Thèse de Adel Almoukhalalati (LCPQ), mai 2016

Applications Of Variational Perturbation Theory In Relativistic Molecular Quantum Mechanic

Abstract : The father of relativistic quantum mechanics P. A. M. Dirac predicted that, the more realistic version of quantum mechanics that he established would not offer much more when compared to the non-relativistic formulation of quantum mechanics when applied to ordinary atomic and molecular systems.

When the relativistic quantum theory was around forty years old, people had started to recognize how important relativistic effects can be even for the study of atomic and molecular systems. Relativistic effects are manifested via the contraction of atomic s and p orbitals, the expansion of atomic d and f orbitals, and spin-orbit coupling. A classical example on the importance of relativistic effects is the band structure of metallic gold for which non-relativistic calculations will lead to an overestimation of the 5d-6p gap predicting a UV absorption band which is compatible with a metal that looks like silver.

The thesis focuses on the atomic and molecular calculations within the 4-component relativistic framework. In particular, the use of the variational perturbation theory in relativistic framework.

The perturbation theory in quantum mechanics, is based on partitioning the Hamiltonian H into zeroth-order Hamiltonian H_0 and V that forms the perturbation through a parameter lambda. In many-body (Rayleigh-Schrödinger) perturbation theory, we have an exact solution of the Hamiltonian H_0, whereas in the variational perturbation theory, we assume to have an optimized energy for any value of the parameter lambda.

The thesis contains two principal projects, the first project concerns the description of the electron correlation in the relativistic framework. In this project, we focused on the perturbative approach to derive the relativistic formulas necessary for the energy in two-electron atoms.

The correlation energy is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation. The exact eigenvalue is not available, but in the non-relativistic domain the best solution is a full CI for a given basis.

Our main goal, in this project, will be to show that the best solution of the wave equation for the embedded Dirac-Coulomb Hamiltonian, is not a Full CI, as in the non-relativistic case, but a MCSCF which uses a CI development in positive-energy orbitals only, but which keeps rotations between the positive and negative energy orbitals to optimize the projection operator. !divAbstract

The second project concerns a study of the effects of the nuclear volume in the vibrational spectra of diatomic molecules. In the early 80s, The group of Professor Eberhardt Tiemann in Hanover used the rotational spectroscopy with high resolution to study a series of diatomic molecules containing heavy atoms like lead in order to establish spectroscopic constants (R_e bond length, vibrational frequency omega_e etc.) with a great precision. A molecule AB has several isotopomers according to isotopes atoms A and B, and it was well known at that time only the spectrum of each isotopomer is slightly different because of the mass differences between each isotope of the atoms A and B. Prof. Tiemann and his collaborators discovered that we must also take into account the difference in nuclear volume of each isotope.We provide an independent check on previous experimental and theoretical studies of nuclear volume effects in rotational spectroscopy, notably re-derivation of theory and benchmark previous calculations by 4-component relativistic state of the art correlated calculations.

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