  This work completes the construction of purely algebraic version of the
theory of non-linear quantum chemistry methods. It is shown that at the heart
of these methods there lie certain algebras close in their definition to the
well-known Clifford algebra but quite different in their properties. The most
important for quantum chemistry property of these algebras is the following :
for a fixed number of electrons the corresponding sector of the Fock space
becomes a commutative algebra and its ideals are determined by the order of
excitations from the Hartree-Fock reference state. Quotients of this algebra
can also be endowed with commutative algebra structures and quotient
Schr{\"o}dinger equations are exactly the couple cluster type equations.
Possible computer implementation of multiplication in the aforementioned
algebras is described. Quality of different polynomial approximations of
configuration interaction wave functions is illustrated with concrete examples.
Embedding of algebras of infinitely separated subsystems in algebra of the
united system is discussed
