Ultracold quantum gases usually require a trapping potential to hold the cloud
of atoms together. An exception is the case of dipolar quantum gases which
can form self-bound droplets, but
recent experiments [\ref{cabreraScience17},\ref{semeghini}] found that also binary Bose mixtures
can form such self-bound ultra-dilute liquid droplets. These experiments confirmed beyond-mean-fiel
d (BMF) predictions
[\ref{petrovPRL15},\ref{petrovPRL16}] which used Lee-Huang-Yang corrections in the local
density approximation [\ref{larsenAnnPhys63}],
see also Ref.[\ref{cikojevicPRB18}]. In our work, we
proceed beyond the beyond-mean-field approximation, and study liquid
Bose mixtures of $^{39}$K using a variational {\em pair density} functional theory,
the hypernetted-chain Euler Lagrange (HNC-EL) method. The HNC-EL methods
accounts for correlations non-perturbatively by using a multi-component
Jastrow-Feenberg ansatz [\ref{hebenstreitPRA16}]. We focus on the case
of a mixture of uniform density, as it would be realized inside large saturated droplets.
Forschungs-