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Relativistic Coupled Cluster Theory – in Molecular Properties and in Electronic Structure

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Relativistic Coupled Cluster Theory – in Molecular Properties and in Electronic Structure A Thesis submitted for the degree of Doctor of Philosophy Relativistic Coupled Cluster Theory { in Molecular Properties and in Electronic Structure Avijit Shee Universit´eToulouse III { Paul Sabatier Contents Acknowledgements ii 1. Introduction1 2. Publication List3 I. Electronic Structure4 3. Brief Overview of Relativistic Electronic Structure Theory5 3.1. Relativistic Molecular Hamiltonian . .5 3.1.1. Elimination of Spin . .8 3.1.2. 4-component to 2-component . 10 3.2. Relativistic Mean Field Theory . 11 3.3. Relativistic Correlation Methods . 14 3.4. Single Reference Methods: . 17 3.4.1. Relativistic CI: . 17 3.4.2. Relativistic MP2: . 18 3.4.3. Relativistic Coupled Cluster . 19 3.5. Multi-Configurational Methods: . 20 3.5.1. MCSCF . 21 3.5.2. CASPT2 and NEVPT2 . 21 3.5.3. DMRG . 21 3.6. Approximate Correlation methods . 22 4. Application of Relativistic Correlation Methods 24 4.1. Comparison of Computational Cost Between Relativistic and Non-Relativistic Methods: . 24 4.2. Improvements to the Computational Scheme: . 25 4.3. Papers . 27 4.3.1. Paper I : A theoretical benchmark study of the spectro- scopic constants of the very heavy rare gas dimers ..... 27 4.3.2. Paper II : 4-component relativistic calculations of L3 ion- ization and excitations for the isoelectronic species UO2+, + OUN and UN2 (manuscript)................... 37 i Contents II. Molecular properties 52 5. Molecular Properties with ab-initio Relativistic Methods 53 5.1. Molecule Under External field . 54 5.2. Effect of Spin-Orbit coupling on molecular properties . 56 5.2.1. Electronic g-tensor . 56 5.2.2. NMR shielding . 57 5.2.3. Parity Violation . 59 5.3. Coupled Relativistic and Correlation Effect on Molecular properties: . 60 5.4. Numerical vs Analytical approach . 61 6. Implementation of the 4c Analytical Coupled Cluster Gradient 64 6.1. Time Reversal Symmetry . 65 6.2. Double Group Symmetry . 65 6.3. Paper III: Analytic Gradient at the 4-component Relativistic Cou- pled Cluster Level with Inclusion of Spin-Orbit Coupling (manuscript) 67 7. Summary and Future Perspective 80 III. Appendices 82 A. Integrals in Kramer's Pair Basis 83 B. Sorting of Integrals in Double Group Symmetry-DPD Scheme 91 ∗ C. C4 Character Table 94 IV. Additional Paper 95 ii Acknowledgements First and foremost, I would like to thank my supervisor Trond Saue. I would like to mention his help to settle me down in this very much `french' country, will remember several nights of watching sports together, going for lunch among many other instances, in the non-scientific side. On the scientific side, I learnt from him what sort of theoretical rigour one should follow. Other than that his programming tips, even latex tips were very helpful. Next I like to thank Debashis Mukherjee, with whom I started working in electronic structure theory back in Kolkata, India. Countless hours of discussion with him taught me very basic to very sophisticated aspects of that domain, and that teaching has not finished yet. I like to thank Luuk Visscher to be a collaborator in the gradient project. Several Skype sessions with him was not only helpful in terms of technicalities, but also it gave me hope, confidence and enthusiasm to finish that project. I like to thank Sangita Sen. We started working on a MRCC project together, which was successful on the one hand, but on the other it has not been exploited to its full capacity so far. Hopefully, there will be time to make the full use of that theory. Apart from that, several hours of `inexperienced' scientific discussion with her helped me to gain a lot of experiences, taught me several things which are otherwise difficult to grasp. From the DIRAC family I would like to thank specially Stefan Knecht, for his collab- oration in one article and, Radovan Bast for technical advices and for his ideas about modern coding practice. Very friendly and lively atmosphere of our annual DIRAC meeting always makes me feel that science can also be a joyous experience. Fantastic people associated with it help to create that atmosphere. Thanks to all of them. I will like to thank all the past and present members of the LCPQ group, specially to Adel and Nathaniel for their patience to help me in french language related sufferings. Also, I like to thank the members of my previous group in IACS - Rahul Maitra, Debalina Sinha, Shubhrodeep Pathak, Avijit Sen, Pradipta Samanta, for giving me memories to cherish. I will like to mention the names of few other scientists, short term work or discussions with whom were very enlightening - Daniel Crawford, Ed Valeev, Marcel Nooijen, Toru Shiozaki, Lan Cheng, Matthias Hanauer among few others. Thanks to Ranajoy, Pratyay, Jishnu, Anjan, Vivek, Biplab, my friends back in India (though a few of them are living in abroad). Life always feels better around them. Last but not the least I want to mention my younger brother, elder sister and my parents for their silent support and help. There are no words to thank them. iii 1. Introduction \You see, we're in a funny position: It's not that we're looking for the theory, we've got the theory { a good, good candidate { but we're in the step in the science that we need to compare the theory to experiment by seeing what the consequences are and checking it. We're stuck in seeing what the consequences are, and it's my aim, it's my desire to see if I can work out a way to work out what the consequences of this theory are (LAUGHS). It's a kind of a crazy position to be in, to have a theory that you can't work out the consequences of ... I can't stand it, I have to figure it out. Someday, maybe. Let George Do It". ....Richard P. Feynman in `The Pleasure of Finding Things Out' With the crowning success in explaining some enigmatic behaviour of heavy element chemistry [50], relativity made firm its foothold in electronic structure theory in the early 1980s. Nowadays it has widespread application, for instance, in the domain of lan- thanide and actinide chemistry - for the studies of single molecule magnets, luminescent complexes etc. Lanthanide and actinide compounds normally consist of several open- shells, which is considered as a strong correlation problem. The effect of relativity and correlation is certainly not additive [77], and one needs to develop a correlation method within the framework of relativity (meaning based on the Dirac equation). But, the over- whelming computational cost of a relativistic correlation method precludes the routine use of them, which has made its application domain somewhat limited. It is therefore necessary not only to develop new methodology and computational techniques, but also to study the existing methodologies for the challenging problems. This thesis includes studies of such kind along with implementation and use of new methods. In the first part of this thesis I will provide some background of relativistic electronic structure theory. I will mostly be focusing on the intricacies pertaining to the domains which are not really present in the non-relativistic version. Hamiltonians and different methodologies - starting from the mean field theory to the sophisticated most recent correlation theories will be touched upon. Thereby, one will be able to see the breadth and scope of this field. I will emphasize the theoretical details of the Coupled Cluster theory, since a large part of the discussions in this thesis will revolve around that method. In the next chapter, I will first discuss why relativistic theories are so much more expensive and what measures had been undertaken to deal with that problem. Nonethe- less, as already discussed, it is of paramount interest to study relativity and correlation together and we will get into the business of studying problems where both of them have major roles to play. We have studied the problem of heavy and superheavy rare gas dimers. This study has provided benchmark spectroscopic constants for these systems. Moreover, it has shown at least one example where the interplay between the Gaunt interaction and correlation can be very dramatic indeed. We have continued our study 1 1. Introduction of this effect on a different problem, namely the simulation of X-ray spectroscopy for actinides. As X-ray spectroscopy probes the core-region of a molecule, relativity has to be considered from the outset. Besides, one needs to carefully treat open-shells present in such problems since they lead to various physical effects e.g, Auger shifts and the Coester-Kronig effect. We have implemented Relativistic Single Reference Open Shell Møller-Plesset perturbation theory for this purpose. Another important aspect of relativistic theories is that they provide the most natural framework for studying electromagnetic interactions, especially for magnetic properties. In the second part of this thesis we will move into the discussion of molecular proper- ties. In the first chapter of this part we will theoretically analyze the role of spin-orbit coupling in various molecular properties. This analysis helped us to conclude that Spin Orbit Coupling (SOC) has a great deal to do in both magnetic and electric properties. While in the case of magnetic properties the origin of the problem lies fundamentally in the nature of the interaction, for electric properties it hinges on the change of bonding behaviour with and without the consideration of spin-orbit coupling. In numerous situ- ation SOC is not at all a perturbation, and then one should include SOC at the zeroth order of the Hamiltonian. The effect of Spin-Orbit coupling is ubiquitous in nature. Therefore various approximate ways of treating it has been tried over the years.
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