A molecular theory of liquid phase vibrational energy relaxation (VER) [S. A. Adelman , Adv. Chem. Phys. 84, 73 (1993)] is applied to study the temperature T and density rho dependencies of the VER rate constant k(T,rho)=T-1(-1), where T-1 is the energy relaxation time, of model Lennard-Jones systems that roughly simulate solutions of high-mass, low-frequency dihalogen solutes in rare gas solvents; specifically the I-2/Xe, I-2/Ar, and ICI/Xe solutions. For selected states of these systems, the theory's assumptions are tested against molecular dynamics (MD) results. The theory is based on the expression T-1=beta(-1)(omega(l)), where omega(l) and beta(omega) are, respectively, the solute's liquid phase vibrational frequency and vibrational coordinate friction kernel. The friction kernel is evaluated as a cosine transform of the fluctuating force autocorrelation function of the solute vibrational coordinate, conditional that this coordinate is fixed at equilibrium. Additionally, the early-time decay of the force autocorrelation function is approximated by a Gaussian function which is exact to order t(2). This Gaussian approximation permits evaluation of T-1 in terms of integrals over equilibrium solute-solvent pair correlation functions. The pair correlation function formulas yield T-1's in semiquantitative agreement with those found by MD evaluations of the Gaussian approximation, but with three orders of magnitude less computational effort. For the isothermal rho dependencies of k(T,rho), the theory predicts for all systems that the Gaussian decay time tau is nearly independent of rho. This in turn implies that k(T,rho) factorizes into a liquid phase structural contribution and a gas phase dynamical contribution, yielding a first-principles form for k(T,rho) similar to that postulated by the isolated binary collision model. Also, the theory predicts both "classical" superlinear rate isotherms, and "nonclassical" sublinear isotherms similar to those recently observed by Troe and co-workers for azulene relaxation in supercritical fluids. The isochoric T dependencies of k(T,rho) are studied in the range 300 to 1000 K. For none of the solutions are the rate isochores found to accurately conform to either Arrhenius or Landau-Teller kinetics.