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The Electron Density

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Topological Approaches to the Chemical Bond

Part of the book series: Theoretical Chemistry and Computational Modelling ((TCCM))

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Abstract

In this chapter we analyze the basic scalar field in the topological theories of the chemical bond, the electron density, \(\rho \). After a succinct revision of the general properties of the \(\rho \) field, the topology that it induces in real space through its attractor basins is presented. This gives rise to an atomic partition, which is generally known as the Quantum Theory of Atoms in Molecules (QTAIM). In this Chapter, the types and properties of the different critical points of \(\rho \) are reviewed, and the physical roots of the partition are examined. As it will be shown, only when QTAIM basins are used several basin observables, like the atomic kinetic energy, can be defined consistently. In many instances, we also need to know the joint probability of finding not only one electron at a point, but a pair of electrons at a couple of positions, for instance. This leads us to conclude the chapter by presenting the general formalism of reduced densities and density matrices, and to use them to partition the energy in the interacting quantum atoms (IQA) approach or to construct localization and delocalization descriptors.

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Notes

  1. 1.

    It is interesting to recall that in the beginnings of quantum mechanics a significant debate arose related to the difficulty to find relativistic wave equations with positive definite density. For instance, the Klein–Gordon equation [40] \((\Box ^2 - m^2c^4) \Psi = 0 \) (The D’Alambertian operator is defined as \(\Box ^2 = \nabla ^2 - (1/c^2) \partial ^2/\partial t^2\)), which was the first relativistic equation to be proposed, has solutions with \(\rho >0\) and solutions with \(\rho <0\). The problem was finally solved after a revolution that proposed the existence of particles with proper positive energy as well as of a new set of \(E<0\) counterparts (antiparticles).

  2. 2.

    A similar demonstration for multielectron systems would integrate all but the coordinates that are averaged.

  3. 3.

    Recall that for any function f, \(\left[ \hat{f}, \hat{\boldsymbol{p}}\right] = i \hslash \boldsymbol{\nabla }f\). Using the commutator rule \([AB,C]=A[B,C]+[A,C]B\) it can also be deduced that:

    $$\begin{aligned} \left[ \hat{H}, \hat{\boldsymbol{r}} \right] = [\hat{T}+\hat{V}]= \frac{\hat{\boldsymbol{p}}}{m}[\hat{\boldsymbol{p}},\hat{\boldsymbol{r}}]=-\frac{i\hslash \hat{\boldsymbol{p}}}{m}\qquad \qquad \qquad {(3.84)} \end{aligned}$$

    since \([\hat{V},\hat{r}]=0\).

  4. 4.

    Recall from Combinatory Logic that: \( \left( N \atop m\right) =\frac{N!}{m!(N-m)!}\) , gives us the number of n-tuples in the system, so that the number of pairs is given by \( \left( N \atop 2\right) =\frac{N!}{2!(N-2)!}=\frac{N(N-1)}{2}\). If we know consider ordered pairs (e.g. \(AB\ne BA\)), we have twice as many, i.e. \(N(N-1)\). More generally, for ordered n-tuples (which allow m! permutations), we will have: \(\frac{N!}{(N-m)!}\). Note that different normalizations have been proposed depending on the use of ordered or unordered pairs.

  5. 5.

    Recall that given two events A and B, the conditional probability of A given B, P(A|B), is given by:

    $$\begin{aligned} P(A|B)=\frac{P(A\cap B)}{P(B)} \end{aligned}$$
    (3.148)

    where \(P(A\cap B)\) is the probability of the joint event A and B.

  6. 6.

    The first equality refers to the fact that the total probability must be equal to one by definition, which in our case, means summing up all the occupation probabilities. The second expression gives us the average occupation of a given \(\varOmega \) from the statistics of a random variable \(\bar{x}=\sum _i x_i P(X_i)\).

  7. 7.

    The concept of entropy in information theory provides the “uncertainty” in a given event. For example, a normal coin has maximum entropy (we have maximum probability of having either heads or tails). At the other end, a two-face coin has minimum entropy. Mathematically, this is measured by Shannon’s entropy:

    $$\begin{aligned} H(X)=-\sum _{i=1}P(x_i)\text {log} P(x_i)\qquad \qquad \qquad {(3.171)}, \end{aligned}$$
    (3.171)

    where the variable X can have \(x_1\ldots x_n\) values with \(P(X_1)\ldots P(x_n)\) probabilities.

  8. 8.

    You are probably acquainted with the variance of a discrete variable measured n times: \(S^2=\frac{(\sum x-\bar{x})^2}{n}\). For a random variable X taking values \(x_1\ldots x_n\) with probabilities \(P_1\ldots P_n\), \(S^2=(\sum _i x_i-\bar{x})^2P(X_i)\). Hence, using Eq. 3.170: \( \sigma ^2(\varOmega )= \sum _i (n-N_\varOmega )^2P_n(\varOmega )= \sum _i (n^2 P_n(\varOmega ))+N_\varOmega ^2-2N_\varOmega \sum _i (n P_n(\varOmega )=\sum _i (n^2 P_n(\varOmega ))-N_\varOmega ^2\).

  9. 9.

    Develo** Eq. 3.173 and introducing Eq. 3.172: \(2D_{\varOmega ,\varOmega }=\sum _n n^2P_n(\varOmega )-\sum _n nP_n(\varOmega )=\sigma ^2(\varOmega )+N_\varOmega ^2-N_\varOmega \). Reorganizing, we obtain Eq. 3.174.

  10. 10.

    From Eqs. 3.173 and 3.174: \(D_{\varOmega ,\varOmega }=N_\varOmega (N_\varOmega -1)=n(n-1)\).

  11. 11.

    \( \sigma ^2(\varOmega ) = \int _\varOmega \int _\varOmega (\rho (\boldsymbol{x}_1) \rho (\boldsymbol{x}_2) + \rho (\boldsymbol{x}_1) h_{xc}(\boldsymbol{x}_1, \boldsymbol{x}_2)) d \boldsymbol{x}_1 d \boldsymbol{x}_2-N_{\varOmega }^2 +N_{\varOmega }\) with \(\int _\varOmega \int _\varOmega \rho (\boldsymbol{x}_1) \rho (\boldsymbol{x}_2) d \boldsymbol{x}_1 d \boldsymbol{x}_2=N_{\varOmega }^2\).

  12. 12.

    \(\sum _B F_{\varOmega _A,\varOmega _B}= \int _{\varOmega _A} \rho (\boldsymbol{x}_1) d\boldsymbol{x}_1 \sum _B \int _{\varOmega _B} h_{xc}(\boldsymbol{x}_2| \boldsymbol{x}_1)d \boldsymbol{x}_2 = N_A \int _{\mathbb R^3} h_{xc}(\boldsymbol{x}_2| \boldsymbol{x}_1)d \boldsymbol{x}_2 =-N_A\)

  13. 13.
    $$\begin{aligned} \int _\varOmega \int _\varOmega \rho (\boldsymbol{x}_1)\rho (\boldsymbol{x}_2) d\boldsymbol{x}_1 d\boldsymbol{x}_2-\sum _i\sum _j\int _\varOmega \int _\varOmega (\phi _i^*(\boldsymbol{x}_1)\phi _i(\boldsymbol{x}_2))(\phi _j^*(\boldsymbol{x}_1)\phi _j(\boldsymbol{x}_2)) d\boldsymbol{x}_1 d\boldsymbol{x}_2= N_\varOmega ^2-\sum _iS_{ij}^2(\varOmega ) \qquad \qquad \qquad {(3.180)} \end{aligned}$$
    (3.180)
  14. 14.

    Isopycnic transformations are linear orbital transformations that leave the first-order density matrix invariant. They make it possible to localize not only the Hartree-Fock spin-orbitals, but also the natural spin-orbitals.

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Authors and Affiliations

  1. Departamento de Química Física y Analítica, Universidad de Oviedo, Oviedo, Spain

    Ángel Martín Pendás

  2. Laboratoire de Chimie Théorique, Sorbonne Université, Paris, France

    Julia Contreras-García

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  1. Ángel Martín Pendás
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Correspondence to Ángel Martín Pendás .

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Martín Pendás, Á., Contreras-García, J. (2023). The Electron Density. In: Topological Approaches to the Chemical Bond. Theoretical Chemistry and Computational Modelling. Springer, Cham. https://doi.org/10.1007/978-3-031-13666-5_3

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