Ivan Infante, Vrije Universiteit, Amsterdam, NL, 2006 (pdf, 4.1 - Dirac

166 Pages·2006·4.1 MB·English

VRIJE UNIVERSITEIT
Computational Studies in Actinide
Chemistry
ACADEMISCH PROEFSCHRIFT
ter verkrijging van de graad Doctor aan
de Vrije Universiteit Amsterdam,
op gezag van de rector magnificus
prof.dr. L. M. Bouter,
in het openbaar te verdedigen
ten overstaan van de promotiecommissie
van de faculteit der Exacte Wetenschappen
op dinsdag 3 oktober 2006 om 13.45 uur
in het auditorium van de universiteit,
De Boelelaan 1105
door
Ivan Antonio Carlo Infante
geboren te Clusone, Itali¨e
promotor: prof.dr. E.J. Baerends
copromotor: dr. L. Visscher
CONTENTS
1 Relativistic Quantum Chemistry 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The spectroscopy of small molecules . . . . . . . . . . . . . . . . . . . 3
1.3 The treatment of nuclear waste . . . . . . . . . . . . . . . . . . . . . . 4
1.4 This thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Theory and Methodology 9
2.1 The origin of the Dirac equation . . . . . . . . . . . . . . . . . . . . . 9
2.2 The properties of the Dirac Equation . . . . . . . . . . . . . . . . . . . 11
2.3 TheSolutionoftheOne-ElectronDiracEquationinthefieldofaNucleus 13
2.4 The Many-Electron Dirac Equation . . . . . . . . . . . . . . . . . . . . 14
2.5 The Dirac-Coulomb-Hartree-Fock approach . . . . . . . . . . . . . . . 15
2.6 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Basis set considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 Transformed Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.9 Electron correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.10 Coupled-cluster theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.11 Fock-space Coupled Cluster . . . . . . . . . . . . . . . . . . . . . . . . 25
2.12 The Intermediate Hamiltonian Fock-space Coupled Cluster . . . . . . 27
2.13 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.14 The QM/MM approach . . . . . . . . . . . . . . . . . . . . . . . . . . 31
iv CONTENTS
I Spectroscopy of Small Molecules 35
3 The importance of spin-orbit coupling and electron correlation in
the rationalization of the ground state of the CUO molecule 37
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.2 Spin-Orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.3 Verification of the computed values . . . . . . . . . . . . . . . . 45
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 The electronic structure of UO revisited 51
2
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.1 Orbital composition and UO+ . . . . . . . . . . . . . . . . . . 57
2
4.4.2 Basis-set convergence . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.3 Active space convergence . . . . . . . . . . . . . . . . . . . . . 61
4.4.4 Analysis of the excited states: scalar relativistic results . . . . 63
4.4.5 Analysis of the excited states: the spin orbit (SODC) case . . . 66
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 Supplementary Materials . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 On the Performance of the Intermediate Hamiltonian Fock-Space
Coupled–Cluster Method on Linear Triatomic Molecules: the Elec-
tronic Spectra of NpO+, NpO2+ and PuO2+ 77
2 2 2
5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4.1 Electronic structure of the ground state of NpO3+ and PuO4+ 81
2 2
5.4.2 Electronic spectrum of NpO+ and PuO2+ . . . . . . . . . . . . 85
2 2
5.4.3 Potential Energy Curves . . . . . . . . . . . . . . . . . . . . . 92
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
CONTENTS v
5.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
II Part II : DFT calculations on aqueous systems 95
6 A QM/MM study on the aqueous solvation of the tetrahydrox-
ouranylate [UO F ]2− complex ion 97
2 4
6.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4.1 Structureofthe[UO F (H O)]2− inthegasandsolventphases.
2 4 2
MM modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4.2 Electronicstructureofthe[UO F (H O)]2−complexinthegas-
2 4 2
phase. The difference between hexa and hepta coordination.. . 105
6.4.3 Electronic structure of the HEXA and HEPTA complexes in
the solvent phase. . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7 A QM/MM study on the aqueous solvation of the tetrahydrox-
ouranylate [UO (OH) ]2− complex ion 113
2 4
7.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.4.1 The isolated [UO (OH) ]2− complex . . . . . . . . . . . . . . . 117
2 4
7.4.2 The isolated [UO (OH) ]2− + H O complex. . . . . . . . . . . 118
2 4 2
7.4.3 The embedded [UO (OH) ]2− + H O complex . . . . . . . . . 122
2 4 2
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.6.1 Setup for the QM/MM partitioning . . . . . . . . . . . . . . . 126
Summary 129
Samenvatting 135
Acknowledgments 141
List of Publications 145
vi CONTENTS
Bibliography 147
1
CHAPTER
Relativistic Quantum Chemistry
If I understand Dirac correctly, his meaning is this: there is no God, and Dirac is his
Prophet.
Wolfgang Pauli
1.1 Introduction
In the phenomena that occur in everyday life we intuitively apply a series of notions
whichareatthebaseoftheNewtonianphysics. When,forexample,wecrossastreet
and the traffic light is on red, we can quickly estimate the speed of the car that is
comingonourdirection, providedthatdoesnotsuddenlyaccelerate. Ifthemeasured
time to reach our position is long enough, then we can safely cross the road. Our
guess is based on a certain description of the dynamics of the car that turns out to
be the same for us and for the car’s driver. The laws that describe this motion are
said to be invariant under a Galilei transformation over all the inertial frames. The
theory that explains the movement of these macroscopic objects is known as classical
mechanics.
Macroscopicobjectscanbepushedtomoveatveryhighspeed, andiftheirveloc-
ities come close to the speed of light, then classical mechanics is no longer capable of
2 Relativistic Quantum Chemistry
predicting their trajectories. Einstein developed a new theory where space and time
are not two distinct entities, like we intuitively expect, but are closely linked. The
equations of motion in this new domain are not invariant under a Galilei transfor-
mation, but under a Lorentz transformation. We can explain what happens to these
very fast objects making paradoxical examples. One of these is the well known twins
paradox, in which one of two brothers is an astronaut who travels on a spaceship at
speedoflightandtheother,lessadventurousbrother,preferstoliveinaslowmoving
object, i.e. the earth. At the end of his trip, the first brother is younger than his
brother,becauseofthephenomenonoftimecontraction1. Afast-movingpersonages
more slowlydue tothe relationthat connects the coordinates oftwo different frames:
t0 = √t−(xv/c2) (in the approximation we move along the x axis). The dynamics that
1−v2/c2
explains the behavior of fast moving objects is called the special theory of relativity.
Atmicroscopicscale,itismoredifficulttopredicttrajectoriesbecauseofthedual
wave-particle nature that each object intrinsically possesses. The Heisenberg princi-
pleteachesusthatspeedandpositioncannotbeknownexactlyatthesametime,the
moreaccuratelyoneismeasured,thebiggeristheuncertaintyonthemeasurementof
the other. The equations of motion of microscopic objects can be deduced from the
Schr¨odinger equation and the theory that explains their behavior is called quantum
mechanics. In the last forty years, the evolution of computer power has helped in
finding an approximate solution to the Schr¨odinger equation for microscopic systems
of increasing complexity. In particular, the movement of electrons can be decoupled
from the much slower nuclei (Born-Oppenheimer approximation), in order to predict
the properties of molecular systems, by looking, for example, at their atomic vibra-
tions,attheelectronicenergylevels,oratthefeaturesofthevalenceelectrons. Inthe
beginning,itwasimportanttoassessthevalidityofthequantummechanicalformulae
in order to interpret or match experimental data. Currently, quantum chemistry has
achieved a degree of precision so high that it can be used as a powerful tool to pre-
dictthebehaviorofachemicalsystem, andanticipatetheoutcomeofanexperiment,
which is the ultimate goal.
A very small object can also move at velocities close to the speed of light. For
example,anelectronmovinginthevicinityofa“heavy”nucleuscanincreaseitsspeed
so much that its mass can change significantly and the volume (orbital) in which it
moves deforms. This can affect certain properties of a system. It has been shown
that very precise calculations on the H molecule demand a relativistic description.
2
1The twins paradox is more complicated than it looks. For simplicity, we considered the ”stan-
dard”explanation,inwhichwesupposethatthespaceshipmotionisnotaccelerated.
1.2 The spectroscopy of small molecules 3
The combination of quantum mechanics and the special theory of relativity is known
as relativistic quantum mechanics. Usually, the relativistic contribution is observed
for elements with high atomic number, because its magnitude increases as Z2. For
heavier elements the radial distribution of the s and p orbitals shows a contraction
andthedandf orbitalsaremorediffuse. Furthermore,atthesevelocities,theorbital
angularmomentuml andthespinmomentumscannotbeseparated,butarecoupled
into a single quantity, the total angular momentum j = l + s. Under the influence
of this coupling, the energy levels of an electron in an atom depend on the main
quantum number n and the total angular momentum j.
In many ordinary cases relativity does not manifest itself to a great extent and
canbesafelyneglected(especiallyforlightatoms),butinothersituationsrelativistic
corrections must be included for a satisfactory description of a system.
Ionization potentials, electron affinities, bond lengths, activation energies can all
be described in the relativistic framework depending on the desired accuracy. There
are many situations in which relativistic effects play a major role, and in this thesis I
want to focus on two different cases: the spectroscopy of small molecules containing
an actinide element and the analysis of the interaction between a solute molecule -
which includes an heavy element - and a solvent.
1.2 The spectroscopy of small molecules
Actinide molecules contain one (or more) heavy atoms, with atomic number ranging
from90to103. Thespectroscopyofthesemoleculesremainsachallengeforcomputa-
tional chemists because the manifoldof accessible states is so dense, that anaccurate
description can be achieved only by describing on equal footing both the dominant
relativistic effects and the electron correlation. Nowadays, relativity can be taken
into account at a very high accuracy by solving the 4-component Dirac equation or
by using a pseudo-relativistic one-component (scalar) or two-component (spin-orbit)
equation. The reliability of molecular property calculations, depends, thus, on the
descriptionoftheelectroncorrelation. Thecorrelationenergyisdefinedasthediffer-
ence between the exact energy of the system and the Hartree-Fock energy, in which
the electron-electron interaction is included in a mean-field approximation (MFA).
Correlationenergyisdividedintodynamiccorrelation,generatedfromthemutualre-
pulsionofelectronsasdeviationfromtheMFA,andnon-dynamiccorrelation, arising
from the inadequacy of a single reference determinant to describe a molecular state.
Till this moment, the most widely used approach that accurately describes these
4 Relativistic Quantum Chemistry
twocontributionshasproventobeCASSCF/CASPT2. Thereliabilityofthismethod,
however, is limited by the heavy computational costs that are requested when active
spacesincludingmorethan20orbitalsarechosen. Itisthusnecessarytoexplorenew
methodsthatremovethisbottleneckbykeepingthesameorevenincreasingthelevel
of precision in the calculation.
In the last ten years a new method is emerging in atomic physics as the most
powerful tool to evaluate with very high precision the excited spectrum of atomic
systems: the multi-reference Fock-Space Coupled Cluster method (FSCC) [1]. The
attractionforthismethodismultiple: 1)itisaCoupledClusterwavefunctionmodel,
which is considered the most accurate way to compute dynamic correlation energy;
2) it describes on the same ground the non-dynamic correlation energy; 3) it scales
like N6 (in case of CCSD), that is a relatively cheap scaling factor for a post Hartree
Fock method. The drawback is that FSCC suffers of convergence problems for the
presenceofintruderstates. Toovercomethisproblem, thenovelIntermediateHamil-
tonianFSCC(IHFSCC),avariantoftheFSCCmethod,hasbeenrecentlydeveloped.
Despite being very promising tools, a significant disadvantage of FSCC and IHFSCC
is that until this day only few results are available for molecular systems.
In this thesis, one of the goals is to test the validity of single reference and multi-
reference Coupled Cluster methods by analyzing actinide systems of different com-
plexity. The choice has fallen on linear triatomic molecules as start, because they
are small systems with high symmetry. In particular, the UO and CUO molecules
2
have been selected. The infrared spectrum of these systems in argon matrix shows
low energies that were assigned to the U-O asymmetric stretch (and C-U stretch in
the case of CUO). In neon matrix, the same asymmetric stretch lies at much higher
value, shifted of about 130 cm−1. The experimentalists [2], puzzled by this behavior,
suggested a change on the ground state of the trapped molecule, thus implying an
interaction of the actinide molecule with the noble gases. This would be unexpected
as the noble gases are known to have a very small reactivity. These two molecules
havebeenobjectofexperimentalandtheoreticaldebateforyearsandanindisputable
interpretation of their behavior has not been found until today. This thesis aims to
provide an explanation using novel advanced computational models.
1.3 The treatment of nuclear waste
Thechemistryoftheactinidesisattractivemainlybecauseitisusedintheproduction
ofenergybycontrollednuclearfissionchainreactions. Theprocessfromtheextraction