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Functions
is zero (except at infinity) are called as critical points (CPs), CPs can be classified into four types
according to how many eigenvalues of Hessian matrix of real space function are negative.
(3,-3): All three eigenvalues of Hessian matrix of function are negative, namely the local
maximum. For electron density analysis and for heavy atoms, the position of (3,-3) are nearly
identical to nuclear positions, hence (3,-3) is also called nuclear critical point (NCP). Generally the
number of (3,-3) is equal to the number of atoms, only in rarely cases the former can be more than
+
(
e.g. Li2) or less than (e.g. KrH ) the latter.
(3,-1): Two eigenvalues of Hessian matrix of function are negative, namely the second-order
saddle point. For electron density analysis, (3,-1) generally appears between attractive atom pairs
and hence commonly called as bond critical point (BCP). The value of real space functions at BCP
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have great significance, for example the value of and the sign of ∇ ꢀ at BCP are closely related
to bonding strength and bonding type respectively for analogous bonds (The Quantum Theory of
Atoms in Molecules-From Solid State to DNA and Drug Design, p11); I have demonstrated that the
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at BCP can be reliably used to predict hydrogen bond binding energies (J. Comput. Chem, 40,
868 (2019)); local information entropy at BCP is a good indicator of aromaticity (Phys. Chem.
Chem. Phys., 12, 4742 (2020)).
3,+1): Only one eigenvalue of Hessian matrix of function is negative, namely first-order
(
saddle point (like transition state in potential energy surface). For electron density analysis, (3,+1)
generally appears in the center of ring system and displays steric effect, hence (3,+1) is often named
as ring critical point (RCP).
(3,+3): None of eigenvalues of Hessian matrix of function are negative, namely the local
minimum. For electron density analysis, (3,+3) generally appears in the center of cage system (e.g.
pyramid P4 molecule), hence is often referred to as cage critical point (CCP).
The positions of CPs are searched by Newton method, one need to assign an initial guess point,
then the Newton iteration always converge to the CP that is closest to the guess point. By assigning
different guesses and doing iteration for each of them, all CPs could be found. Once searches of CPs
are finished, one should use Poincaré-Hopf relationship to verify if all CPs may have been found,
the relationship states that (for isolated system)
n(3,-3) – n(3,-1) + n(3,+1) – n(3,+3) = 1
If the relationship is unsatisfied, then some of CPs must be missing, you may need to try to
search those CPs by different guesses. However even if the relationship is satisfied, it does not
necessarily mean that all CPs have been found. Notice that the function spaces of ELF/LOL and
Laplacian of are much more complex than , it is very difficult to locate all CPs for these functions,
especially for middle and large system, so, you can stop trying for searching CPs once all CPs that
you are interested in have been found.
The maximal gradient path linking BCP and associated two local maxima of density is termed
as “bond path”, which reveals atomic interaction path for all kinds of bonding. The collection of
bond paths is known as molecular graph, which provides an unambiguous definition of molecular
structure. Bond path can be straight line or curve, obviously for the latter case the length of bond
path is longer than the sum of the distances between BCP and associated two (3,-3) CPs.
Let us see an example. In the complex shown below, the imidazole plane is vertical to
magnesium porphyrin plane, the nitrogen in imidazole coordinated to magnesium. Magenta, orange
and yellow spheres correspond to (3,-3), (3,-1) and (3,+1) critical points, brown lines denote bond
paths.
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