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Correlated band theory implemented as a combination of density functionaltheory with exact diagonalization [DFT+U(ED)] of the Anderson impurity termwith Coulomb repulsion $U$ in the open 14-orbital $5f$ shell is applied toUTe$_2$. The small gap for $U$=0, evidence of the half-filled $j=\frac{5}{2}$subshell of $5f^3$ uranium, is converted for $U$=3 eV to a flat band semimetalwith small heavy-carrier Fermi surfaces that will make properties sensitive topressure, magnetic field, and off-stoichiometry, as observed experimentally.The predicted Kondo temperature around 100 K matches the experimental valuesfrom resistivity. The electric field gradients for the two Te sites arecalculated by DFT+U(ED) to differ by a factor of seven, indicating a strongsite distinction, while the anisotropy factor $\eta=0.18$ is similar for allthree sites. The calculated uranium moment $<M^2>^{1/2}$ of 3.5$\mu_B$ isroughly consistent with the published experimental Curie-Weiss values of2.8$\mu_B$ and 3.3$\mu_B$ (which are field-direction dependent), and thecalculated separate spin and orbital moments are remarkably similar to Hund'srule values for an $f^3$ ion. The $U$=3 eV spectral density is compared withangle-integrated and angle-resolved photoemission spectra, with agreement thatthere is strong $5f$ character at, and for several hundred meV below, the Fermienergy. Our results support the picture that the underlying ground state ofUTe$_2$ is that of a half-filled $j=\frac{5}{2}$ subshell with two half-filled$m_j=\pm\frac{1}{2}$ orbitals forming a narrow gap by hybridization, thendriven to a conducting state by configuration mixing (spin-chargefluctuations). UTe$_2$ displays similarities to UPt$_3$ with its $5f$ dominatedFermi surfaces rather than a strongly localized Kondo lattice system.

UTe$_2$: a nearly insulating half-filled $j=\frac{5}{2}$ $5f^3$ heavy fermion metal