Two new kinds of mathematical symbols with application to several aspects of computational chemistry, related to programming techniques and artificial intelligence, are described. The first one constitutes a family of symbols, related to logical expressions. They are referred to here as logical functions (LFs). The other symbol described in this paper is called a nested summation symbol (NSS). We insist on the quantum chemical usefulness of these symbols, so, application examples of the NSS graphism over some computational quantum chemistry topics are given. The results show how some standard formulae become shorter and easier to be written, generalized, programmed and evaluated. In some NSS expressions and other mathematical applications, the use of such symbols makes the whole problem formulation compact and elegant. Moreover, both symbols constitute a powerful link between mathematical formalism and programming in high level languages: Essentially, NSS are connected with general nested do loops (GNDL) structures, a general programming concept developed in our laboratory, and LFs are related to general logical if sentences. The pedagogical potential of both mathematical devices is also revealed.