  Density functional theory (DFT) calculations of 58 liquid elements at their
triple point show that most metals exhibit near proportionality between thermal
fluctuations between virial and potential-energy in the isochoric ensemble.
This demonstrates a general "hidden" scale invariance of metals making the
dense part of the thermodynamic phase diagram effectively one dimensional with
respect to structure and dynamics. DFT computed density scaling exponents,
related to the Gr{\"u}neisen parameter, are in good agreement with experimental
values for 16 elements where reliable data were available. Hidden scale
invariance is demonstrated in detail for magnesium by showing invariance of
structure and dynamics. Computed melting curves of period three metals follow
curves with invariance (isomorphs). The experimental structure factor of
magnesium is predicted by assuming scale invariant inverse power-law (IPL) pair
interactions. However, crystal packings of several transition metals (V, Cr,
Mn, Fe, Nb, Mo, Ta, W and Hg), most post-transition metals (Ga, In, Sn, and Tl)
and the metalloids Si and Ge cannot be explained by the IPL assumption. Thus,
hidden scale invariance can be present even when the IPL-approximation is
inadequate. The virial-energy correlation coefficient of iron and phosphorous
is shown to increase at elevated pressures. Finally, we discuss how scale
invariance explains the Gr{\"u}neisen equation of state and a number of
well-known empirical melting and freezing rules.
