Accepted Manuscript

Site-specific Polarizabilities from Analytic Linear-response Theory

Juan E. Peralta, Veronica Barone, Koblar A. Jackson

PII: S0009-2614(14)00420-5DOI: http://dx.doi.org/10.1016/j.cplett.2014.05.045Reference: CPLETT 32190

To appear in: Chemical Physics Letters

Received Date: 1 April 2014Accepted Date: 16 May 2014

Please cite this article as: J.E. Peralta, V. Barone, K.A. Jackson, Site-specific Polarizabilities from Analytic Linear-response Theory, Chemical Physics Letters (2014), doi: http://dx.doi.org/10.1016/j.cplett.2014.05.045

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customerswe are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, andreview of the resulting proof before it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Site-specific Polarizabilities from Analytic Linear-responseTheory

Juan E. Peralta, Veronica Barone, Koblar A. Jackson

Department of Physics and Science of Advanced Materials, Central Michigan University,Mount Pleasant, MI 48859

Abstract

We present an implementation of the partitioning of the molecular polariz-

ability tensor [J. Chem. Phys. 125, 34312 (2006)] that explicitly employs the

first-order electronic density from linear-response coupled-perturbed Kohn-

Sham calculations. This new implementation provides a simple and robust

tool to perform the partitioning analysis of the calculated electrostatic po-

larizability tensor at negligible additional computational effort. Comparison

with numerical results for Si3 and Na20, and test calculations in all-trans-

polyacetylene oligomeric chains up to 250 A long show the potential of this

methodology to analyze the electric response of large molecules and clusters

to electric fields.

∗Corresponding authorEmail address: juan.peralta@cmich.edu (Juan E. Peralta)

Preprint submitted to Elsevier May 21, 2014

1. Introduction

The response of molecules and clusters to external stimuli such as elec-

tric and magnetic fields provides a powerful tool to probe their electronic

structure. In particular, the electrostatic dipole polarizability tensor, α,

characterizes the second-order electronic response to uniform static electric

fields and it is proven to be a useful property to understand the basic physics

and chemistry of clusters both experimentally and theoretically.[1, 2, 3, 4]

From the practical viewpoint, there is also a growing interest in understand-

ing how nanoscale systems behave upon the action of electric fields with the

ultimate goal of tuning their properties for device applications.[5, 6]

The calculation of α in molecules and clusters can be routinely carried

out with a variety of available electronic structure programs. Although there

are a number of approximations based on wavefunction theory that can yield

accurate polarizabilities in small and medium systems, density functional

theory (DFT) has emerged as the only computationally affordable choice

for large systems. Importantly, DFT has been shown to yield accurate po-

larizabilities in systems such as metal clusters and polymers.[7, 8, 9, 10] In

addition to the general interest in determining the polarizability from elec-

tronic structure calculations, computational methods have been proposed

that allow for an evaluation of the contributions from the different molec-

ular fragments to the overall electronic response. Partitioning techniques

can be used to decompose the calculated polarizability into contributions

from molecular constituents for physical interpretation.[11, 12, 13, 14] Van

Alsenoy and coworkers have employed the Hirshfeld partitioning to analyze

the molecular polarizability using a finite differences scheme in a variety of

molecular systems and clusters.[15, 16, 17, 18] This methodology allows the

2

response of different atomic sites of a cluster or molecule to be quantified

based on the analysis of changes in the electric dipole upon the action of

small finite electrostatic fields. Jackson et al. have proposed a similar site-

specific partitioning scheme and utilized it to analyze the structural and size

dependence of the polarizability in Si and Na clusters.[19, 20, 21] In a recent

work, Zeng et al. have employed these same ideas to partition the molecular

hyperpolarizability into site-specific local and nonlocal contributions.[22]

In this work we present the implementation of the partitioning analysis

of Krishtal et al.[15] and Jackson et al.[19] using explicitly the first-order

density from linear-response coupled-perturbed Kohn-Sham (CPKS) calcu-

lations. This new implementation provides a streamlined and robust tool

to perform the site-specific analysis of the calculated electrostatic polariz-

ability tensor at virtually no additional computational effort. The main

advantage of this implementation is its ability to treat very large systems

where the finite differences method presents numerical problems. We show

that our implementaion is numerically equivalent to the original[19] by di-

rect comparison of a site-specific analysis for the Si3 and Na20 clusters.

We also show proof-of-concept calcuations in all-trans-polyacetylene (TPA)

oligomeric chains up to 250 A long. These types of extended systems il-

lustrate cases where our current implementation of site-specific analysis

presents an clear advantage over the finite differences version.

2. Theory and Implementation

We start with the general expression for the electronic polarizability

tensor α of a finite system. The Cartesian components of α can be expressed

3

as (atomic units will be used herein)

αij =dµidFj

∣∣∣∣F =0

, (1)

where µ =∫r n(r) d3r is the electronic dipole moment and F is the external

uniform electrostatic field. To make connection with linear-response theory,

it is convenient to write α in terms of the first-order density n(1)

j (r) =

dn(r)/dFj |F =0,

αij =∫ri n

(1)

j (r)d3r , (2)

with ri the Cartesian components of the position operator r. Following

Ref. 19, the site-specific analysis is carried out by splitting the integral in

Eq. 2 into atomic contributions,

αAij =∫

(ri −RAi )n(1)

j (r)wA(r)d3r , (3)

and a charge transfer (CT) term,

αCTij =∑A

RAi

∫n(1)

j (r)wA(r)d3r , (4)

where wA(r) is an atomic weight function, RA is the nuclear position of

atom A, and the sum runs over all atoms in the system. The atomic weight

functions are chosen such that they satisfy∑

AwA(r) = 1 and therefore it

is straightforward to show that

αij = αCTij +∑A

αAij , (5)

where now the total polarizability is explicitly written in terms of atomic

plus charge transfer contributions. In Ref. [19] the atomic weights are based

on a Voronoi decomposition of space, which uses wA = 1 at all points in

space that are closer to the position of atom A than to any other atom,

4

and wA = 0 otherwise. While this this choice creates abrupt boundaries

between atoms these boundaries are “fuzzy” in the Hirshfeld partitioning

scheme. The Voronoi approach is clearly inappropriate in heteronuclear

systems where simple proximity is insufficient to identify a point in space

with a particular atom. It is important to mention that both the atomic αA

and charge transfer αCT contributions are origin-independent.

The first-order densities n(1)

j (r) are obtained from solving the first-order

CPKS equations in the presence of three independent infinitesimal (first-

order) perturbations h(1)

j (r) = rj that represent the first-order potential cor-

responding to an applied external uniform electric field. It should be noted

that this procedure is standard in many quantum chemistry packages and

therefore no details will be given here.[23, 24] With the first-order density

matrix [P (1)

j ]µν in the atomic orbital basis set φµ(r) obtained from solving the

CPKS equations, the first-order density n(1)

j (r) =∑

µν [P (1)

j ]µνφµ(r)φν(r)

is used to evaluate Eq. 3 and Eq. 4. It should be pointed out that al-

though the Hirshfeld partitioning has been revised and improved recently

[25, 26] to allow for direct comparison between different codes in this work

the weight functions wA are chosen following the original Hirshfeld partition-

ing method.[27, 28] A potential drawback of this approach is that it is not

suitable for ionic or strongly polar species where the promolecular density is

not readily available since typically in most codes neutral promolecular den-

sities are used.[25] However, our approach can be easily extended to include

other partitioning schemes by changing the weight functions wA. The entire

site-specific analysis was implemented in an in-house version of Gaussian

09.[29]

Calculations involving Si and Na atoms reported in this work were car-

ried out using the all-electron basis set employed by Jackson[19, 30] for

5

the calculation of static polarizabilities, which consists of 16 s-type and p-

type primitives contracted into six s-type and five p-type atomic orbitals,

respectively,and four d-type uncontracted atomic orbitals. The standard

6-31G**[31] basis was employed for for C and H (TPA oligomers). No

symmetry constraints (“nosymm” keyword in Gaussian) were used, and the

numerical integration was performed using a pruned grid of 99 radial shells

and 590 angular points per atomic shell (“grid=ultrafine”) for both, the

self-consistent procedures and the site-specific analysis. No relativistic ef-

fects were included. The Si3 and Na20 geometrical structures were obtained

from Jackson et al.[19, 30] The oligomeric TPA structures C2nH2n+2 were

generated using the MP2 optimized geometrical data from Pino and Scuse-

ria for the periodic TPA monostrand. [32] Structural data, as well as all

calculated polarizabilities are available as Supplementary Material.

3. Results and Discussion

We have performed consistency checks of our implementation of the site-

specific analysis by comparing to numerical results for Si3 and the Na20

clusters obtained using the NRLMOL code[33, 34] and the same generalized

gradient energy functional of Perdew, Burke, and Erzerhof (PBE). The re-

sults are summarized in Table 1 and Table 2. Atomic and CT contributions

are very close for both systems, with small discrepancies of less than 3%

in the total polarizability that can be attributed to the numerical versus

analytical differentiation procedures. The small differences in the atomic

and CT contributions can be related to the different implementations of the

atomic weights.

We have also performed test calculations in TPA oligomers of increas-

ing length, up to 250 A. These are particularly interesting since such long

6

Table 1: Site-specific isotropic polarizabilities (inbohr3) for the Si3 cluster. Si2 and Si3 are equiv-alent by symmetry.

CPKS Numericala

Si1 21.73 20.38Si2,3 23.24 22.79CT 36.17 41.67Total 104.38 107.63

a Numerical differentiation calculationswere performed with NRLMOL us-ing a finite electric field strength of0.001 a.u.

finite systems pose a challenge for numerical differentiation with respect to

uniform electric fields: On the one hand the field strength needs to be small

enough for the associated potential energy to be considered a small pertur-

bation over the entire length of the oligomer, while on the other hand, such

small fields imply a very stringent (and therefore computationally demand-

ing) convergence criteria of the self-consistent energy calculations. Calcula-

tions using NRLMOL indicate that using a perturbation size of 10−5 au this

problem arises for TPA oligomers of about 31 units of length (or approxi-

mately 80 A). This is therefore a case where analytic linear-response theory

presents a clear advantage over numerical differentiation methods.

In Fig. 1 we show the principal contributions to αA as a function of the

position for all C atoms in the n=41 oligomer using the PBEh hybrid density

functional approximation.[35, 36] The edge atoms (left-side of the plot) show

a larger atomic polarizability. In particular, the αAzz component shows a large

alternation between adjacent atoms. This can be understood in terms of the

different hybridization of the C atoms towards the edge determined by the

C–C bond alternation given by the TPA structure (1.3667 A and 1.4298 A).

7

Table 2: Site-specific isotropic polarizabilities (inbohr3) for the Na20 cluster.

CPKS Numerical a

Na1 29.38 26.48Na2 33.94 32.02Na3 42.25 42.04Na4 24.03 19.24Na5 25.03 20.91Na6 4.99 −0.50Na7 29.18 26.16Na8 23.99 19.1Na9 33.98 32.53Na10 44.22 44.29Na11 25.03 20.86Na12 55.83 58.26Na13 38.88 38.57Na14 2.77 1.07Na15 45.58 46.31Na16 14.25 8.34Na17 45.60 46.73Na18 50.35 51.3Na19 38.99 38.67Na20 55.93 58.37CT 1129.49 1165.78Total 1793.69 1796.52

a Numerical differentiation calculationswere performed with NRLMOL us-ing a finite electric field strength of0.001 a.u.

8

αAxx and αAyy show a smaller variation with position, spanning only a few

units from the edge. This site dependence of the atomic polarizability for

atoms in different environments is a powerful feature of the site-specific

analysis that also allows the identification of screening effects in atomic

clusters.[19, 21, 20] It is important to mention that for the n=41 chain used

in this example, the total isotropic atomic polarizability (including C and H

atoms) accounts for less than 2% of the total polarizability.

The convergence of the (normalized) longitudinal electric polarizability

with increasing oligomeric length has been extensively studied in the liter-

ature (see for instance Ref. 37, and 38). Therefore it is relevant to assess

the convergence of the atomic and CT components of α as a function of the

oligomeric length. In Table 3 we show the calculated isotropic atomic and

CT components for TPA oligomers from n=41 to n=101 (approximately

102 A to 250 A long) using the PBE functional and also its hybrid counter-

part, PBEh. The total polarizability values have a large contribution from

the CT term, which is in fact the dominant contribution in all cases. It is

worth mentioning that it is the longitudinal αCTxx that contributes the most,

with αCTyy more than one order of magnitude smaller and αCTzz exactly zero

by construction. We find that the slow convergence towards the polymeric

limit is mostly due to the slow convergence of the CT contribution, while the

atomic term converges much faster. Interestingly, the atomic contribution

is almost identical for both PBE and PBEh functionals, in contrast to the

CT. The smaller polarizability values calculated with the PBEh functional

as compared to the PBE values can be understood in terms of a simple sum-

over-states second-order perturbation theory and the smaller energy gap of

PBE vs. PBEh (the energy gap is, for instance, 0.6 eV vs. 1.6 eV for PBE

and PBEh, respectively for the n=101 oligomer).

9

Figure 1: Principal components of the atomic polarizabilities of C atoms in the C82H84

oligomer (Cartesian axes orientaion is represented at the left side of the plot). The num-bering in the plot is chosen such that N=1 corresponds to the edge C atom and N=41 tothe central C atom.

4. Concluding Remarks

In this Letter we report the implementation of the site-specific parti-

tioning of the molecular polarizability tensor using explicitly the first-order

density from linear-response CPKS calculations, in contrast to the original

implementation that employs finite differences of the dipole components. By

comparing the results in Si3 and Na20 we have shown that that both ap-

proaches are numerically equivalent. Using the first-order density we were

able to apply the site-specific partitioning scheme to TPA oligomers of up

to 250 A, which are particularly challenging for numerical differentiation

methods. This new methodology will allow the routine evaluation of the

10

Table 3: Isotropic atomic and CT contributions to the total po-larizability per unit n (in bohr3) for C2nH2n+2 TPA oligomers ofincreasing length calculated using the PBE and the hybrid PBEhfunctionals.

nAtomic CT Total

PBE PBEh PBE PBEh PBE PBEh41 5.81 5.75 324.2 172.9 330.0 178.751 5.78 5.73 377.3 186.6 383.1 192.361 5.76 5.71 418.1 196.2 423.9 201.971 5.75 5.70 449.8 203.2 455.6 208.981 5.74 5.69 474.8 208.5 480.6 214.291 5.73 5.68 494.9 212.7 500.7 218.4101 5.72 5.68 511.3 216.1 517.1 221.7

site-specific partitioning of the polarizability in a broad variety of large clus-

ters and complexes and hence it provides a useful analysis tool to interpret

electrostatic dipole polarizability tensor calculations.

5. Acknowledgements

JEP acknowledges support from the US Department of Energy Grant

No. DE-FG02-10ER16203. VB acknowledges the support from NSF-CBET-

1335944 and an award from Research Corporation for Science Advancement.

KAJ was supported by the US Department of Energy Grant No. DE-

SC0001330. KAJ is also grateful for the hospitality of the University of

Minnesota School of Physics and Astronomy, and particularly that of Prof.

Ken Heller, during his sabbatical leave.

Bibliography

References

[1] W. A. de Heer, The physics of simple metal clusters: experimental

aspects and simple models, Rev. Mod. Phys. 65 (1993) 611–676.

11

[2] H. Gould, T. Miller, Recent developments in the measurement of static

electric dipole polarizabilities, in: Stroke, HH (Ed.), Advances In

Atomic, Molecular, and Optical Physics, Vol 51, Vol. 51, 2005, pp.

343–361.

[3] G. Maroulis, Applying Conventional Ab Initio and Density Functional

Theory Approaches to Electric Property Calculations. Quantitative As-

pects and Perspectives, in: Putz, MV and Mingos, DMP (Ed.), Appli-

cations of Density Functional Theory to Chemical Reactivity, Vol. 149

of Structure and Bonding, 2012, pp. 95–129.

[4] A. Sihvola, Dielectric polarization and particle shape effects, Journal of

Nanomaterials 2007, article ID 45090.

[5] P. Szumniak, S. Bednarek, J. Paw lowski, B. Partoens, All-electrical

control of quantum gates for single heavy-hole spin qubits, Phys. Rev.

B 87 (2013) 195307.

[6] A. Kwasniowski, J. Adamowski, Tuning the exchange interaction by

an electric field in laterally coupled quantum dots, J. Phys.: Condens.

Matter 21 (2009) 235601.

[7] A. J. Cohen, Y. Tantirungrotechai, Molecular electric properties: an

assessment of recently developed functionals, Chem. Phys. Lett. 299

(1999) 465–472.

[8] O. Guliamov, L. Kronik, J. M. L. Martin, Polarizability of small carbon

cluster anions from first principles, J. Phys. Chem. A 111 (10) (2007)

2028–2032.

12

[9] D. Xenides, P. Karamanis, C. Pouchan, A critical analysis of the perfor-

mance of new generation functionals on the calculation of the (hyper)

polarizabilities of clusters of varying stoichiometry: Test case the sim-

gen (m+n=7, n=0-7) clusters, Chem. Phys. Lett. 498 (2010) 134–139.

[10] B. Champagne, E. A. Perpete, S. J. A. van Gisbergen, E.-J. Baerends,

J. G. Snijders, C. Soubra-Ghaoui, K. A. Robins, B. Kirtman, Assess-

ment of conventional density functional schemes for computing the po-

larizabilities and hyperpolarizabilities of conjugated oligomers: An ab

initio investigation of polyacetylene chains, J. Chem. Phys. 109 (1998)

10489–10498.

[11] K. E. Laidig, R. F. W. Bader, Properties of atoms in molecules: Atomic

polarizabilities, J. Chem. Phys. 93 (10) (1990) 7213–7224.

[12] L. Gagliardi, R. Lindh, G. Karlstrm, Local properties of quantum chem-

ical systems: The loprop approach, J. Chem. Phys. 121 (10) (2004)

4494–4500.

[13] A. V. Marenich, C. J. Cramer, D. G. Truhlar, Reduced and quenched

polarizabilities of interior atoms in molecules, Chem. Sci. 4 (2013) 2349–

2356.

[14] M. Panhuis, R. Munn, P. Popelier, J. Coleman, B. Foley, W. Blau, Dis-

tributed response analysis of conductive behavior in single molecules,

Proc. Natl. Acad. Sci. U.S.A. 99 (2) (2002) 6514–6517.

[15] A. Krishtal, P. Senet, M. Yang, C. Van Alsenoy, A Hirshfeld parti-

tioning of polarizabilities of water clusters, J. Chem. Phys. 125 (2006)

034312.

13

[16] B. A. Bauer, T. R. Lucas, A. Krishtal, C. Van Alsenoy, S. Patel, Vari-

ation of ion polarizability from vacuum to hydration: Insights from

Hirshfeld partitioning, J. Phys. Chem. A 114 (34) (2010) 8984–8992.

[17] A. Krishtal, P. Senet, C. Van Alsenoy, Origin of the size-dependence of

the polarizability per atom in heterogeneous clusters: The case of AlP

clusters, J. Chem. Phys. 133 (2010) 154310.

[18] N. Otero, C. V. Alsenoy, P. Karamanis, C. Pouchan, Electric response

properties of neutral and charged Al13X (X= Li, Na, K) magic clusters.

a comprehensive ab initio and density functional comparative study,

Comp. Theor. Chem. 1021 (2013) 114 – 123, clusters: From Dimers to

Nanoparticles.

[19] K. Jackson, M. Yang, J. Jellinek, Site-specific analysis of dielectric prop-

erties of finite systems, J. Phys. Chem. C 111 (2007) 17952–17960.

[20] L. Ma, J. Wang, G. Wang, Site-specific analysis of dipole polarizabilities

of heterogeneous systems: Iron-doped Sin (n = 1–14) clusters, J. Chem.

Phys. 138 (2013) 094304.

[21] L. Ma, K. A. Jackson, J. Jellinek, Site-specific polarizabilities as pre-

dictors of favorable adsorption sites on Nan clusters, Chem. Phys. Lett.

503 (2011) 80–85.

[22] Q. Zeng, L. Liu, W. Zhu, M. Yang, Local and nonlocal contributions

to molecular first-order hyperpolarizability: A Hirshfeld partitioning

analysis, J. Chem. Phys. 136 (2012) 224304.

[23] G. J. B. Hurst, M. Dupuis, E. Clementi, Abinitio analytic polarizabil-

ity, first and second hyperpolarizabilities of large conjugated organic

14

molecules: Applications to polyenes C4H6 to C22H24, J. Chem. Phys.

89 (1) (1988) 385–395.

[24] S. M. Colwell, C. W. Murray, N. C. Handy, R. D. Amos, The determi-

nation of hyperpolarisabilities using density functional theory, Chem.

Phys. Lett. 210 (1-3) (1993) 261–268.

[25] P. Bultinck, C. Van Alsenoy, P. W. Ayers, R. Carbo-Dorca, Critical

analysis and extension of the Hirshfeld atoms in molecules, J. Chem.

Phys. 126 (2007) 144111.

[26] T. Verstraelen, P. W. Ayers, V. Van Speybroeck, M. Waroquier,

Hirshfeld-E partitioning: AIM charges with an improved trade-off be-

tween robustness and accurate electrostatics, J. Chem. Theory Comput.

9 (5) (2013) 2221–2225.

[27] F. L. Hirshfeld, Bonded-atom fragments for describing molecular charge

densities, Theor. Chim. Acta 44 (1977) 129–138.

[28] E. Davidson, S. Chakravorty, A test of the Hirshfeld definition of atomic

charges and moments, Theor. Chem. Acc. 83 (1992) 319–330.

[29] Gaussian 09, Revision A.02, M. J. Frisch, G. W. Trucks, H. B. Schlegel,

G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr.,

T. Vreven, G. Scalmani, B. Mennucci, V. Barone, G. A. Petersson, M.

Caricato, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda,

J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai,

X. Li, H. P. Hratchian, J. E. Peralta, A. F. Izmaylov, K. N. Kudin,

J. J. Heyd, E. Brothers, V. Staroverov, G. Zheng, R. Kobayashi, J.

Normand, J. L. Sonnenberg, S. S. Iyengar, J. Tomasi, M. Cossi, N.

15

Rega, J. C. Burant, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross,

V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann,

O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y.

Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G.

Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K.

Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q.

Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A.

Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith,

M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, W.

Chen, M. W. Wong, and J. A. Pople, Gaussian, Inc., Wallingford CT,

2009.

[30] K. Jackson, L. Ma, M. Yang, J. Jellinek, Atomistic dipole moments and

polarizabilities of Nan clusters, n = 2–20, J. Chem. Phys. 129 (2008)

144309.

[31] W. J. Hehre, R. Ditchfield, J. A. Pople, Self-consistent molecular orbital

methods. XII. further extensions of gaussian-type basis sets for use in

molecular orbital studies of organic molecules, J. Chem. Phys. 56 (1972)

2257–2261.

[32] R. Pino, G. E. Scuseria, Importance of chain–chain interactions on the

band gap of trans-polyacetylene as predicted by second-order pertur-

bation theory, J. Chem. Phys. 121 (2004) 8113–8119.

[33] M. R. Pederson, K. A. Jackson, Variational mesh for quantum-

mechanical simulations, Phys. Rev. B 41 (1990) 7453–7461.

[34] K. Jackson, M. R. Pederson, Accurate forces in a local-orbital approach

to the local-density approximation, Phys. Rev. B 42 (1990) 3276–3281.

16

[35] C. Adamo, V. Barone, Toward reliable density functional methods with-

out adjustable parameters: The PBE0 model, J. Chem. Phys. 110 (13)

(1999) 6158–6170.

[36] J. P. Perdew, M. Ernzerhof, K. Burke, Rationale for mixing exact ex-

change with density functional approximations, J. Chem. Phys. 105

(1997) 9982–9985.

[37] K. N. Kudin, R. Car, R. Resta, Longitudinal polarizability of long

polymeric chains: Quasi-one-dimensional electrostatics as the origin of

slow convergence, J. Chem. Phys. 122 (2005) 134907.

[38] M. Frisch, M. Head-Gordon, J. Pople, Direct analytic SCF second

derivatives and electric field properties, Chem. Phys. 141 (2-3) (1990)

189–196.

17

http://ees.elsevier.com/cplett/download.aspx?id=1155061&guid=05d30292-573a-4c3e-8e63-d40eeae26ecf&scheme=1

• Site-­‐specific  static  dipole  polarizabilities  evaluated  from  linear-­‐response.    • The  new  method  provides  a  simple  and  robust  tool  to  analyze  calculated  

polarizabilities.  • Proof-­‐of-­‐concept  calculations  in  Si3  and  Na20,  and  all-­‐trans-­‐  polyacetylene  

oligomeric  chains  up  to  250  Å  long  show  its  potential.