  A fully self-consistent density-functional theory (DFT) with improved
functionals is used to provide a comprehensive account of structural,
electronic, and optical properties of C$_{3}$N$_{4}$ polymorphs. Using our
recently developed van Leeuwen-Baerends (vLB) corrected local-density
approximation (LDA), we implemented LDA+vLB within full-potential
N$^{th}$-order muffin-tin orbital (FP-NMTO) method and show that it improves
structural properties and band gaps compared to semi-local functionals
(LDA/GGA). We demonstrate that the LDA+vLB predicts band-structure and
work-function for well-studied 2D-graphene and bulk-Si in very good agreement
with experiments, and more exact hybrid functional (HSE) calculations as
implemented in the Quantum-Espresso (QE) package. The structural and
electronic-structure (band gap) properties of C$_{3}$N$_{4}$ polymorphs
calculated using FP-NMTO-LDA+vLB is compared with more sophisticated
hybrid-functional calculations. We also perform detailed investigation of
photocatalytic behavior using QE-HSE method of C$_{3}$N$_{4}$ polymorphs
through work-function, band (valence and conduction) position with respect to
water reduction and oxidation potential. Our results show
$\gamma$-C$_{3}$N$_{4}$ as the best candidate for photocatalysis among all the
C$_{3}$N$_{4}$~polymorphs but it is dynamically unstable at `zero' pressure. We
show that $\gamma$-C$_{3}$N$_{4}$ can be stabilized under hydrostatic-pressure,
which improves its photocatalytic behavior relative to water reduction and
oxidation potentials.
