Rijksuniversiteit Groningen Bachelor Research Project Electronic structure of element 123 Koen van der Schoor supervised by Anastasia Borschevsky 2016 Abstract We compare the energies of different electronic configurations to determine the ground state of element 123 using a relativistic coupled cluster approach. We consider some predictions by others and make some predictions ourselves, and conclude that we can limit the possibilities to two configurations, [118]8s26f17d18p1 and [118]8s28p3. We also explore the performance of different computational approaches and different basis sets by doing calculations on the ionization potentials of the group 2 elements. We find that the CCSD and CCSD(T) calculations have a much higher accuracy than the HF calculations, and that there is little difference between the pvdz, pvtz, and pvqz basis sets. We also find that there is little difference between relativistic X2C and 4c calculations, while both are much more accurate than non-relativistic calculations. Furthermore, we investigate the influence of relativity on the first and second ionization potentials of the group 1 elements. We find that relativistic calculations lead to a much higher accuracy for the first IP, but that they do not change much for the second IP. Contents Introduction 2 Method 3 Relativity . .3 Electron correlation . .4 Basis sets . .4 Research project 5 Alkaline earth metals . .5 Alkali metals . .8 Element 123 . 11 Conclusion 13 Bibliography 15 1 Introduction Recently, the International Union of Pure and Applied Chemistry (IUPAC) announced that it decided on the names for elements 113 (nihonium), 115 (moscovium), 117 (tennessine), and 118 (oganesson) [1]. This means that all 118 elements in the periodic table are now named. Of these elements, only the first 94 are known to exist in nature. The elements that do not appear in nature can still be created. This can happen in a particle accelerator or a nuclear reactor, where a beam of atoms is aimed at some other atoms. If two of these atoms collide, it is possible that they \stick" together and form a new atom. Then, the decay of this atom or its decay products can be detected, which verifies that the element was indeed produced. Elements heavier than uranium (Z = 92) are commonly referred to as heavy elements or transuranium elements, and those heavier than lawrencium (Z = 103) are known as superheavy elements. Very little is known about the chemical properties of the heavy and superheavy elements. It takes a lot of money, time, and effort to create them, and their short lifetimes make them difficult to observe. Elements with Z > 118 have never even been produced, which means that all we know about them, which is very little, comes from theory. Because so little is known about the properties of these elements, it is important to have accurate theoretical predictions, so that scientists who do experiments know what they are looking for, and where they might find it. Theoretical predictions are also important to obtain fundamental knowledge about these elements, since not everything can be done with experiments. The most fundamental electronic property of an element is its ground state. Knowing the ground state of an element allows us to place it in the periodic table. This is important, because we can then identify the (lighter) homologs of our new element. Some properties of our new element might be related to the properties of its homologs. It also allows us to predict how the element would interact with other elements, since this depends partly on the electron distribution in the outer shells of the atom. Furthermore, we can predict which compounds the element is likely to form. Lastly, when we have the ground state of an element, we are in a much better position than before to determine its first ionized state. If we have the ground state and the first ionized state, we can determine the ionization potential. We are interested in determining the ground state of element 123, partly because of the reasons listed above, and partly because so far, no elements beyond element 122 have been studied with high accuracy. There already exist predictions for the ground states of the elements with Z > 118. For element 119, Fricke et al. predict a ground state of [118]8s1 using Hartree-Fock calculations [2]. Pyykk¨opredicts the same from his Dirac-Fock calculations [3]. For element 120, Fricke et al. predict a ground state of [118]8s2 using Hartree-Fock calculations [2]. Pyykk¨opredicts the same from his Dirac-Fock calculations [3]. For element 121, Fricke's Hartree-Fock study predicts a 2 1 ground state of [118]8s 8p1=2 [2]. Eliav et al. predict the same [4] from their relativistic coupled cluster calculations, and so does the study by Umemoto and Saito using density functional theory 1 2 1 [5]. For element 122, Fricke et al. predict a ground state of [118]7d3=28s 8p1=2 using Hartree-Fock calculations [2]. Eliav et al. predict the same [6] from their relativistic coupled cluster study. Umemoto and Saito predict from their density functional theory calculations a different ground state, [118]8s28p2 [5]. For element 123, the Hartree-Fock study by Fricke et al. predicts a ground 1 1 2 1 state of [118]6f5=27d3=28s 8p1=2 [2]. Umemoto and Saito predicts the same, using density functional theory calculations [5]. 2 Method The ground state of an element is the state in which it has its lowest energy. To determine this state, we need to find a number of possible electron configurations, and then determine the energy of each of these configurations. The configuration with the lowest energy is then probably the ground state. To determine a possible electron configuration, we can look at lighter elements that are in the same column in the periodic table as our element, and guess that their configurations will look alike. We can also look at elements with a lower atomic number than our element, and add electrons to different shells. When determining the energy of a certain electron configuration of an atom theoretically, there are roughly three things to take into account: the treatment of relativity, the treatment of electron correlation, and the choice of basis sets. Relativistic effects are small in light elements, but start to become important for elements heavier than iron (Z = 26). In multi-electron atoms, electrons interact both with the nucleus and with each other. The interaction between the electrons is known as electron correlation. Finally, we can describe the atomic orbitals in different ways, depending on our choice of basis sets. Relativity It is possible to do either relativistic or non-relativistic calculations. For most systems, non- relativistic calculations are sufficient. However, for heavy and superheavy elements relativistic effects start to play an increasingly larger role, which means that we must use relativistic calcu- lations. Nevertheless, doing non-relativistic calculations is still useful, because it shows how large the influence of relativistic effects on the energy and other properties of an atom is. We will now loosely follow an explanation by Pershina [7]. Currently the best way to describe a many body system is the Dirac-Coulomb-Breit (DCB) Hamiltonian X X 1 h = h (i) + + B DCB D r ij i i<j ij where hD(i) is the one-electron Dirac operator 2 n hD(i) = c~αi~pi + c (βi − 1) + V (i) where the ~α and β are the four-dimensional Dirac matrices, V n is the nuclear attraction operator, which takes the finite nucleus size into account (as opposed to treating the nucleus as a pointlike particle), and Bij is the Breit term " # 1 (~αi~αj) (~αi~rij)(~αj~rij) Bij = − 1 + 3 2 rij rij The operators in the DCB Hamiltonian, also referred to as the Dirac equations, are 4 x 4 matrix operators. The corresponding wave function is therefore a four-component (4c) vector. Methods that use this Hamiltonian are therefore referred to as 4c methods. In our calculations we use the DIRAC program [8], which does not implement the Breit term, so we have the hDC Hamiltonian. By default, the DIRAC program does relativistic calculations. To do non-relativistic calculations, we set the speed of light in the program to a practically infinitely high value. An approximation to the 4c methods are the two-component, or 2c, methods. In the 2c ap- proximation, the positronic and electronic solutions of the Dirac-Hartree-Fock calculations are decoupled. This reduces the number of matrix elements in the Hamiltonian to interactions solely among electrons and nuclei. So, by using 2c methods, computational time and effort can be saved. In our research, we use the recently developed X2C approach [9]. This method accurately reproduces 4c results, and requires less computational effort. 3 Electron correlation In multi-electron atoms, the electrons are influenced not only by the nucleus, but also by each other. This interaction is called electron correlation. To make accurate predictions of the atomic energies we need to take this into account. In our research we use two different approaches: the (Dirac-)Hartree-Fock (DHF) approach and the coupled cluster approach. The Hartree-Fock approach does not treat electron correlation, but the coupled cluster approach does. In the Hartree-Fock theory, the wave function of a system containing multiple orbitals is taken to be the product of the Slater determinants of the individual spin orbitals (the spatial orbitals multiplied with a spin function). It is assumed that an electron does not see all the other electrons individually, but rather feels the effect of the average positions of all the other electrons together. The Hartree-Fock equations are iterative, that means, to solve the Schr¨odingerequation for a multi-electron system using Hartree-Fock theory, one starts with an initial guess, and in each iteration the wavefunction is adjusted to minimize the energy.