We use the generating functional method for the matrix elements of second quantization operators to obtain a high-temperature expansion of the thermodynamic potential of a quantum system. This method permits isolating irreducible parts of matrices, including the particle-density matrices. We derive an equation for the full unary density matrix, which is equivalent to the variational principle for the thermodynamic potential. The thermodynamic functions and the density matrix can thus be found in the framework of the same variational problem.