In atom-based thermochemistry, (ABT), state functions are referenced to free atoms, as opposed to the thermochemical convention of referencing to elements in their standard state. The shift of the reference frame reveals previously unrecognized linear relationships between the standard atomization enthalpies Delta H-at degrees(g) of gas-phase diatomic and triatomic molecules and Delta H-at degrees(s) of the corresponding solids for large groups of materials. For 35 alkali and coinage-metal halides, as well as alkali metal hydrides, Delta H-at degrees(s) = 1.1203 Delta H-at degrees(g) + 167.0 kJ mol(-1) is found; the standard error is SE = 16.0 kJ mol(-1), and the correlation coefficient is R = 0.9946. The solid coinage-metal monohydrides, CuH(s), AgH(s), and AuH(s), are predicted to be unstable with respect to the formation from the metals and elemental hydrogen by an approximately constant standard enthalpy of formation, Delta H-f degrees(s) approximate to +80 +/- 20 kJ mol(-1). Solid AuF is predicted to be marginally stable, having Delta H-f degrees(s) = -60 +/- 50 kJ mol(-1) and standard a Gibbs energy of formation Delta(f)G degrees(s) approximate to -40 +/- 50 kJ mol(-1). Triatomic alkaline-earth dihalides MX2 obey a similar linear relationship. The combined data of altogether 51 materials obey the relationship Delta H-at degrees(s) = 1.2593 Delta H-at degrees(g) + 119.9 kJ mol(-1) with R = 0.9984 and SE = 18.5 kJ mol(-1). The atomization enthalpies per atom of 25 data pairs of diatoms and solids in the groups 14-14, 13-15, and 2-16 are related as Delta H-at degrees(s) = 2.1015 Delta H-at degrees(g) + 231.9 kJ mol(-1) with R = 0.9949 and SE = 24.0 kJ mol(-1). Predictions are made for the GeC, GaSb, Hf-2, TIN, BeS, MgSe, and MgTe molecules and for the solids SiPb, GePb, SnPb, and the thallium pnictides. Exceptions to the rule, such as SrO and BaO, are rationalized. Standard enthalpies of sublimation, Delta H-subl degrees = Delta H-at degrees(s) - Delta H-at degrees(g), are calculated as a linear function of Delta H-at degrees(g) profiting from the above linear relationships, and predictions for the Delta H-subl degrees of the thallium pnictides are given. The validity of the new empirical relationships is limited to substances where at least one of the constituent elements is solid in its standard state. Reasons for the late discovery of such relationships are given, and a meaningful ABT is recommended by using Delta H-at degrees as an important ordering and reference state function.