The valency interaction formula (VIF) method is given a broader and more general interpretation in which these simple molecular structural formulas implicitly include all overlaps between valence atomic orbitals even for interactions not drawn in the VIF picture. This applies for VIF pictures as one-electron Hamiltonian operators as well as VIF pictures as one-electron density operators that constitute a new implementation of the VIF method simpler in its application and more accurate in its results than previous approaches. A procedure for estimating elements of the effective charge density-bond order matrix, $$($) over bar (mu nu), from electron configurations in atoms is presented, and it is shown how these lead to loop and line constants in the VIF picture. From these structural formulas, one finds the number of singly, doubly, and unoccupied molecular orbitals, as well as the number of molecular orbitals with energy lower, equal, and higher than -(1)/E-2(h), the negative of the hydrogen atom's ionization energy. The VIF results for water are in qualitative agreement with MP2/6311++G3df3pd, MO energy levels where the simple VIF for water presented in the earlier literature does not agree with computed energy levels. The method presented here gives the simplest accurate VIF pictures for hydrocarbons. It is shown how VIF can be used to predict thermal barriers to chemical reactions. Insertion of singlet carbene into H-2 is given as an example. VIF pictures as one-electron density operators describe the ground-state multiplicities of B-2, N-2, and O-2 molecules and as one-electron Hamiltonian operators give the correct electronegativity trend across period two. Previous implementations of VIF do not indicate singly occupied molecular orbitals directly from the pictorial VIF rules for these examples. The direct comparison between structural formulas that represent electron density and those that represent energy is supported by comparison of a simple electronegativity scale, chi(D) = N/n(2), with well-known electronegativity scales of Pauling, Mulliken, and Allen. This scale comes from the method used to calculate P-mu mu for sp(3) hybridized period-two elements and is comparable to electronegativity because it has the same form as < 1/r > for hydrogenic orbitals. It therefore provides a physical basis for the representation of one electron density and Hamiltonian operators by the same VIF picture.