  In comets the atomic oxygen green to red-doublet emission intensity ratio
(G/R ratio) of 0.1 has been used to confirm H$_2$O as the parent species
producing oxygen emission lines. The larger ($>$0.1) value of G/R ratio
observed in a few comets is ascribed to the presence of higher CO$_2$ and CO
relative abundances in the cometary coma. We aim to study the effect of CO$_2$
and CO relative abundances on the observed G/R ratio in comets observed at
large ($>$2 au) heliocentric distances by accounting for important production
and loss processes of O($^1$S) and O($^1$D) in the cometary coma. Recently we
have developed a coupled chemistry-emission model to study photochemistry of
O($^1$S) and O($^1$D) atoms and the production of green and red-doublet
emissions in comets Hyakutake and Hale-Bopp. In the present work we applied the
model to six comets where green and red-doublet emissions are observed when
they are beyond 2 au from the Sun. In a water-dominated cometary coma and with
significant ($>$10%) CO$_2$ relative abundance, photodissociation of H$_2$O
mainly governs the red-doublet emission, whereas CO$_2$ controls the green line
emission. If a comet has equal composition of CO$_2$ and H$_2$O, then ~50% of
red-doublet emission intensity is controlled by the photodissociation of
CO$_2$. The G/R ratio values and green and red-doublet line widths calculated
by the model are consistent with the observation. Our model calculations
suggest that in low gas production rate comets the G/R ratio greater than 0.1
can be used to constrain the upper limit of CO$_2$ relative abundance provided
the slit-projected area on the coma is larger than the collisional zone. If a
comet has equal abundances of CO$_2$ and H$_2$O, then the red-doublet emission
is significantly ($\sim$50%) controlled by CO$_2$ photodissociation and thus
the G/R ratio is not suitable for estimating CO$_2$ relative abundance.
