# The total position-spread tensor: Spin partition

## Abstract

The Total Position Spread (TPS) tensor, defined as the second moment cumulant of the position operator, is a key quantity to describe the mobility of electrons in a molecule or an extended system. In the present investigation, the partition of the TPS tensor according to spin variables is derived and discussed. It is shown that, while the spin-summed TPS gives information on charge mobility, the spin-partitioned TPS tensor becomes a powerful tool that provides information about spin fluctuations. The case of the hydrogen molecule is treated, both analytically, by using a 1s Slater-type orbital, and numerically, at Full Configuration Interaction (FCI) level with a V6Z basis set. It is found that, for very large inter-nuclear distances, the partitioned tensor growths quadratically with the distance in some of the low-lying electronic states. This fact is related to the presence of entanglement in the wave function. Non-dimerized open chains described by a model Hubbard Hamiltonian and linear hydrogen chains H{sub n} (n ≥ 2), composed of equally spaced atoms, are also studied at FCI level. The hydrogen systems show the presence of marked maxima for the spin-summed TPS (corresponding to a high charge mobility) when the inter-nuclear distance is about 2 bohrs.more »

- Authors:

- Laboratoire de Chimie et Physique Quantiques - LCPQ/IRSAMC, Université de Toulouse (UPS) et CNRS (UMR-5626), 118, Route de Narbonne, 31062 Toulouse Cedex (France)
- (Spain)
- Institut für Chemie und Biochemie - Physikalische und Theoretische Chemie, Freie Universität Berlin, Takustr. 3, D-14195 Berlin (Germany)
- Dipartimento di Chimica Industriale “Toso Montanari”, Università di Bologna, Viale Risorgimento 4, I–40136 Bologna (Italy)

- Publication Date:

- OSTI Identifier:
- 22416209

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 142; Journal Issue: 9; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ATOMS; CONFIGURATION INTERACTION; DISTANCE; ELECTRONS; FLUCTUATIONS; HAMILTONIANS; HYDROGEN; MOBILITY; MOLECULES; PARTITION; POSITION OPERATORS; QUANTUM ENTANGLEMENT; SPIN; TENSORS; WAVE FUNCTIONS

### Citation Formats

```
El Khatib, Muammar, E-mail: elkhatib@irsamc.ups-tlse.fr, Evangelisti, Stefano, E-mail: stefano@irsamc.ups-tlse.fr, Leininger, Thierry, E-mail: Thierry.Leininger@irsamc.ups-tlse.fr, Brea, Oriana, E-mail: oriana.brea@uam.es, Departamento de Química, Facultad de Ciencias, Módulo 13, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Fertitta, Edoardo, and Bendazzoli, Gian Luigi, E-mail: gianluigi.bendazzoli@unibo.it.
```*The total position-spread tensor: Spin partition*. United States: N. p., 2015.
Web. doi:10.1063/1.4913734.

```
El Khatib, Muammar, E-mail: elkhatib@irsamc.ups-tlse.fr, Evangelisti, Stefano, E-mail: stefano@irsamc.ups-tlse.fr, Leininger, Thierry, E-mail: Thierry.Leininger@irsamc.ups-tlse.fr, Brea, Oriana, E-mail: oriana.brea@uam.es, Departamento de Química, Facultad de Ciencias, Módulo 13, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Fertitta, Edoardo, & Bendazzoli, Gian Luigi, E-mail: gianluigi.bendazzoli@unibo.it.
```*The total position-spread tensor: Spin partition*. United States. doi:10.1063/1.4913734.

```
El Khatib, Muammar, E-mail: elkhatib@irsamc.ups-tlse.fr, Evangelisti, Stefano, E-mail: stefano@irsamc.ups-tlse.fr, Leininger, Thierry, E-mail: Thierry.Leininger@irsamc.ups-tlse.fr, Brea, Oriana, E-mail: oriana.brea@uam.es, Departamento de Química, Facultad de Ciencias, Módulo 13, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Fertitta, Edoardo, and Bendazzoli, Gian Luigi, E-mail: gianluigi.bendazzoli@unibo.it. Sat .
"The total position-spread tensor: Spin partition". United States.
doi:10.1063/1.4913734.
```

```
@article{osti_22416209,
```

title = {The total position-spread tensor: Spin partition},

author = {El Khatib, Muammar, E-mail: elkhatib@irsamc.ups-tlse.fr and Evangelisti, Stefano, E-mail: stefano@irsamc.ups-tlse.fr and Leininger, Thierry, E-mail: Thierry.Leininger@irsamc.ups-tlse.fr and Brea, Oriana, E-mail: oriana.brea@uam.es and Departamento de Química, Facultad de Ciencias, Módulo 13, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid and Fertitta, Edoardo and Bendazzoli, Gian Luigi, E-mail: gianluigi.bendazzoli@unibo.it},

abstractNote = {The Total Position Spread (TPS) tensor, defined as the second moment cumulant of the position operator, is a key quantity to describe the mobility of electrons in a molecule or an extended system. In the present investigation, the partition of the TPS tensor according to spin variables is derived and discussed. It is shown that, while the spin-summed TPS gives information on charge mobility, the spin-partitioned TPS tensor becomes a powerful tool that provides information about spin fluctuations. The case of the hydrogen molecule is treated, both analytically, by using a 1s Slater-type orbital, and numerically, at Full Configuration Interaction (FCI) level with a V6Z basis set. It is found that, for very large inter-nuclear distances, the partitioned tensor growths quadratically with the distance in some of the low-lying electronic states. This fact is related to the presence of entanglement in the wave function. Non-dimerized open chains described by a model Hubbard Hamiltonian and linear hydrogen chains H{sub n} (n ≥ 2), composed of equally spaced atoms, are also studied at FCI level. The hydrogen systems show the presence of marked maxima for the spin-summed TPS (corresponding to a high charge mobility) when the inter-nuclear distance is about 2 bohrs. This fact can be associated to the presence of a Mott transition occurring in this region. The spin-partitioned TPS tensor, on the other hand, has a quadratical growth at long distances, a fact that corresponds to the high spin mobility in a magnetic system.},

doi = {10.1063/1.4913734},

journal = {Journal of Chemical Physics},

number = 9,

volume = 142,

place = {United States},

year = {Sat Mar 07 00:00:00 EST 2015},

month = {Sat Mar 07 00:00:00 EST 2015}

}