  The symmetry of the wave function describing the Cooper pairs is one of the
most fundamental quantities in a superconductor but its measurement in the
iron-based superconductors has proved to be very difficult. The complex
multi-band nature of these materials makes the interplay of superconductivity
with spin and orbital dynamics very intriguing, leading to very material
dependent magnetic excitations, and pairing symmetries. Here we use
first-principles many-body method, including ab initio determined two-particle
vertex function, to study the spin dynamics and superconducting pairing
symmetry in a large number of iron-based superconductors. In iron compounds
with high transition temperature, we find both the dispersive high-energy spin
excitations, and very strong low energy commensurate or nearly commensurate
spin response, suggesting that these low energy spin excitations play the
dominate role in cooper pairing. We find three closely competing types of
pairing symmetries, which take a very simple form in the space of active iron
$3d$ orbitals, and differ only in the relative quantum mechanical phase of the
$xz$, $yz$ and $xy$ orbital contributions. The extensively discussed s$^{+-}$
symmetry appears when contributions from all orbitals have equal sign, while
the opposite sign in $xz$ and $yz$ orbitals leads to the $d$ wave symmetry. A
novel orbital antiphase $s^{+-}$ symmetry emerges when $xy$ orbital has
opposite sign to $xz$ and $yz$ orbitals. We propose that this orbital-antiphase
pairing symmetry explains the puzzling variation of the experimentally observed
superconducting gaps on all the Fermi surfaces of LiFeAs. This novel symmetry
of the order parameter may be realized in other iron superconductors.
