Nanoscience & Technology Open Access

We have carried out the present ab initio study by using upto- date computational concepts through our newly developed Numerical Finite-Difference analysis via DFTB method. Significantly, to differentiate between zero (translational and rotational motion) and non-zero (vibrational motion) energy eigenvectors, we have accurately predicted the gradient ds=0.01 a.u. at the Potential Energy Surface (PES) of the re-optimized gold atomic structures Au_3-58 at ΔE = 0. With this small energy gap, we can see how the interaction of the different atomic motions was varied with 3N and 3N-6 degrees of freedom, and also they are very independent to each other in gold atomic clusters. Moreover, the non-zero eigenvalues are quite stable whereas the values of the zero-eigen values are varying over several orders of magnitude.

Keywords: Gold Atomic Clusters; Density-Functional Tight- Binding (DFTB) approach; Finite-Difference Method; Force Constants (FCs); Energy Eigen Vectors

In general, nanoclusters 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, for that reasons, many new periodic tables can thus be envisioned classifying differently-sized clusters of the same material as new elements [1]. Potential applications are enormous, ranging from devices in nano-electronics and nano-optics to applications in medicine and materials [2].

The vibrational properties of clusters and small particles have been studied very intensively, and vital for understanding and describing the atomic interactions in the cluster [3, 4, 5, 6, 7, 8, 9]. 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) [10, 11, 12, 13, 14, 15]. For this reason, a better understanding of the rules governing the vibrational properties of nanostructured materials is of high technological and fundamental interest. The vibrational properties play a major role in structural stability.

The structural determination of metal nanoparticles of their vibrational (phonon) density of states have been calculated by Huziel E. Sauceda and Ignacio L. Garzon. Specific heat capacity is an important thermodynamic property and is directly related to the structural stability, identification and energy of substances [9, 16]. Most recently, Huziel E. Sauceda and Ignacio L. Garzon calculated vibrational properties and specific heat of core-shell Ag-Au icosahedral nanoparticles [7].

Moreover, the big challenges were to differentiate in between translational, rotational motion and vibrational motion of energy eigen vectors, nevertheless, we have accurately predicted the gradient ds=0.01 a.u. over single point Potential Energy Surface (PES) that is after the re-optimization. With this the energy gap, one can see the interaction of the different atomic motions are very less and they are independent.

We use our numerical finite-difference approach along with Density Functional Tight-Binding (DFTB) method. Overall, for a better understanding and to visualize, we have given an example of a cluster Au₁₉ and the detailed information discussed in the results and discussion section [17].

The DFTB is based on the density functional theory of Hohenberg and Kohn in the formulation of Kohn and Sham [18, 19, 20]. In addition, the Kohn-Sham orbital’s Ψi(r) of the system of interest are expanded in terms of atom-centered basis functions $\left\{{\varphi }_m\left(r\right)\right\}$ ,
$$\begin{array}{ccc}{\psi }_i(r)=& {\displaystyle \sum _m^{}{c}_{im}{\varphi }_m(r)},m=j& (1)\end{array}$$
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 coeffcients $C_{im}$ . Index m describes the atom, where ${\varphi }_m$ is centered and it is angular as well as radially dependant. The ${\varphi }_m$ is determined by self-consistent DFT calculations on isolated atoms using large Slater-type 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
$$\begin{array}{cc}{\displaystyle \sum _m{c}_{im}\left(H_{mn}-{\epsilon }_i{S}_{mn}\right)=0.}& \left(2\right)\end{array}$$
Here, C_im's are expansion coefficients, ${\epsilon }_i$ is for the single-particle energies (or where ${\epsilon }_i$ 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
$$\begin{array}{cc}\begin{array}{cc}\begin{array}{ccc}{H}_{mn}=& \langle \begin{array}{ccc}{\varphi }_m& \left|\widehat{H}\right|& {\varphi }_n\end{array}\rangle ,& S_{mn}=\end{array}& \langle {\varphi }_m|{\varphi }_n\rangle .\end{array}& (3)\end{array}$$
They depend on the atomic positions and on a well-guessed density p(r). By solving the Kohn-Sham equations in an effective one particle potential, the Hamiltonian Ĥ is defined as
$$\begin{array}{ccc}\begin{array}{cc}\widehat{H}{\psi }_i(r)=& {\epsilon }_i{\psi }_i(r),\end{array}& \begin{array}{cc}\widehat{H}=& \begin{array}{cc}\widehat{T}+& V_{eff}(r)\end{array}.\end{array}& (4)\end{array}$$
To calculate the Hamiltonian matrix, the effective potential Veff has to be approximated. Here $\widehat{T}$, being the kinetic-energy operator ${\displaystyle \sum \left(\widehat{T}=-\frac{1}{2}{\nabla }^2\right)}$ and V_eff (r) being the effective Kohn-Sham potential, which is approximated as a simple superposition of the potentials of the neutral atoms,
$$\begin{array}{cc}{V}_{eff}\left(r\right)={\displaystyle \sum _i{V}_j^0\left(\left|r-R_j\right|\right).}& \left(5\right)\end{array}$$
$v_j^o$ 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,
$$E_{tot}\simeq {\displaystyle \sum _i{\in }_i-}{\displaystyle \sum _j{\displaystyle \sum _{m_j}^{occ}{\epsilon }_{j{m}_j}}+\frac{1}{2}{\displaystyle \sum _{j\ne j^{\text{'}}}\left(\left|R_j-R_{j^{\text{'}}}\right|\right),}}\begin{array}{cc}{\epsilon }_B\equiv {\displaystyle \sum _i^{occ}{\epsilon }_i-{\displaystyle \sum _j{\displaystyle \sum _{m_j}^{occ}{\epsilon }_{j{m}_j}}}}& \left(6\right)\end{array}$$
Here, the majority of the binding energy (${\epsilon }_B$ ) is contained in the difference between the single-particle energies ${\epsilon }_i$ of the system of interest and the single particle energies of the isolated atoms (atom index j, orbital index $U_{j{j}^{\text{'}}}\left(\left|R_j-R_{j^{\text{'}}}\right|\right)$ is determined as the difference between ${\epsilon }_B$ and ${\epsilon }_B^{SCF}$ for diatomic molecules (with $E_B^{SCF}$ 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 [18, 20, 21].

In our case, we have calculated the numerical first-order derivatives of the forces ($F_{i\alpha }$ ;$F_{j\beta }$ ) instead of the numerical-second-order derivatives of the total energy ($E_{tot}$ ). In principle, there is no difference, but numerically the approach of using the forces is more accurate,
$$\begin{array}{cc}\frac{1}{M}\frac{{\partial }^2{E}_{tot}}{\partial R_{i\alpha }\partial R_{j\beta }}=\frac{1}{M}\frac{1}{2ds}\left[\frac{\partial }{\partial R_{i\alpha }}\left(-F_{j\beta }\right)+\frac{\partial }{\partial R_{j\beta }}\left(-F_{i\alpha }\right)\right]& \left(7\right)\end{array}$$
Here, F is a restoring forces which is acting upon the atoms, ds is a differentiation step-size and M represents the atomic mass, for homonuclear case. The complete list of these Force Constants (FCs) is called the Hessian H, which is a ($3N\times 3N$ ) matrix. Here, i is the component of (x, y or z ) of the force on the j'th atom, so we get 3N.

In order to fully exploit the potential applications of cluster based nanomaterials, it is necessary to gain full control of the cluster size, shape and structure. Nevertheless, in this article, we present an in-depth study to select the very accurate gradient (ds) with respect to the bond length fluctuations over the potential energy surface (PES), and how one must be careful to extract the vibrational frequency and the translational, rotational frequency of the re-optimized neutral gold cluster at ΔE = 0. Here, we have given an example of a cluster $(A{u}_N,N=19)$ .

At first, we tried to find a scheme which will allow to discriminate between the Hessian eigenvalues which correspond to the vibrational eigenvectors and the Hessian eigenvalues which correspond to translational and rotational motion of the atoms in the clusters. For all the cluster structures, $(A{u}_N,N=3-58)$ ) this was simply done by comparing the eigenvalues, it was possible to numerically distinguish between zero eigenvalues - translation and rotation and non-zero-eigenvalues - vibration. The method has been described with the details in section 2.

We found that the gradients $[ds=\pm 0.01a.u]$ at the equilibrium coordinate values of the optimized clusters, for an interpolation n = 1, which has been implemented within the scheme developed by M. Dvornikov [22] along with DFTB method, which is a reasonable value and allowed to discriminate between the translational, rotational motion (Zero eigenvalues) and the vibrational motion (Non-Zero-eigenvalues) of the atoms of the Hessian eigenvalues. The desired set of system eigenfrequencies ($3N-6$ ) is obtained by a diagonalization of the symmetric positive semidefinite Hessian matrix. The vibrational frequency of the optimized neutral gold cluster which will be useful to predicts many properties.

Consider that if the whole problem was solved analytically, six out of the 3N eigenvalues were exactly zero. Moreover, the non-zero eigenvalues have to be positive, because by assumption, the structures correspond to minima on the PES. However, since the energy calculations as well as the differentiation and the diagonalization are realized through numerical algorithms, the zero eigenvalues will not exactly be equal to zero. But still, they should be very small compared to the non-zero eigenvalues. For single structures, we now tried to set up the differentiation parameters ds (gradient) and n (interpolation) in such a way that we would get a positive semidefinite Hessian with six roots, i.e. with six eigenvalues which are as close as possible to zero.

We found, that reasonable step-sizes should be in the proximity of ds=0.01 atomic units. For all the clusters unto 58 from 3, i.e. clusters which are optimized up to high accuracy, smaller values of ds lead to zero-eigenvalues which are even closer to zero. However, if one applies such a very small step-sizes (ds = 0.001 a.u.)on bigger structures, the resulting Hessian might not longer be positive semidefinite and the results might become inaccurate. In addition to that it was not possible to numerically discriminate between the zero and the non-zero eigenvalues anymore (see Figure 1). Our new strategy proceeds on the assumptions, that the numerically optimized structures are almost exact, that the Hessian changes very little around the minimum and that the differentiation scheme works in principle.

If the energy absorbed when bond breaks, at the same time, the energy released when bond forms. When increase the bond length then bond strength will become weaker but if we bring closer the bond length to each other, as a result, the bond strength will become stronger. We can observe attraction with a shared electrons as well as repulsion due to nuclei and electron shell. At gradient ds=0.01 atomic units, the forces in the atom are repulsions between electrons and attraction between electrons and protons. The neutrons play no significant role. Mainly, the Coulomb forces are occurred due to attraction between electrons and nuclei, repulsion between electrons and between nuclei. The force between atoms is given by a sum of all the individual forces, and the fact that the electrons are located outside the atom and the nucleus in the center.

When two atoms come very close, the force between them is always repulsive, because the electrons stay outside and the nuclei repel each other. Unless both atoms are ions of the same charge (e.g., both negative) the forces between atoms is always attractive at large internuclear distances r. Since the force is repulsive at small r, and attractive at small r, there is a distance at which the force is zero. This is the equilibrium distance at which the atoms prefer to stay. The interaction energy is the potential energy between the atoms. It is negative if the atoms are bound and positive if they can move away from each other. The interaction energy is the integral of the force over the separation distance, so these two quantities are directly related. The interaction energy is a minimum at the equilibrium position. This value of the energy is called the bond energy, and is the energy needed to separate completely to infinity (the work that needs to be done to overcome the attractive force.) The strongest the bond energy, the hardest is to move the atoms, for instance the hardest it is to melt the solid, or to evaporate its atoms.

On Au₁₉ atomic cluster, the non-zero eigenvalues are quite stable whereas the values of the zero-eigenvalues are varying over several orders of magnitude. Nevertheless, Figure 1 shows that how exactly it differs with the log of ω values with respect to ds values. To be noticed that the negative sign occur due to the log term.

In conclusion, we strongly confirm that our novel approach with the numerical finite-difference method will be much helpful to extract the very accurate and the existing stored energy in terms of vibrational spectrum for any type of metallic atomic clusters and make use of it for the different technological applications, even to compare with the theoretical as well as the experimental results.

I am grateful and much thankful to Prof. M. Springborg, because this most important work has been carried out under his supervision, at the beginning of the project itself.

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