The Symmetry Landscape of Cosmic Molecules: From TMC-1 to Orion KL and IRC+10216

1. Introduction

Symmetry is one of the principal organizing concepts of molecular physics and chemistry [1,2,3,4,5,6,7,8,9,10]. Molecular point groups govern degeneracies, rotational and vibrational selection rules, nuclear-spin statistics, and the allowed tensorial properties of molecules [11,12,13,14]. In astronomical spectroscopy, symmetry is indispensable for molecular identification, yet it is rarely treated as a statistical descriptor of an entire molecular inventory.

Taurus Molecular Cloud 1 (TMC-1) is a cold, quiescent dark-cloud filament whose cyanopolyyne peak is exceptionally rich in unsaturated carbon chains, radicals, anions, cyanopolyynes, and aromatic nitriles [15,16,17,18]. The GOTHAM inventory provides a uniform set of fitted column densities for 102 detected species, including 75 main isotopic species [14,17]. The original version of the present study used these 75 species to construct species-weighted and column-density-weighted symmetry distributions. We revealed that mirror and axial symmetry prevail in the symmetry distribution of molecules constituting TMC-1.

A single cold cloud, however, cannot determine whether the observed concentration of mirror and axial symmetry is specific to low-temperature carbon-chain chemistry or reflects a more general property of rotationally detected astronomical molecules. We therefore extend the analysis to two strongly contrasting environments. Orion KL is a chemically rich high-mass star-forming region containing a hot core, compact ridge, plateau, and extended-ridge components [19,20,21]. The Herschel/HIFI survey identified emission from 39 molecules represented by 79 isotopologues across 480-1907 GHz. IRC+10216 is the archetypal carbon-rich circumstellar envelope around CW Leonis; a recent complete 90-116 GHz survey identified 43 spectroscopic species, which reduce to 23 distinct main chemical species after isotopologues are collapsed [22,23,24].

The central questions are therefore: (i) how strongly are the three molecular corpora concentrated in a small set of point groups; (ii) how common are reflection symmetry, proper rotational axes, and linearity; and (iii) which aspects of the observed pattern remain stable when the physical environment and dominant chemistry change? The comparison also permits a sharper assessment of the roles of low temperature, chain-growth chemistry, molecular stability, conformational structure, and rotational-spectroscopic selection.

The paper is organized as follows. Section 2 describes the three molecular corpora, the conventions used for point-group and conformer assignment, and the species- and abundance-weighted statistical measures. Section 3 presents the symmetry distributions for TMC-1, Orion KL, and IRC+10216; compares their symmetry statistics and physical conditions; examines the relevant hierarchy of molecular energy scales; and discusses the structural and observational origins of mirror and axial symmetry. Section 4 interprets the principal findings, with particular emphasis on the robustness of reflection symmetry, the environmental dependence of molecular linearity, and the influence of rotational-spectroscopic selection. Finally, Section 5 summarizes the main conclusions and outlines the broader implications of the observed symmetry landscape.

2. Materials and Methods

2.1. Molecular Corpora

Three survey-defined corpora are used. For TMC-1, the corpus consists of the 75 main-isotopic species in the GOTHAM molecular inventory [14]. Carbon-13 and deuterium-substituted species are excluded because isotopic substitution can lower nuclear-framework symmetry without changing the underlying bonding topology [11,12]. For Orion KL, the 39 molecular species of the Herschel/HIFI HEXOS analysis are counted once each, irrespective of the number of detected isotopologues. For IRC+10216, isotopologues in the 3-mm survey are collapsed to 23 distinct main chemical species [21]. These choices make the three corpora conceptually comparable as species inventories.

The TMC-1 data contain a uniform set of fitted total column densities and therefore permit both species-weighted and column-density-weighted symmetry statistics. The Orion KL and IRC+10216 surveys probe multiple excitation and spatial components and are not combined here into directly comparable abundance-weighted totals [19,21,22]. Their principal cross-source comparison is therefore species-weighted.

2.2. Column-Density Weighting for TMC-1

Mathematically, the column density $A_i$ of the molecule numbered i is given by Eq. 1:

$$A_i={\int }_{LOS}{n}_i\left(x\right)dx,$$

(1)

where abbreviation LOS means “line of sight” - the imaginary line extending from the observer through the molecular cloud, $n_i\left(x\right)$ is the local number density of molecular species i at position x along the line of sight, and dx is an element of distance along the line of sight, measured in cm. Therefore, $\left[A_i\right]={cm}^{-2}$. For example, $\left[A_i\right]={10}^{14}{cm}^{-2}$ means that, on average, an imaginary column with a cross-sectional area of ${cm}^2$, extending through the observed cloud, contains ${10}^{14}$ molecules of species i. Column density is not the local concentration and does not directly give the total number of molecules in the entire cloud. It is the number density integrated along the line of sight. If the gas were homogeneous over a path length, then $A_i=n_{i}L.$In our paper, the measured column density serves as an observational proxy for the amount of each molecular species in the telescope beam.

Methyl-bearing molecules whose A- and E-spin components were fitted separately were recombined conceptually and counted as a single chemical species; their published total molecular column density was not counted twice.

$$A_{\mathrm{t}\mathrm{o}\mathrm{t}}={\displaystyle {\sum }_{i=1}^{75}}{A}_i=8.06479\times {10}^{14}{\mathrm{c}\mathrm{m}}^{-2}.$$

(2)

Here $A_i$ denotes the fitted total column density of species i, defined with Eq. 1 (see Table 1). This sum is a property of the selected GOTHAM corpus and must not be interpreted as the total molecular column density of TMC-1, because several abundant small species, including $\mathrm{C}\mathrm{O}$, fall outside the analyzed frequency range or require different excitation treatment [14].

2.3. Point-Group Assignment and Symmetry-Element Convention

Point groups were assigned to the equilibrium nuclear geometries represented by the spectroscopic molecular carriers. Linear heteronuclear chains were assigned to $C_{\mathrm{\infty }v}$. Planar molecules possessing only their molecular plane were assigned to $C_s$. Structures possessing a twofold axis and two vertical reflection planes were assigned to $C_{2v}$, whereas trigonal pyramidal or methyl-top structures with a threefold axis and vertical reflection planes were assigned to $C_{3v}$. The complete assignments are supplied in Appendix A.

The identity operation E was omitted because it is possessed by every molecule and does not distinguish structures. A literal count of all geometrically distinct symmetry elements is not finite for linear point groups: an ideal $C_{\mathrm{\infty }v}$ molecule possesses an infinite-order principal axis and infinitely many vertical planes. We therefore count the presence of a symmetry-element type rather than every geometrically distinct plane.

No restriction to mirror-containing point groups was imposed during classification. Mirror-free groups, including $C_1$, $C_n$, $D_n$, and $S_n$ , were admissible; the occurrence of only $C_{\mathrm{\infty }v}$, $C_s$, $C_{2v}$, and $C_{3v}$is an empirical outcome of the adopted molecule-by-molecule equilibrium-structure assignments.

2.4. Species- and Abundance-Weighted Measures

To characterize the occurrence of symmetry properties in the molecular corpora, we distinguish between symmetry-element prevalence and point-group fraction. This distinction is necessary because different symmetry-element types may coexist in the same molecule, whereas point-group classes are mutually exclusive.

Let $I_{i\alpha }$ be an indicator that equals unity if molecule i possesses a symmetry-element type $\alpha$ and zero otherwise:

$$I_{i\alpha }=\left\{\begin{array}{ll}1,& \mathrm{if} \mathrm{species} i \mathrm{possesses} \alpha ,\\ 0,& \mathrm{otherwise}.\end{array}\right.$$

(3)

Here $i=1,\dots ,k$, where k is the total number of molecular species in the analyzed corpus. The index α denotes a symmetry-element type, such as a mirror plane $\sigma$, a proper rotational axis $C_n$, a linear axis $C_{\infty }$, an inversion centre i, or a finite improper rotational axis $S_n$. The indicator $I_{i\alpha }$ records only the presence or absence of a specified symmetry-element type. Thus, each molecular species is counted only once for a given α, irrespective of how many geometrically distinct elements or symmetry operations of that type it possesses. The number of molecular species possessing symmetry-element type α is: $n_{\alpha }={\displaystyle {\sum }_{i=1}^k}{I}_{i\alpha }$, and the corresponding species prevalence is:

$$f_{\alpha }={\displaystyle \frac{n_{\alpha }}{k}}={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^k}{I}_{i\alpha }}{k}} .$$

(4)

Thus, $f_{\alpha }$ represents the fraction of molecular species in the corpus possessing symmetry-element type α. Both $n_{\alpha }$ and $f_{\alpha }$ are dimensionless. For example, if every species in a corpus possesses a mirror plane, then $n_{\sigma }=k$ and $f_{\sigma }$ = 1. For TMC-1, the available column-density data also permit an abundance-weighted characterization. Let $A_i$ denote the fitted total column density of species i, and let $A_{tot}$ be the summed column density of all species included in the analysed corpus: $A_{tot}={\displaystyle {\sum }_{i=1}^k}{A}_i$. The total column density contributed by species possessing symmetry-element type α is: $A_{\alpha }={\displaystyle {\sum }_{i=1}^k}{A}_i{I}_{i\alpha }$ and the corresponding abundance prevalence is:

$$F_{\alpha }={\displaystyle \frac{A_{\alpha }}{A_{tot}}}={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^k}{A}_i{I}_{i\alpha }}{{\displaystyle {\sum }_{i=1}^k}{A}_i}}$$

(5)

Here $A_{\alpha }$ and $A_{tot }$ have units of column density, namely cm⁻², whereas $F_{\alpha }$ is dimensionless. Accordingly, $f_{\alpha }$ answers the question: what fraction of the molecular species possesses symmetry-element type α? By contrast, $F_{\alpha }$ answers: what fraction of the summed molecular column density is contributed by species possessing symmetry-element type α? It is important that symmetry-element types are not mutually exclusive. A single molecular species may simultaneously possess several symmetry-element types. For example, a molecule belonging to $C_{3v}$ possesses both a threefold proper rotational axis and mirror planes. Consequently, ${\sum }_{\alpha }{f}_{\alpha }\ne 1$ in general, and similarly ${\sum }_{\alpha }{F}_{\alpha }\ne 1$. Therefore, $f_{\alpha }$ and $F_{\alpha }$ are prevalences of individual symmetry properties and should not be interpreted as components of a probability distribution over symmetry-element types. Table 2 supplies $n_{\alpha }$, $f_{\alpha }$, and $F_{\alpha }$ for the TMC-1 corpus.

Point groups are treated differently because they constitute mutually exclusive classes. Let $n_G$ be the number of molecular species assigned to point group G. Since every species is assigned to one and only one point group, ${\sum }_G{n}_G=k$. The species fraction of point group G is therefore defined as:

$$p_G={\displaystyle \frac{n_G}{k}}$$

(6)

and hence ${\sum }_G{p}_G=1$. For TMC-1, the corresponding abundance-weighted point-group fraction is (see Table 1):

$$P_G={\displaystyle \frac{{\sum }_{i\in G}{A}_i}{{\displaystyle {\sum }_{i=1}^k}{A}_i}}$$

(7)

Obviously ${\sum }_G{P}_G=1$ is true. The same weighting definitions were applied directly to mutually exclusive point-group classes. The Shannon entropy H, normalized Shannon entropy $H_{norm}$ and inverse Simpson effective number of occupied point groups ${\tilde{D}}_2$ were calculated as:

$$H=-{\sum }_g{p}_{g}ln{p}_g , H_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}={\displaystyle \frac{H}{\mathrm{l}\mathrm{n}4}}, {\tilde{D}}_2={\displaystyle \frac{1}{{\sum }_g{p}_g^2}}.$$

(8)

The analyzed corpus is a defined census rather than a random sample drawn from it. The reported statistics are consequently descriptive; no null-hypothesis p-values are assigned.

2.5. Conformer Convention

The primary analysis is a species-level equilibrium-structure census, not a conformer-resolved population analysis. This distinction is minor for small rigid molecules but important for flexible complex organics in warm sources. A species can have a mirror-symmetric lowest-energy conformer while thermally populated higher conformers have lower symmetry, including C1. Accordingly, the reported 100% mirror-plane result for Orion KL is explicitly a nominal species-level equilibrium classification and must not be read as a statement that every populated conformer is mirror symmetric.

3. Results

3.1. Distribution of Symmetries in TMC-1

The distributions of point groups and symmetry-element types among the molecular species detected in TMC-1 are presented in Table 1 and Table 2.

Table 1. Species- and abundance-weighted point-group distributions (TMC-1). 

Point group, G	Species, ${\mathit{n}}_{\mathit{G}}$	Species fraction,${\mathit{p}}_{\mathit{G}}=\frac{{\mathit{n}}_{\mathit{G}}}{\mathit{k}}$	${\sum }_{\mathit{i}}{\mathit{A}}_{\mathit{i},}$ ${\mathit{c}\mathit{m}}^{-2}$	Abundance fraction,${\mathit{P}}_{\mathit{G}}$
$C_{\infty v}$	33	44.00%	3.81245 × ${10}^{14}$	47.27%
$C_s$	26	34.67%	6.4593 × ${10}^{13}$	8.01%
$C_{2v}$	8	10.67%	3.5473 ×${10}^{13}$	4.40%
$C_{3v}$	8	10.67%	3.25168 ×${10}^{14}$	40.32%
Total	75	100.00%	8.06479 ×${10}^{14}$	100.00%

Table 1. Species- and abundance-weighted point-group distributions (TMC-1). 

Point group, G	Species, ${\mathit{n}}_{\mathit{G}}$	Species fraction,${\mathit{p}}_{\mathit{G}}=\frac{{\mathit{n}}_{\mathit{G}}}{\mathit{k}}$	${\sum }_{\mathit{i}}{\mathit{A}}_{\mathit{i},}$ ${\mathit{c}\mathit{m}}^{-2}$	Abundance fraction,${\mathit{P}}_{\mathit{G}}$
$C_{\infty v}$	33	44.00%	3.81245 × ${10}^{14}$	47.27%
$C_s$	26	34.67%	6.4593 × ${10}^{13}$	8.01%
$C_{2v}$	8	10.67%	3.5473 ×${10}^{13}$	4.40%
$C_{3v}$	8	10.67%	3.25168 ×${10}^{14}$	40.32%
Total	75	100.00%	8.06479 ×${10}^{14}$	100.00%

The linear $C_{\infty v}$ class is the largest species class. $C_{\infty v}$ and $C_s$ together contain 78.67% of all detected main species in TMC-1. The abundance-weighted distribution is strongly influenced by ${NH}_3$: the $C_{3v}$ class contains only 10.67% of species but 40.32% of the summed column density. Linear molecule HCNO is depicted in Figure 1.

Figure 1. Linear molecule HCNO is depicted.

Figure 1. Linear molecule HCNO is depicted.

Table 2. Prevalence of symmetry-element types in the 75-species TMC-1 corpus. 

Symmetry element	Species count	Species prevalence, ${\mathit{f}}_{\mathit{\alpha }}$	Abundance prevalence, ${\mathit{F}}_{\mathit{\alpha }}$
Mirror plane, σ	75	100.00%	100.00%
Linear axis, $C_{\infty }$	33	44.00%	47.27%
Threefold axis, $C_3$	8	10.67%	40.32%
Twofold axis, $C_2$	8	10.67%	4.40%
Inversion centre, i	0	0	0
Finite improper axis, $S_n$	0	0	0

Table 2. Prevalence of symmetry-element types in the 75-species TMC-1 corpus. 

Symmetry element	Species count	Species prevalence, ${\mathit{f}}_{\mathit{\alpha }}$	Abundance prevalence, ${\mathit{F}}_{\mathit{\alpha }}$
Mirror plane, σ	75	100.00%	100.00%
Linear axis, $C_{\infty }$	33	44.00%	47.27%
Threefold axis, $C_3$	8	10.67%	40.32%
Twofold axis, $C_2$	8	10.67%	4.40%
Inversion centre, i	0	0	0
Finite improper axis, $S_n$	0	0	0

Table 2 supplies the species prevalence $f_{\alpha }$ of a specified symmetry property within the molecular corpus defined with Eq. 4. Simple calculations yield: $f_{mirror}^{TMC1}=1, f_{axial}^{TMC1}=0.653, f_{linear}^{TMC1}=0.44.$ Here, each molecular species is counted only once in $f_{axial}^{TMC1}$, regardless of how many symmetry axes or individual rotational operations it possesses. These quantities are species-weighted prevalences, not abundance-weighted fractions.

The species-weighted point-group entropy is $H=1.206$, $H_{norm}=0.870$, and the inverse Simpson effective number is ${\tilde{D}}_2=2.97.$ The abundance-weighted values are $H=1.060$, $H_{norm}=0.765$, and $D_2=2.54.$

3.2. Distribution of the Symmetries in Orion KL

Now in Table 3 we supply the data calculated for Orion KL molecular cloud.

Let us calculate the species prevalence f of a specified symmetry property within the molecular corpus of Orion KL, introduced with Eq. 4: $f_{mirror}^{OrionKL}=1, f_{axial}^{OrionKL}=0.795, f_{linear}^{OrionKl}=0.513$.

The point-group entropy of the Orion KL species census is $H=1.158, H_{norm}=0.835$ with ${\tilde{D}}_2=2.77.$The corpus is therefore somewhat more concentrated than TMC-1, primarily because the linear $C_{\infty v}$ class exceeds one half of all species. The Orion KL source is not characterized by a single temperature. The HIFI analysis separates a hot core, compact ridge, plateau, and extended ridge, and the hot core exhibits the broadest distribution of rotational temperatures [19]. Representative hot-core gas temperatures are of order 100-300 K, with individual molecular analyses extending above this range. Thus, Orion KL provides the most direct high-temperature contrast to TMC-1.

3.3. Distribution of Symmetries in IRC+10216

Now we supply the data established for IRC+10216 molecular cloud.

Table 4. Species symmetries and species fractions established for IRC+10216. 

Point group	Species, n	Species fraction, ${\mathit{p}}_{\mathit{G}}=\frac{{\mathit{n}}_{\mathit{G}}}{\mathit{k}}$
$C_{\infty v}$	18	78.3%
$C_{2v}$	3	13.0%
$C_s$	1	4.35%
$C_{3v}$	1	4.35%
Total	23	100%

Table 4. Species symmetries and species fractions established for IRC+10216. 

Point group	Species, n	Species fraction, ${\mathit{p}}_{\mathit{G}}=\frac{{\mathit{n}}_{\mathit{G}}}{\mathit{k}}$
$C_{\infty v}$	18	78.3%
$C_{2v}$	3	13.0%
$C_s$	1	4.35%
$C_{3v}$	1	4.35%
Total	23	100%

Species prevalence $f_{\alpha }$ of a specified symmetry property within the molecular corpus IRC+10216, calculated with Eq. 4 is: $f_{mirror}^{IRC}=1, f_{axial}^{IRC}=0.957, f_{linear}^{IRC}=0.783.$

The point-group entropy is H = 0.730, $H_{norm}=0.527$ with ${\tilde{D}}_2=1.58$, making IRC+10216 by far the most structurally concentrated of the three survey corpora. The dominance is almost entirely the result of the linear $C_{\infty v}$class.

IRC+10216 is radially stratified rather than isothermal. The gas is very warm in the inner circumstellar envelope and cools outward; observations indicate temperatures near 20 K at tens of arcseconds from the star and approximately 15 K or lower farther out. Many long carbon chains are produced predominantly in the outer envelope. IRC+10216 is therefore not a simple ‘hot’ counterpart to TMC-1; its importance here is that it isolates a chemically distinct, carbon-rich chain-growth network.

3.4. Comparative Symmetry Statistics

Now we compare the data established for TMC-1, Orion-KL and IRC+10216 molecular clouds. The comparison is presented in Table 5. The established tendencies are summarized as follows:

$$f_{linear}: 44.0\mathrm{\%}\to 51.3\mathrm{\%}\to 78.3\mathrm{\%}$$

$$f_{axial}: 65.3\mathrm{\%}\to 79.5\mathrm{\%}\to 95.7\mathrm{\%}$$

(9)

$$f_{mirror}:100\mathrm{\%}\to 100\mathrm{\%}\to 100\mathrm{\%}$$

3.5. Physical Properties of TMC-1, Orion KL and IRC+10216 Molecular Clouds

The three objects occupy very different physical regimes. TMC-1 is a cold, nearly static dark cloud; Orion KL is a hot, dense and dynamically disturbed star-forming region; IRC+10216 is an expanding circumstellar envelope whose density and temperature decrease strongly with distance from the central star [26,27,28,29,30,31,32,33,34]. Thermodynamic properties of these clouds are summarized in Table 6. For comparison, the molecular-hydrogen density $n\left({\mathrm{H}}_2\right)$ is used and the thermal gas pressure is calculated from

$$P_{\mathrm{t}\mathrm{h}}=n_{\mathrm{t}\mathrm{o}\mathrm{t}}{k}_{\mathrm{B}}{T}_{kin},$$

(10)

where $T_{kin }$is the kinetic temperature of the cloud. If only $n\left({\mathrm{H}}_2\right)$ is quoted, a useful approximation including helium is supplied by Eq. 11:

$$n_{\mathrm{t}\mathrm{o}\mathrm{t}}\simeq 1.2\hspace{0.17em}n\left({\mathrm{H}}_2\right),$$

(11)

and therefore:

$${\displaystyle \frac{P_{\mathrm{t}\mathrm{h}}}{k_{\mathrm{B}}}}\simeq 1.2\hspace{0.17em}n\left({\mathrm{H}}_2\right)T_{kin}.$$

(12)

The values below should be regarded as representative rather than spatially uniform. The mass density was estimated using $\rho \simeq 2.8m_{\mathrm{H}}n\left({\mathrm{H}}_2\right),$ where the factor 2.8 approximately accounts for helium.

A representative set of parameters for chemical modelling of TMC-1 is $T_{\mathrm{k}\mathrm{i}\mathrm{n}}\simeq 10\hspace{0.25em}\mathrm{K}, n\left({\mathrm{H}}_2\right)\simeq 2\times {10}^4{\mathrm{c}\mathrm{m}}^{-3}.$ These values are commonly adopted in molecular-excitation and chemical calculations for TMC-1 [14,15,16].

Orion KL cannot be characterized by a single density and temperature. It contains the hot core, compact ridge, extended ridge, plateau and high-velocity outflow components. For the hot core, representative values are $T_{\mathrm{k}\mathrm{i}\mathrm{n}}\simeq 200\mathrm{K},n\left({\mathrm{H}}_2\right)\simeq {10}^7{\mathrm{c}\mathrm{m}}^{-3}.$ The literature describes the hot core as a compact region with temperature close to 200 K and density around ${10}^7{\mathrm{c}\mathrm{m}}^{-3}$ [31]. Molecular surveys give approximately 100-300 K for the hot core and 100-150 K for the compact ridge. IRC+10216 is fundamentally different: it is not a static molecular cloud but the expanding, carbon-rich envelope of the AGB star CW Leo. Its expansion velocity is approximately $v_{\mathrm{e}\mathrm{x}\mathrm{p}}\simeq 14.5\hspace{0.25em}{\mathrm{k}\mathrm{m}\hspace{0.17em}\mathrm{s}}^{-1},$ and its mass-loss rate is approximately $\dot{M}\simeq \left(1.5-2.0\right)\times {10}^{-5}\hspace{0.17em}{M}_{\odot }\hspace{0.17em}{\mathrm{y}\mathrm{r}}^{-1}$ [32,33,34]. For an approximately stationary spherical wind, we estimate:

$$\rho \left(r\right)={\displaystyle \frac{\dot{M}}{4\pi r^2{v}_{\mathrm{e}\mathrm{x}\mathrm{p}}}},$$

(13)

so that $n\left({\mathrm{H}}_2,r\right)={\displaystyle \frac{\dot{M}}{4\pi r^2{v}_{\mathrm{e}\mathrm{x}\mathrm{p}}\hspace{0.17em}2.8m_{\mathrm{H}}}}.$ Using $\dot{M}=2\times {10}^{-5}\hspace{0.17em}{M}_{\odot }\hspace{0.17em}{\mathrm{y}\mathrm{r}}^{-1}$, we estimate:

$$n\left({\mathrm{H}}_2,r\right)\simeq 1.5\times {10}^7{\left({\displaystyle \frac{{10}^{15}\hspace{0.25em}\mathrm{c}\mathrm{m}}{r}}\right)}^2{\mathrm{c}\mathrm{m}}^{-3}.$$

(14)

Thus, approximately,

$$n\left({\mathrm{H}}_2\right)\simeq \left\{\begin{array}{ll}1.5\times {10}^9{\mathrm{c}\mathrm{m}}^{-3},& r={10}^{14}\mathrm{c}\mathrm{m},\\ 1.5\times {10}^7{\mathrm{c}\mathrm{m}}^{-3},& r={10}^{15}\mathrm{c}\mathrm{m},\\ 1.5\times {10}^5{\mathrm{c}\mathrm{m}}^{-3},& r={10}^{16}\mathrm{c}\mathrm{m},\\ 1.5\times {10}^3{\mathrm{c}\mathrm{m}}^{-3},& r={10}^{17}\mathrm{c}\mathrm{m}.\end{array}\right.$$

(15)

The gas temperature also decreases outward and is commonly represented by several radial power laws rather than one constant value. Close to the star it can exceed 1000 K; in the intermediate molecular envelope it is tens to a few hundred Kelvin; in the outer envelope it approaches approximately 10-30 K [32,33,34]. The thermodynamic parameters of clouds are summarized in Table 6.

The aforementioned data support the idea that in the thermodynamic sense, all three objects are extraordinarily good classical ideal gases. The ideal-gas approximation requires the mean intermolecular distance to be much larger than the molecular dimensions. The mean separation is approximately: $\mathcal{l}\simeq n^{-1/3}.$ Typical values are ${\mathcal{l}}_{\mathrm{T}\mathrm{M}\mathrm{C}-1}\sim 4.0\times {10}^{-2}\mathrm{c}\mathrm{m},$ and, even for the dense Orion hot core, ${\mathcal{l}}_{\mathrm{O}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{n}}\sim 5\times {10}^{-3}\mathrm{c}\mathrm{m}=50\mu \mathrm{m}.$ These distances are enormous compared with a molecular diameter of order $d\sim {10}^{-8}\mathrm{c}\mathrm{m}.$ Consequently, ${\displaystyle \frac{d}{\mathcal{l}}}\ll 1,$ and excluded-volume and ordinary intermolecular-potential corrections to $P=n{k}_{\mathrm{B}}T$ are completely negligible. Quantum degeneracy is also negligible because $n{\lambda }_{\mathrm{t}\mathrm{h}}^3\ll 1$, where thermal de Broglie wavelength ${\lambda }_{th}$ is given by: ${\lambda }_{th}={\displaystyle \frac{h}{\sqrt{2\pi {mk}_{B}T}}}$, where m is the mass of the molecule, and h is the Planck constant. Thus, TMC-1, Orion KL and IRC+10216 molecular clouds may be considered as ideal gases [26,27,28,29,30,31,32,33,34]. However, these gases exist under very different thermal conditions.

3.6. Hierarchy of Molecular Energy Scales in TMC-1, Orion KL and IRC+10216

It is instructive to establish the hierarchy of energies in TMC-1, Orion KL and IRC+10216 molecular clouds. Comparison of the energy scales associated with chemical bonding $E_{chem}$, thermal translational motion $E_{therm}$, intermolecular interactions $E_{int}$, and the entropy contribution $TS$ to the Helmholtz free energy is supplied in Table 7. Typical covalent bond energies are of order 1–10 eV per bond, with several eV per bond being characteristic of many C–C, C–H, C–N and N–H bonds. Thus, chemical bonding lies orders of magnitude above the translational thermal scale in all three environments.

The translational entropy per molecule was calculated with the Sackur–Tetrode expression. Molecular clouds are extraordinarily dilute, so the volume available per molecule is enormous. Consequently, the translational entropy per molecule can be tens of $k_B.$

Thus, we conclude, that the hierarchy of energies summarized by Eq. 17 holds for all of addressed molecular clouds.:

$$E_{chem}\gg TS>E_{therm}\gg E_{int}$$

(17)

This interrelation actually may be described in the terms of dimensionless constants, as it is demonstrated in Appendix D.

3.7. Structural Origin of Mirror Symmetry

Now we address the central finding of our investigation, namely prevalence of the mirror symmetry within the total list of molecules constituting molecular clouds. Hierarchy of energies established with Eq. 17 has an important implication for molecular symmetry. Molecular geometry and point-group symmetry are primarily established on the electronic and chemical-bond energy scale of electronvolts. By comparison, the characteristic translational thermal energy is only about 10⁻³ eV in TMC-1 and about 2.6 × 10⁻² eV in the Orion KL hot core. Thus, ordinary thermal motion is far too weak to reorganize stable molecular bonding structures simply through thermal agitation. In the inner envelope of IRC+10216 the thermal scale becomes substantially larger, but chemical bond energies normally remain dominant. Thus, chemical structure of the molecules dictates their symmetry.

The universal occurrence of a mirror plane is striking, but it has a direct structural interpretation, which is rooted in two very different properties of the mirror symmetry:

TMC-1 is rich in linear carbon chains and in nonlinear molecules dominated by sp and $s{p}^2$ bonding. Linear molecules possess planes containing their molecular axis. In nonlinear unsaturated molecules, conjugation favors coplanarity because torsion diminishes overlap between neighboring $p$ orbitals. Rings and aromatic nitriles in the corpus are likewise predominantly planar. Thus, a simple structural sequence connects the characteristic chemistry of TMC-1 to the observed symmetry: unsaturation promotes conjugation, conjugation promotes planarity, and planarity produces reflection symmetry. Consider the molecule, which atoms/nuclei belong to the same plane $\Pi$. If every nucleus lies in one plane $\Pi$, reflection through maps each nuclear coordinate onto itself:

$$\left(x,y,0\right)\to \left(x,y,-0\right)=\left(x,y,0\right)$$

(18)

Therefore, the nuclear configuration is invariant under that reflection, irrespective of how asymmetric the arrangement within the plane may be. So, an exactly planar molecule has at least the symmetry operations $\left\{E,\sigma \right\}$ and therefore belongs at minimum to $C_s$ .

The second reason explaining the abundance of mirror symmetry is not related to the planarity of the molecules. Consider the ammonia molecule ${NH}_3$ depicted in Figure 2.

Ammonia molecule is not planar, it possesses a trigonal pyramidal geometry. However, it also demonstrates the mirror plane symmetry. It should be emphasized that the mirror symmetry is compatible with any bond angle ($\theta ={107.8}^0$ for ammonia).

For many TMC-1 molecules belonging to Abelian point groups - particularly $C_s \mathrm{a}\mathrm{n}\mathrm{d} C_{2\upsilon }$, their mirror-reflection operations commute with all other operations of the molecular point group. An especially important case is reflection in the molecular plane, conventionally denoted ${\sigma }_h$. For many rigid planar molecules, this operation is central and therefore commutes with all point-group operations. Nevertheless, that property must not be extended to every kind of mirror plane, particularly the vertical planes ${\sigma }_{\nu }$ of $C_{3\upsilon }$ and higher groups $C_{n\upsilon }$

The result should nevertheless not be universalized to all interstellar environments. Saturated species containing several tetrahedral centers commonly possess nonplanar equilibrium structures or multiple conformers and often belong to $C_1$. Such molecules are prominent in warm, oxygen-rich sources where grain-surface chemistry and thermal desorption are efficient. A comparison with hot molecular cores would therefore provide a direct test of whether mirror-plane prevalence follows the balance between unsaturated gas-phase and saturated grain-surface chemistry.

3.8. Origin of Axial Symmetry

The prevalence of axial symmetry in the three molecular corpora should be considered in relation to their very different physical and chemical environments. TMC-1 is a cold dark cloud with a characteristic kinetic temperature of approximately 10 K; Orion KL contains hot, dense and dynamically disturbed molecular gas, with the hot-core component reaching temperatures of approximately 200 K; whereas IRC+10216 is an expanding carbon-rich circumstellar envelope characterized by strong radial gradients of temperature and density [22,23]. Nevertheless, linear molecules constitute a substantial fraction of all three molecular inventories. Their species fractions are 44.0% in TMC-1, 51.3% in Orion KL, and 78.3% in IRC+10216. Thus, the occurrence of axial symmetry cannot be attributed exclusively to the low-temperature conditions of TMC-1.

In TMC-1, the abundance of $C_{\infty v}$ species primarily reflects the well-known carbon-chain character of the cloud. Barrierless radical-neutral and ion-neutral reactions can extend existing unsaturated carbon chains while retaining an approximately linear, predominantly sp-hybridized carbon skeleton [27,28,33]. Once a linear molecular backbone has been formed, subsequent addition reactions may preserve its geometry. Axial symmetry can therefore be inherited through a sequence of chemical reactions rather than being independently selected at the level of the final molecular product. From this viewpoint, molecular symmetry provides a compact structural descriptor of the history and topology of the underlying reaction network.

The Orion KL molecular inventory demonstrates that a large linear fraction can also occur under much warmer conditions. Its chemistry differs profoundly from that of TMC-1: thermal desorption from dust grains, high-temperature gas-phase reactions, shocks, and the coexistence of several physical components all contribute to the observed molecular population. Nevertheless, many abundant small molecules and unsaturated species remain linear. Heating from approximately 10 K in TMC-1 to approximately 200 K in the Orion KL hot core increases the characteristic translational thermal energy according to $E_{therm}={\displaystyle \frac{3}{2}}{k}_{B}T$. from approximately 1.3 × 10⁻³ eV to 2.6 × 10⁻² eV per molecule. Both values remain much smaller than typical covalent bond energies, which are of the order of several electronvolts. Consequently, the higher temperature of Orion KL strongly affects excitation, reaction rates, desorption and chemical pathways, but ordinary thermal translational motion is still insufficient to destroy the bonding skeleton of a stable molecule or directly reorganize its equilibrium point-group symmetry.

IRC+10216 provides an even more revealing comparison. It is neither a cold quasi-static cloud nor a hot molecular core, but an expanding carbon-rich circumstellar envelope in which temperature and density decrease strongly with distance from the central star. The particularly high fraction of linear species, 78.3% in the corpus considered here, is consistent with its carbon-rich chemistry and the efficient production of carbon chains, cyanopolyynes, radicals and related unsaturated species. The dominance of linear molecules therefore persists across a large radial range of physical conditions. This again suggests that the observed axial symmetry is primarily associated with the chemical architecture of the molecular network rather than with a particular value of the ambient temperature.

The comparison of the three sources consequently suggests the hierarchy supplied by Eq. 17 for the dilute molecular gas under the representative conditions considered here, although the quantitative ratios vary substantially among the sources. This hierarchy helps distinguish molecular structure from molecular motion. Translational and rotational excitation can change strongly with temperature without changing the equilibrium molecular point group, whereas alteration of the molecular skeleton requires energies on the chemical scale.

Entropy requires a separate qualification. The absolute quantity TS should not be compared directly with bond or thermal energies as an intrinsic molecular energy scale. In an extremely dilute gas the translational entropy per molecule may be large because of the enormous configurational volume available to each molecule. The physically relevant quantity when alternative molecular structures or reaction products are compared is instead the entropy difference entering the free-energy difference, $\mathsf{\Delta }F=\mathsf{\Delta }U-T\mathsf{\Delta }S$. Thus, any proposed thermodynamic preference for one molecular symmetry over another must ultimately be formulated in terms of $\mathsf{\Delta }F$, rather than the absolute entropy of the molecular gas. In the present work we therefore do not interpret the observed symmetry distributions as direct equilibrium minimization of free energy.

The finite $C_{3v}$ class requires additional caution. In TMC-1 it combines chemically different species, including ammonia and methyl-terminated chains and nitriles. Its large abundance-weighted contribution is strongly influenced by ${NH}_3$ and therefore should not be interpreted as evidence for a universal chemical preference for threefold symmetry. More generally, abundance-weighted symmetry statistics can be dominated by one or several exceptionally abundant molecular carriers. Leave-one-species-out analyses, particularly exclusion of ${NH}_3$ in the TMC-1 corpus, would therefore provide a useful robustness test in future extensions of the present study.

Taken together, TMC-1, Orion KL and IRC+10216 suggest that the prevalence of axial symmetry is better understood as a consequence of molecular architecture and chemical-network inheritance than as direct thermodynamic selection of symmetry itself. The particularly strong representation of linear molecules in chemically and thermodynamically different environments indicates that axial symmetry may emerge repeatedly because important astrochemical reaction pathways naturally construct and preserve linear unsaturated molecular skeletons. In this sense, the symmetry distribution of an astronomical molecular inventory may retain information about its chemical formation history.

3.9. Low-Temperature Conformational Selection in TMC-1

TMC-1 has characteristic kinetic and rotational temperatures of only several Kelvin. Low temperature suppresses excited conformers whenever a planar or axial ground-state structure is separated from a distorted conformer by an energy $\Delta E$. In an equilibrium approximation, their population ratio contains $\mathrm{e}\mathrm{x}\mathrm{p}\left(-\Delta E/k_{B}T\right)$. Even modest conformational energy differences can therefore sharpen the population toward the lowest-energy geometry. TMC-1 is not in global chemical equilibrium, however, and this argument concerns conformer populations rather than the formation abundance of different chemical species.

3.10. Rotational-Spectroscopy Selection

The present result characterizes a rotationally detected corpus. Pure rotational spectroscopy normally requires a permanent electric dipole moment. Inversion symmetry forces the permanent dipole to vanish, and several other highly symmetric structures are likewise nonpolar. Such molecules are absent or strongly underrepresented in radio inventories even when they are chemically present. Conversely, $C_{\mathrm{\infty }v}$, $C_{nv}$ and $C_s$ structures can support permanent dipoles along an axis or within a mirror plane, making them favorable rotational targets.

This limitation is especially relevant to the absence of inversion centers and to the apparent universality of mirror symmetry. Correcting it rigorously would require a chemically complete modeled population, including nonpolar species, followed by simulation of the survey selection function. Such a correction lies outside the descriptive purpose of the present first study. The point-group and element distributions should therefore not be described as an unbiased distribution of every molecular entity in TMC-1.

3.11. Limitations and Future Work

Our limitations define the interpretation. First, the corpus is frequency- and sensitivity-limited and omits important molecular carriers. Second, column densities derived for different spatial distributions and excitation conditions are used as abundance proxies; their sum is not a direct count of molecules throughout the entire cloud. Third, some radicals and floppy molecules can exhibit vibronic, large-amplitude or conformational effects that complicate assignment of a single rigid equilibrium point group. Fourth, the finite indicator convention records the presence of an element type and does not enumerate the infinitely many planes of a linear molecule.

The most informative extensions are: comparison with an oxygen-rich hot core; separate analysis of isotopic substitution as symmetry breaking; formula-matched comparison of structural isomers; sensitivity analysis excluding the most abundant species; and integration of symmetry labels into an astrochemical reaction network to determine whether dominant reactions preserve or destroy molecular symmetry. The relation of the distribution of symmetries to the complexity of molecular clouds should be addressed [35]. The Ramsey graph analysis of the symmetry of the molecules constituting the molecular clouds seems to be promising [36].

3.12. Kinematic/Geometric Interpretation of the Exceptional Role of Axial and Mirror Symmetries

The exceptional prevalence of proper axial and mirror symmetries found in the three molecular corpora suggests a possible connection with the fundamental geometry of rigid-body transformations [37,38]. Chasles’ theorem states that every orientation-preserving displacement of a rigid body in three-dimensional Euclidean space can be represented as a rotation about an axis combined with a translation parallel to the same axis; such a transformation is known as a screw displacement (see Appendix E). Thus, axial rotation occupies a fundamental position in the kinematics of a rigid body. Mirror symmetry cannot, however, be identified with the translational part of Chasles’ theorem [37,38]. A translation preserves orientation, whereas a reflection reverses it. A broader geometrical interpretation is therefore more appropriate. Any Euclidean rigid transformation may be written as:

$${\overrightarrow{r}}^{\prime }=Q\overrightarrow{r}+\overrightarrow{a},Q\in \mathrm{O}\left(3\right),$$

(19)

where the orthogonal transformation Q belongs either to the proper component, namely: $\mathrm{d}\mathrm{e}\mathrm{t} Q=+1,$or to the improper component, namely: $\mathrm{d}\mathrm{e}\mathrm{t} Q=-1.$ Proper rotations represent the former class, whereas reflections provide a fundamental representative of the latter. In this sense, proper axial symmetry and mirror symmetry correspond to the two fundamental orientation classes of three-dimensional orthogonal transformations. Equivalently, the orthogonal group O may be decomposed as:

$$\mathrm{O}\left(3\right)=\mathrm{O}⁺\left(3\right)\cup \mathrm{O}⁻\left(3\right),$$

(20)

where ${\mathrm{O}}^{+}\left(3\right)=\mathrm{S}\mathrm{O}\left(3\right),{\mathrm{O}}^{-}\left(3\right)=\left\{Q\in O\left(3\right):detQ=-1\right\}$, with the two components distinguished by: $\mathrm{d}\mathrm{e}\mathrm{t} \mathrm{Q}=+1 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}\mathrm{e}\mathrm{t} \mathrm{Q}=-1.$

This observation suggests a geometrical interpretation of their exceptional occurrence in molecular clouds. Proper axial and mirror symmetries are not merely two arbitrarily selected molecular symmetry elements: together they reflect the fundamental dichotomy of three-dimensional rigid geometry into orientation-preserving and orientation-reversing transformations. Chasles’ theorem further singles out axial rotation as the rotational ingredient from which every proper rigid-body displacement can be constructed when supplemented by translation along the appropriate axis [37,38].

This geometrical correspondence does not by itself establish a physical mechanism selecting mirror- or axis-symmetric molecules. Nevertheless, it suggests that their observed predominance may have a deeper kinematic interpretation: the dominant molecular symmetry elements are precisely those associated with the elementary transformation classes underlying the geometry and motion of rigid bodies/molecules in three-dimensional Euclidean space. This correspondence is kinematic rather than causal. It does not explain why particular molecular structures are chemically formed or spectroscopically detected, but it places proper rotations and orientation-reversing transformations within the general geometry of three-dimensional rigid-body transformations. The rigid-motion argument shows that mirror reflections and proper rotations are also geometrically privileged operations: they belong to a minimal and particularly natural set of transformations from which Euclidean rigid motions can be constructed (see Appendix E).

3.13. Symmetry of Molecules and Noether’s Theorem

The exceptional prevalence of mirror planes and proper rotational axes in the molecular inventories considered here suggests a possible connection with the fundamental geometrical symmetries governing the motion of a stable molecule, emerging from the Noether theorem. Such a connection should, however, be formulated with care. In particular, molecular point-group operations are symmetries of an equilibrium molecular configuration, whereas Noether’s theorem concerns continuous symmetries of the dynamical laws. The two notions are related, but they should not be identified. Mirror reflection is a discrete spatial transformation and therefore does not, by itself, generate a conserved quantity through the standard form of Noether’s theorem. In particular, it would not be correct to identify molecular mirror symmetry directly with conservation of linear momentum. Nevertheless, reflection possesses an exceptional geometrical role because Euclidean translations and rotations can be generated by compositions of reflections (see Appendix E). Accordingly, reflections may be regarded as elementary generators from which the continuous Euclidean motions associated with linear- and angular-momentum conservation can be constructed.

The resulting conceptual hierarchy of symmetries contains three distinct levels. At the first level, the molecular point group characterizes internal geometrical invariances of the equilibrium molecular structure. At the second level, translations and rotations describe the possible rigid motions of the molecule in physical space; in this sense the argument is closely related to the Chasles-type decomposition of rigid-body motion and to the fact that Euclidean isometries can be generated by reflections. At the third level, the continuous invariance of the dynamical laws under translations, rotations, and time translations gives, through Noether’s theorem, conservation of linear momentum, angular momentum, and energy, respectively.

This hierarchy suggests a possible physical interpretation of the strongly uneven symmetry distribution observed in TMC-1, Orion KL, and IRC+10216. The dominant internal symmetry elements, namely: mirror planes and proper rotational axes belong to the simplest geometrical operations closely connected with rigid-body kinematics. Proper rotations are directly related to the continuous rotational symmetry underlying angular-momentum conservation, whereas reflections have a privileged generating role in Euclidean geometry because their compositions produce both translations and rotations. The persistence of a stable molecular species in time is additionally embedded in time-translation-invariant dynamics and hence in energy conservation.

This argument should be regarded as a structural and kinematical interpretation rather than as a proof that Noether conservation laws cause particular molecular point groups to be abundant. Molecular point-group symmetry is ultimately determined by electronic structure, bonding, energetics, and formation pathways, while the observed astronomical inventory is also affected by chemical kinetics and spectroscopic selection. Nevertheless, the coincidence is noteworthy: the two classes of spatial symmetry that dominate the investigated molecular corpora (reflections and proper rotations) are also elementary operations intimately associated with the geometry of rigid motion, while the temporal persistence of molecular structure is naturally considered within time-translation-invariant dynamics. This may indicate that the observed restriction of molecular symmetry reflects not only chemical constraints but also the exceptionally fundamental geometrical status of these transformations.

4. Discussion

4.1. Reflection Symmetry Is Robust Across the Three Corpora

The most conspicuous comparative result is the persistence of a mirror plane in every nominal species-level assignment. This result survives a transition from the cold TMC-1 inventory to the chemically rich hot-core environment of Orion KL and to the carbon-rich circumstellar chemistry of IRC+10216. It therefore cannot be attributed simply to the approximately 10 K temperature of TMC-1.

This conclusion must nevertheless be formulated carefully. A planar equilibrium nuclear framework automatically has the molecular plane as a reflection plane, but planarity is not the same concept as reflection symmetry in an arbitrary three-dimensional structure. More importantly, warm sources can populate multiple conformers, and a flexible molecule may have both mirror-symmetric and $C_1$ conformers. The robust empirical statement is therefore that the detected species inventories are overwhelmingly composed of carriers whose standard equilibrium structures contain reflection symmetry.

4.2. Linearity Is Strongly Environment-Dependent

Unlike reflection symmetry, linearity changes substantially among the three sources. The linear fraction rises from 44.0% in TMC-1 to 51.3% in Orion KL and to 78.3% in IRC+10216 [23]. The associated decline in point-group Shannon entropy in IRC+10216 shows that this is not a small perturbation but a strong concentration of the molecular inventory into a single structural class. The IRC+10216 result has a direct chemical interpretation. Ultraviolet-driven chemistry in the outer envelope produces $C_{2}H$ and related radicals, followed by rapid gas-phase reactions that lengthen unsaturated carbon chains. The same network produces families such as $C_{4}H-C_{6}H,$ $C_{3}N-C_{5}N$ and ${HC}_{3}N-{HC}_{5}N$ while preserving a common linear skeleton. Axial symmetry can therefore be inherited through reaction-network topology rather than selected independently as a thermodynamic property of the final molecule.

4.3. Temperature Is Not a Sufficient Control Parameter

The three sources separate temperature from chemistry. TMC-1 is cold, with characteristic gas temperatures near 10 K. Orion KL contains molecular components with characteristic temperatures of order 100-300 K and is the strongest high-temperature test. IRC+10216 spans a wide radial temperature range, while many of its linear carbon chains are associated with the cold outer envelope. The persistence of reflection symmetry in all three corpora therefore argues against a one-parameter relation in which low temperature itself produces mirror symmetry. Low temperature can still influence conformer populations. If two conformers differ in energy by ΔE, an equilibrium population ratio contains the Boltzmann factor, namely: $exp\left(-{\displaystyle \frac{\mathsf{\Delta }\mathrm{E}}{k_{B}T}}\right)$ and cold gas can strongly suppress a higher-energy distorted conformer. This affects the geometry populated within a given chemical species, but it is not by itself an explanation for the chemical abundances of different species.

4.4. Rotational-Spectroscopic Selection

All three inventories are constructed primarily from rotational spectroscopy. A permanent electric dipole moment is normally required for strong pure rotational transitions. Inversion symmetry forces the permanent dipole to vanish, and other highly symmetric structures can likewise be nonpolar. Consequently, centrosymmetric molecules and other nonpolar carriers are strongly underrepresented in radio molecular inventories.

This bias is particularly relevant to the apparent absence of inversion centres and improper axes. It does not, however, trivially require a mirror plane: polar molecules can belong to point groups without reflection symmetry. The repeated mirror-plane result is therefore not equivalent to the usual statement that radio surveys favour polar molecules, although the two effects cannot presently be separated quantitatively.

4.5. Thermodynamic Symmetry Factors Do Not Explain the Trend

Two thermodynamic factors should be considered: interrelation between the number of degrees of freedom of molecules and interrelation between the symmetry of a molecule and its entropy. Consider TMC-1 molecules. These molecules exist under low temperatures, under which vibrational degrees of freedom are excluded. Thus, linear molecules of TMC-1 built of three atoms possessing axial symmetry possess five degrees of freedom, compared to six degrees of freedom, inherent for planar molecules containing a triple of atoms. Hence, the energy of linear 3-atomic molecules is smaller than that of planar molecules. This factor may promote formation molecules in TMC-1, but it could not be decisive. Indeed, we have already seen that the energy of thermal motion in molecular clouds is much smaller than the energy of chemical bonds. However, thermal energies of molecules in the inner molecular envelope IRC+10216 are already not negligible.

Now address the entropy argument. The observed concentration of symmetric structures should not be interpreted as a universal entropic preference for symmetry. The rotational partition function contains the rotational symmetry number ${\sigma }_{rot}$ in the denominator, so increased rotational symmetry reduces rotational entropy by a term proportional to $k_{B}ln{\sigma }_{rot}$. The corresponding free-energy contribution has the opposite sign from a naive “entropy favors symmetry” argument and is small compared with chemical bond energies at interstellar temperatures. Formation kinetics, reaction topology, survival, and observability are more plausible controlling factors.

4.6. A Useful Structural Hierarchy

Across the three corpora a common hierarchy appears: reflection symmetry is the broadest property, proper axial symmetry is less universal, and perfect linearity is more restrictive. The numerical hierarchy becomes progressively stronger from TMC-1 through Orion KL to IRC+10216.

$$reflection symmetry\supset axial symmetry\supset linearity$$

(21)

The fact that mirror symmetry remains nearly invariant while linearity changes strongly suggests that these two observations should not be forced into a single explanation. Linearity is demonstrably sensitive to reaction-network chemistry; the near-universality of reflection symmetry may have a more general structural or observational origin.

4.7. Similarity to the Pauling Rules

The strongly uneven distribution of molecular symmetries revealed in the three astronomical corpora suggests that molecular-cloud chemistry preferentially occupies a restricted subset of the geometrically possible molecular structures. In particular, simple symmetry elements—most notably reflection planes and proper rotational axes, especially the axial symmetry of linear species—dominate the observed inventories, whereas higher-order rotational symmetries and more elaborate combinations of symmetry elements are rare or absent. This empirical restriction of structural possibilities bears a qualitative resemblance to the Pauling rules for ionic crystals. Pauling’s rules do not enumerate all geometrically conceivable arrangements of atoms; rather, they identify a comparatively small set of local structural principles that strongly constrain which coordination environments and connectivities are chemically realizable and stable. In an analogous, but not identical, sense, the symmetry statistics of interstellar and circumstellar molecules indicate that astrochemical structure formation is not distributed uniformly over the space of molecular symmetries. Instead, bonding constraints, energetic stability, reaction pathways, and the chemical composition of the environment appear to funnel molecular structures toward a limited repertoire of relatively simple symmetry classes. The analogy with the Pauling rules should therefore be understood as methodological rather than literal: both cases illustrate how fundamental chemical constraints drastically reduce an otherwise enormous space of geometrically admissible structures. The observed prevalence of simple molecular symmetries may consequently be viewed as a statistical structural rule of astrochemical matter, whose microscopic origin remains to be established.

We propose therefore the following heuristic principle: Principle of Symmetrical Simplicity. Among the geometrically admissible structures of matter, chemically realized structures tend to occupy a restricted subset characterized by comparatively simple and recurrent symmetry patterns, because the requirements of bonding, energetic stability, and feasible formation pathways strongly reduce the space of possible geometrical configurations.

In molecular clouds, this principle is manifested statistically by the strongly non-uniform distribution of molecular point groups and symmetry elements. The observed molecular inventories are not spread uniformly over the space of possible molecular symmetries. Instead, they are strongly concentrated in a small number of relatively simple symmetry classes, with reflection symmetry and proper axial symmetry playing an exceptional role, whereas high-order rotational axes and more elaborate combinations of symmetry elements are rare or absent.

A closely related structural economy is embodied in the Pauling rules for ionic crystals [39,40]. The Pauling rules demonstrate that the enormous number of geometrically conceivable arrangements of ions is drastically restricted by local requirements involving coordination, electrostatic valence, polyhedral connectivity, and chemical identity [39,40]. Thus, although molecular symmetry statistics and the Pauling rules concern different structural levels and arise from different microscopic mechanisms, they express a common organizing idea: chemical matter explores only a small and highly constrained part of geometrically possible configuration space.

The proposed Principle of Symmetrical Simplicity generalizes this observation in symmetry language. It does not state that nature always maximizes symmetry. Indeed, maximal symmetry and symmetrical simplicity are fundamentally different concepts. A structure possessing numerous high-order symmetry operations may be highly symmetric but not symmetry-simple in the present sense. Rather, symmetrical simplicity denotes the preferential realization of a limited repertoire of elementary, robust, and chemically compatible symmetry motifs. Accordingly, the principle may be summarized schematically as: geometrical possibility → chemical constraints → restricted structural space → recurrent simple symmetry.

From this perspective, the Pauling rules may be regarded as a classical local manifestation of structural restriction in ionic solids, whereas the symmetry distributions found in molecular clouds provide a statistical molecular manifestation of the same broader tendency. The Principle of Symmetrical Simplicity is therefore proposed not as a replacement for the Pauling rules, nor as a universal theorem, but as a higher-level heuristic principle connecting chemical constraints with the restricted symmetry landscape of physically realized matter.

5. Conclusions

The principal result of the present study is that molecular symmetry is distributed highly unevenly within the astronomical molecular inventories considered here. The detected molecules do not populate the possible symmetry classes in anything approaching a uniform manner. Instead, their symmetry landscape is strongly concentrated in a small family of point groups, with mirror symmetry and proper axial symmetry occupying an exceptional position. This pronounced asymmetry in the distribution of symmetries is observed in all three chemically and physically distinct environments investigated: TMC-1, Orion KL, and IRC+10216.

The most striking result concerns reflection symmetry. Under the adopted species-level equilibrium-structure convention, every molecular species in all three corpora possesses at least one mirror plane. Thus, reflection symmetry is not merely common but is the only symmetry property found universally throughout the analysed molecular inventories. Its persistence from the cold TMC-1 cloud to the much warmer and dynamically complex Orion KL environment, and further to the radially stratified circumstellar envelope IRC+10216, demonstrates that its prevalence cannot be explained solely by low temperature. Mirror symmetry therefore emerges from the present census as the most robust structural symmetry of detected cosmic molecules.

Proper axial symmetry constitutes the second dominant feature of the symmetry landscape. The fraction of species possessing a nontrivial proper rotational axis is 65.3% in TMC-1, 79.5% in Orion KL, and 95.7% in IRC+10216. Within this axial population, linear molecules are particularly important: their fractions increase from 44.0% in TMC-1 to 51.3% in Orion KL and 78.3% in IRC+10216. In contrast to the universality of mirror symmetry, the substantial source-to-source variation of axial and especially linear symmetry indicates a strong dependence on the chemical environment. The extreme concentration of IRC+10216 in the linear $C_{\infty v}$ class is naturally associated with its carbon-rich chemistry and with reaction networks that efficiently construct and preserve unsaturated linear molecular skeletons. The combined results therefore reveal a clear structural hierarchy: reflection symmetry ⊃ axial symmetry ⊃ linearity, but, more importantly, they demonstrate that the symmetry distribution of astronomical molecular inventories is intrinsically strongly nonuniform. Mirror and axial symmetries dominate, whereas other symmetry types are comparatively weakly represented or, within the present corpora, absent. This unevenness should itself be regarded as a significant characteristic of the molecular populations rather than as an incidental feature of individual molecules.

The physical origins of the two dominant symmetries need not be identical. The prevalence of axial and linear structures can be related relatively directly to molecular architecture and to the topology of astrochemical reaction networks, particularly carbon-chain growth. The near-universality of mirror symmetry appears to be more general and may reflect a combination of molecular structural constraints, chemical formation pathways, conformational selection, and observational selection. In particular, the present inventories are constructed predominantly from rotational spectroscopy and therefore cannot be regarded as unbiased censuses of all molecules present in the corresponding astronomical environments.

We therefore propose that the degree of nonuniformity of molecular symmetry distributions, and especially the exceptional prevalence of mirror and axial symmetry, constitutes a new structural descriptor of astronomical molecular populations. In this interpretation, the symmetry landscape of a molecular cloud or circumstellar envelope is not merely a classification of molecular geometries: it may retain information about the chemical pathways through which its molecular inventory was formed and preserved. Extending this approach to additional molecular environments and incorporating symmetry directly into astrochemical reaction networks should reveal whether the dominance of mirror and axial symmetry is a general organizing feature of molecular matter in the Universe.