Electronic structure calculation of atoms and molecules, in the past few decades has largely been dominated by density functional methods.  This is primarily  due to the fact that this can account for  electron  correlation  effects  in a rigorous,  tractable manner  keeping the computational  cost  at a manageable  level.  With recent advances in methodological development, algorithmic progress as well as computer technology, larger physical, chemical and biological systems are amenable to quantum mechanical calculations than ever before. Here we report the development of a new method for accurate reliable description of atoms, molecules within the Hohenberg-Kohn-Sham density functional theory (DFT). In a Cartesian grid, atom-centered localized basis set, electron density, molecular orbitals, two-body potentials are directly built on the grid.  We employ a Fourier convolution  method  for  classical  Coulomb  potentials  by  making  an  Ewald-type  decomposition technique in terms of short- and long-range interactions. One-body matrix elements are obtained from  standard  recursion  algorithms  while two-body  counterparts  are  done  by  direct  numerical integration. A systematic analysis of our results obtained on various properties, such as component energy,  total energy,  ionization energy,  potential energy  curve,  atomization  energy,  etc., clearly demonstrates that the method is capable of producing quite accurate and competitive (with those from other methods in the literature) results.  In brief, a new variational DFT method is presented for atoms and molecules, completely in Cartesian grid.