Chapter 7 Quantum-Mechanical Theory of Atoms in Molecules: A Relativistic Formulation Jerzy Cioslowski Department of Chemistry and CSIT Florida State University, Tallahassee, FL 32306-3006, USA Jacek Karwowski Instytut Fizyki, Uniwersytet Mikolaja Kopernika Grudzi¸ adzka 5, 87-100 Toru´ n, Poland 1. INTRODUCTION Analysis of chemical and physical phenomena often calls for parti- tioning of electronic properties into contributions from atoms, bonds, and molecular fragments [1]. Such a partitioning requires a theoretical prescription for discerning atoms in molecules (AIMs) that is not pro- vided by the conventional formulation of quantum mechanics. However, AIMs can be rigorously defined as open subsystems with properties that satisfy well-known quantum-mechanical relationships, such as the virial and Ehrenfest theorems. A nonrelativistic theory of AIMs that incorpo- rates such a definition has been formulated by Bader [2]. Central to this theory is the generalization of Schwinger’s principle of stationary action [3] that results in the identification of AIMs as unions of nuclei and dis- joint domains in the Cartesian space (called atomic basins) bordered by surfaces of zero flux in the electron density gradient [2, 4-9]. In this chapter, we arrive at a relativistic definition of AIMs by inves- tigating the Hamilton’s principle applied to the Lagrangian density of a 1