Research article Open Access
Bonding Forces and Energies on the Potential Energy Surface (PES) of the Optimized Gold Atomic Clusters at the Gradient (ds = 0.01 a.u.)
K. Vishwanathan*
Physical and Theoretical Chemistry, University of Saarland, 66123, Saarbrucken, Germany
*Corresponding author: K. Vishwanathan, Physical and Theoretical Chemistry, University of Saarland, 66123, Saarbrucken, Germany, Tel: + 49-0151-63119680; Email: @
Received: September 27, 2018; Accepted: October 31, 2018; Published: November 14, 2018
Citation: Vishwanathan K (2018) Bonding Forces and Energies on the Potential Energy Surface (PES) of the Optimized Gold Atomic Clusters at the Gradient (ds = 0.01 a.u.). Nanosci Technol 5(2): 1-4. 10.15226/2374-8141/5/2/00159. DOI: 10.15226/2374-8141/5/2/00159
AbstractTop
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 Au3-58 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
Introduction
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].
Theoretical and Computational Procedure
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 { ϕ m ( r ) } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq aHvpGzdaWgaaWcbaGaamyBaaqabaGcdaqadaqaaiaadkhaaiaawIca caGLPaaaaiaawUhacaGL9baaaaa@3D96@ ,
ψ i (r)= m c im ϕ m (r) ,m=j (1) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeWaaa qaaiabeI8a5naaBaaaleaacaWGPbaabeaakiaacIcacaWGYbGaaiyk aiabg2da9aqaamaaqahabaGaam4yamaaBaaaleaacaWGPbGaamyBaa qabaGccqaHvpGzdaWgaaWcbaGaamyBaaqabaGccaGGOaGaamOCaiaa cMcaaSqaaiaad2gaaeaaa0GaeyyeIuoakiaacYcacaWGTbGaeyypa0 JaamOAaaqaaiaacIcacaaIXaGaaiykaaaaaaa@4D73@
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 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGPbGaamyBaaqabaaaaa@38C9@ . Index m describes the atom, where ϕ m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaad2gaaeqaaaaa@38DB@ is centered and it is angular as well as radially dependant. The ϕ m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaad2gaaeqaaaaa@38DB@ 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
m c im ( H mn ε i S mn )=0. ( 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeGaaa qaamaaqafabaGaam4yamaaBaaaleaacaWGPbGaamyBaaqabaGcdaqa daqaaiaadIeadaWgaaWcbaGaamyBaiaad6gaaeqaaOGaeyOeI0Iaeq yTdu2aaSbaaSqaaiaadMgaaeqaaOGaam4uamaaBaaaleaacaWGTbGa amOBaaqabaaakiaawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaaWcba GaamyBaaqab0GaeyyeIuoaaOqaamaabmaabaGaaGOmaaGaayjkaiaa wMcaaaaaaaa@4BF2@
Here, Cim's are expansion coefficients, ε i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaaaa@38B6@ is for the single-particle energies (or where ε i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaaaa@38B6@ are the Kohn-Sham eigenvalues of the neutral), and the matrix elements of Hamiltonian Hmn and the overlap matrix elements Smn are defined as
H mn = ϕ m | H ^ | ϕ n , S mn = ϕ m | ϕ n . (3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeGaaa qaauaabeqabiaaaeaafaqabeqadaaabaGaamisamaaBaaaleaacaWG TbGaamOBaaqabaGccqGH9aqpaeaadaaadaqaauaabeqabmaaaeaacq aHvpGzdaWgaaWcbaGaamyBaaqabaaakeaadaabdaqaamaaHaaabaGa amisaaGaayPadaaacaGLhWUaayjcSdaabaGaeqy1dy2aaSbaaSqaai aad6gaaeqaaaaaaOGaayzkJiaawQYiaiaacYcaaeaacaWGtbWaaSba aSqaaiaad2gacaWGUbaabeaakiabg2da9aaaaeaadaaadaqaaiabew 9aMnaaBaaaleaacaWGTbaabeaakmaaeeaabaGaeqy1dy2aaSbaaSqa aiaad6gaaeqaaaGccaGLhWoaaiaawMYicaGLQmcacaGGUaaaaaqaai aacIcacaaIZaGaaiykaaaaaaa@5733@
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
H ^ ψ i (r)= ε i ψ i (r), H ^ = T ^ + V eff (r) . (4) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeWaaa qaauaabeqabiaaaeaadaqiaaqaaiaadIeaaiaawkWaaiabeI8a5naa BaaaleaacaWGPbaabeaakiaacIcacaWGYbGaaiykaiabg2da9aqaai abew7aLnaaBaaaleaacaWGPbaabeaakiabeI8a5naaBaaaleaacaWG PbaabeaakiaacIcacaWGYbGaaiykaiaacYcaaaaabaqbaeqabeGaaa qaamaaHaaabaGaamisaaGaayPadaGaeyypa0dabaqbaeqabeGaaaqa amaaHaaabaGaamivaaGaayPadaGaey4kaScabaGaamOvamaaBaaale aacaWGLbGaamOzaiaadAgaaeqaaOGaaiikaiaadkhacaGGPaaaaiaa c6caaaaabaGaaiikaiaaisdacaGGPaaaaaaa@54BB@
To calculate the Hamiltonian matrix, the effective potential Veff has to be approximated. Here T ^ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaecaaeaaca WGubaacaGLcmaaaaa@3790@ , being the kinetic-energy operator ( T ^ = 1 2 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabqaSqaam aabmaabaWaaecaaeaacaWGubaacaGLcmaacqGH9aqpcqGHsisldaWc aaqaaiaaigdaaeaacaaIYaaaaiabgEGirpaaCaaabeqaaiaaikdaaa aacaGLOaGaayzkaaaabeqab0GaeyyeIuoaaaa@410B@ and Veff (r) being the effective Kohn-Sham potential, which is approximated as a simple superposition of the potentials of the neutral atoms,
V eff ( r )= i V j 0 ( | r R j | ). ( 5 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeGaaa qaaiaadAfadaWgaaWcbaGaamyzaiaadAgacaWGMbaabeaakmaabmaa baGaamOCaaGaayjkaiaawMcaaiabg2da9maaqafabaGaamOvamaaDa aaleaacaWGQbaabaGaaGimaaaakmaabmaabaWaaqWaaeaacaWGYbGa eyOeI0IaamOuamaaBaaaleaacaWGQbaabeaaaOGaay5bSlaawIa7aa GaayjkaiaawMcaaiaac6caaSqaaiaadMgaaeqaniabggHiLdaakeaa daqadaqaaiaaiwdaaiaawIcacaGLPaaaaaaaaa@4EB4@
v j o MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaDa aaleaacaWGQbaabaGaam4Baaaaaaa@3900@ is the Kohn-Sham potential of a neutral atom, r j =r R j MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGQbaabeaakiabg2da9iaadkhacqGHsislcaWGsbWaaSba aSqaaiaadQgaaeqaaaaa@3CED@ is an atomic position, and R j MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGQbaabeaaaaa@37E7@ 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 i i j m j occ ε j m j + 1 2 j j ' ( | R j R j ' | ), ε B i occ ε i j m j occ ε j m j ( 6 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWG0bGaam4BaiaadshaaeqaaOGaeS4qISZaaabuaeaacqGH iiIZdaWgaaWcbaGaamyAaaqabaGccqGHsislaSqaaiaadMgaaeqani abggHiLdGcdaaeqbqaamaaqahabaGaeqyTdu2aaSbaaSqaaiaadQga caWGTbWaaSbaaWqaaiaadQgaaeqaaaWcbeaaaeaacaWGTbWaaSbaaW qaaiaadQgaaeqaaaWcbaGaam4BaiaadogacaWGJbaaniabggHiLdGc cqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaamaaqafabaWaaeWaae aadaabdaqaaiaadkfadaWgaaWcbaGaamOAaaqabaGccqGHsislcaWG sbWaaSbaaSqaaiaadQgadaahaaadbeqaaiaacEcaaaaaleqaaaGcca GLhWUaayjcSdaacaGLOaGaayzkaaGaaiilaaWcbaGaamOAaiabgcMi 5kaadQgadaahaaadbeqaaiaacEcaaaaaleqaniabggHiLdaaleaaca WGQbaabeqdcqGHris5aOqbaeqabeGaaaqaaiabew7aLnaaBaaaleaa caWGcbaabeaakiabggMi6oaaqahabaGaeqyTdu2aaSbaaSqaaiaadM gaaeqaaOGaeyOeI0YaaabuaeaadaaeWbqaaiabew7aLnaaBaaaleaa caWGQbGaamyBamaaBaaameaacaWGQbaabeaaaSqabaaabaGaamyBam aaBaaameaacaWGQbaabeaaaSqaaiaad+gacaWGJbGaam4yaaqdcqGH ris5aaWcbaGaamOAaaqab0GaeyyeIuoaaSqaaiaadMgaaeaacaWGVb Gaam4yaiaadogaa0GaeyyeIuoaaOqaamaabmaabaGaaGOnaaGaayjk aiaawMcaaaaaaaa@8484@
Here, the majority of the binding energy ( ε B MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadkeaaeqaaaaa@388F@ ) is contained in the difference between the single-particle energies ε i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaaaa@38B6@ of the system of interest and the single particle energies of the isolated atoms (atom index j, orbital index U j j ' ( | R j R j ' | ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaaBa aaleaacaWGQbGaamOAamaaCaaameqabaGaai4jaaaaaSqabaGcdaqa daqaamaaemaabaGaamOuamaaBaaaleaacaWGQbaabeaakiabgkHiTi aadkfadaWgaaWcbaGaamOAamaaCaaameqabaGaai4jaaaaaSqabaaa kiaawEa7caGLiWoaaiaawIcacaGLPaaaaaa@443B@ is determined as the difference between ε B MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadkeaaeqaaaaa@388F@ and ε B SCF MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aa0 baaSqaaiaadkeaaeaacaWGtbGaam4qaiaadAeaaaaaaa@3AFB@ for diatomic molecules (with E B SCF MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaDa aaleaacaWGcbaabaGaam4uaiaadoeacaWGgbaaaaaa@3A1E@ 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].
Structural Re-Optimization Process
In our case, we have calculated the numerical first-order derivatives of the forces ( F iα MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGPbGaeqySdegabeaaaaa@3979@ ; F jβ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGQbGaeqOSdigabeaaaaa@397C@ ) instead of the numerical-second-order derivatives of the total energy ( E tot MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWG0bGaam4Baiaadshaaeqaaaaa@39D1@ ). In principle, there is no difference, but numerically the approach of using the forces is more accurate,
1 M 2 E tot R iα R jβ = 1 M 1 2ds [ R iα ( F jβ )+ R jβ ( F iα ) ] ( 7 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeGaaa qaamaalaaabaGaaGymaaqaaiaad2eaaaWaaSaaaeaacqGHciITdaah aaWcbeqaaiaaikdaaaGccaWGfbWaaSbaaSqaaiaadshacaWGVbGaam iDaaqabaaakeaacqGHciITcaWGsbWaaSbaaSqaaiaadMgacqaHXoqy aeqaaOGaeyOaIyRaamOuamaaBaaaleaacaWGQbGaeqOSdigabeaaaa GccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGnbaaamaalaaabaGaaGym aaqaaiaaikdacaWGKbGaam4CaaaadaWadaqaamaalaaabaGaeyOaIy labaGaeyOaIyRaamOuamaaBaaaleaacaWGPbGaeqySdegabeaaaaGc daqadaqaaiabgkHiTiaadAeadaWgaaWcbaGaamOAaiabek7aIbqaba aakiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiabgkGi2cqaaiabgkGi 2kaadkfadaWgaaWcbaGaamOAaiabek7aIbqabaaaaOWaaeWaaeaacq GHsislcaWGgbWaaSbaaSqaaiaadMgacqaHXoqyaeqaaaGccaGLOaGa ayzkaaaacaGLBbGaayzxaaaabaWaaeWaaeaacaaI3aaacaGLOaGaay zkaaaaaaaa@6C14@
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×3N MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaad6 eacqGHxdaTcaaIZaGaamOtaaaa@3B2C@ ) matrix. Here, i is the component of (x, y or z ) of the force on the j'th atom, so we get 3N.
Results and Discussion
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) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadg eacaWG1bWaaSbaaSqaaiaad6eaaeqaaOGaaiilaiaad6eacqGH9aqp caaIXaGaaGyoaiaacMcaaaa@3E1E@ .

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=358) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadg eacaWG1bWaaSbaaSqaaiaad6eaaeqaaOGaaiilaiaad6eacqGH9aqp caaIZaGaeyOeI0IaaGynaiaaiIdacaGGPaaaaa@3FCB@ ) 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.
Gradient at Potential Energy Surface (PES)
We found that the gradients [ds=±0.01a.u] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaads gacaWGZbGaeyypa0JaeyySaeRaaGimaiaac6cacaaIWaGaaGymaiaa dggacaGGUaGaamyDaiaac2faaaa@41FD@ 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 ( 3N6 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaad6 eacqGHsislcaaI2aaaaa@3932@ ) 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.
Figure 1: Au19(C1): The energy gap in between Zero [below] and Non-Zero (3N-6) [above] Eigen Values at
Bonding Forces and Energies
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.
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