PhD TUTORIAL Theoretical methods for small-molecule ro-vibrational spectroscopy Lorenzo Lodi and Jonathan Tennyson University College London, Department of Physics and Astronomy, Gower Street, London WC1E 6BT, UK E-mail: j.tennyson@ucl.ac.uk Abstract. Solution of the first principles equations of quantum mechanics is providing an increasingly accurate and predictive approach for solving problems involving atoms and small molecules. A general introduction to the methods used for the ab initio calculation of rotational-vibrational spectra of small molecules is presented, with a strong focus on triatomic systems. The use of multi-reference electronic structure methods to compute molecular potential-energy and dipole- moment surfaces are discussed. Issues related to the construction of such surfaces and the inclusion of corrections due to relativistic and non-Born-Oppenheimer effects are reviewed. The derivation of exact, internal-coordinate nuclear-motion effective Hamiltonians and their solution using a discrete-variable-representation are discussed. Sample results for the water molecules are used throughout the tutorial to illustrative the theoretical and numerical issues in such calculations. PACS numbers: 31.15.-p; 33.15.Mt; 33.20.-t; 33.20.Vq; Contents 1 Introduction 3 2 Basic theoretical concepts 4 2.1 The molecular Hamiltonian ........................ 4 2.1.1 The Born-Oppenheimer perturbational approach ........ 4 2.1.2 The Born-Huang variational approach .............. 6 2.1.3 Criticisms and further developments ............... 8 2.1.4 The idea of molecular shape .................... 8 2.2 Relativistic effects ............................. 9 2.2.1 Quantum electrodynamics ..................... 9 2.2.2 The one-electron Dirac equation ................. 9 2.2.3 The spin-free Dirac equation ................... 10 2.2.4 The many-electron Dirac equation ................ 10 2.2.5 The Pauli Hamiltonian ....................... 11 2.2.6 The many-electron Pauli Hamiltonian .............. 11 2.2.7 The Douglas-Kroll-Hess Hamiltonian ............... 11 2.2.8 Quantum electrodynamics corrections .............. 12 2.3 Some indicative examples ......................... 12