Evolution Mechanism of Metallic Dioxide MO2 (M = Mn, Ti) from Nanorods to Bulk Crystal: First-Principles Research

Using first-principle calculations, the surface energy, cohesive energy, and electronic properties of α-MnO2 and rutile TiO2 nanorods and microfacets were investigated and clarified to, in the first instance, determine the evolution mechanism. The results show that the surface energies of α-MnO2 nanorods and microfacets conform to function 1.0401 Jm + N × 0.608 Jm, while the surface energies of the rutile TiO2 nanorods and microfacets are governed by a 1.0102 × 1.1997 rule. Their electronic properties, such as the Mulliken population and Mulliken charge, can only be normalized by their surface areas to attain a linear function. Meanwhile, the surface energy of α-MnO2 with the nanostructure closely conforms to the function for normalized Mulliken population and Mulliken charge as f(x) = 102.9 × x + 0.101 with an R value of 0.995. Thus, our research into the evolution mechanism affecting the surface effect of nanometer materials will be useful for investigating the intrinsic mechanism of the nanometer effect and doping process of metallic dioxide catalysts.


Introduction
TiO 2 , which is a vital inorganic functional nanomaterial, has been widely used in down-flop pigments, ultraviolet screening, photoelectric conversion, photocatalysis, and so on [1].MnO 2 is a popular and cost-effective material for the removal of pollutants in air, water, and industry [2].Both have been widely investigated and improved to enhance their catalytic performance, such as by doping with metallic elements [1,3], incorporation into carbon nanotubes [4], and manufacturing with a nanometer structure [5].Especially, in the nanocrystallization process, the TiO 2 and MnO 2 nanometer materials exhibit additional surface and nanometer effects although they have the same components and skeleton units as the bulk morphology.Both have been successfully applied to catalytic redox for some pollutants.But they exhibit different catalytic capabilities for the same reactant in somewhere.To remove arsenite oxidation, the arsenite (As III ) is oxidized to As V for more than 0.4 hours by manganese dioxide, and the Mn-As bond length is from 0.271 nm to 0.34 nm [6].For the rutile TiO 2 , however, it is found that the Ti-As bond length is from 0.283 nm to 0.405 nm, and the adsorption energy of As V on TiO 2 (110) is greater than that on MnO 2 [7].Regarding the decomposition of CO, Chen et al. [8] indicated that the CO adsorbed onto the anatase TiO 2 resulted in a moderate adsorption energy (about 0.3 eV) and a positive shift of the C-O stretching frequency (about +44 cm −1 ) whereas the CO could no longer be adsorbed onto the MnO 2 [9].Considering the adsorption of O 2 onto MnO 2 , the oxygen reduction reaction can occur either in solution [10] or in air [11].Meanwhile, Petrik and Kimmel [12] stated that O 2 could be adsorbed onto rutile TiO 2 only at very low temperatures.Their different catalytic activities have attracted the attention of many researchers.Barnard et al. [13] had modeled the electronic properties of TiO 2 nanoparticles and pointed out that the free energy of surface would keep constant after the sizes of nanoparticles were larger than 100 nm [14].After studying a series of low stoichiometric surfaces, they found the effects of edges and corners were omitted when the nanoparticles were larger than ∼2 nm, and constructed the morphology of rutile TiO 2 only composed by {110} Miller index [15].Nevertheless, Deringer and Csányi [16] and Tompsett et al. [17] discovered that nanorods of rutile TiO 2 and -MnO 2 have the same equilibrium geometric morphologies, with a structure consisting mainly of {100} and {110} Miller indexes.Furthermore Hummer et al. [18] pointed out that the surface energies of TiO 2 were dependent with edges and corners of nanocrystal at particle size ≤3 nm.At present, former researches do not identify the intrinsic mechanism between bulk and nanorods although they have studied the nanoscale morphology of rutile TiO 2 and -MnO 2 for a long time.But such an intrinsic mechanism plays a vital role in the design and optimization of metallic oxide nanomaterials.In a previous paper [19], it have been stated that there may be an optimal -MnO 2 nanorod, which has a surface energy suitable for promoting enhanced surface activity, together with an appropriate degree of cohesive energy for maintaining structural stability.As an extension of previous work, the present study further sets out to investigate the evolution mechanism of bulk and nanorods of MO 2 (M = Mn, Ti) metallic dioxide.

Simulation Models and Method
To elucidate the nanometer effect of metallic dioxide MO 2 (M = Mn, Ti) with a nanostructure, several models of MO 2 (M = Mn, Ti) in crystal, bulk surface, nanorod, and microfacet topological configurations were constructed and studied systematically according to their stoichiometric proportions.Their corresponding simulated models are shown in Figures 1 and 2. Regarding the rutile TiO 2 model construction, only two prominent and stable Miller index planes, such as {100} and {110}, are considered.In the present study, the -MnO 2 and rutile TiO 2 nanorod models were constructed based on the experimental results obtained by  and Deringer and Csányi [16].All the simulation models are shown in Figure 2.For the rutile TiO 2 (100) bulk surface, there are triple units of {100} Miller index slabs, while, for the {110} bulk surfaces, there are double {110} Miller index slabs, as shown in Figures 2(b) and 2(c), respectively.A microfacet rutile TiO 2 [(100 × 110)] supercell structure containing only double units of {100} and {110} Miller index slabs, which had been proven to exhibit a similar catalysis performance as a nanostructure [19,22], was built as a bulk surface with nanometer morphologies only, as shown in Figure 2(d).Regarding the nanorod (NR) models, all of them were combined with only {110} and {100} Miller index slabs, as shown in Figures 2(e)-2(g).(In our future work, further Miller index slabs will be considered to represent a more complicated situation.)The smallest nanorod addressed in the present study, that is, (NR(1)), consisting of two {110} and one {100} Miller index slab unit, was built as a Ti 32 O 64 supercell, as shown in Figure 2(e).The second rutile TiO 2 nanorod (NR(2)) contains two units each of {100} and {110} Miller index slabs to construct a Ti 52 O 104 supercell (Figure 2(f)).The largest rutile TiO 2 nanorod contains triple {110} and double {100} Miller index slab units named NR(3), which form a Ti 88 O 176 supercell (Figure 2(g)).Similar way of constructed configuration is forced to -MnO 2 nanorods.The latter is the largest that can be handled in the calculation limits of our computer cluster.All these primitive nanorods can be regarded as being free nanomaterials in morphologies by using their periodic boundary conditions and transitional symmetry.The purpose of such constructions is to investigate the surface effect of different Miller indices over several models.All the bulk surface models were calculated assuming slabs with a minimum thickness of 14 Å.In all the bulk surface and nanorod models, a separating vacuum distance of at least 12 Å was used to distance the slabs from their periodic image.For the first step, all of the models were not terminated by hydrogenation as followed by Barnard et al. 's report [13].The complicated surface models of metallic dioxide MO 2 (M = Mn, Ti) will be studied in our further research.To distinguish the difference between -MnO 2 and rutile TiO 2 in similar configurations, every simulated model was labeled "M" or "T" to represent the -MnO 2 and rutile TiO 2 series, respectively.For example, the (100) bulk surface for -MnO 2 was labeled M(100), while T(100) represents the (100) bulk surface for rutile TiO 2 .This convention is used for every model shown in Figures 1 and 2.
Based on the calculated sets of Deringer and Csányi [16], all the above -MnO 2 and rutile TiO 2 simulation models were relaxed by applying the following process: a first-principles pseudopotential plane-wave method, based on density functional theory, was implemented in the Cambridge Sequential Total Energy Package (CASTEP) code [23].The electronic structure was calculated using the Generalized Gradient Approximation (GGA) devised by Perdew et al., with a Tkatchenko-Scheffler approach (TS) being used for the dispersion corrections [16].The PBE + U exchangecorrelation function has been demonstrated to give a good description of the defect properties in -MnO 2 [17].All calculations were performed in a ferromagnetic spin polarized configuration, while effects of more complex magnetic orders were left for future work due to their low energy scale.For the rutile TiO 2 , however, Deringer and Csányi [16] pointed out that adding  terms caused the results to steadily be worsened, in much the same way as in [22,24].In the present study, therefore,  correction was not applied to any of the rutile TiO 2 models.All the subsequent calculations were performed based on the equilibrium lattice constants obtained without cell relaxation using a cutoff of 500 eV, which was more precise than previous papers [13][14][15]18].This included the recalculation of the energy for the bulk unit cell so that all the comparative energies could be obtained.A minimum of 8 × 1 × 1 -points were used in the Brillouin zone of the conventional cell and scaled appropriately for supercells.All the atomic positions in these primitive cells were relaxed in spin polarized situation according to the total energy and force using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm scheme [25], based on the cell optimization criteria (a root mean square (RMS) force of 0.03 eV/ Å, a stress of 0.05 GPa, and a displacement of 0.001 Å).The convergence criteria for the self-consistent field (SCF) and energy tolerances were set to 1.0 × 10 −6 and 5.0 × 10 −5 eV/atom, respectively.Previous study [10] has demonstrated that a good description of the structural stability, band gaps, and magnetic interactions can be obtained when PBE +  is applied with the fully localized limit, which is therefore also used in the present study. = 2.0 eV is employed for -MnO 2 .Table 1 lists the calculated lattice parameters for -MnO 2 obtained from PBE + .These results are within 1.8% of the theoretical [17] and experimental [20] parameters, but the common tendency for PBE +  to overestimate the unit cell volume is evident.Regarding the values listed for rutile TiO 2 in Table 2, the results are also similar to those of theoretical [16] and experimental [21] reports on DFT + TS.Therefore the calculated sets are appropriate for investigating the surface effects of MO 2 (M = Mn, Ti).  3 and  4, the surface energy  surface for -MnO 2 and rutile TiO 2 obtained via PBE +  and DFT + TS calculations, respectively, is shown.The surface energy is calculated by taking the difference between the energy of a constructed slab and the same number of -MnO 2 or rutile TiO 2 formula units in the bulk [19]:

Results and Discussion
where  total is the energy of a surface or nanorod model containing  formula crystal units,  b is the total energy of the crystal, and  is the surface area of the simulated models, where the bulk surface and microfacet contain two surfaces by their periodic boundary condition.The results are shown in Figure 3 (as well as in Tables 3 and 4).In [19], it is shown that the  surface values of the -MnO 2 (100) and (110) bulk surfaces are equal to 0.6503 Jm −2 and 0.6794 Jm −2 , respectively, which are similar to the results reported by Tompsett et al. (0.64 Jm −2 and 0.75 Jm −2 , resp.) [17].Regarding the rutile TiO 2 , the  surface value for the T(100) is equal to 1.2492 Jm −2 , which is a little larger than that (1.00 Jm −2 ) obtained by Deringer and Csányi [16].Furthermore, the  surface value of the T(110) is equal to 1.1503 Jm −2 , which is similar to that (1.10 Jm −2 ) obtained by Ramamoorthy et al. [26], but a little larger than the 0.81 Jm −2 obtained by Lindan et al.
[27] and the 0.80 Jm −2 obtained by Deringer and Csányi [16].However, none of them influence the trend and the internal law governing the surface energy in this manuscript.
The results are shown in Figures 3 and 4. For -MnO 2 , it is found that the  surface value increases according to a trend of  surface M(100) <  surface M(110) <  surface MNR(I) <  surface MNR(II) <  surface MNR(III) <  surface M[(100 × 110)], as shown in Figure 3(a) (labeled by A).Given that the surface energies of M(100) and M(110) are nearly equivalent, the abscissa has no physical significance.Therefore, it assumes that the abscissa value of M(100) and M(110) is 2.5 and that of M[(100 × 110)] is 9, which is greater than that of MNR(III) by 3 points, as shown in Figure 3(a) (labeled by B).It is found that the surface energies of -MnO 2 of different morphologies fall in line with the relationship among them, implying the existence of an internal correlation.Furthermore, the sequence of M[(100 × 110)] does not correspond to nanorod(III), because other nanorods may exist.Upon closely analyzing their geometric structure, it is found that all the -MnO 2 nanorods and microfacets are composed of two Miller indexes, for example, the (110) and (100) microsurfaces.They differ only in the numbers of the ( 110) and (100) microsurfaces.If it is hypothesized that the average of the values of  surface M(100) and  surface M(110) is one component element of the surface energy for the nanorods and microfacets, their surface energies exhibit some linear relationship.This has not been observed previously.Their formulized relationship can be explained as follows: one constant parameter of the surface energy, 1.0401 Jm −2 , labeled , and another constant parameter of the surface energy, 0.6648 Jm −2 labeled , which is equal to the average of  surface M(100) and  surface M(110), that is, (0.6593 Jm −2 + 0.6974 Jm −2 )/2 = 0.6648 Jm −2 .The correlation function for the surface energy of the nanorods and microfacets to the average value of the surface energy  for M(100) and M(110) is given as From the results of analysis, the surface energy of -MnO 2 nanorods and microfacets has a quantization character.Therefore, it is needed to determine whether there is the same for rutile TiO 2 .To do so, the surface energies of the rutile TiO 2 are shown in Figure 3(b) and in Table 4.The trend in the surface energy  surface for rutile TiO 2 from bulk surface → nanorod → microfacet was found to be different from that of -MnO 2 .Their surface energies were found to be similar to each other.The difference between them is very small, as shown in Figure 3(b).For example, the largest  surface is for T(100), which is equal to 1.2492 Jm −2 .The smallest is for T(110), which is equal to 1.1503 Jm −2 .Their difference is only 0.0989 Jm −2 .Furthermore, the differences between the microfacet and nanorods are obviously very small.Therefore, the trend in the surface energy for the rutile TiO 2 in a bulk surface and nanorods assumes a horizontal line, as shown in Figure 3(b).In deep analysis, the microfacet and nanorod models are also composed of two Miller indexes, for example, (110) and (100) microsurfaces.And it takes the average of  surface T(100) (1.2492 Jm −2 ) and  surface T(110) (1.1503 Jm −2 ) as one constant   , where   is equal to (1.2492 Jm −2 + 1.1503 Jm −2 )/2 = 1.1997Jm −2 .It is found that the relationship between the surface energy of microfacets/nanorods and constant   can also be fitted by linear regression, as shown in Figure 4.The surface energy function is given by  surface TNR(1) = 1.0102 × 1.1997 Jm −2 ,  surface TNR(2) = 1.0317 × 1.1997 Jm −2 ,  surface TNR(3) = 1.0347 × 1.1997 Jm −2 , and  surface T[(100 × 110)] = 0.9879 × 1.1997 Jm −2 .This evolution character of the surface energy of rutile TiO 2 is different from that for -MnO 2 .The quantization character for -MnO 2 nanorods is a positive integer, while that for rutile TiO 2 is equal to 1.However, they all have a quantization phenomenon in their surface energies.

Evolution Character of Cohesive Energy.
The cohesive energy represents the work that is required for a crystal to be decomposed into atoms, which in turn denotes the stability of the respective simulation model.Here, the  cohesive value for several -MnO 2 or rutile TiO 2 models has been calculated from the following equation [19]: where  and  represent the number of M (M = Ti or Mn) and O atoms in the respective morphologies of rutile TiO 2 or - total denotes the total energy of the M  O  models, and  M gas and  O gas are the energies of the gaseous M (M = Ti or Mn) and O atoms, respectively.Before optimizing the gaseous atoms, a 10 × 10 × 10 ( Å3 ) vacuum box is constructed and a single atom is placed, such as Ti, Mn, or O, in the center of the box to be relaxed and thus to obtain its global minimum energy.The results are given as  Mn gas = −588.1855eV and  O gas = −432.2548eV, as given in Table 3.For rutile TiO 2 , the results are  Ti gas = −1594.3577eV and  O gas = −431.9368eV, wherein the difference of energy for oxygen in -MnO 2 and rutile TiO 2 was originated from their different calculated sets in previous part of Simulation Models and Method.The results are given in Table 4.A previous study [19] has discussed the evolution of cohesive energy for -MnO 2 and found that the structural stability of the nanorods and microfacet is lower than that of the crystal and bulk surfaces.The present study will examine the relationship between the bulk surface, nanorods, and microfacet.If only the absolute values of the cohesive energy are considered, it can only determine that the structural stability of nanorods and microfacet is lower than that of bulk surfaces and crystals, as is already known.Analyzing their geometric morphologies, it is found that they are also composed by two Miller indexes, for example, the (110) and (100) microsurfaces.This paper applies the same treatment to the average value of cohesive energy  a-cohesive for M(110) and M(100) to establish a basic standard, which is equal to −4.6611 eV/atom (where  cohesive M(110) = −4.6487eV/atom and  cohesive M(100) = −4.6735eV/atom) in Figure 5. Considering the ratio   of the cohesive energy for microfacets and nanorods to  a-cohesive , it is found that every instance of  is nearly equal to 1.09, where   1 =  a-cohesive / cohesive MNR(I) = 1.1048,   2 =  a-cohesive / cohesive MNR(II) = 1.0958,   3 =  a-cohesive / cohesive MNR(III) = 1.0907, and   4 =  a-cohesive / cohesive M[(100 × 110)] = 1.0942.It is found that all the correlation constants are equal to 1.09, as shown in Figure 6.This trend, which presents as a horizontal line, is different from the trend in the surface energy.Obviously, a quantization phenomenon can be seen.Regarding the cohesive energy of rutile TiO 2 , the difference is found to be very small, unlike the case of -MnO 2 .For example, the most stable structure is the rutile TiO 2 crystal, for which the cohesive energy is equal to −7.8669 eV/atom.The least stable structure is TNR(1), for which the cohesive energy is equal to −7.7255 eV/atom.The difference between them is only equal to 0.1414 eV/atom.The second stable structure is the bulk surface T(100), for which the cohesive energy is equal to −7.7885 eV/atom.The cohesive energies of the bulk surface T(110), microfacets [(100 × 110)], and TNR(3) are very similar, being −7.7750 eV/atom, −7.7712 eV/atom, and −7.7779 eV/atom, respectively.The difference in their cohesive energies is only 0.0067 eV/atom, which may be regarded as being the calculation error, so that they can all be regarded as having the same structural stability.Regarding the trend in their cohesive energies, shown in Figure 5, they closely approximate to each other.Applying the same treatment to the cohesive energy of rutile TiO 2 , it can also take the average value of cohesive energy  a-cohesive for T(110) and T(100) to be a basic standard, which is equal to −7.7817 eV/atom (where  cohesive T(110) = −7.7750eV/atom and  cohesive T(100) = −7.7885eV/atom).Considering the ratio   of the cohesive energy for the nanorods/microfacets to  a-cohesive , it is found that every instance of  is nearly equal to 1.00, where   1 =  a-cohesive / cohesive TNR(1) = 1.0073,   2 =  a-cohesive / cohesive TNR(2) = 1.0037,   3 =  a-cohesive / cohesive MNR(3) = 1.0049, and   4 =  a-cohesive / cohesive T[(100 × 110)] = 1.0013.All the normalized parameters are nearly equal to 1.00, as shown in Figure 6.Then, the evolution character of the cohesive energy for -MnO 2 and rutile TiO 2 is abstracted absolutely.In line with the evolution of the geometric morphologies in the growth of nanorods, the ratio of their cohesive energies divided by  a-cohesive is found to be nearly equal to 1.

Evolution Character of Electronic Structure.
From the above analysis, it is found that, from the bulk surface to nanorods, and even to microfacet with a nanometer structure, the surface and cohesive energies of -MnO 2 and rutile TiO 2 exhibit some quantization phenomena.It is well known that the surface and cohesive energies are derived from the  geometric or electronic structures.Therefore, in the next section of this paper, we study the evolution of the geometric and electronic structures by applying Mulliken analysis to reveal whether there is any quantization phenomenon corresponding to those energies.The Mulliken population  A-B between atoms A and B and the Mulliken charge  A are defined as follows [21]: where  V and  V are the density and overlap matrices, respectively, and   is the weight associated with the calculated -points in the Brillouin zone.Usually, the magnitude and sign of (A) characterize the ionicity of atom A in the supercell, while  A-B can be used to approximate the average covalent bonding strength between atoms A and B. It is known that all the bulk surface, nanorod, and microfacet models originate from their crystal.As a result, the area of the crystal can be regarded as being infinite.To determine the intrinsic mechanism of the evolution character on the surface and cohesive energies, it sets a bond length  A-B , Mulliken population  A-B , and Mulliken charge  A of the crystal as the base values.These base values are equal to the average value of the bond length  A-B , the Mulliken population  A-B , and the Mulliken charge  A in -MnO 2 or rutile TiO 2 crystal, respectively.Then, the bond length variance (Δ) 2 is set equal to ( A-B −  A-B ) 2 to elucidate the influence of the other geometric morphologies, such as the bulk surface, nanorods, and microfacet by their growth.The Mulliken population variance (Δ A-B ) 2 is equal to ( A-B −  A-B ) 2 .However, the Mulliken charge for -MnO 2 or rutile TiO 2 consists of two parts, namely, the lost charge of metallic elements Mn or Ti and the reception charge of the oxygen elements.Therefore, the Mulliken charge variance (Δ A ) 2 is equal to the sum of (Δ M ) 2 and (Δ O ) 2 .In this case,  A-B ,  A-B ,  Mn , and  O for -MnO 2 are equal to 1.9255, 0.3925, −0.59, and 1.05, respectively.For rutile TiO 2 ,  A-B ,  A-B ,  Mn , and  O are equal to 1.9714, 0.5050, −0.66, and 1.31, respectively.To identify the quantization phenomenon of the surface effect, the summation of ∑(Δ) 2 , ∑(Δ A-B ) 2 , and ∑(Δ A ) 2 by their surface area  is normalized.Although the size and shape of nanocrystal have been set as a function of its free energy [15], the surface area  is the vital factor to affect the chemical performance of nanomaterials.Then the surface area  is chosen to be a normalized parameter in this paper.The detailed data is exhibited in Tables 5 and 6.
The final results are shown in Figure 7. Figure 7 shows that the normalized parameters ∑(Δ) 2 /s, ∑(Δ A-B ) 2 /s, and ∑(Δ A ) 2 /s for the bulk surface, nanorod, and microfacet models for -MnO 2 and rutile TiO 2 fluctuate considerably and exhibit irregularities.Regarding the electronic properties of the bond strength and atomic charge, the largest is for M(100) or T(110) to -MnO 2 or rutile TiO 2 , respectively, as shown in Figures 7(a) and 7(d), whose the surface energy is the smallest for each model.In the case of -MnO 2 , there is an intrinsic law conforming to the configuration evolution, such as the linear correlation between MNR(I) and MNR(III), shown in Figures 7(b) and 7(c).Upon plotting ∑(Δ A-B ) 2 / and ∑(Δ A ) 2 /s for MNR(I)-MNR(III) and M[(100 × 110)], along with the quantization number  of the abscissa value of 1, 2, 3, and 6, as shown in Figure 8, it is found that they exhibit a linear correlation with  2 of 0.989 and 0.992, respectively.It is well known that the value of the surface energy sometimes reflects the catalytic performance   of materials, in which case the evolution in the normalized Mulliken population and the Mulliken charge of -MnO 2 nanorods and microfacet are very similar to those of the surface energy, as shown in Figures 3 and 4, thus confirming how the surface catalytic performance of nanomaterials is mainly controlled by the electronic structure.Furthermore, we can assume that the surface energy ( surface ) is a function of the normalized Mulliken population (∑(Δ A-B ) 2 /s) and normalized Mulliken charge (∑(Δ A ) 2 /).Their summation (∑(Δ A-B ) 2 / + ∑(Δ A ) 2 /) is shown in Figure 9.It noted an absolute linear relationship, as shown in Figures 9(a), 9(b), and 9(c), where the function of  surface versus (∑(Δ A-B ) 2 /s +∑(Δ A ) 2 /s) is () = 102.9×  + 0.101 with an  2 value of 0.995.This indicates that the surface effect in nanomaterials differs from that in bulk materials.Regarding rutile TiO 2 , a near-horizontal correlation between TNR(1) and TNR(3) is shown in Figures 7(e) and 7(f).Because all the nanorods and microfacets are composed of {100} and {110} Miller indexes, it can be assumed that their electronic structure originates from that of the (100) and (110) bulk surfaces.These were treated in the same way as their surface and cohesive energies.If we hypothesize that the average value of ∑(Δ A-B ) 2 / and ∑(Δ A ) 2 /s for M(100) and M(110) is one component element contributing to the evolution character of nanorods and microfacet, their electronic structures can be explored and some linear function can be found.First, we abstracted the average value  of ∑(Δ A-B ) 2 /s and ∑(Δ A ) 2 /s in M(100) and M(110) to be  M ( A-B ) = 0.04535 and   ( A ) = 0.004559, respectively.Second, we abstracted the quantization number by the quotient of the average value of  divided by the corresponding value of ∑(Δ A-B ) 2 /s and ∑(Δ A ) 2 /s for the nanorod and microfacet models.The same treatment was applied to rutile TiO 2 , wherein the average value   of ∑(Δ A-B ) 2 /s and ∑(Δ A ) 2 / in T(100) and T(110) is found to be   M ( A-B ) = 0.06711 and   M ( A ) = 0.000545, respectively.The results are shown in Figure 10 and in Tables 5 and 6. Figure 10 shows that there is an obviously linear correlation in the evolution from nanorod to microfacet for -MnO 2 , regardless of the Mulliken population and Mulliken charge.After linear fitting, they obtain a function () = −0.5921*  + 4.011 for ∑(Δ A-B ) 2 /s and () = −0.42* +3.461 for ∑(Δ A ) 2 /, respectively, where the quantization number  or  is a positive integer.This evolution law is the same as that of the surface energy shown in Figure 3.For the rutile TiO 2 , regardless of the Mulliken population and Mulliken charge, there is another evolution law which is different from that for -MnO 2 .After linear fitting, a near-horizontal line for ∑(Δ A-B ) 2 / and ∑(Δ A ) 2 / in the evolution from nanorod to microfacet for rutile TiO 2 is obtained, while the quantization number  or  is close to 1, which is the same as the surface energy shown in Figure 3.This abstracts the quantization phenomenon in an electronic structure and its relationship with the surface energy.

Discussion
It is well known that metallic oxides offer great potential for application to catalysts, not only for clean energy applications but also for pollution mitigation.Typically applied dioxides are -MnO 2 and rutile TiO 2 .Their nanometer structure is ideal for attaining the greatest catalytic action.However, there are only a few valid theories that can be used to guide their design.The determination of the intrinsic mechanism of the surface effects and the correlation with the bulk surface or crystal has attracted the attention of many researchers.Deringer and Csányi [16] and Tompsett et al. [17] determined the geometric configuration of -MnO 2 and rutile TiO 2 nanorods by applying the Wulff construction method, the results of which were corroborated by experiment.Hummer et al. [18] compared the discrepancy between the total calculated nanoparticles surface energies and the summed energies of the constituent faces for rutile TiO 2 and inferred that they uncorrelated with each other as the discrepancy was large.However, they are not able to identify the contributors to the surface effect.In the present study, the surface energies of the -MnO 2 nanorods exhibit a quantization phenomenon.Following the growth of the nanorods, the surface energies of the nanorods are defined by the function  surface = 1.0401Jm −2 + ×0.608Jm −2 , where  is a positive integer and is a maximum value of 6.Then, considering only the surface energies of the -MnO 2 nanorods, the optimal structure is identified.Considering their stability, they also obey the following law:  a-cohesive / cohesive MNR ≈ 1.09.The interaction between the surface energy and cohesive energy in the quantization phenomenon also conforms to the following commonly held view: the growth of the surface energy of nanometer materials will adversely affect their structural stability.This evolution character of the -MnO 2 nanorods differs from that of rutile TiO 2 .It is found that the surface energies of rutile TiO 2 nanorod and microfacet conform to another function:  surface TNR ≈ 1.0102 × 1.1997 Jm −2 , which means that the surface energy will remain nearly constant during the growth of rutile TiO 2 nanomaterials.This phenomenon indicates that the surface effect would have a similar impact on a catalyst consisting of rutile TiO 2 nanorods, ignoring their morphologies.Regarding their structural stability, it is found that their cohesive energy conforms to the following rule:  a-cohesive / cohesive TNR ≈ 1.00.This phenomenon indicates that the rutile TiO 2 nanorods will exhibit a better structural stability during the manufacturing process, relative to -MnO 2 nanorods.With further analysis, their quantization phenomenon originates from the evolution character of the electronic structure in terms of the difference in the bond strength and the atomic charge, rather than the geometric configuration.From the previous analysis, the surface energies of -MnO 2 nanorods and microfacet are increased straightly with the summation of ∑(Δ A-B ) 2 /s and ∑(Δ A ) 2 /s, but they keep constant for rutile TiO 2 .
Mulliken population and charge are originated from the valence electrons of component elements from their formulas [28].In other words, if we enhance the valence electrons of -MnO 2 catalysts by doping process, their surface activity would be improved because of the increased surface energy.So it is not hard to understand why their doping elements for MnO 2 catalysts are Pt [29], Pd [29], Ag [30], Nb [31], Fe [32], and so on, which are translation metals with abundance of valence electrons instead of metalloid elements.But for rutile TiO 2 nanorods and microfacet, their surface energies are fluctuating smoothly with the summation of ∑(Δ A-B ) 2 /s and ∑(Δ A ) 2 /s.So if we dope the rutile TiO 2 catalysts with translation metals, such as Fe, V, and Cr [33][34][35], their improved effect would be very limited because the catalytic performance of rutile TiO 2 is sensitive to its change of energy gap [36].Then the doping processes for rutile TiO 2 catalysts are used for the metalloid elements, such as N [37], Sn [38], and S [39], which affects the internal bonding orbitals.Conclusively limited by their surface energies of rutile TiO 2 nanomaterials, the optimized way to enhance the catalytic performance of rutile TiO 2 is that doping technology appending nanofabrication instead of single nanofabrication method.Then our investigation has a vital significance to understand and help the optimized processed of metallic oxides catalysts.Space limitations mean that, within the scope of this paper, we have not been able to address other Miller indexes for -MnO 2 or rutile TiO 2 nanorods, such as {112}, {211}, and {111} for the -MnO 2 nanorods and {101} for the rutile TiO 2 nanorods, as mentioned by Deringer and Csányi [16] and Tompsett et al. [17], respectively, although their proportions are smaller than those of the (100) and (110) Miller indexes for -MnO 2 or rutile TiO 2 nanorods.However, we do not think that this flaw influences the significance of this paper, given that we began by identifying the evolution mechanism of the metallic oxidation of MO 2 (M = Mn, Ti) nanorods and microfacet, which have a correlation with their bulk surfaces and structures.The overall evolution character of metallic oxidation MO 2 (M = Mn, Ti) nanorods and their other nanometer structures will be revealed and addressed in our future research.Furthermore, the evolution mechanism between a nanometer structure and bulk surface will be useful for investigating the intrinsic mechanisms of nanoeffects.

Conclusion
The evolution mechanism of metallic dioxide MO 2 (M = Mn, Ti) from nanorods to bulk crystal has been investigated by first-principles calculation.The results of the investigation show the following: (1) The surface energies of -MnO 2 and rutile TiO 2 nanorods and microfacets have a quantization phenomenon.For -MnO 2 , it is found that the surface energy conforms to the function:  =  +  × , where  is equal to 1.0401 Jm −2 ,  is equal to 0.6648 Jm −2 , and  is equal to a positive integer of no more than 6.For rutile TiO 2 , the surface energy conforms to another function:  surface TNR ≈ 1.0102 × 1.1997 Jm −2 , which remains constant regardless of the geometric structure of the rutile TiO 2 nanorods.
(2) The cohesive energies of the -MnO 2 and rutile TiO 2 microfacets and nanorods also have a quantization phenomenon.For -MnO 2 , it is found that the cohesive energy conforms to the function  a-cohesive / cohesive MNR ≈ 1.09, where  a-cohesive is equal to the average of  cohesive M(110) and  cohesive M(100).For rutile TiO 2 , the cohesive energy conforms to  a-cohesive / cohesive TNR ≈ 1.00, where  a-cohesive is equal to the average value of  cohesive T(110) and  cohesive T(100).
(3) The electronic properties of -MnO 2 and rutile TiO 2 nanorods and microfacet also exhibit a quantization phenomenon.After being normalized by their surface area, the Mulliken population and Mulliken charge variance of -MnO 2 exhibit a linear function as () = 0.5921 *  + 4.011 for ∑(Δ A-B ) 2 /s and () = 0.42 *  + 3.461.However, the Mulliken population and Mulliken charge variance of rutile TiO 2 exhibit a nearly horizontal line in the evolution from nanorod to microfacet.

Figure 8 :
Figure 8: Schematic curve of Normalizing Mulliken population and Mulliken charge of -MnO 2 in different morphologies with nanostructure.

Figure 9 : 2 MFigure 10 :
Figure 9: Schematic curve of surface energy versus Normalizing Mulliken population or Mulliken charge of -MnO 2 in different morphologies with nanostructure.
3.1.Test of Potential.The value of the  parameter for our PBE +  calculations is determined by ab initio calculations.

Table 2 :
Predicted DFT + TS, experimental and theoretical lattice parameters for rutile TiO 2 .