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How can we theoretically calculate the boiling point of water at given pressure (other subtle parameters as well, if any)? What is the most accurate (minimum discrepancy with experimental value) computation that can analytically predict the boiling point of water?

Possibly we need to invoke quantum mechanics for this. I anticipate that many answer would say that EXAXCT prediction is computationally infeasible, but please give the outline in algorithmic form regardless of computational cost. My main goal is to learn how quantum mechanics can be applied to phenomena which can be observed by layman. Other example where quantum mechanics is used to predict physical properties of small molecules from first principle are also welcome.

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The boiling point (i.e. the saturated vapor pressure as a function of temperature) is described by the Clausius–Clapeyron equation which is a consequence of classical thermodynamics. Of course if we start investigating why certain thermodynamic potentials used in that equation have to be such and such then eventually we'll need quantum physics to explain the size of atoms and molecules. But there are more direct commonly observed macroscopic consequences of quantum physics which include photovoltaics, lasers, superconductivity. These seem to be closer to what you are looking for.

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The following article seems relevant.

Simulating Fluid-Phase Equilibria of Water from First Principles. MJ McGrath et al. J. Phys. Chem. A 2006, 110 (2), 640-646. VLab eprint.

Their abstract:

Efficient Monte Carlo algorithms and a mixed-basis set electronic structure program were used to compute from first principles the vapor-liquid coexistence curve of water. A water representation based on the Becke-Lee-Yang-Parr exchange and correlation functionals yields a saturated liquid density of 900 kg/m3 at 323 K and normal boiling and critical temperatures of 350 and 550 K, respectively. An analysis of the structural and electronic properties of the saturated liquid phase shows an increase of the asymmetry of the local hydrogen-bonded structure despite the persistence of a 4-fold coordination and decreases of the molecular dipole moment and of the spread of the lowest unoccupied molecular orbital with increasing temperature.

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