The linear response function χ(r,r′): another perspective

Samir Kenouche, Jorge I. Martínez-Araya

Research output: Contribution to journalArticlepeer-review


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 ⟨χrξ| is the linear functional that corresponds to any ket |ψ⟩, the value ⟨r|ψ⟩. In condensed writing ⟨χrξ|⟨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 ⟨χrξ| 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 χrξ(r) proceed as nascent delta function, turning into a real delta function in the limit where ξ→0. In this regard, χrξ(r) acts as a limit of an integral operator kernel in a convolution integration procedure.

Original languageEnglish
JournalJournal of Mathematical Chemistry
Early online date22 Feb 2024
Publication statusE-pub ahead of print - 22 Feb 2024


  • Conceptual DFT
  • Dirac distribution
  • Linear response function

ASJC Scopus subject areas

  • General Chemistry
  • Applied Mathematics


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