Shannon Entropy of Chemical Elements

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Shannon Entropy of Chemical Elements

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Szymon Łukaszyk^*

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16 November 2023

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17 November 2023

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Abstract

Hund's rule of maximum multiplicity is a powerful empirical tool for determining the electron population of electronic shells. It was recently discovered that it is also the rule of minimum Shannon entropy. This study extends these findings to the Aufbau rule and discusses them from the perspective of emergent dimensionality, providing a monotonically increasing entropy function. We observed that some elements that violate the Aufbau rule have the same entropies in the actual and Aufbau electron configurations. On the other hand, lower entropies of all actual elements' configurations are associated with higher or equal numbers of unpaired electrons in these configurations compared to the Aufbau configurations. The only exception to this rule is palladium.

Keywords:

Subject: Physical Sciences - Atomic and Molecular Physics

1. Introduction

It was recently reported [1] that the real driving force behind Hund’s rule of maximum spin multiplicity appears to be the second law of infodynamics [2]

\frac{d S_{i n f o}}{d t} \leq 0,

(1)

where

S_{i n f o}

in this time derivative is the information entropy proportional to Shannon entropy

H = \sum_{k = 1}^{n} p_{k} {log}_{b} (\frac{1}{p_{k}}), b \in R_{+} ∖ {1},

(2)

of a discrete random variable that attains maximum

{log}_{b} (n)

if the events are equiprobable (i.e., if the probabilities

p_{k} = 1 / n

) and vanishes for certain and impossible events1 In other words, electron populations within an orbital minimize Shannon entropy (2).

It is now generally accepted [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] that information in the universe evolves, decreasing the information entropy

S_{i n f o}

. Assuming that the total entropy of the universe S is constant and is the sum of the information entropy and the physical entropy

S_{p h y s}

, we obtain [1]

\frac{d S_{i n f o}}{d t} + \frac{d S_{p h y s}}{d t} = 0 .

(3)

This study extends the findings of [1] to the Aufbau rule and discusses them from the perspective of emergent dimensionality [5,9,10,11,12,13,14,17]. Other applications of Shannon entropy in chemistry [18,19,20,21,22] and in particular in the context of hydrogen [23,24,25] are known from the state of the art.

2. Hund’s rule and Shannon entropy

A simple but inventive procedure for determining Shannon entropy of an electron population was disclosed in [1]. Any set of N electrons satisfies

N = N_{↑} + N_{↓}

, where

N_{↑}

and

N_{↓}

denote, respectively, the number of up- and down-spin electrons. Thus, the probabilities of finding ↑ and ↓ electrons within the set of N electrons are

p_{↑} = N_{↑} / N

and

p_{↓} = N_{↓} / N

Definition 1

(Electron set entropy). An electron set entropy is

\begin{matrix} H & = - p_{↑} {log}_{b} (p_{↑}) - p_{↓} {log}_{b} (p_{↓}) = \\ = {log}_{b} (N) - \frac{N_{↑}}{N} {log}_{b} (N_{↑}) - \frac{N_{↓}}{N} {log}_{b} (N_{↓}) . \end{matrix}

(4)

This definition was disclosed in [1] but is given here for clarity Electrons are fermions, so electron sets populating chemical elements’ orbitals must satisfy the Pauli exclusion principle. For the s orbital, for example, which can accommodate a maximum of

N = 2

electrons, the corresponding probabilities are

p_{↑} = 1 \cup p_{↓} = 1

if the orbital accommodates one electron and

p_{↑} = p_{↓} = 1 / 2

if the orbital accommodates two electrons. Thus, possible electron set entropies (4) are

H = {0, {log}_{b} (2)}

. The author of [1] assumes that the electron set entropies of states s¹ and s² are the same, but this assumption is unjustified, as we shall discuss later.

In general, any orbital can store up to

N_{m a x} = 4 ℓ + 2, ℓ \in N_{0}

(5)

electrons, where

ℓ = 0, 1, \dots, n - 1

is the angular momentum quantum number and n is the principal quantum number. Thus, the situation becomes more diverse for orbitals with larger ℓ as multiple sets with different electron set entropies and numbers of unpaired electrons are possible, as shown in Figure 1, Figure 2, Figure 3 and Figure 4. In general, the number of distinct electron sets allowed by the Pauli exclusion principle and available for an orbital is given by triangular numbers as

T = \frac{1}{2} \{\begin{matrix} ⌊\frac{N + 2}{2}⌋ (⌊\frac{N + 2}{2}⌋ + 1) & 1 \leq N \leq \frac{N_{m a x}}{2} \\ ⌊\frac{N_{m a x} - N + 2}{2}⌋ (⌊\frac{N_{m a x} - N + 2}{2}⌋ + 1) & \frac{N_{m a x}}{2} \leq N \leq N_{m a x} \end{matrix},

(6)

and therefore for

N = {1, N_{m a x} - 1, N_{m a x}}

, only one distinct electron set is possible. If there is more than one electron set available, nature selects this electron set that maximizes the number of unpaired electrons, so the electron set entropy is

\begin{matrix} H = 0 & iff 1 \leq N \leq \frac{N_{m a x}}{2}, \\ 0 < H < l o g_{b} (2) & iff \frac{N_{m a x}}{2} + 1 \leq N \leq N_{m a x} - 1, \\ H = l o g_{b} (2) & iff N = N_{m a x} . \end{matrix}

(7)

However, as shown in [1], for

N_{m a x} / 2 + 1 \leq N \leq N_{m a x} - 2}

, nature also selects this electron population among the allowed ones, which minimizes the orbital Shannon entropy. This rule of populating the sublevels can be stated in the simple theorem illustrated in Figure 5.

Theorem 1

(Orbital entropy). For any orbital capable of storing

N_{m a x}

(5) electrons, storing N electrons, and populated to maximize the number of unpaired electrons (i.e. according to Hund’s rule), the orbital Shannon entropy amounts

H (N) = \{\begin{matrix} 0 & N \leq N_{m a x} / 2 \\ \begin{matrix} {log}_{b} (N) - \frac{N_{m a x} / 2}{N} {log}_{b} (\frac{N_{m a x}}{2}) \\ - (\frac{N - N_{m a x} / 2}{N}) {log}_{b} (N - \frac{N_{m a x}}{2}) \end{matrix} & N \geq N_{m a x} / 2 \end{matrix},

(8)

where for

N_{m a x} / 2 + 1 \leq N < N_{m a x}

we call the entropy intermediate and for

N = N_{m a x}

pure.

Proof.

According to Hund’s rule, for

N \leq N_{m a x} / 2

the number of unpaired electrons is equal to

L = N

, while for

N > N_{m a x} / 2

is equal to

L = N_{m a x} - N

. For

N \leq N_{m a x} / 2

electrons can freely populate available

N_{m a x} / 2

sublevels and therefore

H = \frac{N_{↑}}{N} {log}_{b} (\frac{N_{↑}}{N}) = {log}_{b} (1) = 0

(the same for ↓). For

N > N_{m a x} / 2

the electrons will begin to repopulate the available

N_{m a x} / 2

sublevels following the Pauli exclusion principle up to

N = N_{m a x}

, where

H = {log}_{b} (N) - {log}_{b} (\frac{N}{2}) = {log}_{b} (2)

. This completes the proof. □

Some researchers postulate that certain elements are exceptions to Hund’s rule. Chromium, for example, having the atomic number

Z = 24

is between vanadium (

Z = 23

, electron configuration [Ar]3d³4s²) and manganese (

Z = 25

, [Ar]3d⁵4s²) within the periodic table of elements. Thus, it should have electron configuration [Ar]3d⁴4s², following Hund’s rule, but instead, it has [Ar]3d⁵4s¹, as one electron from 4s moves to 3d to make it more stable. But it is not Hund’s rule that is violated. It is the Aufbau rule. Hund’s rule governs the electron population of a solitary orbital only.

3. Aufbau rule and Shannon entropy

The Aufbau or Madelung energy ordering rule is another powerful empirical tool for predicting the electron configurations of chemical elements corresponding to the ground state. It correctly predicts the electron configurations of most of the elements. However, about twenty chemical elements (with only two nondoubleton sets of consecutive ones) violate the Aufbau rule, leading to intriguing exceptions and anomalies. Chromium and copper violations are attributed to a delicate balance of electron-electron repulsion and the energy gap between the 3d and 4s orbitals. Only palladium exhibits double electron promotion with all ten electrons filling the 4d orbital and therefore is often called a "double anomaly". Palladium also has the smallest differential entropy [19]. This exceptional behavior is attributed to the compactness of 3d orbitals and complex electron interactions. Furthermore, there are no chemical elements that have orbital f⁸ in their electron configurations, although the Aufbau rule predicts f8 for gadolinium and curium (

Z = {64, 96}

). The elements that violate the Aufbau rule are listed in Table 1, which shows their actual electron configurations and the configurations predicted by the Aufbau rule.

We note that for

Z \geq 105

, actual ground states are predicted. Thus, other elements, such as darmstadtium (

Z = 110

) and roentgenium (

Z = 111

) may also violate the Aufbau rule. Furthermore, the electron configuration of nickel (

Z = 28

) is disputed, but we included it in the list of exceptions.

Similarly to orbitals, we can calculate the number of unpaired electrons of a chemical element, as shown in Figure 6 for

0 \leq Z \leq 118

. We can also calculate the Shannon entropy of a chemical element. In two ways.

First, we can consider an element as an electron set and calculate its entropy using Definition 1, as shown in Figure 7. For example, the electron configuration of oxygen is [He]2s²2p⁴→ 2·s²+p⁴ and thus

N = 8

N_{↑} = 5

N_{↓} = 3

, and

H_{E} = \frac{5}{8} {log}_{2} (\frac{8}{5}) + \frac{3}{8} {log}_{2} (\frac{8}{3}) \approx 0.9544

However, the entropy according to this definition does not increase monotonically and, in addition, palladium has a lower entropy in the Aufbau configuration. Therefore, noting that the entropies of independent systems are additive quantities, we provide another, monotonically increasing entropy.

Definition 2.

An element’s Shannon entropy is the sum of the orbital entropies (8) in the electron configuration of this element, neglecting the principal quantum number n.

The principal quantum number represents the average distance of electrons from the nucleus, and therefore its meaning in the context of the emergent dimensionality is restricted. Only the Pythagorean theorem introduces negative distances (as non-principal square roots), and distance non-negativity is not an axiom of diffuse metrics [26], such as the Łukaszyk–Karmowski metric [9].

For example, for oxygen is [He]2s²2p⁴→ s²s²p⁴. Since

H (s^{2}) = {log}_{b} (2)

, and

H (p^{4})

is given by the relation (8), the entropy for oxygen is equal to

\begin{matrix} H = H (s^{2}) + H (s^{2}) + H (p^{4}) = \\ = {log}_{b} (2) + {log}_{b} (2) + {log}_{b} (4) - \frac{3}{4} {log}_{b} (3) - \frac{4 - 3}{4} {log}_{b} (4 - 3) = \\ = 4 {log}_{b} (2) - \frac{3}{4} {log}_{b} (3), \end{matrix}

(9)

and the number of unpaired electrons

L = 4

. Similarly, for chromium (

Z = 24

)

[Ar] 3 d^{5} 4 s^{1} \to 3 \times s^{2} + 2 \times p^{6} + d^{5} + s^{1}

, so its entropy is

H = 3 {log}_{b} (2) + 2 {log}_{b} (2) + 0 + 0 = 5 {log}_{b} (2)

and the number of unpaired electrons

L = 5 + 1 = 6

Elements’ entropies

H {(Z)}_{act}

and

H {(Z)}_{Auf}

for

0 \leq Z \leq 108

(Hassium is the heaviest element with known properties) are shown in Figure 8 based on actual electron configurations and configurations obtained using the Aufbau rule. Furthermore, Table 1 lists the entropies and numbers of unpaired electrons of the Aufbau rule violators.

The Aufbau rule reflects the periodicity of the elements within the windows defined by the noble gases’ atomic numbers, as shown in Figure 6, Figure 7 and Figure 8. Subsequent atomic orbitals form the Janet’s sequence (OEIS A167268) that can be generated by the relation

N (l) = 4 ℓ (l) + 2

(5) with

ℓ (l) = {0, 0, 1, 0, 1, 0, 2, 1, 0, 2, 1, 0, \dots}

given by

ℓ (l) = ⌊\frac{1}{4} {⌈\sqrt{4 l}⌉}^{2}⌋ - l, l \in N, (OEIS A 216607),

(10)

where

⌊x⌋

and

⌈x⌉

are floor and ceiling functions2.

The atomic numbers of noble gases

Z_{NG}

form the sequence OEIS A018227 that can be generated using formulas

Z_{NG +} (k) = Z_{NG} {(k - 1)}_{+} + A_{+} (k), or

(11)

Z_{NG +} (k) = \frac{2 k^{3} + 12 k^{2} + 25 k - 6 + {(- 1)}^{k} (3 k + 6)}{12},

(12)

where

A (k)

is the sum of the elements of a given period of the periodic table (OEIS A093907) that can be generated using formulas

\begin{matrix} A_{+} (k) & = Z_{NG} (k) - Z_{NG} (k - 1), or \\ A_{+} (k) & = 2 {(⌊\frac{k}{2}⌋ + 1)}^{2}, or \\ A_{+} (k) & = \frac{{(2 k + 3 + {(- 1)}^{k})}^{2}}{8}, \\ A_{+} (k) & = \frac{1}{2} (k^{2} + \frac{6 k + 5}{2} + {(- 1)}^{k} \frac{2 k + 3}{2}) . \end{matrix}

(13)

Figure 9. The periodic table of chemical elements arranged according to their entropy. The Aufbau rule violators (red symbols). Spin-only-contributors (yellow frames). Entropy-only-contributors (green frames). Spin-and-entropy-contributors (blue frames).

Z = {64, 96}

are exceptional spin-only-contributors.

Z = {29, 47, 79}

are exceptional spin-and-entropy-contributors having pure entropy.

Z = 46

is an exceptional entropy-only-contributor, which is not a spin-contributor.

Z = {64, 96}

are exceptional spin-only-contributors.

Z = {29, 47, 79}

are exceptional spin-and-entropy-contributors having pure entropy.

Z = 46

is an exceptional entropy-only-contributor, which is not a spin-contributor.

However, the sequence (10) is also valid for

l \in Z

, as shown in Table 2. In particular

N (l) = 2

for

\begin{matrix} l = \{\begin{matrix} {⌈\frac{k}{2}⌉}^{2} \cap ⌈\frac{k}{2}⌉ (⌈\frac{k}{2}⌉ \pm 1) & iff k \geq 0 \\ - {⌈\frac{k}{2}⌉}^{2} & iff k < 0 \end{matrix}, \end{matrix}

(14)

and thus, 2’s generated by positive l’s are denser, as these l’s include pronic numbers (OEIS A002378) and

k (k \pm 1) = \hat{k} (\hat{k} \mp 1) \geq 0

for

\hat{k} - k

. A mirror situation occurs for imaginary

k = a i \in I

a \in Z

, in which case

N (l) = 2

for

\begin{matrix} l = \{\begin{matrix} - {⌈\frac{a}{2}⌉}^{2} \cap - ⌈\frac{a}{2}⌉ (⌈\frac{a}{2}⌉ \pm 1) & iff a \geq 0 \\ {⌈\frac{a}{2}⌉}^{2} & iff a < 0 \end{matrix} . \end{matrix}

(15)

and 2’s generated by negative l’s are denser, as shown in Figure 10.

Figure 10. Janet’s sequence (red) and its reflection (green).

Furthermore, although the sequences (13) are equivalent to each other

\forall k \in Z

their elements no longer represent the sums of the elements of a given period of the periodic table if

k < 0

. Also, the sequences (11) and (12) are different for

k < 0

, as shown in Figure 11.

Figure 11. The atomic numbers of noble gases

Z_{NG}

generated by the OEIS formula (12) (red) and recurrence formula (11) (blue).

Figure 11. The atomic numbers of noble gases

Z_{NG}

generated by the OEIS formula (12) (red) and recurrence formula (11) (blue).

A simple conclusion of Theorem 1 (cf. Figure 5) is that the entries s¹, p¹-p³, d¹-d⁵, f¹-f⁷, etc. of the configuration of an element do not contribute to the entropy of the element, but only to the number of unpaired electrons; entries s², p⁶, d¹⁰, f¹⁴, etc., contribute with

{log}_{b} (2)

to the entropy but do not contribute to the number of unpaired electrons; and the remaining entries contribute both to entropy and to the number of unpaired electrons. Therefore, we can introduce the following definitions.

Definition 3.

An entropy-only-contributor is the element that has entropy

H = l {log}_{b} (2)

, where

l \in N

and the number of unpaired electrons

L = 0

These elements are defined by

Z_{EOC} (m) = 4 \sum_{l = 1}^{m} ℓ (l) + 2 m = 4 \sum_{l = 1}^{m} ⌊\frac{{⌈\sqrt{4 l}⌉}^{2}}{4}⌋ - 2 m^{2},

(16)

where

ℓ (l)

is given by (10) (cf. Figure 6) and l increases from

l = 1

for

Z = 2

l = 19

for

Z = 118

, etc.

Definition 4.

A spin-only-contributor is the element that has the number of unpaired electrons greater than the preceding element and entropy

H = l {log}_{b} (2)

, where

l \in N

These elements range between

Z_{EOC} (m)

(16) and

\begin{matrix} max (Z_{SOC} (m)) = \\ = 4 \sum_{l = 1}^{m} ⌊\frac{{⌈\sqrt{4 l}⌉}^{2}}{4}⌋ - 2 m^{2} - 2 m - 1 + 2 ⌊\frac{{⌈\sqrt{4 (m + 1)}⌉}^{2}}{4}⌋, \end{matrix}

(17)

Definition 5.

A spin-and-entropy-contributor is the element that has the number of unpaired electrons lower than the preceding element and entropy

H \neq l {log}_{b} (2)

, where

l \in N

These elements range between

max (Z_{SOC} (m))

(17) and

Z_{EOC} (m)

(16) and are responsible for soft transitions between the entropy steps shown in Figure 8 and Figure 13.

Thus, for the Aufbau electron configurations each element is either a spin-only-contributor, an entropy-only-contributor, or a spin-and-entropy-contributor. There are only four exceptions to this rule for actual electron configurations, which occur for orbital d, namely

Z = {29, 47, 79}

which are spin-and-entropy-contributors having entropy equal to

l {log}_{b} (2)

. For example, d¹⁰ in silver’s configuration [Kr]4d¹⁰5s¹ makes it entropy equal to

H (47) = 9 {log}_{b} (2)

(krypton has entropy

H (36) = 8 {log}_{b} (2)

) and s¹ makes it a spin-contributor. Furthermore, palladium is an entropy-only-contributor instead of being a spin-and-entropy-contributor and

Z = {64, 96}

(orbitals f) are spin-only-contributors instead of being spin-and-entropy-contributors. Furthermore, all Aufbau rule non-singleton violators, have entropies higher or equal than those of the preceding exceptional elements. These results strengthen the meaning of the Aufbau rule.

The fact that hydrogen is a spin-only-contributor, while helium is an entropy-only-contributor suggests a different arrangement of the periodic table of elements shown in Figure 9, where period terminating entropy-only-contributors define the separations of the elements. The outline of the extended periodic table is shown in Figure 12 for

Z \leq 721762

. From the 3^rd period, the left-hand rows of this periodic table increase by four every two periods

{7, 11, 15, \dots}

, which represents the linear part of the curve.

Figure 12. The outline of the periodic table of elements arranged to isolate pure-entropy-contributors for

Z \leq 721762

(cf. Figure 9).

Figure 12. The outline of the periodic table of elements arranged to isolate pure-entropy-contributors for

Z \leq 721762

(cf. Figure 9).

Figure 13. Shannon entropy of elements

1 \leq Z \leq 361

as given by Hund’s and Aufbau rules (red), and its approximations

{(Z - 1)}^{5 / 8}

(green) and

{(Z - 1)}^{ln (2)}

(blue). Inset for

1 \leq Z \leq 721762

Figure 13. Shannon entropy of elements

1 \leq Z \leq 361

as given by Hund’s and Aufbau rules (red), and its approximations

{(Z - 1)}^{5 / 8}

(green) and

{(Z - 1)}^{ln (2)}

(blue). Inset for

1 \leq Z \leq 721762

Figure 13 shows the element entropies for

l = 29

, yielding

Z = 218

. This value is well beyond any considerable physical limit, although elements as heavy as

Z = 164

(or even

Z = 204

) are predicted to exist in nature [27] due to doubly-magic nuclei [27,28]. However, the Compton wavelength of an atom becomes smaller than the Planck length for

Z > 238

[10,14], which is physically implausible because the Planck area is the smallest area required to encode one bit of information [10,29,30,31]. Figure 13 illustrates a trend of increasing the entropy of the elements. We conjecture that this trend continues until the element’s half-life is equal to Planck time and for

b = 2

is bounded by

{(Z - 1)}^{ln (2)}

(cf. the blue curve in the inset of Figure 13).

4. Conclusions

We observed that the following holds for the 21 chemical elements violating the Aufbau rule:

for 13 of them, the entropy is lower for the actual and Aufbau configurations; the remaining ones have the same entropies in actual and Aufbau configurations;
6 of them have the number of unpaired electrons different from that predicted by the Aufbau rule, including palladium, the only element with a lower number of unpaired electrons;
the first nondoubleton set $44 \leq Z \leq 47$ contains four elements having lower entropies and the same numbers of unpaired electrons with the exception of $Z = 46$ having lower number of unpaired electrons;
the second nondoubleton set $89 \leq Z \leq 93$ contains five elements having the same entropies and numbers of unpaired electrons.

Furthermore, for Aufbau electron configurations, each element contributes either to the number of unpaired electrons but not to entropy, to entropy with

{log}_{b} (2)

but not to the number of unpaired electrons, or both to the number of unpaired electrons and to the intermediate entropy, wherein for actual configurations only

Z = {29, 47, 79}

violate this rule. Overall, these results strengthen the meaning of the Aufbau rule and invite further research. Perhaps the Aufbau rule violators can be explained within the framework of assembly theory [8,15,16].

We finally note that the Shannon entropy H of hydrogen is zero, not

{log}_{b} (2)

. In emergent dimensionality [5,9,10,11,12,13,14,17], neither one nor two electrons exist. Therefore, s¹ is populated before s² not because one electron is less than two electrons, but because the Shannon entropy for one electron

H = 0

is lower than the Shannon entropy

H = {log}_{b} (2)

for two electrons.

Acknowledgments

I truly thank my godson, Wawrzyniec Bieniawski, for the state-of-the-art input. I thank Andrzej Tomski for finding the analytical formula for the number of distinct electron sets available for an orbital (6). I thank my partner and friend, Renata Sobajda, for her prayers. I truly thank my wife Magdalena Bartocha for her unwavering support and motivation.

Data Availability Statement

The public repository for the code written in MATLAB computational environment is given under the link https://github.com/szluk/Evolution_of_Information (accessed on November 03, 2023).

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1	In the latter case $0 ln (1 / 0)$ is not defined. It is, by convention, taken as 0.
2	Or alternatively by OEIS A167268.

Figure 1. Orbital p.

p^{k} \leftrightarrow {k, 0}

, for

1 \leq k \leq 3

p^{4} \leftrightarrow {2, {log}_{2} (4) - 3 {log}_{2} (3) / 4}

p^{5} \leftrightarrow {1, {log}_{2} (5) - 3 {log}_{2} (3) / 5 - 2 {log}_{2} (2) / 5}

p^{6} \leftrightarrow {0, {log}_{2} (2)}

Figure 1. Orbital p.

p^{k} \leftrightarrow {k, 0}

, for

1 \leq k \leq 3

p^{4} \leftrightarrow {2, {log}_{2} (4) - 3 {log}_{2} (3) / 4}

p^{5} \leftrightarrow {1, {log}_{2} (5) - 3 {log}_{2} (3) / 5 - 2 {log}_{2} (2) / 5}

p^{6} \leftrightarrow {0, {log}_{2} (2)}

Figure 2. Orbital d.

d^{k} \leftrightarrow {k, 0}

, for

1 \leq k \leq 5

d^{10} \leftrightarrow {0, {log}_{2} (2)}

Figure 2. Orbital d.

d^{k} \leftrightarrow {k, 0}

, for

1 \leq k \leq 5

d^{10} \leftrightarrow {0, {log}_{2} (2)}

Figure 3. Orbital f.

f^{k} \leftrightarrow {k, 0}

, for

1 \leq k \leq 7

f^{14} \leftrightarrow {0, {log}_{2} (2)}

Figure 3. Orbital f.

f^{k} \leftrightarrow {k, 0}

, for

1 \leq k \leq 7

f^{14} \leftrightarrow {0, {log}_{2} (2)}

Figure 4. Orbital g.

g^{k} \leftrightarrow {k, 0}

, for

1 \leq k \leq 9

g^{18} \leftrightarrow {0, {log}_{2} (2)}

Figure 4. Orbital g.

g^{k} \leftrightarrow {k, 0}

, for

1 \leq k \leq 9

g^{18} \leftrightarrow {0, {log}_{2} (2)}

Figure 5. Orbital g. The number of unpaired electrons L (red, rescaled) and associated orbital entropy H (green) as functions of the number of the orbital electrons N.

Figure 6. Number of unpaired electrons of chemical elements (green) showing the Aufbau rule violators (red) and elements of the same L (blue). Five elements (

Z = {24, 41, 42, 64, 96}

) have L higher than that predicted by the Aufbau rule. Only palladium (

Z = 46

) has a lower L.

Figure 6. Number of unpaired electrons of chemical elements (green) showing the Aufbau rule violators (red) and elements of the same L (blue). Five elements (

Z = {24, 41, 42, 64, 96}

) have L higher than that predicted by the Aufbau rule. Only palladium (

Z = 46

) has a lower L.

Figure 7. Electron set entropy of chemical elements (green curve) showing the Aufbau rule violators (red x) and elements of the same entropy (blue +). Five elements (

Z = {24, 41, 42, 64, 96}

) have electron entropy lower than that predicted by the Aufbau rule. This function does not increase monotonically and

H_{E} {(46)}_{act} > H_{E} {(46)}_{Auf}

Figure 7. Electron set entropy of chemical elements (green curve) showing the Aufbau rule violators (red x) and elements of the same entropy (blue +). Five elements (

Z = {24, 41, 42, 64, 96}

) have electron entropy lower than that predicted by the Aufbau rule. This function does not increase monotonically and

H_{E} {(46)}_{act} > H_{E} {(46)}_{Auf}

Figure 8. Shannon entropy of chemical elements (green) showing Aufbau rule violators (red) and elements of the same entropy (blue). Thirteen elements (

Z = {24, 28, 29, 41, 42, 44, 45, 46, 47, 64, 78, 79, 96}

including palladium) have entropy lower than that predicted by the Aufbau rule. Furthermore,

H {(Z)}_{act} \leq H {(Z)}_{Auf}, \forall Z

Figure 8. Shannon entropy of chemical elements (green) showing Aufbau rule violators (red) and elements of the same entropy (blue). Thirteen elements (

Z = {24, 28, 29, 41, 42, 44, 45, 46, 47, 64, 78, 79, 96}

including palladium) have entropy lower than that predicted by the Aufbau rule. Furthermore,

H {(Z)}_{act} \leq H {(Z)}_{Auf}, \forall Z

Table 1. Chemical elements violating Aufbau rule.

		Electron configuration		Unpaired electrons		Shannon entropy		Electron set entropy
	Z	actual	Aufbau	$L {(Z)}_{act}$	$L {(Z)}_{Auf}$	$H {(Z)}_{act}$	$H {(Z)}_{Auf}$	$H_{E} {(Z)}_{act}$	$H_{E} {(Z)}_{Auf}$
Cr	24	[Ar]3d⁵4s¹	[Ar]3d⁴4s²	6	4	$5 {log}_{b} (2)$	$6 {log}_{b} (2)$	0.9544	0.9799
Ni	28	[Ar]3d⁸4s² (or [Ar]3d⁹4s¹)	[Ar]3d⁸4s²	2		${log}_{b} (2^{37 / 9} 3^{2} 5^{- 5 / 9})$	${log}_{b} (2^{9} 3^{- 3 / 8} 5^{- 5 / 8})$	0.9963
Cu	29	[Ar]3d¹⁰4s¹	[Ar]3d⁹4s²	1		$6 {log}_{b} (2)$	${log}_{b} (2^{46 / 9} 3^{2} 5^{- 5 / 9})$	0.9991
Nb	41	[Kr]4d⁴5s¹	[Kr]4d³5s²	5	3	$8 {log}_{b} (2)$	$9 {log}_{b} (2)$	0.9892	0.9961
Mo	42	[Kr]4d⁵5s¹	[Kr]4d⁴5s²	6	4	$8 {log}_{b} (2)$	$9 {log}_{b} (2)$	0.9852	0.9934
Ru	44	[Kr]4d⁷5s¹	[Kr]4d⁶5s²	4		${log}_{b} (2^{54 / 7} 5^{- 5 / 7} 7)$	${log}_{b} (2^{10} 3 5^{- 5 / 6})$	0.9940
Rh	45	[Kr]4d⁸5s¹	[Kr]4d⁷5s²	3		${log}_{b} (2^{11} 3^{- 3 / 8} 5^{- 5 / 8})$	${log}_{b} (2^{61 / 7} 5^{- 5 / 7} 7)$	0.9968
Pd	46	[Kr]4d¹⁰	[Kr]4d⁸5s²	0	2	$9 {log}_{b} (2)$	${log}_{b} (2^{12} 3^{- 3 / 8} 5^{- 5 / 8})$	1	0.9986
Ag	47	[Kr]4d¹⁰5s¹	[Kr]4d⁹5s²	1		$9 {log}_{b} (2)$	${log}_{b} (2^{73 / 9} 3^{2} 5^{- 5 / 9})$	0.9997
La	57	[Xe]5d¹6s²	[Xe]4f¹6s²	1		$12 {log}_{b} (2)$		0.9998
Ce	58	[Xe]4f¹5d¹6s²	[Xe]4f²6s²	2		$12 {log}_{b} (2)$		0.9991
Gd	64	[Xe]4f⁷5d¹6s²	[Xe]4f⁸6s²	8	6	$12 {log}_{b} (2)$	${log}_{b} (2^{15} 7^{- 7 / 8})$	0.9887	0.9937
Pt	78	[Xe]4f¹⁴5d⁹6s¹	[Xe]4f¹⁴5d⁸6s²	2		${log}_{b} (2^{100 / 9} 3^{2} 5^{- 5 / 9})$	${log}_{b} (2^{16} 3^{- 3 / 8} 5^{- 5 / 8})$	0.9995
Au	79	[Xe]4f¹⁴5d¹⁰6s¹	[Xe]4f¹⁴5d⁹6s²	1		$13 {log}_{b} (2)$	${log}_{b} (2^{109 / 9} 3^{2} 5^{- 5 / 9})$	0.9999
Ac	89	[Rn]6d¹7s²	[Rn]5f¹7s²	1		$16 {log}_{b} (2)$		0.9999
Th	90	[Rn]6d²7s²	[Rn]5f²7s²	2		$16 {log}_{b} (2)$		0.9996
Pa	91	[Rn]5f²6d¹7s²	[Rn]5f³7s²	3		$16 {log}_{b} (2)$		0.9992
U	92	[Rn]5f³6d¹7s²	[Rn]5f⁴7s²	4		$16 {log}_{b} (2)$		0.9986
Np	93	[Rn]5f⁴6d¹7s²	[Rn]5f⁵7s²	5		$16 {log}_{b} (2)$		0.9979
Cm	96	[Rn]5f⁷6d¹7s²	[Rn]5f⁸7s²	8	6	$16 {log}_{b} (2)$	${log}_{b} (2^{19} 7^{- 7 / 8})$	0.9950	0.9972
Lf	103	[Rn]5f¹⁴7s²7p¹	[Rn]5f¹⁴6d¹7s²	1		$17 {log}_{b} (2)$		0.9999

Z - the same H and L. Z - lower H, higher L. Z - palladium: higher

H_{E}

, lower H, lower L. Electron set entropy calculated for

l o g_{2}

Table 2. Extended Aufbau rule. Periodic table for

1 \leq k \leq 7

Table 2. Extended Aufbau rule. Periodic table for

1 \leq k \leq 7

k	$l_{m i n}$	$l_{m a x}$	$N (l) = 4 ⌊\frac{1}{4} {⌈\sqrt{4 l}⌉}^{2}⌋ - 4 l + 2$						$A (k)$	$Z_{NG} (k)$
$- 9$	$- 25$	$- 21$	2		$- 2$	$- 6$	$- 10$	$- 14$	$- 30$	10
$- 8$	$- 20$	$- 17$			$- 2$	$- 6$	$- 10$	$- 14$	$- 32$	$- 40$
$- 7$	$- 16$	$- 13$	2		$- 2$	$- 6$	$- 10$		$- 16$	8
$- 6$	$- 12$	$- 10$			$- 2$	$- 6$	$- 10$		$- 18$	$- 24$
$- 5$	$- 9$	$- 7$	2		$- 2$	$- 6$			$- 6$	6
$- 4$	$- 6$	$- 5$			$- 2$	$- 6$			$- 8$	$- 12$
$- 3$	$- 4$	$- 3$	2		$- 2$				0	4
$- 2$	$- 2$				$- 2$				$- 2$	$- 4$
$- 1$	$- 1$		2						2	2
0	0		2						2	0
1	1		2						2	2
2	2	3	2					6	8	10
3	4	5	2					6	8	18
4	6	8	2				$10_{3}$	6	18	36
5	9	11	2				$10_{6}$	6	18	54
6	12	15	2			$14_{3}$	$10_{2}$	6	32	86
7	16	19	2			$14_{6}$	$10_{?}$	6	32	118
8	20	24	2		18	14	10	6	50	168
9	25	29	2		18	14	10	6	50	218
10	30	35	2	22	18	14	10	6	72	290
11	36	41	2	22	18	14	10	6	72	362

The subscripts indicate the number of exceptions to the Aufbau rule. Setting 18 instead of 22 yields

Z = 238

, the conjectured limit on the heaviest physically significant element.

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Shannon Entropy of Chemical Elements

Abstract

1. Introduction

2. Hund’s rule and Shannon entropy

3. Aufbau rule and Shannon entropy

4. Conclusions

Acknowledgments

Data Availability Statement

References

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