Emergent Scalar and Vector Fields in Quantum Chemical Topology

Abstract

Several potentially useful scalar and vector fields that have been scarcely or even never used to date in Quantum Chemical Topology are defined, computed, and analyzed for a few small molecules. The fields include the Ehrenfest force derived from the second order density matrix, which does not show many of the spurious features encountered when it is computed from the electronic stress tensor, the exchange-correlation (xc) potential, the potential acting on one electron in a molecule, and the additive and effective energy densities. The basic features of the topology of some of these fields are also explored and discussed, paying attention to their possible future interest.

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, ${\boldsymbol{\nabla }}f\left( \varvec{r} \right)$ and ${\boldsymbol{\nabla }}^{t} f\left( \varvec{r} \right)$ represent the gradient of the scalar field $f\left( \varvec{r} \right)$ in row and column forms, respectively, ${\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}f\left( \varvec{r} \right)$ is the 3 × 3 array of the second derivatives of $f\left( \varvec{r} \right)$, and ${\boldsymbol{\nabla }}^{t} f\left( \varvec{r} \right){\boldsymbol{\nabla }}g\left( \varvec{r} \right)$ is the 3 × 3 array that results from the matrix product of the column vector ${\boldsymbol{\nabla }}^{t} f\left( \varvec{r} \right)$ with the row vector ${\boldsymbol{\nabla }}g\left( \varvec{r} \right)$. All the integrals involving the molecular orbitals (MOs) were evaluated using the McMurchie-Davidson algorithm as implemented in the PROMOLDEN code.

The xc potential, $\rho^{\text{xc}} \left( {\varvec{r}_{1} ,\varvec{r}_{2} } \right)$, can be written in terms of a set of m real MOs ${\varphi }_{i}$ ($i = 1, \ldots ,m$) in the form

$$\varvec{\rho}^{\text{xc}} \left( {\varvec{r}_{1} ,\varvec{r}_{2} } \right) = \sum\limits_{p \ge q}^{m} \sum\limits_{r \ge s}^{m} \lambda_{pq,rs} {\varphi }_{p} \left( {\varvec{r}_{1} } \right){\varphi }_{q} \left( {\varvec{r}_{1} } \right){\varphi }_{r} \left( {\varvec{r}_{2} } \right){\varphi }_{s} \left( {\varvec{r}_{2} } \right),$$

(6.25)

where the array λ is symmetric in the (pq) and (rs) pairs. To compact the notation, we will collect the pair of indices p and q with $p \ge q$ in a single index i, and the product ${\varphi }_{p} \left( \varvec{r} \right){\varphi }_{q} \left( \varvec{r} \right)$ will be represented as $\phi_{i} \left( \varvec{r} \right)$. Then

$$\rho^{\text{xc}} \left( {\varvec{r}_{1} ,\varvec{r}_{2} } \right) = \sum\limits_{i,j} \lambda_{ij} \phi_{i} \left( {\varvec{r}_{1} } \right)\phi_{j} \left( {\varvec{r}_{2} } \right).$$

(6.26)

Although it is not strictly necessary, it is convenient for our purposes to diagonalize λ and express $\rho^{\text{xc}} \left( {\varvec{r}_{1} ,\varvec{r}_{2} } \right)$ in the form

$$\rho^{\text{xc}} \left( {\varvec{r},\varvec{r}_{2} } \right) = \sum\limits_{i}{\eta}_{i} G_{i} \left( \varvec{r} \right)G_{i} \left( {\varvec{r}_{2} } \right),$$

(6.27)

where ${\eta}_{i}$ are the eigenvalues of λ,

$$G_{i} \left( \varvec{r} \right) = \sum\limits_{j} d_{ji} \phi_{j} \left( \varvec{r} \right),$$

(6.28)

and $\left( {d_{1i} ,d_{2i} , \ldots } \right) \equiv {\varvec{d}}_{i}$ is the ith eigenvector of λ. This is the usual way of proceed in the PROMOLDEN program to facilitate the numerical evaluation of all the integrals that appear within the Interacting Quantum Atoms (IQA) method. Substituting 6.27 into Eq. 6.5 we have

$$V_{\text{xc}} \left( \varvec{r} \right) = \sum\limits_{i}{\eta}_{i} G_{i} \left( \varvec{r} \right)\int {d\varvec{r}_{2} \frac{{G_{i} \left( {\varvec{r}_{2} } \right)}}{{|\varvec{r} - \varvec{r}_{2} |}}} = \sum\limits_{i}{\eta}_{i} G_{i} \left( \varvec{r} \right)V_{{G_{i} }} \left( \varvec{r} \right).$$

(6.29)

The gradient of $V_{\text{xc}} \left( \varvec{r} \right)$ is

$${\boldsymbol{\nabla }}V_{\text{xc}} \left( \varvec{r} \right) = \sum\limits_{i} \eta_{i} G_{i} \left( \varvec{r} \right){\boldsymbol{\nabla }}V_{{G_{i} }} \left( \varvec{r} \right) + \sum\limits_{i} \eta_{i} V_{{G_{i} }} \left( \varvec{r} \right){\boldsymbol{\nabla }}G_{i} \left( \varvec{r} \right)$$

(6.30)

The three components of ${\boldsymbol{\nabla }}G_{i} \left( \varvec{r} \right)$ can be obtained simply by deriving Eq. 6.28. On the other hand, from the definition of $V_{{G_{i} }} \left( \varvec{r} \right)$ in Eq. 6.29 we have

$${\boldsymbol{\nabla }}V_{{G_{i} }} \left( \varvec{r} \right) = - \int {d\varvec{r}_{2} \frac{{G_{i} \left( {\varvec{r}_{2} } \right)\left( {\varvec{r} - \varvec{r}_{2} } \right)}}{{|\varvec{r} - \varvec{r}_{2} |^{3} }}}$$

(6.31)

From the definition of $F_{\text{xc}} \left( \varvec{r} \right)$ in Eqs. 6.12 and 6.27 we also have

$$\varvec{F}_{\text{xc}} \left( \varvec{r} \right) = - \sum\limits_{i} \eta_{i} G_{i} \left( \varvec{r} \right)\int {d\varvec{r}_{2} \frac{{G_{i} \left( {\varvec{r}_{2} } \right)\left( {\varvec{r} - \varvec{r}_{2} } \right)}}{{|\varvec{r} - \varvec{r}_{2} |^{3} }}} = \sum\limits_{i} \eta_{i} G_{i} \left( \varvec{r} \right){\boldsymbol{\nabla }}V_{{G_{i} }} \left( \varvec{r} \right),$$

(6.32)

so that

$${\boldsymbol{\nabla }}V_{\text{xc}} \left( \varvec{r} \right) = {\varvec{F}}_{\text{xc}} \left( \varvec{r} \right) + \sum\limits_{i} \eta_{i} V_{{G_{i} }} \left( \varvec{r} \right){\boldsymbol{\nabla }}G_{i} \left( \varvec{r} \right).$$

(6.33)

It is important to note that, according to the above equations, $\varvec{F}_{\text{xc}} \left( \varvec{r} \right) \ne {\boldsymbol{\nabla }}V_{\text{xc}} \left( \varvec{r} \right)$. The Hessian of $V_{\text{xc}} \left( \varvec{r} \right)$ is obtained by simply deriving Eq. 6.33:

$${\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}V_{\text{xc}} \left( \varvec{r} \right) = \varvec{H}_{\text{xc}} \left( \varvec{r} \right) + \sum\limits_{i} \eta_{i} \left[ {{\boldsymbol{\nabla }}^{t} G_{i} \left( \varvec{r} \right){\boldsymbol{\nabla }}V_{{G_{i} }} \left( \varvec{r} \right) + V_{{G_{i} }} \left( \varvec{r} \right){\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}G_{i} \left( \varvec{r} \right)} \right],$$

(6.34)

where $\left( {\varvec{H}_{\text{xc}} } \right)_{\alpha \beta } \equiv \left( {\partial F_{{{\text{xc,}}\alpha }} /\partial \beta } \right)$. From Eq. 6.32 one has

$$\varvec{H}_{\text{xc}} = \sum\limits_{i} {\eta_{i} \left[ {{\boldsymbol{\nabla }}^{t} V_{{G_{i} }} \left( \varvec{r} \right){\boldsymbol{\nabla }}G_{i} \left( \varvec{r} \right) + G_{i} \left( \varvec{r} \right){\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}V_{{G_{i} }} \left( \varvec{r} \right)} \right],\quad {\text{with}}}$$

(6.35)

$${\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}V_{{G_{i} }} \left( \varvec{r} \right) = \int {d\varvec{r}_{2} \frac{{G_{i} \left( {\varvec{r}_{2} } \right)}}{{|\varvec{r} - \varvec{r}_{2} |^{5} }}\left[ {3\left( {\alpha - \alpha_{2} } \right)\left( {\beta - \beta_{2} } \right) - \delta_{\alpha \beta } |\varvec{r} - \varvec{r}_{2} |^{2} } \right].}$$

(6.36)

Regarding the Ehrenfest force, $\varvec{F}_{e} \left( \varvec{r} \right)$, calling $\left( \varvec{H} \right)_{e,\alpha \beta } \equiv H_{e,\alpha \beta } = \left( {\partial F_{e,\alpha } /\partial \beta } \right)$, one easily obtains from Eq. 6.10

$$\varvec{H}_{e} \left( \varvec{r} \right) = \rho \left( \varvec{r} \right){\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}V_{\text{mep}} \left( \varvec{r} \right) + {\boldsymbol{\nabla }}^{t} V_{\text{mep}} \left( \varvec{r} \right){\boldsymbol{\nabla }}\rho \left( \varvec{r} \right) + \varvec{H}_{\text{xc}} \left( \varvec{r} \right)$$

(6.37)

The exact relationship between $\varvec{F}_{e} \left( \varvec{r} \right)$ and $\varvec{f}_{\text{PAEM}} \left( \varvec{r} \right) = \rho \left( \varvec{r} \right)\varvec{F}_{\text{PAEM}} \left( \varvec{r} \right)$, obtained by explicity computing the gradient ${\boldsymbol{\nabla }}\left( {V_{\text{xc}} /\rho } \right)$ that appears in Eq. 6.19, is

$$\varvec{f}_{\text{PAEM}} \left( \varvec{r} \right) = \varvec{F}_{e} \left( \varvec{r} \right) - \frac{{V_{\text{xc}} \left( \varvec{r} \right)}}{{\rho \left( \varvec{r} \right)}}{\boldsymbol{\nabla }}\rho \left( \varvec{r} \right) + \sum\limits_{i} \eta_{i} V_{{G_{i} }} \left( \varvec{r} \right){\boldsymbol{\nabla }}G_{i} \left( \varvec{r} \right).$$

(6.38)

As a consequence of the last term in Eq. 6.38 the expression for $\varvec{H}_{\text{PAEM}}$, defined as $\left( {\varvec{H}_{\text{PAEM}} } \right)_{\alpha \beta } = \left( {\partial \,\varvec{f}_{{\alpha ,{\text{PAEM}}}} \left( \varvec{r} \right)/\partial \beta } \right)$ is a little bit cumbersome:

$$\begin{aligned} \varvec{H}_{\text{PAEM}} & = \varvec{H}_{e} - \frac{{V_{\text{xc}} }}{\rho }{\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}\rho - \frac{{\left( {{\boldsymbol{\nabla }}^{t} \rho } \right)F_{\text{xc}} }}{\rho } + \frac{{V_{\text{xc}} }}{{\rho^{2} }}{\boldsymbol{\nabla }}^{t} \rho {\boldsymbol{\nabla }}\rho \\ & \quad - \frac{{{\boldsymbol{\nabla }}^{t} \rho }}{\rho }\sum\limits_{i} \eta_{i} V_{{G_{i} }} {\boldsymbol{\nabla }}G_{i} + \sum\limits_{i} \eta_{i} \left[ {\left( {{\boldsymbol{\nabla }}^{t} G_{i} } \right)\left( {{\boldsymbol{\nabla }}V_{{G_{i} }} } \right) + V_{{G_{i} }} {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}G_{i} } \right]. \\ \end{aligned}$$

(6.39)

The gradient and Hessian of $E_{\text{eff}} \left( \varvec{r} \right)$, Eq. 6.23, are given by

$${\boldsymbol{\nabla }}E_{\text{eff}} = {\boldsymbol{\nabla }}G - \rho {\boldsymbol{\nabla }}V_{\text{mep}} - V_{\text{mep}} {\boldsymbol{\nabla }}\rho - {\boldsymbol{\nabla }}V_{\text{xc}} ,$$

(6.40)

$$\begin{aligned} {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}E_{\text{eff}} & = {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}G - \rho {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}V_{\text{mep}} - {\boldsymbol{\nabla }}^{t} V_{\text{mep}} {\boldsymbol{\nabla }}\rho - V_{\text{mep}} {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}\rho \\ & \quad - {\boldsymbol{\nabla }}^{t} \rho {\boldsymbol{\nabla }}V_{\text{mep}} - {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}V_{\text{xc}} . \\ \end{aligned}$$

(6.41)

where ${\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}V_{\text{xc}} \left( \varvec{r} \right)$ is given by Eq. 6.34 The gradient and Hessian of $E_{\text{add}} \left( \varvec{r} \right)$, Eq. 6.20, are even more complicated:

$$\begin{aligned} {\boldsymbol{\nabla }}E_{\text{add}} & = {\boldsymbol{\nabla }}G - \rho {\boldsymbol{\nabla }}V_{\text{mep}}^{\text{nuc}} - V_{\text{mep}}^{\text{nuc}} {\boldsymbol{\nabla }}\rho \\ & \quad - \frac{1}{2}\left[ {\rho {\boldsymbol{\nabla }}V_{\text{mep}}^{\text{ele}} + V_{\text{mep}}^{\text{ele}} {\boldsymbol{\nabla }}\rho + \varvec{F}_{\text{xc}} } \right] - \frac{1}{2}\sum\limits_{i} \eta_{i} V_{{G_{i} }} {\boldsymbol{\nabla }}G_{i} , \\ \end{aligned}$$

(6.42)

$$\begin{aligned} {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}E_{\text{add}} & = {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}G - \rho {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}V_{\text{mep}}^{\text{nuc}} - {\boldsymbol{\nabla }}^{t} V_{\text{mep}}^{\text{nuc}} {\boldsymbol{\nabla }}\rho - V_{\text{mep}}^{\text{nuc}} {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}\rho - {\boldsymbol{\nabla }}^{t} \rho {\boldsymbol{\nabla }}V_{\text{mep}}^{\text{nuc}} \\ & \quad - \frac{1}{2}\left[ {{\varvec{H}}_{\text{xc}} + \rho {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}V_{\text{mep}}^{\text{ele}} + {\boldsymbol{\nabla }}^{t} V_{\text{mep}}^{\text{ele}} {\boldsymbol{\nabla }}\rho + V_{\text{mep}}^{\text{ele}} {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}\rho + {\boldsymbol{\nabla }}^{t} \rho {\boldsymbol{\nabla }}V_{\text{mep}}^{\text{ele}} } \right] \\ & \quad - \frac{1}{2}\sum\limits_{i} \eta_{i} \left[ {{\boldsymbol{\nabla }}^{t} G_{i} {\boldsymbol{\nabla }}V_{{G_{i} }} + V_{{G_{i} }} {\boldsymbol{\nabla }}^{t} {\boldsymbol{\nabla }}G_{i} } \right]. \\ \end{aligned}$$

(6.43)

To shorten Eqs. 6.39–6.43, the dependence on r of every magnitude has been avoided.