17.1.2 B: Becke (1988) gradient-corrected exchange energy

A. D. Becke, Phys. Rev. A 38 (1988) 3098.

$$K = k({\rho+\hat \rho\over 2}) +k({\rho-\hat \rho\over 2})$$ (7)

where we define for each of the $\alpha$ and $\beta$ spin densities

$$x(\rho) = \rho^{-\frac43} \left\vert\nabla\rho\right\vert$$ (8)

$$k(\rho) = - \rho^\frac43 \left[ {3\over 2} \left({3\over 4\pi}\right)^{1/3} +\beta {x(\rho)^2 \over 1+6\beta x(\rho) \sinh^{-1}x(\rho)} \right]$$ (9)

with $\beta = 0.0042$.