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Atoms are indistinguishable particles which can be transformed one into another by the elements of a group G,1 which corresponds to the internal symmetry of their periodic system. We construct a molecular periodic system using G and bosonic creation operators. The vectors li> = b+ ~o> correspond to various atoms where lo> is the vacuum state vector, and b+i is the creation operator for atom i. The annihilation operator is b;; boson symmetry requires fbrbJ = b+ rb+ J = 0, [bi,b+ j] = 1. State vectors lijk ... > correspond to molecules. They can be recast as a direct sum of irreducible representations whose vectors are (often) linear combinations of individual molecular states. The one-particle operator P1 of the lie algebra of G is I:i,j P1(i;j) b+pj" We have tested our systems by plotting a variety of tabulated experimental data along principle axes for atomic and for diatomic and triatomic molecular multiplets.