  We employ density functional theory to calculate the self consistent
electronic structure, free energy and linear source-drain conductance of a
lateral semiconductor quantum dot patterned via surface gates on the 2DEG
formed at the interface of a $GaAs-AlGaAs$ heterostructure. The Schr\"{o}dinger
equation is reduced from 3D to multi-component 2D and solved via an
eigenfunction expansion in the dot. This permits the solution of the electronic
structure for dot electron number $N \sim 100$. We present details of our
derivation of the total dot-lead-gates interacting free energy in terms of the
electronic structure results, which is free of capacitance parameters.
Statistical properties of the dot level spacings and connection coefficients to
the leads are computed in the presence of varying degrees of order in the donor
layer. Based on the self-consistently computed free energy as a function of
gate voltages, $V_i$, and N, we modify the semi-classical expression for the
tunneling conductance as a function of gate voltage through the dot in the
linear source-drain, Coulomb blockade regime. Among the many results presented,
we demonstrate the existence of a shell structure in the dot levels which (a)
results in envelope modulation of Coulomb oscillation peak heights, (b) which
influences the dot capacitances and should be observable in terms of variations
in the activation energy for conductance in a Coulomb oscillation minimum, and
(c) which possibly contributes to departure of recent experimental results from
the predictions of random matrix theory.
