Accepted Manuscript Site-specific Polarizabilities from Analytic Linear-response Theory Juan E. Peralta, Veronica Barone, Koblar A. Jackson PII: DOI: Reference:
S0009-2614(14)00420-5 http://dx.doi.org/10.1016/j.cplett.2014.05.045 CPLETT 32190
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
1 April 2014 16 May 2014
Please cite this article as: J.E. Peralta, V. Barone, K.A. Jackson, Site-specific Polarizabilities from Analytic Linearresponse Theory, Chemical Physics Letters (2014), doi: http://dx.doi.org/10.1016/j.cplett.2014.05.045
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Site-specific Polarizabilities from Analytic Linear-response Theory 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 polarizability tensor [J. Chem. Phys. 125, 34312 (2006)] that explicitly employs the first-order electronic density from linear-response coupled-perturbed KohnSham calculations. This new implementation provides a simple and robust tool to perform the partitioning analysis of the calculated electrostatic polarizability tensor at negligible additional computational effort. Comparison with numerical results for Si3 and Na20 , and test calculations in all-transpolyacetylene 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 author Email address:
[email protected] (Juan E. Peralta)
Preprint submitted to Elsevier
May 21, 2014
1. Introduction The response of molecules and clusters to external stimuli such as electric 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 understanding 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 polarizabilities in systems such as metal clusters and polymers.[7, 8, 9, 10] In addition to the general interest in determining the polarizability from electronic structure calculations, computational methods have been proposed that allow for an evaluation of the contributions from the different molecular 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 sitespecific 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) calculations. This new implementation provides a streamlined and robust tool to perform the site-specific analysis of the calculated electrostatic polarizability 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 direct 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 illustrate 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) dµi αij = , dFj F =0 where µ =
R
(1)
r n(r) d3 r 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 , Z αij =
3 ri n(1) j (r)d r ,
(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, A αij =
Z
3 (ri − RiA ) n(1) j (r)wA (r)d r ,
(3)
and a charge transfer (CT) term, CT αij
=
X
RiA
Z
3 n(1) j (r)wA (r)d r ,
(4)
A
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 P functions are chosen such that they satisfy A wA (r) = 1 and therefore it is straightforward to show that CT αij = αij +
X
A αij ,
(5)
A
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 (firstorder) perturbations h(1) j (r) = rj that represent the first-order potential corresponding 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 [Pj(1) ]µν in the atomic orbital basis set φµ (r) obtained from solving the P (1) CPKS equations, the first-order density n(1) µν [Pj ]µν φµ (r)φν (r) j (r) = is used to evaluate Eq. 3 and Eq. 4. It should be pointed out that although 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 partitioning 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 densities 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 carried 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 ptype 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 effects were included. The Si3 and Na20 geometrical structures were obtained from Jackson et al.[19, 30] The oligomeric TPA structures C2n H2n+2 were generated using the MP2 optimized geometrical data from Pino and Scuseria 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 sitespecific 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 results 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 increasing length, up to 250 ˚ A. These are particularly interesting since such long 6
Table 1: Site-specific isotropic polarizabilities (in bohr3 ) for the Si3 cluster. Si2 and Si3 are equivalent by symmetry.
CPKS Si1 Si2,3 CT Total a
21.73 23.24 36.17 104.38
Numericala 20.38 22.79 41.67 107.63
Numerical differentiation calculations were performed with NRLMOL using a finite electric field strength of 0.001 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 perturbation over the entire length of the oligomer, while on the other hand, such small fields imply a very stringent (and therefore computationally demanding) convergence criteria of the self-consistent energy calculations. Calculations 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 approximately 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 component shows a large a larger atomic polarizability. In particular, the αzz
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).
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Table 2: Site-specific isotropic polarizabilities (in bohr3 ) for the Na20 cluster.
CPKS Na1 Na2 Na3 Na4 Na5 Na6 Na7 Na8 Na9 Na10 Na11 Na12 Na13 Na14 Na15 Na16 Na17 Na18 Na19 Na20 CT Total a
29.38 33.94 42.25 24.03 25.03 4.99 29.18 23.99 33.98 44.22 25.03 55.83 38.88 2.77 45.58 14.25 45.60 50.35 38.99 55.93 1129.49 1793.69
Numerical
a
26.48 32.02 42.04 19.24 20.91 −0.50 26.16 19.1 32.53 44.29 20.86 58.26 38.57 1.07 46.31 8.34 46.73 51.3 38.67 58.37 1165.78 1796.52
Numerical differentiation calculations were performed with NRLMOL using a finite electric field strength of 0.001 a.u.
8
A and αA show a smaller variation with position, spanning only a few αxx yy
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 literature (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 counterpart, 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 CT that contributes the most, worth mentioning that it is the longitudinal αxx CT more than one order of magnitude smaller and αCT exactly zero with αyy zz
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 sumover-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 C82 H84 oligomer (Cartesian axes orientaion is represented at the left side of the plot). The numbering in the plot is chosen such that N=1 corresponds to the edge C atom and N=41 to the central C atom.
4. Concluding Remarks In this Letter we report the implementation of the site-specific partitioning 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 approaches 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 polarizability per unit n (in bohr3 ) for C2n H2n+2 TPA oligomers of increasing length calculated using the PBE and the hybrid PBEh functionals.
n 41 51 61 71 81 91 101
Atomic PBE PBEh 5.81 5.75 5.78 5.73 5.76 5.71 5.75 5.70 5.74 5.69 5.73 5.68 5.72 5.68
CT PBE PBEh 324.2 172.9 377.3 186.6 418.1 196.2 449.8 203.2 474.8 208.5 494.9 212.7 511.3 216.1
Total PBE PBEh 330.0 178.7 383.1 192.3 423.9 201.9 455.6 208.9 480.6 214.2 500.7 218.4 517.1 221.7
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• • •
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.
