## Abstract

In this paper, we propose a conceptual approach to assign a “mathematical meaning” to the non-local function χ(r,r^{′}). Mathematical evaluation of this kernel remains difficult since it is a function depending on six Cartesian coordinates. The idea behind this approach is to look for a limit process in order to explore mathematically this non-local function. According to our approach, the bra ⟨χr_{′}^{ξ}| is the linear functional that corresponds to any ket |ψ⟩, the value ⟨r^{′}|ψ⟩. In condensed writing ⟨χr_{′}^{ξ}|⟨r|ψ⟩=⟨r^{′}|ψ⟩, and this is achieved by exploiting the sifting property of the delta function that gives it the sense of a measure, i.e. measuring the value of ψ(r) at the point r^{′}. It is worth noting that ⟨χr_{′}^{ξ}| is not an operator in the sense that when it is applied on a ket, it produces a number ψ(r=r^{′}) and not a ket. The quantity χr_{′}^{ξ}(r) proceed as nascent delta function, turning into a real delta function in the limit where ξ→0. In this regard, χr_{′}^{ξ}(r) acts as a limit of an integral operator kernel in a convolution integration procedure.

Original language | English |
---|---|

Journal | Journal of Mathematical Chemistry |

Early online date | 22 Feb 2024 |

DOIs | |

Publication status | E-pub ahead of print - 22 Feb 2024 |

## Keywords

- Conceptual DFT
- Dirac distribution
- Linear response function

## ASJC Scopus subject areas

- General Chemistry
- Applied Mathematics