To explore the features associated with small reduced gradients, we first examine
plots of s versus ρ (Fig. 18.1). These plots were generated by evaluating the B3LYP
[56, 57] density and reduced gradients on cuboid grids, with a 0.1 a.u. step size, for
each molecule or dimer. To provide even more sampling of the small low-density,
low-gradient regions in hydrogen-bonded complexes, additional calculations were
performed for water and formic acid dimers with a much denser 0.025 a.u step size.
Plotting s versus ρ, as in Fig. 18.1, reveals the basic pattern of intramolecular
interactions. Methane (Fig. 18.1a) illustrates the typical covalent bond pattern. The
top left-side points (small density and large reduced gradient) correspond to the
exponentially-decaying tail regions of the density, far from the nuclei. The points
on the bottom right side (density values of ca. 0.25 a.u. and low reduced density
gradient) correspond to the C–H covalent bonds. Covalent bonds have their characteristic BCP in the electron density, corresponding to s = 0. Regions near the
nuclei have larger density values and appear beyond the right edge of the plot. The
plot has an overall shape of the form aρ
−1/3 because atomic and molecular densities
are piecewise exponential. The results for water are very similar, the only difference
being that the covalent bonds lie at higher density values, past the edge of the plot.
In Fig. 18.1b–d, we consider six examples of chemical systems displaying various
types of non-covalent interactions. Plots of s versus ρ for these systems all exhibit a
new feature: one or more spikes in the low-density, low-gradient region, a signature
of non-covalent interactions. This is the basis of NCI.
18.2.2 The Density Second Eigenvalue
According to the divergence theorem [58], the sign of the Laplacian of the density,
∇
2 ρ, indicates whether the net gradient flux is entering, ∇
2 ρ < 0, or leaving,
∇
2
ρ > 0, an infinitesimal volume around a reference point. Hence, it highlights
whether the density is concentrated or depleted at that point, relative to the surrounding environment. To differentiate between different types of weak interactions
one cannot resort to the sign of the Laplacian itself (as is common within AIM
theory) since it is dominated by the principle axis of variation and is positive for all
closed-shell interactions [59].
To understand bonding in more detail, it is often useful to decompose the
Laplacian into the contributions along the three particular axes of maximal variation. These components are the three eigenvalues, λ i of the electron-density Hessian
matrix, such that, ∇
2
ρ = λ 1 + λ 2 + λ 3 , (λ 1 < λ 2 < λ 3 ). At points with zero gradient,
analysis of the Hessian eigenvalues is analogous to determining the signature of the
critical point. Thus, at nuclei (cusps interpreted as maxima of ρ), all the eigenvalues
are negative, while at the center of cages or holes (minima of ρ) all the eigenvalues
are positive. In the remaining points of space λ 3 > 0, λ 1 < 0, and λ 2 can be either
18 A Complete NCI Perspective: From New Bonds to Reactivity
499
plots of s versus ρ (Fig. 18.1). These plots were generated by evaluating the B3LYP
[56, 57] density and reduced gradients on cuboid grids, with a 0.1 a.u. step size, for
each molecule or dimer. To provide even more sampling of the small low-density,
low-gradient regions in hydrogen-bonded complexes, additional calculations were
performed for water and formic acid dimers with a much denser 0.025 a.u step size.
Plotting s versus ρ, as in Fig. 18.1, reveals the basic pattern of intramolecular
interactions. Methane (Fig. 18.1a) illustrates the typical covalent bond pattern. The
top left-side points (small density and large reduced gradient) correspond to the
exponentially-decaying tail regions of the density, far from the nuclei. The points
on the bottom right side (density values of ca. 0.25 a.u. and low reduced density
gradient) correspond to the C–H covalent bonds. Covalent bonds have their characteristic BCP in the electron density, corresponding to s = 0. Regions near the
nuclei have larger density values and appear beyond the right edge of the plot. The
plot has an overall shape of the form aρ
−1/3 because atomic and molecular densities
are piecewise exponential. The results for water are very similar, the only difference
being that the covalent bonds lie at higher density values, past the edge of the plot.
In Fig. 18.1b–d, we consider six examples of chemical systems displaying various
types of non-covalent interactions. Plots of s versus ρ for these systems all exhibit a
new feature: one or more spikes in the low-density, low-gradient region, a signature
of non-covalent interactions. This is the basis of NCI.
18.2.2 The Density Second Eigenvalue
According to the divergence theorem [58], the sign of the Laplacian of the density,
∇
2 ρ, indicates whether the net gradient flux is entering, ∇
2 ρ < 0, or leaving,
∇
2
ρ > 0, an infinitesimal volume around a reference point. Hence, it highlights
whether the density is concentrated or depleted at that point, relative to the surrounding environment. To differentiate between different types of weak interactions
one cannot resort to the sign of the Laplacian itself (as is common within AIM
theory) since it is dominated by the principle axis of variation and is positive for all
closed-shell interactions [59].
To understand bonding in more detail, it is often useful to decompose the
Laplacian into the contributions along the three particular axes of maximal variation. These components are the three eigenvalues, λ i of the electron-density Hessian
matrix, such that, ∇
2
ρ = λ 1 + λ 2 + λ 3 , (λ 1 < λ 2 < λ 3 ). At points with zero gradient,
analysis of the Hessian eigenvalues is analogous to determining the signature of the
critical point. Thus, at nuclei (cusps interpreted as maxima of ρ), all the eigenvalues
are negative, while at the center of cages or holes (minima of ρ) all the eigenvalues
are positive. In the remaining points of space λ 3 > 0, λ 1 < 0, and λ 2 can be either
18 A Complete NCI Perspective: From New Bonds to Reactivity
499