Physical Science & Biophysics Journal Density Function Thermal Scaling and Physical Significance of Shape Functions Phy Sci & Biophy J Density Function Thermal Scaling and Physical Significance of Shape Functions Carbó-Dorca R* University of Girona, Catalonia, Spain *Corresponding author: Carbó-Dorca R, University of Girona, Catalonia, Spain, Tel:+34972405121; E-mail: ramoncarbodorca@gmail.com Abstract The solid-state concept of thermal voltage is used to define a homothetic scaling of the quantum mechanical one-particle reduced density function. This scaling might be used to construct a temperature dependent quantum density. Once defined such scaling, named here as thermal scaling, it is simple to use it with a precise temperature, adapting such scaling matching the associated shape function. The temperature achieving this equality is termed Shape Temperature S T and, if N is the number of particles of a given quantum object, one can demonstrate that the simple equality: 12 S T N  holds. Furthermore, Shape Temperature can be associated to a characteristic Shape Frequency max S  , via Wien’s law, which yields equality: max 0.7 , S N THz   linking number of particles with frequency. Keywords: Electronic Density Function; Shape Function; Thermal Voltage; Density Function Scaling; Thermal Scaling; Shape Temperature; Shape Frequency Introduction The definition of shape functions, as was proposed by Parr and Bartolotti [1] in 1983, corresponds to a simple homothecy of the first order reduced quantum density functions. The homothetic parameter being related to the density function inverse Minkowski norm, which coincides with the inverted number of particles, associated to the quantum object attached to it. Indeed, as many authors have been using shape functions, see for instance the review [2] and included references, in many theoretical applications like quantum similarity [3,4], such an elementary scaling transformation, when performed on the electronic density function, yields undoubtedly quite an interesting chemical concept. Among other properties, shape functions possess a unit Minkowski norm and as the attached density functions are definite non-negative functions. Both characteristics are sufficient to accept shape functions as one-electron continuous probability density functions. The present paper tries to connect shape functions with a recent definition of a general density function homothetic scaling, the so called thermal scaling [5]. Thus, first thermal scaling of density functions will be defined, using an original solid-state concept: The thermal voltage [6]. After this, it is essentially a matter of simple study to deduce how thermal scaling and shape functions might be related. Finally, based on thermal scaling, some Perspective Volume 1 Issue 1 Received Date: September 04, 2017 Published Date: September 22, 2017