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materials composed purely of alkali atoms are expected to closely mimic the free - electron gas or jellium models as they have only one very delocalized electron outside of a noble - gas shell . because of the relative simplicity of pure alkali systems they are often viewed as good systems for benchmarking quantum - mechanical methods and for investigating the transition from localized to intinerent electronic behavior  @xcite .
this is especially true for density - functional - based calculations since the original approximations to density - functional theory  @xcite were based upon analytical expressions derived from exchange - only treatments of the the free - electron gas .
for example slater derived an expression for a local potential which reproduced the hartree - fock trace of the free electron gas  @xcite .
later , kohn and sham derived an expression that reproduced the fermi energy of the free electron gas  @xcite .
the latter expression also reproduces the total exchange - energy of a free - electron gas ( see for example ref .
the factor of @xmath1 difference in the kohn - sham and slater approximations also lead to the use of the x@xmath2 approximations  @xcite and approximations along the lines of this method continue to be investigated as an attractive means for developing approximate element dependent functionals that permit fully analytic implementation of density functional theory  @xcite .
such analytic implementation is computationally very efficient and has been used for structure optimization of icosahedral fullerenes containing more than two thousand atoms using triple - zeta quality basis set  @xcite .
the ability to accurately reproduce polarizabilities of large systems is of significant importance to many forefront research areas in computational chemistry , materials science and quantum physics .
the ability to accurately determine polarizabilities is required to account for hydrogen bonding , van der waal s interactions ,  @xcite solvation effects ,  @xcite and a materials dielectric response . as discussed in ref .
@xcite , the same interactions or matrix elements required for an accurate determination of phenomena that are directly mediated by polarizabilities also determine a variety of transition rates which include spontaneous emission , stimulated absorption and emission , and foerster - energy transfer rates .
such rates are of direct importance to the problem of many photovoltaic applications .
the radiative transition rates must also be quantified for applications to quantum - control of matter or any type of light - mediated manipulation of molecular- and cluster- materials .
derivatives of the polarizability tensor with respect to normal - mode displacement also determine the intensity of the raman shifts of a given molecule or cluster  @xcite .    because of the central role that polarizabilities play in materials science and chemistry
there have been significant efforts aimed at experimentally validating theoretically predicted polarizabilities .
however , such experiments are themselves very difficult to interpret for a variety of reasons . from the standpoint of comparison to experiments on bulk systems
, the polarizability of an array of nonoverlapping polarizable molecules may be approximated from the clausius - mossotti relation .
such comparisons have been performed with some success on very idealized systems such as fullerene molecules  @xcite .
however , for pure - metal clusters the polarizability may not be determined from such a means because the individual clusters would coalesce into the the bulk material if placed upon a lattice .
as such the preferred experimental approach for measurement of metal - cluster polarizabilities rely upon the electrostatic equivalent of a stern - gerlach experiment in which a beam of metal clusters traverse a nonuniform electric field and are deflected due to the the induced polarization of the cluster  @xcite .
again , for simple systems such a fullerene molecules quantitative agreement between theory and experiment has been achieved . however , for metallic clusters deviations exist between different experiments and also between experiment and theory .
generally , it appears that the theoretically predicted polarizabilties display a more monotonic and smooth behavior than the experimental measurements .
moreover , earlier comparisons suggested that the experimental polarizabilities tended to be larger than theory .
such discrepancies are now largely understood to be due to temperature dependent corrections that depend upon the permanent dipole of the clusters .
however , some discrepancies still exist and it is not understood why .
one possibility that we attempt to investigate here is that different low - lying geometries of a cluster may have significantly different polarizabilites or anisotropies in their polarizabilities
.    experimental measurements of polarizability of li clusters containing up to 22 atoms have been reported  @xcite . here , we present and compare our density functional predictions of polarizabilities of li clusters with the avialable experimental data .
the calculations are performed for clusters in the size range 8 - 22 as number of studies have already addressed polarizabilities up to size 8  @xcite . rather than concentrate on the _ ground state _ for each size regime we have adopted a different approach that relies on the generation of several low - energy structures .
the polarizability of each structure is subsequently calculated to determine whether conformer - induced changes in polarization are expected to be observable .
the determination of lowest energy structure of clusters is a nontrivial task . to obtain the candidate structures for the ground state of li clusters ,
we make use of a recently determined database of sodium clusters  @xcite .
this database is generated using a global optimization technique such as the basin hopping algorithm using the interatomic potential determined within the kohn - sham dft framework .
this numerical ks dft provides very good description of sodium clusters  @xcite . using starting geometries from this database and bachelet - hamman - schluter pseudo potential method  @xcite
we quickly generate low - energy structures for li clusters using conjugate - gradient technique .
a set of few lowest - energy geometries are then selected for further optimization at the all - electron level until forces on all atoms are below the 0.001 hartree / bohr .
for each of these equilibrium geometries we have then performed six additional calculations with electric fields turned on .
the included fields are ( e,0,0 ) , ( -e,0,0 ) , ( 0,e,0 ) , ( 0,-e,0 ) , ( 0,0,e ) and ( 0,0,-e ) with e=0.001 atomic units . from the standard relation between the dipole
moment , polarizability , and applied electric field : @xmath3 we determine the polarizability tensor using a central differencing method .
additional discussion of this procedure may be found in ref .
@xcite . from the above expression it is possible to determine the central - difference off - diagonal elements ( @xmath4 ) from either @xmath5 and @xmath6 or @xmath7 and @xmath8 . in the limit of a vanishingly small - field or an exactly harmonic polarizability tensor
, the values obtained in the two different ways would agree perfectly ( regardless of issues related to basis sets ) .
this exact relation provides one way of estimating whether the applied electric field is small enough to accurately determine the polarizability tensor .
we find that the off - diagonal elements are small but agree to approximately 1 - 5% .
of course , the size of the off - diagonal element depends on the original choice of coordinate system .
our original coordinate system uses the principal axes of inertia . to determine the principal polarizability axes we then diagonalize the polarizability tensor and rotate the molecule so that the molecular x , y and z axes coincide with the predicted principle polarizability axes . as a final check we have ascertained for several cases that the predicted polarizabilities are accurate and uncontaminated by higher - order polarizabilities . for these tests , we perform an analysis similar to that of quong and pederson for the c@xmath9 and benzene polarizabilites  @xcite . that is , we have determined that the polarizabilities obtained from either energies or dipole moments agree .    to perform the electronic - structure calculations we have used the nrlmol suite of codes due to pederson _
et al . _
this all - electron method uses gaussian - type orbitals to represent both the core , valence and unoccupied electrons .
a highly accurate variational integration mesh  @xcite is used to calculate matrix elements required for the secular equation .
the coulomb potential is calculated analytically ( in terms of incomplete gamma functions )  @xcite on this mesh .
the exchange - correlation potential and energy kernel is also calculated analytically on this mesh . for the work discussed here we use the perdew - burke - ernzerhof  @xcite generalized gradient approximation .
the basis sets here have been specifically optimized for this functional . for li
, we use a total of ten single - gaussians to construct 5 contracted s - type orbitals , 3 contracted p - type orbitals and 1-contracted d - type orbital .
each d - type orbital used also contributes a spherical s - type orbital with an @xmath10 prefactor .
this basis set is used for optimizing the geometry .
.gaussian decay parameter , @xmath11 in bohr@xmath12 , and contraction coefficients , c(1s ) and c(2s ) , used in this work . as discussed in the text
, the default basis set also contains single gaussians . for the s functions we use @xmath13-@xmath14 .
the p functions use @xmath15-@xmath16 .
the d functions use @xmath15 . for the polarizability calculations , we also include @xmath14 and @xmath17 for the p- and d- functions respectively . [ cols= " > , > , > " , ]
in this paper we have discussed a relatively automated procedure for investigating polarizabilities of clusters as a function of size and conformation .
this procedure has been tested on lithium clusters in the n=8 - 22 range due to availability of experimental data .
while the binding energy per atom and average polarizability show very little nonmonotonic behavior in this range , there are significant departures from isotropic polarizabilities in these clusters . as expected from jellium models it is only for n=20 , the polarizability tensor is found to be significantly isotropic .
the calculated polarizabilities show smoother decrease with increase in cluster size than the experimental values .
the spin polarized calculations indicate that all even size clusters as expected have zero moments ( no unpaired electron ) .
however , a few odd size clusters are found to show higher ( 3 @xmath18 ) spin states .
we thank dr .
p. calaminici and dr .
g. maroulis for encouraging us to complete the present work for this issue .
this work is supported in part by the national science foundation through crest grant , by the university of texas at el paso ( utep startup funds ) and in part by the office of naval research , directly ( onr 05pr07548 - 00 ) and through the naval research laboratory .
computational support was provided by the dod high performance computing modernization office .
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we present a detailed investigation of static dipole polarizability of lithium clusters containing up to 22 atoms . 
 we first build a database of lithium clusters by optimizing several candidate structures for the ground state geometry for each size . 
 the full polarizability tensor is determined for about 5 - 6 isomers of each cluster size using the finite - field method . 
 all calculations are performed using large gaussian basis sets , and within the generalized gradient approximation to the density functional theory , as implemented in the nrlmol suite of codes . 
 the average polarizability per atom varies from 11 to 9 within the 8 - 22 size range and show smoother decrease with increase in cluster size than the experimental values . while the average polarizability exhibits a relatively weak dependence on cluster conformation , significant changes in the degree of anisotropy of the polarizability tensor 
 are observed . 
 interestingly , in addition to the expected even odd ( 0 and 1 @xmath0 ) magnetic states , our results show several cases where clusters with an odd number of li atoms exhibit elevated spin states ( e.g. 3 @xmath0 ) .    received 30 may 2007    density - functional theory , nrlmol , polarizability , lithium clusters    32.10.dk,33.15.kr,42.65.an,36.40.cg , 30.40.vz