Carbon-bearing molecules, particularly CO, have been widely used as tracers of molecular gas in the interstellar medium (ISM). In this work, we aim to study the properties of molecules in diffuse, cold environments, where CO tends to be under-abundant and/or sub-thermally excited. We performed one of the most sensitive (down to $mathrm{tau_{rms}^{CO} sim 0.002}$ and $mathrm{tau_{rms}^{HCO^+} sim 0.0008}$) sub-millimeter molecular absorption line observations towards 13 continuum sources with the ALMA. CO absorption was detected in diffuse ISM down to $mathrm{A_v< 0.32,mag}$ and hcop was down to $mathrm{A_v < 0.2,mag}$, where atomic gas and dark molecular gas (DMG) starts to dominate. Multiple transitions measured in absorption toward 3C454.3 allow for a direct determination of excitation temperatures $mathrm{T_{ex}}$ of 4.1,K and 2.7,K, for CO and for hcop, respectively, which are close to the cosmic microwave background (CMB) and provide explanation for their being undercounted in emission surveys. A stronger linear correlation was found between $mathrm{N_{HCO^+}}$ and $mathrm{N_{H_2}}$ (Pearson correlation coefficient P $sim$ 0.93) than that of $mathrm{N_{CO}}$ and $mathrm{N_{H_2}}$ (P $sim$ 0.33), suggesting hcop being a better tracer of H$_2$ than CO in diffuse gas. The derived CO-to-h2 conversion factor (the CO X-factor) of (14 $pm$ 3) $times$ 10$^{20}$ cm$^{-2}$ (K kms)$^{-1}$ is approximately 6 times larger than the average value found in the Milky Way.
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