Shannon, Rényi, Tsallis Entropies and Onicescu Information Energy for Low-Lying Singly Excited States of Helium

Abstract

Knowledge of the electronic structures of atomic and molecular systems deepens our understanding of the desired system. In particular, several information-theoretic quantities, such as Shannon entropy, have been applied to quantify the extent of electron delocalization for the ground state of various systems. To explore excited states, we calculated Shannon entropy and two of its one-parameter generalizations, Rényi entropy of order $\alpha$ and Tsallis entropy of order $\alpha$, and Onicescu Information Energy of order $\alpha$ for four low-lying singly excited states (1s2s ${}^1{S}^e$, 1s2s ${}^3{S}^e$, 1s3s ${}^1{S}^e$, and 1s3s ${}^3{S}^e$ states) of helium. This paper compares the behavior of these three quantities of order 0.5 to 9 for the ground and four excited states. We found that, generally, a higher excited state had a larger Rényi entropy, larger Tsallis entropy, and smaller Onicescu information energy. However, this trend was not definite and the singlet–triplet reversal occurred for Rényi entropy, Tsallis entropy and Onicescu information energy at a certain range of order $\alpha$.

1. Introduction

The electronic structure, a unique feature of an atomic or molecular system, is fundamental for further understanding of other properties. To characterize an electronic structure, one critical aspect is whether the electron distribution is localized or delocalized [1,2]. Originated from the information theory and later extended to the position space, the information-theoretic (IT) quantities provide a possible quantitative account of the extent of electron localization. For instance, the more delocalized is the electron distribution, the larger is the Shannon entropy (S) [3,4,5,6,7,8,9,10,11,12,13,14]. Moreover, Rényi entropy of order $\alpha$ ($R_{\alpha }$) [15,16,17] and Tsallis entropy of order $\alpha$ ($T_{\alpha }$) [18,19,20,21] are the one-parameter generalizations of Shannon entropy and offer extended descriptions. In addition, Onicescu Information Energy of order $\alpha$ ($E_{\alpha }^{\mathrm{O}}$) [22,23] is related to the frequency moment of the electron density [24,25,26]. These IT quantities have been applied to various atomic and molecular systems [4,5,6,7,8,9,10,11,12,13,14,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80] and potentials [81,82,83,84,85,86,87,88,89,90,91]. Further background details of these IT quantities were summarized by Ilić and Stanković [92] and by us [80].

However, a large part of the reports focus on the ground state, to the best of our knowledge. Even though some results for Rydberg states [10,40,46,62,63,64,81] are available, behaviors of these IT quantities for excited states remain to be explored. In particular, for the helium atom, Shannon entropy generally increases for higher excited states [10,11,12] but several exceptions are also found [10]. To examine the trend in IT quantities for increasing excitations, we calculated Shannon, Rényi, and Tsallis entropies and Onicescu Information Energy for four low-lying singly excited states (1s2s ${}^1{S}^e$, 1s2s ${}^3{S}^e$, 1s3s ${}^1{S}^e$, and 1s3s ${}^3{S}^e$ states) of helium, subsequent to our previous benchmark report for the ground state helium [80]. This paper compares the behavior of these quantities of order 0.5 to 9 for the ground and four excited states of helium.

2. Theoretical Methods

2.1. Determination of Ground and Singly Excited States of Helium via Hylleraas Wave Functions

The non-relativistic Hamiltonian of helium atom (in atomic units) is

$$H=-\frac{1}{2}{\nabla }_1^2-\frac{1}{2}{\nabla }_2^2-\frac{2}{r_1}-\frac{2}{r_2}+\frac{1}{r_{12}},$$

where subscripts 1 and 2 are indexes of electrons, and $r_{12}$ is the distance between the two electrons. Featuring the explicit $r_{12}$ term, Hylleraas wave functions [93,94,95] take the electron correlation into account [96,97] and are written as

$${\psi }_{mnk}(r_1,r_2,r_{12})=\sum _{mnk}{C}_{mnk}(e^{-{\beta }_1{r}_1}{e}^{-{\beta }_2{r}_2}{r}_1^m{r}_2^n{r}_{12}^k+e^{-{\beta }_1{r}_2}{e}^{-{\beta }_2{r}_1}{r}_2^m{r}_1^n{r}_{12}^k)$$

for singlet states, and

$${\psi }_{mnk}(r_1,r_2,r_{12})=\sum _{mnk}{C}_{mnk}(e^{-{\beta }_1{r}_1}{e}^{-{\beta }_2{r}_2}{r}_1^m{r}_2^n{r}_{12}^k-e^{-{\beta }_1{r}_2}{e}^{-{\beta }_2{r}_1}{r}_2^m{r}_1^n{r}_{12}^k)$$

for triplet states. $m, n, k$, and $\omega$ are zero or positive integers and $m+n+k\le \omega$ so $\omega$ determines the size of basis set. Besides, ${\beta }_1$ and ${\beta }_2$ are variational parameters. For singlet states with arbitrary ${\beta }_1$ and ${\beta }_2$, and also for triplet states with ${\beta }_1\ne {\beta }_2$, $m\ge n$ and $\omega$ starts from zero. On the other hand, for triplet states with ${\beta }_1={\beta }_2$, $m>n$ and $\omega$ counts from one. The restrictions above are summarized in

Table 1. Restrictions on $m, n$ and $\omega$.

Spin Multiplicity Of The State	Relation between ${\mathit{\beta }}_1$ and ${\mathit{\beta }}_2$	Restriction on m and n	$\mathit{\omega }$ Starts from
singlet	${\beta }_1\ne {\beta }_2$
singlet	${\beta }_1={\beta }_2$	$m\ge n$	$\omega \ge 0$
triplet	${\beta }_1\ne {\beta }_2$
triplet	${\beta }_1={\beta }_2$	$m>n$	$\omega \ge 1$

.

2.2. Definitions for Electron Density

In the position space, Shannon, Rényi, and Tsallis entropies and Onicescu informational energy are defined in terms of the electron density. Nonetheless, different definitions of electron density exist in the literature and are described as follows. For a N-electron wave function $\Psi ({\stackrel{\rightharpoonup }{x}}_1,{\stackrel{\rightharpoonup }{x}}_2,\cdots ,{\stackrel{\rightharpoonup }{x}}_N)$, the spinless one-electron density [98], or N-normalized one-electron density, $\rho \left(\stackrel{\rightharpoonup }{r}\right)$ is defined as

$$\rho \left({\stackrel{\rightharpoonup }{r}}_1\right)\equiv N\int {|\Psi ({\stackrel{\rightharpoonup }{x}}_1,{\stackrel{\rightharpoonup }{x}}_2,\cdots ,{\stackrel{\rightharpoonup }{x}}_N)|}^{2}d{s}_{1}d{\stackrel{\rightharpoonup }{x}}_2\cdots d{\stackrel{\rightharpoonup }{x}}_N,$$

where ${\stackrel{\rightharpoonup }{x}}_i$ is the spin $s_i$ and space ${\stackrel{\rightharpoonup }{r}}_i=(r_i,{\theta }_i,{\varphi }_i)$ coordinates of the ith electron. Integration of this electron density equals to the number of electrons N in the system as

$$\int \rho \left(\stackrel{\rightharpoonup }{r}\right)d\stackrel{\rightharpoonup }{r}=N.$$

On the other hand, the one-normalized one-electron density $\sigma \left(\stackrel{\rightharpoonup }{r}\right)$ is defined by dividing $\rho \left(\stackrel{\rightharpoonup }{r}\right)$ by N as [99]

$$\sigma \left(\stackrel{\rightharpoonup }{r}\right)\equiv \frac{\rho \left(\stackrel{\rightharpoonup }{r}\right)}{N}.$$

Therefore, in contrast to the former choice of normalizing the electron density $\rho \left(\stackrel{\rightharpoonup }{r}\right)$ to N, $\sigma \left(\stackrel{\rightharpoonup }{r}\right)$ is normalized to one as

$$\int \sigma \left(\stackrel{\rightharpoonup }{r}\right)d\stackrel{\rightharpoonup }{r}=1.$$

This $\sigma \left(\stackrel{\rightharpoonup }{r}\right)$ is also called shape factor [99,100], shape function [99,101], unit-normalized density [27], 1-normalized density [29], or density per particle [102,103]. In this work, we computed the radial part of one-normalized one-electron density $\sigma \left(r_1\right)$ according to Calais and Lowdin as [104]

$$\begin{array}{cc}\hfill \sigma \left(r_1\right)\equiv \frac{\rho \left(r_1\right)}{N}& =\int |\psi (r_1,r_2,r_{12}){|}^{2}d{\stackrel{\rightharpoonup }{r}}_2\hfill \\ & =\frac{2\pi }{r_1}{\int }_0^{\infty }{r}_{2}d{r}_2{\int }_{|r_1-r_2|}^{r_1+r_2}{r}_{12}d{r}_{12}{|\psi (r_1,r_2,r_{12})|}^2.\hfill \end{array}$$

2.3. Information-Theoretic Quantities in the Position Space

In this work, we examined four information-theoretic quantities, including Shannon entropy (S), Rényi entropy of order $\alpha$ ($R_{\alpha }$), Tsallis entropy of order $\alpha$ ($T_{\alpha }$), and Onicescu informational energy of order $\alpha$ ($E_{\alpha }^{\mathrm{O}}$). These quantities originated from probability and information theory, and were later extended to the position space by taking the electron density as the argument. Further background details of these IT quantities are summarized in our previous work [80]. The information-theoretic quantities for one-normalized one-electron density $\sigma \left(r\right)$ are defined as

$$\begin{array}{cc}\hfill S\left[\sigma \right(r\left)\right]& \equiv -{\int }_0^{\infty }\sigma \left(r\right)ln\sigma \left(r\right)4\pi r^{2}dr,\hfill \\ \hfill R_{\alpha }\left[\sigma \left(r\right)\right]& \equiv \frac{1}{1-\alpha }ln\left[{\int }_0^{\infty }\sigma {\left(r\right)}^{\alpha }4\pi r^{2}dr\right],\hfill \\ \hfill T_{\alpha }\left[\sigma \left(r\right)\right]& \equiv \frac{1}{\alpha -1}\left[1-{\int }_0^{\infty }\sigma {\left(r\right)}^{\alpha }4\pi r^{2}dr\right],\hfill \\ \hfill E_{\alpha }^{\mathrm{O}}\left[\sigma \left(r\right)\right]& \equiv \frac{1}{\alpha -1}{\int }_0^{\infty }\sigma {\left(r\right)}^{\alpha }4\pi r^{2}dr.\hfill \end{array}$$

The integrals were solved by 600-point Gauss–Laguerre quadrature.

Although not calculated in this work, the information-theoretic quantities for N-normalized one-electron density $\rho \left(r\right)$ are defined similarly as

$$\begin{array}{cc}\hfill S\left[\rho \right(r\left)\right]& \equiv -{\int }_0^{\infty }\rho \left(r\right)ln\rho \left(r\right)4\pi r^{2}dr,\hfill \\ \hfill R_{\alpha }\left[\rho \left(r\right)\right]& \equiv \frac{1}{1-\alpha }ln\left[{\int }_0^{\infty }\rho {\left(r\right)}^{\alpha }4\pi r^{2}dr\right],\hfill \\ \hfill T_{\alpha }\left[\rho \left(r\right)\right]& \equiv \frac{1}{\alpha -1}\left[1-{\int }_0^{\infty }\rho {\left(r\right)}^{\alpha }4\pi r^{2}dr\right],\hfill \\ \hfill E_{\alpha }^{\mathrm{O}}\left[\rho \left(r\right)\right]& \equiv \frac{1}{\alpha -1}{\int }_0^{\infty }\rho {\left(r\right)}^{\alpha }4\pi r^{2}dr,\hfill \end{array}$$

Formulas for conversion between IT quantities using the two different densities are [27,29,32,55,56,57,59,72]

$$\begin{array}{cc}\hfill S\left[\rho \right(r\left)\right]& =S\left[\sigma \right(r\left)\right]N-NlnN,\hfill \\ \hfill R_{\alpha }\left[\rho \left(r\right)\right]& =R_{\alpha }\left[\sigma \left(r\right)\right]+\frac{\alpha }{1-\alpha }lnN,\hfill \\ \hfill T_{\alpha }\left[\rho \left(r\right)\right]& =T_{\alpha }\left[\sigma \left(r\right)\right]N^{\alpha }+\frac{1-N^{\alpha }}{\alpha -1},\hfill \\ \hfill E_{\alpha }^{\mathrm{O}}\left[\rho \left(r\right)\right]& =E_{\alpha }^{\mathrm{O}}\left[\sigma \left(r\right)\right]N^{\alpha }.\hfill \end{array}$$

3. Results and Discussion

All values are in atomic units. We first examined the accuracy of calculated energy to confirm the quality of wave functions used for calculating other quantities. Moreover, the self consistency of our information-theoretic quantities were examined. Then, we compared and analyzed the behavior of the IT quantities for the ground and singly excited states.

3.1. Accuracy of Presented Energy and Information-Theoretic Quantities

3.1.1. Energy E

In our previous benchmark work for the ground state helium [80], we let ${\beta }_1={\beta }_2$ and changed only one parameter. In contrast, here for singly excited states, we varied ${\beta }_1$ and ${\beta }_2$ independently (

Table 2. Variational parameters ${\beta }_1$ and ${\beta }_2$ for the ground and low-lying singly excited states of helium with size of basis set $\omega =15$.

State	${\mathit{\beta }}_1$	${\mathit{\beta }}_2$	State	${\mathit{\beta }}_1$	${\mathit{\beta }}_2$
1s1s ${}^1{S}^e$	2.00	2.00
1s2s ${}^1{S}^e$	1.19	2.80	1s2s ${}^3{S}^e$	2.09	0.55
1s3s ${}^1{S}^e$	0.74	1.94	1s3s ${}^3{S}^e$	0.78	1.90

) with size of basis set $\omega =15$ (444 terms). Our energies satisfactorily agree with other high precision results [105] (

Table 3. Energy E and Shannon entropy S for the ground and low-lying singly excited states of helium with size of basis set $\omega =15$.

State	E	S
$\mathbf{1}{\mathit{s}}^{\mathbf{2}}{}^{\mathbf{1}}{\mathit{S}}^{\mathit{e}}$	−2.903724371	2.70510285
Drake [105]	−2.9037243770341195
Lin and Ho [52]	−2.9037243768	2.7051028
Ou and Ho [11]	−2.903724371	2.70510285
Restrepo Cuartas and Sanz-Vicario [10]	−2.903605	2.705
$\mathbf{1}\mathit{s}\mathbf{2}\mathit{s}{}^{\mathbf{1}}{\mathit{S}}^{\mathit{e}}$	−2.145974042	5.49196878
Drake [105]	−2.145974046054419(6)
Lin and Ho [106]	−2.145974042
Ou and Ho [11]	−2.145974012	5.49196837
Katriel et al. [107]	−2.1459740
Restrepo Cuartas and Sanz-Vicario [10]	−2.145967	5.492
$\mathbf{1}\mathit{s}\mathbf{2}\mathit{s}{}^{\mathbf{3}}{\mathit{S}}^{\mathit{e}}$	−2.175229377	5.23597814
Drake [105]	−2.17522937823679130
Lin and Ho [106]	−2.17522937822
Katriel et al. [107]	−2.1752294
Restrepo Cuartas and Sanz-Vicario [10]	−2.175229	5.236
$\mathbf{1}\mathit{s}\mathbf{3}\mathit{s}{}^{\mathbf{1}}{\mathit{S}}^{\mathit{e}}$	−2.061271958	6.76932178
Drake [105]	−2.061271989740911(5)
Lin and Ho [106]	−2.06127196
Restrepo Cuartas and Sanz-Vicario [10]	−2.061270	6.769
$\mathbf{1}\mathit{s}\mathbf{3}\mathit{s}{}^{\mathbf{3}}{\mathit{S}}^{\mathit{e}}$	−2.068 689 066	6.605 067 28
Drake [105]	−2.068 689 067 472 457 19
Lin and Ho [106]	−2.068 689 06
Restrepo Cuartas and Sanz-Vicario [10]	−2.068 689	6.605

), demonstrating the reliability of our calculations.

3.1.2. Shannon Entropy S, Rényi Entropy $R_{\alpha }$, Tsallis Entropy $T_{\alpha }$ and Onicescu Information Energy $E_{\alpha }^{\mathrm{O}}$

Our Shannon entropy well agreed with the literature [10,11,52,80] (Table 3). Moreover, we tested the self consistency by seeing if our values satisfy the analytic relations between them as follows. Let us take the 1s2s ${}^1{S}^e$ state for example (Figure 1 and Figure 2). First, by definition, both Rényi entropy $R_{\alpha }$ and Tsallis entropy $T_{\alpha }$ recover Shannon entropy when the order $\alpha$ goes to one as

$$\begin{array}{c}\hfill \begin{array}{c}\underset{\alpha \to 1}{lim}{R}_{\alpha }=S,\hfill \\ \underset{\alpha \to 1}{lim}{T}_{\alpha }=S.\hfill \end{array}\end{array}$$

(1)

Our numerical values obey this requirement because the curves of Rényi entropy and Tsallis entropy intersect each other at the value of Shannon entropy when order goes to one (Figure 1).

Second, the sum of Tsallis entropy $T_{\alpha }$ and Onicescu informational energy $E_{\alpha }^{\mathrm{O}}$ equals to $1/(\alpha -1)$ as

$$T_{\alpha }+E_{\alpha }^{\mathrm{O}}=\frac{1}{\alpha -1}$$

(2)

This phenomenon is best seen in Figure 2. When the order $\alpha$ decreases toward 0.5, $T_{\alpha }$ increases, $E_{\alpha }^{\mathrm{O}}$ decreases, and their sum $1/(\alpha -1)$ tends to $-2$. Furthermore, when the order is close $\alpha$ to one, Onicescu informational energy diverges, but the sum of Tsallis entropy and Onicescu informational energy still equals $1/(\alpha -1)$.

Third, Rényi entropy $R_{\alpha }$, Tsallis entropy $T_{\alpha }$ and Onicescu informational energy $E_{\alpha }^{\mathrm{O}}$ are related by the following relations: [18,80]

$$\begin{array}{cc}\hfill R_{\alpha }& =\frac{ln[1+T_{\alpha }(1-\alpha )]}{1-\alpha },\hfill \\ \hfill R_{\alpha }& =\frac{ln\left[E_{\alpha }^O(\alpha -1)\right]}{1-\alpha }.\hfill \end{array}$$

Our numerical values fulfill these analytic relations between them.

3.2. Comparison of Shannon Entropy S of the Ground and Four Singly Excited States

We first analyzed the behavior of Shannon entropy, from which Rényi entropy and Tsallis entropy were generalized. As shown in Table 3, the higher is the energy, the larger is the Shannon entropy [10,11,12]. However, although this trend was observed for the presented S-type states of helium, several exceptions have been reported if the states to be compared have different total orbital angular momentum L [10].

On the other hand, Shannon entropy reflects the extent of electron delocalization [10,11,12]. In particular, for a given configuration, the electron density of the triplet state is shift toward and more concentrated to the origin than that of the singlet (Figure 3). As a result, the triplet state has a lower Shannon entropy than the singlet. Restrepo Cuartas and Sanz-Vicario called this phenomenon the Entropic Hund’s first rule [10].

3.3. Comparison of Rényi Entropy $R_{\alpha }$, Tsallis Entropy $T_{\alpha }$ and Onicescu Information Energy $E_{\alpha }^{\mathrm{O}}$ of the Ground and Four Singly Excited States

Now, we move on to Rényi entropy $R_{\alpha }$, Tsallis entropy $T_{\alpha }$, and Onicescu Information Energy $E_{\alpha }^{\mathrm{O}}$. For simplicity, we consider the singlet states or the triplet states individually, and then compare the singlet with the triplet for a given configuration in the next subsection.

Within either the series of singlets (1s${}^2$ ${}^1{S}^e$, 1s2s ${}^1{S}^e$ and 1s3s ${}^1{S}^e$ states) or the series of triplets (1s2s ${}^3{S}^e$ and 1s3s ${}^3{S}^e$ states), higher excited states had larger Rényi entropy $R_{\alpha }$ (Figure 4a) and Tsallis entropy $T_{\alpha }$ (Figure 4b), and smaller Onicescu Information Energy (Figure 4c) throughout order $\alpha$ from 0.5 to 9. Furthermore, for a given IT quantity, the curves of different configurations gradually separate when the order $\alpha$ decreases toward $0.5$. One possible explanation is that the electron distribution of higher excited states oscillated more (Figure 3), and this behavior may be highlighted at a certain range of order $\alpha$.

3.4. Singlet–Triplet Reversal of Rényi entropy $R_{\alpha }$, Tsallis entropy $T_{\alpha }$ and Onicescu Information Energy $E_{\alpha }^{\mathrm{O}}$

Here we compare the IT quantities of the singlet state with those of the corresponding triplet state. At fist glance, for a given configuration, the singlet state generally had a larger Rényi entropy, larger Tsallis entropy and smaller Onicescu Information Energy than the triplet state (Figure 4). However, this ordering was not definite and may even be reversed. To be more specific, the difference between an IT quantity of the singlet state and that of the corresponding triplet state are in Figure 5a for the 1s2s and in Figure 5b for the 1s3s configuration. The singlet–triplet reversal occurred at $1.71\le \alpha \le 2.2$ for the 1s2s and at $1.52\le \alpha \le 2.8$ for the 1s3s configuration.

Moreover, the difference between Onicescu Information Energy of the singlet state and that of the triplet state has opposite sign to the difference for Tsallis entropy (Figure 5). This behavior can be readily verified by

$$\begin{array}{c}{E}_{\alpha }^{\mathrm{O}}\left(\mathrm{singlet}\right)-E_{\alpha }^{\mathrm{O}}\left(\mathrm{triplet}\right)\hfill \\ =[\frac{1}{\alpha -1}-T_{\alpha }\left(\mathrm{singlet}\right)]-[\frac{1}{\alpha -1}-T_{\alpha }\left(\mathrm{triplet}\right)]\hfill \\ =-[T_{\alpha }\left(\mathrm{singlet}\right)-T_{\alpha }\left(\mathrm{triplet}\right)].\hfill \end{array}$$

Selected values for Rényi, Tsallis entropy and Onicescu Information Energy for the low-lying singly excited states of helium are in

Table 4. Selected values for Rényi entropy of order $\alpha$ ($R_{\alpha }$) for the low-lying singly excited states of helium with size of basis set $\omega =15$. Values of $R_1$ are set to Shannon entropy. Full data are in Table S1.

$\mathit{\alpha }$	1s2s		1s3s
	${}^{\mathbf{1}}{\mathit{S}}^{\mathit{e}}$	${}^{\mathbf{3}}{\mathit{S}}^{\mathit{e}}$	${}^{\mathbf{1}}{\mathit{S}}^{\mathit{e}}$	${}^{\mathbf{3}}{\mathit{S}}^{\mathit{e}}$
0.5	7.82736064	7.37538422	1.00515457 × 101	9.75036390
1	5.49196878	5.23597814	6.76932178	6.60506728
2	2.46576126	2.46773641	2.51349859	2.51571175
3	1.70778650	1.70278837	1.74051392	1.74026803
4	1.33674386	1.32941893	1.36502603	1.36403409
9	6.37201665 × 10−1	6.28474643 × 10−1	6.60262552 × 10−1	6.58729039 × 10−1

,

Table 5. Selected values for Tsallis entropy of order $\alpha$ ($T_{\alpha }$) for low-lying singly excited states of helium with size of basis set $\omega =15$. Values of $T_1$ are set to Shannon entropy. Full data are in Table S2.

$\mathit{\alpha }$	1s2s		1s3s
	${}^{\mathbf{1}}{\mathit{S}}^{\mathit{e}}$	${}^{\mathbf{3}}{\mathit{S}}^{\mathit{e}}$	${}^{\mathbf{1}}{\mathit{S}}^{\mathit{e}}$	${}^{\mathbf{3}}{\mathit{S}}^{\mathit{e}}$
0.5	9.81658692 × 101	7.79050690 × 101	3.02575822 × 102	2.59995972 × 102
1	5.49196878	5.23597814	6.76932178	6.60506728
2	9.15055847 × 10−1	9.15223459 × 10−1	9.19015588 × 10−1	9.19194622 × 10−1
3	4.83571213 × 10−1	4.83406163 × 10−1	4.84612118 × 10−1	4.84604549 × 10−1
4	3.27290267 × 10−1	3.27156002 × 10−1	3.27781851 × 10−1	3.27765306 × 10−1
9	1.24236085 × 10−1	1.24180846 × 10−1	1.24364781 × 10−1	1.24356940 × 10−1

and

Table 6. Selected values for Onicescu Information Energy of order $\alpha$ ($E_{\alpha }^{\mathrm{O}}$) for low-lying singly excited states of helium with size of basis set $\omega =15$. $E_{\alpha }^{\mathrm{O}}$ is undefined for $\alpha =1$. Full data are in Table S3.

$\mathit{\alpha }$	1s2s		1s3s
	${}^{\mathbf{1}}{\mathit{S}}^{\mathit{e}}$	${}^{\mathbf{3}}{\mathit{S}}^{\mathit{e}}$	${}^{\mathbf{1}}{\mathit{S}}^{\mathit{e}}$	${}^{\mathbf{3}}{\mathit{S}}^{\mathit{e}}$
0.5	−1.00165869 × 102	−7.99050690 × 101	−3.04575822 × 102	−2.61995972 × 102
2	8.49441529 × 10−2	8.47765409 × 10−2	8.09844111 × 10−2	8.08053779 × 10−2
3	1.64287869 × 10−2	1.65938368 × 10−2	1.53878810 × 10−2	1.53954502 × 10−2
4	6.04306538 × 10−3	6.17733033 × 10−3	5.55148184 × 10−3	5.56802665 × 10−3
9	7.63914364 × 10−4	8.19153809 × 10−4	6.35218223 × 10−4	6.43059145 × 10−4

, respectively, and the full data are in Tables S1–S3.

Additionally, singlet–triplet reversal has also been reported for the interacting repulsive potential energy between two electrons in singly-excited states in quantum dots under certain confinement conditions [107,108], and in singly-excited states of helium-like ions under screened Coulomb interacting potentials with certain screening strengths and atomic numbers Z [109].

4. Conclusions

Generally, a higher excited state of helium had a larger Shannon entropy, larger Rényi entropy of order $\alpha$, larger Tsallis entropy of order $\alpha$, and smaller Onicescu information energy of order $\alpha$. This behavior possibly reflects the more delocalized nature of electron distribution of higher excited states. However, for Rényi entropy, Tsallis entropy and Onicescu information energy, this trend was not definite and singlet–triplet reversal occurred at a certain range of order $\alpha$. In the future, the knowledge of bounds and asymptotic analysis of these quantities may provide additional understanding of this behavior [110,111,112,113,114,115,116].

On the other hand, these IT quantities not only reflect the extent of electron delocalization, as discussed in this article, but also have numerous applications. For instance, Rényi entropy has been widely used to examine quantum entanglement [117,118,119,120,121,122,123]. Furthermore, Tsallis statistics is relevant to certain systems of ultracold atoms [124,125,126,127]. Moreover, these IT quantities are applied to investigate electron correlation [30,31,33,39,72,128,129,130,131,132,133,134,135,136] and various molecular properties as well [28,36,45,48,49,59,60,61,72,73]. Calculated with the highly correlated Hylleraas wave functions, the results in this work then serve as a useful and reliable reference for the various applications mentioned above.

As for a theoretical application, our present calculations would also play a role to test the Collins Conjecture [137]—correlation energy (the difference between the exact energy and Hartree–Fock energy) is directly proportional to entropy—when one could extend the calculations to other helium-like ions. Thus, such a conjecture was tested by Dehesa and coworkers [138,139] to be valid for von Neumann entropy in helium iso-electronic sequence. It would be of interest to test the Collins Conjecture for Shannon, Renyi, or Tsallis entropy to be valid for helium sequence using highly correlated wave functions.

Finally, it should be mentioned that, to the best of our knowledge, there is no formula relating entropy to the quantum number of any given excited state, in a manner similar to the formula relating quantum defect to the effective quantum number for such an excited state. Nevertheless, this is an interesting issue and it is worthwhile to explore further in future investigations.