# High-Temperature Heat Capacity of the \({\text {Na}}_{2}\) Molecule

## Abstract

The heat capacity of the \({\text {Na}}_{2}\) molecule is investigated in the 2000 K–6000 K temperature range. At these temperatures, the classical treatment is appropriate. The contributions from the ground state and all the excited states dissociating to the 3s + 3s and 3s + 3p limits are taken into account. The results are compared with other known data and are discussed under the influence of unbound rovibrational and electronic excited states.

## Keywords

Excited states Heat capacity High temperatures Unbound states## 1 Introduction

Difficulties in the calculation of thermophysical properties at high temperatures are widely known. The NIST-JANAF Table [1] have many inaccuracies, for example, due to neglecting electronic excited states as in the case of the BBr molecule [2]. Spectacular problems with the partition functions of some molecules obtained by the summation of rovibrational levels were reported by Babou et al. [3]

Heat capacity is a thermodynamic quantity which is very sensitive to all kinds of approximations (because of the second derivative in its relation with the partition function), making it a challenge to obtain reliable results. The other quantities can often be reliably calculated even if serious approximations, such as neglecting the rovibrational coupling, are used. It is particularly easy to calculate entropy [2, 4, 5, 6].

The unbound states (both metastable and scattering) also significantly influence thermodynamic quantities; the heat capacity is again the most sensitive. The recent tendency is to use all the unbound states in thermodynamic calculations [7, 8, 9]. Fortunately, in the case of the classical approach, it is the simplest option available but at high temperatures one also needs to know the purely repulsive PECs (potential energy curves) to describe all the interactions between the atoms.

This article presents the heat capacity of the \({\text {Na}}_{2}\) molecule in the temperature range of 2000–6000 K. This molecule was chosen because the results for this weakly bound molecule are vulnerable to various factors and approximations and because the results known so far do not agree.

Formulas (and calculations) of the internal partition functions and the internal heat capacities are in atomic units and the resulting heat capacities are given in \(\text{ J }\cdot {(\text{ molK })^{-1}}\). All the calculations were performed with a Wolfram Mathematica computing system.

The rest of the article is organized as follows: Sect. 2 presents the methods for the classical calculation of heat capacity and the PECs used in those calculations. Section 3 contains the results and the discussion, and finally Sect. 4 contains the concluding remarks.

## 2 Method

### 2.1 Calculation of Heat Capacity

*R*is the contribution to the heat capacity resulting from the translational motion, and \(\beta =1/(k_{B}T)\) is the inverse temperature.

*V*) is calculated with the bound states approach [10]

### 2.2 The PECs of the \({\text{ Na }}_{2}\) Molecule

In the present calculations all states with the 3s + 3s and 3s + 3p dissociation limits were taken into account in an exact or approximate manner. The two states with the limit 3s + 3s are \(\mathrm {X}^{1}{\Sigma }_{g}^{+}\) and \(1^{3}{\Sigma }_{u}^{+}\). The excited states with the 3s + 3p limit are \(^{3}{\Pi }_{u}\), \(^{1}{\Sigma }_{u}^{+}\), \(^{3}{\Sigma }_{g}^{+}\), \(^{1}{\Pi }_{u}\), \(^{1}{\Sigma }_{g}^{+}\), \(^{1}{\Pi }_{g}\), \(^{3}{\Pi }_{g}\), \(2^{3}{\Sigma }_{u}^{+}\) [12]. There are also higher lying excited states but at the temperatures considered here they can be neglected.

As discussed later the accuracy of the PECs at low and high inter-atomic distances is crucial so it is important to use *ab initio* PECs fitted to actually calculated energies as in Zhang et al. [13], not ones which are based only on spectroscopic constants (which describes best the vicinity of the PEC minimum and are often empirical) as in Biolsi [9] and Song et al. [4].

In the present study, the analytical PECs fitted to the *ab initio* energies were used.

For the ground \(\mathrm {X}^{1}{\Sigma }_{g}^{+}\) state the PEC from Zhang et al. [13] (CBS) was used. This publication also reports the first excited state (\(1^{3}{\Sigma }_{u}^{+}\)) but it has the wrong short-range limit (\(-\,\infty \) instead of \(\infty \)).

Parameters of the fits to Konowalow data with Eq. 5

\(1^{3}{\Sigma }_{u}^{+}\) | \(^{3}{\Pi }_{u}\) | \(^{1}{\Sigma }_{u}^{+}\) | \(^{3}{\Sigma }_{g}^{+}\) | \(^{1}{\Pi }_{g}\) | |
---|---|---|---|---|---|

\(a_{0}\) | 1360.491 653 | 3.747 024 542 | 1.144 429 410 | 1.730 230 037 | 4.593426118r |

\(a_{1}\) | - 0.0 069 549 798 197 | 0.0 | 0.0 | 0.0 | 0.0 |

\(a_{2}\) | - 0.04 091 745 793 | - 0.08 899 981 630 | - 0.03 926 042 290 | - 0.01 069 642 562 | 0.0 |

\(a_{3}\) | 0.5 715 556 185 | 0.0 | 0.0 | 0.0 | 0.0 |

\(a_{4}\) | - 2.549 438 894 | 0.04 048 723 955 | 0.000 568 355 | - 0.002 487 249 959 | 0.0 |

\(a_{5}\) | 9.238 779 840 | 0.0 | 0.0 | 0.0 | 0.0 |

\(a_{6}\) | - 20.79 211 128 | 0.0 | 0.0 | 0.0 | 0.0 |

\(a_{7}\) | 29.03 994 607 | 0.0 | 0.0 | 0.0 | 0.0 |

\(a_{8}\) | - 22.58 787 238 | 0.0 | 0.0 | 0.0 | 0.0 |

\(a_{9}\) | 7.539 412 612 | 0.0 | 0.0 | 0.0 | 0.0 |

\(\beta _{1}\) | 3.018 139 348 | 0.7 650 584 306 | 0.2 120 271 489 | 0.3 555 527 001 | 1.076 476 159 |

\(\beta _{2}\) | 0.4 307 030 965 | 0.3 129 811 451 | 0.2 310 664 856 | 0.2 140 515 091 | 0.0 |

The ground state heat capacity based on the bound states (\(C_{p}^{B}\)), exclusion of negative contribution (\(C_{p}^{NE}\)), and unbound states (\(C_{p}^{U}\)); the upper entries—the results based on the CBS PES, the lower entries—the results based on the IRM PES compared to the Song [4] results

T (K) | \(C_{p}^{B}\) | \(C_{p}^{NE}\) | \(C_{p}^{U}\) | Song |
---|---|---|---|---|

2000 | 30.58 | 35.68 | 35.33 | \(\sim 39\) |

31.19 | 36.82 | 36.49 | ||

2500 | 27.41 | 32.67 | 32.22 | \(\sim 37\) |

27.73 | 33.29 | 32.89 | ||

3000 | 25.42 | 30.55 | 30.03 | \(\sim 35\) |

25.59 | 30.86 | 30.40 |

## 3 Results

*Heat capacity—ground and the first excited state* Comparing different approaches one can see that for the ground state at high temperatures inclusion of the unbound states plays a very significant role (Table 2, the upper entries—the results are based on the CBS PES, for the lower entries—the results are based on the IRM PES). At 2500 K, the difference between the bound and unbound heat capacities is 4.81 \(\text{ J }\cdot {(\text{ molK })^{-1}}\); Biolsi [9] also reports a significant discrepancy but in this case the method of comparison is not specified. The results in Table 2 were restricted to 3000 K to minimize the influence of the excited states.

The difference between the IRM [4] and CBS PECs is also notable (the lower and upper entries): for the bound version of the heat capacity it is 0.61 \(\text{ J }\cdot {(\text{ molK })^{-1}}\) at 2000 K and for the unbound the corresponding discrepancy is 1.16 \(\text{ J }\cdot {(\text{ molK })^{-1}}\). It means that the whole range is important for the results, and this fact is a guideline for PEC construction—it is important to provide accurate results not only in the vicinity of the minimum but also at the high and low inter-atomic distances. Note that Musiał et al. [12] and Zhang et al. [13] calculate the energies for inter-atomic distances only down to around \(2a_{0}\) but for Konowalow’s results the lowest distance is \(\hbox {3.8a}_{0}\) (and highest distance is \(\hbox {15a}_{0}\) while the integration had to be done to \(\hbox {20a}_{0}\) or more for fully converged results). This problem is also important for shallow excited states. The low lying \(1^{3}{\Sigma }_{u}^{+}\) state of \({\text {Na}}_{2}\) will, especially, be affected: the correct partition function for this state is negative at higher temperatures so that, in case of inaccuracies, even a wrong sign of contribution is possible. This can cause a serious problem for heat capacity which is a very sensitive quantity for all kind of effects [2, 8, 15].

For comparison, Table 2 shows the results of Song et al. [4] (approximate values read out from the plot) based on the IRM ground state PEC which differs because of two reasons—neglecting the unbound contribution and the separation of rotations and vibrations.

Partition functions of \(1^{3}{\Sigma }_{u}^{+}\) based on the PECs of Konowalow and Zhang—infinite wall approximated. The results of Mies and Julienne are also given

T (K) | Konowalow fit | Zhang—infinite wall | Mies Julienne |
---|---|---|---|

2000 | \(-\,5.612\cdot 10^{3}\) | \(6.911\cdot 10^{3}\) | \(1.316\cdot 10^{3}\) |

6000 | \(-\,1.297\cdot 10^{5}\) | \(-\,3.216\cdot 10^{5}\) | \(-\,1.216\cdot 10^{5}\) |

The heat capacities based on the ground and the first excited state (\(1^{3}{\Sigma }_{u}^{+}\)): bound states versions (\(C_{p}^{B}\)) based on Konowalow (K) excited state PEC; negative contribution excluded versions (\(C_{p}^{NE}\)), and unbound \(C_{p}^{U}\) versions. The NIST and Biolsi’s results are also given

T (K) | \(C_{p}^{B}\) K | \(C_{p}^{NE}\) K | \(C_{p}^{U}\) K | NIST | Biolsi |
---|---|---|---|---|---|

2000 | 30.87 | 37.46 | 33.06 | 32.78 | 35.13 |

2500 | 27.62 | 33.93 | 28.64 | 31.70 | 31.64 |

3000 | 25.57 | 31.37 | 25.38 | 34.42 | 29.17 |

3500 | 24.27 | 29.65 | 22.94 | 40.75 | 27.46 |

4000 | 23.41 | 28.49 | 20.99 | 47.51 | 26.26 |

4500 | 22.82 | 27.69 | 19.33 | 52.18 | 25.36 |

5000 | 22.41 | 27.12 | 17.81 | 54.01 | 24.67 |

5500 | 22.11 | 26.71 | 16.35 | 53.09 | – |

6000 | 21.88 | 26.40 | 14.90 | 49.94 | 23.65 |

*Heat capacity—ground and all excited states* The results obtained with all excited states are summarized in Table 5. The values here are generally closer to the NIST ones, whereas Biolsi’s results are much lower at most temperatures.

In order to look for a possible source of the discrepancy, the results of this study were recalculated removing some of the electronic states. This test was done at a temperature of 6000 K. The results for the ground and the first excited state followed the Biolsi pattern, so the \(^{3}{\Pi }_{g}\) state that is missing in Biolsi’s calculations was suspected. Removal of the \(^{3}{\Pi }_{g}\) state gives a value of 58.04, removing all states which were approximated (i.e., leaving three excited states) yields 52.80. Removing the two lowest states dissociating to the 3s + 3p limit (\(^{3}{\Pi }_{u}\), \(^{1}{\Sigma }_{u}^{+}\)) from the calculation yields 46.63.

The heat capacities with all electronic states and all unbound contributions based on Konowalow (K) first excited state PECs

T (K) | \(C_{p}^{U}\) K | NIST | Biolsi |
---|---|---|---|

2000 | 33.63 | 32.78 | 35.16 |

2500 | 31.60 | 31.70 | 31.78 |

3000 | 34.13 | 34.42 | 29.52 |

3500 | 40.94 | 40.75 | 28.09 |

4000 | 49.61 | 47.51 | 27.19 |

4500 | 56.83 | 52.18 | 26.57 |

5000 | 60.39 | 54.01 | 26.14 |

5500 | 60.12 | 53.09 | – |

6000 | 57.19 | 49.94 | 25.54 |

The final conclusion about this behavior is that, in general, the higher excited states (dissociating to the 3s + 3p limit) increase the heat capacity again. Note that it is not possible to attribute this to any given electronic states because heat capacity is not additive with respect to the partition functions of the electronic states (the reason for that is the logarithm in Eq. 1), meaning it is not possible to split the total heat capacity between the heat capacities of the electronic states. Because of the second derivative in the definition of heat capacity, it can also be concluded that all higher excited states have similar second derivatives.

The specific source of the discrepancy for the low values at high temperatures in Biolsi’s results must be then be attributed generally to the differences in the higher electronic states. The difference in the partition functions of electronic states, which can seem unimportant, can lead to a big difference in the second derivative.

*T*is the temperature in Kelvin. The difference between the calculated points and the values of this equation is in the range of 0.01 \(\%\) to 0.3 \(\%\).

## 4 Conclusions

Heat capacity at high temperatures is a thermophysical quantity which is very sensitive to the accuracy of the partition function of all the electronic states. The exact calculations require knowledge of the exact PECs of many electronic states, as in case of weakly bounded molecules as the \({\text {Na}}_{2}\) molecule. In light of the present results, it is crucial to have more exact results than the old Konowalow data and to use analytical representations which are more sophisticated than the Hulburt–Hirschfelder potential used by Biolsi which is known to be unsatisfactory at low inter-atomic distances [16]. Both the present results and those of Biolsi certainly suffer from inaccuracies at the low inter-atomic part of PECs so that it is important that the PECs are calculated at very small inter-atomic distances. If the results at even higher temperatures are to be calculated reliably, the electronic states dissociating to the 3s+4s limit will also have to be taken into account, but unfortunately analytical PECs are not available for these.

## Notes

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