An application of the Dirac delta function

When may such a concept as the Dirac delta function be useful Here is an example. Let us imagine that we have (in 3D space) two molecular charge distributions Pa(i ) and psif). Each of the distributions consists of an electronic part and an nuclear part. 

How can such charge distributions be represented mathematically There is no problem in mathematical representation of the electronic parte, they are simply some functions of the position r in space -pei. t ) and —pel.B( ) for each molecule. The integrals of the corresponding electronic distributions yield, of course, —Na and —Nb (in a.u.), or minus the number of the electrons (minus, because the electrons carry negative charge). How do we write the nuclear charge distribution as a function of r There is no way to do it without the Dirac delta function. With the function our task is simple  

We put delta functions with the intensities equal to the nuclear charges in the nuclear positions. For neutral molecules d r and /pnud,B(r) dV have 

Thus the Dirac delta function enables us to write the total charge distributions and their interactions in an elegant way  

Tb demonstrate the difference, let us write the electrostatic interaction of the two charge distributions with and without the Dirac delta functions. The first gives the following expression 

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