Predicting Atomic Decay Rates Using an Informational-Entropic Approach

Abstract

We show that a newly proposed Shannon-like entropic measure of shape complexity applicable to spatially-localized or periodic mathematical functions known as configurational entropy (CE) can be used as a predictor of spontaneous decay rates for one-electron atoms. The CE is constructed from the Fourier transform of the atomic probability density. For the hydrogen atom with degenerate states labeled with the principal quantum number n, we obtain a scaling law relating the n-averaged decay rates to the respective CE. The scaling law allows us to predict the n-averaged decay rate without relying on the traditional computation of dipole matrix elements. We tested the predictive power of our approach up to n = 20, obtaining an accuracy better than 3.7% within our numerical precision, as compared to spontaneous decay tables listed in the literature.

Appendix: Fourier Transforms and Numerical Methods

1.1 A.1 Computation of Fourier Transforms

In what follows, we describe the derivation of the Fourier transform of an arbitrary function written in spherical coordinates as Φ _{n ℓ m} (r, 𝜃, ϕ) = CY _{ℓ m} (𝜃, ϕ)R(r). Our derivation reproduces results in Ref. [22]. In Ref. [22], the derivation is dedicated to atomic wave functions with radial part R _{n ℓ} (r) with special form $Ce^{-\gamma r}r^{\ell }L_{n+\ell }^{2\ell + 1}(2\gamma r)$, where $L_{n+\ell }^{2\ell + 1}$ is the generalized Laguerre polynomial. Here, we obtain the Fourier transform for a general radial function R _{n ℓ} (r) and carry out the Fourier transform numerically.

Consider general functions in spatial and momentum coordinates Φ _{n ℓ m} (x, y, z) and $\tilde {{\Phi }}_{n\ell m}(k_{x},k_{y},k_{z})$, respectively, related to each other by a Fourier transform

$$ F(\mathbf{k})=\tilde{{\Phi}}_{nlm}(k_{x},k_{y},k_{z}) = \iiint e^{-i\vec{k}\cdot\vec{x}}{\Phi}_{n\ell m}(x,y,z)dxdydz. $$

(12)

We start by writing (x, y, z) and (k _x , k _y , k _z ) in spherical coordinates,

$$ \begin{aligned} x &= r\sin\theta\cos\phi && k_{x} = k\sin\alpha\cos\beta \\ y &= r\sin\theta\sin\phi && k_{y} = k\sin\alpha\sin\beta \\ z &= r\cos\theta && k_{z} = k\cos\alpha, \end{aligned} $$

(13)

and expand $\vec {k}\cdot \vec {x}$ in (12) as:

$$\begin{array}{@{}rcl@{}} \vec{k}\cdot\vec{x}& = & kr(\sin\theta\cos\phi\sin\alpha\cos\beta\\ &&+ \sin\theta\sin\phi\sin\alpha\sin\beta\\ &&+ \cos\theta\cos\alpha)\\ &= & kr\left( \sin\theta\sin\alpha\cos(\phi-\beta)+\cos\theta\cos\alpha\right). \end{array} $$

(14)

The general atomic wave function can be written as

$$ {\Phi}_{n\ell m}(r,\theta,\phi) = (A e^{\pm i m\phi})(B P_{\ell}^{m}(\cos\theta))(Ce^{-\gamma r}r^{\ell}L_{n+\ell}^{2\ell+ 1}(2\gamma r)), $$

(15)

where A, B, C, and γ are constants independent of coordinates. Reference [22] provides a detailed derivation for this specific form of the radial function.

We are interested only in the general radial form, which is:

$$ {\Phi}_{n\ell m}(r,\theta,\phi) = (A e^{\pm i m\phi})(B P_{\ell}^{m}(\cos\theta))(R_{n\ell}(r)). $$

(16)

Using (14), we get

$$\begin{array}{@{}rcl@{}} &&\tilde{{\Phi}}_{n\ell m}(k,\alpha,\beta) \!= \! (A e^{\pm i m\phi})(B P_{\ell}^{m}(\cos\theta))(R_{n\ell}(r))\\ && \! \!= \! \! AB \!\iiint \!e^{-ikr\sin\theta\sin\alpha\cos(\phi-\beta)\pm im \phi-ikr\cos\theta\cos\alpha}P_{\ell}^{m}R_{n\ell}(r)r^{2}\sin\theta drd\theta d\phi\\ && \! \!= \! \!A \!B \! \!{\int}_{0}^{\infty} \! \!R_{n\ell}(r)r^{2}dr \! \!{\int}_{0}^{\pi} \! \!e^{-ikr\cos\theta\cos\alpha}P_{\ell}^{m} \!(\cos \!\theta) \!\sin \!\theta d\theta \!{\int}_{0}^{2\pi} \! \!e^{-ikr\sin\theta\sin\alpha\cos(\phi-\beta)\pm im\phi}d\phi. \end{array} $$

(17)

Consider first the ϕ integral:

$$ I_{1} = {\int}_{0}^{2\pi} e^{-ikr\sin\theta\sin\alpha\cos(\phi-\beta)\pm i m \phi}d\phi. $$

(18)

Introducing ϕ − β = ω,

$$\begin{array}{@{}rcl@{}} I_{1} &=&{\int}_{\beta}^{2\pi-\beta} e^{-ikr\sin\theta\sin\alpha\cos\omega\pm i m(\omega+\beta)}d\omega\\ &=& e^{\pm i m\beta}{\int}_{0}^{2\pi}e^{-ib\cos\omega\pm im\omega}, \end{array} $$

(19)

where b = kr sin 𝜃 sin α. The limits of integration can be changed to (0, 2π) due to the cyclic property of the integrand. Using the integral expression of the Bessel function

$$ J_{n}(x) = \frac{1}{2\pi}{\int}_{0}^{2\pi}e^{\frac{in\pi}{2}}e^{in\tau-ix\cos\tau}d\tau, $$

(20)

and since $e^{\frac {in\pi }{2}}=(i)^{n}$, we can write (18) as

$$ I_{1} = 2\pi(-i)^{m} e^{\pm im\beta}J_{m}(b). $$

(21)

Equation (17) becomes

$$ \tilde{{\Phi}}_{n\ell m}(k,\alpha,\beta) = 2\pi AB e^{\pm im\beta}(-i)^{m}{\int}_{0}^{\infty}R_{n\ell}(r)r^{2}dr {\int}_{0}^{\pi}e^{-ikr\cos\theta\cos\alpha}P_{\ell}^{m}(\cos\theta)J_{m}(b)\sin\theta d\theta. $$

(22)

Consider now the integral

$$ I_{2} = {\int}_{0}^{\pi}e^{-ikr\cos\theta\cos\alpha}P_{\ell}^{m}(\cos\theta)J_{m}(b)\sin\theta d\theta. $$

(23)

First, use the generating function defined as

$$ (1-2tx+t^{2})^{-\nu}\equiv\sum\limits_{\ell= 0}^{\infty}C_{\ell}^{\nu}(x)t^{\ell}, $$

(24)

to write the generating function of the Legendre polynomials as

$$ \sum\limits_{\ell= 0}^{\infty}P_{\ell}(x)t^{\ell} = \frac{1}{\sqrt{1-2tx+t^{2}}}, $$

(25)

where $P_{\ell }(x)=C_{\ell }^{1/2}$ is the coefficient for ν = 1/2.

Apply the operator $(-1)^{m}(1-x^{2})^{\frac {m}{2}}\frac {d^{m}}{dx^{m}}$ to both sides and using the definition of the associate Legendre polynomials,

$$\begin{array}{@{}rcl@{}} \sum\limits_{\ell= 0}^{\infty}P_{\ell}^{m}(x)t^{\ell}=(-1)^{m}(1-x^{2})^{\frac{m}{2}}(2m-1)!(1-2tx+t^{2})^{-\frac{2m + 1}{2}}t^{m}, \end{array} $$

whose right-hand side is simply:

$$(-1)^{m}(1-x^{2})^{\frac{m}{2}}(2m-1)!\sum\limits_{\ell= 0}^{\infty}C_{\ell}^{m+\frac{1}{2}}(x)t^{\ell+m}. $$

Equating powers of t we obtain the identity:

$$ P_{\ell}^{m}(x) = (-1)^{m}(1-x^{2})^{\frac{m}{2}}(2m-1)!C_{\ell-m}^{m+\frac{1}{2}}(x). $$

(26)

Using the identity from Ref. [23],

$$ {\int}_{0}^{\pi} e^{iz\cos\theta\cos\alpha}J_{\nu-\frac{1}{2}}(z\sin\theta\sin\alpha)C_{\mu}^{\nu}\left( \cos\theta\right)\sin^{\nu+\frac{1}{2}}\theta d\theta = \left( \frac{2\pi}{z}\right)^{1/2}i^{\mu}\sin^{\nu-\frac{1}{2}}\alpha C_{\mu}^{\nu}(\cos\alpha)J_{\nu+\mu}(z), $$

(27)

and writing z = −kr, $\nu =\frac {1}{2}+m$, μ = ℓ − m, we get

$$\begin{array}{@{}rcl@{}} I_{2} &=& {\int}_{0}^{\pi}e^{-ikr\cos\theta\cos\alpha}P_{\ell}^{m}(\cos\theta)J_{m}(b)\sin\theta d\theta\\ &=&(-1)^{m}(2m-1)!{\int}_{0}^{\pi}e^{-ikr\cos\theta\cos\alpha}C_{\ell-m}^{m+\frac{1}{2}}(\cos\theta)J_{m}(b)\sin^{\frac{m}{2}+ 1}\theta d\theta\\ &=&(2m-1)!(i)^{\ell-m}\left( \frac{2\pi}{-kr}\right)^{\frac{1}{2}}(1-\cos^{2}\alpha)^{\frac{m}{2}} C_{\ell-m}^{m+\frac{1}{2}}\left( \cos\alpha\right)J_{\ell+\frac{1}{2}}\left( -kr\right)\\ &=&(-1)^{m}(i)^{\ell-m}\left( \frac{2\pi}{-kr}\right)^{\frac{1}{2}} P_{\ell}^{m}\left( \cos\alpha\right)J_{\ell+\frac{1}{2}}\left( -kr\right). \end{array} $$

(28)

The total Fourier transform is:

$$ \tilde{{\Phi}}(k,\alpha,\beta) = AB{\int}_{0}^{\infty}R_{n\ell}(r)r^{2}(-1)^{m}(i)^{\ell-m}\left( \frac{2\pi}{-kr}\right)^{\frac{1}{2}}P_{\ell}^{m} \left( \cos\alpha\right) J_{\ell+\frac{1}{2}}\left( -kr\right)dr. $$

(29)

Neglecting imaginary phases and irrelevant constants, and using expressions for A and B from spherical harmonics we obtain:

$$ \tilde{{\Phi}}(k,\alpha,\beta)\propto \sqrt{\frac{(2\ell+ 1)(\ell-m)!}{(\ell+m)!}}{\int}_{0}^{\infty}R_{n\ell}(r)r^{2}(-kr)^{-\frac{1}{2}} P_{\ell}^{m}\left( \cos\alpha\right)J_{\ell+\frac{1}{2}}\left( -kr\right)dr; $$

(30)

or, grouping factors of − 1,

$$ \tilde{{\Phi}}(k,\alpha,\beta)\propto (-1)^{\ell}\sqrt{\frac{(2\ell+ 1)(\ell-m)!}{(\ell+m)!}}{\int}_{0}^{\infty}R_{n\ell}(r)r^{2}(kr)^{-\frac{1}{2}} P_{\ell}^{m}\left( \cos\alpha\right)J_{\ell+\frac{1}{2}}\left( kr\right)dr. $$

(31)

As we noted in Section 3.2, this expression can also be written in terms of the spherical Bessel function

$$ \tilde{{\Phi}}(k,\alpha,\beta)\propto \sqrt{\frac{(2\ell+ 1)(\ell-m)!}{(\ell+m)!}}{\int}_{0}^{\infty}R_{n\ell}(r)r^{2} P_{\ell}^{m}\left( \cos\alpha\right)j_{\ell}\left( -kr\right)dr. $$

(32)

Let us look at a simple example where ℓ = 1 and m = 0. Equation (30) becomes:

$$\begin{array}{@{}rcl@{}} \tilde{{\Phi}}(k,\alpha,\beta)&\propto& \cos\alpha{\int}_{0}^{\infty}R_{n1}(r)r^{2}(-kr)^{-\frac{1}{2}} J_{\frac{3}{2}}\left( -kr\right)dr\\ &\propto&\cos\alpha{\int}_{0}^{\infty}R_{n1}(r)r^{2}\left( \frac{\sin(kr)}{(kr)^{2}}-\frac{\cos(kr)}{kr}\right)dr\\ &\equiv & F(k)\cos\alpha, \end{array} $$

(33)

which introduces a cos α factor in the Fourier transform, leading to the same symmetry in coordinate and momentum space. The modal fraction is then,

$$\begin{array}{@{}rcl@{}} \tilde{f}(k,\alpha,\beta) &=& \frac{|F(k)\cos\alpha|^{2}}{\left( |F(k)\cos\alpha|^{2}\right)_{max}}\\ &=& \cos^{2}\alpha \frac{|F(k)|^{2}}{\left( |F(k)|^{2}\right)_{max}}\\ &\equiv& \tilde{f}(k,\alpha)=\tilde{f}\cos^{2}\alpha, \end{array} $$

(34)

since the maximum mode must be given by | cos α| = 1.

The configurational entropy is (integrating over the azimuthal coordinate),

$$\begin{array}{@{}rcl@{}} S &=& -2\pi\iint \tilde{f}(k,\alpha)\log\tilde{f}(k,\alpha) k^{2}\sin\alpha dkd\alpha\\ &=& -2\pi\iint \tilde{f}(k)\cos\alpha^{2}\log\left( \tilde{f}(k)\cos^{2}\alpha\right)k^{2}\sin\alpha dkd\alpha\\ &=& -2\pi \left( \frac{2}{3}{\int}_{0}^{\infty} \tilde{f}(k)\log\left( \tilde{f}(k)\right)k^{2}dk-\frac{4}{9}{\int}_{0}^{\infty}\tilde{f}(k)k^{2}dk\right). \end{array} $$

(35)

1.2 A.2 Numerical Procedures

Numerical procedures for this paper consist of two main parts: computation of the integral defined in (8) to obtain the Fourier transform for the probability density function, and computation of the integral defined in (3) to obtain the CE. There are four important parameters that affect substantially the accuracy of the numerical procedures: the step sizes Δr and Δk, and the limits of integration for r and k. The optimal step size and interval of integration for the radial variable r should be tackled first, as the density function of the hydrogen atom can be generated analytically using the generalized Laguerre polynomial in (15).

Given that the probability density function for a given state {n, ℓ, m} has n − ℓ − 1 nodes, we search for the zeroes of the probability density function until we find n − ℓ − 1 roots, except the origin. We stop the integration in r when the probability density function drops below 10^− 7 after the peak following the last node. Call this value r _∞ . We set the number of steps as N = 2⁹. The step size is then r _∞ /N, which we verified yields stable results.

We compute the Fourier transform using Δk = 0.1/r _∞ . To determine k _∞ , we first locate the peaks of $\tilde {f}(k)k^{2}$. We then set k _∞ as the value of k when the amplitude of $\tilde {f}(k)k^{2}$ drops to 1% of its last peak.

Based on these parameters and the trapezoid approximation, the numerical integrations yield stable results with an error in the k integral controlled by Δk², which is smaller than 10^− 4 for small n and 10^− 8 for larger n.