq.(2.32) is an eigenvalue equation, except for the overlap matrix S. By requiring that the basis is orthogonal, S vanishes. Since F depends on C, which needs to be optimised, through the orbitals, the process of solving Eq.(2.32) is done iteratively. This is done by optimizing the coefficients and the Fock matrix diagonalized, until some desired convergence is reached with respect to set of coefficients used in constructing the Fock matrix and those resulting from the diagonalization, the so-called self-consistent field (SCF) method. With the HF method the ground state solution and molecular orbitals are obtained. How- ever, electron correlation effects are not accurately included and hence there is a need to in- corporate correlation effects, especially when performing a calculation that involves excited states.

2.2.2 Configuration interaction method The configuration interaction (CI) method [45] is one of the most straight forward method for including electron correlation effects. It is built as an improvement of the Hatree-Fock solution. In the CI equation, the electronic wave function is constructed as

ϕCI = a0ϕSD + aiϕi, (2.33) i ∑ where ϕSD is the initial single-determinant Hatree-Fock wave function, the ai’s are the expansion coeﬃcients, and the ϕi’s are the Slater determinants that are obtained by distributing the electrons between the occupied and unoccupied (virtual) spin orbitals. Applying the variational principle to Eq. (2.33) leads to the secular equation for the CI coeﬃcients,

Ha = Ea (2.34) where the diagonal matrix E contains the CI energy, a contains the CI coefficients ai and H is the Hamiltonian matrix H = ϕ H ϕ . (2.35) ij ⟨ i| el| j⟩ Eq. (2.34) has as many solutions as there are configurations in the CI coefficients, with the lowest energy solution corresponding to the ground state, the second lowest to the first excited state, etc. The size of the CI expansion of Eq. (2.33) scales by the number of orbitals included in the calculation as well as electron excitations that are allowed outside the HF wave function. In CI singles (CIS), CI singles and doubles (CISD), only single excitation and single and double excitations, respectively, are allowed, and so on. If all possible excitations outside the HF wave function are performed, the approach is called full CI (FCI). The FCI method scales exponentially with the number of electrons and orbitals and hence it is computationally feasible only for systems with few electrons [45]. In larger electronic systems there is a need to truncate the FCI expansion. The most common method is the multi-reference CI (MRCI) method [45]. Terms that have an insignificant contribution to the total energy of the system are truncated, while recovering a significant amount of the electron correlation. In truncating the configuration space, it is important to distinguish between static correlation, which is associated with electrons avoiding each other on permanent a basis, like those occupying different spatial orbitals, and dynamic correlation which is the “instant” correlation between electrons, like those occupying the same spatial orbital. By retaining the dominant configurations of the FCI expansion, static correlation is treated and these configurations are referred to as the reference configuration of the CI wave function. By including all configurations in the reference space as well as those generated by excitations out of the reference space, the MRCI wave function is created.

11 A typical approach in reducing the computational time for the CI calculation to achieve faster convergence is by using orbitals that diagonalize the density matrix of the orbitals used in the CI calculation. The corresponding eigenvectors of the diagonal matrix are called natural orbitals (NOs). An MRCI calculation that is carried out with such orbitals is compact and suitable to be implemented in an electron scattering calculation.

2.3 Electronic scattering calculations

The ﬁxed-nuclei electron scattering calculations were performed using an algebraic variational method known as the Complex Kohn variational method [47]. The basic concepts, in the case of one particle elastic scattering, are illustrated in Section 2.3.2, and then application to a more realistic case, like an electron-molecular ion scattering which is studied in this work, is also brieﬂy discussed. First we discuss the basic scattering process by a spherically symmetric short range potential and how parameters of resonances are determined from the scattering data. + These parameters are essential in reactions studied here, like the MN reaction of He + H− in + paper III and the DR of H2O in paper IV.

2.3.1 Basic scattering theory and resonance Reactions, similar to the ones studied in this thesis can be classiﬁed as scattering processes, and as such, we can analyse the initial system (reactants), of mass which is related to the asymptotic wave function Ψin and the ﬁnal system (products), which is related to the asymptotic wave function Ψout . The outgoing system is related to the incoming system by the S-operator [48],

Ψout = SˆΨin. (2.36)

Theoretically, the S-operator contains all information about the reaction, hence the expectation value of Sˆ can help compute observables of the reaction. Sˆ can also be expressed in term of the transition operator, Tˆ, Sˆ = 1ˆ 2πiT,ˆ (2.37) − whose matrix elements are related to the scattering amplitude, f(k, k′) for an incoming state with momentum k and scattered state with momentum k′ [48],

2 f(k, k′) = (2π) µ k′ Tˆ k , (2.38) − ⟨ | | ⟩ where mu is the mass of the system. The first term in Eq. (2.37) is for the case when there in no scattering takes place and the second term constitutes the transition operator, Tˆ. The mathematical basis for defining the scattering process is discussed for the case of a simple potential V(q), which is spherically symmetric, hence the scattering in each partial wave, l, can be considered separately. Here q is the separation distance between incoming particle and the target. We additionally assume the potential to be short range (i.e. qV (q) 0 1 → as q ). For such a potential, the Hamiltonian operator, Hˆ = 2 + V (q), will commute → ∞ 2µ∇ 2 with the operator L and Lz [49]. Thus the Schrödingerequation (2.7) is separable in spherical coordinates. The symmetry about the incident direction (taken along z-axis) makes the total wavefunction, Ψ(q, θ), to be independent of the azimuthal angle φ. For a short range potential, the solutions are sought from the Schrödingerequation, subject to the bondary conditions:

ikq ikz e Ψ(q, θ)q e + f(E, θ) , (2.39) →∞ ∼ q

12 where f(E, θ) = f(k, k′) (achieved by choosing the momentum k along z-axis) is the scattering amplitude and the dependence on the azimuthal angle, φ, has been omitted due the cylindrical symmetry of the potential. The wave function can be expanded as,

1 ∞ Ψ(q, θ) = A ψ (q)P (cos θ), (2.40) q ℓ ℓ ℓ ∑ℓ=0

where the constant Aℓ is the expansion coefficient. Here Pℓ(cos θ) are the well known Legen- dre polynomials [50] and ψℓ(q) is the radial wave function which satisfies the radial Schrödinger equation [51], 1 d2 ℓ(ℓ + 1) + ψ (q) + V (q)ψ (q) = Eψ (q). (2.41) 2µ dq2 2µq2 ℓ ℓ ℓ [ ] A power series expansion of ψℓ(q) will show that there are two kinds of solutions, one that ℓ+1 is regular at the origin, and behaves like ψℓ(q) q , as q 0 and an irregular solution that ℓ → → behaves like ψℓ(q) q− , as q 0 [49]. Subject to the boundary condition ψℓ(0) = 0, the solution to Eq. 2.41→ has the asymptotic→ form [26, 52],

ψ (q) α (q) S (E)β (q) (2.42) ℓ → ℓ − ℓ ℓ

where αℓ(q) and βℓ(q) are solutions of the incoming and outgoing waves, respectively. Sℓ(E) is the partial wave scattering matrix (S-matrix). It can be interpreted as the eigenvalue to the S-operator [48]. Since Sˆ is unitary, each of the eigenvalues will have modulus one and hence can be written as an exponent of an imaginary number,

2iηℓ(E) Sℓ(E) = e , (2.43) where ηℓ(E) is the phase shift [52]. For a totally elastic collision, where incident and outgoing electron has a wave number k, the scattering matrix is one-by-one and the cross section is

∞ σ(E) = σℓ(E), (2.44) ∑ℓ=0 where π σ (E) = (2ℓ + 1) 1 S (E) 2. (2.45) ℓ k2 | − ℓ | In the electron scattering process, the electron can be temporarily captured by the molecular ion into a short-lived (resonant) exited neutral state. This happens at certain energies, known as resonant energies, Eres. The energy can be expressed as a complex eigenvalue [52], iΓ E = E , (2.46) res − 2 where Γ is the autoionization width and by analysing the time evolution factor for the wave function at the energy given by Eq. 2.46 it can be shown that it is related to the lifetime, τ, of a resonant state by 1 τ = . (2.47) Γ The existence of a quantum resonant state (often referred to as a quasi-bound state) is often depicted by observing distortion of the continuum due to the interaction with an otherwise bound state. If the wave function for the bound state and the continuum state belong to the same internal state of the system, it is referred to as a shape resonance, while if the two states

13 asstesapvraini h lsi rs eto.Tu h hs hf a eexpressed be can shift phase the Thus section. cross form elastic Breit-Wigner the the in in variation sharp the causes where ( term The operator the where Schrödinger parti- radial spinless wave a partial of as The scattering written of V(r). be case potential can a symmetric for equation spherically described short-range, easily a are by method cle the of method concepts variational basic The Kohn Complex 2.3.2 noticed. be can an [53]) for line-shapes sum Beutler-Fano H (so-called eigen-phase reaction profiles of scattering of value section elastic local cross shows elastic the which (H on (a), cation depends water 2.2 thus the Fig. with in colliding illustrated electron is This Γ. by increase will shift at energy, phase the that implies This states. two [52] driving the as the of resonance coupling resonance Feshbach shape the a a by for In driven a while is barrier, 2.1(b). called potential breakup is the the and through it 2.1(a) tunnelling Figs. quantum system in is the mechanism illustrated are of resonances states of internal types different to belong Z iue2.1: Figure Energy )tres onfunctional Kohn A targets. 0) ̸= ae ntecmlxntr fteeeg ftesse,teSmti suulyexpressed usually is S-matrix the system, the of energy the of nature complex the on Based η ℓ (0) E Z/r stebcgon hs hf.A at At shift. phase background the is = ceai ersnaino a hp eoac n b ehahresonance. Feshbach (b) and resonance Shape (a) a of representation Schematic E sicue ognrlz otesatrn yete eta ( neutral either by scattering the to generalize to included is res n o atte“up cusi eemndb h uonzto width, autoinization the by determined is occurs “jump” the fast how and , L ˆ sdfie as defined is q L ˆ η ℓ = ( S E ℓ − ( = ) E 1 2 2 = ) O dr d η η + I 2 ℓ (0) ℓ 2 (0) sdt iiietevrainlprmtr,i defined is parameters, variational the minimize to used , { e + + ( ( E E E E Lφ l − ˆ tan + ) ( ,wt hr aito tarsnneeeg.The energy. resonance a at variation sharp with ), − − l 2 2 ssoni i.22() hr ieetresonance different where (b), 2.2 Fig. in shown is 1) + (a) l r O E E ( 2 14 r + res res 0 = ) E e + π − + − + 1 = hneeg spsigtruharesonance a through passing is energy when [ −

Z Energy i i , r 2 2 E Γ Γ 2( .Tesaeo h lsi rs section cross elastic the of shape The ). } res + E e res V h -arxwl aeapl,which pole, a have will S-matrix the , 2 Γ iη ( r − ℓ (0) ) ( E − E ) ) Feshbach , k ] 2 2 . . eoac.Tetwo The resonance. q Z )o ionic or 0) = (2.48) (2.50) (2.51) (2.49) (b) as ∞ I[φ (r) = φ (r) Lˆ φ (r) = φ (r)Lφˆ (r)dr. (2.52) l ⟨ l | | l ⟩ l l ∫0 This functional is deﬁned such that I[φl(r)] = 0 if φl(r) represents the exact solution. The boundary conditions built into the basis functions, φl(r) are

φl(0) = 0 (2.53) φ (r ) F (kr) + λG (kr), (2.54) l → ∞ ∼ l l

Where Fl and Gl are two linearly independent solutions of Eq. (2.50), for the case when V (r) = 0 and λ is a variational linear parameter. If, in Eq. (2.51), Z = 0, Fl(r) and Gl(r) are the incoming ̸ t and outgoing Coulomb functions, respectively. If a trial wave function, φl(r) = φl(r) is used instead, this functional will diﬀer from zero. We deﬁne the residual between̸ the exact wave function and the trial wave function as

δφ (r) = φ (r) φt(r), (2.55) l l − l and the following boundary conditions are assumed;

δφl(0) = 0 (2.56) δφ (r ) = δλG (kr). (2.57) l → ∞ l δλ = λ λt, and λt is the variational parameter. Inserting the trial wave function into Eq. (2.52) yields −

t ∞ ˆ I[φl(r)] = I[φl(r) + δφl(r)] = φl(r)Lδφl(r)dr (2.58) ∫0 ∞ ∞ ∞ + δφl(r)Lδφˆ l(r)dr + φl(r)Lφˆ l(r)dr + δφl(r)Lφˆ l(r)dr. ∫0 ∫0 ∫0

The last two terms of Eq. (2.59) will vanish due to the fact that Lφˆ l(r) = 0. Integrating the t ﬁrst term by parts, and by matching the exact solution, φl(r), and the trial function, φl(r) at r = 0 and r = we arrive at ∞ ∞ k φ (r)Lδφˆ (r)dr = W δλ, (2.59) l l 2 ∫0 where W is the Wronskian, deﬁned as d d W = F (r) G (r) G (r) F (r). (2.60) l dr l − l dr l By inserting the trial wave function, Eq. (2.59) can now be written as

2 ∞ ∞ λ = λt + φt(r)Lφˆ t(r)dr δφt(r)Lδφˆ t(r)dr. (2.61) kW l l − l l ∫0 ∫0 Eq. (2.61) is known as the Kato’s identity and it provides a stationary principle for approxi- mating λ in the following way, where we have kept only ﬁrst order terms of Eq. (2.61),

2 ∞ λs = λt + φt(r)Lφˆ t(r)dr. (2.62) kW l l ∫0

15 If we choose a trial wave function, that contains only linear trial coeﬃcients, Eq. (2.62) can be solved explicitly. The trial wave function can have the form m t t φl(r) = fl(r) + λ gl(r) + ciϕi(r) (2.63) i=1 m∑+1

= ciϕi(r), i=0 t ∑ where c0 = λ , ϕ0(r) = gl(r), cm+1 = 1, ϕm+1(r) = fl(r). (2.64) 2 Here the set ϕi, with iϵ[1, m], are square integrable (L ) functions and f (r ) F (kr) (2.65) l → ∞ ∼ l g (r ) G (kr). (2.66) l → ∞ ∼ l t The coeﬃcients λ and ci are determined by applying the following variational conditions; ∂λs ∂λs = t = 0. (2.67) ∂ci ∂λ Substituting the trial wave function of Eq. (2.63) into Eq. (2.62) and taking the derivative with respect to the ci coeﬃcients gives

∞ ˆ t ϕi(r)Lφl(r)dr = 0, i = 1, . . . , m. (2.68) ∫0 Taking the derivative with respect to λt instead gives,

∞ ˆ t gl(r)Lφl(r)dr = 0, i = 1, . . . , m. (2.69) ∫0 By deﬁning all the ci’s by a vector c, Eq. (2.68)and Eq. (2.69) can be expressed in compact form 1 c = M− s, (2.70) − Where the matrix M contains the elements ∞ Mij = ϕi(r)Lϕˆ j(r)dr, i, j = 0, . . . , m, (2.71) ∫0 and s is a vector containing the elements

∞ si = ϕj(r)Lfˆ l(r)dr, i = 0, . . . , m. (2.72) ∫0 By substituting Eq. (2.70) into Eq. (2.62), the stationary value of λs can be obtained

s 2 ∞ 1 λ = f (r)Lfˆ (r) sM− s . (2.73) kW l l − [∫0 ] In order to avoid the matrix M from being singular at real energies, gl(r) is chosen to be an + outgoing complex valued function, hl (r), deﬁned as i [F (kr) iG (kr)] h+(r) = l − l , (2.74) l √k 1 then the Wronskian simpliﬁes to W = − . Eq. (2.73) reduces to an expression for the T-matrix, √k iδl λ = Tl = e sin δl [48] s ∞ 1 T = 2 f (r)Lfˆ (r) sM− s . (2.75) l − l l − [∫0 ] The matrix M, whose inverse is generally non-singular at real energies [54], is now symmetric and complex, hence the method is called the complex Kohn method.

16 Generalization to electron molecular ion scattering The electron- molecular ion scattering calculations are performed in the framework of the fixed nuclei approximation [55]. The trial wave function has to be appropriately chosen in this case, since the interaction potential is non-spherical. Such a wave function has to be suitable for describing both the interaction region and asymptotically. Usually, in scattering calculations, the boundary conditions imposes that such a wave function is finite-valued at the origin and behaves like an outgoing wave asymptotically. For a multi channel electron-molecular ion scattering, the trial wave function is expressed as [47] Φ = Aˆ ϕ F + Θ , (2.76) a { a′ a′a} a ∑a′ where the first summation in the RHS is running over all the energetically open n-electron target states. ϕa′ is the target state function and Fa′a is the wavefunction of the scattered electron. The operator A antisymmetrizes product ϕa′ Fa′a. The subscripts (a′, a) refer to the (final, incident) channels. The second term contains a set of square integrable (n + 1)-electron functions that are used to represent polarization and correlation effects which are not included in the first term. It can be expanded as

a Θa = dµΘµ, (2.77) µ ∑

where Θµ are the n + 1 conﬁguration state functions (CSFs).

The channel continuum functions, Fa′a, are further expanded as [47]

Ylm(ˆr) F = [f a′ (k r)δ δ δ + T a′a ga′ (k r)] + ca′aϕ (r), (2.78) a′a l a′ ll0 mm0 a′a ll0mm0 l a′ r i i i ∑lm ∑

a′ a′ where fl and gl are the continuum wave functions representing the incoming and outgoing wave for the scattering electron, Ylm is the normalized spherical harmonic, while ϕi are the set of square integrable functions . By energy conservation, the channel momenta, ka′ is determined as 2 k = 2(E Ea ), (2.79) a′ − ′ where E is the total energy and Ea′ is the energy of the target molecular ion corresponding to the a′a state χa′ . The coeﬃcients Tll0mm0 are the T-matrix elements and are the fundamental dynamical quantities of interest, from which quantities like the eigenphase sum and the scattering cross section can be calculated. In a realistic electron-molecular ion collision event, the interaction potential, V (r), is not spherical and thus the S-matrix (or T-matrix) will have both the l and m indices as shown by Eq. (2.78). The T-matrix, thus described above, is related to the scattering matrix, S-matrix by the simple relation T a′a = 1 Sa′a . (2.80) ll0mm0 − ll0mm0 By diagonalizing the scattering matrix, the eigen-phase sum can be calculated at a ﬁxed R as

η(E; R) = ηi(E; R), (2.81) i ∑

17 where the index “i” here represents all the quantum numbers. To obtain the autoionization width, Γ(R) and the energy position, Eres(R) of a resonant state at an internuclear distance R, the eigenphase sum is ﬁtted to a Breit-Wigner formula [56]

(0) 1 Γ 2 η(E; R) = η (E; R) + η (E; R) = tan− + a + bE + cE , (2.82) res 2(E E ) ( − res )

where the the parameters Eres, Γ, a, b, and c, which depend on R, are optimized in the ﬁt. The potential energy of the resonant state is the obtained by adding the resonant energy to the potential energy of the ion. + Fig. 2.2 shows the results that are obtained for one of the states of the H2O molecule. Here the internuclear distances between the H atoms and the O are 1.8 a0 and 2.0 a0, respectively, while the angle ∠HOH is 108.8◦. In Fig. 2.2(a) the eigenphase sum shows a sudden shift of π whenever it hit a resonance. The electron scattering cross section is shown in Fig. 2.2(b) and it is also showing a sudden decrease whenever it reaches a resonance. In the neighbourhood of such a resonance the eigenphase sum is ﬁtted to Eq. (2.82) to obtain Eres(R) and Γ(R).

2 (a) (b) 1.5

103

1 ) 2 0 0.5

-0.5

-1 Cross section (a Eigenphase sum (rad)

-1.5 102 -2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 Collision energy (eV) Collision energy (eV) + 1 Figure 2.2: The elastic scattering of H2O + e− in A’ symmetry. In (a) the eigenphase sum is displayed as a function of energy and in (b) the corresponding elastic cross section is shown.

18 2.4 Adiabatic to diabatic transformation

The greatest challenge in trying to solve the coupled nuclear differential equation (2.16) is the numerical instability caused by the nature of the non-adiabatic couplings. The non-adiabatic couplings that depend on the internuclear distance, R, may vary rapidly in a region where two states come close to each other. Also, the presence of the term Fa leads to a differential equation that contains both first and second order derivatives whose numerical evaluation is non-trivial [44, 57]. A basic approach is to transform the electronic basis into a basis (diabatic basis) that has the off-diagonal elements of the nuclear kinetic energy operator that are equal to zero or are very small, ∂ F d (R) = ϕd(R, r) ϕd(R, r) = 0. (2.83) ij ⟨ i |∂R| j ⟩ ∼ In performing this transformation, the electronic states no longer become eigenstates of the electronic Hamiltonian. Thus the diabatic potential energy curves do not obey the non-crossing rule and therefore when one follows the character of a diabatic state through an avoided crossing, the character of the state will be preserved as illustrated for a diatomic case in Fig. 2.3. Landau [22] and Zener[23] were the first to independently study the two state crossing of potential energy curves. However, in reality there might be more than two states (often infinite) involved and hence there are many approaches developed to perform the transformation. They can be loosely categorised into strict diabatic and quasidiabatic transformation and they are further discussed below.

φ a φ a 2 2 d φ d φ 2 1 Energy

φ d φ d 1 2 φ a φ a 1 1

Rx Internuclear distance Figure 2.3: Schematic diagram of potential energy curves in the vicinity of an avoided crossing a point Rx. The black (solid) lines are the adiabatic potential energy curves for two states (ϕ1 a d and ϕ2) and the red (dashed) lines are the diabatic potential energy curves for two states (ϕ1 d and ϕ2).

19 Quasidibatisation In the quasidibatic representation, only part of the non-adiabatic coupling is removed. The method used in the thesis is to perform diabatization by configurational tracking, where the electronic diabatic state is followed before and after the avoided crossing by examining the CI-coefficients of the adiabatic states. + In the DR study of H2O , there are an infinite number of Rydberg states that converge to the ground ionic state. Since this is a polyatomic system, a strict diabatization can not be achieved [18]. A quasi-diabatization is performed by removing states dominated by the configuration of the form (1a’)2(2a’)2(3a’)2(4a’)2(1a”)1(nλ)1, where n is an integer and λ represents the symmetry of the molecular orbital. These states are the bound Rydberg or continum states that have the electronic configuration similar to the ion, but with an additional electron in a higher orbital (nλ). In addition to tracking the configuration, at short internuclear distances, the electron scattering calculation is used to determine accurate energy positions of the resonant state potential curve relative to the ion potential energy. Fig 2.4 shows a slice of the potential 1 energy surfaces for the water molecule in A′ symmetry, displaying the potential energies of the quasidiabatic (resonant) states and their corresponding adiabatic potential energy surfaces are displayed below the ion. The points are the resonant energy positions determined by the electron scattering calculation.

0.3

0.2

0.1

Energy (Hatree) 0

-0.1

-0.2 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

R2 (a0)

Figure 2.4: Potential energy curves for water. The internuclear distance between one H atom and O is ﬁxed at 1.908 a0 and the other is varied as shown in the x-axis in the ﬁgure. The angle is frozen at 108.9◦ .

Strict Diabatisation Strict diabatization is only possible for states of diatomic systems [18]. A considerable simpli- ﬁcation is made by assuming that a ﬁnite set, say M, adiabatic coupled together [41]. It is also assumed that F a = 0 for i M, j > M. (2.84) ij ≤

20 a d The adiabatic states, ϕi (R, r), are then transformed to diabatic states, ϕi (R, r), using

M d a T ϕi (R, r) = ϕj (R, r)Tij (R), (2.85) j=1 ∑ where Tij is an element of the orthogonal transformation matrix, T. A similar transformation is performed of the nuclear wavefunction

M d a T χi (R, r) = χj (R, r)Tij (R), (2.86) j=1 ∑ in order to preserve the total wavefunction. By inserting the diabatic nuclear wave function (2.86) in the Schr¨odingerequation for the nuclear motion (2.16) we obtain the coupled Schr¨odingerequation in the strictly diabatic basis as

1 ∂2 χd + Vdχd = E1χd, (2.87) 2µ ∂R2 where the diabatic potential energy matrix are obtained from

Vd = TTVaT. (2.88)

The diabatic matrix elements are

V d = ϕd H ϕd , (2.89) ij ⟨ i | el| j ⟩ which in contrast to the adiabatic potential matrix elements, contains oﬀ-diagonal elements. The diagonal elements are the diabatic potential energy curves, similar to the ones shown in Fig. 2.3, and these curves can cross each other. The oﬀ-diagonal elements are called the electronic couplings and compared to the non-adiabatic couplings, they vary smoothly with R. + + This is the approach employed in the study of MN of H + H− and He + H− of papers I and III, respectively. If we assume that only two states are coupled together in the crossing region, a two-by-two strict transformation can also be performed. Here the transformation matrix, T, is

cos[γ(R)] sin[γ(R)] T = . (2.90) sin[γ(R)] cos[γ(R)] (− ) The rotational angle, γ(R) is obtained from integrating the ﬁrst derivative coupling element [41]

∞ γ(R) = Fij(R′)dR′. (2.91) ∫R To ensure that far away from the crossing region the diabatic and adiabatic states are the same, a scaling factor is imposed, such that

1 0 T = ,R R , (2.92) 0 1 ≪ x ( ) and 0 1 T = ,R R . (2.93) 1 0 ≫ x (− )

21 Thus the non-adiabatic coupling elements are such that

πFij(R) F˜ij(R) = . (2.94) ∞ 2 R Fij(R′)dR′ Mead and Truhlar [41] have pointed out that∫ a strict diabatic representation, in the sense discussed above, is only possible for a diatomic molecule. In the studies carried out in this + + thesis a strict diabatisation is performed in the MN studies of H + H− and He + H−. In + studying the DR process in H2O , which is a triatomic system, a quasidiabatic approach is + employed. A Landau-Zener [22, 23] semi-classical analysis is carried out for MN of H + H− + and Be + H−, and here a two-by-two transformation is performed.

22 Chapter 3

Nuclear Dynamics

This chapter details the models used to study the nuclear dynamics of the two different processes. The previous chapter discussed how the relevant states and couplings are obtained and these are used as input parameters for modelling the nuclear motion. The cross section for the molecular collision processes is then calculated after solving the nuclear Schrödinger + equation. For the DR process of H2O , a time-dependant wave packet method, based on + the time-dependant Schrödingerequation is employed. For the MN studies of H + H− and + He + H− the time-independent Schrödingerequation is used. The semi-classical Landua-Zener + + model has also been used to study MN of H + H− and Be + H−.

3.1 Time-dependant wave propagation

A wave packet is a superposition of stationary wave functions, such that it is localized and normalizable [58]. The advantages of using time dependant wave packets are that it can be used for any system, whether free or bound, the time evolution of the process can be followed carefully and details can be analysed at any given time, and thus we can intuitively follow the collision process in time. The time-dependant wave packets are all based on solving the time-dependant Schrödingerequation, ∂ i Ψ = Hˆ Ψ. (3.1) ∂t + In the study of the DR process of H2O , we have used the internal degrees of freedom; R1 and R2 for the two hydrogen-oxygen bonds distances and θ for the angle the hydrogen bonds make with the oxygen atom. First a 1D study is systematically carried out by fixing two coordinates at equilibrium, R2 =1.9086a0 and θ = 108.9◦ and varying R1 (“asymmetric stretch”) or varying both R1 and R2, equidistantly, while fixing θ = 108.9◦ (“symmetric stretch”) . The other degrees of freedom will be added in steps until all three degrees are varied and multi-dimensional wave packets are propagated.

3.1.1 1D wave dynamics By direct integration of the time-dependant Schrödingerequation, a wave packet is propagated with local complex potentials [59]. The equation is of the form ∂Ψ(R, t) 1 i = T + V (R) i Γ(R) Ψ(R, t), (3.2) ∂t − 2 ( ) with the initial condition [59] b Γ(R) Ψ(R, t = 0) = χ0(R). (3.3) √ 2π 23 1 d2 Here T = is the nuclear kinetic energy operator,V(R) and Γ(R) are the potential −2µ dR2 energy and autoinization width of the resonant state. χ0(R) is the initial vibrational state of theb ion. This vibrational wavefunction is obtained numerically by using a finite difference method [60]. Here the ion potential energy and wavefunction is defined on a grid of the coordinate R, such that R(j) = R(0) + R(j 1), j = 1 . . . , n. (3.4) △ − where R(n) is at the end of the grid. Then the second order derivative is approximated as

d2 1 χ (R(j)) ≈ χ (R(j + 1)) 2χ (R(j)) + χ (R(j 1)) . (3.5) dR2 ( R)2 { − − } △ The time-independent Schr¨odingerequation for the nuclear vibrational motion of the ion then becomes tri-diagonal eigenvalue problem to be solved. Likewise, the kinetic energy operator is approximated as 1 1 T Ψ(R(j), t) Ψ(R(j + 1), t) 2Ψ (R(j), t) + Ψ (R(j 1), t) . (3.6) ≈ −2µ ( R)2 { − − } △

Givenb the initial wave function at t = t0, the wave function at any time t can be expressed as i(t t0)H i tH Ψ(Ri, t) = e− − Ψ(Ri, t0) = e− △ Ψ(Ri, t0). (3.7) The term multiplying the initial wave packet in Eq. (3.7) is known as the evolution operator and there are several numerical approximations designed to evaluate this term. In this work to perform the wave propagation, the Cranck-Nicholson propagator [61] is used to propagate wave packets in one dimension. In this approach, the evolution operator is approximated by the Cayley form; 1 1 i tH i tH − 2 △ e− △ (3.8) ≈ 1 1 + i tH 2 △ Although this is a first order approximation, it is a stable and unitary approximation [62]. The wave packet is then propagated on the resonant state potential energy surface. Fig. 3.1 1 shows a wave packet propagated on the first resonant state of the H2O potential curve in A′ symmetry. As can be seen from the figure, the wave packet spreads out in R as the time increases. For calculating the total cross section, the transition amplitude is calculated by the Fourier transform method [63]. In this method, the transition amplitude is calculated as

k iEt Tj(E) = lim Ψj(Ri, t)e dt, (3.9) t 0 2πµ → ∫ {√ } where k is the momentum of the dissociating fragments. From the transition amplitude, the cross section for dissociating a along a single state is given by [59, 64],

2π3 σ (E) = g T (E) 2 (3.10) j E | j | where g is the multiplicity ratio of the neutral and the ionization continuum. To obtain the total cross section a summation over all contributing states is performed.

24 1e-06 t=0 t=50 t=200 t=500 t=1000 t=10000 8e-07

6e-07 2 (R,t)| Ψ | 4e-07

2e-07

0 0 2 4 6 8 10 12 14 16

R (a0)

1 Figure 3.1: Wave Propagation on the ﬁrst resonant state potential curve of H2O, in A′ symmetry, for diﬀerent times

3.1.2 The MCTDH method The Multi-Conﬁgurational Time-Dependant Hatree (MCTDH) program [65] is designed to study wave packets that are propagated in multiples degrees of freedom. In our work it will be applied to study the dissociation mechanism of H2O in two and three degrees of freedom. We intend to study the reaction including all three degrees of freedom. In this method, the nuclear wave packet is expressed as separable terms. For a two-dimensional case, say the nuclear coordinates are R1 and R2, and the nuclear wave function is written as

nR1 nR2 R1 R2 ψ(R1,R2, t) = Aij(t)ϕi (R1, t)ϕj (R2, t). (3.11) i=1 j=1 ∑ ∑

Ri Here Aij are known as the MCTDH expansion coefficient and ϕi are the single particle functions (SPF’s) for each degree of freedom. Each SPF can be represented in terms of a finite basis known as a primitive basis, which is a product of a time-dependant function and a time- independent function; NRi Ri Ri Ri ϕi (Ri, t) = cik (t)χk (Ri). (3.12) ∑k=1 Thus for each of the two radial coordinates and the angular motion separate basis functions are + used. For the DR study of H2O , a discrete variable representation (DVR) of sine functions for the radial motion and Legendre functions for the angular motion have been used. In order for the MCTDH algorithm to work efficiently, the system’s Hamiltonian need to be expressed as a sum of products of single coordinate operators. The kinetic energy operator used, using internal coordinates with θ = 109.8◦ frozen, in the study is of the form

25 2 ˆ 1 ∂ 1 ∂ cos θ ∂ ∂ T = 2 , (3.13) −2µOH ∂R1 − 2µOH ∂R2 − mO ∂R1 ∂R2

where µOH is the reduced mass of O and H atoms and the mass of oxygen is mO. Using the POTFIT [65] program in the MCTDH package, the potential energy surfaces and the autoionization widths are efficiently fitted to the desired product format. The wave packets are initiated in the resonant state and the initial condition is similar to Eq. (3.3), but here the coordinates are multi-dimensional and the vibrational wave function is calculated by using a relaxation in imaginary time. A complex absorbing potential (CAP) [66] is placed in the asymptotic region of the resonant state and by analysing the flux absorbed by the CAP, the DR cross section is obtained. The flux operator is approximated by

Fˆγ = i[H,Wˆ γ], (3.14) with Hˆ as the system’s Hamiltonian and Wγ is the CAP added to the resonant state potential in the asymptotic region and γ is a specific reaction channel as defined by Jäckle and Meyer [66]. The probability for dissociating into a specific channel HO(ν) + H is calculated by

2 2 ∞ iEτ S (E) = Re g (τ)e− dτ , (3.15) | γν | π ∆(E) 2 γν | | {∫0 } where ∞ g (τ) = dt Ψ(t + τ) Pˆ W Pˆ Ψ(t) , (3.16) γν ⟨ | γν γ γν| ⟩ ∫0 and Pˆγν is the projection operator over the internal vibration states of HO, Pˆ = φ φ . (3.17) γν | ν⟩⟨ ν| The total cross section for dissociating to a state is then given by [67] 2π3 σ(E) = g S (E) 2, (3.18) E | γν | ν ∑ where g is the multiplicity ratio of the ion plus free electron and the neutral ion-pair state.

3.2 Time-independent nuclear scattering

The nuclear dynamics of the MN studies are carried out using the time-independent Schr¨odinger + + equation. This method is applied to study the MN reaction of H + H− in paper I and He + H− of paper III. The time-independent scattering of nuclei is similar to the scattering discussed in section 2.3.1, except for the fact that now we have inelastic, multichannel scattering. The asymptotic form of the wave function is similar to Eq. 2.39. However the scattering amplitude will depend on the incoming and outgoing channel. Thus the nuclear wave function is expanded into radial wavefunctions and Legendre polynomials

1 ∞ χ (R) = A ψ (R)P (cos θ), (3.19) i R l i,l l ∑l=0 where Al is a constant that is chosen to fulﬁl the boundary condition, Pl(cos θ) are the well known Legendre polynomials and ψi,l(R) is the radial wavefunction, satisfying the equation

1 d2 l(l + 1) M + ψ (R) + V dψ (R) = Eψ (R). (3.20) −2µ dR2 2µR2 i,l ij j,l i,l j=1 [ ] ∑ 26 In order to numerically integrate the radial equation, the log derivative is introduced,

1 yl = ψl(R)ψl− (R), (3.21) where ψl(R) is a square matrix containing all the terms ψi,l(R). It can be expressed as a linear combination of the incoming and outgoing wave solution [68], for R Rρ, where Rρ is some radius, for which beyond the matrix Vd is diagonal. Thus we have ≥

ψl(R) = αl(R) + βl(R)Sl, (3.22) where Sl is the scattering matrix [48]. Using this expression, the radial Schr¨odingerequation (3.20) is transformed to a more numerically stable matrix Riccati equation [69]

2 y′l + Ql + y l = 0, (3.23) where l(l + 1) Q = 2µ(E1 Vd) 1. (3.24) l − − R2 The Riccati equation is integrated using the Johnson [69] algorithm out to the point where the diabatic potential energy curves have reached their asymptotic form. For the MN reactions we asy are studying, the asymptotic limit is set at R = 50a0. The scattering amplitude can then be computed from the scattering matrix using the well known multi-channel formula [48] 1 fij(θ, E) = (2ℓ + 1) (Sij,ℓ δij) Pℓ (cos θ) (3.25) 2i kikj − ∑ℓ √ Here ki and kj are the wave numbers of the ﬁnal and initial states (channels), respectively, and Sij,ℓ is the scattering matrix for scattering from j to i for a ﬁxed angular momentum, ℓ. The partial cross section can also be obtained as

π 2 σij,ℓ = 2 (2ℓ + 1) δij Sij,ℓ , (3.26) kj | − | and the total cross section as ∞ σij = σij,ℓ (3.27) ∑ℓ=0 The diﬀerential cross section is obtained from the scattering amplitude as [48]

dσij ki 2 = fij (θ, E) . (3.28) dΩ kj | |

3.3 Semi-classical Landau-Zener studies

In a Landua-Zener (LZ) [22, 23] model, the nuclear dynamic are studied semi-classically. It is assumed that only two states are interacting. We have applied this method to study the results of paper II. The Hamiltonian is expressed in the form

Tˆ + U (R) U (R) Hˆ = 11 12 , (3.29) U (R) Tˆ + U (R) ( 21 22 ) where U11(R), and U22(R) are the diabatic potential energy curves, while U12(R) and U21(R) are the oﬀ-diagonal elements that corresponds to the electronic couplings.

27 In the model, it is assumes that U11(R) U22(R) = kR, where k is a constant, i.e. the diﬀerence between the two diabatic states is− a linear function of the internuclear separation. The electronic coupling are assumed to be constant,

U12(R) = U21(R) = H12. (3.30)

In the LZ model, it is assumed that the transition region is small and only the characteristics of the region are of importance. The probability, p, of transition from one diabatic state to the other is given by [22, 23] 2πH2 p = exp 12 , (3.31) − v k ( x ) where vx is the radial velocity at the curve crossing. There are several empirical methods that have been applied in trying to approximate electronic couplings and some of them are discussed + and applied in ref. [70]. One of the system studied here is the MN of H + H−, where in the adiabatic representation there are two covalent states converging to the n =2 limit and three 1 + converging to n =3 limit of the Σg symmetry [26]. However, according to Lewis [71], in the crossing region only two states are involved in the change of character from ionic to covalent. Using the energy diﬀerence of the adiabatic states, at the avoided crossing, Rx, the electronic couplings, H12, can be approximated by [21]

1 2 1 a a 2 H12 = Vi (Rx) Vi 1(Rx) , (3.32) 2 − − { i } ∑ [ ] a where Vi is the adiabatic potential energy curves obtained by ab initio methods as outlined in ref. [26]. Also a two-by-two strict diabatisation using the transformation matrix (2.90) has been performed and the electronic couplings are the oﬀ-diagonal elements of the diabatic potential energy matrix (2.88). These two methods of obtaining the electronic couplings have been applied to obtain the so-called “APS” and “ATD” couplings of paper II. The collision cross section is evaluated by [72] π σ(E) = 2 (2l + 1)Pℓ(E), (3.33) ki ∑ℓ where Pℓ(E) is the multi-state Landau-Zener probability [21]. The probability is evaluated by adding all two-state crossing probabilities when going in and going out as can be analysed from Fig. 1.1.

28 Chapter 4

Discussion of attached papers

4.1 Paper I: Diﬀerential cross section for mutual neu- + tralization of H + H−

The mutual neutralization reaction of

+ H + H− H(n) + H(1), (4.1) → where H represents any of the hydrogen isotopes, is here studied ab initio and fully quantum mechanically using diabatic states that were previously calculated by Stenrup et al. [26]. The work is here extended to include other hydrogen isotopologues; deuterium and tritium. Ad- ditionally, not only the total cross section but also diﬀerential cross section is computed. In this study, all possible collision combinations that can arise from the isotopes are considered, + + + namely; identical nuclei species (H + H−,D + D−, and T + T−) and non-identical nuclei + + + species (D + H−,T + H−, and T + D−). In section 4.1.1 the equations relevant to collision of identical nuclei species are derived. This reaction is of interest in the modelling of the early universe where in the formation of primordial H2, the MN reaction ( 4.1) is a competing reaction and hence a reliable data of the its cross section is desirable [6, 8, 9]. The ITER fusion experiment has a neutral beam heating based on H− ion source [4, 73]. To model the H− density in the low temperature plasma of the ion source, the measurement of the Hα/Hβ balmer line ratio has been suggested [3, 4, 74]. These reaction has to be studied for other isotopes as well since deuterium is known to be present at the divetor section of the plasma [3].

4.1.1 Scattering of nuclei species When considering identical particles, care has to be taken in the coherent addition of the scattering amplitude as the total wavefunction has to be symmetric or antisymmetric with respect to exchange of nuclei, depending on whether its a half integer spin (fermions) or integer + spin (boson) particle. As pointed out by Masnou-Seeuws and Salin [75] the symmetric [fij (E, θ)] and antisymmetric [fij−(E, θ)] scattering amplitudes can be expressed as [26]

+/ di ex f − (E, θ) = f (E, θ) f (E, π θ) , (4.2) ij ij ± ij − where the direct and exchange scattering amplitudes are [75] 1 f di (E, θ) = f g (E, θ) + f u (E, θ) (4.3) ij 2 ij ij 1 f ex (E, θ) = [f g (E, θ) f u (E, θ) ]. ij 2 ij − ij [ ] 29 Since H2,D2 and T2 are homonuclear diatomic molecules, and thus the charges in the electronic Hamiltonian, Hel, are equal, the gerade and ungerade symmetries of the electronic states will be taken into consideration. Thus the subscripts ”g” and ”u” in Eq. (4.4) denote the gerade and ungerade manifolds, respectively. The hydrogen and tritium quasimolecules are fermions with a half integral spin and the deuterium quasimolecule is a boson with integral spin. The spin multiplicity factors are derived in appendix A. In computing the diﬀerential cross section, using Eq. (3.28), we combine the symmetric spin with the antisymmetric spatial wave function and the antisymmetric spin with the symmetric spatial wave function such that for H2 and T2 , 1 f (E, θ) 2 = f g (E, θ) + f u (E, θ) + f g (E, π θ) f u (E, π θ) 2 | ij | 16| ij ij ij − − ij − | 3 + f g (E, θ) + f u (E, θ) f g (E, π θ) + f u (, π θ) 2. (4.4) 16| ij ij − ij − ij − |

For D2 , we have 1 f (E, θ) 2 = f g (E, θ) + f u (E, θ) + f g (E, π θ) f u (E, π θ) 2 | ij | 6| ij ij ij − − ij − | 1 + f g (E, θ) + f u (E, θ) f g (E, π θ) + f u (, π θ) 2. (4.5) 12| ij ij − ij − ij − | By putting Eq. (3.25) in Eq. (4.4) and Eq. (4.4) and expanding using properties of Legendre polynomials [ Pℓ( cos θ) = Pℓ(cos θ) if ℓ is even, Pℓ( cos θ) = Pℓ(cos θ) if ℓ is odd] [50], the diﬀerential cross section− is − −

dσij 3 ki g,odd u,even 2 (θ) = fij (θ, E) + fij (θ, E) dΩ 4 kj | |

1 ki g,even u,odd 2 + fij (θ, E) + fij (θ, E) , (4.6) 4 kj | | and

dσij ki g,odd u,even 2 (θ) = fij (θ, E) + fij (θ, E) dΩ 3kj | |

2ki g,even u,odd 2 + fij (θ, E) + fij (θ, E) , (4.7) 3kj | | for the fermions and bosons, respectively. Here even/odd refer to a summation over even and odd angular momentum quantum numbers, ℓ, in Eq. (3.25). The total cross section for the different particles is obtained by integrating Eq. (4.6) or Eq. (4.7) over the unit sphere. For scattering of non-identical particles, there is no inversion symmetry. There is no differential cross section for the mutual neutralization reaction studied here, reported in literature. Also isotope dependence on the total cross section and differential cross section has not previously been reported. Fig 4.1 shows the differential cross section as a function of scattering angle for all isotopes colliding at 0.1 eV. For collisions between like isotopes, the backward scattering is higher than the forward scattering, whereas for non-identical particles the forward scattering is dominant. The irregular oscillations in the differential cross section are reflecting contributions of high angular momenta, ℓ. An increase in collision energy causes the motion to exhibit a more classical behaviour and hence as shown in Fig 4.2, the oscillations are longer and with decreasing amplitudes.

30 10-11 T++T- T++D- D++D- T++H- H++H- D++H- 10-12 /sr) 2

10-13

10-14

10-15 Differential cross section (cm

10-16 0 20 40 60 80 100 120 140 160 180 Scattering angle (θo)

Figure 4.1: Diﬀerential cross sections for mutual neutralization of all isotopes at 0.1 eV collisions.

10 meV -11 10 100 meV 1.0 eV -12 10.0 eV

/sr) 10 2 100.0 eV 10-13

10-14

10-15

10-16 Differential cross section (cm 10-17

10-18 0 20 40 60 80 100 120 140 160 180 Scattering angle (θo)

+ Figure 4.2: Diﬀerential cross sections for mutual neutralization H + H− at diﬀerent collision energies.

31 Furthermore, the forward and backward scattering are more pronounced and the diﬀerential cross section becomes more symmetric with increasing energy. By integrating the expressions for the diﬀerential cross section over the unit sphere, we obtain expression for the total cross sections and in the case of H2 it is identical with the cross section expression derived in ref. [26]. Fig. 4.3 shows the total cross section for all hydrogen isotopes in the energy range 0.001 eV to 100 eV.

T++T- + - -11 T +D 10 1.0e-14 + - D +D + - 9.0e-15 T +H D++H- + - 8.0e-15 H +H ) 2 10-12 7.0e-15 6.0e-15 20 40 60 80 100

10-13 Cross section (cm

10-14

0.001 0.01 0.1 1 10 100 Collision energy (eV) Figure 4.3: The total cross section mutual neutralization for all hydrogen isotopes.

The Coulombic interaction character of the cross section is observed in all isotopes at col- 1 lision energies less than 1.0 eV, where the cross section is proportional to E− , according to Wigner’s threshold law [20]. At high collision energies, the cross sections, for the lighter isotopes is higher while at low energies the heavier isotopes have a larger cross section. This is also illustrated in Table 4.1, where the ratio of the cross section for the heavier isotopes to that of H2 are greater than one at low energy but less than one for all isotopes at 100 eV.

T2 TD D2 TH DH 0.001 eV 1.472 1.375 1.296 1.154 1.104 0.01 ev 1.473 1.381 1.302 1.162 1.104 0.1 eV 1.487 1.385 1.310 1.155 1.111 1.0 eV 1.518 1.400 1.288 1.154 1.100 10.0 eV 1.317 1.205 1.139 1.062 1.056 100.0 eV 0.729 0.716 0.774 0.872 0.919

Table 4.1: Ratios of the total cross section for the heavier isotopes to that of H2.

Using electronic coupling elements proposed by Janev [76], a Landau-Zener model as described in section 3.3, the total cross section of mutual neutralization of all hydrogen isotopes has been studied. The calculation in the energy range 0.001 eV and 100 eV, for all diﬀerent isotopes, are shown Fig. 4.4. It is interesting to note that even by such a simple semi-empirical model and a similar trend as the ab initio study is also observed. The partial cross section for the contribution of the n = 2 and n = 3 channels is shown in Fig 4.5. The ﬁnal state distribution for all isotopes have also been computed and the n = 1 channel 6 is contributing with a ratio of less than 10− for all energies. At low energies, the n = 3 channel

32 T++T- 1.0e-14 + - 10-11 T +D D++D- 8.0e-15 T++H- D++H- + - 6.0e-15 H +H

) -12 2 10 4.0e-15 20 40 60 80 100

10-13 Cross section (cm

10-14

0.001 0.01 0.1 1 10 100 Collision energy (eV) Figure 4.4: The total cross section for all isotopes calculated by the Landa-zener model, using electronic couplings proposed by Janev [76]. dominates, which implies that the MN reaction of all isotopes is driven by the non-adiabatic coupling at Rx = 35a0. With an increase in energy, which lowers the centrifugal barrier, the n = 2 channel start to contribute signiﬁcantly.

10-10 T++T- + - -11 T +D 10 D++D- T++H- + - 10-12 D +H H++H- ) 2 10-13

10-14

10-15 Cross section (cm

10-16

10-17

10-18 0.001 0.01 0.1 1 10 100 Collision energy (eV) Figure 4.5: Partial cross section for the n = 2 and n = 3 channels for all hydrogen isotopes, using the Landau-Zener model

At low energy, the cross section for the heavier isotope is larger while at high energy the heavier isotope have the smallest cross section.. The isotopic dependance is in agreement with + what is observed by Olson et al. [77] in the study of the MN process of He + H−, which shows that the lighter isotope has a larger cross section at high energy. However this is not in + good agreement with a study of the MN process in Li + H− [78], where they observe a mass independent cross section at high energy.

33 4.2 Paper II: Landau-Zener studies of mutual neutral- + + ization in collisions of H + H− and Be + H−.

The discussion of the results on paper I entailed a fully quantum study using electronic states that were computed by Stenrup et al. [26]. To carry out the study, a strict diabitization of the 1 + 1 + seven Σg and six Σu states of H2 were considered, using the non-adiabatic couplings. In the Landau-Zener multi-state model, we assume that only two states are successively coupled at each avoided crossing distance. Only one covalent state is coupled [71]. Using the adiabatic electronic states for H2 and the non-adiabatic couplings computed by Stenrup et al. [26], a 1 + two-by two adiabatic-diabatic transformation (2.90) has been carried out for the Σg states that converge to n = 2 and n = 3. As pointed out in the previous section, the states converging 6 to n = 1 contribute 10− to the total cross section, hence here it has has been neglected . The non-adiabatic couplings used are displayed in Fig. 4.6.

0.2

) -0.2 -1

0 n=2

-0.4

-0.6

Coupling strength (a -0.8

-1 n=3 -1.2 10 15 20 25 30 35 40 Intenuclear distance (a0)

1 + Figure 4.6: Radial couplings among the states converging to n = 2 and n = 3 for the Σg adiabatic states of H2.

To ensure that far way from the crossing region the adiabatic and diabatic states coincide, the rotational angle, γ(R), as a function of the internuclear distance is obtained for n = 2 and n = 3. Rotational angle is obtained by integrating the scaled non-adiabatic couplings according to Eq. (2.94) and they are displayed as shown in Fig. 4.7. Once the rotational angle is known, the transformation matrix, Eq. (2.90), can be constructed and the adiabatic states transformed to diabatic states, which cross and the are shown for the two crossing regions, together with the adiabatic states in Fig 4.8. The electronic couplings which are refereed to as “ATD” in paper II are also shown in Fig 4.9. These are the oﬀ-diagonal elements of the diabatic potential energy matrix, and they vary smoothly with R, compared with the non-adiabatic coupling. The diﬀerence in energy, at the curve crossing of the adiabatic states is also used and the electronic coupling and this method is refereed to as “APS” in paper II. The cross section is very sensitive to the coupling elements used and it is observed that the cross section calculated using the “ATD” method is larger than the one for using the “APS” method. In the paper, in addition to the couplings from a fully quantum study discussed above, several other empirical couplings have been used to compute the total cross section. The empirical couplings proposed by Janev [76] seems to give a better cross section when compared with cross section from the fully quantum study.

34 1.6

1.4

1.2

0.8

0.6 Angle (radians) n=2 n=3

0.4

0.2

0 5 10 15 20 25 30 35 40 45 50 Intenuclear distance (a0)

1 + Figure 4.7: The rotational angle for transforming the Σg adiabatic states for n = 2 and n = 3 in H2.

-0.545 -0.56 - + + H - H + -0.55 H + H -0.58

-0.6 -0.555 H(1) + H(3)

-0.62 -0.56 H(1) + H(2) -0.64 -0.565 -0.66 Energy (Hatree) Energy (Hatree) -0.57 -0.68 -0.575 -0.7

-0.72 -0.58 5 10 15 20 25 30 20 25 30 35 40 45 50 Internuclear distance (a0) Internuclear distance (a0) (a) (b)

1 + Figure 4.8: Adiabatic (solid lines) and diabatic (dashed lines) potential energy curves for Σg states of H2. The adiabatic states that converge to the n = 2 limit are shown in (a) and that converge to n = 3 are in (b).

0.012

0.01 n=2

0.008

0.006

0.004

Energy (Hatree) 0.002

0 n=3

-0.002 5 10 15 20 25 30 35 40 45 50 Internuclear distance (a0)

Figure 4.9: The electronic couplings for n = 2 and n = 3 states of H2.

35 4.3 Paper III: Studies of HeH: DR, RIP, VE, DE, PI, MN,...

The interaction of small atomic ions is of fundamental interest to the physics of atomic collision. Due to the nuclear size and number of electrons of such systems, it is relatively possible to perform a “good” theoretical model. They are also fundamentally important for their abun- + dance in nature and their involvement in the early universe chemistry. The system of H + H− discussed in paper I and II is such a system. Another system that is of interest is the HeH system. It is believed that the HeH+ ion played a crucial role in the gas evolution in the early universe [8]. The focus in this paper has been to compute “reliable” resonant states of the HeH system. A full conﬁguration interaction method is used to perform the structure calculation which determines the potential energy curves of HeH and HeH+. The non-adiabatic couplings between the states were analytically calculated, using the MESA code [79]. Work is ongoing to compute the non-adiabatic couplings at small distances (< 5 a0), where autoionisation have to be taken into consideration. The electron scattering calculation is also carried out to determine the autoionization widths of the resonant states . Once this data is obtained a series of reactions could be studied, namely; dissociative recombination (DR), resonant ion-pair formation (RIP), vibrational excitation (VE), dissociative excitation (DE), penning ionisation (PI) and mutual neutralization (MN). The main goal, so far, has been considering states that are relevant for the MN reaction. Amongst the states computed, eleven resonant states of 2Σ+ symmetry of HeH were obtained above the ground state of HeH+. These are the states relevant for the MN reaction and they are shown in Fig. 2 of paper III. As opposed to the H2 states in paper I, the HeH resonant states lie above the ground state of HeH+ and thus at small distances, there is a possibility of the resonant states to couple to the ionization continuum. Fig. 4 of paper III shows the autoinisation widths of the resonant states and the non-adiabatic couplings among the states converging to the n = 3 limit. To study the nuclear dynamics, an adiabatic to diabatic transformation is performed of the resonant states. The adiabatic potential energy matrix is complex and of the form Va(R) = V(R) + W(R), (4.8) where V(R) is a diagonal real matrix containing the adiabatic potential energy,

a V1 (R) 0 0 a ··· 0 V2 (R) 0 V(R) =  a ··· . (4.9) 0 0 V3 (R)  . . . ···.   . . . ..     The complex matrix W(R) is of the form [80], Γ (R) Γ (R)Γ (R) Γ (R)Γ (R) 1 1 2 1 3 ··· i Γ2(R)Γ1(R)Γ2(R) Γ2(R)Γ3(R)  √ √ ··· . (4.10) −2 Γ3(R)Γ1(R) Γ3(R)Γ2(R)Γ3(R) √ . . √ . ···.   . . . .. √ √    Γk(R) is the autoinization width for state k. The approximation here is that when the non- adiabatic couplings are computed, the resonant states are treated as bound states. Using the non-adiabatic couplings, the transformation matrix T(R), deﬁned in section 2.4, is obtained and the adiabatic potential matrix is transformed to a diabatic potential matrix Vd(R); Vd(R) = TT(R)Va(R)T(R) + TT(R)W(R)T(R). (4.11)

36 + A ﬁrst step towards studying the MN mechanism in He + H− collisions have been performed using the log-derivative method. The numerical equation is solved from 0.5 a0 to 50 a0. The total cross section for the reaction is then evaluated. So far autoinization has not been included and in Fig. 4.10 the cross section in the energy range 0.1 eV to 500 eV is compared with other merged beam experimental results [27, 32, 33]. At low energy, the cross section obtained seems to be larger than the experimental cross section by Peart et al. [27], but lower than the results by Olamba and co-workers [32]. However it seems to reasonably agree in the energy range 60 eV to 200 eV. It would be fascinating to look at the comparison again when autoinization is included. The results is also compared with other theoretical results [34, 35], which were performed at higher energies and there is not much a range to do a fairly good comparison, except that at about 200 eV, they are equal with the value obtained by Chibisov et al. [34].

Olamba 1996

Peart 1994

Gaily 1970

Chibisov 1997

Ermolaev 1992

-13 )

10 2

Present calculation

-14

10 Cross section (cm

0.1 1 10 100 1000

Energy (eV)

+ Figure 4.10: Total cross section for the MN reaction of He + H−. The present result is compared with previous theoretical and experimental results.

The branching ratios of what states a dominating in the reaction are also investigated. In Fig. 4.11 they are plotted and are compared with a recent experiment performed by Urbain [81], using merged-beam technique.

He(1s2s S)+H

0.5

He(1s2s S)+H

He(1s2p P)+H

0.4 He(1s2p P)+H

He(1s3s S)+H

0.3 3

He(1s3p P)+H

He(1s3d D)+H

0.2

He(1s3d D)+H Branching ratios

0.1

0.0

1E-3 0.01 0.1 1 10

Energy (eV)

+ Figure 4.11: Branching ratios for mutual neutralization pf He + H−, compared with ratios obtained by merged beam experiment.

4.4 Paper IV: Theoretical study of the mechanism of + H2O dissociative recombination

+ The H2O ion is known to be present in interstellar medium [82, 83] as well as in dense molecular clouds [84, 85] and thus its destruction is of importance in understanding the evolution of molecular clouds. Another breakdown of the cation takes place in atmospheres of icy moons [86]. Dissociative recombination of the water cation is such a mechanism that can lead to the break- + down. DR of H2O has been experimentally studied over the years [87–89], where the DR cross section has been measured as well as the branching ratios and final state distributions. It has been observed that at low collision energies, the three-body breakup dominates with almost 60%. A similar multi-bond fragmentation is found in many polyatomic ions. The main goal of paper IV is to take a first step towards a theoretical study of the DR of + H2O . What is the main driving mechanism for the reaction, is it the direct or indirect mechanism? Using simplified models for both the direct and indirect mechanism, the contributions to the DR total cross section are computed. I have focused on the study of the direct mechanism. + This study is carried out to investigate the role of the direct mechanism in DR of H2O . Here time dependent wave packets are propagated on the complex potential energy surfaces of the resonant states relevant to the process. A Multi-Reference Configuration Interaction (MRCI) method is used to compute the electronic bound states of the ion and the neutral molecule. In carrying out the study, first a one dimensional “asymmetric stretch” of one of the O-H bonds is studied. The other O-H bond is fixed at R = 1.9082a0 and the H-O-H angle is fixed at θ = 108.9◦. We also consider a “symmetric stretch” in where both the O-H bonds are varied simultaneously, while the angle frozen at 1 ′ 1 ′′ 3 ′ 3 ′′ θ = 108.9◦. In Cs symmetry, there are electronic states of A , A , A and A symmetries. In 1 1 3 3 1 1 3 3 C2v symmetry, there are electronic states of A1, A2, A1 , A2, B1, B2, B1 , B2 symmetries.

38 To compute the energy positions of the resonant states above the ion and also the autoinisation widths, electron scattering calculations have been performed. The auto-ionization widths for “asymmetric stretch” are shown in Fig. 4.12. Similarly, widths and resonant states for the symmetric stretch in both Cs and C2v symmetries are obtained.

0.0012 1 0.003 3 1 A' 1 A' 2 1A' 3 1 2 3A' 3 A' 3 A' 4 1A' 3 1 4 3A' 0.001 5 A' 0.0025 5 A'

0.0008 0.002

0.0006 0.0015 Width (Hatree) Width (Hatree) 0.0004 0.001

0.0002 0.0005

0 0 1.5 2 2.5 3 3.5 4 4.5 5 1.5 2 2.5 3 3.5 4 4.5 5 R2 (a0) R2 (a0) (a) (b) 0.00045 0.0003 1 1A" 1 3A" 2 1A" 2 3A" 0.0004 0.00025 0.00035

0.0003 0.0002

0.00025 0.00015 0.0002 Width (Hatree) Width (Hatree) 0.00015 0.0001

0.0001 5e-05 5e-05

0 0 1.5 2 2.5 3 3.5 4 4.5 5 1.5 2 2.5 3 3.5 4 4.5 5 R2 (a0) R2 (a0) (c) (d)

1 3 1 Figure 4.12: Autoionization widths for the “asymteric strech” in (a) A′, (b) A′, (c) A′′ and 3 (d) A′′ symmetries.

The resonant states, bound states and resonant energy positions for the symmetric stretch are shown in figure 4.13. The solid lines are the potential surfaces in Cs symmetry, while the dashed lines are the surfaces in C2v symmetry. The black solid lines are the ground and first excited state of the ion. As can be seen from the figure, above the ground state of the ion, the electronic scattering calculation has been performed to obtain the energy positions of the resonant states and they are shown as dots. It is worth noting that at larger distances, the states in C2v and Cs symmetries have equal values for the potential energy. Using a one-dimensional wave propagation, the Cranck-Nicholson propagator [61] numerical methods is used. For calculating the total cross section, using the transition amplitude,the Fourier transform method, outlined in section 3.1, is used. The total cross section for the 3 “symmetric strecth” is Cs shown in Fig. 4.15 (a). At low energy, the A′ symmetry dominates, while in the energy range 1 eV to 3 eV, the 3A” symmetry is also contributing significantly. The cross section is also compared with the total cross section in C2v symmetry, shown in Fig. 4.15 (b). A similar study is done for the asymmetric stretch mode. The total cross section for “symmetric stretch” in both symmetries and the “asymmetric stretch” mode is shown in Fig. 4.15,

39 0.2 1 1 1 A' / 1 B2 21A' / 11A 0.175 1 1 1 3 A' / 2 A1 41A' / 31A 0.15 1 1 1 5 A' / 2 B2 0.125

0.1

0.075

0.05 Energy (Hatree) 0.025

-0.025

-0.05 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 R2=R1 (a0)

Figure 4.13: A one dimensional slice of the potential energy surface for the resonant states, in + the symmetric stretch mode, relevant for the DR of H2O in Cs (in dashed colored lines) and C2v (in solid colored lines). Here the angle θ = 108.9◦, is ﬁxed while bond lenghts, R1 = R2 are varied equidistantly.

Total Total 1 1 -16 1A' -16 A + B 10 10 3 1 3 2 3A' A + B 1 1 1 2 1A" A + B 3 2 3 1 3A" A + B -18 -18 2 1

) 10 ) 10 2 2

10-20 10-20

10-22 10-22 Cross section (cm Cross section (cm

10-24 10-24

10-26 10-26 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Collision energy (eV) Collision energy (eV) (a) (b)

Figure 4.14: direct DR cross section in (a) Cs and (b) C2v, symmetries. In the C2v symmetry, states that are similar to the same symmetry, at large distances, with the Cs symmetry, have been added together. where it is also compared with experimental results. At lower energy, it is evident that the direct DR mechanism is not mainly important and as seen in paper IV, here the driving mechanism is mainly the indirect process. The indirect process has been studied using a simpliﬁed model [38, 90], and at low energy, the cross section obtained is higher than the experimental results. However in the model, autoinisation is not included. The cross section for the indirect DR mechanism goes to zero at 0.4757 eV, and this is at the ﬁrst excited state of the vibrational state of the ion.

40 10-12 Total Cs Total Cs-symmetric -13 Total C2v-symmetric 10 exp.(a) exp.(b) exp.(c) 10-14 ) 2 10-15

10-16

10-17 Cross section (cm 10-18

10-19

10-20 0.01 0.1 1 10 Collision energy (eV) Figure 4.15: Total direct DR cross section for the one-dimensional wave propagation on the H2O resonant states, points are results from experiments; a) ref [87], b) ref [89], c) ref [88].

41 Chapter 5

Conclusion and outlook

Although a large number of chemical reactions are believed to be taking place adiabatically, there is still some reactions, like the ones studied here, that do not fit into this picture. The non- adiabatic effects are found to play a significant in driving the dynamics for mutual neutralisation and dissociative recombination reaction. The transition between the states relevant for these processes is induced by the non-adiabatic couplings. By computing the reactions cross section, these reactions have been studied quantum mechanically and semi-classically to investigate their significance at low energy and to also study the significance of the contributing states to the cross section by calculating the branching ratios. + The study of mutual neutralization of H + H− has been a continuation of a study carried out before to compute the total cross section. Here the differential cross section has been computed, to investigate the dependence of the cross section on the scattering angle. With increasing energy, the scattering differential cross section gets larger at θ 0◦ and θ 180◦. For like isotopes, the forward scattering is dominant, while for unlike isotopes→ the backward→ scattering dominates. Isotope effects have also been studied and at low energy (< 10 eV) the heavier isotope has the largest cross section, while at higher energies, the lighter isotope has the highest cross section. This is somehow understood from looking at the branching ratios, where the significance of the n = 2 channel is greater for lighter isotopes. A future outlook on this project is to understand the fast oscillations of the differential cross section, which seem to be energy dependant. + The mutual neutralization reaction for He + H− has been studied so far without including autoinization. Autoinization widths will couple the resonant states considered here to the ionization continuum, at short distances, hence it is anticipated that this effect may reduce the total cross section obtained. The next goal on this project is to study the nuclear dynamics with the inclusion of autoinization. The study of isotope dependence of the total cross section as well as the differential cross section of the reaction is also a future objective. The cross section obtained so far seem very comparable with the experimental results. More interesting are the branching ratios, which are also comparable with the recent experiment. A further investigation into the other chemical processes involving the HeH system, outlined in the results discussion of section 4.3 is also a future objective. + The dissociative recombination reaction of the H2O cation has been studied, so far using simplified models. Including one-dimension, the direct mechanism has been investigated. Still there is a significant contribution to the cross section from the direct mechanism. The simplified model for the indirect mechanism gives a larger cross section than the experiments. A further investigation of the indirect mechanism, using a model which somehow accounts for autoinization is proposed for the future work. Also, the next goal on this project is to study two-body and three-body breakup. Here the Multi-Configuration Time-Dependant Hatree (MCTDH)

42 method will be employed to implement the wave propagation. Furthermore, since in all the calculations done, the states have been treated as uncoupled to each other. A further study, including coupled states, where a complete description of both the direct and indirect mechanism is studied in a three-body breakup mechanism is the ﬁnal goal.

43 Appendix A

Deriving spin multiplicity factors for bosons and fermions

1 1. For H and T the nuclear spin, I = 2 , is half integer Hence for two identical particles we 1 1 have I = and I = . Thus 1 2 2 2 I = 0,..., I + I = 0, 1. (A.1) | 1 2| I = 0 implies antisymmetric spin while I = 1 is symmetric spin. Hence the spin multiplicity is 2I + 1 = 3, 1 (A.2) Thus multiplicity factors for the total wavefunction will be 1/4 for antisymmetric and 3/4 symmetric.

2. for D the nuclear spin, I = 1, is an integer. For two particles there spins are I1 = 1 and I2 = 1, thus I = 0,..., I + I = 0, 1, 2. (A.3) | 1 2| I = 0, 2 implies antisymmetric spin while I = 1 is symmetric spin. Hence the spin multiplicity is 2I + 1 = 5, 3, 1 (A.4) Thus multiplicity factors for the total wavefunction will be 1/3 for antisymmetric and 2/3 symmetric.

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