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We report first-principles investigations of the coloration of WO$_3$ uponcharge insertion, using sodium tungsten bronze (Na$_x$WO$_3$) as a modelsystem. Our results explain well the systematic color change of Na$_x$WO$_3$from dark blue to violet, red-orange, and finally to golden-yellow as sodiumconcentration $x$ increases from 0.3 to unity. Proper accounts for both theinterband and the intraband contributions to the optical response are found tobe very important for a detailed understanding of the coloration mechanism inthis system.

First-principles theory of coloration of WO$_3$ upon charge insertion