Efficient method to calculate total energies of large nanoclusters

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by  M. Yu

Efficient method to calculate total energies of large nanoclusters
M. Yu1,3, R. Ramprasad2, G. W. Fernando1 and Richard M. Martin3
1Department of Physics, University of Connecticut, Storrs, CT 06269
2Department of Chemical, Materials and Biomolecular Engineering,
University of Connecticut, Storrs, CT 06269 and
3Department of Physics, University of Illinois at Urbana-Champaign, IL 61801
(Dated: February 2, 2008)
We present an approach to calculate total energies of nanoclusters based on first principles es-
timates. For very large clusters the total energy can be separated into surface, edge and corner
8
energies, in addition to bulk contributions. Using this separation and estimating these with direct,
0
first principles calculations, together with the relevant chemical potentials, we have calculated the
0
total energies of Cu and CdSe tetrahedrons containing a large number of atoms. In our work we
2
considerpolyhedralclusters sothatin additionourwork providesdirect information onrelaxation.

n For Cu the effects are very small and the clusters vary uniformly from very small to very large
a sizes. ForCdSethereareimportantvariations insurfaceandedgestructuresforspecificsizes; nev-
J ertheless, the approach can be used to extrapolate to large non-stoichiometric clusters with polar

2 surfaces.

] PACSnumbers: 31.15.A-,61.50.Ah,68.35.Md
i
c
s
- I. INTRODUCTION valuescalculatedforasmallnumberofwelldefinedclus-
l
r ters. This approach provides crucial information needed
t
m totreataccuratelytheseclustersaswellasgeneralresults
Nanoscience provides an ideal platform in the search
that can be used for other classes of clusters.
. fornovelmaterialswithdesirableandtunableproperties.
t
a Nanocrystals(NCs)ofvarioussizesandshapeshavebeen
m
found to exhibit a wide variety of physical and chemical
II. METHODOLOGY
- properties1,2,3,4,rarelyseeninbulkmaterials. Synthesiz-
d
ing nanoparticlesofagivensize andshapeis notoriously
n A. Calculational method for clusters
difficult and has become a key focus area due to techni-
o
c callysignificantpropertiesthatdependonthesize/shape
[ of the cluster. Growth of such clusters is governed by Surface energies have been extracted from first prin-
bothkineticsandthermodynamics5. IfaNChasahighly ciples slab (total energy) calculations by several groups
1 (see for example, Refs.11,12). Besides that, a first prin-
v symmetric crystal structure (such as zincblende), it is
ciples wedge-shaped approachhas also been proposedto
5 likely that when synthesizing, the crystal will grow with
calculate the surface energies of polar surfaces10. In the
3 no preferred direction of growth. On the other hand, if
4 thecrystalstructurehasapreferredaxisofsymmetry,as present work, our goal is somewhat different compared
0 in the hexagonal wurtzite structure which has a unique to the above. We will use energies obtained from first
. principles calculations on polyhedral nanoclusters to es-
1 polar axis, preferential growthalong this axis can be ex-
0 pected6,7. Energies associated with various facets (i.e., timate total energies of larger nanoclusters. The energy
8 with different surface orientations) will also play a key can be written as
0
role during growth. The total energy, which depends on
v: these facet orientations, will determine the stability of a Etpootly =XσαAα+XǫβLβ +Xcγ +XNiµi, (1)
i nanocluster having a given size and shape8,9,10. Calcu- α β γ i
X
latingtheseenergiesassociatedwithlargenanocrystalsis
r where, σα, ǫβ and cγ denote surface, edge and corner
a anontrivialtask. Traditional“bruteforce”ordirectfirst energiesrespectively,N isthenumberofatomsoftypei,
i
principles methods become quite laborious as the size of
and µ is the corresponding chemical potential. The last
i
the cluster increases.
term in Eq. (1) contains the bulk energy if the chemical
First principles methods have provided reliable total potentialssatisfytheconditionsgiveninsectionIIB.For
energies for atoms and molecules; this is also true in a stoichiometric cluster the total energy is independent
solids,providedthatthenumberofatomsinagivenunit of the individual chemical potentials and is well-defined
cell is relativelysmall. When this number becomes large relative to the energy of the bulk crystal.
(say more than a thousand), then it becomes computa- In the present work, a least-squares fit for the total
tionally prohibitive to carry out first principles calcula- energies of several small clusters were used to estimate
tions. Naturally,itisdesirabletodevelopalternatetech- the surface, edge and corner energies (as parameters).
niquestoobtainqualitytotalenergiesofsystemscontain- If the above energies can be evaluated in a computa-
ingalargenumberofatoms. Thisworkisfocusedonob- tionally efficient way, then the total energy of a cluster
taining such total energy estimates using first principles (polyhedron), containing a (substantially) large number
2
ofatoms,canbeexpressedalgebraicallyasinEq.(1). We where ∆H is the formation energy. The bulk values,
f
demonstrate that using accurate density functional the- Ebulk(Cd)andEbulk(Se),areobtainedfromtotalenergy
tot tot
ory(DFT)basedestimatesoftotalenergiesofafewsmall calculations of pure Cd and pure Se (per atom, in their
clusters, it is possible to estimate the above (surface, equilibrium structures) separately, while Ebulk(CdSe) is
tot
edge, corner) contributions (as well as chemical poten- the total energy of a CdSe pair in the bulk.
tials)andthenusethem(asparameters)inlargerclusters Followingwellknownprocedures13,wecansetbounds
to estimate total energies. The calculations were based listed below for the chemical potentials of the individual
on the local density approximationwithin the DFT13 as species in CdSe:
implemented in the local orbital SIESTA code14. Norm-
conserving nonlocal pseudopotentials of the Troullier- Etboutlk(Cd)+∆Hf ≤µCd ≤Etboutlk(Cd) (6)
Martins type15 were used to describe all the elements. Ebulk(Se)+∆H µ Ebulk(Se). (7)
For a specific shape, the cluster can be defined by one tot f ≤ Se ≤ tot
characteristic length ℓL0, so that the surface and edge The right hand side of the first inequality represents the
terms can be expressed as Aα = aαℓ2, Lβ = bβℓ1 re- fact that, µCd in the cluster must be smaller than the
spectively, with aα and bβ being constants. Therefore, (pure Cd) bulk value Etboutlk(Cd), since otherwise, Cd
Eq. (1) may be written in the following form; mustphaseseparate. Thelefthandsideofthisinequality
follows from the fact that for µ , one can use the same
(Etpootly −XNiµi)/ℓ2 argument and utilize Eq. (5) toSoebtain the following:
i
=Xσαaα+Xǫβbβ/ℓ+Xcγ/ℓ2. (2) 0≤−µSe+Etboutlk(Se)=µCd−Etboutlk(Cd)−∆Hf. (8)
α β γ
From the above arguments, it appears that we can
We labelthe energyexpressiononthe left sideofEq.(2)
onlyevaluatetheindividualchemicalpotentials,µ and
Cd
as “terminationenergy” since it is the added energy due
µ , within the range given above. However, for a spe-
Se
to the presence of surfaces, edges and corners. For ex-
cificfamilyofnon-stoichiometricclustershavingthesame
ample,foratetrahedralstructureboundedbyfour(111)
shape, we demonstrate below that the total energy can
surfaces. (see Fig. 1), The characteristic size is ℓL (ℓ
0 be determined withno knowledgeofthe chemicalpoten-
being an integer and L the nearest neighbor distance
0 tials.
alonganedge),representinganedgeofatriangular(111)
facet. The tetrahedral structure includes four equivalent
(111) surfaces, six equivalent (111) (111) edges, and
−
fourequivalent(111) (111) (111)corners. Thischoice
− −
enables us to work with well defined (111) surfaces, as
well as equivalent corners, edges and surfaces. In this
case, the termination energy in Eq. (2) turns out to be
1 1
(Etpootly −XNiµi)/ℓ2 = √3σL20+6ǫL0ℓ +4cℓ2.(3)
i
In later discussions, we use the above equation for both
FCC Cu clusters and zincblende CdSe clusters.
B. Non-stoichiometric case
Oneoftheexampleswehavechosentostudyisthebi-
FIG.1: CdSeZincblende(tetrahedral) structureboundedby
nary compound CdSe. When the cluster is stoichiomet-
four (111) surfaces terminated by Cd atoms. Small dots rep-
ric, the number of Cd atoms will be equal to the num- resent Cd atoms, while large dots represent Se atoms. The
ber of Se atoms, and the sum of the chemical potentials, characteristicsizeisℓL0(ℓbeinganinteger,withℓ=6shown
µ +µ = total energy per CdSe pair, can easily be here), representingan edge of a triangular (111) facet.
Cd Se
evaluated from bulk total energy calculations. In a non-
stoichiometric case, the number of Cd atoms is different
from the number of Se atoms, and some energy terms, We first show that the total energy of any CdSe
such as the surface energies,depend on the separate val- (zincblende) tetrahedron, Epoly, bounded by similar
tot
ues of µ or µ , instead of their sum. To examine this
Cd Se (111) facets as shown in Fig. 1, can be expressed in a
further,wesetlowerandupperboundsfortheindividual
slightly different form of Eq. (1), i.e.,
chemical potentials (pertaining to bulk CdSe) as
Etboutlk(CdSe) = µCd+µSe (4) Etpootly(CdSe)−NCdµCd−NSeµSe
= Ebulk(Cd)+Ebulk(Se)+∆H , (5) =√3σL2ℓ2+6ǫL ℓ+4c. (9)
tot tot f 0 0
3
Here L is the nearest neighbor distance along an edge
0
of the tetrahedron.
Combining with Eq. (4), we obtain:
Epoly(CdSe) N Ebulk(CdSe)
tot − Se tot
=(N N )µ +√3σL2ℓ2+6ǫL ℓ+4c.(10)
Cd− Se Cd 0 0
However, note that
1 3
N N = ℓ2+ ℓ+1, (11)
Cd Se
− 2 2
whichresultsfromasimplecountoftheatomsinatetra-
hedron having an edge of length ℓL (ℓ being an inte-
0
gerandL the nearestneighbordistancealonganedge).
0
This result clearly shows that the difference in the num-
ber of Cd and Se atoms arises from surfaces (ℓ2), edges
(ℓ1)orcorners(ℓ0),andleadstothe followingimportant
simplification:
FIG. 2: The termination energy (see text for definition) con-
Etpootly(CdSe)−NSeEtboutlk(CdSe) tribution scaled by ℓ2 for Cu clusters as a function of the
1 3 inverse of the characteristic size (1/ℓ) of the tetrahedrons.
=(2µCd+√3σL20)ℓ2+(2µCd+6ǫL0)ℓ+(µCd+4c). Y-interceptisproportionaltosurfaceenergy(σ),slopeispro-
portional to edge energy(ǫ), and curvature is proportional to
(12)
corner energy(c). The solid squares represent the values cal-
The significance of Eq. (12) is that, even in this non- culated using DFT while theopen triangles represent thees-
timated values from our algebraic expression. The solid line
stoichiometric case, it is possible to estimate the total
is a least squares fit for the DFTpoints.
energies independent of the chemical potentials. This is
because in the above equation, the coefficients of ℓ2, ℓ
andthe constanttermactasstraightforwardparameters
to be estimated. Since there is no volume (L3) term on
L = 2.545˚A after geometric relaxation), bounded by
the right side of Eq. (12), neither µ nor µ will have 0
Cd Se
four, equivalent (111) facets. The relevant chemical po-
a direct effect on the final total energy to be predicted.
Note that Ebulk(CdSe) has an unambiguous value, as tential,µCu,canbeobtainedfromafccbulktotalenergy
the total enetrogty per CdSe pair in the (bulk) zincblende calculation. The first principles total energies, Etpootly, for
structure. Now the parameters, such as surface energy, the tetrahedronsareobtainedafterfully relaxingagiven
edge energy, and corner energy, can be fitted using sev- cluster.
eral, known (DFT based) total energy values from small We find it is more useful to focus on the energy term
clusters. Finally, in order to predict the total energies of (Epoly N µ ), since this represents a termination
tot − Cu Cu
large polyhedrons, the algebraic expression in Eq. (12) energy to the bulk chemical potential contribution due
can be utilized as previously. to the presence of surfaces, edges and corners. The ter-
minationenergiesscaledbythesquareofacharacteristic
length ℓ2, i.e., (Epoly N µ )/ℓ2, for different sized
III. RESULTS tot − Cu Cu
clusters(ℓ=2toℓ=8)calculatedusingDFTareshown
as solid squares in Fig. 2. To estimate surface, edge
Test results from pure (fcc) Cu as well as (zincblende) andcornerenergies,totalenergiesofafew smallclusters
CdSe clusters show that this scheme is reliable to a high (ℓ=2,3,4,5,6)wereusedinaleastsquaresfitaccording
degree of accuracy. One of the significant results of the to Eq.(3). The estimated σ, ǫ andc were 0.1164eV/˚A2,
present study is our ability to calculate energies of nan- 0.2949 eV/˚A, and 1.0047 eV respectively. A previous
oclustersthatarenon-stoichiometricandthathavepolar DFT calculation has reported the (111) fcc Cu surface
surfaces. Furthermore, we will demonstrate that the to- energy σ = 0.1213eV/˚A2 16 while the experimental re-
talenergiescanbeevaluatedindependently ofindividual sult for the surface energy of Cu is 0.1144eV/˚A2 17.
(111)
chemical potentials (Eq. (6) (7)). Hence our calculated σ value is in good agreement with
previous theoretical and experimental values. Using the
three parameters for surface, edge and corner energies,
A. FCC Cu we now estimate the termination energy as a function
of cluster size, as shown in Fig. 2. The estimated total
We begin our discussion with pure fcc Cu clusters, energies for the two larger clusters (ℓ = 7 and ℓ = 8)
which are regular tetrahedral clusters having a charac- are carried out using the algebraic expression shown in
teristic length, L = ℓL (ℓ being a positive integer, Eq. (3).
0
4
Epoly/ℓ2 (eV)
ℓ (L0) NCu DFtoTt algebraic
2 10 -3041.43
3 20 -2704.60
4 35 -2662.65
5 56 -2726.96
6 84 -2840.92
7 120 -2981.98 -2981.95
8 165 -3139.40 -3139.41
TABLE I: Total energies for tetrahedral fcc Cu clusters
The energies predicted using the algebraic expression
are in excellent agreement with the ‘exact’ calculations
FIG. 3: The termination energy contribution (see text) di-
from first principles as evident from the results for ℓ=7
vided by ℓ2 for CdSe clusters as a function of the inverse of
and ℓ=8 tetrahedrons in Table I and Fig. 2. The above
the characteristic size. The squares correspond to DFT en-
result clearlyprovides further supportfor our method of
ergies while the lines are least squares fits. The solid line
estimation.
corresponds to the minimal µ , while the dashed line cor-
Cd
responds to the maximal µ . The open triangles represent
Cd
energies obtained from ouralgebraic calculation after fitting.
B. Tetrahedral, zincblende based CdSe clusters
The second system in our discussion is a non-
stoichiometric, zincblende CdSe cluster bounded by four
tributions from the total energy of the cluster as a func-
equivalent (111) facets terminated by Cd atoms. As de- tion of 1/ℓ. The algebraic values of Epoly obtained from
scribed in the Methodology section and Fig. 1, the char- tot
Eq. (3) for ℓ = 7,8, are indicated on the fitted curves,
acteristic size is ℓL with L =4.342˚A. The bulk values,
0 0 alongwithothervalues resultingfromdirectDFT calcu-
Ebulk(Cd)andEbulk(Se),areobtainedfromtotalenergy
tot tot lations. Ingeneraltheterminationenergydependsonthe
calculations of pure Cd (in hcp structure) and pure Se
choice of µ , as evident from Fig. 3, which is required
(in trigonal structure) separately, while Ebulk(CdSe) is Cd
tot to compare these surfaces with other surfaces. However,
thetotalenergyofaCdSepair(inzincblendestructure).
the predicted values for the total energy (Eploy) of the
Through explicit calculations, we obtain Ebulk(Cd) = tot
tot ℓ = 7,8 clusters relative to other polyhedra with the
1467.17 eV;Ebulk(Se) = 256.97 eV;Ebulk(CdSe) =
− tot − tot same shape can be found using Eq. (10), which yields
1724.84 eV. Therefore, the formation energy is calcu-
− the same value for different values of µ (see Table II;
lated to be ∆H = 0.7 eV (from Eq. (5)). Cd
f − i.e., the algebraic value Epoly/ℓ2 = 4032.30eV for the
In Table II, DFT based total energies for polyhedrons tot −
ℓ=7 polyhedronforallthe valuesofµ .) This valueis
from ℓ = 2 to ℓ = 8 are shown. Using DFT based Cd
comparable to the energy obtained from the direct DFT
small clusters energies (for ℓ = 2,3,4,5,6), we have
calculations, which is an advantage since it shows that
obtained the necessary fitting parameters involving sur-
we can obtain certain energies independent of µ , as
face energy, edge energy, and corner energy, which vary Cd
within the intervals 50 61 meV/˚A2, 227 268meV/˚A, discussed earlier.
− −
and 39 214 meV respectively. We also see that cer-
−
tain clusters undergo noticeable reconstructions, while Epoly/ℓ2 (eV)
others do not. Surface energy varies within the inter- ℓ (L0) NCu NSe DFtoTt algebraic
val 50 61 meV/˚A2, showing an average of surface en-
2 10 4 -3920.98
−
ergies within the reconstructed (for example ℓ = 4,5) 3 20 10 -3542.94
and non-reconstructed (for example ℓ = 6) structures, 4 35 20 -3528.42
which is reasonably lower than previous DFT calcula- 5 56 35 -3644.61
tions( σ 75 95 meV/˚A2 11). Surface reconstructions 6 84 56 -3821.40
∼ −
have been studied before using various methods, such as 7 120 84 -4032.21 -4032.30
the tight binding method18. Here we have observed size 8 165 120 -4263.30 -4263.30
dependent,surfacereconstructionofthe CdSesystemby
using DFT combined with the local density approxima- TABLEII:CdSe(zincblende)totalenergiesforvariouscluster
tion. sizes.
In Fig. 3, for different Cd chemical potentials, we
plot the scaled termination energy, (Epoly N µ
N µ )/ℓ2 by subtracting the chemictaolt p−otenCtidalCcdon−-
Se Se
5
IV. DISCUSSION ablyoriginatefromthechangesinstateoccupationsnear
theFermienergy,withthelargestchangesapparentlyoc-
For polyhedral nanoclusters of fcc Cu, calculations curring for the edges. Also note that as the cluster size
from 10 to 165 atoms show that the energies are well L , the cornerandedgecontributions become small
→∞
described by this form even for small clusters. Thus the compared to the surface energies and thus providing a
energies for all sizes can be determined efficiently based way of estimating the latter for large clusters.
on calculations for small clusters and we propose that Thetotalenergiesaredescribedwellbyaleastsquares
this is a useful approachfor metals. fit, carried out using the results for 5 clusters with 14
WhentestingthisapproachforCdSepolyhedronsthat to 140 atoms, which accurately determines the energies
are Cd terminated and non-stoichiometric, one encoun- for clusters with 204 and 285 atoms. However, the di-
ters energy contributions that depend on the individual vision into three separate contributions as surface, edge
chemical potentials, as seen from Eq. (1) which defines and corner energies, is not as well determined due to
Epoly. These chemical potentials are subject to upper the (possible reconstruction-induced)variations between
tot
and lower bounds13 (see inequalities (6), (7)). However, the different clusters. The estimated surface energy rep-
it turns out that the total energies of even the non- resents an average of different reconstructed and unre-
stoichiometric CdSe polyhedra considered here can be constructed clusters, which is reasonably lower than the
obtainedwithoutknowingtheindividualchemicalpoten- value reported in previous DFT calculations, as pointed
tials. Thisissufficienttoextrapolatetolargesizeforthis out in the results section.
familyofclustersindependentofchemicalpotentialsand
using only directly calculated total energies. The reason
forthe aboveisthatthe totalenergyofsuchclusterscan
V. CONCLUSION
be calculated from well defined bulk energies by adding
surface, edge and corner termination terms; these termi-
nation terms appear as mere parameters that scale with In summary, we have presented an efficient method
the dimensions of the cluster and can be estimated from for calculating total energy of large nanoclusters using a
DFT calculations of small clusters with similar topolo- parametrized,algebraicformwithparametersfittedfrom
gies. In addition, the surface, edge and corner termina- small, first principles based, nanocluster calculations.
tiontermscanbeusedforotherclustersbyincludingthe The method appears to work quite well for pure met-
chemical potentials in the way given in Eq. (12). als. Even for non-stoichiometric, semiconducting clus-
For CdSe, we find an overall trend similar to that for ters with polar surfaces,this approachprovides a way of
Cu; however, there are deviations from a smooth curve estimating total energies of large clusters, important in-
for the energies as a function of size. The deviations are formationsuch as relaxationenergies that are specific to
likelytobe associatedwith reconstructionsthatareseen the chosen clusters, as well as surface, edge and corner
forcertainnanoclustersizes. Thesereconstructionsprob- energies that can be used for other clusters.
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