Tuned Liquid Dampers (TLDs) have been installed in large engineering structures to suppress unwanted motions. They function by allowing fluid to slosh in a tank mounted rigidly to the structure which contain devices for dissipating energy. In this paper, the TLD is comprised of a rectangular tank fitted with a number of vertical slatted screens to provide damping when the fluid is in motion. The rectangular tank TLD is coupled to a simple mechanical model for the displacement of an externally-forced structure of large mass. The influence of the fluid motion in the rectangular tank is included in this model through two components of the complex-valued net horizontal force provided by the fluid on the tank, namely the added mass and damping coefficients. These depend on both the forcing frequency and the magnitude of the tank displacement. The calculation of these key coefficients is performed by developing an analytical solution to a linearised boundary-value problem representing the forced motion of a rectangular tank having an arbitrary configuration of vertical slatted screens. The tank problem is formulated using classical water wave theory with linearised boundary conditions both on the free surface and across the screens. These latter linearised screen conditions are designed to accurately capture both the added inertia effects of a slatted screen and the damping effects from an equivalent non-linear turbulent drag law, whose successful implementation is reported in Crowley and Porter (in press). Advantage is also taken of the linearised theory used to demonstrate key qualitative features of TLD systems analytically. Numerical results are shown to compare very well with experimental results for particular screen arrangements. Different screen configurations are then considered to indicate general criteria for ‘optimising’ the TLD performance, by reducing overall displacement across all forcing frequencies by altering the number, placement and porosity of the slatted screens in the tank.