RHF and DFT Study of the Molecular and Electronic Properties of ( SiO 2 ) n and ( GeO 2 ) n Nanoclusters

The study of nanoclusters has attracted a lot of scientific research over the years. This class of materials are important because they bridge the gap between bulk materials and molecular structures. Silicon and Germanium oxides have many applications in semiconductor technology and nanotechnology. In this research work, molecular and electronic properties of Silicon and Germanium dioxide nanoclusters are studied. The results obtained reveal the comparative advantages and disadvantages of using any of the two oxides for particular applications. Restricted Hartree Fock and Density Functional Theory computations of the molecular and electronic properties of (SiO2)n and (GeO2)n nanoclusters (n = 1,..,6) are studied. Silicon dioxide clusters are found to have higher thermal energies and lower average bond lengths and are thus more stable than Germanium dioxide clusters. At n = 1, both nanoclusters are non-polar, but gradually become more polar with increase in n. The average polarizability, molecular hyperpolarizability and total thermal energies of the nanoclusters increases with increase in molecular size. Computed values of the electron affinities for (SiO2)n clusters agree with experimental results. Some of the most intense Infra Red vibrational motions observed in both molecules are anti-symmetric stretching of Si=O/GeO and chain in plane, symmetric stretching of the Si=O/Ge-O bonds and chain in plane and symmetric twisting/breathing of the chain(s) in plane. The two nanoclusters are also Raman active at some frequencies.


Introduction
Silicon (Si) and Germanium (Ge) are two Group IV elements which crystallize in the diamond cubic crystal structure.Germanium, unlike Silicon, does not exist in its pure form on Earth.Silicon dioxide is one of the most common compounds on Earth.Silicon and Germanium dioxides exist in crystalline and amorphous phases.Germanium dioxide unlike Silicon dioxide is soluble in water.(Janke, 2008).
Silicon dioxide is suitable for the fabrication of semiconductor devices due to its chemical stability and good electrical insulation properties.Also, Silicon dioxide nanoparticles have high surface areas which are highly desirable as catalytic support (Sahnoun et al, 2005).Silicon and its compounds have dominated the semiconductor industry over the years for a number of reasons.Firstly, the existence of a stable Silicon oxide which serves as a dielectric on device surfaces has played an important role in favor of Silicon.Secondly, the scarcity and cost of Germanium has also favored the dominance of Silicon.However, with recent advances in semiconductor technology, the use of SiO 2 as a dielectric is becoming challenging.This is due to the fact that as the dielectric thickness reduces, it approaches the point where electron tunneling becomes an important effect (Janke, 2008).
Germanium is being considered extensively as an alternative desirable material.This is because of its higher low field mobility -a factor of two higher for electrons and four for holes compared with Silicon.Germanium dioxide is similar to Silicon dioxide in many aspects and occurs in three stable forms at ambient temperature (Ivana et al, 2014).The properties of Germanium are less understood compared to those of Silicon.This is largely due to the dominance of Silicon technology for the past decades and difficulties in transferring techniques which have been very successful in Silicon to Germanium.Problems have also been encountered in modeling Germanium and its oxides.Despite these problems, the study of Germanium compounds and its defects is an area of active research interest (Lee, 2007).
Nanoparticles attract much scientific interest as they effectively bridge the gap between bulk materials and atomic or molecular structures.Group-IV nanocrystals have emerged as a promising group of materials that extends the realm of application of bulk diamond, Silicon, Germanium and related materials beyond their traditional boundaries.Over the last two decades of research, their potential for application in areas such as optoelectronics and memory devices has been progressively unraveled.Silicon dioxide thin films have been used extensively in electronics, optics and surface treatment fields.Also, Silicon nanoparticles have good potentials for biochemical applications in controlled drug delivery and tissue engineering (Nikolaos et al, 2016).
In view of the importance of these nanoparticles, it is imperative to carry out a comparative study of the two oxides.This study is aimed at revealing the comparative advantages and disadvantages of using any of the two oxides for particular applications.RHF and DFT are used to carry out a comparative analysis of the structural, electronic and vibrational properties of some small Silicon and Germanium dioxide nanoclusters.

Methods
The windows version of Gaussian 9W Revision A.02-SMP is used.Geometry optimization is carried out at Restricted Hartree Fock (RHF) and Density Funtional Theory (DFT) levels of theory.Geometry optimizations locate minima on the potential energy surface, thus predicting equilibrium structures of the molecular systems.The input molecular structures are built using GaussView 5.0.8 (Frisch et al, 2009).
The structures are completely optimized and studied at the RHF level of theory using the basis set 6-31G(d,p).The molecules are then studied at DFT level of theory using Becke 3 Lee Yarr Parr (B3LYP) method which is a cost effective method for inclusion of electron correlations with the same basis set (Gurku & Ndikilar, 2012;Galadanci et al, 2015).The DFT method has proven to be one of the most accurate methods for the computation of the electronic structure of solids (Jamal, 2014).In addition, vibrational frequencies for optimized molecular structures are also calculated.The optimized geometry of the molecules has no imaginary vibrational frequencies which imply that minimum potential energy is found for each molecule.
The Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) energies are extracted from the eigenvalues and used to compute the Ionization Potential (IP), Electron Affinity (EA), energy band gap (or global hardness) E g , chemical potential( ), chemical hardness ( ) and electrophilicity index ( ) (Taura et al, 2017).

Optimized Molecular Structures
The optimized structures are shown in Fig. 1.The stable optimized structures of Silicon oxides have two double linear bonds between Silicon atoms and Oxygen atoms; unlike the Germanium oxides that have single linear bonds.Fig. 1 shows that small silicon oxide and Germanium oxide clusters can exist as ring or chain structures.It is expected that as the cluster size increases other three dimensional structures may occur with more structural isomers (Yuhai & Tersoff, 1999).The mean optimized bond lengths for the common bonds in both nanocluster types are computed and presented in Table 1.  1 shows that the mean Si-O bonds are longer and hence weaker than the side Si=O bonds.The change in mean bond length with molecular size is very insignificant.The mean bond length of Si-O at both levels of theory best agree with the experimental value of 1.6 (Yuhai & Tersoff, 1999).At B3LYP level of theory, greater mean bond lengths are predicted for both nanoclusters.The Ge-O bonds within the sub chains in the nanoclusters are longer and weaker than the linear side Ge-O bonds.The chain Ge-O bonds within the sub chains are longer compared to the equivalent Si-O chains.Thus, from the values of the mean bond lengths, it can be concluded that Silicon dioxide nanoclusters are more stable than those of Germanium.X-ray diffraction gives the Ge-O bond length as 1.74 (Micoulaut et al, 2006) which agrees with values obtained in Table 1.

Dipole Moments, Quadrupole Moments, Polarizabilities and Total Thermal Energies
The first derivative of the energy with respect to an applied electric field gives the dipole moments of the molecules.It is given as a three-dimensional vector in the Gaussian output file and is a measure of asymmetry in the molecular charge distribution (Robertson, 2004).Total dipole moments (µ) are shown in Table 2.It is deduced from Table 2 that both nanoclusters at n=1 are non polar, but gradually become more polar with increase in n.Also, RHF values are slightly lower than B3LYP values.
The total thermal energies (E 0 ), averages polarizability (<α>) and molecular hyperpolarizability (β mol ) of the clusters at both levels of theory are also computed and presented in Table 2.These molecular properties are essential in understanding many physical phenomena, molecular simulation and modeling of fundamental processes, search of new optical materials and thermodynamic properties of molecules.The relationship between the molecular structure and non-linear optical properties of molecules can be deduced from hyperpolarizability values.Hyperpolarizability is a measure of the nonlinear optical activity of a molecule and thus molecules with large values can produce second order optical effects (Ejuh & Ndjaka, 2013) This shows that as the molecular size increases, the potential of the molecules to have non-linear optical properties also increases.The values obtained for the polarizability of SiO 2 are higher compared to those obtained from ab initio coupled perturbed Hartree-Fock (CPHF) method (Bacskay, 1983).Table 2 shows large values of hyperpolarizability and thus these nanoclusters can serve as good Non-Linear Optical (NLO) materials.Important parameters influencing hyperpolarizability include donor-acceptor system, nature of substituents, conjugated π system and the influence of planarity (Arulmozhi & Madhavan , 2015).
Table 2 shows that total thermal energy decreases from RHF to B3LYP for both nanocluster molecules.Hence, inclusion of electron correlation brings a reduction in total thermal energies.The thermal energies also increase with increase in the size of the molecules.Generally, the Silicon dioxide clusters have higher thermal energies compared to Germanium.Hence, they are more stable, as earlier deduced from the mean bond lengths.
Quadrupole moments (Table 3) are second order approximations of the total electron distribution and give an approximate idea of the molecular shape.Table 3 shows that the molecules are slightly elongated along XX axis, with increase in elongation as the size of the molecule increases for n = 2,...,6.For n = 1, the molecules are slightly elongated along the ZZ axis.Electron correlation slightly reduces the molecular elongation.The average quadrupole moments at RHF level of theory for SiO 2 (-24.15Debye-Ang) and GeO 2 (-29.12Debye-Ang) are higher than that of Carbon dioxide (Battaglia et al, 1981).

Electronic Properties and Reactivity Indices
Large values of HOMO and LUMO energies play an important role in the electronic, optical and chemical properties of molecules (Ejuh et al, 2017).Electronic systems with larger HOMO-LUMO energy band gap (E gap ) are less reactive compared to those with a smaller band gap (Arjunan et al, 2011) Table 4 shows that the RHF method predicts slightly higher values for the Ionization Potential (IP) and Energy band gap compared to the B3LYP method.Computed values of the electron affinities for (SiO2) n clusters agree with experimental results observed at 4.66 eV photon energy (Robertson, 2004).However, the observed electron affinities within the range 4.0 -4.5 eV for the GeO 2 /Ge interface differs from the computed values.This could be due to the contribution from the interface.The computed energy band gaps for Silicon and Germanium dioxide nanoclusters are lower compared to experimental values.However, the predicted band gaps of the nanoclusters are comparable to those of the elements, Germanium (0.66 eV) and Silicon (1.12 eV) (Ay & Aydinli, 2004) .
The chemical potential and chemical hardness are key indicators of the overall reactivity of a molecule and are the most fundamental descriptors of charge transfer during a chemical reaction.The nature of electronegativity, hardness and the electrophilicity index are fundamentally qualitative descriptors as they are not observable.They occur but cannot be seen.Hence, the possibility of experimental determination of such descriptors is not yet possible (Nazmul & Dulal, 2012).The predicted chemical potentials do not change with increase in nanocluster size at both levels of theory.This shows that if there is an interface between any two nanoclusters in Table 4, there will be no significant flow of particles between them.RHF predicts far more higher values for chemical hardness for both molecules compared to DFT.This suggest that the neglect of electron correlation in the computational method erroneously depicts a higher resistance to change in electron distribution.Electrophilicity is an intrinsic property of molecules and signifies the energy lowering process on soaking electrons from donors (Nazmul & Dulal, 2012).The electrophilicity index measures the stabilization in energy when the system acquires an additional electronic charge from the environment.The computed electrophicity index for the nanoclusters are not significantly different for both nanoclusters at both levels of theory.

Electrostatic Potential Derived Charges
The mean electrostatic potential derived charges using the CHelpG scheme of Breneman are computed and shown in Table 5.These charges can be used to study electrostatic interaction energies in the different molecules.Electrostatic potential maps for a given molecule explicitly show regions of electronegativity and electropositivity that are not easily inferred from inspection of the parent charge density distribution.From a potential map, energetically favorable Coulombic interactions of a molecule with its neighbors can readily be understood, since these will involve the overlap of electropositive with electronegative regions, to give at least a partial cancellation (Stewart & Craven, 1993).Ge O 1 1.264757 -0.632378 0.950423 -0.475211 1.496560 -0.748280 1.141633 -0.570816 2 1.383725 -0.691863 1.034084 -0.517042 1.556195 -0.747875 1.127695 -0.563847 3 1.439765 -0.719883 1.076988 -0.538493 1.624682 -0.812341 1.206259 -0.603133 4 1.491125 -0.745562 1.089062 -0.594531 1.655614 -0.827807 1.187429 -0.593714 5 1.494880 -0.747440 1.121364 -0.560682 1.674502 -0.837218 1.198165 -0.537027 6 1.509132 -0.754566 1.254990 -0.566618 1.686429 -0.843214 1.204615 -0.602307 The bulk positive charge in both molecules resides with the Silicon or Germanium atoms and negative charge resides mainly with the Oxygen atoms as expected.This is due to the high electro-negativity of the Oxygen atoms.B3LYP values for both molecules are slightly lower in magnitude.Also, the Germanium oxides have higher mean electrostatic potential derived charges.It is noticed that O atoms in the chain are slightly more electronegative than those at the sides (Ejuh et al, 2015).

Vibrational Frequencies and Assignments
The vibrational spectra of nanoclusters at ground state were also predicted.These frequency calculations are valid only at stationary points on the potential energy surface, thus our computations were performed on the optimized structures of the molecules.Infra Red and Raman intense vibration frequencies and spectra are determined at both levels of theory.
Raw frequency calculations computed at the Hartree-Fock level contain known systematic errors due to the neglect of electron correlation, resulting to overestimates of about 10-12%.Therefore, it is usual to scale frequencies predicted at the HF level by an empirical factor of 0.8929.Use of this factor has been demonstrated to produce very good agreement with experiment for a wide range of systems.Our values in this study must be expected to deviate even a bit more from experiment because of the choice of a medium-sized basis set around .For B3LYP , a scale factor of 0.9613 is used (Frisch et al, 2009).
Table 6 and 7 show respectively some common intense IR and Raman vibrational frequencies for the clusters.The computed IR vibrations for Silicon dioxide agree with experimental work as twisting, anti-symmetric bending and symmetric stretching are observed respectively at 448 cm -1 , 826 cm -1 , 1026 cm -1 (Wang et al, 1996).Fig. 2 shows some selected IR spectra for the studied molecules.The similarity between the spectra of both molecules at a given n is clearly shown in Fig. 2 with similar peaks.Experiments have shown that the IR spectra of GeO 2 has two prominent peaks at 560 cm -1 and 870 cm -1 , corresponding to asymmetric stretching of the Ge-O bonds (Micoulaut et al, 2006).et al. (2011) observed intense Raman peaks for SiO 2 at 521 cm -1 and 1100 cm -1 .Experiments show that the Raman spectra of crystalline polymorphs of GeO 2 have three strong peaks at 173 cm -1 , 701 cm -1 and 873cm -1 corresponding respectively to asymmetric stretching of Ge-O-Ge bonds, Oxygen breathing and symmetric stretching twisting of bonds.This agrees with the common vibrations observed in this work.

Conclusion
The mean bond lengths and total thermal energies predict that Silicon dioxide clusters are more stable than the corresponding Germanium dioxides.The polarity of the molecules increases with increase in size.The quadrupole moments and polarizabilities of the two types of nanoclusters were also studied.There is a great similarity in the electronic properties of the two nanocluster molecules.The bulk positive charges reside mainly with the Silicon and Germanium atoms and the highly electronegative Oxygen atoms carry the bulk negative charge.
It would be interesting to study and determine the structural diversification of the nanoclusters.The chain and linear structures can be examined to see which are more stable and have more favorable electronic and molecular properties for use.Also, understanding of the structural hierarchy of the small silicon and germanium oxide clusters will provide good models to test theories and help in understanding the bulk properties of many materials based on these oxides.This understanding may also aid the design of new materials with novel structural building blocks which may have many applications in nanotechnology.

Figure 1 .
Figure 1.Optimized Structures of the Nanocluster Molecules Some Sampled IR Spectra at B3LYP

Table 2 .
. Accurate knowledge of atomic polarizabilities is very important in many applications including development of the next-generation atomic time and frequency standards, optical cooling and trapping schemes, quantum information with neutral atoms, study of fundamental symmetries, thermometry and other macroscopic standards, study of cold degenerate gases, study of long-range interactions, atomic transition rate determinations, and benchmark comparisons of theory and experiment.Dipole Moments, Average Polarizabilities, Hyperpolarizabilities and Total Thermal Energies

Table 4 .
Electronic Properties and Reactivity Indices

Table 5 .
Mean Electrostatic Potential Derived Charges on different atoms

Table 7 .
Some common Raman Intense Vibrational Frequencies (cm -1 ) (a) Si=O/Ge-O linear bond symmetric stretching in plane