One of the drawbacks of ECPs is that the valence orbitals which are
employed do not exhibit the radial nodal structure which naturally
arises when all-electron methods are employed.
The ab initio model potential (AIMP) method, first
suggested by Schwartz and Switalski[29]
and more extensively developed by Bonifacic and
Huzinaga[30,31,32,33],
attempts to ameliorate this unsavory feature of ECPs.
The fundamental idea of the AIMP method is that a projection operator,
, constructed from an appropriate set of atomic core orbitals,
can be appended to the Hamiltonian in order to project the atomic core
solutions out of the valence wavefunction. In their 1973 work, Bonifacic
and Huzinaga present the modified single-particle atomic Hamiltonian
where 
is the number of expansion terms for the effective potential,
is the number of frozen frozen atomic shells, 
, 
, and
are expansion parameters, and 
is the projection operator
for a particular frozen shell of angular momentum l and principle quantum
number n.
This single-particle Hamiltonian is then employed in conjunction with the
Coulomb operator to produce an atomic, multi-electron Hamiltonian,
The various parameters of the single-particle Hamiltonian are then optimized to reproduce the desired valence atomic orbital energies and orbital shapes when the Hamiltonian is applied to the atomic system.
The more recent implementation due to Barandiaran, Seijo, and Huzinaga[34] employ the atomic Cowan-Griffin-Hartree-Fock (CGHF) orbitals in the construction of their projection operators, and in their optimization of the various parameters which are a part of the one-electron Hamiltonian. Because the CGHF method deals entirely with scalar-type wavefunctions, the resultant single-particle functions are not j-dependent, and therefore may be used in the construction of such scalar operators as are found in 5.176. In order to optimize the various parameters associated with the Hamiltonian, an all-electron basis set must be used not only in the construction of the CGHF wavefunction, but for the atomic AIMP optimization process. Once the form of the model potential has been optimized, the valence basis set is truncated by removing the Gaussian primitives with the highest radial exponents. The truncation is performed to such an extent as to assure that the outer orbital shape and orbital eigenvalues of the valence orbitals may still be reproduced by application of the AIMP method to the atomic system. The resultant truncated basis set can often be reduced one half the size of the original, all-electron basis without significantly compromising the quality of the predictions.
