Defining the Temperature of an Isolated Molecule

The microcanonical temperature of an isolated molecule is derived in terms of Boltzmann and Gibbs volume entropies within the quantum harmonic vibrational approximation. The effects of the entropy functional choice and various approximations are examined. The difference between Boltzmann and Gibbs volume temperatures is negligible for molecules as small as ten atoms. However, it is significant for smaller systems, opening a way to probe them experimentally. A simple, analytical expression of the temperature as a function of the vibrational energy is provided, allowing predictions with a 3% margin of error. The microcanonical temperature is discussed and exemplified with polycyclic aromatic hydrocarbon molecules and other molecules of astrophysical interest.


I. Introduction
Any excess of vibrational energy in a molecule above its zero-point level may induce physical-chemical reactions like isomerization, dissociation, or rearrangement.The role of such vibrational energy excess is particularly relevant when a light-excited molecule transfers the photon energy to the ground state without emitting radiation.The classic example is the vision mechanism, where a complex sequence of events is triggered in the rhodopsin protein after the retinal chromophore converts photon into vibrational energy. 1 We often say that a molecule with large vibrational energy is hot.However, if we ask what the molecule's temperature is, the question sounds odd.The fundamental thermochemistry of a canonical system tells that the environmental temperature determines the vibrational excess through ( ) where EZP is the vibrational zero-point energy, Te is the environmental temperature, kB is the Boltzmann constant, h is the Planck constant, and the sum runs over the vibrational degrees k with frequency k.Nevertheless, an isolated molecule cannot count on the environment to thermalize.
The vibrational energy excess will be trapped there until it is irradiated or induces a chemical reaction.
A hot, isolated molecule is a microcanonical system with constant total energy, and, different from the canonical problem, we want to determine ( ) m TE , the microcanonical temperature as a function of the energy.Defining the temperature of such an isolated molecule could be worth doing, 2 especially if we aim at describing large systems.Indeed, the temperature of isolated molecules and small finite systems has been under scrutiny for decades in diverse fields.
2][13] Indeed, defining such a temperature may be helpful even for understanding the photophysics of chromophores in vacuumlike hydrophobic cavities of proteins. 14n the one hand, the temperature of isolated systems has been addressed pragmatically using phenomenological approaches for applied purposes.Berry and Smirnov, 10 for instance, started from a reasonable estimate for the heat capacity of isolated clusters to derive their temperature.Another common approach has been to derive the canonical temperature for a canonical ensemble with kinetic averages equivalent to those of the microcanonical system. 3dersen, Bonderup, and Hansen 15 followed this way to derive a high-energy approximation for the molecular temperature.D'Hendecourt and co-workers 11 estimated the properties of molecules in the interstellar medium from the observed temperature, also using an equivalent canonical approach.Most of these works, however, either do not rest on a solid statistical basis or have a limited validity domain.
On the other hand, more statistical-mechanics-oriented works have provided much knowledge on small, isolated systems.We know, for instance, that the microcanonical temperature can be defined independently of the probe. 168][19][20] Nevertheless, these works usually do not allow for a practical estimate of the temperature in applied cases of interest.This paper aims to bridge this gap between theory and applications by deriving a microcanonical temperature for an isolated molecule that can be directly used in diverse applied fields, has an extended validity domain, and is based on solid statistical mechanics grounds.

II. Theoretical model
To define the microcanonical temperature of an isolated molecule, we must go back to the principles of statistical thermodynamics.For a system with total energy E, the temperature is where ( ) SE is the entropy, which is a function of the number of microstates at energy E. Later, we will discuss what functional of E is the most adequate when dealing with a small system far from the thermodynamical equilibrium.
If we approximate our isolated molecule by an ensemble of N quantum harmonic oscillators, the total molecular energy is where nk is the number of quanta deposited in vibrational mode k, and  denotes a particular ensemble   k n .Thus, to know the number of microstates, we must determine the number of solutions of Eq. ( 3).This means the number of ways we can distribute different values of nk among the vibrational modes, keeping E constant.Despite the simplicity of this formulation, it is a formidably tricky problem to solve analytically.We can approach it numerically, but the number of microstates is so large that it may become unattainable to compute even for a molecule with only six atoms.
Fortunately, as I will also show later (Section E.3), we can get an approximated solution for the number of microstates by supposing the ensemble of vibrational frequencies   k  can be replaced by a single average frequency  .The degenerated problem, which simplifies to , 2 where k k Mn =  , is a very well-known statistical-mechanics textbook example. 21,22

Modeling an isolated molecule
We can consider an isolated molecule as an ensemble of weakly coupled quantum harmonic oscillators."Weakly coupled" means that energy can flow between vibrational modes, but the vibrational energy is still approximately given by Eq. ( 3).This energy flow is crucial for the molecule to ensure that all microstates sharing the same energy are equally probable.For instance, suppose the molecule has 1 = 300 cm -1 and 2 = 600 cm -1 , a microstate with n1 = 2 and n2 = 0 should be as likely to occur as another one with with n1 = 0 and n2 = 1 (assuming that nk for k > 2 is the same in both microstates).Although weak coupling is assumed, it is never explicitly considered.
This quantum harmonic vibrational mode will not describe strongly anharmonic modes, like intramolecular hydrogen bonds or internal rotations (like those methyl groups are prone to).
We will also not discuss temperature associated with the energy allocated in the translational and rotational modes.Even though these components are relevant when probing the isolated molecule, 23 they go beyond the defined scope of this work.

B. Reviewing the statistical mechanics' concepts
Before proceeding with the discussion, it is worth revisiting some fundamental concepts from statical mechanics. 18The basic quantities we need are the density of states and the integrated density of states In these equations, E is the energy of a system with a Hamiltonian ( ) . Z comprises all external control variables like volume or fields, and  describes the microscopic state (in our case,  collects the quantum numbers nk in Eq. ( 3)).In Eq. ( 5),  is the Dirac function and, in Eq. ( 6),  is the unit-step function.The trace Tr denotes the integral over the Hilbert space.
The entropy can be defined in different ways.For regular systems in the thermodynamic limit ( ) N → , they all tend to converge to the same result.Nevertheless, when dealing with an isolated molecule, the number N of degrees of freedom is small, and we cannot assume such equivalence.Among several possibilities, we will discuss two entropy formulations, the Boltzmann entropy 24 ( ) ( ) and the Gibbs volume entropy 18,24,25 ( ) ( ) In Eq. ( 7),  is a small energy constant.
With either entropy formulation, the microcanonical temperature can be calculated with Eq. ( 2) and the heat capacity with 1 .

T C E
C.

Counting microstates
We can directly count the microstates contributing to either ( ) the entropy, and evaluate the derivative in Eq. ( 2) numerically to get the Boltzmann ( ) ( ) T and the Gibbs volume temperatures.The superscript (n) denotes the numerical approach to distinguish from other approaches discussed below.To apply the numerical procedure, we note that the molecular energy given by Eq. (3) changes by no less than the energy of the slowest normal mode 1  .Thus, the number of microstates with energy between where ( ) The integrated density of states is

D. Solving the degenerated problem
As mentioned, the degenerated problem is used below (Section E) as an approximation for the molecular problem.Besides that, it is also helpful to check some basic hypotheses and approximations underlying the temperature definition, namely: • the equivalence between Gibbs volume and Boltzmann entropy; • the extension of discrete into continuous variable needed to compute derivatives; • the effect of Stirling's approximation, which is commonly used to simplify the results of the microstates' counting; • the impact of the 2 N approximation, also commonly applied to simplify the results.
In the following two subsections, we will discuss the solution of the degenerated problem in the Boltzmann and Gibbs volume formulations.Although the solution in the Boltzmann formulation discussed in the previous section is well known, 21,22 I am not aware of any demonstration using the Gibbs volume formulation so far.Subsection 3 compares both results and shows how the Boltzmann temperature to Gibbs volume temperature for large N.

The Boltzmann formulation
The number of microstates satisfying Eq. ( 4) is the number of ways of allocating quanta in N vibrational modes.Thus, the number of microstates between E and Eh  + is the number of multisets of length M on N elements: ) The dependence on EM is implicit in M. Taking h  = in Eq. ( 7) allows directly compute the Boltzmann entropy as ) The extension of this equation from a discrete to a continuous variable ( M EE → ) allows using Eq.(2) to get where  is the digamma function defined as ( ) ( ) and  is the gamma function.The superscript (e) indicates this is an extended-to-continuous approximation.
Then, with Eq. ( 2), we get where the superscript (s) indicates Stirling's approximation.Finally, for large systems, we have 2 N , and we get the well-known result ( ) Schwabl 26 offers an alternative approach to computing B T .Instead of using combinatorial to estimate , he rewrites this quantity in terms of the Fourier transform of the Dirac function in Eq. ( 5).The resulting integral is evaluated via the Laplace method, which requires N →.
This approximation once more yields ( ) l B T .

The Gibbs volume formulation
To solve the degenerated problem in the Gibbs volume formulation, we proceed along the same lines as in the previous section, but employing the integrated density of states in Eq. ( 6).For the quantum harmonic oscillator, this quantity is the number of microstates with energy between 0 and E, ( ) ( ) where the sum runs over the density of states defined in Eq. ( 13) and is evaluated using the hockeystick identity. 27Thus, the Gibbs volume entropy is Replacing it in Eq. ( 2) gives the extended-to-continuous ( M EE → ) approximation for the microcanonical temperature Alternatively, with Stirling's approximation in Eq. ( 20), we have ( ) T in Eq. ( 9) gives the heat capacity ( ) 22 22  1 ln .42

Comparing Boltzmann and Gibbs volume formulations
Let us now compare the previous results for the degenerated problem for a particular example of a system composed of six quantum harmonic oscillators, all with frequency 1015 cm -1 .
Its number of states ( ) and the integrated density of states ( ) grow monotonically.The energy resolution is determined by , which for the degenerated problem is h .For typical photoexcited systems, we may expect vibrational energy excess of about 2 to 4 eV.These values are much bigger than the vibrational frequency (0.14 eV in our example).Thus, we can compute the temperature by numerical differentiation of the entropy (Eq.( 2)), using either ( ) tend to the same values, 17 as we can see comparing Eqs. ( 18) and (22).
We also can see in Figure 1-bottom that the extended-to-continuous approximations ( ) In the Boltzmann formulation, it is usual to assume that 2 N , yielding ( ) l B T in Eq. (18).
Although this approximation is fully justified for large systems, it has a major conceptual implication: it shifts the system from a Boltzmann to a Gibbs volume description, as made evident by Eqs. ( 18) and ( 22), where we can see that ( ) ( ) ls BG TT = .
Several authors, 21,28,29 starting from the Boltzmann formulation, already apply the large N approximation at the level of the number of microstates, Eq. (13).Because this approximation reduces Eq. ( 13) to Eq. ( 19), they inadvertently work in the Gibbs volume formulation.

Defining the average frequency
My hypothesis to compute the temperature of an isolated molecule is that there is an average frequency  for the N-degenerated problem that is a good approximation for the isolated molecule, composed of N different frequencies.I tested two possibilities,  given as the arithmetic and the harmonic mean 1 11  .
The rationale for adopting A  is the direct analogy to the solution of the degenerated problem.Note, for instance, that Eq. ( 22) can be rewritten as Thus, it is natural to search for an approximated solution of the non-degenerated problem with the same functional form.On the other hand, for a given E, there are many more microstates involving changes in the number of quanta in the low-frequency than in the high-frequency modes.Thus, it may be helpful to weigh the average frequency toward the low-frequency modes.If we take as weight the maximum number of quanta we can add to a specific vibrational mode, max / providing a rationale for adopting the harmonic mean of the frequencies.
In comparing these two possibilities, we will see in Sections 2 and 3 that the arithmetic mean does better than the harmonic mean.

Test case: the temperature of H2O2
Let us now work out a specific example of a small, isolated molecule, H2O2, for which we can count the microstates and compute the temperature numerically.H2O2 has four atoms and N = 6 vibrational modes with harmonic frequencies 346, 956, 1330, 1469, 3704, and 3709 cm -1 (computed with the geometry, frequency, noncovalent, extended tight-binding (GFN1-xTB) method 30 ).The arithmetic mean frequency is A  = 1919 cm -1 , and its harmonic mean frequency is H  = 1015 cm -1 .Up to 4 eV, it has 169,501 microstates, distributed as a function of E as shown in  for the N = 6 degenerated system with the harmonic mean frequency of H2O2, cm -1 .
The Boltzmann and Gibbs volume temperatures of the isolated H2O2 computed by numerical differentiation of the entropy (Eq.( 2)) are shown as dots in Figure 2, middle and bottom.
The approximated results using the arithmetic mean frequency in the 6-degenerated problem are All approximated results are excellent in the region above 1 eV.The agreement deteriorates in the low-energy region near the zero-point energy.The exception is Stirling's approximation with arithmetic mean frequency, which also describes the low energy well.The extended-tocontinuous approximation does not hold the correct behavior at low energies, delivering too hightemperature predictions.

Model validation
The degenerated system with arithmetic mean frequency gave an excellent approximation for the temperature of the isolated H2O2.Nevertheless, is this a general result or just a coincidence?
To check this point, I have computed the Gibbs volume temperature for 30 arbitrary systems ( 10with N = 6, 10 with N = 7, and 10 with N = 8) with frequencies randomly distributed between 300 and 3000 cm -1 .The systems with N equal 7 and 8 do not correspond to molecules because N must be at N − .Nevertheless, they are useful to assess the errors.The systems with N = 8 have about 10 9 microstates up to 4 eV (see Eq. ( 19)).N = 9, which correspond to Nat = 5, is already unaffordable.For each energy value, the relative error between the numerical ( ) temperatures and the temperatures of the equivalent degenerated system using Stirling's approximation was evaluated.These errors are shown in Figure 3 for temperatures computed with arithmetic mean frequencies.Most of the error is in the low-energy region, and they are independent of the system size.
The mean relative errors in Table 1 confirm this picture. ( )s G T with arithmetic mean frequency has a mean relative error of 3% no matter N. The mean relative error of ( ) s B T also with arithmetic mean frequency is a bit bigger, 8%.The mean relative errors with harmonic means are always larger than those with arithmetic mean.They are also dependent on the system size, reducing for bigger N. It is not shown in the table, but most of the error when using the harmonic mean frequency is

III. Discussion
A.

Leading terms in the molecular temperature
The asymptotic expansion of ( ) s G T (Eq. ( 22) at E → gives The leading term is linear in E and does not depend on the particularities of the molecule, only on its size through N.
In cluster studies, it is common to estimate the microcanonical temperature using the phenomenological formula 10,31 2 , where kin E is the mean kinetic energy of the cluster.For a harmonic system, 2 kin EE = and, therefore, Eq. ( 30) corresponds to the leading term in the Gibbs volume temperature expansion, Eq. (29).
The isolated molecule modeled developed here is purely based on a harmonic approximation.Belega and co-workers 31 showed that anharmonic effects could be included in Eq. (30) in by replacing N by an effective Neff factor computed from the reduced averaged kinetic energy in each mode.
If, instead of the Gibbs volume, we start from the Boltzmann temperature ( ) s B T (Eq. ( 17)) the asymptotic expansion at E → gives As expected, ( ) ( ) aa BG TT → when N →.However, for small systems, the leading term depends on the molecular properties through  .Moreover, the ( ) T with the energy.
Andersen, Bonderup, and Hansen 15 derived the microcanonical temperature of a small isolated system starting from the Boltzmann entropy and searching for the average energy of a canonical system equilibrated at the microcanonical temperature.Their result for an ensemble of N (non-degenerated) quantum harmonic oscillators in the high-temperature limit is ( ) Considering that in this limit E hv , their formula matches the leading order of the ( ) T expansion in Eq. (31).
A common assumption in phenomenological models is that the heat capacity (Eq.( 9)) of the isolated molecule does not depend on the temperature, so a simple linear relation 0

E
CT E =+ holds between energy and temperature. 2,8 ccording to Eq. ( 29), such hypothesis is satisfied for a molecule with N vibrational degrees if . 2 3  3 Note that the radiative cooling should bring the molecule to the nonlinear region even if this relation is initially satisfied.

B. Experimental comparison between Boltzmann and Gibbs volume
Currently, there is an ongoing debate on which function is the correct way of computing the entropy for an isolated system.While Hilbert, Hänggi, and Dunkel 18,24 claimed that, among several entropy models, only Gibbs volume fulfils all three laws of thermodynamics simultaneously, Swendsen and Wang 19 argued that Gibbs volume fails to satisfy the postulates of thermodynamics and make incorrect predictions for systems with nonmonotonic energy densities.
(See Ref. 32 for an account of this debate.) Although this discussion goes beyond the scope of this paper, the present results create a way for experimentally testing which entropy, Boltzmann or Gibbs volume, delivers the best theoretical prediction.We could think of a setup where a small, isolated molecular ion is photoexcited in an electrostatic ion trap. 4,6,33 Te laser pulse duration should be such as it allows thermal equilibration before radiative decay starts, but without losing too much energy resolution.
Then, monitoring its radiative cooling would inform on its peak temperature (before any cooling).
As discussed in Section II.E.2, for a four-atoms molecule with a few eV of energy excess, the difference between Boltzmann and Gibbs volume temperatures should be significant enough to cause different color emissions.

C. Temperature of isolated molecules of applied interest
The temperatures ( ) and Echt 34 measured its heat capacity as 12.6  1.4 meV/K.According to Eq. (22), this temperature corresponds to E = 14.7 eV, yielding a theoretical estimate of ( ) s G C = 12.0 meV/K (Eq. ( 23)) in excellent agreement with the experiment.Note that this result is smaller than the high-energy limit, B Nk = 15.0 meV/K (N = 174), due to the nonlinear dependence of the temperature on the energy.
7][8] Since the seminal papers by Sellgren 35 and Leger and Puget, 12 some specific near-infrared (IR) spectral features of interstellar dust have been assigned to polycyclic aromatic hydrocarbons (PAHs). 36These earlier works raised the hypothesis that those IR features arise from small, 5-10 Å molecules instantaneously heated to about 1000 K by UV radiation.
Working on observational data, Leger and co-authors 11,12 estimated a color temperature of 600 K, corresponding to a peak temperature of 950  150 K.Then, using the Einstein heat capacity model and assuming 6-eV photoexcitation, they predicted that the IR features should be due to carbon species with about 60 atoms.energies taken from the literature: anthracene, 37 naphthalene (neutral and cation), phenanthrene, pyrene (neutral and cation), 38 coronene, dicoronylene, ovalene, 39 and fullerene. 40 can revisit this assignment by using the microcanonical temperature derived here.Let us assume a peak temperature of 950  150 K and that the arithmetic mean frequency is 1100  100 cm -1 .Let us also allow a broader variation in the UV excitation, 5  1 eV.With ( ) s G T given by Eq. ( 22), we can estimate N and the number of atoms Nat.Letting all parameters vary according to  37 for 10 6 random systems.This mean number of atoms agrees with the previous calculations.
Nevertheless, as we can see in the density plot at the bottom of Figure 4, it also indicates that a large variety of possible molecular systems falls within the parameter constraints.The figure also shows a few PAH molecules ranging from small to large sizes.Except for naphthalene neutral and cation, they all fall in the high-density region, as we would expect from PAH mixtures that should occur in the interstellar media. 39,41 his analysis considers only molecules returning ground state via radiationless pathways.If they fluoresce, the peak temperature after re-equilibration in the ground state should correspond to a vibrational excess discounting the energy of the emitted photon.

IV. Conclusions
This paper explores the microcanonical temperature of an isolated molecule.The molecule is treated as an ensemble of quantum harmonic oscillators, and the temperature is derived in the frame of Boltzmann and Gibbs volume entropies.Although the direct solution of the problem seems not to be accessible analytically, we discuss how approximations based on fully degenerated-spectrum systems using average frequencies can adequately describe the molecular temperature.The discussion tracks the effect of several hypotheses and approximations on the results, including the entropy functional, the type of average frequency, the extension of discrete into continuous variables, the use of Stirling's approximation, the assumption of a large number of degrees of freedom N, and the expansion in the leading orders.Table 2 summarizes the several expressions discussed in the text.
The comparison between Boltzmann and Gibbs volume approaches shows that the temperatures converge to the same values for large N.The difference is already negligible for a 10-atoms molecule.However, for smaller molecules, the difference is significant, and it should be possible to probe it experimentally.In particular, we discuss how a common numerical approximation corresponds, in fact, to a shift from the Boltzmann into the Gibbs volume formulation.We also see how the microcanonical temperature relates to previous phenomenological approaches.For a large vibrational energy excess, the common assumption of a linear relationship between temperature and energy (constant heat capacity) is valid.Nevertheless, dissociative and radiative cooling should bring the system to nonlinear regions.
For practical purposes, for a molecule with more than 10 atoms, I recommend using ( ) T .At this point, it is unclear which one delivers the correct answer.
We should also bear in mind that the microcanonical temperature models discussed here are strictly valid for molecules whose vibrational modes can be well-represented by a harmonic approximation.In the future, the present modeling should be extended to anharmonic systems to allow dealing with more general systems, like molecular clusters.
of multisets and the binomial coefficient, respectively.

T
using the gamma function are spot-on.Stirling's approximation in ( ) the result, but its main problem is at the very low energies near the zero-point level.As long we are not dealing with this region, we may favor Stirling's approximation given the algebraic simplicity of( )

Figure 2 -
Figure 2-top.The number of states
middle and the harmonic mean in the bottom figure.For a vibrational energy excess of 3.9 eV, the Boltzmann temperature reaches about 9013 K and the Gibbs volume, 7482 K.The growth of the number of states ( ) E  with E, although monotonic, shows significant fluctuations, especially in the low-energy region.Consequently, the numerically-computed Boltzmann temperature ( ( ) n BT in Figure2) also fluctuates around the 6-degenerated mean value.The integrated density of states ( )E washes out most of these fluctuations and yields a much smoother Gibbs volume temperature function (( )

Figure 3 .
Figure 3. T  Relative error between the numerical temperature and the Stirling's approximated temperature of the equivalent degenerated model with arithmetic mean frequencies computed for 30 random systems with N = 6, 7, and 8. Top: Boltzmann; bottom: Gibbs volume.
the low-energy region.Given the better performance of A  over H  , only the former will be used in the next sections.

4 -
top for a few molecules (GFN1-xTB level).They are all systems of astrochemical interest.The difference between Boltzmann and Gibbs volume temperatures is negligible for any system above 24 vibrational degrees (10 atoms).For isolated C60 at 808 K, Deng
volume (solid lines) and Boltzmann (dashed lines) temperatures for a few molecules (arithmetic mean frequency).N is the number of vibrational modes.Bottom: Density map of possible molecular systems with temperature 950  150 K, UV excitation 5  1 eV, and arithmetic mean frequency 1100  100 cm -1 .The density grows from yellow to blue.The dots illustrate the absorption peak of a few PAH molecules with excitation

20 a
random normal distribution with standard deviation given by the error bars, we arrive at Nat = 65

T
with Stirling's approximation) or equivalently ( ) l B T (the Boltzmann temperature for large N) and arithmetic mean of frequencies, which should predict temperatures within 3% margin of error.For smaller molecules, the temperature can be computed with either ( ) s G T or ( ) s B

Table 1 .
Mean relative error T  between numerical and degenerated-model approximated temperatures for Gibbs volume (G) and Boltzmann (B) formulations with harmonic (H) and arithmetic (A) mean frequencies.

Table 2 .
Summary of the microcanonical temperatures derived under different approximations from Boltzmann and Gibbs volume entropies for a degenerated set of N quantum harmonic oscillators with frequency  .For the non-degenerated problem,  is the arithmetic mean frequency of the spectrum.is the digamma function.For an isolated molecule wellrepresented by harmonic vibrational modes,( )