A new systematic method is presented to construct a correlation energy func
tional in the density-matrix functional theory. A new exact relation, parti
cle-hole symmetry, is presented, which states that the exact functional giv
es the same correlation energy for the first-order reduced density matrix (
1-RDM) and the corresponding hole 1-RDM. Nakatsuji's density equations of t
he first and the second orders together with the decoupling approximation o
f the 3- and 4-RDMs are solved numerically to examine the properties of the
correlation energy functional. This functional, defined as the solution of
the set of equations, satisfies Levy's homogeneous coordinate scaling and
the particle-hole symmetry and reproduces about 95% of the correlation ener
gies of several molecules. By expanding these equations in perturbation ser
ies the leading term of the correlation energy functional is identified. Nu
merical analysis shows that this simplified functional, which contains one
fitting parameter, reproduces the energies of several molecules accurately.
The direction of further study is discussed.