6.5 Summary and Conclusions
We have briefly reviewed here some scalar and vector fields that might play a
significant role in future developments of Quantum Chemical Topology. In this
contribution we have focused on some scalar densities and forces that depend on
the pair density, thus delivering direct information on electron-electron interactions.
Several interesting conclusions have emerged. On the one hand, we have shown
that the gradient of the potential acting on an electron in a molecule (PAEM)
defined by Zhao and Yang is strinkingly similar to the Ehrenfest force. This means
that the absence of a Hellmann-Feynman link between both does not affect qualitatively their shape. Another potentially important discovery is the semiquantitative similarity between the exchange-correlation potential and the electron density. We believe that this reinforces the link between covalency and
topology, paving the way to future studies that definitively link energetic properties
to the electron density topology. Our other energetic scalars, the additive and
effective energy densities, have also shown intriguing connections to several
well-known fields. The former, for instance, integrates to the total molecular
energy, and is locally similar to minus the Hamiltonian kinetic energy density, that
does also integrate to E at stationary points on potential energy surfaces. As the
exchange-correlation potential mimics the density from the energetic side, the
additive energy density mimmics the Laplacian. Future works will determine
the role that these functions may play in chemical bonding theory.
Acknowledgements The authors acknowledge the financial support from the Spanish MICINN,
Project CTQ2012-31174. AGB also acknowledges FICYT for a Ph.D. grant (BP 11-127).
Appendix
Here, we give some details regarding the calculation of the different scalar and
vector fields defined in Sect. 6.2, as well as of their gradients and Hessians. In all of
the following expressions, $f r
ð Þ and $
t f r
ð Þ represent the gradient of the scalar field
f r
ð Þ in row and column forms, respectively, $
t
$f r
ð Þ is the 3 × 3 array of the second
derivatives of f r
ð Þ, and $
t f r
ð Þ$g r
ð Þ is the 3 × 3 array that results from the matrix
product of the column vector $
t f r
ð Þ with the row vector $g r
ð Þ. All the integrals
involving the molecular orbitals (MOs) were evaluated using the McMurchieDavidson algorithm as implemented in the PROMOLDEN code.
The xc potential, q
xc r 1 ; r 2
ð
Þ, can be written in terms of a set of m real MOs u i
(i ¼ 1; . . .; m) in the form
q
xc r 1 ; r 2
ð
Þ¼
X m
p ! q
X m
r ! s
k pq;rs u p r 1
ð Þu q r 1
ð Þu r r 2
ð Þu s r 2
ð Þ;
ð6:25Þ
146
A. Martín Pendás et al.
We have briefly reviewed here some scalar and vector fields that might play a
significant role in future developments of Quantum Chemical Topology. In this
contribution we have focused on some scalar densities and forces that depend on
the pair density, thus delivering direct information on electron-electron interactions.
Several interesting conclusions have emerged. On the one hand, we have shown
that the gradient of the potential acting on an electron in a molecule (PAEM)
defined by Zhao and Yang is strinkingly similar to the Ehrenfest force. This means
that the absence of a Hellmann-Feynman link between both does not affect qualitatively their shape. Another potentially important discovery is the semiquantitative similarity between the exchange-correlation potential and the electron density. We believe that this reinforces the link between covalency and
topology, paving the way to future studies that definitively link energetic properties
to the electron density topology. Our other energetic scalars, the additive and
effective energy densities, have also shown intriguing connections to several
well-known fields. The former, for instance, integrates to the total molecular
energy, and is locally similar to minus the Hamiltonian kinetic energy density, that
does also integrate to E at stationary points on potential energy surfaces. As the
exchange-correlation potential mimics the density from the energetic side, the
additive energy density mimmics the Laplacian. Future works will determine
the role that these functions may play in chemical bonding theory.
Acknowledgements The authors acknowledge the financial support from the Spanish MICINN,
Project CTQ2012-31174. AGB also acknowledges FICYT for a Ph.D. grant (BP 11-127).
Appendix
Here, we give some details regarding the calculation of the different scalar and
vector fields defined in Sect. 6.2, as well as of their gradients and Hessians. In all of
the following expressions, $f r
ð Þ and $
t f r
ð Þ represent the gradient of the scalar field
f r
ð Þ in row and column forms, respectively, $
t
$f r
ð Þ is the 3 × 3 array of the second
derivatives of f r
ð Þ, and $
t f r
ð Þ$g r
ð Þ is the 3 × 3 array that results from the matrix
product of the column vector $
t f r
ð Þ with the row vector $g r
ð Þ. All the integrals
involving the molecular orbitals (MOs) were evaluated using the McMurchieDavidson algorithm as implemented in the PROMOLDEN code.
The xc potential, q
xc r 1 ; r 2
ð
Þ, can be written in terms of a set of m real MOs u i
(i ¼ 1; . . .; m) in the form
q
xc r 1 ; r 2
ð
Þ¼
X m
p ! q
X m
r ! s
k pq;rs u p r 1
ð Þu q r 1
ð Þu r r 2
ð Þu s r 2
ð Þ;
ð6:25Þ
146
A. Martín Pendás et al.