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We recently proposed a new approach that permits to apply the methods of Quantum Chemistry to infinite periodic systems. This is based on the transformation of the topology of a finite fragment of the periodic system into a Clifford Torus, and a redefinition of the distances in this new manifold. The standard Gaussian orbitals, that are so common in Quantum-Chemistry calculations, are also redefined, in order to obtain smooth orbitals whose behavior is compatible with the periodicity of the torus. We call these new functions “Clifford Gaussians”. Finally, after a set of calculations on systems having increasing size, an extrapolation to a system having infinite size is performed, in order to obtain the crystal properties at the thermodynamic limit. In the present paper, we apply this formalism to the study of two model systems: the free particle in a ring (a 1D system), and a pair of electrons in a square (2D) or a cube (3D). In the first case, for which the spectrum of the Hamiltonian is analytically known, we show that a set of  s -type Clifford Gaussians is able to reproduce the exact eigenvalues of the system with an extremely high accuracy, suggesting that the Clifford Gaussians, under some circumstances, could form a complete basis set for the torus. In the case of the second system, that had already be studied by our group by using a large number of ordinary Gaussians, we show that two Clifford Gaussians only are able to reproduce the behavior of the system with a good accuracy. In particular, they are able to describe the Wigner localization of the electrons that occurs at very low density. We believe that this formalism could open a new approach for the study of infinite periodic systems within the powerful and well established framework of Quantum Chemistry.

Clifford Gaussians as Atomic Orbitals for periodic systems: one and two electrons in a Clifford Torus