3.4.4 LDMs-as a Tool to Evaluate the Quality of Basis Sets
and of the Levels of Theory
Another potential use of the LDM is in assessing the quality of basis sets and/or the
different new density functional theory (DFT) functionals, for example. The fundamental assumption is that the closer a given level of theory is to another the closer
the corresponding LDMs will be and the smaller the Frobenius distance.
As an exploratory investigation, only Hartree-Fock (HF) results on four small
molecules (CH 4 , CH 3 OH, H 2 O, and NH 3 ) are considered here in conjunction with a
variety of standard basis sets. Since the Hartree-Fock method is variational, the
lower the energy the better the quality of the corresponding basis set. In the set of
basis sets investigated in this work, the lowest energy is obtained at the HF/cc-pvqz
level of theory, which implies that it is the best level of theory used as the comparison standard.
Figure 3.9 displays the correlation between the total energy and the LDM
Frobenius distance from the best result (HF/cc-pvqz) for each of the four studied
molecules. The plots show that the general trend is that the lower the energy the
smaller the distance from the best result.
This new proposal is elaborated in detail elsewhere [25]. Clearly much more
numerical corroboration is needed before claiming a definitive usefulness of LDMs
in evaluating and comparing the quality of basis sets and/or levels of theory.
3.5 Closing Remarks
As stressed above, the first Hohenberg–Kohn (HK) theorem [35, 36] by establishing a unique functional mapping between the ground-state electron density and
both the external potential and the total number of electrons fixes the Hamiltonian.
The electron density then, through the intermediacy of the time-independent
Shrödinger equation, determines the eigenstates and eigenvalues uniquely. Once the
eigenstates are fixed all properties of the ground and excited state are also fixed. It is
not surprising then that powerful descriptors can be extracted from the electron
density. The pioneering work of Paul Popelier in his Quantum Topological
Molecular Similarity (QTMS) approach [97–103], whereby Euclidean similarity
distances between molecules in a molecular set are defined on the basis of difference in the sum of their bond critical points properties, led the way for others like us
to follow his step. While our approach is different than QTMS, since it rests on a
full atomic level description of electron localization and delocalization in each
molecule in the set while QTMS is based on a full bond-by-bond level of analysis,
what QTMS and the analysis of LDMs have in common is their basis in the
topological partitioning of the electron density and its characteristic gradient vector
field, as suggested in 1981 by I. Dmitriev in the opening quotation of this chapter.
Both QTMS and LDMs analyses are traceable to physical quantities derived from
82
C.F. Matta et al.
and of the Levels of Theory
Another potential use of the LDM is in assessing the quality of basis sets and/or the
different new density functional theory (DFT) functionals, for example. The fundamental assumption is that the closer a given level of theory is to another the closer
the corresponding LDMs will be and the smaller the Frobenius distance.
As an exploratory investigation, only Hartree-Fock (HF) results on four small
molecules (CH 4 , CH 3 OH, H 2 O, and NH 3 ) are considered here in conjunction with a
variety of standard basis sets. Since the Hartree-Fock method is variational, the
lower the energy the better the quality of the corresponding basis set. In the set of
basis sets investigated in this work, the lowest energy is obtained at the HF/cc-pvqz
level of theory, which implies that it is the best level of theory used as the comparison standard.
Figure 3.9 displays the correlation between the total energy and the LDM
Frobenius distance from the best result (HF/cc-pvqz) for each of the four studied
molecules. The plots show that the general trend is that the lower the energy the
smaller the distance from the best result.
This new proposal is elaborated in detail elsewhere [25]. Clearly much more
numerical corroboration is needed before claiming a definitive usefulness of LDMs
in evaluating and comparing the quality of basis sets and/or levels of theory.
3.5 Closing Remarks
As stressed above, the first Hohenberg–Kohn (HK) theorem [35, 36] by establishing a unique functional mapping between the ground-state electron density and
both the external potential and the total number of electrons fixes the Hamiltonian.
The electron density then, through the intermediacy of the time-independent
Shrödinger equation, determines the eigenstates and eigenvalues uniquely. Once the
eigenstates are fixed all properties of the ground and excited state are also fixed. It is
not surprising then that powerful descriptors can be extracted from the electron
density. The pioneering work of Paul Popelier in his Quantum Topological
Molecular Similarity (QTMS) approach [97–103], whereby Euclidean similarity
distances between molecules in a molecular set are defined on the basis of difference in the sum of their bond critical points properties, led the way for others like us
to follow his step. While our approach is different than QTMS, since it rests on a
full atomic level description of electron localization and delocalization in each
molecule in the set while QTMS is based on a full bond-by-bond level of analysis,
what QTMS and the analysis of LDMs have in common is their basis in the
topological partitioning of the electron density and its characteristic gradient vector
field, as suggested in 1981 by I. Dmitriev in the opening quotation of this chapter.
Both QTMS and LDMs analyses are traceable to physical quantities derived from
82
C.F. Matta et al.