Medical, Pharma, Engineering, Science, Technology and Business

Physical and Theoretical Chemistry, University of Saarland, Germany

- *Corresponding Author:
- Vishwanathan K Physical and Theoretical Chemistry

University of Saarland

66123 Saarbrucken, Germany

**Tel:**496813020

**E-mail:**[email protected]

**Received Date**: January 24, 2017; **Accepted Date:** February 27, 2017; **Published Date**: March 10,
2017

**Citation: **Vishwanathan K, Springborg M (2017) Effect of Size, Temperature,
and Structure on the Vibrational Heat Capacity of Small Neutral Gold Clusters. J
Material Sci Eng 6: 325. doi: 10.4172/2169-0022.1000325

**Copyright:** © 2017 Vishwanathan K, et al. This is an open-access article
distributed under the terms of the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided
the original author and source are credited.

**Visit for more related articles at** Journal of Material Sciences & Engineering

The vibrational heat capacity Cvib of a re-optimized neutral gold cluster was investigated at temperatures 0.5-300 K. The vibrational frequency of an optimized cluster was revealed by small atomic displacements using a numerical finite-differentiation method. This method was implemented using density-functional tight-binding (DFTB) approach. The desired set of system Eigen frequencies (3N -6) was obtained by diagonalization of the symmetric positive semi definite Hessian matrix. Our investigation revealed that the Cvib curve is strongly influenced by temperature, size, and structure and bond-order dependency. The effect of the range of interatomic forces is studied; especially the lower frequencies make a significant contribution to the heat capacity at low temperatures. In addition to that, we have exactly predicted the vibrational frequencies (ωi) which occur between 0.55 to 370.72 cm-1, depending on the nanoparticle morphology at T=0 for small neutral gold clusters AuN=3-20. This result has been proved and confirmed by the size effect values. It was found that beside the particle size, geometric shape, defect structure and an increase in asymmetry of nanoparticles effects on heat capacity. Surprisingly, the Boson peaks are typically ascribed to an excess density of vibrational states for the small clusters. Finally, temperature dependencies of the vibrational heat capacities of the re-optimized neutral gold clusters have been studied for the first time.

Gold atomic clusters; Density-functional tight-binding
(DFTB) approach; Finite-differentiation approximation; Force
constants (FCs); Vibrational density of states (VDOS); Vibrational heat
capacity (C_{vib}); Boson Peaks (BP)

The study of nanostructured materials exhibiting novel properties is one of the most fascinating fields of current research. Small nanomaterial’s are of particular interest because of intriguing characteristics [1-3]. Nanoparticles with smaller dimensions may exhibit different properties in comparison with bulk material. The nanoparticles possess unique physic-chemical, optical and biological proper-ties which can be manipulated suitably for desired applications [4]. The advances in the field of nanoscience and nanotechnology has brought to fore the Nano sized inorganic and organic particles which are finding extensive applications as amendments in medicine and therapeutics, synthetic textiles and food packaging products [5]. The incorporation of engineered nanoparticles into household, personal care, consumer, and industrial products is increasing the exposure of humans and the ecosystems to these materials through production, transportation, storage, use, and disposal. Due to their small sizes, nano-materials (NMs) can enter into cells and interact with cell organelles and/or macromolecules and may thus disrupt the normal cellular functions [6-9]. Various NMs have been observed to show systemic effects when administered into systemic circulation, either intentionally for biomedical therapy or accidentally during environmental exposures, and may even cross the blood/brain barrier [6,8,10].

Clusters are well suited for a rapidly increasing number of applications and they have been an active eld of research for about a quarter of a century. Instead of reviewing all the literature, we refer the reader to the already existing review article by Baletto and Ferrando and also to the book by Wales [11,12]. Clusters can be viewed as solids at the Nano scale; yet molecular cluster chemistry and solid state chemistry have traditionally been considered as separate topics [13]. Nowadays, gold chemistry plays a very important role in Nano electronics and bio-nanoscience [14]. Particularly, gold clusters are of potential relevance to the Nano electronics industry and hence remain the subject of many experimental as well as theoretical studies. They contain edge atoms that have low coordination [15] and can adopt binding geometries that lead to a more reactive electronic structure [16]. Gold in the Nano regime, especially gold Nano crystals have shown size-sensitive reactive properties and are considered to be as promising chemical catalysts [17]. As one of the precious metals, gold has good corrosion resistance and extremely high stability and has been widely researched for biomedical applications [18].

The vibrational properties of clusters and small particles have been studied very intensively [19-26], and are vital for understanding and describing the atomic interactions in the cluster [27-32]. Thermal properties like heat capacity and thermal conductivity as well as many other material properties are strongly influenced by the vibrational density of states (VDOS). For this reason, a better understanding of the rules governing the vibrational properties of nanostructured materials is of high technological and must be given a high priority. The vibrational properties play a major role in structural stability [25,26,33].

The size-dependent properties of metallic clusters are currently of considerable interest, both experimentally [34] and theoretically [24,35]. Although the size effect on specific heat capacity has recently attracted much attention [36], many publications only focus on low temperature [37]. Some of the theoretical studies on the thermodynamic properties of clusters are based on molecular-dynamic simulations [11] from which the caloric curve, the heat capacity of clusters and the phase transitions can be determined. Nano clusters are interesting because their physical, optical and electronic characteristics are strongly size dependent. Often changing the size by only one atom can significantly alter the physical chemical properties of the system [38]. Many new periodic tables can thus be envisioned classifying differently-sized clusters of the same material as new elements. Potential applications are enormous, ranging from devices in nano-electronics and nanooptics [39] to applications in medicine and materials.

Reyes-Nava et al. [40] calculated the heat capacity for a few
NaN systems using the simple many-body Gupta potential, which
approximates the atomic interactions with an analytical description
that does not explicitly include electronic degrees of freedom. It was
found that solid-liquid phase transitions occurring over a certain
temperature range depend critically on the size N of the system. Lee
et al. [41] carried out accurate molecular dynamics simulations. From
a literature search [42], one can see that only a very few theoretical
studies attempt to calculate the thermodynamically properties of the
clusters directly through de-termination of the partition function Z.
Doye and Calvo calculated the partition function for Lennard-Jones
clusters with N ≤ 150 **(Figure 1)** [43].

In this study we combine numerical finite-difference approach and
DFTB method. At T=0, the vibrational frequency of a re-optimized
neutral gold cluster is obtained for optimized cluster of AuN=3-20
which was calculated by Dong and Springborg [44]. The desired set
of system Eigen frequencies (3N-6) is obtained by a diagonalization of
the symmetric positive semi definite Hessian matrix. The effect of the
range of interatomic forces has been studied. We found that the lower
frequencies made an excellent contribution to the heat capacity (to be
able to store significant heat energy) even at low temperatures **(Figure
2)**. This novel and reliable methodology is constructed to explain the
essence of the physical picture.

In our case, we are studying the finite-temperature behavior, which is sensitive to the size of the cluster [45] and the recent interest in planar gold clusters [46,47]. For example, according to Y. Dong and M. Springborg's [44] results, the gold clusters with up to N = 6 have a two-dimensional structure, whereas from N=7, the gold clusters form three-dimensional structures. However, in some of the research studies say that for the clusters with N=7-15, the structure can be either 2D or 3D or even both. However, experimental and theoretical studies have found that planar structures are stable up to around 15 atoms [48,49].

In the present work, we shall at first remember our earlier results on Vibrational Heat Capacity of Gold Cluster Au_{N}=14 at Low
Temperatures [50]. As an extension of that work, we shall subsequently
present new results devoted to the vibrational contributions to the
thermodynamic low-temperature properties of the other clusters.
Nevertheless, the purpose of the last part is to explore which kind of
information can be obtained by studying the heat capacities of the
Au_{N}=14 cluster and, in particular, to see whether the heat capacities can
be correlated to structural and/or energetic properties of the clusters.

The DFTB [51-55] is based on the density functional theory of
Hohenberg and Kohn in the formulation of Kohn and Sham. In
addition, the Kohn-Sham orbitals ψ_{i}(r) of the system of interest are
expanded in terms of atom-centered basis functions {ф_{m}(r)},

(1)

While so far the variational parameters have been the real-space
grid representations of the pseudo wave functions, it will now be the
set of coefficients cim. Index m describes the atom, where ф_{m} is centered
and it is angular as well as radially dependent. The ф_{m} is determined by
self-consistent DFT calculations on isolated atoms using large Slatertype
basis sets.

In calculating the orbital energies, we need the Hamilton matrix elements and the overlap matrix elements. The above formula gives the secular equations

(2)

Here, cim's are expansion coe cients, is for the single-particle
energies (or where are the Kohn-Sham eigenvalues of the neutral),
and the matrix elements of Hamiltonian H_{mn} and the overlap matrix
elements S_{mn} are defined as

(3)

They depend on the atomic positions and on a well-guessed density ρ(r). By solving the Kohn-Sham equations in an effective one particle potential, the Hamil-tonian is defined as

(4)

To calculate the Hamiltonian matrix, the effective potential V_{eff} has to be approximated. Here, being the kinetic-energy operator being the effective Kohn-Sham potential, which
is approximated as a simple superposition of the potentials of the
neutral atoms,

(5)

is the Kohn-Sham potential of a neutral atom, r_{j}=r-R_{j} is an
atomic position, and R_{j} being the coordinates of the j-th atom. The
short-range interactions can be approximated by simple pair potentials,
and the total energy of the compound of interest relative to that of the
isolated atoms is then written as,

(6)

Here, the majority of the binding energy is contained in the difference between the single-particle energiesi of the system of interest and the single-particle energies of the isolated atoms (atom index j, orbital index mj), is determined as the difference between and for di-atomic molecules (with being the total energy from parameter-free density-functional calculations). In the present study, only the 5d and 6s electrons of the gold atoms are explicitly included, whereas the rest are treated within a frozen-core approximation [53,55,56].

**Re-optimization and numerical force constants (FCs)**

The vibrational frequencies of the gold clusters were calculated within the harmonic approximation by diagonalization of the Hessian matrix. The finite-difference method has been implemented within DFTB approach for our calculation (a finite-difference approximation to calculate the force constants). We found a total energy over those gradients were extended for a small displacement ds=(± 0.01) a.u. within the equilibrium coordinates of a previously optimized structure (at T=0) by Dong and Springborg [44]. However, the DFTB method has some difficulties to extract the force constants which are most important for our spectrum calculations. Mainly, to get one set of hessian matrix it is necessary to compute two times for both positive and negative gradients. It is a reasonable value and allowed us to discriminate between the translational, rotational motion (Zeroeigenvalues) and the vibrational motion (Non-Zero-eigenvalues).

In our case, we have calculated the numerical first-order derivatives
of the forces (F_{iα}; F_{jβ}) instead of the numerical-second-order derivatives
of the total energy (Etot). In principle, there is no difference, but
numerically the approach of using the forces is more accurate,

(7)

Here, ds is a differentiation step-size and M represents the atomic mass, for homo-nuclear case. The complete list of these force constants (FCs) is called the Hessian H, which is a (3N × 3N) matrix. Here, i is the component of (x, y or z) of the force on the j'th atom, so we get 3N.

**Calculation of vibrational heat capacity**

Finally, from the calculated vibrational frequencies and with the use of Boltzmann statistics [57-59], we can get the formula to investigate size, structure and temperature effects on the vibrational heat capacity of clusters,

(8)

Here, N is the total number of atoms in a cluster, h is the reduced
Planck's constant, ω_{i} is vibrational frequencies (low(min); high(max)),
k_{B} is Boltzmann's constant and T is the absolute temperature. Naturally,
zero frequencies are excluded from summation (3N-6) in eqn. (8). In
this study, we focus only on the vibrational part of the heat capacity,
i.e., on C_{vib}.

**The Classical Treatment of C _{trans} + C_{rot} + C_{elec}:** The heat capacity of
a system with independent degrees of freedom can be ap-proximated
as a sum of its individual contributions,

(9)

Here, C_{trans}, C_{rot}, C_{vib} and C_{elec} are called as translation, rotational,
vibrational and electronic heat capacity. We have proceeded to calculate
the specific heat C_{vib} contribution due to the vibrational energy. In this
study, we focus only on the vibrational part of the heat capacity, i.e.,
on C_{vib}. The other contributions are treated classically. Most often, the
electronic excitation energies are much larger than k_{T} and we neglected
the electronic partition function.

In this article, we present an in-depth study on the behavior
of vibrational frequency at T=0 K, the heat capacity (C_{vib}) of a reoptimized
neutral gold cluster (Au_{N}, N=3-20) at T=0.5 to 300 K and the
Boson peaks C_{vib}/T^{3} at T=0.5-30 K. Our results are of special interest
since the Boson peaks are thermally activated as well as the local
atomic arrangements (within a cluster or group of strongly correlated
atoms) but still are constrained by surrounding atoms [60]. This is an
interesting point and may shed light on the enhancement of the heat
conductance process mentioned in Physical Review Letters 103, 048301
(2009) [61] and PLOS ONE 8, e58770 (2013) [62]. As a function of
temperature: at low temperatures the Boson peak corresponds to these
local rearrangements with (mutual) reaction of one atom to another
[63].

We applied the method outlined above for studying the properties
of neutral gold atomic clusters with sizes Au_{N}=3-20 [44]. We analyzed
vibrational spectrum and vibrational heat capacity which is being a
striking feature and a classical size effect. Our investigation revealed
that the C_{vib} curve shows a very strong influence of temperature, size
and structure dependency. In addition to that we have also plotted
the C_{vib}(min) and C_{vib}(max) curves separately for the lowest (ω_{min}) and the
highest (ω_{max}) vibrational frequency modes and then studied their
physical behaviors.

We have compared our theoretical results with the theoretical and experimental results calculated by Bishea and Morse [64], Gruene
et al., [65] Mancera [66], Molina et al. [67], and with Nose [68], for
the vibrational spectrum of Au_{N=3,4,5,6,7,19,20} clusters. Interestingly, our
results were in excellent agreement with their results.

**The vibrational frequency (ω _{i}) and properties**

**Figure 1** shows the low (at the least) and the high (at the most)
frequency ranges for cluster Au_{N=3-20}. The lowest and the highest
frequency ranges are 0.55-34.18 cm^{-1} and 165.46-370.72 cm^{-1},
respectively.

The relative importance of high and low frequencies naturally depends on the size, structure and frequency spectra of the clusters. The size is super critical to the physiochemical properties of Au cluster. It is certainly affected by the ratios of dangling bond to overall bulk bonding number. Nanoparticles have a substantial fraction of their atoms on the surface. The surface energy is the (thermodynamically unfavorable) energy of making dangling bonds' at the surface. Atoms at the surface are under-coordinated, and because breaking bonds results in a loss of energy, surface atoms always have higher energy than atoms in the bulk. It is surely expected that such a vibrational spectrum depends on the material, size, and shape of clusters and nanoparticles [22]. Most importantly, the vibrational properties of atomic clusters are a fingerprint of their structures and can be used to investigate their thermodynamic behavior at low temperatures [23].

Of course, there is an increasing amount literature illustrating the
conceptual and practical relevance of two dimensional (2D) systems
with long range interactions [69-71]. The vibrational frequency of FCs
contributions comes from both the symmetric and the asymmetric
stretch and bending modes. We believe that bond-stretching (k_{s}) and
bond-bending (k_{b}) force constants depend on the nearest neighbor
distance obtained from lattice vibrations [72]. It is important to note
that the stretching modes moves to lower frequency (approaching
zero), and the bending mode b moves to higher frequency. The low
frequencies are probably due to the influence of large collective
motions of atoms and the high frequencies due to localized motions of
atoms [73], which diminish bond length fluctuations. The interatomic
interaction that is responsible for the frequency ranges and the variation
of the FCs led to a shift in the mode frequencies [74].

**The vibrational heat capacity C _{vib(min)} for the lowest frequency
ω_{min} at 0.1-30 K**

**Figure 2** has shown C_{vib}(min) for all the clusters. The frequency of
the lowest vibrational state of a nanoparticle is ω_{min}. The asymptotic
behavior of heat capacity at low temperatures takes the form with
respect to α_{i}=ξ_{min}=ω_{min}=T of eqn. (8). The ω_{min} value is determined by
the size, shape, and defect structure of the nanoparticle.

Interestingly, for all the clusters the C_{vib(min)} starting at ranges
(critical temperature, T_{c}) are within the temperature range of about T
=Tc=0.1-3 K. As a result, the effect of the lowest frequency modes is
dominant for all the clusters. This is very clear evidence that shape of
C_{vib(min)} curves differ with respect to the size and structure of the clusters.

The clusters Au_{N} (N=4, 5, 6, 8, 9, 11, 12, 15, 17, 18, 20) have
extremely low ωmin frequencies between 0.55-9.83 cm^{-1}. Which is even
below the range of Far Infrared FIR, IR-C 200-10 cm^{-1}? The starting at
ranges are within the temperature range of about T=Tc=0.1-0.7 K. In
the same manner, for the clusters Au_{N} (N=3, 7, 10, 13, 14, 16, 19) the
frequencies are in between 10.08-34.17 cmare within the temperature range of about T=T_{c}=0.7-3.0 K.. Which is within the range
of Far Infrared FIR, IR-C 200-10 cm^{-1}. Due to this the starting at ranges are within the temperature range of about T=T_{c}=0.7-3.0 K.

In **Figure 2**, particularly the cluster Au_{14} (black line) and Au_{16} (blue
line) both are very similar due to their ω_{min} values, 17.02 and 17.13
cm^{-1}. However, even though they have different sizes, they have the
very same C_{s} symmetry.

Moreover, the C_{vib(Min)} curve shows the asymptotic behavior, rising
smoothly and finally reaching a linear flat at temperatures above
T=15 K for all Au_{N} (N=3-20) clusters. This is scientifically significant
with respect to the cluster size and shape. The minimum vibrational
frequency ω_{min} plays an important role in determining the shape of the
C_{vib} curve at low temperature.

Anomalous behavior of neutral gold clusters: In **Figure 2**, we have
noticed some anomalous behavior in the shoulder of the heat capacity
C_{vib(min)} curves between the clusters Au_{N} (N=10-19). And the same
behavior is exhibited between the clusters Au_{N} (N=4-5), which is due
to the size and structure de-pendency of the clusters. The data reveal an
anomalous contribution to the heat capacity at low temperatures. It was
identified that this anomaly in heat capacity is caused by the effect of
disorder in the cluster size and structure. Probably, gold nanostructures
exhibit anomalous thermal behavior such as the shape transformation
of Nano rods, generation of nanoparticles by heating a mesh etc., at
extremely low temperature [75]. The results of our work within the
numerical model caused us to consider the intrinsic relation between
normal (anomalous) particle diffusion and normal (anomalous) heat
conduction, along with size dependent thermal conductivity.

**The vibrational heat capacity C _{vib}(max) for the highest frequency
ωmax at 10-1000 K**

**Figure 3** has shown C_{vib}(max) for all the clusters. The frequency of
the highest vibrational state of a nanoparticle is ωmax. The asymptotic
behavior of heat capacity at low temperatures takes the form with
respect to α_{i}=ξmax=ωmax/T of eqn. (8). The ωmax value is determined by
the size, shape, and defect structure of the nanoparticle.

Here at C_{vib}(max) curve, we did not see much of a difference on the
shoulder (asymptotic curve) that means they are very much closer to
each other. Of course, here we do not see anomalous behavior as we
have observed in the case of low frequency **(Figure 2)**. The reasons
are, the high frequencies do contribute to the heat capacity when
the temperature increases from 100 to 300 K (C_{vib}(max)increases from
0.04 k_{B/ }atom to 0.11 k_{B}/atom). Most importantly, as this takes place,
vibrational modes with lower frequencies give larger contribution
to the heat capacity at ambient temperatures. However, the high
frequencies do not generate large C_{vib}(max) differences when comparing
different cluster sizes **(Figures 2 and 3)**. The increase in heat capacity
depends on the particle size, shape and the clusters perfection.

In **Figure 3** you can see C_{vib}(max) for all the clusters, and, most
importantly, you can see that C_{vib}(max) starting at ranges are within the
temperature range of about T=10-37 K, where the effect of the highest
frequency modes are dominant for all the clusters. It is evident, that, the
starting points varies with respect to the critical temperature Tc from
which the C_{vib}(max) curve raises either suddenly or smoothly. The overall
temperature ranges are 10-1000 K. The maximum high frequency falls
within the range of Mid Infrared MIR, IR-C 3330-200 cm^{-1} for the
corresponding harmonic frequency ranges of 165.46-370.72 cm^{-1}, with
the one exception of the frequency 165.46 cm^{-1} which is for the AuN
(N=4) cluster.

**Comparison:** The intensity of the enhancement of the specific heat capacity for the lowest vibrational frequency comes in the temperature
range of 0.1-15 K **(Figure 2) **but for the highest vibrational frequency it
covers the temperature span 10-500 K** (Figure 3)**. In both of these cases,
above these maximum temperatures it gets saturated. Nevertheless, the
vibrational heat capacity C_{vib}(min) and C_{vib}(max) is zero at T=0 but there are
variations in the at ranges which is only because of the rst mode of the
vibrational frequencies, and it nally asymptotes towards 0.125 k_{B}/atom
at high temperatures.

**Superposition of the lowest, the middle and the highest frequencies**

The ESI ϯ shows the frequency ranges for all the clusters, AuN
(N=3-20). **Figures 4-7** shows how C_{vib} varies with cluster size, structure
and temperature. These results show that C_{vib}(min) curve for the lowest
frequency, by that we mean very low frequency shoulder, varies for
different clusters with respect to their size and structure. It is almost
identical with the middle C_{vib}(mid) and the high frequency C_{vib}(max) curve.
These results will be important to describe heat transfer at Nano scale as
well as Casimi-Lifshitz forces between clusters due to thermal quantum
fluctuations [76].

**Figure 5:** Only for the exact middle frequency (ω_{mid}) as a function of cluster size (N), Au_{3,5,7,9,11,13,15,17,19} [odd number of clusters] at T=0 K, and the respective modes are 2,5,8,11,14,17,20,23,26. The lowest middle frequency (ω_{min}) 79.22 cm ^{-1} of Au (N = 19), and the highest middle frequency (ω_{max}) 106.38 cm ^{-1} of Au (N=7).

**The lowest frequency:** **Figure 4** shows the very beginning of the
low and the high frequencies, Au (N=5), 0.55 cm^{-1} and Au (N=10),
34.18 cm^{-1}. However, both the frequency values are only for the lowest
frequency range. Here, one of the frequency ranges is not within the
Far Region of IR (Far Infrared FIR, IR-C), 200-10 cm^{-1}.

A very interesting observation that Au (N=5) has very high heat
capacities (C_{vib}(min)= ~ 0.125 k_{B}/atom at 5 K), whereas Au (N=10) has
low heat capacities (C_{vib}(min) ≤ 0.01 kB/atom at 5 K). Comparisons of
the relative low-frequency modes which were produced at close and
distant locations for departures and arrivals at C_{vib}(min). The effects of
temperature on the low-frequency vibrational spectrum and local
structural arrangements, and the low-frequency density of states
distributions reveal that increasingly transverse atoms motions play a
dominant role in controlling the band corresponding to the bending or
transverse oscillations of the nano-particles at low temperatures.

**The exact middle frequency (only possible for the odd sized
clusters): Figure 5** shows that the exact middle frequencies for odd number of clusters, Au_{3,5,7,9,11,13,15,17,19}, out of which Au (N=19), has the
lowest middle frequency,79.22 cm^{-1} and Au (N=7), has the highest
middle frequency, 106.38 cm^{-1} which is the frequency range and the
rest of the odd number of clusters which falls within this range **(Figures
5 and 6)**. Here, the frequency ranges are within the Far Region of IR
(Far Infrared FIR, IR-C), 200-10 cm^{-1}.

The highest frequency: In the same manner, **Figure 7** shows mainly
only the very end of low and high frequencies, Au (N=4), 165.46 cm^{-1} and Au (N=20), 370.72 cm^{-1}. However, both the frequency values are
only for the highest frequency range. Here, the frequency ranges are
within the Far Region of IR (Far Infrared FIR, IR-C), 200-10 cm^{-1} and
also within the Middle Region of IR (Mid Infrared MIR, IR-C), 3330-
200 cm^{-1}. From the **Figures 4-7**, we certainly confirmed that the low
frequency contributions are much higher than the middle and the high
frequency contributions to the heat capacity, C_{vib}. Amazingly, C_{vib} curves
differ significantly with respect to the low/middle/high frequency as a
result of temperature as well as size and structure of the clusters.

**The vibrational spectrum (ω _{i}) of Au_{N=3} and the reliability of
our model**

Bishea and Morse [64] did research on the spectrum of Au_{3}. For
example, they found that the totally symmetric breathing mode in the
excited electronic state had a frequency of 182.9 cm^{-1}. We should expect
the totally symmetric breathing mode in the ground state will have a
somewhat higher frequency, perhaps around 200-250 cm^{-1}. The gold
trimer, Au3, has three normal modes, two of which may be degenerate,
depending on the symmetry. Moreover, additional modes may be in
other experiments. In our case, after the re-optimization the vibrational
frequencies were found to be 19.21, 87.47 and, 246.21 cm^{-1}.

Throughout our calculated vibrational frequencies, we assume
that the ground state of Au_{3} must have either C_{2v }or C_{s} geometry.
We consider the various possible ways that we can arrange the three
atoms in Au_{3}, as the Jahn-Teller distortion will drive it away from the
equilateral D_{3h} configuration [77-79]. For example, A_{u3} without the
spin-orbit coupling (SOC) effects exhibits a Jahn-Teller distortion
towards the C_{2v} symmetry; however, with spin-orbit coupling it
recovers the D_{3h} symmetry.

**An important point:** The very low frequency mode 19.21 cm^{-1} **(Figure 1)**, corresponds to a motion that is not well-approximated as a
harmonic motion, since motion on the potential energy surface along
this coordinate converts the molecule from one equivalent isosceles
triangle to another. As we move along this coordinate, the potential
energy goes up (initially quadratically), then becomes anharmonic,
then reaches a maximum, then descends into a different minimum
where a different gold atom lies at the apex of the isosceles triangle.
This will have implications for the heat capacity (C_{vib}) **(Figure 8)**, which
we have modeled as a pure harmonic oscillator.

With this confirmation of Au_{3} vibrational modes, and with the
help of Gabedit package [80]. Tolerance for principal axis classi cation:
0.00500 in angstrom (A) and Precision for atom position: 0.09399 in
angstrom (A) we went back to the total energies, structures and then
verified their symmetry of global structure optimization which were
predicted by Dong and Springborg [44]. As a result, some of those A_{un} clusters symmetries were differed (For example, Au_{3} (D_{2}), Au_{9} (D_{2v}/D_{2}),
Au_{10} (D_{2}), Au_{11} (Cl), Au_{12} (Cl ), Au_{13} (Cs), Au_{15} (Ci ), Au_{17} (Cl ), Au_{18} (C_{2}),
Au_{19} (Cl), and Au_{20} (Cl)) **(Figures 1-6 and 9-14)**.

The discrepancy at the frequency modes: For the global minimum
energy structure, if the θ >900 then the modes are 24.7, 127.0, 183.1
cm^{-1} and if the θ >600 with acute structure the modes are 69.0, 89.3, 167.1 cm^{-1} again, and if the <600 with acute structure the modes are 5.3,
109.2, 168.6 cm^{-1}, which are Harmonic frequencies for various isomers
of Au3 calculated using PBE/VDB by Mancera and Benoit [66]. So this
gives a confirmation that the vibrational modes varies with respect to
the bond angle and isomers. Even for the experimental values [64] the
modes are 61.9, no value (silent) and 179.9 cm^{-1} as well as, in another
experimental calculation [81], no value (silent), 118 and 172 cm^{-1}.

**The vibrational heat capacity C _{vib} of gold neutral clusters**

**Figures 15-17** shows C_{vib} as a function of temperature for di erent
cluster sizes. Out of AuN=3-20 cluster we have selected some of the 14 special clusters which display different interesting properties. The
vibrational heat capacity C_{vib} has been plotted with respect to α_{i}=ω_{i}/T
and α_{i}=ω_{i} /(0:6950356* T ) of eqn. (8) at temperature, T=0.5-300 K.

**Figure 17:** Au_{5, 6, 9, 12}: C_{vib} per atom as a function of temperature at 0.5-300 K. α_{i}=ω_{i}/(0.6950356*T ) of eqn. (8). and the corresponding Boson peak C_{vib}=T^{3} vs. T.

The C_{vib} curve is a sum of C_{vib} curves, one for each normal mode of vibration. Each of these individual C_{vib} curves is S-shaped, but with the
inflection points (where the curve changes from concave up to concave down) occurring at different temperatures. The temperature of the
inflection point is proportional to the vibrational frequency for that particular normal mode. Since each individual C_{vib} curve is S-shaped,
the sum must be S-shaped (to a degree) also. However, because
our molecules/clusters have some normal modes of vibration with
extremely low vibrational frequencies, some of the individual curves
being summed to give the total C_{vib} curve will have their inflection
points at extremely low temperatures. If we examined the C_{vib} curve at
extremely low temperatures (perhaps lower than we have calculated, at
0.4 K → 0 K), we would see that they start out at with C_{vib}=0, then rise
from C_{vib}=0 (concave up), then go through an inflection point at a low
temperature. All of this follows logically from the fact that the overall
C_{vib} curve for a single molecule/cluster is a sum of C_{vib} curves for each
vibrational mode.

Au_{3} (a planar-triangular C_{2v}): In fact, under careful observation, in
the C_{vib} curve, there is a minute \Jerk/bending", in between 0.15 and
0.25 kB/atom at the temperature range about 0.5 to 10 K **(Figure 8)**.
Thus, there seems to be a strong correlation between the atom located
coordination numbers and the bond angle, which result in a huge
variation, due to the fact that the normal mode energy intervals (19.21;
87.47 and; 246.21 cm^{-1}) are so large which has a great influence on the
stability of the cluster.

Au_{6} (a planar-triangular D_{3h}), Au_{7} (a decahedron D_{5h}) and Au_{8} (a
tetrahedron T_{d}):

In **Figure 15**, the C_{vib} curve starts with the temperature at 0.6 K, and
the motion of the atom at 10 K, and makes a hump-like shape which
is an a fascinating phenomenon to observe. Most importantly, N=6
is a much more rapidly increasing function of T at low temperature
than in the cases of the other cluster sizes (but still C_{vib} rises smoothly
and reaches the size effect value). However, it has more energy stored
at low temperature, but the C_{vib} curve crosses over at about 40 K of
the Au (N=8) and 50 K of the Au (N=7) clusters which demonstrates
its rapidness. Nevertheless, the main reason could be, the minimum
frequency start with degenerate, followed by a double state degenerate,
followed again double state degenerate and some single state
degenerates and finally end with double state degeneracy, which also
provides another confirmation that atoms are located in a periodic and
zigzag arrangements with respect to the nearest neighboring atoms
coordination. In the case of Au_{7} and Au_{8} both C_{vib} starts at 0.5 K and
rises smoothly with asymptotic behavior, and these are the only clusters
which have more double state degeneracy and triplet state degeneracy,
respectively. So the structural dependency of heat capacity is being
confirmed with the triple, double and single state degeneracies.

Interatomic interactions in the clusters reflect a frequency
distribution with a high degree of degeneracy due to the high symmetry
of their lowest energy configurations (the potential energy with respect
to the atomic coordinates). No degeneracy was obtained for the other
gold clusters (except Au_{N} , (N=6; 7; 8)) due to the absence of symmetry
in their minimum energy.

Au_{4} (D_{2h}); Au_{10} (S_{8}); Au_{13} (C_{1}); Au_{15} (C_{1}); and Au_{20} (C_{1}): In **Figure 16**,
it is shown that the C_{vib} curve heat capacity goes sharply up from zero,
without the small region where the heat capacity is very close to zero.
The very low frequency modes begin contributing to the heat capacity
at lower temperatures. It is possible to have very low lying excited
electronic states that can be thermally excited at low temperatures.
If that is the case, there will be an additional contribution to the heat
capacity due to electronic excitation. The heat capacity of the real
substance would be higher than what we have calculated. However, no
experimental data is available for comparison. Surprisingly, for each
size C_{vib} there is a monotonously increasing function of T (which tends asymptotically). The temperature dependence of the individual modes **(Figures 4-7)** led to the total vibrational heat capacity for all the clusters.

The calculated heat capacity curve remains near zero at the lowest temperatures, then begins monotonically rising near 1.75 K. Similar behavior is reflected in all of the clusters that were studied, although the precise location of the transition from near-zero values to the monotonic increase varies. The existence of this low-temperature, near-zero heat capacity regions arises because all of these clusters have a lowest vibrational frequency. The heat capacity only begins to deviate significantly from zero when the thermal energy, kT, has a significant likelihood of exciting the lowest frequency vibrational mode. This behavior is in effect a quantum size effect, resulting from the sparse vibrational density of states.

In addition at low temperatures it is the lowest-energy (lowestfrequency) vibrational states that mainly contribute to heat capacity. Indeed, if the frequency of the lowest vibrational state of a nanoparticle is at a minimum, the asymptotic behavior of heat capacity at low temperatures arises. This is a region T=0.5-1.75 K where the effect of the lowest frequency internal modes are dominant for all the clusters.

**Au _{14}(C_{s}): **In our case, Au

**Figure 16** also shows that interatomic vibrations in heat capacity
can be neglected at around (1.75-0.5 K) and the C_{vib} curve rises gradually
with temperature, which is an exact signature of the vibrational changes.
The smooth change, especially pronounced at low temperatures (1.75-
25 K), is even more interesting.

It is due to the energy increase of the system. For a given size, the
reduction of the heat capacity is more significant at lower temperatures.
As the temperature is raised, the difference between the vibrational
energies becomes progressively more conspicuous. There can never be
any method which explains the difference as long as we retain a simple
harmonic oscillator. However, the C_{vib} curve gives a confirmation that
is temperature dependent. In the quantum limit, T (1.75- 0.5 K) →0, the
C_{vib} curve approaches zero, while in the classical limit, T (300 K) →high
temperature →size effect, the curve moves towards a value of C_{vib}=2.50
kB/atom, which clearly indicates a temperature influenced size effect/
dependency.

**Au _{14} and Au_{14}:** For Au clusters, Koskinen et al., [51] calculated
above the ground states of N=11, 12, 13, and 14, respectively,
corresponding roughly to T=1000 K. Particularly, they have shown
Au-

Moreover, the expected absolute value C_{vib} should be 2.57 k_{B}/atom.
In our case, the difference is only 0.07 k_{B}/atom which are reasonable
within the numerical accuracy. Nevertheless, C_{vib} has been achieved
almost an accurate value, 2.56(389) k_{B}/atom when the temperature is
high enough, 950 K*. But in the Koskinen et al., [51] case, the di erences
were higher 1.48 and 0.49 k_{B}/atom than the expected absolute value
for both 3D liquid and 3D solid phases. This discrepancy is due to the
minimum energy difference of the anionic [51] and neutral clusters
(which are more stable) [44]. So we conclude that the 2D planner
structure is preferential over the 3D Structure.

**Au _{18} (D ):** In our case, Au

Moreover, the expected absolute value (for size-dependent) should
be C_{vib}=2.67 k_{B}/atom for neutral cluster Au_{18}. In our case, the difference
is 0.08 kB/atom from the expected value. Again, nevertheless, C_{vib} has
been achieved almost an accurate value, 2.65(895) k_{B}/atom when the
temperature is high enough, 950 K*. With this we can be sure that the
structure can be D_{1} for Au_{18}. However, the C_{vib} curve shows that the
heat capacity goes sharply up from zero, excluding the small region
where the heat capacity is very close to zero, at T=0.5-1.25 K. The rest
of the clusters size effect has been addressed well.

The acoustic vibrations are more important at low temperatures, because they dominate the heat capacity [20-26]. Low-frequency modes of harmonic systems can be related to the small amplitude of acoustic waves, which are experimentally observed in all elastic bodies. This implies that, at low temperatures, the specific heat is largely determined by the low frequency part of the vibrational spectrum and it is only at high temperatures that a substantial portion of the spectrum comes into play.

The vibrational (Phonon) density of states has been calculated by Sauceda and Garzon [25]. Eigenvectors of normal modes are associated with low and high frequency vibrations. This behavior is caused by the stiffening of the bonds (i.e., the frequency shifts to a higher value as the strength increases). Such a local stiffening of the diagonal terms of the FCs matrix is known to lead to the formation of localized oscillation modes [83]. The nature of the bond is most readily interpreted through the eigenvectors and eigenvalues of this FCs Hessian matrix. The most important contribution is a short-range order in the disordered state of the cluster. FCs (stretching and bending modes) matrices, independent of symmetry and magnitudes, are strongly correlated with the bond lengths. A small change in bond length upon disordering can have a large effect on FCs and vibrational entropy. The general increase in the bond angle in this series indicates an increasing repulsion between the bonds and this is consistent with the increasing bond order.

The contribution of the vibrational free energy is related to the disorder in the FCs. In general, gold clusters are not so strongly disordered, having only minor positional disorder. Whereas the zigzag structure (for nonlinear clusters) has lower energy, in contrast, the zigzag structure within the clusters readily changes into 2D or 3D structures with its nearest neighbors towards better stabilization by multi-coordination [84]. We must remember that each normal mode acts like a simple harmonic oscillator, with a concerted motion of many atoms. The center of mass does not move. All atoms pass through their equilibrium positions simultaneously and normal modes are independent; they do not interact. This means that normal modes do not exchange energy. This is only true in the absence of harmonic terms, which is a theoretical approximation that is never achieved in reality.

For example, if the symmetric stretch is excited, the energy stays
in the symmetric stretch. Asymmetry mainly affects the heat capacity
at low temperatures; fragmentally both the internal energy and the
entropy. Obviously, the C_{vib} curves are fairly different: the lower
frequencies make a larger contribution to heat capacity. The absolute
heat capacity value depends strongly not only on the size of cluster, but
also on their shape and structural ordering.

Au_{5}(C_{2v}); Au_{6}(D_{3h}); Au_{9}(C_{s}) and Au_{12}(C_{1}): The energy gap of
vibrational modes, the influence of the edge atoms and its effects (\
hump" shape) in **Figure 17**. The energy intervals are large in between the
modes which vary with respect to the cluster size, structure, symmetry,
molar mass and the arrangements of bond orders within the clusters.
Particularly, at the end of the last two modes of these following clusters.
Au_{5} (C_{2v}): Mode 8 (224.53 cm^{-1}) and Mode 9 (276.83 cm^{-1}), Au_{6} (D_{3h}):
Mode 10 (178.30 cm^{-1}) and degeneracy Modes 11, 12 (282.99 cm^{-1}),
Au_{9} (C_{s}): Mode 20 (220.93 cm+) and Mode 21 (313.24 cm^{-1}), Au_{12} (C_{1}):
Mode 29 (264.76 cm^{-1}) and Mode 30 (325.89 cm^{-1}). In addition to that
the degrees of degeneracy also played a major role, due to the fact of
motion of the atoms at the temperature 0.5 to 7 K range, the C_{vib} curve
shows a hump like shape which is an excellent phenomena on these
clusters, Au_{5}, Au_{6}, Au_{9} and Au_{12} **(Figure 17)**. Thus, we believe that more
energy can be stored below 7 K. This also could be the cause of the edge
atoms occupying the surface of the clusters.

Thus, in these AuN (N=3-20) cases, frequencies are low and the curve
is squeezed so that this point on the curve falls at a low temperature. If
the frequencies are much higher, the same point on the curve will fall
at a much higher temperature. The heat capacity remains significantly
above zero at lower temperatures. This is because the neutral gold
cluster has much lower vibrational frequencies than most molecules
have. Nevertheless, our investigation revealed, the vibrational heat
capacity curve shows a very strong influence of size, temperature,
structure and bond-length (stretching, bending) dependency **(Figure 18)**.

The Boson peak (BP) C_{vib}/T^{3} vs. T of the neutral gold clusters: As shown in **Figures 8, 15-17**, with respect to the eqn. (8) and the corresponding
Boson peak C_{vib}/T^{3} vs. T were plotted at temperature within the range
T=0.5-30 K. Truly, we are surprised by our observation of a Boson peak
in our nanoparticles. Hao Zhang and Jack F. Douglas [85,86] usually
study the Boson peak from the velocity autocorrelation function, but
they are both concerned about a \excess" contribution to the vibrational
density of states. This feature has been observed in metal nanoparticles
and zeolites and attributed to the coordinated harmonic motions
of groups of atoms in the boundary region of the particle. There are
similarities here to a glass because the surface of a nanoparticle has
many features in common with this class of materials. In cases where
the modes have been resolved, the Boson peak has corresponded to
a ring of oscillating particles, the relatively low mass being related to
the relatively high mass of these modes. Farrusseng and Tuel [87] also
studied the perspectives on zeolite-encapsulated metal nanoparticles
and their applications in catalysis.

One of the universal features of disordered glasses is the “Boson peak”, which is observed in neutron and Raman scattering experiments. The Boson peak is typically ascribed to an excess density of vibrational states. Shintani and Tanaka [88] studied the nature of the boson peak, using numerical simulations of several glass-forming systems. They have discovered evidence suggestive of the equality of the Boson peak frequency to the Io e-Regel limit for “transverse” phonons, above which transverse phonons no longer propagate. Their results indicate a possibility that the origin of the Boson peak is transverse vibrational modes associated with defective soft structures in the disordered state. Furthermore, they suggest a possible link between slow structural relaxation and fast Boson peak dynamics in glass-forming systems.

However, Malinovsky and Sokolov [89] found that the form of a low-frequency Boson peak in Raman scattering is universal for glasses of varying chemical composition. They have shown that from the shape of the Boson peak one can determine the structural correlation function, i.e., the character of violations of ordered arrangement of atoms within several coordination spheres in non-crystalline solids. Most importantly, very recently Milkus and Zaccone found [90] that bond-orientation order is not so important for the boson peak. Whereas a much more important parameter is the local breaking of inversion symmetry.

Indeed, there is similar behavior of heat capacity in glasses and clusters at low temperatures. Glasses have some distribution of interatomic distances and modification of atomic coordination induced by disorder, compared to periodic crystals [91]. In clusters or nanoparticles, it is caused by the reduced atomic coordination of the surface atoms. Because the ratio of surface to volume is large, the number of atoms with reduced coordination is significant. The vibrations of surface atoms enhance the VDOS at low energies and the heat capacity increases.

The Boson peak is usually studied in glasses, where enhancement
of heat capacity is induced by disorder. The modeling of clusters may
be important for understanding the mechanism which lead to this
effect. In glasses, there are different local con gurations of atoms that
may be simulated by isolated clusters. In particular, it was found that
the vibrational density of states (VDOS) exhibit an excessively lowfrequency
contribution. A corresponding low-temperature peak is
observed in the temperature dependence of the specific heat if plotted
as C_{vib}(T )/T^{3}.

Au_{3}(C_{2v}): **Figure 8** shows that the increase in Boson peak curve
C_{vib}=T^{3} from 0-0.00035 k_{B}/atom [violet line] for i=ω_{i}/T and 0-0.001 k_{B}/ atom [green line (Aqua-marine 4)] for α_{i} =i /(0:6950356*T ) found for
T→0→high temperature; 30K, and the maximum deviation at 6.00 K
and 4.9 K, respectively, are due to a strong disorderly nature in the
cluster. In addition to that conformed with the parameter dependency.

Au_{5}(C_{2v}); Au_{6} (D_{3h}); Au_{9}(C_{s}) and Au_{12}(C_{1}): **Figure 17** shows that the
calculated C_{vib}/T^{3} vs. T for Au_{N} (N=5; 6; 9; 12) in the temperature 0.5-10
K range. The increase in Boson peak C_{vib}/T^{3} from 0.88503 k_{B}/atom (the
maximum deviation with higher amplitude at 0.6 K, Au_{5}. Nevertheless,
with the lesser amplitude the deviation occurs at 0.6 at 0.6 K for the
other clusters, Au_{6}, Au_{9} and Au_{12}) found for T →0 to higher temperature,
the rest of the clusters are in the interval between 0.5 to 4 K.

Howsoever, **Figure 16** also shows the calculated C_{vib}/T^{3} vs. T for
Au_{N} (N = 10; 14; 18; 20) in the temperature 0.5-30 K range. The detailed
confirmation is given below

**Au _{10}(S_{8}): **There is strong evidence that the maximum deviation
of the Boson peak C

**Au _{14}(Cs): **Based on the experimental observations, the maximum
in silica glass is placed at about 10 K and 8.5 K for neutral gold cluster
(

**Au _{18}(D_{1}): **The C

**Au _{20}(C_{1}): **The C

**To be noticed: **Let us remember the **Figure 2**, and the vibrational
heat capacity C_{vib}(min) at above 30 K, that at high temperatures it gets
saturated 0.12 k_{B}/atom (as a linear). Similarly, in the Boson peak, C_{vib}/
T^{3} also gets saturated (a linear) at above 30 K, but here it is towards
zero values (0 kB/atom). The lower frequencies certainly make a larger
contribution to heat capacity. Nevertheless, the Boson peaks are highly
visible, i.e., and the strength of the peaks strongly depends on the
atomic coordination number. There are several origins, most of them
related to the thermal fluctuations. If the atomic coordination is low, a
single negative force constant renders the atomic arrangement much
closer to an unstable situation than in the highly coordinated case [60].

**The size effect of the gold neutral cluster Au _{N}=3-20**

Interestingly, from the **Figure 18**, the clusters Au_{16} and Au_{20} possess
the most significant and stable cluster of all other clusters which have
C_{s} and C_{1} symmetry (comparable with Au_{20} tetragonal cluster). In fact,
the cluster Au_{16} has a greater number of single state degeneracy modes.

**Au _{16}: **For the size effect at about temperature 950/950* K, the C

**Au _{20}: **For the size effect at about temperature 950/950* K, the C

However, this minute differences are also clearly seen at
temperature 300/300 K* for the rest of the clusters. The size effect on the heat capacity should be very sensitive to the accuracy because the
absolute differences are small as shown in **Figure 18**.

**Physical parameters influence at the C _{vib} curves and its shapes**

If one carefully observes in **Figures 2-8** and 15-17 they shows very
clearly that the shape of the C_{vib} curves are not only dependent upon
the low and high frequency ranges, but also with respect to the physical
parameters (h, k_{B}). However, along with this we have noticed that the
size dependency values of C_{vib} vary a little bit. Nevertheless, with high
temperature at 950 K/950 K*, our calculated C_{vib} values are overlapping
with the absolute values of C_{vib}.

This confirms the accuracy of the size dependency (**Figure 18**, violet,
blue and red line curve). Nevertheless, near to the room temperature
at about 300 K/300 K*, the C_{vib} values are still very close to the absolute
values.

Most importantly, at both the above mentioned temperatures,
smoothness and the asymptotic behavior of heat capacity C_{vib} for
α_{i}=ω_{i}/T, looks like much better than α_{i}=ω_{i} / (0:6950356*T) of eqn. (8) **(Figures 2, 3 and 8)**. In addition to that, the changes at the shape of
the C_{vib} curves are only due to existing some single state modes, but
many double or triple state degree of degeneracies modes, and they
are the contributing factors at C_{vib}, see for example, cluster sizes Au_{6,7,8} at T=0 K **(Figure 19)**. So with this in view, we can confidently say that
our new methodology is a trustworthy and a successful model, with
which one can take one further in making more accurate experimental
calculations.

**Comparison with the theoretical and experimental results**

The spectral frequencies are the most recent studies covering neutral
clusters are reported using far-infrared multiphoton-dissociation spectroscopy (FIR-MPD) for Au_{N=3,4} [94] and for Au_{N=7,19,20} [65]. In
closing we would like to comment on a trend in the experimentally
observed low-frequency vibrational spectra which was realized recently
by Gruene et al. [65]. This work experimentally investigated neutral
gold clusters (Au_{N=7}, Au_{N=19} and Au_{N=20}) in the gas phase by means of
vibrational spectroscopy, which is inherently sensitive to structure.
However, before they performed experiments, they used Gaussian
package with DFT calculation.

The gold cluster, Au_{N=7,19,20} has 15, 51, 54 normal vibrational modes
(NVM) respectively, some of which may be degenerate, depending on
the symmetry. More-over, some modes may appear in one or another
experiment. We calculated the normal modes based on the structures
which were predicted by Dong and Springborg [44]. However, after
re-optimization the vibrational frequencies were found as mentioned
in the **Figures 2-19**. Our calculated spectrum ranges are the peculiar
values, which are in excellent agreement with the theoretical (DFT)
and experimental results calculated for gold clusters by Gruene et al.
[65]. Their visible modes at IR absorption coefficients, and the (ω_{i})
frequencies are Au_{20} (148 cm^{-1}), Au_{19} (149 and 166 cm^{-1}), and for Au_{7} (165, 186 and 201 cm^{-1}). These values are in excellent agreement with
our calculated value of Au_{20} (148.64 cm^{-1}) and for Au_{19} (within 144.15,
152.87 158.47, 165.28 cm^{-1}) **(Figure 19)**.

However, they have only predicted vibrational frequencies in
between 47 and 220 cm^{-1} wavenumbers. We have calculated even
lower (3.99 cm^{-1}) and higher (370.72 cm^{-1}) wavenumbers than those.
This is because in infrared (IR) absorption spectroscopy the number
of allowed transitions is restricted by selection rules, and thus directly
reflects the symmetry of the particle.

Overall for cluster Au_{N=3-20}, certainly and in particularly, our lowest
minimum frequency value of Au8 cluster is 4.66 cm^{-1} closer to the
harmonic frequency of the Au8 cluster which was calculated by Mancera
and Benoit [66]. Their harmonic frequency is 6.9 cm^{-1}, as a minimum.
Additionally, our highest maximum frequency values are comparable
with Nose S calculated (ω_{i}=221 cm^{-1}) as a highest frequency for Au_{20} Tetrahedron symmetry [68] using Molecular Dynamics Simulations.

Our calculated Au_{20}(C_{1}) cluster's vibrational modes are comparable
and in excellent agreement with the Au_{20} Tetrahedron (T_{d}) symmetry
calculated using density functional theory (DFT) with two different
approximations: the hybrid-B3LYP and GGA-BP86 by Molina et al.
[67]. Their ranges for BLYP and BP86 are 26.42-162.22 cm^{-1} (BLYP)
and 29.13-172.35 cm+ (BP86), respectively. Certainly, the most stable
structure for Au_{20} was found through frequency calculations for the
(T_{d}) Au_{20} structure by Jun Li, Xi Li, Hua-Jin Zhai, and Lai-Sheng Wang
[95,96].

The harmonic frequencies (ω_{i}) for the global minimum energy
structure of Au_{4,5} calculated using PBE/VDB, in cm^{-1} are being reported
by Mancera and Benoit [66], which are in good agreement with some
of our calculated normal modes of Au_{4} (a planar rhombus D_{2h}): 165.46
(167.2) cm^{-1}; 147.12 (152.2) cm^{-1}; 98.00 (96.5) cm^{-1} and 31.63 (34.5) cm^{-1}, as well as, Au_{5} (a planar trapezoid C_{2v}): 132.74 (137.5) cm^{-1}; 47.70
(47.6) cm^{-1} and 43.25 (47.6) cm^{-1}.

Moreover, our verified symmetry and calculated vibrational
frequency modes of Au_{N=3-6} clusters are almost in complete agreement
with Mancera and Benoit [66].

**Au _{N=6-8}: Double and Triple state degeneracy at T= 0**

These clusters are unique among the other clusters, due to their
nature. Very strong and visible evidence of the vibrational heat capacity C_{vib} per atom as a function of temperature is shown in **Figure 15**.

**Au _{N=6} (D_{3h}):** The harmonic frequencies for the Au6 cluster are in
the range of 2.44 cm

**Au _{N=7} (D_{5h}): **The harmonic frequencies for the Au

**Au _{N=8} (T_{d}): **The harmonic frequencies for the Au8 cluster are in
the range of 4.66 cm

We have extracted vibrational frequencies at T=0 K, and
investigated the vibrational heat capacities at a temperature range
0.5-300 K, of the re-optimized gold atomic clusters (Au_{N=3-20}).
Especially, the vibrational modes with lower frequencies are given a
significant contribution to the heat capacity at low temperatures. The
C_{vib} is a monotonic function that tends asymptotically. The vibrational
heat capacity is a strong function of cluster size and temperature,
particularly in the low temperature regime.

Our first step towards the understanding of the so-called finite temperature effects was to take into account the vibrations of clusters. We focused our interest on the cluster size and temperature dependence of the heat capacities and the free energies of the clusters. Finally, we have studied vibrational and thermodynamic properties of the clusters in order to explore the interaction between stability, structure, and heat capacities of clusters. Our approach is worthy of further investigation and would pave a way in realizing numerical values which would allow for an experimental vibrational spectrum and heat capacity, which would prove crucial in development of Nano electronic devices. Nevertheless, our work gives a possible cause for the size, temperature, and structures effect of Au atomic clusters.

Details about the harmonic frequencies and the predicted minima of the global structure optimization of Au clusters can be found in the supplementary material (see the ESIϯ).

A part of this work was supported by the German Research Council (DFG) through project Sp 439/23-1. We gratefully acknowledge their very generous support.

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