  A general method to treat non-Gaussian vacuum wave functionals in the
Hamiltonian formulation of a quantum field theory is presented. By means of
Dyson--Schwinger techniques, the static Green functions are expressed in terms
of the kernels arising in the Taylor expansion of the exponent of the vacuum
wave functional. These kernels are then determined by minimizing the vacuum
expectation value of the Hamiltonian. The method is applied to Yang--Mills
theory in Coulomb gauge, using a vacuum wave functional whose exponent contains
up to quartic terms in the gauge field. An estimate of the cubic and quartic
interaction kernels is given using as input the gluon and ghost propagators
found with a Gaussian wave functional.
