Total position spread

In physics, the total position-spread (TPS) tensor is a quantity originally introduced in the modern theory of electrical conductivity. In the case of molecular systems, this tensor measures the fluctuation of the electrons around their mean positions, which corresponds to the delocalization of the electronic charge within a molecular system. The total position-spread can discriminate between metals and insulators taking information from the ground state wave function. This quantity can be very useful as an indicator to characterize Intervalence charge transfer processes, the bond nature of molecules (covalent, ionic or weakly bonded), and Metal–insulator transition. In physics, the total position-spread (TPS) tensor is a quantity originally introduced in the modern theory of electrical conductivity. In the case of molecular systems, this tensor measures the fluctuation of the electrons around their mean positions, which corresponds to the delocalization of the electronic charge within a molecular system. The total position-spread can discriminate between metals and insulators taking information from the ground state wave function. This quantity can be very useful as an indicator to characterize Intervalence charge transfer processes, the bond nature of molecules (covalent, ionic or weakly bonded), and Metal–insulator transition. The Localization Tensor (LT) is a per electron quantity proposed in the context of the theory of Kohn to characterize electrical conductivity properties. In 1964, Kohn realized that electrical conductivity is more related to the proper delocalization of the wave function than a simple band gap. In fact, he proposed that a qualitative difference between insulators and conductors also manifests as a different organization of the electrons in their ground state where one has that: the wave function is strongly localized in insulators and very delocalized in conductors. The interesting outcome of this theory is: i) it relates the classical idea of localized electrons as a cause of insulating state; ii) the needed information can be recovered from the ground state wave function because in the insulated regime the wave function breaks down as a sum of disconnected terms. It is until 1999 that Resta and coworkers found a way to define the Kohn delocalization by proposing the already mentioned Localization Tensor. The LT is defined as a second order moment cumulant of the position operator divided by the number of electrons in the system. The key property of the LT is that: it diverges for metals while it takes finite values for insulators in the Thermodynamic limit. Recently, the global quantity (the LT not divided by the number of electrons) has been introduced to study molecules and named Total Position-Spread tensor. The total position spread Λ is defined as the second moment cumulant of the total electron position operator, and its units are in length square (e.g. bohr²). In order to compute this quantity, one has to take into account the position operator and its tensorial square. For a system of n electrons, the position operator and its Cartesian components are defined as: Where the i index runs over the number of electrons. Each component of the position operator is a one-electron operator, they can be represented in second quantization as follows: where i,j run over orbitals. The expectation values of the position components are the first moments of the electrons' position.