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We perform a first principles band calculation for quasi-two-dimensionalorganic superconductors $\beta$-(BDA-TTP)$_{2}$I$_{3}$ and$\beta$-(BDA-TTP)$_{2}$SbF$_{6}$. The first principles band structures betweenthe I$_{3}$ and SbF$_{6}$ salts are apparently different. We construct atight-binding model for each material which accurately reproduces the firstprinciples band structure. The obtained transfer energies give the differencessuch as (i) larger dimerization in the I$_{3}$ salt than the SbF$_{6}$ salt,and (ii) different signs and directions of the inter-stacking transferenergies. To decompose the origin of the difference into the dimerization andthe inter-stacking transfer energies, we adopt a simplified model byeliminating the dimerization effect and extract the difference caused by theinter-stacking transfer energies. From the analysis using the simplified model,we find that the difference of the band structure comes mainly from thestrength of dimerization. To compare the strength of the electron correlationhaving roots in the band structure, we calculate the physical propertiesoriginated from the effect of the electron correlation such as the spinsusceptibility applying two particle self-consistent (TPSC) method. We findthat the maximum value of the spin susceptibility of the I$_{3}$ salt is largerthan that of the SbF$_{6}$ salt. Hypothetically decreasing the dimerizationwithin the model of the I$_{3}$ salt, the spin susceptibility takes almost thesame value as that of the SbF$_6$ salt for the same magnitude of thedimerization. We expect that the different ground state between the I$_{3}$ andSbF$_{6}$ salt mainly comes from the strength of the dimerization which isapparently masked in the band calculation along a particular $k$-path.

Enhancement of electron correlation due to the molecular dimerization in organic superconductors $\beta$-(BDA-TTP)$_{2}X$ ($X$=I$_3$, SbF$_6$)