Static and dynamic NMR propertiesof gas-phase xenon

Matti Hanni

Department of PhysicsUniversity of OuluFinland

Academic dissertation to be presented with the assent of the Faculty of Science,University of Oulu, for public discussion in the Auditorium L10, on June 7th,2011, at 12 o’clock noon.

REPORT SERIES IN PHYSICAL SCIENCES Report No. 68OULU 2011 • UNIVERSITY OF OULU

OpponentProfessor Barbara Kirchner, University of Leipzig, Germany

ReviewersProfessor Barbara Kirchner, University of Leipzig, GermanyProfessor Martin Kaupp, Technische Universitat Berlin, Germany

CustosProfessor Juha Vaara, University of Oulu, Finland

ISBN 978-951-42-9456-3ISBN 978-951-42-9457-0 (PDF)ISSN 1239-4327Uniprint OuluOulu 2011

Matti Hanni: Static and dynamic NMR properties of gas-phase xenon

Department of Physics, University of Oulu, P.O.Box 3000, FIN-90014 Universityof Oulu, Finland, Report Series in Physical Sciences No. 68 (2011)

Abstract

This thesis presents computational studies of both the static and dynamic param-eters of the nuclear magnetic resonance (NMR) spectroscopy of gaseous xenon.First, state-of-the-art static magnetic resonance parameters are computed in smallxenon clusters by using methods of quantum chemistry, and second, time-de-pendent relaxation phenomena are investigated via molecular dynamics simula-tions at different experimental conditions. Based on the underlying quantum andclassical mechanics concepts, computational methods represent a procedure com-plementary to experiments for investigating the properties of atoms, molecules,clusters and solids.

Static NMR spectral parameters, chemical shift, shielding anisotropy and asym-metry parameter, nuclear quadrupole coupling, and spin-rotation coupling, arecalculated using different electronic structure methods ranging from the uncorre-lated Hartree-Fock method to correlated second-order Møller-Plesset many-bodyperturbation, complete/restricted active space multiconfiguration self-consistentfield, and to coupled-cluster approaches. The bond length dependence of theseproperties is investigated in the xenon dimer (Xe2). A well-characterized prop-erty in experimental NMR, the second virial coefficient of nuclear shielding, istheoretically calculated by a variety of methods and convincingly verified againstexperimental findings. Here, it is mandatory to include effects from special rel-ativity as well as electron correlation. As a side result, a purely theoretical po-tential energy curve for Xe2, comparable to best experimental ones, is calculated.A pairwise additive scheme is established to approximate the NMR properties indifferently coordinated sites of xenon clusters Xen (n = 2 − 12). Especially thepairwise additive chemical shift values are found to be in close agreement withquantum-chemical results and only a small scaling factor close to unity is neededfor the correct behavior. Finally, a dynamical magnetic resonance property, theexperimental nuclear spin-lattice relaxation rate R1 of monoatomic Xe gas due tothe chemical shift anisotropy (CSA) mechanism is validated from first principles.This approach is based on molecular dynamics simulations over a large range oftemperatures and densities, combined with the pairwise additive approximationfor the shielding tensor. Therein, the shielding time correlation function is seento reflect the characteristic time scales related to both interatomic collisions andcluster formation. For the first time, the physics of gaseous xenon is detailed infull in the context of CSA relaxation.

Keywords: xenon, nuclear magnetic resonance spectroscopy, electronic structure,potential energy curves, spectral parameters, pairwise additivity, molecular dy-namics, chemical shift anisotropy relaxation

To Henna, Helka and Sylvi

“If life seems jolly rotten,there’s something you’ve forgotten”

— Eric Idle

Acknowledgements

The work presented in this thesis was carried out in the NMR Research Group atthe Department of Physics of the University of Oulu during years 2004–2011. Theformer and present head of the Department of Physics, Prof. Jukka Jokisaari andProf. Matti Weckstrom, are thanked for providing an enthusiastic working atmo-sphere as well as facilities at my disposal. Prof. Jukka Jokisaari and the rest of theNMR community in Oulu are especially thanked for creating a spirited and livelyworking environment within the NMR Research Group. Financial support fromMagnus Ehrnrooth Foundation, Oskar Oflunds Stiftelse, Oulu University Schol-arship Foundation and Oulu University Pharmacy Foundation is acknowledged.

I would like to thank the reviewers of my thesis, Prof. Dr. Martin Kaupp andProf. Dr. Barbara Kirchner for their top-notch work. Valuable and helpful com-ments were received, thank you for that.

I would like to thank Dr. Nino Runeberg for the final calculations of Xe2 poten-tial energy curve in article I. The Dirac software with the GIAO feature used in ar-ticle II was kindly provided by Prof. Hans Jørgen Aagaard Jensen and Dr. MiroslavIlias prior to its official release.

Both of my supervisors, Prof. Juha Vaara and Dr. Perttu Lantto, have beenextremely and invaluably important for me throughout my Ph.D. studies. Yourmentoring have always been of exceptionally high quality. I am truly gratefulto both of you. Juha, you have given me both ambitious, yet realistic, projectsand a vast number of excellent tools to tackle them. Thank you so much foryour guidance, vision and efforts. Perttu, you have been and still are my closestnext-door co-worker. Your ability to think downright practically and to shed ascientific light on numerous issues has been ground-breaking at all stages of theresearch included in this thesis.

Teemu, you have always been a great thinker. We have had many good dis-cussions both on issues related to science and other aspects. Jouni, you’ve helpedme on numerous cases in the context of scientific computing. You also have illu-minating views on many topics in life. Jiri, I admire your humble attitude. All ofmy scientific family, including all the past and present members of the NMR Re-search Group in Oulu as well as in the Molecular Magnetism group in Helsinki,is fully acknowledged for excellent team spirit, memorable conference trips and

stimulating scientific and other discussions.A sincere thanks to my parents Eero and Maija for giving me a sceptical, but

optimistic attitude towards life, and for supporting me in countless ways duringmy life. I want to express my gratitude to my siblings Mari, Pekka, Tuomas,and Taneli and their families, for always being there for me and for numerousjoyful events. I am also indebted to my parents-in-law Erkki and Eeva, and to thefamilies of Henna’s siblings Heli, Petri, and Elina, for their immense hospitalityand support.

Dear Henna, you are the love of my life. I am deeply grateful for your infinitelove, understanding, and patience during the years. Our lovely daughters, Helkaand Sylvi, you put a smile on my face every single day of my life. You, my closestones, are truly wonderful. I dedicate this thesis to you.

Oulu, May 2011, Matti Hanni

List of Original Papers

This thesis consists of an introductory part and the following original articles,which are referred to in the text by their Roman numerals:

I Matti Hanni, Perttu Lantto, Nino Runeberg, Jukka Jokisaari, and JuhaVaara,Calculation of binary magnetic properties and potential energy curve in xenondimer: Second virial coefficient of 129Xe nuclear shieldingJournal of Chemical Physics 121, 5908 (2004).

II Matti Hanni, Perttu Lantto, Miroslav Ilias, Hans Jørgen AagaardJensen, and Juha Vaara,Relativistic effects in the intermolecular interaction-induced nuclear magneticresonance parameters of xenon dimerJournal of Chemical Physics 127, 164313 (2007).

III Matti Hanni, Perttu Lantto, and Juha VaaraPairwise additivity in the nuclear magnetic resonance interactions of atomicxenonPhysical Chemistry Chemical Physics 11, 2485 (2009).

IV Matti Hanni, Perttu Lantto, and Juha VaaraNuclear spin relaxation due to chemical shift anisotropy of gas-phase 129Xe,manuscript submitted for publication.

The author of this thesis carried out most of the computational work, includingthe quantum-chemical and molecular dynamics calculations, as well as the anal-ysis of the data, for all the articles. Dr. Perttu Lantto performed the Breit-Paulicalculations in article II. The final electronic structure calculations of the state-of-art potential energy curves utilizing bond basis functions in article I were carriedout by Dr. Nino Runeberg. The first versions of the manuscripts were preparedby the author (except for the chapter concerning Breit-Pauli calculations in arti-cle II), and the articles were finished as team work. The codes used to analyze thepairwise additivity and relaxation processes in articles III and IV, respectively,were programmed by the author.

Abbreviations

ACF autocorrelation functionAE all-electronBPPT Breit-Pauli perturbation theoryBSSE basis set superposition errorCAS/RAS complete/restricted active spaceCCSD coupled-cluster singles and doublesCCSD(T) CCSD with non-iterative triplesCP counterpoise (correction)CSA chemical shift anisotropyDFT density-functional theoryD(H)F Dirac-(Hartree)-FockECP effective core potentialEFG electric field gradientFFT fast Fourier transformGIAO gauge-including atomic orbitalGTO Gaussian-type orbitalHF Hartree-FockMCSCF multiconfigurational self-consistent fieldMD molecular dynamicsMPn nth-order Møller-Plesset many-body perturbation theoryNMR nuclear magnetic resonanceNQC nuclear quadrupole couplingNR nonrelativisticPAA pairwise additive approximationPES/C potential energy surface/curveQC quantum-chemical

RECP relativistic effective core potentialRH Roothaan-Hall equationsRKB restricted kinetic balanceSCF self-consistent fieldSDF spectral density functionSR spin-rotationTCF time correlation functionUKB unrestricted kinetic balance

Contents

AbstractAcknowledgementsList of Original PapersAbbreviationsContents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1 Xenon NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2 Objectives and structure of the thesis . . . . . . . . . . . . . . . . . . 17

2 NMR spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Nuclear shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Nuclear quadrupole and spin-rotation coupling . . . . . . . . . . . 21

3 NMR relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1 Time correlation and spectral density functions . . . . . . . . . . . . 243.2 Redfield relaxation theory . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.1 CSA relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Other relaxation mechanisms in gaseous xenon . . . . . . . . . . . . 283.4 Molecular dynamics simulations . . . . . . . . . . . . . . . . . . . . 29

4 Electronic structure methods . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1 Hierarchy of Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Molecular Hamiltonians in the presence of magnetic field . . . . . . 33

4.2.1 Nonrelativistic case . . . . . . . . . . . . . . . . . . . . . . . . 334.2.2 Relativistic case . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Magnetic tensors from an energy expansion . . . . . . . . . . . . . . 364.3.1 Nonrelativistic case . . . . . . . . . . . . . . . . . . . . . . . . 374.3.2 Relativistic case . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4 One-electron picture . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4.1 Basis sets and pseudopotentials . . . . . . . . . . . . . . . . . 39

4.4.1.1 Kinetic balance . . . . . . . . . . . . . . . . . . . . . 414.5 Many-electron effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5.1 Many-electron wave function . . . . . . . . . . . . . . . . . . 424.5.2 Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.5.2.1 Dirac-Hartree-Fock and Levy-Leblond methods . . 44

4.5.3 Multiconfigurational Hartree-Fock . . . . . . . . . . . . . . . 444.5.4 Coupled-cluster techniques . . . . . . . . . . . . . . . . . . . 464.5.5 Perturbational methods . . . . . . . . . . . . . . . . . . . . . 464.5.6 Density-functional theory . . . . . . . . . . . . . . . . . . . . 47

4.6 Scaling features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.7 Computational software used . . . . . . . . . . . . . . . . . . . . . . 48

5 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.1 Magnetic properties of xenon dimer . . . . . . . . . . . . . . . . . . 50

5.1.1 Potential energy curves . . . . . . . . . . . . . . . . . . . . . 535.1.2 Second virial coefficient of nuclear shielding . . . . . . . . . 55

5.2 Pairwise additive approximation . . . . . . . . . . . . . . . . . . . . 575.3 Computational approach to CSA relaxation . . . . . . . . . . . . . . 60

6 Conclusions and future prospects . . . . . . . . . . . . . . . . . . . . . . . 64References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Original papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

1 Introduction

Computational science has established itself, alongside experimental and theo-retical work, as the third cornerstone of modern molecular research. It providesmethods and algorithms to study the physical properties of matter on theoreticalgrounds, independently of the experimental setup. It should be regarded as bothan application of the fundamental theory and a complement to the experiments.It enables the calculation of the outcomes of theory, and probes its limits in ex-perimental conditions. Nuclear magnetic resonance (NMR) spectroscopy [1, 2],offers a rich phenomenological framework in many scientific disciplines, rangingfrom basic and applied research in physics and chemistry [3–11] to applied areas,such as medicine [12–15] and biology [16–19]. The spectral parameters of NMRcan be studied either experimentally or by computational methods in all phasesof matter.

In NMR spectroscopy, the sample is exposed to a large external magnetic field.Radio-frequency pulses are used to excite and observe transitions between nu-clear Zeeman energy levels, which are due to the interaction of the nuclear mag-netic dipole moment with the external magnetic field. The details of the energylevels arise from the modification of the nuclear Zeeman interaction by the sur-rounding electrons and nuclei. Other degrees of freedom, such as the nuclearand electronic positions, are embedded into the NMR parameters, which canbe studied experimentally or non-empirically with quantum mechanical calcula-tions [20]. Attempts to provide a description of the quantum mechanical behaviorof atoms and molecules usually start from the non-relativistic (NR) Schrodingerand relativistic Dirac equations. A common approximation is to work in the Born-Oppenheimer approximation, i.e., to separate the motion of electrons and nucleibased on their large mass difference, which allows electrons to follow the mo-tion of nuclei instantly. Approximate electronic structure is therefore solved fora given set of fixed nuclear positions at a time. By changing the nuclear posi-tions, also the NMR parameters change. Electron-electron interactions are mostoften described by a Coulomb operator. Besides standard NR approach, differ-ent quasi-relativistic as well as fully relativistic frameworks have been devised todeal with the calculation of NMR parameters [21–23].

Radio-frequency radiation can be used to create a non-equilibrium nuclear

16

spin state. The evolution back to thermal equilibrium state takes place via time-dependent spin-lattice and spin-spin interactions, capable of inducing transitionsbetween nuclear Zeeman spin states. NMR spin relaxation [24] consists of theseprocesses. The time dependence can be accessed in molecular dynamics (MD)simulations. Therein, the potential energy surface (PES) defines the forces be-tween atoms. Subsequently, the positions of the interacting particles at each dis-crete time step are obtained. From the positional data, the NMR properties are ob-tained either by electronic structure calculations of instantaneous MD snapshots(supermolecules), or by computationally much lighter a priori parametrizationsbased on smaller molecular entities present in the system.

1.1 Xenon NMR

Xenon holds a special position in the realm of NMR [3, 25, 26]. 129/131Xe nucleiare used as inert, atomistic and non-toxic probes to study the structure and dy-namics of various host materials. The importance of xenon has been enhanced byoptical pumping methods [27–29], in which the nuclear spin polarization, and,ultimately, the experimental signal-to-noise ratio is increased. A great portion ofthe success of xenon owes to its large and sensitive electron cloud, which readilyresponds to changes in the environment. These are reflected in the NMR param-eters.

Probably the experimentally best-characterized intermolecular interaction ef-fect in NMR is the second virial coefficient [30, 31], σ1, of 129Xe nuclear shieldingin gaseous xenon. It is the first correction on top of the nuclear shielding of anatom in vacuo, describing the pair interactions of the probe atom and the speciesin its surroundings. Similarly, higher-order interactions exist, which are more sig-nificant for complex materials than for the xenon gas studied here. Due to heavycomputational burden, reasonably accurate theoretical evaluations [32–34] of σ1

have become feasible much later than the experiments [35, 36]. In particular forXe, the inclusion of both electron correlation and relativity at a reasonable basisset level is required in quantum-chemical calculations.

Pairwise additive interatomic potential energy functions are used in the simu-lations of materials science, molecular physics, and chemistry [37]. For the xenondimer, both empirical [38, 39] and theoretical [40] ground-state potential energycurves (PECs) are found in the literature. Should the pairwise additive approxi-mation (PAA) be suitable for the interaction-induced NMR parameters as well, isunclear. It would enable a complete and straightforward parametrization of theNMR tensors, carried out in advance for the evaluation of time-dependent NMRproperties. This allows the processing of configurations from molecular simula-tion trajectories without computationally expensive electronic structure calcula-tions on instantaneous simulation snapshots.

First 129/131Xe NMR relaxation experiments in gaseous and liquid xenon dateback to 1950-60s [12, 24]. Much as the static NMR parameters, the xenon NMRrelaxation times also serve as a probe of the local environment in materials re-

17

search. Moudrakovski et al. [41] confirmed that the main relaxation mechanismsin gaseous 129Xe are the relaxation due to the interaction of the nuclear spin androtational angular momentum of the molecule (the spin-rotation, SR, interaction)and the magnetic-field dependent relaxation via the modulation of the local mag-netic field due to chemical shift anisotropy (CSA). The theoretical background forNMR relaxation, inherently giving impetus to the use of MD simulations in thisfield of research, is mainly due to work of Redfield [42, 43].

1.2 Objectives and structure of the thesis

The objective of the thesis is to study both the static and dynamic NMR param-eters of xenon with computational methods. Highly accurate quantum-chemical(QC) calculations are carried out for both the magnetic tensors and PEC, withan emphasis on relativity, and on the systematic evaluation of both one- andmany-electron effects on these properties. PAA is tested against QC results forNMR parameters in the clusters of xenon. Finally, MD simulations of CSA T1

relaxation times for Xe(g) are carried out over a large range of experimental con-ditions. The results in this thesis partially lay the basis for further studies onthe NMR properties of materials in more complex environment, by expandingthe standard computational toolbox of NMR to include easily-grasped concepts,such as the coordination number.

In articles I and II of the thesis, a series of calculations of the NMR shield-ing and quadrupole coupling tensors are presented as a function of interatomicdistance in Xe2. The spin-rotation tensor is discussed in article I. Therein, wealso calculate one of the most accurate, entirely theoretical PECs for Xe2 to date.The basis-set limit for NMR parameters is approached by either incremental in-clusion of basis functions on top of a large Gaussian set of primitives [44] inthe NR framework, or by using a well-defined basis set family [45] in relativis-tic calculations. Electron correlation effects for these properties are systemati-cally studied, ultimately at the coupled cluster singles and doubles with non-iterative triples [CCSD(T)] level of theory. Relativistic effects are taken into ac-count using either a fully relativistic four- or a quasi-relativistic, perturbationalone-component method. Within the latter, the coupling of electron correlationand relativity is evaluated explicitly. Subsequently, an accurate description ofthe temperature dependence of the second virial coefficient of the nuclear shield-ing, σ1(T ), is achieved. The computational confirmation of the original experi-ments [36] emerges from the work presented in articles I and II.

PAA parametrizes a molecular property, such as nuclear shielding, computedwith electronic structure methods at different bond lengths of the Xe dimer. Thus,once parametrized, the computation of an NMR parameter only involves the po-sitions of the nuclei in a system. In article III, PAA is utilized for the NMR param-eters in differently coordinated atomic sites corresponding to different phases ofxenon. The PAA results are compared with direct Hartree-Fock (HF) electronicstructure calculations carried out for the full clusters. A good balance between

18

these results, especially for the chemical shift, is obtained. The coordination num-ber is found to be the fundamental predictive ingredient for the magnitude of thechemical shift at given atomic site. A qualitative explanation of the dependenceof the shift on the electronic ground and excited states is given in terms of theparamagnetic nuclear spin-electron orbit operator.

In article IV, a quantitative account on the 129Xe CSA NMR relaxation is ob-tained on purely theoretical grounds. The scheme employed directly utilizes MDsimulation trajectories, for which the PAA (article III) for the nuclear shieldingtensor is used. The interaction potential responsible for the trajectories in the MDsimulations is obtained from article I, while the binary chemical shift and shield-ing anisotropy property curves from article II are used in the calculation of CSAautocorrelation functions (ACFs) in the context of Redfield relaxation theory.

The layout of the thesis is as follows. In chapters 2 and 3, the basic conceptsof NMR and nuclear spin relaxation are presented, respectively. Chapter 4 out-lines the electronic structure methods needed for the calculation of NMR spectralparameters. In chapter 5, the results from the articles I-IV are summarized. Inthe final chapter 6, conclusions are drawn and future research prospects are pre-sented.

2 NMR spin Hamiltonian

The NMR spin Hamiltonian [1, 46] is used to interpret and/or predict the exper-imental NMR spectrum. It is a phenomenological energy expression, where onlythe magnetic nuclei of the molecular system described by their intrinsic nuclearspins, IK , as well as the uniform magnetic flux density of the NMR spectrometer,B0, are explicitly taken into account. All the other degrees of freedom, such asparticle positions, are averaged over and embedded into the parameters of theNMR Hamiltonian, the NMR tensors. The NMR spin Hamiltonian (in frequencyunits) is written as

HKNMR = − 1

2π

∑K

γKIK · (1− σK) ·B0 +∑K<L

IK · (DKL + JKL) · IL

+∑K

IK ·BK · IK = HNZ +Hσ +HD +HJ +HB (2.1)

where IK(> 12 ) is related to the nuclear magnetic dipole moment as

µK = γK~IK . (2.2)

Here γK is the nuclear gyromagnetic ratio, andσK and BK are the nuclear shield-ing and quadrupole coupling tensor of nucleus K, respectively. The tensors DKL

and JKL are the direct dipole-dipole and indirect spin-spin coupling tensors, re-spectively, between nuclei K and L. The dominant term in Eq. (2.1) is the nu-clear spin Zeeman (NZ) interaction, HNZ = − 1

2π

∑K γKIK ·B0. The Hamiltoni-

ans Hσ , HJ , HB can be regarded as small perturbations arising from the electroncloud, which mediates interactions of the nuclear spin with the external magneticfield and the other nuclear spins, as well as from the interaction of the nuclearquadrupole moment with electric field gradient at the nuclear site, respectively.In addition, HD describes the direct through-space interaction of two nuclearmagnetic dipole moments. When the nuclear quadrupole moment is zero, i.e.,for nuclei with spin quantum number IK < 1, BK vanishes.

The NMR parameters are generally expressed as second-rank tensors. Theyare time- and ensemble averages of all similar nuclei in the sample, and contain

20

information about the local structure and dynamics of the surroundings of thenuclei. Experimentally, a reduction to observable NMR spectral parameters iscarried out according to the motional averaging properties of the phase of matter.In normal isotropic liquids and gases, due to the tumbling of the molecules, thespin Hamiltonian reduces 1 to

HKiso = − 1

2π

∑K

γK(1− σK)IK,ZB0 +∑K<L

JKLIK,ZIL,Z , (2.3)

where the remaining NMR spectral parameters are the shielding and the spin-spin coupling constants, calculated as the traces of the corresponding tensors:

σK =13

TrσK ; JKL =13

Tr JKL (2.4)

σK and JKL define the spectral peak positions and splittings, respectively. Theaverages of the traceless tensors DKL and BK are zero in isotropic phases. Inanisotropic phases, such as the liquid crystalline or solid-state environment, theNMR spectrum is enriched by quadrupolar and dipolar couplings, as well asanisotropic parts of σK and JKL. From QC electronic structure calculations thefull tensors are available, although the detailed description of rovibrational, rela-tivistic, electron correlation and solvent effects is often resource-consuming.

2.1 Nuclear shielding

Nuclear shielding tensor, σK , describes a small deviation of the magnetic fluxdensity from B0 at the site of nucleus K:

BK = (1− σK) ·B0. (2.5)

This deviation is explained by the fact that the magnetic field B0 induces elec-tric currents in the electron cloud, and, subsequently, these circulating currentsmodify the local magnetic field BK . The term −σK · B0 usually decreases thetotal field, hence the name shielding tensor. This means that the positive diamag-netic component of σK dominates over the negative paramagnetic one. This isdue to the quantum-mechanical properties of the electron cloud in ground andexcited electronic states. In all cases, σK produces a very small correction to theprevailing B0 field.

The chemical shift is the difference of the shielding constants (or resonancefrequencies νK) measured for the same nucleus in two different environments. Itis defined as

δK =νK − νK,ref

νK,ref=σK,ref − σK1− σK,ref

≈ σK,ref − σK , (2.6)

1Carried out, for simplicity, in the heteronuclear high-field approximation [46], where only thefirst-order perturbation theory corrections for the energy of the spin system are included [47].

21

where a suitable reference shielding constant σK,ref is chosen to experimentallyaccount for the small changes (of the order of parts per million, ppm) betweenthe nuclei in different environments as depicted in the NMR spectra. For xenon,chemical shift ranges of up to 350 ppm in solution and 7500 ppm in compoundsare observed [12, 48], corresponding to the fact that xenon nuclei are surroundedby a large and easily polarizable electron cloud. For these reasons, the chemicalshift sensitively reflects the properties of various host materials [3, 4, 10, 18, 25, 26,49–51]. For the xenon systems studied in this thesis, the approximation to removethe denominator in Eq. (2.6) produces a maximum relative error of around 1%.

The pair- and higher-order interactions affecting a molecular electromagneticproperty of a non-ideal gas/liquid are classically expressed in a virial expan-sion [30] as a function of the number density of the medium n = N/V and tem-perature T . Particularly for the shielding constant, this procedure yields [31]

σ(n, T ) = σ0 + σ1(T )n+ σ2(T )n2 + · · · . (2.7)

Here, σ0 is the shielding constant in vacuo. The second, linearly density-dependentterm contains σ1(T ), the second virial coefficient of nuclear shielding [31]. Arti-cles I and II focus on the physics of this property, as well as on the PEC andbinary NMR tensors of Xe2. Quantum dynamics of He, Ne and Ar nuclei affectσ1(T ) only a little [52]. Based on this, also the σ1(T ) of the heavier xenon could bedescribed classically. In mixtures of gases [53], there is one linear term describingeach pair interaction of the constituent gas components. Accordingly, also higherinteractions exist in dense and/or ordered systems.

2.2 Nuclear quadrupole and spin-rotation coupling

Nuclear quadrupole coupling (NQC) BK describes the interaction of the nuclearelectric quadrupole moment (NQM) eQK with an electric field gradient (EFG)tensor FK at the nuclear site K. It can be written as

BK =χK

2IK(2IK − 1)(2.8)

where χK is the nuclear quadrupole coupling tensor. It is directly proportionalto the EFG tensor with a relation

χK = −eQKh

FK . (2.9)

The EFG tensor results from the electrostatic potential U as

Fαβ ≡∂

∂αEβ = − ∂2U

∂α∂β, (2.10)

where E is the electric field, and α, β label the coordinates x, y, z. For nuclei withIK > 1, with non-zero EFG at the site of nucleus, the quadrupolar interaction is

22

quite strong and often dominates also NMR relaxation. Since the NQM tensor isexpressible in terms of nuclear spin operators IK [2, 54], the perhaps surprisingform of the quadrupolar interaction,

∑K IK ·BK · IK , is possible.

When a molecule rotates in the sample, the moving electrons and nuclei createa local magnetic field, which is experienced by the nuclear magnetic dipole mo-ment. The spin-rotation tensor, CK , describes the interaction of the nuclear spinswith the total angular momentum of the molecule, L. The phenomenologicalspin-rotation Hamiltonian [55] is written as

HC = −1~∑K

IK ·CK · L. (2.11)

For free molecules in dilute gases, L is a well-defined quantity, thus enablingthe measurement of CK . In liquids, although the local magnetic field causedby the SR interaction is quite large [56], the SR correlation time (vide infra) is,however, roughly an order of magnitude smaller than for simple reorientationalmotion. This hinders the NMR measurement of the spin-rotation interaction. Insolids, the environment is rotationally restricted, which reduces the magnitudeof spin-rotation interaction. Due to these issues, CK is not commonly consideredas an NMR spectral parameter. However, it is meaningful in the context of NMRrelaxation [57] for example for dilute gases and for rotationally free molecules.

NMR tensors involving interactions of two spins

The dipole-dipole coupling tensor, DKL, addresses the direct through-space cou-pling of two nuclear magnetic dipole moments. It can be formulated analyticallywithin classical magnetostatics [58]. The spin-spin coupling tensor, JKL, depictsthe interaction of the two nuclear magnetic dipole moments mediated by the elec-tron cloud surrounding the constituent nuclei. The spin-spin coupling tensor canbe evaluated as a second-order electronic energy derivative [59]. No calculationshave been carried out for either JKL or DKL in this thesis.

3 NMR relaxation

The study of NMR relaxation processes was pioneered by Bloch [60], who devel-oped phenomenological equations describing the time evolution of the macro-scopic nuclear magnetization M = nγK~〈IK〉 in the laboratory coordinate system(X,Y, Z):

dMX

dt= γK(B0MY +B1MZ sinω1t)−

MX

T2

dMY

dt= γK(−B0MX +B1MZ cosω1t)−

MY

T2(3.1)

dMZ

dt= γK(−B1MX sinω1t−B1MY cosω1t)−

MZ −M0

T1

where 〈 〉 denotes ensemble average, n is the number density of nuclei, M0 is theequilibrium value of the magnetization, and T1 and T2 are the longitudinal (spin-lattice) and transverse (spin-spin) relaxation times, respectively. Eqs. (3.1) containthe Larmor precession term consisting the effect of both the static external fieldB0 and the so-called resonant or transverse field B1. In addition, all componentsof the magnetization follow a monoexponential decay, either to zero (MX,Y ) or toa non-vanishing equilibrium value of M0 (MZ). These decays have characteristictime constants, T2 and T1, respectively. The equilibrium value M0 depends on B0

and the magnitude of µ, µ = γK~√IK(IK + 1) , as

M0 =nµ2

3kTB0. (3.2)

In general, NMR relaxation rates R1 = 1/T1 and R2 = 1/T2 depict how rapidly,after a perturbation of the spin system, the thermal spin equilibrium state isreached again. Specifically, T1 and T2 are the time scales involved in the energyexchange of the spin system and the surroundings and loss of phase coherence inthe motion of individual spins, respectively [46].

An advanced description of the NMR relaxation is obtained from Redfield the-ory [42, 43], followed in this thesis. The underlying work for the Redfield theory

24

is described in Refs. [61–66], and accounts similar to that of Redfield were in-dependently given by Bloch [67] and Hubbard [68]. Before entering relaxationtheory, a quick survey on the basic properties of the time correlation and spectraldensity functions, which are essential in the Redfield theory, is presented.

3.1 Time correlation and spectral density functions

The time correlation function (TCF) of two (real) functions A(t) and B(t) is de-fined as

C(t) = 〈A(t′);B(t′ + t)〉 =∫ ∞−∞

A(t′)B(t′ + t) dt′. (3.3)

When A = B, the term autocorrelation function (ACF) is used, and when A 6= B,one talks about cross-correlation function (CCF). As t′ is averaged over, one canchoose t′ = 0, which yields C(t) = 〈A(0);B(t)〉. The normalized ACF of A(t) is

Cnorm(t) =〈A(0);A(t)〉〈A(0);A(0)〉

. (3.4)

By integrating Eq. (3.4) one obtains the correlation time τA that measures the char-acteristic time in which the autocorrelation of A(t) is lost:

τA =∫ ∞−∞

〈A(0);A(t)〉〈A(0);A(0)〉

dt (3.5)

The spectral density function (SDF) is a Fourier transform of the ACF1

J(ω) =∫ ∞−∞

C(t) exp(−iωt) dt (3.6)

Thus, C(t) and J(ω) form a Fourier transform pair [69].One may carry out the calculations directly by replacing the continuous inte-

gration in the definition of TCF [Eq. (3.3)] by a discrete summation of the func-tion values, as represented in Ref. [70]. It is, however, more efficient to use fastFourier transform (FFT) [71] techniques to calculate the TCF for large data sets.In this scheme, one first calculates the FFT of an individual discretized functionA(t), multiplies this in the frequency domain with its complex conjugate, and fi-nally does the inverse FFT in the frequency domain to end up with the ACF. Inarticle IV, an FFT algorithm adopted from Numerical Recipes [72] was used tocalculate the ACFs for the nuclear shielding tensor components from differentsimulation snapshots. The SDF was then acquired by another FFT.

1In this case, ACF has to describe a stationary process of random, or stochastic nature (the Wiener-Khinchin theorem).

25

3.2 Redfield relaxation theory

Redfield theory [42, 43] is a weakly-coupled semi-classical formulation for thetime evolution of nuclear magnetization. It is semi-classical in the sense that onlythe spin system is treated quantum mechanically and the other degrees of free-dom, i.e., the bath is treated classically. The weak interactions between the spinsand bath are depicted by time-dependent NMR tensors, capable of relaxing thesystem towards the thermal equilibrium state. Reviews on Redfield theory – fol-lowed here – are given in Refs. [24, 73, 74]. In this thesis, the CSA relaxation mech-anism is focused on, for which the Redfield theory is detailed in Refs. [75, 76].

The relaxation part of the equation-of-motion of the component ε of the mag-netization M is, within the Redfield theory,

dMε

dt=

∑αα′ββ′

Rαα′ββ′ρ(t)ββ′(Mε)α′α = −Mε −M0

T, (3.7)

where the Redfield relaxation matrix Rαα′ββ′ is written as

Rαα′ββ′ = Jαβα′β′(ωα − ωβ) + Jαβα′β′(ωα′ − ωβ′)

−δαβ∑κ

Jκα′κβ′(ωκ − ωβ′)− δα′β′

∑κ

Jκβκα(ωκ − ωβ). (3.8)

Here α, β, κ label the nuclear spin Zeeman states, and Jαβα′β′(ω) is the Fouriertransform of the TCF 〈Hαβ(0);Hβ′α′(t)〉. In the second equality of Eq. (3.7), theusual connection with the phenomenological Bloch equations [Eqs. (3.1)] is made,enabling the use of longitudinal (T = T1, ε = Z) and transverse (T = T2, ε = X,Y ,and M0 = 0) relaxation times.

The SDF of the perturbing, time-dependent NMR Hamiltonian H(t) is writtenas

Jαβα′β′(ω) =12

∫ ∞−∞〈Hαβ(0);Hβ′α′(t)〉 exp(−iωt) dt. (3.9)

Here, Hαβ(t) is the matrix element of the Hamiltonian between states α and β.For NMR relaxation processes, H(t) is usually decomposed in terms of spher-

ical tensor components

H(t) =2∑l=0

l∑q=−l

(−1)qh(l,−q)(t)A(l,q), (3.10)

where l = 0, 1, 2 is the rank of the tensor. Time dependence is in the NMR inter-action tensor h(l,−q)(t), whereas A(l,q) is a time-independent spin operator, whichalso includes B0. In this approach, the total SDF is written as

Jαβα′β′(ω) =2∑l=0

l∑q=−l

j(ω)A(l,q)αβ A

(l,q)∗α′β′ (3.11)

where the time-dependent terms are collected to j(ω)

j(ω) =∫ ∞

0

〈h(l,−q)(0);h(l,−q)(t)〉 cos(ωt) dt. (3.12)

26

As explained in Ref. [73], the integration limits in Eq. (3.12) arise from the factthat 〈Hαβ(0);Hβ′α′(t)〉 is an even function. Here, j(ω) is the Fourier transform ofthe TCF calculated only for the time-dependent tensorial part of the perturbationHamiltonian.

In the most general case, the full TCF would contain all the ACFs and CCFsof all the time-dependent NMR interaction Hamiltonians, Eqs. (2.1) and (2.11).However, some of the ACFs/CCFs can be excluded directly based on the factthat the NMR parameters with different ranks give a zero CCF, or due to the ex-perimentally observed small magnitude of the involved static NMR parameters,or based on the different time scales of the NMR tensor fluctuations. The cross-correlation effects can be substantial for example for a combination of the CSAand the dipolar coupling mechanisms [24]. While the individual strengths of thecorrelation mechanisms are based on the properties of the corresponding TCF,i.e., the dynamics of the individual NMR tensors, the overall relaxation rate isalso affected by the outcome of the spin operators.

Redfield theory also has shortcomings. It cannot recover the correct equilib-rium population distribution amongst the nuclear spin levels at a finite tempera-ture, but uses a phenomenological correction [77]. This is due to the fact that therates of transitions from α to β and vice versa are equal, i.e., the transition proba-bility Rααββ = Rββαα. This would mean that the population of α will equal thatof β in the thermal spin equilibrium state, in contradiction of the true Boltzmanndistribution of populations. However, a fully quantum mechanical formulationof both the spin system and the bath [1] yields the same factor that is insertedad hoc in Redfield’s quantum-mechanical spin system – classical bath equations.The Redfield theory also assumes a weak spin-bath interaction [74]. Generally,the applicability range of the Redfield theory is presented with the time scales [2]

τc � ∆t� 1Rαα′ββ′

. (3.13)

The former inequality stems from the requirement that changes in the magnetiza-tion have to be recorded at time interval ∆t, which is much longer than the char-acteristic correlation time τc. Computationally, this means that the TCF must bemuch longer than its τc, allowing the integration to infinity (vide infra) [42]. Thelatter inequality between the inverse of the transition probability (∝ relaxationtimes) and ∆t is due to the fact that the weak time-dependent relaxation term inthe equation-of-motion for the magnetization is a lot smaller than the dominantNZ interaction term (Larmor precession term). In other words, one ensures thatthe magnetization does not change too drastically at interval ∆t.

3.2.1 CSA relaxation

The chemical shift anisotropy relaxation rate is unique amongst the nuclear spinrelaxation mechanisms due to its quadratic magnetic field-dependence of the re-laxation rate, R1 = 1/T1 ∝ B2

0 [1]. The CSA relaxation is caused by the time-

27

dependent fluctuations of σK(t), i.e., the time-dependence of the local magneticfield experienced by the nucleiK, BK(t) [see Eq. (2.5)]. The fluctuations can arisefrom the interatomic collisions of the particles, cluster formation, and molecularrotation with respect to B0. Article IV details the underlying physical frameworkof the CSA relaxation mechanism in gas-phase xenon. The molecular origins,in this case, have been claimed to arise besides transient collisions between Xeatoms, from the long-lived (called persistent), weakly-bound Xe2 dimers [78–80].

The relevant CSA relaxation Hamiltonian is rewritten from Eq. (2.1) as

HKNMR(t) = − 1

2πγKIK · [1− σK(t)] ·B0 ≡ HNZ +Hσ(t), (3.14)

Note that the explicit time-dependence is assigned to shielding tensor, whereasthe spin system (IK) is time-independent. In Redfield theory for CSA relaxation,the final expressions for the longitudinal and transverse relaxation rates are writ-ten in terms of the SDF at different frequencies as [75]

R1 =1T1

= 6B20j(ω0) ; R2 =

1T2

= B20 {4j(0) + 3j(ω0)} , (3.15)

where ω0 = γKB0 is the Larmor frequency. A description of the SDF, j(ω), neces-sitates a calculation of the TCFs of the interacting time-dependent CSA Hamilto-nian. One cannot usually directly address the SDF in experiments.2

The shielding tensor has a total of nine spherical components, each composedas a linear combination of the Cartesian tensor components [77]. The sphericaltensor is divided into tensorial ranks 0, 1, and 2. The rank-0 part is a scalar cor-responding to the shielding constant. It shifts the nuclear spin Zeeman levels bya constant amount, and is incapable of causing transitions between them. Therank-1 components cause the often-neglected [84], so-called antisymmetric effecton the CSA relaxation [85, 86]. Antisymmetric shielding tensor components arediscussed within the context of CSA relaxation in detail in Ref. [76]. In particular,approximate formulae for the symmetric as well as antisymmetric Cartesian com-ponents of the shielding tensor for T1 have been given in the literature [87, 88].

The most important, rank-2 (anisotropic but symmetric and traceless) part ofthe CSA perturbation Hamiltonian [second term in Eq. (3.14)] reduces to [75]

Hσ(t) =2∑

q=−2

(−1)qh(−q)(t)A(q), (3.16)

where the label l = 2 is omitted for simplicity. The time-independent parts (A(q))of the Hamiltonian are written as [75]

A(0) = 2B0IZ ; A(±1) = ∓√

62B0I± ; A(±2) = 0, (3.17)

2However, it is possible to obtain j(ω) from NMR relaxation dispersion measurements [81, 82] orwith NMR relaxometry [83].

28

where the last equality follows from the choice B0 = (0, 0, B0) (external fieldalong the Cartesian Z-axis of the laboratory frame). The necessary spherical com-ponents of the shielding tensor are, consequently,

h(0) =12γKσZZ ; h(±1) =

1√6γK(∓σXZ − iσY Z), (3.18)

in terms of the Cartesian shielding tensor components σεκ (ε, κ = X,Y, Z), with idenoting the imaginary unit. The final perturbation Hamiltonian is [75]

H(t) = B0

[2h(0)IZ +

√6

2h(−1)I+ −

√6

2h(1)I−

], (3.19)

where the raising (I+ = IX + iIY ) and lowering (I− = IX − iIY ) spin operatorsare introduced. Thus, Eq. (3.19) implies a calculation of three distinct TCFs, corre-sponding to the spherical shielding tensor components h(0), h(±1) [75]. The totalTCF is proportional to the sum of the individual TCFs of these components:

TCF(t) ∝1∑

q=−1

〈h(−q)(0);h(−q)(t)〉 ; j(ω) = F {TCF(t)} . (3.20)

Thus, a Fourier transformation (denoted by F) finally leads to the CSA spin-lattice and spin-spin relaxation rates, given by Eqs. (3.15).

3.3 Other relaxation mechanisms in gaseous xenon

The spin-rotation interaction is the dominant relaxation mechanism in gaseousxenon at low magnetic fields [89]. Scalar relaxation, linked to the tensor JKL(t),is assumed to be small based on the values of the Xe-Xe spin-spin coupling con-stants calculated in Refs. [90, 91]. The scalar relaxation is also experimentallyfound to be negligible [89, 92]. For nuclei with spins IK > 1 (such as 131Xe), thequadrupolar relaxation usually dominates. Dipolar relaxation is of minor impor-tance in the NMR relaxation of xenon in gaseous state, based on comparison ofanalytical theoretical models and experimental results [93, 94]. If desired, it canbe analytically evaluated from the positional data of an MD trajectory (vide in-fra). The presence of paramagnetic impurities affects the relaxation rates of xenondue to strong interactions between the unpaired electrons and the Xe nucleus, aswitnessed in a mixture of xenon and oxygen [95].

In general, only few experimental NMR methods exist to distinguish betweenthe different relaxation mechanisms [84]. The most obvious way to separate be-tween inter- and intramolecular relaxation mechanisms is to simply dilute thesample. However, this leads to changes in density and viscosity, which also affectthe characteristic relaxation times. To avoid this, the so-called isotopic dilution,i.e., the modification of the abundancies of the isotopes, can be applied to theextent that the inter- and intramolecular interactions (usually the dipole-dipole

29

and spin-spin relaxation mechanisms) affecting relaxation can be separated. Inthe nuclear Overhauser effect [46], heteronuclear dipolar relaxation can be sep-arated out from all other relaxation mechanisms. Moreover, the spin-rotationrelaxation rate increases as the temperature increases, as opposed to all other re-laxation rates [89]. The reorientational correlation of the underlying moleculardynamics is lost more rapidly as the temperature increases [56], thus decreasingthe relaxation rate of the other relaxation mechanisms. The sole exception, theSR interaction, becomes stronger as the temperature increases. This is related tomolecules rotating at larger angular velocity ω,

Iω = L~ (3.21)

at elevated temperatures. Here, I is the moment of inertia tensor of the rotatingmolecule.

Whereas the experimental evaluation of the strength and importance of the in-dividual NMR relaxation processes is a hard task, theoretical modeling of specificNMR relaxation processes, for example by molecular dynamics simulations, isfairly straightforward. The advantages of MD modeling of NMR relaxation [96–101] lie in the easy preparation of the different phases of matter, the possibility ofdetailed analysis in terms of the different relaxation mechanisms, as well as in theindependent calculation of dynamical properties related to the molecular motionand clustering, such as correlation times. In addition, pairwise additive schemes(see article III), can be invoked to ease the computational burden related to the di-rect QC calculation of the instantaneous interaction Hamiltonians, i.e., the NMRtensors.

3.4 Molecular dynamics simulations

MD simulations [70] mimic the dynamics of the atomic/molecular ensemble indiscretized time steps. Although MD is routinely based on pre-parametrized em-pirical force fields, ab initio MD, providing in some cases more realistic interac-tions often within the density-functional theory (DFT) “on the fly”, started its risealready in the 1980s [102].

MD simulations essentially consist of two time-consuming operations: theevaluation of forces between the constituent atoms based on the positional dataand the integration of the particle trajectories to give the new potential and forces.Putting these into equations, Newton’s second law

ai = vi = ri =Fim

(3.22)

connects the acceleration a, velocity v and the position r of particle i with a massm and the force F on it. Forces F arise from the PES, V , as

Fi = −∇Vi = −(∂Vi∂xi

,∂Vi∂yi

,∂Vi∂zi

). (3.23)

30

Many schemes for combined force evaluation and trajectory integration ex-ist, varying in the update schemes of the positions, velocities, and accelerations.Amongst the most popular methods are the leap-frog algorithm [103] and thevelocity-Verlet method [104]. Standard MD simulation techniques also includeperiodic boundary conditions [105] where the usual cubic simulation box or unitcell is replicated in three dimensions to form an infinite system. Once the bound-ary of the simulation cell is crossed, the particle returns from the opposite sideof the box. In the minimum image convention [70], as the particle is close to theboundary, only the interactions due to particles in the closest periodic imageswithin some cut-off distance, are explicitly taken into account.

The canonical NV T ensemble mimics the heat exchange between the systemand its environment, the bath. The isothermal-isobaric NPT ensemble is usedto maintain also the given pressure with realistic fluctuations. As the numberdensity n is the natural density unit in gases, our MD calculations in article IVare first prepared in the NV T (and NPT ) schemes and subsequent productionruns carried out in the microcanonical NV E ensemble. This procedure is due tothe need to collect the most accurate trajectories for the interacting xenon atomsto rigorously provide the TCFs of the CSA Hamiltonian. Notable temperaturecoupling schemes include the Berendsen thermostat [106], and velocity rescal-ing [107], which both essentially suppress the fluctuations of the kinetic energyto a degree to achieve the given temperature, and the Nose-Hoover [108–110]thermostat, which provides the correct canonical NV T ensemble. For NPT , afew common methods are the Berendsen barostat [106], which simply scales bothatomic coordinates and the edge vectors defining the simulation box to reachthe desired pressure and the Parrinello-Rahman [111, 112] method, which pro-duces fairly realistic pressure fluctuations. In article IV, the Berendsen thermo-/barostat [106] and velocity rescaling thermostat [107] were used for preparingthe production runs.

MD simulations provide the most economic approach to incorporate time de-pendence on the molecular properties, such as NMR parameters [24], on experi-mental conditions. The most straightforward way to do this is to use the so-calledsnapshot-supermolecule method [113, 114], in which discrete snapshots of the po-sitions of the atoms in the MD trajectory are collected and properties are usuallyquantum-chemically calculated and averaged over the sampled snapshots. In thefield of NMR relaxation, this method has been used to evaluate for example the2H quadrupolar relaxation rates of dimethyl sulfoxide in water mixture [115, 116],as well as liquid CO2 [117] (17O) and liquid ammonia [118] (14N and 2H). An-other route is to use an analytically fitted EFG (or NQCC) surfaces to describequadrupolar relaxation, as done for example in the case of pure 21Ne in liquidand supercritical states [96, 99]. Particularly in solids periodic NMR methods, us-ing periodic boundary conditions in simulations with the plane-wave basis set– pseudopotential approach [119] or by applying the localized Wannier func-tions [120], show an appealing alternative to the supermolecule methods men-tioned above.

The case of dipolar relaxation is an especially easy one due to the analytic nu-clear position-only dependent formula of the dipole-dipole interaction. On the

31

contrary, no first-principles MD treatments of either the SR or CSA relaxation ingaseous xenon exist to date. This is undoubtedly caused by the fact that both SRand CSA interactions require expensive electronic structure calculations. This isopposite to quadrupolar relaxation, where the interactions can be approximatedfrom electrostatic multipoles [97], or from QC calculations of expectation val-ues [121]. In the former, the partial charges of the molecules generate a fluctu-ating EFG, which interacts with nuclear electric quadrupole moment. However,one can tackle the physics of CSA and SR relaxation in the context of MD byapplying pre-parametrized (in terms of nuclear coordinates), pairwise additiveintermolecular NMR property surfaces (article III). Molecular simulation trajec-tories can then be analyzed in terms of such a pre-parametrized “NMR forcefield” [33, 34, 122–125], without electronic structure calculations of individualsimulation snapshots [121, 126–133].

4 Electronic structure methods

In all first-principles electronic structure calculations, be they wave function-basedor relying on electron density, ultimately three items define the accuracy of the re-sults. They are the selection of molecular Hamiltonian used to describe the phys-ical phenomena, the basis set employed to expand the one-electron states, andfinally the many-body method applied for the electron correlation effects. Thelatter one can be incorporated for example in terms of a combination of Slaterdeterminants or, in DFT, with the exchange-correlation functional. These threeobjects are more or less interconnected to each other, and should be handled withcare in all practical calculations.

4.1 Hierarchy of Hamiltonians

The Hamiltonians used in electronic structure calculations are primarily classi-fied in terms of relativistic and many-electron effects. The term relativistic usedin this thesis refers to different approximations in the treatment of effects aris-ing from special relativity [134]. Coulomb-type operators are used in the NRSchrodinger framework to account for electron-electron interactions. In the rel-ativistic regime, the Breit operator [135] is also used on top of Dirac-CoulombHamiltonian. The fully relativistic four-component wave function includes boththe large and small component terms, and poses more severe demands on the sizeand flexibility of the basis set used in a calculation of a molecular property thanits quasirelativistic or NR counterparts. Due to this, it is often desirable to decou-ple the negative and positive energy components to achieve a two-componentequation for the positive energy states that is more analyzable in terms of famil-iar NR concepts at the quasi-relativistic level. A standard decoupling techniqueis the Foldy-Wouthuysen (FW) transformation [136].

Relativistic four-component methods for the calculation of NMR parametersalmost entirely rely on either Dirac-HF (DHF) [137–141] or DFT [142–144] meth-ods. A single exception for the nuclear shielding tensors of hydrogen and methylhalides used correlated ab initio configuration interaction (CI) and coupled-cluster

33

(CC) wave functions [145]. Certain quasi-relativistic methods [146], in particularthe Breit-Pauli Hamiltonian [147–151], are able to incorporate relativistic effectsin NMR parameters to the leading order in powers of the fine structure constant.Therein, the electron correlation effects using an ab initio wave function can alsobe evaluated [152–154]. A general review on the relativistic calculations of NMRparameters is given in Ref. [23].

4.2 Molecular Hamiltonians in the presence of magnetic field

This section deals with analytic expressions of the tensors σK , CK , and BK forclosed-shell molecules in their electronic ground state. The starting point in boththe relativistic and NR case is the molecular electronic Hamiltonian containingthe external magnetic field. Then, second-order energy expressions for the mag-netic tensors are discussed and finally, the representations of the magnetic tensorsare summarized.

4.2.1 Nonrelativistic case

The NR molecular electronic Hamiltonian1 in the presence of an external mag-netic field [59, 150] is

HNR =1

2me

∑i

[pi + eA(ri)]2 +

e~2me

ge∑i

si ·B + V, (4.1)

where the so-called minimal substitution [58] of the momentum operator, πi =pi + eA(ri), is used. Here, pi = −i~∇i is the momentum operator of electroni, ge is the free-electron g factor [156], and si is the intrinsic dimensionless elec-tron spin. V contains all the familiar Coulombic electron-nucleus and electron-electron interactions. The other operators in (4.1) can be regarded as small per-turbations on top of 1

2me

∑i p

2i + V [59, 157]. A(ri) is the vector potential at the

location ri of electron i:

A(ri) = A0(ri) +∑K

AK(ri) . (4.2)

The magnetic flux density at electron i can be written as

B(ri) = ∇×A(ri) , (4.3)

where the B0-dependent term is, in the Coulomb gauge with∇ ·A0 = 0,

A0(ri) =12B0 × (ri −RO) =

12B0 × riO (4.4)

1The Born-Oppenheimer approximation [155] is used.

34

and the term created by the magnetic nucleus K equals [58]

AK(ri) =µ0~γK

4πIK × riKr3iK

. (4.5)

Here, µ0 is the vacuum permeability, riK is the distance between electron i andnucleus K, and RO is the position of so-called gauge origin. The second term inEq. (4.1) describes the interaction of the electron spin with either external field,resulting in the electron spin Zeeman (SZ) operator, or fields arising from thenuclear magnetic moments.

By substituting the vector potential into Eq. (4.1) one arrives at following op-erator terms in the Hamiltonian

HNR = H(0) +H(k) (4.6)

where H(0) = 12me

∑i p

2i + V is the unperturbed zeroth-order Hamiltonian and

H(k) contains the first- and second-order perturbation operatorsH(1)B0,OZ,H(2)

KB0,DS

and H(1)K,PSO relevant for the tensors calculated in this thesis. These describe the

orbital hyperfine interaction (orbital Zeeman, OZ), electron nuclear Zeeman mod-ification (diamagnetic shielding, DS) and the paramagnetic interaction betweennuclear spin and the electron orbital angular momentum (paramagnetic spin-orbit, PSO), respectively. The first two depend on B0 and can be written as

H(1)B0,OZ =

e

me

∑i

A0(ri) · pi =e

2meB0 ·

∑i

`iO (4.7)

H(2)KB0,DS =

e2

me

∑i

A0(ri) ·AK(ri)

=e2~2me

µo4πγKIK ·

∑i

1 (riO · riK)− riOriKr3iK

·B0, (4.8)

while the last is

H(1)K,PSO =

e

me

∑i

AK(ri) · pi =e~me

µ0

4πγKIK ·

∑i

`iKr3iK

(4.9)

The angular momentum operators for electron i are written with respect to thegauge origin O

`iO = riO × pi ; riO = ri −RO (4.10)

and nucleus K`iK = riK × pi ; riK = ri −RK . (4.11)

4.2.2 Relativistic case

In the atomic core region of the electron cloud, the speed of the electron is sub-stantial, thereby suggesting the use of relativistic methods in electronic structure

35

calculations. For heavy-atom NMR properties, such as nuclear shielding, this ef-fect is pronounced [21, 158, 159], since the properties have a strong dependenceon the core regions of the electron cloud due to the underlying electronic oper-ators. Relativistic effects, in general, are substantial for many molecular proper-ties [160–162].

The starting point of relativistic electronic structure calculations is the Diracequation [163, 164]. It is governed by the one-electron Dirac operator HD. Amore useful operator for electronic structure studies is the many-electron Dirac-Coulomb Hamiltonian, written as [146]

HDC =k∑i=1

HD(i) +e2

4πε0

k∑i,j=1

′ 1rij

(4.12)

HD = cα · πi + βmec2 − eφi(ri) + Vnuc (4.13)

Here k is the number of electrons, φi(ri) is the scalar potential, and Vnuc is the nu-clear potential (a Gaussian charge distribution of the nucleus is often used [165]).Eq. (4.13) contains the Dirac 4 × 4 matrix operators α

αε = ρ⊗ σε ∀ ε ∈ (x, y, z) ; ρ =(

0 11 0

); α0 = β =

(1 00 −1

), (4.14)

formed from the Pauli 2 × 2 spin matrices

σx =(

0 11 0

); σy =

(0 −ii 0

); σz =

(1 00 −1

), (4.15)

which operate on a four-component wavefunction

Ψ =(ψLψS

); ψL =

(ψ1

ψ2

); ψS =

(ψ3

ψ4

). (4.16)

The positive energy one-electron states are primarily due to the large componentof the wave function, ψL, and the negative energy positron states are mostly de-scribed by the small component ψS . Coulombic description of the instantaneouselectron-electron interactions, occurring at infinite speed, is mended through theBreit operator [135], which yields both the magnetic Gaunt term [166] due to themagnetic vector potential created by a moving charge, and the retarded, finite-speed interaction between two electrons. Once added to Eq. (4.12), the resultingoperator is called the Dirac-Coulomb-Breit (DCB) Hamiltonian.

In the relativistic molecular Hamiltonian (4.12) the external field appears lin-early, whereas in the NR case, Eq. (4.1), one has the quadratic form. Thus, mag-netic properties have different analytic expressions in the NR and relativisticframeworks. By expanding the momentum operator πi in Eqs. (4.12-4.13), onearrives at [146]

HDC =k∑i=1

cα · pi +HK,HF +HB0,RZ + βmec2 − eφi(ri) + V (4.17)

36

where V contains the nuclear potential as well as all Coulombic interactions.Here, the relativistic hyperfine and electronic orbital Zeeman operators are iden-tified as

HK,HF = −ceµ0~4π

k∑i=1

∑K

γKIK · (α× riK)r3iK

(4.18)

HB0,RZ = −ce2

k∑i=1

B0 · (α× riO), (4.19)

respectively.The expansion of the momentum operator can be carried out directly in the

DCB framework [167]. However, it is computationally more advantageous to in-clude the effects of Breit interaction at the two- or one-component level insteadof full four-component theory due to relaxed basis set requirements. These ap-proaches require the minimal substitution of momentum followed by the FWtransformation [136] of the DCB Hamiltonian. Then, subsequent expansion interms of nuclear spin and external magnetic field-dependent parts of the Hamil-tonian is carried out. The results of this sc. Breit-Pauli perturbation theory (BPPT)are summarized for the shielding tensor in Refs. [149–151, 168, 169]. A similarmethod, the sc. linear response-elimination of the small component, emergesfrom the work of Melo et al. [170–173]. This leads to, on the one hand, termsthat modify the relativistic wave function (passive terms) and, on the other hand,terms that feature new relativistically modified magnetic operators (active). Whenthe FW transformation and substitution of momentum are carried out vice versa,only the passive terms are recovered [174–176], as pointed out in Ref. [177]. BPPTtype of methods provide the leading-order relativistic effects only, but detail adecomposition of the relativistic terms within the familiar NR concepts in termsof the even powers of the fine structure constant. This provides an analysis pos-sibility for the nuclear shielding tensors.

Other quasi-relativistic methods for the shielding tensor include the varia-tionally stable Douglas-Kroll-Hess method [178–182] and the zeroth-order reg-ular approximation (ZORA) [183, 184]. A couple of promising ways to incor-porate relativistic effects into NMR parameter calculations have appeared re-cently [143, 144, 185–187], wherein the full four-component Dirac equation istransformed exactly to a two-component presentation not at the operator level asusual, but instead in the one- and two-electron matrix elements. Then, the trans-formation is not plagued by the picture change [188], i.e., the complication of hav-ing to adapt the four-component perturbation operators into the two-component(or one-component) framework.

4.3 Magnetic tensors from an energy expansion

To express NMR tensors in a form suitable for electronic structure calculations,one represents the molecular electronic energy as an expansion in the presence

37

B0, in an ensemble of nuclear spins IK [189]

E(B0, {IK}) = E0 + EB0 ·B0 +∑K

EIK· IK +

12B0 ·EB0,B0 ·B0 (4.20)

+∑K

IK ·EIK ,B0 ·B0 +12

∑KL

IK ·EIK ,IL· IL + · · · .

Only the second-order terms are dealt with, since the linear terms vanish forclosed-shell species and the higher-order terms seldom contribute significantlyto NMR properties [190–192], at least at the experimentally available NMR fieldstrengths. The third term, bilinear in B0, refers to the magnetizability tensor,which does not directly enter the NMR spin Hamiltonian. Equating the NMRspin Hamiltonian [Eq. (2.1)] with Eq. (4.20), the shielding and nuclear quadrupoletensors can be written as energy derivatives of the molecular energy. Completelyanalogous expressions can be obtained for any combination of static perturba-tions, leading for example to forces (geometric perturbation) and electrical prop-erties (electric field perturbation). For the spin-rotation tensor, the energy deriva-tive is constructed from an energy expansion similar to (4.20) in the presence ofL.

The expressions of the magnetic tensors can now be derived using the Hellmann-Feynman -theorem [193, 194] together with the second-order perturbation expres-sion for energy [157]:

E(2)0 = 〈0|H(2)|0〉+

∑n6=0

H(1)0n H

(1)n0

E0 − En= H

(2)00 +

12〈〈H(1);H(1)〉〉ω=0 (4.21)

On the second equality, perturbation theory is disguised in the handy notationof response theory [195, 196], where an equality is drawn between the analyti-cal linear response function (at the limit of time-independent perturbation) andthe standard second-order Rayleigh-Schrodinger perturbation theory expression.The first-order energy correction H(1)

00 vanishes for closed-shell molecules in theirelectronic ground state. Thus, the machinery for building up the expressions forthe magnetic tensors both in the NR and relativistic cases is obtained.

4.3.1 Nonrelativistic case

Collecting the terms relevant for the NR shielding tensor in section 4.2.1 and dif-ferentiating with respect to B0 and IK , the NR shielding expression [197] can bewritten as (here using the response function formalism)

σK = σdK + σpK , (4.22)

where the diamagnetic part σdK results from (4.8)

σdK =e2~2me

µ0

4π〈0|∑i

1 (riO · riK)− riOriKr3iK

|0〉, (4.23)

38

and the paramagnetic part σpK from Eqs. (4.7) and (4.9)

σpK =e2~2m2

e

µ0

4π〈〈∑i

`iKr3iK

;∑i

`iO〉〉ω=0. (4.24)

Here, σdK is an expectation value in electronic ground state |0〉. In σpK , due toresponse theory, only the ground-state wave function is explicitly optimized andone has an implicit summation over the excited states |n〉 [157]. The paramagneticpart vanishes for spherically symmetric species, when the gauge origin is at thecenter of symmetry.

In all practical calculations where the external magnetic field is present, the sc.gauge origin problem arises. It stems from the fact that the vector potential is notuniquely defined, whereas the external magnetic field always has to be. Differentrealizations of vector potential using different gauge origins unphysically changethe calculated magnetic properties in any finite basis set. Only at the completebasis-set limit the representation of the quantum-mechanical operators becomesfaithful, i.e., unitary gauge transformations of the wave function do not changethe magnetic observables [198], and the gauge origin problem disappears.

In the presently more or less standard London or gauge-including atomic or-bitals (GIAOs) ansatz [199–205], a suitable exponential phase factor is attachedto each basis function. It transfers the gauge origin from an arbitrary location tothe expansion center of the basis function. This yields magnetic properties inde-pendent of the choice of gauge origin. In all articles I–IV, GIAOs or rotationalGIAOs [206] (the latter for the SR tensor), were used.

The NQC tensor is directly proportional to the EFG tensor [148], see Eqs. (2.8)and (2.9). The corresponding NQC perturbation operator is written as [2]

H(2)K,NQC = − e2

4πε0QK

2IK (2IK − 1)IK ·

∑i

3riKriK − 1r2iK

r5iK

· IK , (4.25)

which yields a quantum mechanical expectation value that is to be added up witha term dependent on the positions of the nuclei (R). Thus, the total quadrupolecoupling is written as

BK =e2

4πε0QK

2IK(2IK − 1)h

∑K 6=L

ZK3RKLRKL − 1R2

KL

R5KL

−〈0|∑i

3riKriK − 1r2iK

r5iK

|0〉

]. (4.26)

Similarly also in the case of spin-rotation tensor [55, 207], both nuclei-dependentand electronic terms arise as

CK = eγK~

2πµ0

4π

∑K 6=L

ZK

(R2KL1−RKLRKL

)R3KL

· I−1eff

+1me〈〈∑i

`iKr3iK

;∑i

`iO〉〉 · I−1eff

]. (4.27)

39

Here, the first term again depends explicitly on R, and the second is a linearresponse function, closely related to the the paramagnetic part of the shieldingtensor in Eq. (4.24). I−1

eff is the inverse of the moment of inertia tensor whenB0 = 0 (in the absence of the Lorentz force acting on the charged particles).

4.3.2 Relativistic case

Following the rules set in the NR case, one collects the terms relevant in Eq. (4.17)and differentiates the energy, to arrive at the following expression for relativisticshielding tensor [139, 159]

σK =12µ0γK~

4π(ec)2〈〈 (αi × riK)

r3iK

; (αi × riO)〉〉ω=0. (4.28)

The notable contrast to the NR shielding expressions [Eqs. (4.23) and (4.24)] isthat in Eq. (4.28) the diamagnetic expectation value is missing. This is due to themomentum operator π, which is not quadratic in (4.17). In the relativistic frame-work, however, the contribution similar to the NR diamagnetic term has beensuggested to arise from the different treatments of positronic negative-energystates [208–211], from the field- and external field- dependent unitary transfor-mations of the Dirac equation [212, 213], or from the matrix treatment of the Diracequation in the field-dependent basis [142, 186]. In the author’s opinion, the onlycompelling reason to resort to division of dia- and paramagnetic terms in σK isto increase analysis possibilities. Even so, the magnitudes of the diamagnetic andparamagnetic terms change among the different approaches [214]. Relativisticcalculation of shielding also poses the gauge origin problem (vide infra). In arti-cle II, GIAOs were used in the calculation of relativistic nuclear shielding of Xe2.For a detailed account on relativistic shielding GIAO theory, see Ref. [141].

For the quadrupole coupling, the nuclear part is formally the same as in the NRcase. The electronic term is now composed from the ground-state four-componentspinor [138] instead of the NR state |0〉. For the spin-rotation tensor, no relativisticanalytic energy derivative expression has been formulated.

4.4 One-electron picture

4.4.1 Basis sets and pseudopotentials

The Schrodinger or Dirac equation is exactly solvable only for the hydrogen-likeone-electron species. For any other kind of systems, many-electron effects compli-cate the calculations. Basis set represents an approximation to the single-electronstates, and partially defines the outcome of any electronic structure calculation.The most common basis functions in modern molecular electronic structure cal-

40

culations are Gaussian-type orbitals (GTOs) [215]

χKµ (r, θ, φ) = Nrle−ζr2Ylml

(θ, φ), (4.29)

where N is the normalization factor, ζ is the exponent of the Gaussian function,and Ylml

(θ, φ) are the spherical harmonics. The GTOs suffer from the inabilityto represent the nuclear cusp2 and the erroneous asymptotic decay of the wavefunction [216]. Both could be better dealt with by using Slater-type orbitals [217].Using GTOs, however, a lot is recovered due to the easy multi-center integralevaluation.

In practical calculations, inner orbitals that show similar properties in an atomand in a molecule, are described by contractions, linear combinations of GTOswith fixed coefficients. This reduces the number of basis functions. In molec-ular calculations, polarization functions are usually invoked to modify orbitalsto meet the requirements of the chemical bonding. Diffuse functions have verysmall exponents, and are needed in situations where the electron cloud is spreadout to a larger degree than in the atomic case. For example anionic systems, ex-cited states as well as polarizability calculations require these functions.

Pseudopotentials or effective core potentials (ECPs) [162, 218] originate fromthe separation of valence and core electron Hamiltonians [219]. Particularly use-ful for large atoms, they are primarily designed to produce the same effectivepotential for valence electrons that would be present in the corresponding all-electron calculation. ECPs do not explicitly include the core electrons in theQC calculation. Thus, the number of basis functions is also reduced. Secondly,relativistic ECPs (RECPs) [220] incorporate relativity. Core polarization poten-tials [221, 222] have been devised to account for the otherwise missing polariza-tion effects between the core and valence regions of the electron cloud.

Molecular magnetic resonance properties require fairly large basis sets sincetheir defining perturbation operators operate both in the valence and core re-gions of the electron cloud. In some cases, this means that one has to mend thedeficiencies in the existing basis sets by uncontracting the inner orbitals and man-ually adding both diffuse and tight functions in the relevant electronic structurecalculation of the property. For the Xe clusters in general, the basis-set require-ments are strict due to the weak van der Waals interaction between xenon atoms.In article I, a NR basis set by Fægri [44] was supplemented by diffuse basis func-tions in the spirit of the IGLO basis set family [223] and utilized in all magneticproperty calculations. For the Xe2 PEC calculation, quasirelativistic ECPs uti-lizing the Stuttgart energy-adjusted eight-valence-electron pseudopotential, aug-mented by core polarization potentials [224], were used. These require a spe-cific correlation-consistent valence basis set [225], which was fully uncontractedand supplemented with bond basis functions [226, 227] in the middle of Xe-Xe“bond”, adapted from Ref. [228]. In article II, good convergence of the shieldingtensor in Xe2 at the DHF level of theory was observed in the well-behaving seriesof relativistic large-component basis sets by Dyall [45]. BPPT calculations werecarried out to determine the coupling between relativistic and correlation effects

2The DHF results with point-like nuclear model and with finite one deviate roughly 30 ppm forxenon atom [159].

41

using the basis sets of Fægri and Almlof [44], which were heavily augmented byeight tight and one diffuse set of exponents in each angular momentum type oc-curring in the atomic ground state, following Ref. [152]. For the jargon-riddlednomenclature of basis sets and their selection in molecular calculations, the readeris instructed to refer to Refs. [229, 230]. Computational scientists benefit from thelarge online basis set portal [231], compatible with many QC software packages.

In articles I and II, the counterpoise (CP) correction [232] was applied to boththe magnetic and energetic binary properties of Xe2, to reduce the basis-set super-position error (BSSE), which arises from the increased variational freedom due tothe overlap of the basis functions of the monomers. To reduce BSSE, dimer prop-erty is compared to the properties of monomers calculated with the full dimerbasis set. A completely different route, not followed here, is to construct a so-called chemical Hamiltonian, which a priori determines a BSSE-free Hamiltonianoperator [233, 234].

4.4.1.1 Kinetic balance

By rewriting the one-electron Dirac equation in a block form as

HD(i)Ψi = EiΨi

HD(i) =(mec

2 + V cσ · pcσ · p −mec

2 + V

); Ψi =

(ψLψS

), (4.30)

a connection between the small and large components of the wave function canbe found at the NR limit as

ψS = − 12cme

σ · pψL. (4.31)

By expanding both ψS and ψL in Gaussian basis {χµ} one obtains the kineticbalance condition [235]

{χSµ} = − 12cme

σ · p{χLµ} (4.32)

Thus, the small-component basis functions are obtained by differentiation of (un-contracted) large component basis functions. In particular in restricted kineticbalance (RKB) [235], the number of positive and negative energy solutions is keptequal, i.e., there is one-to-one correspondence between large and small compo-nent basis functions, achieved by writing a fixed combination of l + 1 and l − 1large-component basis functions for each l-type of small-component basis func-tion. In unrestricted kinetic balance (UKB) [146], two small-component basisfunctions associated with l + 1 and l − 1 are created for each l-type large com-ponent function. In article II, the UKB description is favored over RKB, based onthe results in Refs. [236–238]. Other variations of kinetic balance have appearedrecently [144, 239]. One additional computational burden in the four-component

42

calculation as compared to the NR case arises from the fact that, on top of large-large (LL), also small-small (SS), and small-large (SL) -type two-electron inte-grals [240] have to be evaluated. In practice, however, the effect of includingthe SS integrals was found very small, although time-consuming [240], for theDHF-level shielding in Xe2 in article II.

4.5 Many-electron effects

A vast pool of methods has been created to take into account the many-electroneffects, i.e., electron correlation. Two main paths can be identified amongst thewave function theories: the first is to use standard Rayleigh-Schrodinger pertur-bation theory on top of a HF theory as in the Møller-Plesset (MP) schemes andthe other is to create a linear combination of Slater determinants as in the CI the-ory [241]. Mixed schemes have appeared, such as the complete active space per-turbation theory (CASPT) approach [242], which uses perturbation theory on topof carefully selected active space of Slater determinants. A third route is estab-lished in DFT [243]. Methods that explicitly include electron-electron distancesin the basis set have also been proposed [244]. However, these lack practicalityin molecular property calculations due to complicated integrals. In the followingchapters dealing with ab initio wave function theories, the main focus is in the NRtheory, and relativistic formulation is referred to, when necessary.

4.5.1 Many-electron wave function

Contrary to the simple Hartree product of one-electron atomic orbitals used as amodel many-electron wave function in the early days of QC, a more reasonablek-electron wave function is the Slater determinant [245]

Ψ(1, 2, . . . , k) =1√k!

∣∣∣∣∣∣∣∣∣Φ1(x1) . . . Φk(x1) . . . Φk(x1)Φ1(x2) . . . Φk(x2) . . . Φk(x2)

.... . .

.... . .

...Φ1(xk) . . . Φk(xk) . . . Φk(xk)

∣∣∣∣∣∣∣∣∣ , (4.33)

where x refers to generalized coordinate containing both spatial (r) and spin co-ordinates. Each spin-orbital

Φk(xi) ={φk(ri)αφk(ri)β

(4.34)

occupies an electron with spin either up (α) or down (β) in the restricted HFformalism. This means that Eq. (4.33) obeys the Pauli exclusion principle, whichstates that no two electrons can occupy the same spin-orbital. Hence, Ψ(1, 2, . . . , k)is antisymmetric with respect to the interchange of two electrons.

43

4.5.2 Hartree-Fock

The HF method [246, 247] usually serves as a starting point for more comprehen-sive treatments of electronic structure in atoms and molecules. It has the advan-tage of both yielding qualitatively correct results for many properties and beingcomputationally lightweight (HF scales as M4, where M is the number of basisfunctions). In HF, a single Slater determinant is used as a wave function. More-over, HF-MOs are expressed as linear combinations of atomic orbitals (LCAO)

φk(ri) =∑µ

ckµχKµ (ri), (4.35)

and iteratively solved, in the effective form of Roothaan-Hall (RH) equations [248,249], in the self-consistent field (SCF) approach. In the SCF method, ckµ are thecoefficients to be variationally optimized. For the restricted HF method (closed-shell case) the RH equations can be written as [248, 249]

Fc = Scε, (4.36)

where F is the Fock matrix in the one-particle basis {χµ}, containing the matrixelements

Fµν = 〈χµ|h(1)|χν〉+Nocc∑k

∑ητ

ck∗η ckτ{2〈χµχη|g12|χνχτ 〉− 〈χµχη|g12|χτχν〉} (4.37)

where the summation index k runs over the doubly occupied orbitals. Herehµν = 〈χµ|h(1)|χν〉 is the one-electron matrix that contains the effect of both thekinetic energy operator of electrons and the nuclear potential. The density ma-trix is defined as Pητ = 2

∑Nocck ck∗η c

kτ . The two-electron Coulomb and exchange

matrices can be identified as

J(c)µν =∑ητ

Pητ 〈χµχη|g12|χνχτ 〉 (4.38)

K(c)µν =12

∑ητ

Pητ 〈χµχη|g12|χτχν〉, (4.39)

respectively, with g12 =∑i6=j e

2(4πε0r12)−1. One has to resort to an iterative pro-cedure, as F depends on the solutions c. First one obtains a set of trial orbitalsto form the coefficient matrix c. The overlap matrix Sµν = 〈χµ|χν〉 and hµν arecalculated once and for all for a given basis set. Thus, only the two-electron ma-trices are iteratively optimized by varying the coefficients c until a predefinedthreshold accuracy is reached for the MO energy matrix ε. This naive proceduremight, however, cause the SCF energy to oscillate. The direct inversion in theiterative subspace (DIIS) method [250] overcomes some of the convergence prob-lems by replacing F by a linear combination of the Fock matrices of the previousiterations. Further numerical and algorithmic techniques include the integral pre-screening as well as the direct/semi-direct SCF [251] schemes, invoked to providespeed-up in the SCF approach.

44

4.5.2.1 Dirac-Hartree-Fock and Levy-Leblond methods

Dirac-Hartree-Fock (DHF) theory is the relativistic counterpart of the NR HFmethod. The starting point in the DHF calculation is the relativistic many-electronHamiltonian in Eq. (4.12). In contrast to the NR HF method, the electron spin is anintrinsic property of the four-spinor one-electron states involved, and not a multi-plicative factor as in Eq. (4.34). First DHF calculations for atoms in the RH matrixformalism were performed in the 1960s [252, 253]. At that time, the problem of“variational collapse”, i.e., the issue that DHF electronic energy is not strictly anupper bound to the exact energy due to the negative energy solutions, was, to alarge degree, related to the lack of feasible four-component basis sets. This turnedthe focus on two-component methodology. Dirac, however, suggested alreadyin 1930 that the negative energy states are completely filled in accordance withthe Pauli exclusion principle, thereby validating the variational approach. Rigor-ous DHF calculations of the NMR parameters started in the 1990s [138, 139], dueto both advances in the computer power and in the methodology. Only a sin-gle study of electron-correlated four-component relativistic treatment of shield-ing tensor exists to date [145]. This in mind, one is tempted, within the ab initiorealm, to sum up the relevant correlation and relativistic shielding tensors ob-tained at different levels of theory, as illustrated in article II. Contrary to the caseof shielding, for NQC a plethora of correlated relativistic wave functions are be-ing used [254, 255].

The Levy-Leblond (LL) method [256] offers a four-component NR Hamilto-nian, which also incorporates the electron spin naturally. It is therefore empha-sized that spin is not a “relativistic” property, but strictly a quantum-mechanicalproperty of an electron. In article II, shielding and quadrupole coupling calcu-lations were performed with both DHF [using the DC Hamiltonian of Eq. (4.12)]and LL methods.

4.5.3 Multiconfigurational Hartree-Fock

One can construct M spin orbitals from a basis set with M basis functions. Thus,solution of closed-shell Roothaan-Hall SCF equations yieldsNocc = k/2 occupiedMOs, and M − k/2 unoccupied, virtual MOs. These can be used in the correlatedpost-HF calculations, such as in the multiconfigurational Hartree-Fock (MCHF)(or MCSCF) or CI approaches.

The CI expansionθ =

∑k

CkΨk, (4.40)

mixes individual Slater determinants Ψk with weights Ck. A full CI (FCI) wavefunction, although impractical for anything but the smallest molecules, is exact inthe NR framework for a given basis set. The term correlation energy is defined asthe difference between the exact energy and the energy resulting from complete

45

basis set HF calculation [257, 258].3 Truncated CI methods pose strict require-ments for the choice of the reference HF determinant due to optimization of onlythe {Ck}. To overcome some of the difficulties, multireference CI methods [251]use a mixture of several determinants as a reference space and include all excita-tions to a specific order from each of the determinants. No CI calculations wereperformed in this thesis.

The MCSCF wave function is a linear combination of finite number of Slaterdeterminants, for which both weights {Ck}, and coefficients {ck} (in contrast toCI methods) are variationally [241] optimized. In short, a trial wave functioncontains a set of numerical parameters {κ}, which are optimized according to

∂E({κ})∂{κ}

= 0 ∀ {κ} (4.41)

to reach a minimum in the energy functional.Although MCSCF is conceptually rigorous, it is not computationally light [241],

nor a “black box” type of method [259]. In this thesis, complete active space SCF(CASSCF) calculations fail to reproduce correlation effects for the binary chemi-cal shift δ(R) close to the Xe2 equilibrium distance, due to the small active spaceand too few correlated electrons, as shown in article I. The more general restrictedactive space SCF (RASSCF) wave function requires a further non-trivial divisionof the active orbitals into three subspaces.

CASSCF/RASSCF approaches are particularly useful for the calculation of thevalence properties that include a large fraction of static correlation in terms ofnear-degenerate low-lying excited states, described to a certain degree by the HFvirtual orbitals. One example of static correlation case would be the dissociationof a molecular complex. In this context, the question of size-extensivity [260]arises, linked to the linear scaling of the properties of an ensemble of atoms andmolecules in terms of the number of structural subunits. Size-consistency [261]is the practical manifestation of size-extensivity for the molecular energy at thedissociation limit, as in

EAB(R→∞) = EA + EB , (4.42)

whereEAB andEA (EB) are the energies of the fragmentAB and fragmentA (B),and R is the distance between fragments A and B. Dynamical correlation, whicharises from the r−1

ij term in Hamiltonian describing the instantaneous interactionsbetween electrons, governs the properties of Xe2. This is both due to fact that Xe2

is already qualitatively well-described by only one Slater determinant in termsof HF theory, and that there are no low-lying excited states in Xe2 [262]. In thisrespect, the dynamically correlated perturbative or CC -based methods are morefeasible than multiconfigurational approaches.

3In practice, the correlation energy is often considered to be the difference between the energiesfrom a correlated and a HF calculation using the same finite basis set.

46

4.5.4 Coupled-cluster techniques

CC methods have been pioneered by Cizek et al. [263–265], based on Ref. [266].The CC wave function is an exponential ansatz

ΨCC = eTΨk, (4.43)

where the cluster operator T =∑µ tµτµ includes a linear combination of excita-

tion operators τµ associated with amplitudes, strengths of the excitation, tµ. Theexponential operator eT often operates to a HF wavefunction. One can expandthe CC wave function as

ΨCC = (1 + T1 + T2 + T 21 + T1T2 + T 2

2 + · · ·+ TN )Ψk, (4.44)

where TN includes all N -tuple electronic excitations from the given referencewave function Ψk. When all excitations in the expansion are considered, it be-comes identical to FCI. Contrary to the truncated CI methods, truncated CC ex-hibits size-extensivity due to the exponential operator. The Schrodinger equationcan be written in terms of “configuration” operators C:

eTΨk =N∑i

CiΨk, (4.45)

where lowest-order operators are

C0 = 1 ; C1 = T1 (4.46)

C2 = T2 +12T 2

1 ; C3 = T3 + T1T2 +16T 3

1 . (4.47)

Here, for example C2 contains both the effect of simultaneous excitation of twoelectrons described by the connected T2 operator, and that of the two indepen-dent one-electron excitations described by the disconnected T 2

1 operator. Themost important excitations are double excitations [251], and, hence, the CCSDmethod [267] is significant as well as widely used. Higher order methods arevery accurate, but lack feasibility due to the high computational cost. In CC, theenergy, ECC, and the wave function, i.e., the amplitudes are calculated iteratively,albeit not variationally [251]. CCSD(T) method [268] suitably recovers some of theconnected triple (T3) excitations missing in CCSD in a non-iterative, perturbativemanner. It is considered as the de facto “gold” standard of electron correlationeffects in QC. CCSD and CCSD(T) methods were utilized in articles I and II notonly for the PEC, but also for the NMR tensors. The success of CCSD(T) is largelydue to its ability to handle both dynamic and, increasingly with higher-order ex-citations, also static correlation in a straightforward, black box manner.

4.5.5 Perturbational methods

Of the perturbational schemes used in quantum chemistry, the Møller-Plesset(MP) [269] type of methods are one of the most popular ones. Although the MP

47

perturbation series has been proven to be generally divergent [270, 271], the MP2method in particular often serves as the first, although in many cases overesti-mating, approximation of electron correlation effects.

The MP scheme creates corrections to a specific order both in the energy andthe wave function based on standard time-independent Rayleigh-Schrodingerperturbation theory [272]. The unperturbed Hamiltonian H0 is the HF Hamil-tonian, written as a sum of Fock operators

HHF =N∑i=1

f(i). (4.48)

The perturbation operator is the difference between the full electronic Hamilto-nian and the uncorrelated HF Hamiltonian:

H(1) = H −HHF. (4.49)

Subsequently, time-independent perturbation theory [272] gives the first-ordercorrection to wave function

ψ(1)0 =

∑n

anψ(0)n ; an =

H(1)n0

E0 − En

relevant for the second-order (MP2) energy correction4

E(2)0 = H

(2)00 +

∑n

′anH(1)0n = H

(2)00 +

∑n

′H(1)0n H

(1)n0

E0 − En. (4.50)

The first-order energy correction equals the HF energy. Generally, low-level MPtreatments also come as a side product of the more accurate CCSD(T) method.Hence, MP2 as such should not be emphasized too much, if the calculations canbe carried out with more accurate CC models, as in articles I and II. However,the direct four-component relativistic MP2 treatment [274] is used to calculate theNQC tensors at different internuclear distances of Xe2 in article II.

4.5.6 Density-functional theory

DFT [243] uses the electron density as the basic building block in the electronicstructure calculations. DFT yields a rough estimate of the electron correlationeffects in the QC calculation of atomic and molecular properties, particularly incases where HF fails, and post-HF methods are computationally too heavy. DFT,however, poses a problem in terms of the hitherto unknown exchange-correlationfunctional. Thereby it lacks a systematic way to improve calculations in terms

4This follows the Wigner’s 2n+1 rule [273], which states that the order-2n+1 energy can be calcu-lated from an order-n wave function.

48

of many-electron effects, as opposed to ab initio methods presented earlier. Yet,Perdew and co-workers [275] have studied the steps to incorporate a series of DFTapproximations for the exchange-correlation functional. In this thesis, only a fewDFT calculations, using the BHandHLYP [276] functional in the BPPT frameworkfor the shielding tensors in Xe2, were carried out in article II. In addition, in arti-cle III, the excitation energies of the clusters Xen (n = 2–12) were estimated usingthe B3LYP [277, 278] functional.

4.6 Scaling features

The typical scaling features of the different ab initio theories are presented in Ta-ble 4.1. Whereas the HF – MP2 – CCSD – CCSD(T) progression of methods worksfor the NMR tensors, particularly σ in light systems of one-determinantal naturecontaining roughly ten atoms at maximum, the MCSCF or DFT do not follow thisline. Indeed, DFT is successfully used for example in the case NMR parametersof transition metals [279], while the MCSCF methods, in general, are well-suitedfor systems containing static correlation. The pros and cons of the methods arealso briefly outlined in this table.

Table 4.1. Scaling features of the standard nonrelativistic methods used to cal-culate the NMR parameters in this thesis (modified after Ref. [157]).

method scalinga pros and consHF M4 single-reference, no correlation, inexpensiveMP2 M5 single-reference, easy (dynamical) correlationCCSD M6 single-reference, expensiveCCSD(T) M7 single-reference, expensive, gold standardMCSCF expZb multi-reference, static correlation, non-black boxDFT M3...4 empirical, inexpensive

a M is the number of basis functions.b Z is the number of active orbitals for example in the CASSCF calculation.

4.7 Computational software used

In this thesis, the QC calculations of NMR tensors were mainly performed withthe DALTON [280] program package in the NR framework, and with DIRAC [281]in the relativistic realm. Also, other programs such as GAUSSIAN [282] andCFOUR [283] (formerly ACESII-MAB), were used for NMR tensor calculations.

49

The zero-point vibrational correction to the binary NMR tensors were evaluatedwith VIBROT program [284]. The final PEC calculations in article I were per-formed with MOLPRO software package [285]. Excitation energies in article IIIwere calculated with TURBOMOLE [286]. All MD simulations in article IV wereperformed with GROMACS [287]. Apart from the academic and commercial soft-ware used, the author developed the analysis codes used in articles III and IV,drawing from the existing experience in the field of computational NMR in theresearch group.

5 Summary of results

5.1 Magnetic properties of xenon dimer

Xenon is a widely used NMR probe atom of various host materials. In articles I-II, a thorough study of the binary NMR tensors σK , CK , and BK as functionsof the internuclear distance R of Xe2 is performed. These binary NMR proper-ties constitute first steps into the direction of computational materials researchby introducing the intermolecular interaction effects in xenon. In addition, thesecurves serve as building blocks for the route from microscopic to bulk properties,as described in articles I and II in the case of the second virial coefficient of 129Xeshielding, and in article IV for the 129Xe spin-lattice relaxation time due to chemi-cal shift anisotropy. In article I, the binary NMR tensors were calculated at the NRlevel of theory, and in article II, the studies were extended to both relativistic andmore sophisticated correlated methods, as well as to their coupling. The ultimategoal in both of these articles was the theoretical confirmation of the experimentaltemperature dependence of the second virial coefficient of nuclear shielding (videinfra) [36].

In Table 5.1, the results are outlined for the NMR tensors at three internucleardistances. Fig. 5.1 illustrates the binary chemical shift curve as a function of theinternuclear distance. For the absolute xenon NMR shielding constant, the effectsof special relativity are substantial [159, 236]. This suggests that the NMR prop-erties in Xe2, such as the binary chemical shift, also grave for a relativistic treat-ment [21, 288], albeit the qualitative features of the binary NMR properties are al-ready captured at the NR level of theory (see articles I-II). This is seen in Table 5.1,and in Fig. 5.1 particularly for binary chemical shift δ(R) = σ (Xe atom)−σ (Xe2).However, the fine details of the curves are crucial to grasp the correct dependenceon R, and require more accurate methods. This is extremely important when oneattempts to compare to and verify experimental findings. These in mind, the ad-

51

ditive approximation of the chemical shift and shielding anisotropy,

δ∆σ

}≈ DHF + CORRNR + CORRBPPT

CORRNR = [CCSD(T)−HF]NR (5.1)CORRBPPT = [BHandHLYP−HF]BPPT

as well as the nuclear quadrupole coupling,

B‖ ≈ DMP2 + CORRNR∗

CORRNR∗ = [CCSD(T)−MP2]NR, (5.2)

yields the best currently accessible binary NMR property curves. In the case ofδ and ∆σ, the results build on top of the four-component DHF, contrary to thecase of NQC, where a relativistic correlated theory (DMP2) is available [274]. Forthe latter, the remaining NR correlation effects are included as the difference ofCCSD(T) and MP2 results. Also, more sophisticated methods for relativistic cor-related EFG calculations are emerging [289–291]. The NR electron correlationeffects for all NMR parameters are investigated under the HF – MP2 – CCSD– CCSD(T) progression of methods [see Table 5.1]. This is natural, since thesemethods are all single-determinantal (well-suited for Xe2) and can handle the dy-namical correlation. The CASSCF and RASSCF results deviate from this behavior,since they, with low number of active orbitals, mostly describe static correlation,which is of minor importance in this particular case. The final estimated NR cor-relation effect for δ and ∆σ is estimated as the difference of results from CCSD(T)and HF methods. Finally, the coupling of the relativity and the correlation is esti-mated at the full BPPT level of theory [149–151] using the difference of DFT withBHandHLYP functional [276], and HF theory. All these effects are summed up ata given Xe-Xe distance R. The preferred QC method when comparing with theexperiments would be the fully relativistic correlated wave function theory [145].For the SR coupling, no relativistic theory is available, thus prohibiting the out-lined additive approach. It can be seen from Fig. 5.1(bd) and from Table 5.1 thatthe BPPT method [149–151] offers a respectable alternative for the fully relativisticDHF method in the calculation of binary chemical shift, with the largest discrep-ancies at the smallest internuclear distances. However, with the recent GIAO-DHF theory [141] utilized in article II, a rigorous gauge-including method fornuclear shielding tensor is available at the fully relativistic level, albeit with ahigh computational cost.

In conclusion, the physics behind the binary NMR curves is well-explained inarticles I and II. These curves are fitted to a functional form, which is practical dueto the heavy computational burden related to the treatment of both electron cor-relation and relativity within a sufficiently large basis set in the QC calculation ofbinary NMR properties of xenon clusters. As mentioned earlier, these curves areimportant for both virial coefficients and relaxation rates. In the latter study (ar-ticle IV), the high computational demands rule out the supermolecule-snapshotapproach, and the PAA approach is used instead.

52Ta

ble

5.1.

Cal

cula

ted

129X

ebi

nary

chem

ical

shif

t(δ

,in

ppm

),th

ean

isot

ropy

of129X

esh

ield

ing

tens

or(∆σ

,in

ppm

),131X

enu

clea

rqu

adru

pole

coup

ling

tens

orco

mpo

nent

alon

gth

eX

e-X

ein

tern

ucle

arve

ctor

(B‖,

inkH

z),

and

the

perp

endi

cula

rco

mpo

nent

ofth

e129X

esp

in-r

otat

ion

coup

ling

tens

or(C⊥

,in

kHz)

,in

Xe 2

atth

ree

diff

eren

tint

ernu

clea

rdi

stan

cesR

.

Non

rela

tivi

stic

calc

ulat

ions

Rel

ativ

isti

cca

lcul

atio

nsa

Prop

.R

(A)

LLa

HFb

MP2

bC

CSD

bC

CSD

(T)b

CC

SD(T

)a,c

DH

FD

MP2

HF

BPPT

dBe

ste

δ3

490.

148

9.3

518.

351

2.2

517.

551

2.0

547.

3-

555.

657

0.5

4.36

2728

.929

.030

.731

.031

.531

.031

.6-

32.6

33.4

60.

40.

40.

40.

40.

40.

50.

5-

0.5

0.5

∆σ

374

4.3

743.

178

0.8

773.

078

1.2

767.

685

4.6

-82

1.6

879.

34.

3627

46.2

46.4

48.6

49.4

50.1

45.9

51.7

-49

.851

.16

1.7

1.7

1.6

1.7

1.7

0.8

1.7

-1.

80.

6B ‖

f3

3488

035

240

3317

033

790

3347

033

060

3921

035

360

-35

660

4.36

2722

2022

3812

4414

9813

7712

6823

7410

23-

1157

624

26-1

06-7

5-9

4-1

0726

-105

--9

3g

C⊥h

3-

1.08

41.

142

1.13

01.

142

1.12

5-

--

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3627

-0.

030

0.03

20.

033

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30.

031

--

--

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0.00

00.

000

0.00

00.

000

0.00

0-

--

aFo

rLL

and

rela

tivi

stic

calc

ulat

ions

,see

arti

cle

II.

bFI

I3s2

p2d1

fba

sis

set(

see

arti

cle

I).

cC

CSD

(T)c

alcu

lati

onw

ith

ala

rger

FIV

+d3s

2pba

sis

set(

see

arti

cle

II).

Unp

ublis

hed

resu

lts.

dA

sum

ofcl

ose-

to-b

asis

set-

limit

GIA

OH

FN

Rva

lue

and

the

BPPT

lead

ing-

orde

rpe

rtur

bati

onal

rela

tivi

stic

term

sca

lcul

ated

atth

eH

Fle

velo

fthe

ory

inar

ticl

eII

.The

BPPT

cont

ribu

tion

sar

eob

tain

edw

ith

aco

mm

onga

uge

orig

inpl

aced

atth

em

id-p

oint

ofth

ein

tern

ucle

arse

para

tion

.e

Eq.(

5.1)

forδ

and

∆σ

,Eq.

(5.2

)forB ‖

.See

arti

cles

Iand

IIfo

rin

form

atio

nco

ncer

ning

the

basi

sse

ts.T

heba

sis

setF

IV+d

3s2p

used

for

NR

calc

ulat

ions

.f

Wit

h131X

equ

adru

pole

mom

enteQ

=−

0.11

4×10−

28

m2

(Ref

.[29

2]).

gA

tlon

gin

tern

ucle

ardi

stan

ces,

the

NQ

Csh

ows

nega

tive

valu

es.T

his

isno

tmer

ely

anar

tifa

ctof

the

coun

terp

oise

corr

ecti

onpr

oced

ure

used

here

,sin

ceth

isef

fect

isse

enw

ithi

nal

lcor

rela

ted

theo

ries

.h

Nei

ther

rela

tivi

stic

nor

LLm

etho

dsar

eav

aila

ble

for

the

SRco

uplin

gte

nsorC

.

53

Figure 5.1. Calculated 129Xe binary chemical shift δ(R) as a function of the internucleardistance in Xe2 at different levels of theory: (a) DHF and (b) difference of nonrelativis-tic (NR) LL and DHF, as well as the error made using NR Hartree-Fock corrected withleading-order BPPT relativistic effects, instead of DHF. (c) Relativistic DHF correctedwith NR correlation effects (CORRNR) as well as the cross-coupling of relativity andcorrelation in terms of BPPT theory (CORRBPPT). (d) Deviation from (c) at less com-plete levels of approximation.

5.1.1 Potential energy curves

Potential energy surface presents the variation of the total energy of the molecularentity with respect to nuclear coordinates. As such, PES is used to model molec-ular interactions and chemical reactions. For xenon gas, the pair interactions arethe most important ones. Even for liquids and solids, the Xe-Xe internuclear dis-tances are only a little shorter than in the case of Xe2, as observed in a recenthigh-pressure study of xenon [293] and in solid xenon measurements [294]. Inaddition to the gas phase, the direct use of the Xe2 NMR property curves also inliquid and solid phases should be at least qualitatively feasible. Analytical modelpair potentials [295, 296] produce the qualitative features of the dimer PEC; thesteep potential wall, long attractive tail, and shallow potential well. Realistic po-

54

tentials are obtained either through a fit to experimental bulk properties or withhighly accurate QC methods.

The ground state PEC of the xenon dimer has been the subject of extensiveinvestigation both empirically [38, 39] and theoretically [40, 228, 297, 298], butno quantitative first-principles results have been presented. In article I, a reli-able ground state Xe2 PEC, approaching the accuracy of its empirical counter-parts [38, 39], was calculated. It employs the correlation-consistent cc-pVQZ basisset [225] within the CCSD(T) theory, scalar relativistic effects in terms of RECPswith the inclusion of CPP [224], as well as additional mid-bond basis functions(BF) [228]. The resulting first-principles PEC is used to model the pairwise Xe-Xeinteractions in the MD simulation of CSA relaxation rates in article IV. The im-provements in the equilibrium distance Re and the potential well depth, De, us-ing various first-principles methods, are summarized in Table 5.2 and comparedagainst existing values from empirical [38, 39] and theoretical [40, 228] Xe2 PECs.

Table 5.2. Comparison of the values of equilibrium distance Re, and the depthof the potential well De, for selected potential energy curves of Xe2.a

Method Theoryb Basis set re (A) De (K) Ref.empirical 4.363 282.3 [38]empirical 4.366 282.9 [39]CCSD(T) RECP aug-cc-pVQZ+BFf 4.420 263.4 [228]CCSD(T) RECP (Basis 2 in Ref. [40]) 4.525c 225.1c [40]CCSD(T) NRECP modified cc-pVQZd 4.522 228.8 e

CCSD(T) RECP modified cc-pVQZd 4.473 237.4 e

CCSD(T) RECP+CPP modified cc-pVQZd 4.429 256.8 e

CCSD(T) RECP+CPP modified cc-pVQZd + BFf 4.382 283.1 e

a For details of the ECP/CPP parameters and the basis sets, see Refs. [224]and [225].b Relativistic (R) or NR effective core potentials. CPP stands for the corepolarization potential.c Spin-orbit corrections included.d [16s12p5d4f 3g/7s7p5d4f 3g] basis described in detail in article I.e Article I.f {3s3p2d2f 1g} set of bond functions from Ref. [228].

It was found out in article I that NR all-electron (AE) methods are not accu-rate enough for Xe2 PEC. This is due to the inadequate basis set expansion of thevalence region of Xe2, the lack of relativistic effects in PEC, and the correlation ef-fects beyond CCSD level of theory (the best AE method at the time of writing ar-ticle I). On the contrary, the aforementioned CCSD(T)+RECP+CPP+cc–pVQZ+BFrecipe, superior to earlier theoretical predictions [40, 228], was observed to work

55

well for Xe2 PEC. Three points should be emphasized: the effect of scalar rel-ativistic effects for Re and De are -0.05 A and 8.6 K, as calculated between theNR and relativistic ECP at the cc-pVQZ basis set level, while the effect of CPP isaround -0.04 A and 19.4 K for the same parameters. The largest effect arises fromthe use of BFs (differences are -0.05 A and 26.3 K). Final values are Re = 4.382 Aand De = 283.1 K, which are almost identical to the best empirical results [38, 39].One should note, however, that the additional stabilizing effect of hitherto miss-ing relativistic spin-orbit effects in article I have been estimated to -0.001 A and8 K for Re and De in Ref. [40].

Moreover, good-quality dispersion coefficients were retrieved from a fit to theattractive tail of the Xe2 PEC in article I, as in Ref. [40]. These might be of inter-est in terms of semi-empirical PECs [38, 39], as additional parameters for simpleanalytic MD potentials, and in dispersion-corrected DFT [299, 300] or MP2 [301].

5.1.2 Second virial coefficient of nuclear shielding

Classical virial expansions [30] can be used to categorize the effect of pair- andhigher-order interactions for an electromagnetic property in a medium of densityn. In the virial expansion of σ [Eq. (2.7)], the linear term in n contains the secondvirial coefficient of the nuclear shielding [31], σ1(T ). It is an NMR property thatdescribes the effect of pair interactions between the probe atom and its surround-ings at a given temperature T [302]. σ1(T ) can be written as [31]

σ1(T ) = −4π∫ ∞

0

δ(R) exp[−V (R) / (kT )]R2 dR. (5.3)

where δ(R) = σXe (free) − σXe (Xe2) is the binary chemical shift function calcu-lated as the difference of the shielding constants in a free atom and in the xenondimer Xe2 at internuclear distance R, V (R) is the Xe-Xe PEC, and k is the Boltz-mann constant. QC calculations provide the R-dependence of both the chemicalshift and interaction potential, a feature not easily accessible for δ(R) in the exper-iments [303]. Thus, the first-principles computation of σ1(T ) for the xenon gasyields basic QC computational tools and concepts to study how the structure ofmaterials affects the nuclear shielding.

The virial coefficients of the pure rare gases and gas mixtures have been stud-ied both experimentally [35, 36, 302, 304] and theoretically [32–34, 52, 305] fora long time. Although accurate σ1(T ) for xenon have been measured decadesago [35, 36], the improvements on the theoretical side have been slow. This is dueto the high computational demands posed by the dispersive nature of weakly-bound Xe2, which requires a good description of electron correlation and rela-tivistic effects in both δ(R) and V (R).

Jameson et al. [32] computed the σ1(T ) of 129Xe(g) by scaling (using polarizabil-ities) the QC calculated 39Ar shielding in Ar2, based on the similarity of the raregas species. Later, Kantola et al. [33] used DFT to calculate the binary chemicalshift δ(R) to yield σ1(T ). Large deviation between calculations using functionals

56

SVWN [306] and BPW91 [307–309] was found. In the light of more sophisticatedcalculations presented in articles I and II, DFT is not a viable choice for the cal-culation of δ in σ1(T ). Jameson et al [34] revisited σ1(T ) by calculating a NR abinitio Xe–Xe shielding surface. In article I, σ1(T ) of 129Xe was found to deviateonly a little from the experiment [36] at the low-temperature range. This was pri-marily due to absence of relativistic effects in the binary chemical shift. A partialcorrection followed in II by the DHF-based calculation of δ(R), using the approx-imation of Eq. (5.1). As a result, the computed temperature dependence of σ1 isin excellent agreement with the experimental results [36], as seen in Fig. 5.2. Themaximum deviation between the experiments and the best σ1(T ) values from ar-ticle II is of the order of 1.1 ppt A3 ∧= 0.03 ppm/amagat. Thus, the experimentalaccuracy is reached roughly 30 years after the original experiments.

Figure 5.2. The progress in the quantum-chemical description of the temperature de-pendence of the second virial coefficient of 129Xe nuclear shielding, σ1(T ). The circlesdenote a NR DFT calculation for Xe δ using the BPW91 functional [33] together with afitted form of the Aziz-Slaman potential [38]. The triangles and squares are the best NRand relativistic results from articles I and II, respectively, the latter with the approxima-tion of Eq. (5.1). Both of these use the highly accurate theoretical potential from article I.The solid line without symbols describes the experimental values from Ref. [36].

57

5.2 Pairwise additive approximation

PAA offers a tempting alternative to QC calculations to compute the energies aswell as properties of atoms, molecules, clusters and solids. In more detail, it re-duces the full quantum-mechanical many-body problem of interacting particlesinto a pairwise additive one. PAA has been routinely and successfully used inthe case of potential energy functions providing estimates of various bulk prop-erties in different phases of matter [37]. In approximating QC NMR properties,the applications of PAA have been scarce. Contrary to the PAA nitrogen andproton nuclear shielding in solid ammonia [310], convincing results were ob-tained for pairwise additively evaluated 2H EFG both in water and in the mix-ture of dimethyl sulfoxide and water [311], as well as for the shielding of protonin deuterated H2O [312]. Jameson and de Dios [32] applied PAA for the shieldingof Ar3, based on the results obtained for argon dimers. Moreover, the additivityof Xe–Xe shielding contributions for Xe atom in Xe3 was estimated at differentbending angles, with encouraging results [313]. Subsequently, the shielding ofxenon surrounded by six or eight Ne atoms in a circle was also pairwise addi-tively well-reproduced [313]. These in mind, PAA should be applicable for thesecond-order molecular properties, such as the nuclear shielding of 129Xe.

PAA is particularly appealing for xenon due to its use as a guest atom in thedetermination of structure and properties of various host materials [4, 25, 26].One benefits from the use of pre-parametrized binary property curves in compu-tational NMR, since the calculational cost of the PAA is negligible as compared toany QC calculation. Pre-parametrization is a fit of QC data of interacting xenonatoms at internuclear distances R into a suitable functional form containing anensemble of parameters. In the case of interaction-induced NMR properties ofXe2, one possibility is to fit the parameters {A, ci} of a functional1

f(R) = A/Rg(R) ; g(R) = c0 + c1R+ c2R2 + · · · . (5.4)

PAA is the direct utilization of f(R) for Xe-Xe pairs in different environments.In article III, two different kinds of parametrizations of the NMR properties

have been evoked in Xen (n = 2–12) clusters [38, 314, 315] (for illustration, seeFig. 2 in article III). Apart from the dimer structure, which corresponds to theequilibrium distance of a PEC from a fit to empirical bulk properties of xenon [38],the structural data is obtained from Lennard-Jones optimized clusters [314, 315].These structures have a wide range of differently coordinated xenon atoms, andthus, can approximately describe the Xe gas, as well as the different local struc-tures present in liquid and solid phases of xenon. The first approach, PAA(bin),parametrizes the NMR properties of Xe2 at the HF level of theory. The second,PAA(eff), fits the parameters {A, ci} of Eq. (5.4), in order to reproduce the QC HFprincipal components of the NMR tensors in all the clusters Xen (n = 2–12).

The use of HF theory for the validation of PAA in article III is justified by thesmall qualitative differences in the binary curves between the HF and more so-

1The same functional form has been used for all binary NMR properties in articles I-IV, with theexception of the sign change in NQC that required a more flexible parametrization with h(R) =

A/Rg(R) − B/Rq(R), q(R) = d0 + d1R + d2R2 + · · · .

58

phisticated theories (article II). The other, more practical limitation is that QCCCSD(T) calculations, let alone fully relativistic ones, are not feasible for thelargest Xe clusters. However, in article IV, the PAA of δ and ∆σ has to be ap-plied at the best possible level of theory by using Eq. (5.1). This is due to theultimate comparison against the experimental CSA relaxation rates [79, 80, 89].

In order to provide insight for the research of complex materials, one has to un-derstand the underlying physical features of NMR tensors in the model systemsof small Xe clusters and relate them to familiar materials and structure param-eters, as well as to the sensitivity of Xe NMR tensors. The total chemical shiftδ = σ (free Xe atom) − σ

(Xesite

n

)turns out to have a monotonically growing de-

pendence on the coordination number Z [Fig 5.3(a)], and an inverse relationshipto excitation energies [Fig 5.3(c)], the latter due to the NR paramagnetic shieldingterm.

Figure 5.3. (a) The QC calculated and PAA(eff) 129Xe chemical shifts δ as a functionof the coordination number Z. (b) The QC dia- and paramagnetic NR contributionsto the chemical shift as functions of Z. (c) 129Xe chemical shift δ as a function of thesinglet excitation energy min(∆Ex,∆Ey,∆Ez), where ∆Ea corresponds to an excitedstate coupled to the ground state by rotation around the a axis. (d) Results for three- tosix-coordinated species magnified. (n is the total number of atoms in the cluster).

59

As seen in Fig 5.3(b), the NR diamagnetic part of the chemical shift (δd) re-mains more intact in the molecular formation than the paramagnetic term, δp[Eq. (4.24)]. Thus, δp reflects the total chemical shift to a larger degree than δd. Zis the profound determining factor for δ, since the chemical shift is nearly con-stant for different Xe sites sharing an equal Z in the range of 3 – 6 despite thechanges in the excitation energy [Fig 5.3(d)]. δ is also related to the local densityof states in the valence region of the electron cloud via the matrix elements of thePSO operator, Eq. (4.9).

The qualitative effect of both Z and δp has been discussed for the highly-coordinated cases of Xe12 and Xe@C60 in article III. In the former case, Z ex-plained the high δ. In Xe@C60, based on the same arguments, one would ex-pect high δ. On the contrary, the chemical shift in Xe@C60 is relatively small,based on both experimental [316] and computational [153] work. The low-energyexcited states of Xe@C60, which would imply high δ, are largely delocalized tothe fullerene cage, and, hence, offer small PSO matrix elements 〈0| `iXe

r3iXe|n〉 giving

small contributions to paramagnetic δ. The high-energy excited states, however,have significant electron density at the Xe atom, related to the PSO matrix ele-ments, and thus contribute heavily to δ. This suggests low δ, completely in linewith the earlier theoretical predictions by Adrian [317]. Despite the predictivepower of Z, one should not judge the magnitude of δ based on only it, as firmunderstanding of the underlying quantum-mechanical operators is needed.

Fig. 5.4(a) implies a systematically increasing scaling factor between QC andPAA δ values. Both PAA schemes work well for δ in reproducing the QC results.For ∆σ in Fig. 5.4(b), the results are more scattered, due to the involvement of the(often near-canceling) differences of the shielding tensor components in separatedirections.

Figure 5.4. Pure dimer-based binary [PAA(bin)] and effective binary [PAA(eff)] pairwiseadditive compared to quantum chemical (QC) results for (a) 129Xe chemical shift δ =

σ (free Xe atom)−σ`Xesite

n

´, and (b) the anisotropy of the 129Xe shielding tensor ∆σ , in

clusters Xen (n = 2− 12).

60

Increasing number of outliers is obtained for the individual components ofthe NQC tensors (see article III). The PAA approaches presented here for NMRproperties of xenon clusters could be extended to the evaluation of properties incomplex materials. While the PAA(bin) is by its construction suitable for low-density gases and liquids, the PAA(eff) offers a more meaningful route to NMRproperties of highly-coordinated liquid crystalline and solid-like phases.

5.3 Computational approach to CSA relaxation

Nuclear spin relaxation provides detailed information on dynamics of molecularsystems and materials [24]. In particular, in medicine one benefits from the use ofrelaxation time-weighted magnetic resonance imaging (MRI) of tissues [318, 319],supporting diagnostics [320, 321]. Also, T2 relaxation times are used to quan-tify metabolic compounds in the human brain [322–324]. The relaxation times of129/131Xe are, in the same manner as the static NMR spectral parameters, sensi-tive to the changes in the local environment and thus, are efficient probes of thestructure and dynamics of for example liquids and liquid crystals [8], cavities inporous solids [6], and biological tissues [12], to name a few application areas.

To acquire fundamental knowledge on microscopic relaxation mechanisms,a simple model system of 129Xe(g) was studied in article IV, with an efficientfirst-principles computational approach on the nuclear spin relaxation due to thechemical shift anisotropy mechanism. The Redfield relaxation theory [42, 43] wasapplied. It is formulated in terms of time-dependent NMR tensors, capable ofrelaxing the nuclear spin system into thermal equilibrium. MD simulations pro-vide an accessible way to incorporate the time-dependence of these tensors. Theexperimental temperature and number density conditions [79, 80, 89] were pre-pared by using NV T/NPT MD simulations. By subsequent NV E simulationone interferes as little as possible the underlying microscopic trajectories of inter-acting xenon atoms. The positions of xenon atoms at discrete time steps are thenextracted from the trajectories. Subsequently, one faces the problem of calculat-ing the NMR properties from these positions. QC methods based on simulationsnapshots [121, 126–133] become reasonable in cases, where the tensors responsi-ble for the relaxation are not crucially dependent on either electron correlation orrelativistic effects. Both in the snapshot and PAA-based methods, the samplinginterval of the ACF has to be short enough to monitor all the relevant microscopicrelaxation processes. This is interrelated to the statistical sampling of atoms, i.e.,both to the size of the atomic cluster and the number of central atoms chosen. Inarticle IV, the ACF is sampled at maximum 220 times with a sampling intervalof 1 fs, which results to ACF of length ca. 1 ns. These in mind, for the CSA re-laxation mechanism in 129Xe(g), the only feasible choice appears to be pairwiseadditive evaluation of the shielding tensor at discrete time steps, which has beenproven to work well especially for δ in the low-density low-coordinated xenongas (article III). As explained in section 3.2, CSA R1 = 1/T1 is determined by theSDF j(ω), the Fourier transformation of the shielding ACF.

The density dependence of R1 is presented in Fig. 5.5. The CSA R1 results ob-

61

tained from the computational first-principles approach show an excellent agree-ment with the experiment [41], with the latter results showing a more scatteredpattern than the computed ones. The main conclusion is that, for the first time, adirect and comprehensive set of first-principles-computed CSA relaxation rates,quantitatively comparable to experimental values [41] at wide temperature anddensity range, has been obtained for gas-phase xenon.

Figure 5.5. Density dependence (in units of amagat, amg) of the calculated and exper-imental (Ref. [89]) chemical shift anisotropy spin-lattice relaxation times in the xenongas at 295 K.

One can choose the QC method used in the calculation of both the MD simula-tion potential, V (R), between the Xe atoms, as well as the binary chemical shift,δ(R), by using different physical parametrizations of these functions. The effectsof these choices for CSA T1 are gathered in Table 5.3. In all cases in this table, thelength of the shielding autocorrelation function is 524.288 ps with a time step of1 fs. All relaxation times presented here are calculated based on the Lorentzianfit of the plateau of the SDF in the frequency domain. The details of the physicsin V (R) and δ(R) are crucial in the first-principles calculation of CSA T1. Onlythe most accurate V (R) (see for example article I and Ref. [38] for details) areplausible, and the use of unrealistic potentials such as Lennard-Jones is not rec-ommended. Concerning δ(R), essentially all the NR methods fail in reproducingT1, pointing out that relativistic treatment of δ(R) is mandatory.

62

Table 5.3. Influence of the potential energy function and the physical contentof the binary nuclear shielding tensor parametrization on the calculated 129Xechemical shift anisotropy relaxation time T1 in gaseous xenon at 295 K and99.8 amg.

Potential Switcha T1 (s) Binary shieldingb T1 (s)Ownc on 4380 R, C, RCCd 4700e

off 4040 R, C 4680Own (0.5 fs MD step) off 4120 R, NC 4630LJf off 3010 NR, C 5870Aziz-Slaman (Ref. [38]) on 4220 NR, NC 5690

a Whether the switching function is used to eliminate the potential truncationartifacts, on = switching function used, off = no switching function.b Switch function used along with our best potential, article I. Exactly the samemicroscopic positional data are used to calculate each of these T1 values withdifferent parametrizations of the shielding tensor.c Article I. Potential energy resulting from a counterpoise-correctedCCSD(T)/aug-cc-pVQZ/relativistic large-core effective core potentialcalculation featuring core polarization corrections as well as bond basisfunctions.d R=relativistic, NR=nonrelativistic, C=correlated, NC=uncorrelated, RCC =including the coupling between relativity and correlation. R, C, RCC: fullEq. (5.1); R, C: Breit-Pauli perturbation theory coupling between relativity andcorrelation omitted; R, NC: DHF shielding; NR, C: CCSD(T) shielding; NR, NC:HF shielding. These designations refer to the inclusion/omission of terms inEq. (5.1).e The seeming discrepancy between T1 values of 4700 s and 4380 s reflect theerror margins, ±680 s, involved in the theoretical evaluation of T1.f The potential of article I fitted to the Lennard-Jones functional form.

As described in detail in Fig. 5.6, the extended shielding ACF represents twodecays in different time scales. These fast and slow decays are related to the bi-nary collisions of the xenon atoms [41], and to the existence of more long-living,persistent dimers [78–80]. The existence of two distinct species in xenon gas isverified by the use of lifetime TCF [325]. Moreover, Fourier transformation of thebiexponential CSA ACF reveals two different plateaus in j(ω), corroborating theinitial observation of persistent dimers [78] in terms of two different relaxationrates due to different molecular processes. In conclusion, a comprehensive first-principles account on the CSA relaxation in the low-density gaseous xenon overa wide range of experimental conditions [41, 79, 80] is presented, with a possi-bility to distinguish between different molecular species contributing to the CSArelaxation. Well-justified extensions to other molecular species, phases of matter,as well as to different relaxation mechanisms could be carried out in the future.

63

Figure 5.6. Simulated (a) shielding ACF from the extended trajectory of length ca.1 ns, together with the fitted mono- and biexponential decays and (b) the correspond-ing direct and two analytically Fourier-transformed spectral density functions (SDFs)of gaseous Xe at 1 amg and 295 K. The existence of persistent [78–80] (in addition totransient) dimers is suggested. In (b), the vertical dotted lines depict the 129Xe Larmorfrequencies ω0 ≈ 6.0×108 rad/s and 1.0×109 rad/s (B0 of 8.0 T and 14.1 T, respectively).

6 Conclusions and future prospects

This thesis focuses on two major issues in the realm of xenon NMR using moderncomputational physics. The first objective is to study the binary NMR parame-ters of Xe2 at the state-of-the-art QC level of theory, using both relativistic andnonrelativistic wave function theories, and, to a minor extent, density-functionaltheory. Various approximations on the inclusion of the different physical effectsof the wave function theories were probed in the context of the Xe2 binary NMRproperty curves, with an emphasis on the progression of NR electron correla-tion methods. As a side project, one of the most accurate theoretical potentialenergy curves for Xe2 was calculated, with the fitted set of dispersion coeffi-cients possibly relevant in the studies of large molecular complexes in terms ofdensity functional theory. Not straightforwardly expandable to larger rare gasclusters or other molecular complexes, these methods represent the current bestknowledge within the ab initio treatment of molecular magnetic properties in theXe2 model system. Nevertheless, full parametrization of the NMR property sur-faces of xenon clusters and other molecular xenon-containing complexes wouldbe of interest in the computational modeling of the NMR properties especially insolids at different temperature and pressure conditions [293, 326] and in differ-ent molecular liquids, such as liquid crystals [125]. First steps in this directionwould naively involve the QC computation of both the binary chemical shift of129Xe and other nuclei and the corresponding PECs between them to yield thesecond (and possibly higher) virial coefficients of shielding for all the gas com-ponents in mixtures of rare and other gases. On the methodological side, thiswould preferably require a correlated ab initio relativistic shielding theory, whichcurrently, however, is not a practical alternative. Thus, one must resort to additiveapproximations for the interaction-induced NMR properties.

Secondly, the chemical shift anisotropy relaxation times of gas-phase 129Xeare calculated from first principles within Redfield relaxation theory over a largerange of experimental conditions. This is performed in a systematic and control-lable manner by changing the physics of both the binary chemical shift and thatof the potential energy curve. Therein, the feasible and efficient pairwise additiveapproximation of the nuclear shielding tensor is incorporated into molecular dy-namics simulations, to construct the time autocorrelation function of the spherical

65

shielding tensor components. The defining entity of the CSA relaxation rate, thespectral density function, is investigated directly, free from approximations suchas the extreme narrowing conditions or the monoexponential decay of the cor-responding time correlation function. Expansion to other molecular and atomicentities, phases of matter, and to different relaxation mechanisms can and willbe carried out in the future. In gas-phase xenon, this particular method directlyreveals the different relaxation time scales related to the global and local motionsof the constituent atoms and molecules, presuming that the simulation trajectoryis long enough and sampled sufficiently. A similar extension to macromolecularscale with a suitable parametrization of the NMR force field [327] could also becarried out. In addition, within the first-principles method, it is possible to ob-tain correlation times related to the specific relaxation mechanisms investigatedby means of the relaxation dispersion curves [81, 82] or NMR relaxometry [83].These correlation times could be of use in approximate, multi-exponential mod-els of NMR relaxation. In other solid and liquid phases of matter, which are notas crucially dependent on the electron correlation and the relativity as xenon is,the outlined approach for first-principles computational modeling of NMR re-laxation could be connected with both the dispersion-corrected potential energyapproaches [300, 301] and periodic density-functional-theory-based descriptionof NMR properties [119, 120, 328, 329]. Finally, the first-principles approach, witha proper account of the low-temperature features of NMR relaxation, could inthe more distant future characterize novel superconducting materials by match-ing the to-be-computed temperature-dependent relaxation profiles to the existingexperimental ones [330, 331].

For the experiment-driven computational work presented in the thesis, the fu-ture looks promising. The small systems are already adequately described bythe existing methods built up into the computational software, and for the largersystems, a set of well-motivated approximations can be invoked to reduce thecomputational burden. The study of complicated physical phenomena, as forexample in the case of NMR relaxation, graves for a well-calibrated mixture ofquantum-mechanical and classical modeling tools.

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Original papers

I Reproduced with permission from the Journal of Chemical Physics.M. Hanni, P. Lantto, N. Runeberg, J. Jokisaari, and J. Vaara,Calculation of binary magnetic properties and potential energy curve in xenon dimer:Second virial coefficient of 129Xe nuclear shielding.Journal of Chemical Physics 121, 5908 (2004).Copyright (2004) American Institute for Physics.

II Reproduced with permission from the Journal of Chemical Physics.M. Hanni, P. Lantto, M. Ilias, H. J. Aa. Jensen, and J. Vaara,Relativistic effects in the intermolecular interaction-induced nuclear magnetic reso-nance parameters of xenon dimer.Journal of Chemical Physics 127, 164313 (2007).Copyright (2007) American Institute for Physics.

III Reproduced by permission of the Physical Chemistry Chemical Physics OwnerSocieties.M. Hanni, P. Lantto, and J. Vaara,Pairwise additivity in the nuclear magnetic resonance interactions of atomic xenon.Physical Chemistry Chemical Physics 11, 2485 (2009).Copyright (2009) Royal Society of Chemistry.

IV Reproduced by permission of the Physical Chemistry Chemical Physics OwnerSocieties.M. Hanni, P. Lantto, and J. Vaara,Nuclear spin relaxation due to chemical shift anisotropy of gas-phase 129Xe,Submitted manuscript.Copyright (2011) Royal Society of Chemistry.