Molecular fingerprints based on Jacobi expansions of electron densities

Abstract

Molecular fingerprint indices are derived from Jacobi expansion of molecular electron density in a ball. Electron density computed in the linear combination of atomic orbitals framework, at any computational level, is expanded in terms of suitable Jacobi polynomials times regular spherical harmonics. The procedure is applicable to calculations with either Gaussian or Slater-type orbitals. A very efficient algorithm previously reported for Canterakis–Zernike expansions is shown to be also applicable for this type of expansions. The procedure has been implemented in the DAMQT suite for the analysis of electron density, which facilitates the application to densities obtained with standard packages for molecular structure calculations. Fingerprints derived from Jacobi expansions are compared with other derived from Canterakis–Zernike expansions in some pharmacological molecules.

Appendix

To compute derivatives of functions $J_{kl}^{m}$ with respect to Cartesian coordinates, it is better to express them in terms of unnormalized real regular harmonics, $z_{l}^{m}(\mathbf {r}/r^{*})$:

$$\begin{aligned} J_{kl}^{m}(\mathbf {t}) = \sqrt{\frac{2(k+l)+3}{{r^{*}}^{3}}} \ P_{k}^{(0,2+2l)}(2t-1) \ \mathcal{{N}}_{{\rm lm}}^\Omega \ z_{l}^{m}(\mathbf {t}) \end{aligned}$$

(24)

where $\mathbf {t}\equiv \mathbf {r}/r^{*}$, $t \equiv |\mathbf {t}|$, and the unnormalized real regular harmonics are related to real spherical harmonics by:

$$\begin{aligned} z_{l}^{m}(\mathbf {t}) = \ t^{l} \ \mathcal {Z}_{l}^{m}(\theta ,\phi ) / \mathcal{{N}}_{{\rm lm}}^\Omega \end{aligned}$$

(25)

$\theta$ and $\phi$ being the corresponding angular coordinates of $\mathbf {t}$, and $\mathcal{{N}}_{{\rm lm}}^\Omega$, the angular normalization factor:

$$\begin{aligned} \mathcal{{N}}_{{\rm lm}}^\Omega = \left[ \frac{(2L+1) \ (L-|M|)!}{2 \ (1+\delta _{M0}) \ \pi \ (L+|M|)!}\right] ^{1/2} \end{aligned}$$

(26)

Taking into account that $\frac{\partial t}{\partial x} = \frac{\partial (r/{r^{*}})}{\partial x} = \frac{1}{{r^{*}}} \ \frac{x}{r}$, and using the chain rule: $\frac{\partial }{\partial x} = \frac{{\rm d} t_{x}}{{\rm d} x} \ \frac{\partial }{\partial t_{x}} = \frac{{\rm d}}{{\rm d} x}\left( \frac{x}{{r^{*}}}\right) \ \frac{\partial }{\partial t_{x}} = \frac{1}{{r^{*}}} \ \frac{\partial }{\partial t_{x}}$, the derivatives are given by:

$$\begin{aligned} \frac{\partial J_{kl}^{m}(\mathbf {t})}{\partial x}=\, & {} \sqrt{2(k+l)+3} \ \mathcal{{N}}_{{\rm lm}}^\Omega \ \frac{1}{{r^{*}}} \ \left[ \frac{x}{r} \ \frac{{\rm d} P_{k}^{(0,2+2l)}(2t-1)}{{\rm d} t} \ z_{l}^{m}(\mathbf {t}) \right. \nonumber \\&\quad +\, \left. P_{k}^{(0,2+2l)}(2t-1) \ \frac{\partial z_{l}^{m}(\mathbf {t})}{\partial t_{x}} \right] \end{aligned}$$

(27)

The derivatives of the unnormalized Jacobi polynomials, $P_{k}^{(0,2+2l)}$, with respect to variable t are given by—see [49] eq 8.961.4:

$$\begin{aligned} {P' \ }_{k}^{(0,2+2l)}(2t-1) \equiv \frac{\partial P_{k}^{(0,2+2l)}(2t-1)}{\partial t} = (k+2l+3) \ P_{k-1}^{(1,3+2l)}(2t-1) \end{aligned}$$

(28)

and a recursion formula can be found for them again with the aid of eq 8.961.2 of ref [49], namely:

$$\begin{aligned}&k \ (k+l+1) \ (k+2l+3) \ {P' \ }_{k+1}^{(0,2+2l)} \nonumber \\=\, & {} (2k+2l+3) \ \bigl [ (k+l+1) \ (k+l+2) \ (2t-1) - (l+1) \ (l+2) \bigr ] \ {P' \ }_{k}^{(0,2+2l)} \nonumber \\&\quad -\, k \ (k+l+2) \ (k+2l+3) \ {P' \ }_{k-1}^{(0,2+2l)} \end{aligned}$$

(29)

where the argument of the polynomials has been omitted to facilitate reading. Expressions for derivatives with respect to y and z are obtained by just changing names of variables in Eq. (27).

On the other hand, as it has been reported [55], the derivatives of the regular spherical harmonics with respect to Cartesian coordinates can be computed in a straightforward way in terms of these harmonics.