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

Tungsten trioxide is one of the most extensively studied electrochromic materials. Here we report density functional theory (DFT) investigations of the coloration mechanism of WO$_3$ upon charge insertion. Our results explains very well the systematic change in color of Na$_x$WO$_3$ from blue to golden-yellow with increasing sodium concentration. We find that proper accounts for the free-carriers contribution to the optical response are critical for a quantitative understanding of the coloration mechanism in this system. We thank Dr. Yong Zhang for his helpful discussion. We thank Dr. M. D. Jones for his assistance in coding. We acknowledge the computational support provided by the Center for Computational Research at the University at Buffalo, SUNY.