3.2. YACHAYWASI (Casa del Saber)
3.2.1 LOS MAESTROS
The Quantum Theory of Atoms in Molecules (hereafter referred to as QTAIM) is a methodology developed by Bader [61, 62, 63, 64] to describe the quantum nature of molecules, from which many properties can be quantitatively described. QTAIM relies heavily on quantum observables such as the electron density in order to do this.
2.8.1 Critical Points
In QTAIM a Critical Point (CP) is the point at which the first derivative (i.e. the gradient) of the electron density, ρ(r), is zero. There are many different types of CPs which are distin-guished from each other by the direction of second derivative of the electron density. In 3-D space there are nine second derivatives of ρ(r) which can be arranged in the Hessian Matrix (Eqn 2.71).
The Hessian matrix is real and symmetric and can therefore be diagonalised, which is equivalent to rotating the coordinate system r(x, y, z )→r(x’, y’, z’ ) where the new axes x’, y’, z’ are the principal curvature axes of the CP [64]. The Hessian in its diagonal form (Λ) is shown in Eqn 2.72 where λ1, λ2 & λ3 are eigenvalues.
In addition to being eigenvalues of the Hessian, λ1, λ2& λ3are also the curvatures of the density with respect to the principle axes, x’, y’ & z’ at the Critical Point.
CPs are classified by their rank (ω), which is the number of non-zero curvatures of ρ(r), &
signature (σ), the sum of the signs of the curvatures; denoted CP(ω, σ) [64]. The value of the rank of a CP is nearly always 3 for a stable system. In total there are four types of stable CPs;
Nuclear Critical Point (NCP), Bond Critical Point (BCP), Ring Critical Point (RCP) & Cage Critical Point (CCP). NCPs (3,-3) are found at local maxima of the electron density (negative curvatures along the principle axis x’, y’ & z’) and correspond to the position of atoms, whilst CCPs (3,+3) are found at the opposite, at a local minimum. Perhaps of most interest is the BCP (3,-1) where the second derivative is negative in two directions and positive in the third, perpendicular to the bond path.
The Poincar´e-Hopf (PH) relationship [62] (Eqn 2.73) outlines the number and type of critical points that can coexist in a molecule or crystal:
nN CP − nBCP + nRCP− nCCP =
1 (Isolated molecules) 0 (Infinite Crystals)
(2.73)
Figure 2.8 shows an example of different critical points in the Ce(II)Cp3 system. If the PH relationship is applied to the system in Figure 2.8 where there are 45 BCPs, 18 RCPs, 3 CCPs
& 31 NCPs it can be seen that the relationship is satisfied as an isolated molecule.
Figure 2.8: Critical Points in the [Ce(II)Cp3]– system, green spheres indicate BCPs, red spheres RCPs and blue spheres CCPs.
The electron density at critical points can yield some important properties about a molecule.
Attention is now focused on BCPs, generally speaking a bond can be considered to be covalent if the electron density at the BCP, ρbcp, is 0.20 a.u. or higher, whilst lower than 0.10 a.u. is indicative of ionic bonding. Furthermore it has been shown that ρbcp can be directly related to the binding energy of the bond [64].
There are many other properties that can be obtained from a QTAIM analysis, these can be categorised as topological properties, those obtained as a result of the deformations of the electron density, or as integrated properties, calculated through the integration of the electron density.
2.8.2 Topological Properties
The value of the electron density at a critical point (ρbcp) is an example of a topological property obtained from QTAIM analysis. Other useful topological properties include the Laplacian at
the critical point (∇2ρbcp) & the bond path.
The Laplacian is a second order differential operator defined as the divergence of the gradient, in this case, of the electron density. In section 2.8.1 the Hessian was defined (Eqn 2.71), The Laplacian of the electron density at the critical point can be defined using the eigenvalues of the Hessian (Eqn 2.74).
∇2ρ(r) = ∂2ρ(r)
∂x2 + ∂2ρ(r)
∂y2 +∂2ρ(r)
∂z2 = λ1+ λ2+ λ3 (2.74)
∇2ρbcpis therefore the sum of the curvatures of the density at the critical point. At a BCP(3,-1), two of these are negative whilst the third is positive. The magnitude of these numbers presents information about the bonding associated with that particular BCP. In a covalent bond, the two negative curvatures tend to be larger than the one positive curvature and so ∇2ρbcp< 0, con-versely in a bond considered ionic the positive curvature is dominant and ∇2ρbcp> 0. However it should be noted that this is a guide rather than a rule, for instance in strongly polar bonds because of the accumulation of electron density in all directions the Laplacian can be either sign.
The bond path is a line of locally maximum density which connects two nuclei. It is important to note that in QTAIM the bond path is not necessarily the same as a bond. The point along the bond path where the electron density is at a minimum is the BCP.
2.8.3 Integrated Properties
In addition to topological properties, more information can be obtained by also calculating the integrated properties. For instance, integrating the electron density over an atomic basin of atom A gives the number of electrons localised on A, NA [65]. NA can be related back to the atomic charge of atom A, q(A), which is the difference between the atomic number, ZA, and NA and is shown in Eqn 2.75.
q(A) = ZA− NA (2.75) NA & q(A) provide information about where the electrons are located in the molecules.
Further useful properties can also be obtained by integrating the electron pair density over an atomic basin to calculate the localisation index, λ(A), or integrating over a pair of atomic basins to obtain the delocalisation index, δ(A, B).
λ(A) is a measure of the number of electrons localised on atom A, whilst δ(A, B) is a measure of the number of electrons shared between atoms A & B. When A & B are bonded, and in the absence of charge transfer, then δ(A, B) can be considered as the bond order or index.