  A method is proposed for the calculation of vibrational quantum and thermal
expectation values of physical properties from first principles. Thermal lines
are introduced: these are lines in configuration space parametrized by
temperature, such that the value of any physical property along them is
approximately equal to the vibrational average of that property. The number of
sampling points needed to explore the vibrational phase space is reduced by up
to an order of magnitude when the full vibrational density is replaced by
thermal lines. Calculations of the vibrational averages of several properties
and systems are reported, namely the internal energy and the electronic band
gap of diamond and silicon, and the chemical shielding tensor of L-alanine.
Thermal lines pave the way for complex calculations of vibrational averages,
including large systems and methods beyond semi-local density functional
theory.
