  The symmetry of the superconducting order parameter in quasi-two-dimensional
BEDT-TTF organic superconductors is a subject of ongoing debate. We report ab
initio density functional theory calculations for a number of organic
superconductors containing $\kappa$-type layers. Using projective Wannier
functions we derive parameters of a common low-energy Hamiltonian based on
individual BEDT-TTF molecular orbitals. In a random phase approximation
spin-fluctuation approach we investigate the evolution of the superconducting
pairing symmetry within this model and point out a phase-transition between
extended $s + d_{x^2-y^2}$ and $d_{xy}$ symmetry. We discuss the origin of the
mixed order parameter and the relation between the realistic molecule
description and the widely used dimer approximation. Based on our ab initio
calculations we position the investigated materials in the obtained molecule
model phase diagram and simulate scanning tunneling spectroscopy experiments
for selected cases. Our calculations show that many $\kappa$-type materials lie
close to the phase transition line between the two pairing symmetry types found
in our calculation, possibly explaining the multitude of contradictory
experiments in this field.
