Information theory and thermal properties of an extended cosine hyperbolic potential model

Abstract

This study presents the information-theoretic measures and molar thermodynamic properties for an extended cosine hyperbolic potential. The analytic expressions for the Fisher information in both position and momentum spaces are derived. The Shannon entropy for both position and momentum spaces are also derived. The Cramér-Rao bound and Beckner-Bialynicki-Birula-Mycielski (BBM) inequality are tested and confirmed, presenting the model as a good fit for the study of information theory. The study of thermodynamic properties is applied to phosphorus (P₂), potassium (K₂), potassium bromide (KBr), and silicon monoxide (SiO) molecules using specific analytical equations. The results for molar enthalpy (H), molar entropy (S), molar Gibbs free energy (G), and molar heat capacity (C_p) for the four molecules across a temperature range of 0 K to 6000 K are numerically obtained. The predicted results demonstrate excellent consistency with experimental data obtained from the National Institute of Standards and Technology (NIST) database. The discrepancies observed indicate minor variations in the model’s accuracy, providing reliable predictions for the molar thermodynamic properties of the molecules. The performance of the model validates its suitability for studying information theory and accurately representing thermal properties.

Introduction

The development of information theory by Claude Shannon provides a mathematical framework that quantifies, encodes, transmits, and analyzes information, addressing the fundamental limits of data compression and communication accuracy^1–3. This information theory is critically studied in the science as a quantitative way of assessing the content of information in a system or probability distribution. In physics and quantum mechanics, it helps to analyze uncertainty, complexity, and the structure of wave functions or probability distributions. The most powerful concepts in information-theoretic measures are Fisher information and Shannon entropy. Fisher information quantifies the sensitivity of a probability distribution to small changes in its parameters. It is widely used in quantum mechanics to study the localization of wave functions. Shannon entropy, on the other hand, measures the average uncertainty, accounting for the information content of a probability distribution. In recent developments, Shannon entropy and Fisher information have emerged as powerful tools for analyzing diatomic molecules, offering insights beyond conventional observables like energy and angular momentum. Shannon entropy captures the overall spread of the electronic density, while Fisher information highlights local variations and sharp features. Studies using analytically solvable potentials like the Deng-Fan-Eckart models⁴ show that these measures reflect electron localization, delocalization, and molecular confinement, and satisfy key uncertainty relations. By complementing traditional quantum and spectroscopic analyses, information-theoretic approaches provide a nuanced understanding of chemical bonding, electron correlations, and the response of molecules to external fields, establishing a versatile framework for probing molecular structure and dynamics^5–7. These theoretical measures can be calculated from the solutions of the energy levels/wave functions. The energy levels serve as a foundation for the computation of many quantities in quantum mechanics, including thermodynamic properties. The importance of thermodynamic properties as physical quantities in science and engineering cannot be overstated. These properties are crucial for understanding, developing, and improving processes across diverse scientific and engineering disciplines. They provide vital insights into the energy, stability, and behaviour of substances under different conditions. These quantities have significantly enhanced the understanding of phenomena such as fluorescence microscopy, protein activity, phase transitions, material synthesis, dissolution, and adsorption^8–12. The thermal properties include molar entropy, molar enthalpy, molar Gibbs free energy, and molar heat capacity​. Different authors have reported these properties individually or collectively under different potential models^13,14. In Ref¹⁵., Eyube et al. reported the molar enthalpy and molar Gibbs free energy for three different molecules. These authors employed the improved Pὃschl-Teller oscillator to obtain the energy levels. The two properties were calculated using the Poisson summation formula. The calculated results of the three models showed little deviation from the observed results. The authors reported deviations of 0.8178%, 3.1939%, and 0.5312% for molar enthalpy of P₂​, N₂​, and ICl molecules, respectively. They also reported deviations of 0.3865%, 0.4360%, and 0.405% for the molar Gibbs free energy of P₂​, N₂​, and ICl gaseous molecules. Emeje et al.¹⁶ reported the molar enthalpy, heat capacity​ at constant pressure, and molar Gibbs free energy of nitrogen and iodine molecules for a modified shifted Morse potential function. These authors examined the thermal properties of their proposed model over a temperature range of 0 K to 6000 K. The results were found to align with the observed results in the NIST database. To assess the accuracy of their results, they calculated the absolute average percentage deviation of the calculated results from the observed results. Their study reported percentage deviations of 0.031% and 0.034% for molar enthalpy of I₂​ and N₂​, respectively. They also reported deviations for molar heat capacity and molar Gibbs free energy as 0.073%, 0.153%, and 0.032%, 0.008% for I₂ and N₂, respectively. Similarly, Horchani and Jelassi¹⁷ reported the molar entropy of CsO, CsF, and CsCl molecules for the improved Tietz oscillator. The authors provided the full equation for the energy levels of the improved Tietz oscillator and used it to calculate the vibrational partition function. The calculated results were presented graphically in comparison with observed data. To assess the accuracy of their model, they calculated the percentage deviation of the analytical results from the observed results, reporting deviations of 0.228%, 0.267%, and 0.284% for CsO, CsF, and CsCl, respectively.

In another study, Horchani et al.¹⁸ reported the molar enthalpy of CsO, CsF, and CsCl for the shifted Tietz-Wei potential model. The authors first obtained solutions of the radial Schrödinger equation for their model. The molar enthalpy was then studied in detail by deriving expressions to generate values for comparison. The calculated results for the three molecules agreed with the observed results. The average relative deviations were reported as 1.72%, 1.52%, and 2.86% for CsO, CsF, and CsCl, respectively. Jia et al.¹⁹ reported the molar entropy and molar Gibbs free energy of the nitrogen dimer for a modified Rosen-Morse oscillator. Without providing the energy levels of the modified Rosen-Morse potential, the authors derived the vibrational partition function and used it to compute expressions for entropy and Gibbs free energy. Using contributions from rotational, translational, and vibrational entropies and Gibbs free energy, the authors calculated molar entropy and molar Gibbs free energy for the modified Rosen-Morse potential. Their results showed that these quantities can be accurately predicted for temperatures from 0 K to 6000 K. In Ref²⁰., the molar entropy of I₂​, SiC, CP, and F₂​ molecules for the modified shifted Morse potential were reported. The authors explicitly calculated the expression for molar entropy and generated numerical values, which agreed with observed results with minimal differences. The reported percentage deviations were 0.012%, 0.004%, 0.004%, and 0.010% for I₂, CP, F₂, and SiC, respectively. Recently, Onate et al.²¹ calculated molar enthalpy and molar heat capacity at constant pressure​ for F₂​, I₂​, CsO, and CsF molecules for the symmetric trigonometric Rosen-Morse plus Pὃschl-Teller potential. Their model reproduced results with smaller percentage deviations.

Following the above and other literature, it is evident that different models yield different results for any given molecule. Thus, it is important to identify a model with minimal percentage deviation^22,23. Motivated by the interest in theoretical quantities and molar thermal properties for molecular systems, this study examines Fisher information, Shannon entropy, and molar thermodynamic properties of an extended cosine-hyperbolic-type potential function for several molecules. The extended cosine-hyperbolic-type potential is mathematically written as²⁴

1

In Eq. (1), the parameters and are potential parameters that defines the applications of the potential. To study this potential for different molecules, we followed the condition for diatomic molecular potential energy function as²⁴

2

From the Eq. (2), αis given as

3

where c is the speed of light, μis the reduced mass of the particle, is the dissociation energy and the harmonic vibrational frequency. The extended cosine hyperbolic type potential has energy levels of the form²⁴

4

and the radial wave function as.

5

where,

6

Theoretic measures

The section deals with the computations of the theoretic measures. The major theoretic measure to be studied in the work are Fisher information and Shannon entropy. These two quantities have been examined under different models where the characteristics of the models’ parameters on the quantities are verified^25–27.

Fisher information

Fisher information for both position space and momentum space respectively are given as^28–30

7

where γ (p)and γ (p)are probability densities for position and momentum spaces respectively. The Fisher information for the position space can be obtained using the wave function in Eq. (7) but the Fisher information for the momentum can be obtained by taking the Fourier transform of the radial wave function. Thus, in the momentum space, we relate the probability density and the wave function as

8

where Φ (p)is the Fourier transform of R(r). With the position probability density, a transformation of the form and a derivative as

9

the Fisher information for the position space becomes

10

Using integral of the form

11

The Fisher information for the position space finally becomes

12

For the momentum space,

13

where Then,

14

With change of variable of the form x=γ p,

15

Using integral of the form

16

Then, we have

17

Shannon entropy

The Shannon entropy for both the position space and momentum space respectively are given as^31,32

18

In the position space where the Shannon entropy becomes

19

Substituting for the probability density, the above equation becomes

20

Defining an integral of the form

21

The analytic equation for the position space Shannon entropy becomes

22

In the momentum space, where the Fourier transform of R(y)is considered, the Shannon entropy becomes

23

On simplifying the integral following previous steps, the analytic equation for the momentum Shannon entropy becomes

24

The partition function

The computation of the thermodynamic properties relies on the partition function^33,34. The partition function is a tool used to express the relevant thermodynamic functions such as the S, H, G and C_p. For molar thermodynamic properties, the molar partition function is a contribution of the vibrational part, rotational part and translational part. These three-partition functions will be given one after the other.

Vibrational partition function

The vibrational partition function is given as^35–53

25

where β = 1/k_BT, k_B is the Boltzmann constant, T is the absolute temperature and ν_max is the upper bound vibrational quantum number obtained from the first derivative of the energy level. By expression, the upper bound vibrational quantum state for Eq. (4) is

26

.

Following the energy levels in Eq. (4), Eq. (5) turns out to be

27

To evaluate the summation in Eq. (27), it is convenient to use the modified Poisson summation formula^54,55 which is accurate and simple as it gives the approximate value of the summation. The study will consider the lowest order approximation. Thus, the quantum correction terms will not be considered. The summation in Eq. (27) can then be written as

28

Equation (28) can be simplified using the formula

29

In other to evaluate the definite integral in Eq. (29), we define a variable of the form where Using the transformation and input Eq. (28) into Eq. (29), we have

30

where we have used the following for simplicity.

31

Equation (30) can fully be simplified to obtain the complete vibrational partition function using maple software program to have

32

The rotational and translation partition functions

By regarding a diatomic molecule as a rigid rotor, and neglecting the interactions of the molecules as it is considered to be very weak, the rotational partition function and translational partition function respectively are given as

33

34

where V is the volume of the gas, m is the mass of the gas molecule, is the rotational characteristic temperature and τtakes the value 1 and 2 for heteronuclear and homonuclear molecules respectively. The total partition function is the product of Eqs. (12), (33) and (34). Thus

35

The thermodynamic properties

At this point, the various thermodynamic properties can now be calculated. It should be noted that the experimental values for each of the molar thermodynamic property is a combination of three contributions from the vibrational, rotational and translational parts.

Enthalpy: The molar enthalpy H is the sum of the vibrational enthalpy the rotational enthalpy and the translational enthalpy given as

36

where

37

38

39

The R is a universal gas constant whose numeric value equals 8.3144598 J⋅mol^− 1⋅K^− 1.

Molar entropy

The molar entropy is mathematically given as

40

The molar entropy in Eq. (40) is a contribution of the vibrational, rotational and translational entropies given as

41

where the vibrational entropy is

42

The rotational entropy is given as

43

and the translational entropy is given as

44

where p is the gas pressure.

Molar Gibbs free energy: The molar Gibbs free energy is given by

45

Substituting Eq. (35) into Eq. (45), after some mathematical simplification, the Gibbs free energy in Eq. (45) becomes

46

Molar heat capacity at constant pressure: This is given by

47

The heat capacity at constant pressure is a combination of the contributions from vibrational part and the rotational part. The rotational part incorporates the translational part. Hence, the heat capacity at constant pressure becomes

48

where

49

50

To access the accuracy of the calculated results for the proposed model, we calculate the average absolute percentage deviation using

51

Discussion

In the computation of the numerical values for the four molecules, the value of is taken as 1 based on the study of Onate et al. in ref¹⁵. The NIST result represents the experimental data. The four molar thermodynamic properties such as C_p, G, H and S for phosphorous molecule (P₂), potassium molecule (K₂), potassium bromide (KBr) and silicon oxide (SiO) are reported in this study. For numerical computations, we used the following constants: For P₂, and μ =15.4869amu. For K₂, and μ =19.55amu.For KBr, and μ =0.0381amu. For SiO, and μ =0.0981amu. The analytic equations for the molar thermal properties are calculated explicitly in Eqs. (16), (21), (26), and (28) for H, S, G and C_p at constant pressure respectively. The predicted results obtained from the four analytic equations for the four molecules are analyzed in relation to the observed data obtained from NIST⁵⁶ data base and reported in Figs. 1, 2, 3 and 4. The effect of temperature on heat capacity at constant pressure for P₂, K₂, KBr and SiO molecules are shown in Figs. 1a, b, c and d respectively. The heat capacity varies smoothly with temperature and shows good agreement between the calculated curves and the NIST data confirming the reliability of the model across the full temperature range (0 to 6000 K). This shows the gradual activation of the rotational and the vibrational degrees of freedom with temperature rise. In Fig. 1(a), there is rapid increase in the heat capacity at low temperatures specifically below 1000 K. Above the 1000 K, the heat capacity grows gradually and approaches a plateau. This reflects that most accessible molecular modes are fully excited. As the vibrational contributions becomes constant, the heat capacity tends to flattening at high temperatures which reflects the characteristic of diatomic molecules. In Fig. 1 (b), an increase in temperature corresponds to a decrease in the heat capacity. The decrease in heat capacity of K₂ with increasing temperature is mainly due to its weak molecular bonding. As temperature rises, the higher vibrational states are rapidly occupied and approach the dissociation limit, leaving few bound states to store additional thermal energy. Consequently, the internal energy becomes less responsive to temperature, leading to a reduction in heat capacity. This behaviour reflects the gradual weakening of the K₂ bond and the onset of thermal dissociation, which is typical for weakly bound alkali dimers. For potassium bromide (KBr) in Fig. 1 (c), a rise in temperature leads to an increase in heat capacity. The variation is smooth and monotonic as the heat capacity increases slowly with temperature, showing that more vibrational modes get activated at higher temperatures. The rate of increase diminishes slightly at very high temperatures reflecting Dulong-Petit behaviour. KBr is an ionic molecule with stronger binding which maintains increase in thermal capacity as the vibrational modes constantly contributes at high temperatures. In Fig. 1(d), the heat capacity of SiO has a sharp rise at low temperature followed by a gentle saturation at higher temperatures due to a strong covalent bonding. At the initial state, low temperature vibrational activation dominates and high temperature featured quasi-classical. Thus, reflecting typical molecular behaviour where rotational and vibrational modes are increasingly excited with temperature. The analytic Cp for each molecule is consistent with observed data obtained from NIST database. However, the analytic results for each molecule at different temperatures records small disparity from the NIST result. Thus, the average absolute percentage deviation is calculated using Eq. (51) where E stands for the NIST result, C represents the analytic result while N is the number of the observed data. This is to check the overall accuracy of the analytic result for all the temperatures examined. P₂ has deviation of 0.0081%, K₂ has 0.0086, KBr recorded 0.0336% while SiO recorded 0.0246%. The deviations showed that the model has the highest performance for P₂, followed by K₂ then SiO and least KBr. Figure 2 (a, b, c, and d), present Gibbs free energy as a function of temperature for P₂, K₂, KBr and SiO respectively. For all the four molecules, the Gibbs free energy decreases as the temperature rises. This variation justifies the equation G=H - ST. The formula shows that an increase in temperature reduces the Gibbs free energy. The decrease is normal as the product of temperature and entropy (T·S) grows larger, making G smaller (becomes more negative). As temperature increases, the G decreases because, the entropy contributions dominate at high temperature. There is a steady variation of the G with temperature. In all the molecules i.e. Figures 2 (a), 2 (b), 2(c), and 2 (d), the analytic Gibbs free energy decreases consistently with increase in temperature. The negativity of the slope indicates that the thermal disorder stabilizes the systems thermodynamically. The closeness of the calculated result to the NIST result demonstrates consistency of the formulation of the partition function. Despite the uniformity in the curves of the four molecules, the typical molecular dimers, P₂ and K₂ show smoother declines while KBr and SiO show stronger curvature. This reflects larger vibrational contributions from ionic/covalent bonding. The calculated results are found to be in perfect agreement with the observed data (NIST result) for the four molecules studied. However, there are minor differences arising from different model assumptions. Thus, we calculate the average absolute percentage deviation to ascertain the fitness of the model for the various molecules. The average absolution percentage deviation for P₂, K₂, KBr and SiO stand at 0.0110%, 0.0007%, 0.0106% and 0.0119% respectively. This model has the best performance for K₂, followed by KBr, P₂ and finally SiO. In the computation with experimental comparison, the G data obtained from NIST database are scaled relative to a given H data expressed by the equation The represents the experimentally measured G, while is the molar enthalpy at T=298.15K and pressure of 1 bar for gaseous molecules. Thus, the theoretical result of the scaled G is given by The G is purely negative for the temperature range of 0 K to 6000 K studied. Figures 3a–c, and d respectively, show the effect of temperature on enthalpy. The Figures indicate that the enthalpy of all four molecules increases almost linearly with temperature and agrees perfectly with the NIST data. This trend is expected since enthalpy is obtained from the temperature integral of the heat capacity. Because the heat capacities change only slightly over the temperature range considered, the enthalpy curves rise in an approximately linear manner. From a physical point of view, this shows that the supplied thermal energy is mainly used to excite translational, rotational, and vibrational motions, without any sudden structural changes or phase transitions. The somewhat steeper slopes observed for KBr and SiO are associated with their higher heat capacities, which result from stronger bonding and greater vibrational contributions. Overall, the smooth increase in enthalpy suggests that energy is absorbed gradually as temperature increases. Due to computational errors, there are small differences between the NIST result and the analytic result, leading to the computations of average absolute percentage deviation. Using Eq. (51), the percentage deviation obtained as 0.0178%, 0.0069%, 0.0129% and 0.0095% for P_2, K₂, KBr and SiO respectively for the enthalpy. Figures 4a–c, and d respectively, illustrate the variation of entropy with temperature for all four molecular systems. The calculated results show good agreement with the NIST data. In each case, entropy increases continuously as temperature rises, which is physically expected since higher thermal energy allows the molecules to explore a greater number of accessible microscopic states, thereby increasing disorder in the system. At low temperatures, the entropy increases more rapidly. This initial rise is mainly associated with the activation of rotational and vibrational modes, which are largely frozen out at very low temperatures. As these degrees of freedom become thermally accessible, the number of available molecular configurations grows quickly, leading to a sharp increase in entropy. With further increase in temperature, the growth of entropy becomes more gradual. This slower rise occurs because most of the energetically accessible states are already populated, so additional heating produces only a modest increase in disorder. Differences in entropy among the molecules can be understood in terms of their mass and bonding characteristics. KBr and SiO exhibit relatively higher entropy values, reflecting their heavier masses and stronger vibrational contributions, which provide a larger density of states. In contrast, P₂ shows lower entropy due to its lighter mass and simpler molecular structure. Although K₂ is also a diatomic molecule, its weak bonding and anharmonic vibrational behaviour influence how entropy evolves with temperature. Overall, the smooth and monotonic increase of entropy with temperature indicates a gradual population of molecular energy levels, with no evidence of abrupt transitions within the studied range. This behaviour confirms that thermal disorder develops progressively as temperature increases and further supports the consistency of the present theoretical model with standard thermodynamic expectations. Generally, there are computational error, leading to differences between the NIST results and analytic values at each temperature which are very small, suggesting good agreement of the analytic results with the NIST results. The percentage deviation recorded for entropy of the four molecules are 0.0053%, 0.0032%, 0.0092% and 0.0077% for P₂, K₂, KBr and SiO respectively.

Fig. 1.

(a): Heat capacity against temperature for phosphorus dimer. (b): Heat capacity against temperature for potassium dimer. (c): Heat capacity against temperature for potassium bromide. (d): Heat capacity against temperature for silicon monoxide.

Fig. 2.

(a): Gibbs free energy against temperature for phosphorus dimer. (b): Gibbs free energy against temperature for potassium dimer. (c): Gibbs free energy against temperature for potassium bromide. (d): Gibbs free energy against temperature for silicon monoxide.

Fig. 3.

(a): Enthalpy (H) against temperature (T) for phosphorous dimer. (b): Enthalpy (H) against temperature (T) for potassium dimer. (c): Enthalpy (H) against temperature (T) for potassium bromide. (d): Enthalpy (H) against temperature (T) for silicon monoxide.

Fig. 4.

(a): Entropy (S) against temperature (T) for phosphorous dimer. (b): Entropy (S) against temperature (T) for potassium dimer. (c): Entropy (S) against temperature (T) for potassium bromide. (d): Entropy (S) against temperature (T) for silicon monoxide.

Table 1 shows the effect of the screening parameter on both the Fisher information for position space and momentum space. In the position space, an increase in the screening parameter leads to a decrease in the Fisher information, suggesting more delocalization of the wave function in the position space. However, the momentum Fisher information increases as the screening parameter increases, indicating more localization of the wave function. The product of the Fisher information ranges from 10.231741 to 45.856548. The highest product is obtained with the lowest value of the screening parameter, while the lowest product is obtained for the highest value of the screening parameter. The product is a measure related to the uncertainty principle and complexity, and decreases with the screening parameter, reflecting a reduction in the total information with an increase in the screening parameter. The inverse behaviour between the position space and momentum space is consistent with the information trade-off between the position and momentum domains due to Fourier duality. The Fisher information as a function of the screening parameter satisfies both the uncertainty principle and the Cramer–Rao bound. Table 2 shows the effects of the parameter on both the Fisher information for position space and momentum space. An increase in the parameter significantly results in a decrease in Fisher information for the momentum space and a slow decrease in the position space. As increase, the wave function spread over a larger spatial region and becomes smoother. This reduces the magnitude of the spatial gradients of the probability density ρ (r). Since the position-space Fisher information is directly sensitive to the gradients, there is reduction in sharpness and an increase in the delocalization that decreases the Fisher information in the position space​. At the same time, the smoother spatial distribution suppresses high-momentum components in the Fourier-transformed wave function. The momentum-space probability density γ (p) also becomes smoother, with weaker gradients. Consequently, the momentum-space Fisher information decreases as well. The product of the Fisher information drops sharply, indicating a shift in confinement behaviour due to a changing interaction strength. The high values in the momentum space show a highly localized wave function in the momentum space at low parameter values. Despite the decrease in both spaces, the product of the Fisher information still remains above the minimum. The results in this Table failed to satisfy the uncertainty principle because of the decrease in the conjugate spaces but obeyed the Cramer-Rao inequality since the product of the Fisher information is above the minimum bound. Table 3 shows the effects of the parameter on both the Fisher information for position space and momentum space. The effect of on the Fisher information is the same as that of on the Fisher information. However, the position space Fisher information with are higher than the position space Fisher information with Similarly, the momentum space Fisher information with are higher than the momentum space Fisher information with The product of the Fisher information falls rapidly with increasing suggesting the spread of the wave functions due to a decrease in confinement as the effective potential changes. Table 4 presents Shannon entropy for both the position space and momentum space as a function of the screening parameter. An increase in the screening parameter leads to a decrease in the position space Shannon entropy and an increase in the momentum space Shannon entropy. This shows that the position space wave function becomes more localized with a stronger screening parameter, while the momentum distribution spreads out as a consequence of the uncertainty principle. The sum of the entropies increases monotonically with an increase in the screening parameter, suggesting overall delocalization in the phase space. This characteristic conforms with the entropy-based uncertainty relation known as the Białynicki-Birula and Mycielski (BBM) inequality and indicates that stronger confinement in the position space leads to greater uncertainty in momentum. Table 5 presents the relationship between Shannon entropy in position space and momentum space with the parameter As increases, both Shannon entropies decrease. This shows that the parameter has the same sensitivity to both the Shannon entropy for position space and momentum space. Increasing typically strengthens the attractive part of the interaction or deepens the potential well. This causes the particle to become more tightly bond and makes the wave function concentrate more around the equilibrium region. Thus, the position space probability density becomes narrower which decreases the global spread. This results to decrease in the Shannon entropy for the position space. A stronger confinement in position space generally increases momentum uncertainty. As increases: the momentum density often becomes more structured and less uniformly spread, making the probability weight concentrates into a smaller effective region of momentum space. This reduces the global disorder of the momentum distribution, leading to decrease in the Shannon entropy for the momentum space. Despite the decrease in both entropies, the BBM inequality is satisfied even though the uncertainty principle is not obeyed. This also shows that the wave function becomes more sharply localized in the position space and surprisingly concentrated in the momentum space. The behaviour goes against the standard uncertainty trade-off that usually demonstrates localization in one domain leading to spreading in the other. This also shows that the momentum distribution becomes more peaked, indicating that the particle’s kinetic energy becomes narrower as the system transits towards more bound-like behaviour. In mathematical terms, the strong confinement causes the wave function to decay more rapidly in space, resulting in a smooth and narrow Fourier transform in the momentum space. The system thus has greater information content with lower uncertainty, making the system more predictable in both the position space and momentum space. As the parameter increases and the entropies become smaller, quantum fluctuations are suppressed, which may resemble a semiclassical limit where the system acts more classically. Table 6 presents the relationship between Shannon entropy in position space and momentum space with the parameter An increase in leads to an increase in the position space but a decrease in the momentum space. This shows that a localized density function corresponds to a delocalized density function. This aligns with the usual entropic behaviour for normal systems. The Shannon entropy here captures global uncertainty and exhibits complementary behaviour in the position space and momentum space.

Table 1.

Fisher information in position space I(ρ )and in momentum space I(γ ) against the screening parameter αwith and .

α	I(ρ )	I(γ )
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0	3.869847 2.096430 1.484879 1.174111 0.984967 0.856915 0.763871 0.692773 0.636361 0.590281 0.551763	11.849705 12.992775 13.971001 14.816686 15.554497 16.203493 16.778581 17.291562 17.751885 18.167194 18.543735	45.856548 27.238448 20.745250 17.396430 15.320669 13.885010 12.816677 11.979129 11.296605 10.723743 10.231741

Table 2.

Fisher information in position space I(ρ )and in momentum space I(γ ) against the parameter with α =0.15cmand .

	I(ρ )	I(γ )
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0	0.351716 0.210199 0.163487 0.136891 0.118544 0.104646 0.093540 0.084356 0.076584 0.069895	164.69934 137.93551 124.55179 115.90599 109.63626 104.77582 100.83988 97.552939 94.744351 92.301581	57.92740 28.99388 20.36259 15.86643 12.99667 10.96437 9.432516 8.229206 7.255932 6.451409

Table 3.

Fisher information in position space I(ρ )and in momentum space I(γ ) against the parameter with α =0.25cmand .

	I(ρ )	I(γ )
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0	1.196392 1.105632 0.990734 0.863866 0.735051 0.611771 0.499010 0.399574 0.314533 0.243695	63.553611 53.402687 45.502881 39.234627 34.177720 30.039012 26.608951 23.734528 21.301903 19.224967	76.03503 59.04371 45.08125 33.89347 25.12238 18.37699 13.27814 9.483706 6.700158 4.685030

Table 4.

Shannon entropy in position space S(ρ )and in momentum space S(γ ) against the screening parameter αwith and .

α	S(ρ )	S(γ )
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0	2.145289 1.521436 1.195696 0.993444 0.859292 0.768063 0.705973 0.664591 0.638383 0.623533 0.617320	2.171742 2.538226 2.912732 3.289236 3.664062 4.034935 4.400447 4.759748 5.112349 5.458003 5.796625	4.317031 4.059662 4.108428 4.282680 4.523355 4.802999 5.106420 5.424339 5.750732 6.081536 6.413945

Table 5.

Shannon entropy in position space S(ρ )and in momentum space S(γ ) against the parameter with α =1.5cmand .

	S(ρ )	S(γ )
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0	0.638006 2.575030 6.157443 3.391153 2.590704 2.123919 1.795269 1.540743 1.331421 1.151679	4.573720 3.109115 2.540851 2.228884 1.979993 1.771104 1.590093 1.429769 1.285502 1.154118	5.211725 5.684145 8.698294 5.620037 4.570697 3.895023 3.385361 2.970513 2.616922 2.305797

Table 6.

Shannon entropy in position space S(ρ )and in momentum space S(γ ) against the parameter with α =1.5cmand .

	S(ρ )	S(γ )
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0	1.496983 1.681436 1.871080 2.064506 2.260310 2.457108 2.653546 2.848318 3.040172 3.227919	3.490642 3.208617 2.964353 2.751165 2.563895 2.398497 2.251739 2.121013 2.004182 1.899477	4.987626 4.890053 4.835433 4.815671 4.824206 4.855604 4.905285 4.969332 5.044354 5.127396

Conclusion

This study developed accurate analytic models for the information theory and molar thermodynamic properties. The analytic equations for the Fisher information and Shannon entropy incorporate the parameters of the model. The results for the Fisher information and Shannon entropy proved the validity of the model in the study of information theory since the quantities exhibited consistent duality between position and momentum spaces, reflecting uncertainty principle and offering insight to the localization and delocalization of molecular wave functions. The molar thermodynamic properties like Cp, G, H, and S are applied to some molecules such as P₂, K₂, KBr, and SiO molecules over a temperature range of 0 K to 6000 K. The analytic results show strong agreement with experimental data from the NIST database, with average absolute percentage deviations within the acceptable bounds. The thermal behaviour of each molecule was interpreted based on its structural and spectroscopic characteristics, highlighting phenomena such as vibrational saturation and anharmonicity in K₂. These results confirm the robustness of the modeling approach and its capability to accurately describe molecular thermodynamic behaviour across a broad temperature spectrum. However, some minor discrepancies were observed at different temperatures, which may be due to approximations in vibrational, rotational, or electronic contributions not fully captured in the analytic expressions. However, the models assume ideal gas behaviour, which may not hold under high pressure or condensed phase conditions. This model performs better for phosphorous dimer than the modified hyperbolical-type potential reported in ref⁵⁷. for the four thermal properties. While the present model recorded deviation of 0.053%, 0.0178%, 0.0110% and 0.0081% for entropy, enthalpy, Gibbs free energy and heat capacity of P₂, the modified hyperbolical-type model recorded 0.3795%, 0.8939%, 0.3895, and 0.6978%.

Author contributions

***Conceptualization, Supervision*** : Pradeep Kumar Singh***Formal Analysis*** : Makus Ahmes, Chou-Yi Hsu and Yusufbay Yusupov***Software: *** Ibrahim Mahariq***Writing original draft*** : Doniyor Jumanazarovh.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.