On the performance of long-range-corrected density functional theory and reduced-size polarized...

On the Performance of Long-Range-Corrected DensityFunctional Theory and Reduced-Size Polarized LPol-nBasis Sets in Computations of Electric Dipole(Hyper)Polarizabilities of p-Conjugated Molecules

Angelika Baranowska-Ła� czkowska,*[a] Wojciech Bartkowiak,[b] Robert W. G�ora,[b]

Filip Pawłowski,[a] and Robert Zale�sny*[b]

Static longitudinal electric dipole (hyper)polarizabilities are

calculated for six medium-sized p-conjugated organic

molecules using recently developed LPol-n basis set family

to assess their performance. Dunning’s correlation-consistent

basis sets of triple-f quality combined with MP2 method

and supported by CCSD(T)/aug-cc-pVDZ results are used to

obtain the reference values of analyzed properties. The

same reference is used to analyze (hyper)polarizabilities

predicted by selected exchange-correlation functionals,

particularly those asymptotically corrected. VC 2012 Wiley

Periodicals, Inc.

DOI: 10.1002/jcc.23197

Introduction

Progress in quantum chemistry algorithms, accompanied by

rapid development of computer resources, provides nowadays

the community of theoretical chemists with very efficient tools

to accurately predict electric properties of molecules. The use of

highly correlated methods combined with flexible, sufficiently

diffuse set of basis functions is highly recommended in theoreti-

cal investigations aiming at the accurate evaluation of electric

properties, especially in the case of nonlinear optical effects.

Among the most popular basis sets used in calculations of elec-

tric properties of isolated molecules and their complexes are the

multiply augmented correlation consistent polarized valence

multiple-f x-aug-cc-pVXZ basis sets of Dunning and co-

workers,[1–3] widely known for the high accuracy of results they

yield, particularly when used in Coupled Cluster (CC) or other

elaborate treatments of electron correlation.[4–7] However, the

use of the larger augmented correlation consistent sets in the

CC electric property calculations is still limited to small and me-

dium-sized systems due to high computing cost of those meth-

ods, scaling with the number of basis functions N as N6–N7.

Thus, for larger systems one can either use the Kohn–Sham

formulation of the density functional theory (KS-DFT), or use

the so-called property-oriented medium-sized basis sets, opti-

mized for calculation of a given type of properties. Although

the primary advantage of using these two approaches is the

obvious and often substantial decrease of computing cost, the

most striking risk is a possible deterioration in the accuracy of

results. Careful selection of methods and basis sets is thus nec-

essary to avoid large errors in the computed values.

A number of reduced-size as well as large property-oriented

basis sets is reported in the literature, to name a few, let us

mention the works by Maroulis and coworkers,[8–13] and

Rappoport and Furche[14] (see also the review article by David-

son and Feller[15]). Another group of property-oriented basis

sets is the Pol sets family developed by Sadlej and co-

workers,[16–23] and designed for accurate or moderately accu-

rate description of systems in external static or dynamic elec-

tric fields. In this project, we investigate the performance of

Møller-Plesset second-order perturbation theory (MP2), and

most common exchange-correlation functionals used within

the KS-DFT framework, in combination with the recently devel-

oped polarized LPol-n (n¼ds, dl) basis sets,[23] belonging to

the Pol family of property-oriented basis sets. The performance

of the above methods is addressed by comparison to the

CCSD and CCSD(T) estimates, whereas the results produced by

LPol-n sets are compared with those obtained using aug-cc-

pVXZ sets (X¼D,T).

In contrast to the Dunning sets, in which the diffuse func-

tions are optimized for negative ions, the LPol-n sets are

derived considering an explicit field dependence of Gaussian

type orbitals resulting in the property-adapted field-independ-

ent basis functions. Our earlier studies revealed that the LPol-n

sets can compete with the larger all-purpose Dunning’s basis

sets in the accuracy of determined static and dynamic

[a] A. Baranowska-Ła� czkowska, F. Pawłowski

Institute of Physics, Kazimierz Wielki University, Plac Weyssenhoffa 11,

PL–85072 Bydgoszcz, Poland

[b] W. Bartkowiak, R. W. G�ora, R. Zale�sny

Theoretical Chemistry Group, Institute of Physical and Theoretical Chemistry,

Wrocław University of Technology, Wyb. Wyspia�nskiego 27, PL–50370

Wrocław, Poland

E-mail: [email protected] or [email protected]

Contract grant sponsor: Foundation for Polish Science; contract grant

number: Homing Plus/2010-1/2

Contract grant sponsor: National Science Centre; contract grant

number: DEC-2011/01/D/ST4/03149.

VC 2012 Wiley Periodicals, Inc.

Journal of Computational Chemistry 2013, 34, 819–826 819

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(non)linear electric and optical properties of isolated mole-

cules,[23–25] as well as linear and nonlinear interaction-induced

electric properties of hydrogen-bonded complexes.[26,27] For

instance, the calculations performed for linear HCHO…(HF)p (p

¼ 1–9) complexes, carried out within the finite field HF SCF,

MP2, CCSD, and CCSD(T) approximations, indicated an excel-

lent overall performance of the LPol-n sets.[26] It was achieved

using relatively compact polarized sets, resulting in significant

reduction of the computing time and virtually without any no-

ticeable deterioration of the results. The LPol-n sets were also

used in the evaluation of interaction-induced electric proper-

ties of linear (HCN)q (q ¼ 2–4) complexes within the aforemen-

tioned approximations.[27] Among the conclusions of that

work were the following: The basis set superposition error was

negligible at all employed levels of theory for all the LPol-n

sets, and already the smallest among them (LPol-ds) gave reli-

able estimations of the interaction-induced dipole moments,

linear polarizabilities, and first hyperpolarizabilities of the inves-

tigated complexes. Only recently, the LPol-n sets have been

successfully used in the evaluation of the interaction-induced

electric properties in CO–Ne complex.[28] The LPol-n and modi-

fied LPol-n (MLPol-n) basis sets were also shown to correctly

describe interaction energies in small van der Waals com-

plexes.[29,30] However, so far the applications of LPol-n sets

were limited to relatively small molecules and complexes.

Hence, the primary aim of this study is to further analyze their

performance in computing the static electric dipole (hyper)po-

larizabilities of medium-sized p–conjugated organic molecules.

As it was mentioned earlier, the use of KS-DFT in calcula-

tions of electric dipole (hyper)polarizabilities often results in

significant deterioration of the accuracy (particularly in the

case of hyperpolarizabilities). Indeed, as it was shown over a

decade ago by Champagne et al., most of the exchange-corre-

lation functionals available at that time were unable to quanti-

tatively predict the response of electronic density to electric

field perturbation.[31–33] This erroneous behavior, often referred

to as the DFT overshoot problem, stems from the so-called

self-interaction error. Within the Hartree-Fock theory, the spuri-

ous Coulomb interaction in one-electron system is perfectly

canceled by the corresponding exchange integral; the same is

not true, however, in the case of DFT due to an approximate

form of exchange-correlation functional. A partial remedy to

this problem is to include the exact Hartree–Fock exchange

potential at long range. Functionals augmented in this way

belong to the long-range corrected (LC) class of exchange-cor-

relation functionals.[34–36] Preliminary results, including those

published by some of us, show in some cases a significant

improvement over conventional exchange-correlation function-

als as far as electronic (hyper)polarizabilities are concerned

(both resonant and nonresonant).[37–39] Although the elec-

tronic contributions to resonant and nonresonant (hyper)polar-

izabilities have been studied extensively using LC-DFT,[38,40–46]

more numerical data are required to draw general conclusions,

as the improvement brought by the use of the long-range cor-

rection varies from system to system. In this study, we use the

recently developed LPol-n basis sets using several levels of

theoretical approximations including post-HF methods (MP2,

CCSD) with an emphasis on the LC-DFT performance. The lat-

ter approach, in conjunction with the property-oriented basis

sets, might constitute a promising alternative to methods hav-

ing their roots in the wavefunction theory.

Methodology

In the presence of the static uniform external electric field (E),

the a Cartesian component of the total molecular dipole

moment (la) may be expressed as a Taylor series:

la ¼ l0a þ

Xb

aabð0; 0ÞEb þ1

2!

Xbc

babcð0; 0; 0ÞEbEc

þ 1

3!

Xbcd

cabcdð0; 0; 0; 0ÞEbEcEd þ… ð1Þ

where l0a is the a component of dipole moment, aab(0;0),

babc(0;0,0), and cabcd(0;0,0,0) are the frequency-independent

components of linear polarizability, the first hyperpolarizability

and the second hyperpolarizability, respectively. In this study,

we compute only the diagonal components of a, b, and c ten-

sors for molecules presented in Figure 1. Their geometries

were optimized at the MP2/cc-pVTZ level of theory (the final

RMS gradient was below 10�5 Hartree/Bohr) and oriented in

such a way that the dipole moment vector was set parallel to

Cartesian z-axis (cf. Fig. 1).

In calculations of electric properties, we consider only the

response of electron density and neglect the effects arising

from a change of geometry in the presence of external electric

field. The electronic contribution to the (hyper)polarizabilities

was evaluated fully numerically based on the differentiation of

the energy or dipole moment with respect to an external elec-

tric field using the Rutishauser–Romberg procedure to remove

contaminations from higher order derivatives.[47,48]

The set of functionals selected for this study includes the

generalized gradient approximation (GGA) BLYP; hybrid

Figure 1. Structure of studied molecules and their orientation in Cartesian

coordinate system. Brown, red, blue and white color represents carbon, ox-

ygen, nitrogen and hydrogen atoms, respectively.

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B3LYP,[49] and the LC functionals: CAM-B3LYP,[50] LC-xPBE,[51,52]

and LC-BLYP.[34] This set was also augmented with hybrid

meta-GGA M06-2X functional proposed by Truhlar et al.[53] The

numerical integrations during DFT calculations were done

using a pruned (99,590) point grid. The SCF convergence was

set at 10�11 on the RMS of the density matrix. Frozen-core

approximation was assumed in all post-Hartree–Fock calcula-

tions. We have used two basis set families: Dunning’s correla-

tion-consistent basis sets up to triple-f quality[1,2] as well as

recently developed LPol-ds and LPol-dl basis sets.[23]

All calculations were performed with the Gaussian suite of

programs (using default definitions of functionals including

those asymptotically corrected),[54] the Gamess US[55] and the

Molcas package.[56–58]

Results and Discussion

All molecules studied in this work, presented in Figure 1, were

the subject of an earlier study by Bishop et al.[59] The authors

chosen this set of molecules to validate predictions of the

two-state valence-bond charge-transfer model. 1-formyl-6-

hydroxy-1,3,5-triene (in Fig. 1 labeled as 1) and 1,1-diamino-

6,6-dinitrohexa-1,3,5-triene (3) are valence structures, whereas

1-ammoniohexa-1,3,5-triene-6-carboxylate (2) is an example of

charge-transfer molecule. The remaining molecules: 4,4-(p-

methylpirydyl)-1,1-dicyano-1,3-butadiene, 4-methylpyridone,

and 4-nitroaniline (4–6, respectively) are also of the valence

character, but they contain an aromatic

ring and hence constitute a different

group of molecules than 1 and 3. The

whole set covers a broad range of bond

length alternation (BLA) patterns which,

as it was shown by many authors, deter-

mines to a large extent the sign and

magnitude of nonlinear optical response

of p-conjugated compounds. In fact,

intentional changes of BLA by rational

choice of donor and acceptor groups

were often a common route to maximize

first and second hyperpolarizabilities[60,61]

or even two-photon absorption cross-

section.[62] The electron correlation

effects will have different impact on the

magnitudes of electric dipole (hyper)po-

larizabilities of covalent and charge-

transfer structures. This is the rationale

behind using the molecules in question as guinea pigs for

assessment of the performance of recently proposed exchange-

correlation functionals and LPol basis sets. It is a priori known

that DFT will not perform equally well for all the studied sys-

tems. However, our aim is to put this assumption on the quan-

titative basis. The CC approach with single and double excita-

tions (CCSD) is the reference method used for assessment

purposes. In some instances, we also estimated the perturbative

correction for triple excitations. To validate the adequacy of sin-

gle-reference wavefunctions used in electronic structure calcula-

tions, we computed the s1 diagnostic: [63,64]

s1 ¼ffiffiffiffiffiffiffiffiffiffiffit1 � t1

Ncorr

r; (2)

where t1 is the vector of the coupled-cluster single-excitation

amplitudes and Ncorr is the number of correlated electrons. In

all instances, we obtained the value of s1 below 0.02 what

indicates reliability of single-reference post-Hartree–Fock meth-

ods used in this study.

It follows from recent studies that the BLA in (quasi)linear

p–conjugated chains may be satisfactorily predicted by the

CAM-B3LYP functional.[65,66] To check to what extent our con-

clusions might be affected by the choice of the set of geome-

tries, we have performed an additional geometry optimization

of molecule 3. It follows from Table 1 that despite the changes

in property values due to differences in geometry (MP2 vs.

CAM-B3LYP), the trend is the same in both cases. It should not

be overlooked that one of possible sources of differences in

property values in Table 1 is due to orientation convention

[molecules are rotated in such a way that the total dipole

moment (both geometry and l are determined at the very

same level of theory) is parallel to cartesian z-axis]. In what fol-

lows, we use a set of geometries optimized at the MP2/

cc-pVTZ level of theory.

The data presented in Table 2 confirm that the electron cor-

relation effects are substantial for all the studied molecules.

They are particularly reflected in large changes of first and

Table 1. Diagonal components of electric dipole (hyper)polarizabilities

of system 3 computed at two different geometries.

CAM-B3LYP geometry MP2 geometry

CAM-B3LYP B3LYP CAM-B3LYP B3LYP

azz 354 363 347 360

bzzz �3023 �2056 �4096 �3138

All property values were determined using the aug-cc-pVDZ basis sets.

All values are given in atomic units.

Table 2. Diagonal components of electric dipole (hyper)polarizabilities of systems 1–6 computed

at various levels of theoretical approximation using the aug-cc-pVDZ basis set.

1 2 3 4 5 6

azz

RHF 246.77 250.94 308.09 363.38 115.71 131.53

MP2 235.65 280.22 360.44 402.09 121.29 146.17

CCSD 225.50 273.15 340.33 393.69 121.34 139.77

CCSD(T) 228.21 282.50 344.69 394.89 122.56 143.24

bzzz

RHF �1166.0 2347.8 �2132.8 6 1 2629.3 312.7 �878.7

MP2 �3917.1 6 1 5652.9 6 1 �8481.6 6 1 �3147.0 6 1 93.7 �1433.6

CCSD �3345.4 6 12 6786.6 6 2 �9615.2 6 22 �4697.4 6 7 121.8 6 2 �1299.7 6 2

CCSD(T) �3419.8 6 9 7593.5 6 1 �10859.2 6 17 �6053.9 6 6 100.8 6 2 �1448.9 6 3

czzzz � 10�3

RHF 128 134 114 �76 6 1 14 62

MP2 414 239 271 6 2 �589 6 1 17 121 6 1

CCSD 428 6 9 429 6 2 704 6 46 �407 6 1 15 114 6 1

CCSD(T) 470 6 9 542 6 2 963 6 98 �268 6 1 18 134 6 2




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second hyperpolarizabilities on passing from the HF to the

MP2 method. More accurate treatment of electron correlation

in CCSD and CCSD(T) methods leads to relatively small

changes of linear polarizability of the studied compounds.

However, in the case of hyperpolarizabilities, the changes are

much more significant. These findings bring no surprise because

already in the seminal paper of Sim et al.[67]on para–nitroaniline

(pNA, 6), the authors observed large increase of b value on

including the electron correlation effects. More recently, Ham-

mond and Kowalski[7] reported the coupled-cluster response

theory (CCSD/aug-cc-pVDZ) estimate of 1659.6 au for longitudi-

nal first hyperpolarizability of pNA (of Cs symmetry) which could

be compared to our finite-field estimate of 130062 au. One

should note, however, that different geometries were used, and

orbitals were relaxed in our calculations.[7] Schematic represen-

tation of the influence of electron correlation effects on the

properties is presented in Figure 2. Downwards and upwards

arrows represent the decrease and increase of property values,

respectively. It follows from this figure that electron correlation

effects are included in not particularly systematic manner, and

in majority of cases have different effect on a, b, and c. For

example, on passing from CCSD to CCSD(T) (arrows on the

right), we see the increase of a, b, and c of 3 but the same is

not true in case of 4. The only molecule for which the same

pattern holds in case of all properties is the one denoted as 6.

Perhaps, the most striking feature is the sign reversal of first

hyperpolarizability of compound 4 when using post-HF

Figure 2. Schematic representation of the influence of electron correlation

effects on electric dipole (hyper)polarizabilities (a, b, c) of molecules 1–6.

Left, middle and right arrows represent change in the absolute property

value on passing from HF to MP2, from MP2 to CCSD, and from CCSD to

CCSD(T) level of theory, respectively. [Color figure can be viewed in the

online issue, which is available at wileyonlinelibrary.com.]

Table 3. Diagonal components of electric dipole (hyper)polarizabilities of systems 1–6 computed at various levels of theoretical approximation using

the aug-cc-pVTZ basis set.

1 2 3 4 5 6

azz

RHF 247.12 251.59 307.97 363.80 115.70 131.79

(246.82) (251.55) (308.39 6 0.01) � (115.68) (132.00)

MP2 236.03 281.40 360.41 402.43 121.31 146.68

(235.88) (281.47) (360.93 6 0.04) � (121.57) (146.84)

B3LYP 270.75 290.21 359.50 382.79 121.59 154.44

CAM-B3LYP 258.16 277.40 346.95 379.84 120.06 146.41

BLYP 283.83 – 378.12 391.15 125.06 164.69

LC-BLYP 244.30 264.03 333.23 376.58 118.37 139.06

M06–2X 253.23 273.03 343.15 378.10 119.45 144.56

LC-xPBE 244.89 267.10 334.45 376.04 117.89 139.49

bzzz

RHF �1131.2 2336.7 �2043.0 2645.2 6 0.5 311.7 �858.0

(�1165.3 6 0.1) (2360.7 6 0.7) (�2122.8 6 1.7) – (308.6 6 0.6) (�868.8 6 0.2)

MP2 �3880.5 6 0.2 5638.3 �8349.6 6 0.8 �3190.9 6 0.7 84.6 �1403.8 6 0.6

(�3930.6 6 0.3) (5694.1 6 6.5) (�8518.2 6 4.9) – (83.7 6 1.7) (�1416.9 6 0.2)

B3LYP �2230.5 4288 6 1 �3137.7 1623.6 253.0 �1571.8

CAM�B3LYP �2341.5 3401.0 �4096.5 6 0.6 1127.5 206.4 �1330.7

BLYP �2197.8 – �2425.5 6 0.1 1883.3 298.2 �1769.2

LC-BLYP �2299.8 6 0.2 3599.9 6 0.8 �4992.4 6 0.2 354.2 6 0.4 173.3 �1146.5

M06–2X �2411.3 3568.4 �4235.8 767.3 210.2 �1237.0

LC-xPBE �2413.7 3668.1 6 0.9 �5065.6 6 0.7 237.0 6 0.3 150.7 6 0.4 �1135.4 6 0.2

czzzz � 10�3

RHF 128 141 115 6 1 �70 15 63

(132) (144 6 3) (121 6 3) – (17) 65

MP2 413 246 6 1 266 6 2 �588 6 1 19 122

(422) (259 6 2) (319 6 64) – (22 6 1) (127 6 1)

B3LYP 202 1228 6 56 69 75 35 109

CAM-B3LYP 238 205 6 2 103 �68 25 100

BLYP 181 – 105 199 55 110

LC-BLYP 253 174 188 �212 6 1 17 88

M06–2X 249 200 88 �80 22 93

LC-xPBE 262 174 6 2 185 �204 17 86

LPol-ds values given in parentheses. All values are given in atomic units.



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methods. Due to the substantial electron correlation effects,

one should expect that 4 will be a challenge for most conven-

tional exchange-correlation functionals. Indeed, none of the

chosen functionals is able to reproduce the sign of first hyper-

polarizability of 4 (cf. Table 3). Slightly better performance of

the long-range-corrected functionals is observed for this mole-

cule in the case of longitudinal component of c. A comparison

of the values of second hyperpolarizability of 4 computed

using the aug-cc-pVDZ and the aug-cc-pVTZ basis sets shows

only insignificant differences between the two. Thus, the value

obtained using the CCSD(T)/aug-cc-pVDZ method is our best

estimate of czzzz for molecule 4. It follows from Tables 2 and 3

that both LC-xPBE and LC-BLYP underestimate this property

roughly by 50%. Consequently, similarly to short polymethinei-

mine oligomers,[68] we identify molecule denoted as 4 as par-

ticularly challenging for the DFT.

In the case of other studied molecules, the sign of bzzz and

czzzz is reproduced by all employed exchange-correlation func-

tionals. However, the quantitative differences are large, and it

is difficult to draw any general conclusions considering the

magnitudes of bzzz and czzzz determined using various

exchange-correlation functionals. The relative unsigned errors

with respect to the CCSD(T)/aug-cc-pVDZ reference (d [%] ¼[(P�PCCSD(T))/(PCCSD(T))]�100%, where P¼azz, bzzz, czzzz) are pre-

sented for all molecules in Figure 3. It follows from this figure

that the smallest errors are obtained for LC-BLYP and LC-wPBE.

In the case of 1–3, the long-range-corrected functionals bring

some improvement which, on the other hand, is very system

dependent. Moreover, their performance is not equal in the

case of bzzz and czzzz. All molecules studied here are medium-

sized, but 5 and 6 are the smallest in the whole set. Therefore,

one should not expect a substantial improvement on using

the LC functionals in comparison to their conventional coun-

terparts for these systems. However, the data reported in Table

3 show the opposite. For instance, one finds rather significant

changes in bzzz and czzzz values for 5 on passing from the BLYP

to LC-BLYP functional, and the latter predicts czzzz value quite

close to our best estimate (CCSD(T)/LPol-ds, cf. Table 4).

We now turn our attention to the basis set extension

effects. Considering the differences between the MP2 results

obtained in the aug-cc-pVDZ and the aug-cc-pVTZ basis sets,

increase of the basis set cardinal number X from D to T practi-

cally does not affect the linear polarizabilities. The correspond-

ing differences are well below 1% for all investigated systems.

Although in the case of first and second hyperpolarizabilities,

this change is slightly larger (about 1–2%), only for system 5 it

is approaching 10%, and the aug-cc-pVDZ basis set can be

safely recommended for semiquantitative estimation of molec-

ular linear polarizabilities. Conversely, some caution has to be

taken when using it in evaluation of nonlinear properties.

Hence, whenever high accuracy of results is needed, it is rec-

ommended to use basis set of at least the aug-cc-pVTZ quality.

The effect of double augmentation with diffuse function has

also been studied for molecule 5 at the MP2 level of theory.

The changes in diagonal components of azz, bzzz, and czzzz on

passing from aug-cc-pVDZ to d-aug-cc-pVDZ are 0.3, 7, and

27%, respectively. Although in case of first and second hyper-

polarizability the differences are non-negligible, one should

not overlook that (i) even for medium-sized molecules, such

diffuse basis set leads to convergence problems, (ii) this effect

is expected to be far less pronounced for basis sets containing

higher angular momentum atomic orbitals.

Performance of the LPol-ds and -dl basis sets is considered

on the basis of the results presented in Tables 3 and 4. Missing

data in Table 3 are due to the difficulties in obtaining

Figure 3. Relative unsigned error in linear polarizability (top), first (middle)

and second hyperpolarizability (bottom) (d [%] ¼ ((P�PCCSD(T))/

(PCCSD(T)))�100%, where P¼azz, bzzz,czzzz, respectively) with respect to the

CCSD(T)/aug-cc-pVDZ reference. The plot is based on the data presented

in Table 2.



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numerically stable solutions of the Hartree-Fock and Kohn-

Sham equations. As the LPol-ds and LPol-dl results are very

close to each other in all cases, we present here mostly the

values obtained in the smaller LPol-ds set. At the MP2 level of

approximation, the agreement of the longitudinal electric

polarizability azz values obtained in LPol sets with those

yielded by the aug-cc-pVTZ basis set is excellent for all sys-

tems, with relative differences well below 0.5%. However, as it

was already mentioned, for this property already the aug-

cc-pVDZ basis set proves to be a reasonable choice for semi-

quantitative predictions at the MP2 level. The LPol-ds performs

better than aug-cc-pVDZ set in the case of molecules 1, 2, and

6, whereas aug-cc-pVDZ is closer to the aug-cc-pVTZ reference

results for systems 3 and 5. Regarding the first hyperpolariz-

ability bzzz, the LPol-ds and aug-cc-pVDZ perform practically

equally well for molecules 1, 2, 3, and 6. However, for system

5, the corresponding improvement on going from aug-cc-

pVDZ to LPol-ds basis set is indeed substantial, with the error

in the order of 1% in the case of the LPol-ds set versus over

10% in the case of aug-cc-pVDZ result. The use of LPol-dl set

leads in system 6 to a similar agreement with the aug-cc-pVTZ

result (differences below 2%).

In the case of second hyperpolarizability czzzz, the MP2/LPol-

ds approximation produces results about 2–5% (15–20%) dif-

ferent from the corresponding aug-cc-pVTZ values for systems

1, 2, and 6 (systems 3 and 5). However, it has to be noticed

that the aug-cc-pVTZ results are probably still far from the ba-

sis set limit, and for a complete discussion of the performance

of LPol-n sets results obtained in larger, possibly multiply-aug-

mented, correlation consistent basis sets are required. As could

be anticipated, in the case of RHF and DFT methods similar

conclusions are drawn; however, for DFT method the choice of

basis set is much less important than the appropriate choice

of exchange correlation functional, which can be critical even

for qualitative predictions. Finally, we would like to point out

that, considering the above observations, the CCSD(T)/LPol-ds

results reported for system 5, in particular the longitudinal lin-

ear polarizability and first hyperpolarizability, are at present

the most accurate results reported for this molecule.

Conclusions

The present contribution reports on the results of calcula-

tions of electric dipole (hyper)polarizabilities of several me-

dium-sized organic p-conjugated molecules using the recently

developed LPol family of basis sets. To test their perform-

ance, Dunning’s correlation-consistent basis set of triple-fquality is used. The most important observations regarding

the assessment of LPol basis sets can be summarized as fol-

lows. The LPol-ds and LPol-dl basis sets yield results very

close to each other, and thus the natural choice for routine

calculations is the smaller LPol-ds set. The longitudinal com-

ponents of linear polarizabilities obtained using the LPol-ds

set are in excellent agreement with the aug-cc-pVTZ values

for all studied molecules and at all considered levels of

approximation. The LPol-ds values of the axial components of

first hyperpolarizabilities are also close to those obtained

with the larger aug-cc-pVTZ basis set, and in the case of 4-

methylpyridone (5) a substantial improvement of the result is

observed when passing from the aug-cc-pVDZ to the LPol-ds

basis set. Considering the second hyperpolarizability, the

results obtained with LPol-ds set are in a slightly worse

agreement with aug-cc-pVTZ estimates, however, it has to be

stressed that due to a slow saturation of angular correlation

for this property, the aug-cc-pVTZ values might be still quite

far from the basis set limit.

As a part of this study, we have also tested the quality of

hyperpolarizabilities predicted by several exchange-correlation

functionals against the values obtained at the state-of-the-art

CCSD and CCSD(T) methods. As it turns out, there is no gen-

eral trend neither in the case of first nor in the case of second

hyperpolarizability; the employed functionals either underesti-

mate or overestimate (often dramatically) the properties in

question. Long-range-corrected exchange-correlation function-

als bring some improvement; however, their performance

depends strongly on the studied system. One of the molecules

in the investigated set has been found to be particularly chal-

lenging for the employed functionals as none of them has

been successful in predicting even the correct sign of first and

second hyperpolarizabilities. As far as nonlinear electric dipole

properties are concerned, this proves that even in the case of

medium-sized molecules, LC-DFT does not appear to predict

Table 4. Diagonal components of electric dipole (hyper)polarizabilities

of systems 1–6 computed at various levels of theoretical approximation

using the LPol-ds and LPol-dl basis sets.

azz bzzz czzzz � 10�3

5

RHF/LPol-ds 115.68 308.6 6 0.6 17

MP2/LPol-ds 121.57 83.7 6 1.7 22 6 1

CCSD/LPol-ds 121.40 110.1 6 1.3 19 6 1

CCSD(T)/LPol-ds 122.73 81.7 6 1.6 22 6 1

BLYP/LPol-ds 125.07 297.9 59

LC-BLYP/LPol-ds 118.33 171.5 18

B3LYP/LPol-ds 121.57 252.4 38

CAM-B3LYP/LPol-ds 120.03 204.3 27

M06–2X/LPol-ds 119.61 205.9 25

LC-xPBE/LPol-ds 117.79 147.8 18

RHF/LPol-dl 115.66 311.6 6 0.4 17

MP2/LPol-dl 121.59 86.1 6 1 21

BLYP/LPol-dl 124.99 303.0 59

LC-BLYP/LPol-dl 118.27 173.7 18

B3LYP/LPol-dl 121.51 256.0 38

CAM-B3LYP/LPol-dl 119.97 207.5 27

M06–2X/LPol-dl 119.61 209.3 25

LC-xPBE/LPol-dl 117.77 151.3 18

6

RHF/LPol-ds 132.00 �868.8 6 0.2 65

MP2/LPol-ds 146.84 �1416.9 6 0.2 127 6 1

BLYP/LPol-ds 165.03 �1773.3 114

LC-BLYP/LPol-ds 139.25 �1157.6 90

B3LYP/LPol-ds 154.75 �1581.2 112

CAM-B3LYP/LPol-ds 146.65 �1341.5 103

M06–2X/LPol-ds 144.66 �1251.4 97

LC-xPBE/LPol-ds 139.64 �1148.2 88

RHF/LPol-dl 131.97 �868.7 66 6 1

MP2/LPol-dl 146.83 �1413.6 6 0.1 127 6 1




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these properties satisfactorily. This conclusion is in line with

what was observed by Hammond and Kowalski.[6,7] As sug-

gested by these authors, perhaps a further reparametrization

of long-range-corrected exchange-correlation functionals is

necessary to obtain more reliable results within the KS-DFT

framework.

Acknowledgments

Part of the work, regarding the study of performance of newly

developed basis sets, was supported by the Homing Plus pro-

gramme (Homing Plus/2010-1/2), cofinanced from European Re-

gional Development Fund within Innovative Economy Operational

Programme.

The allocation of computing time granted by Wroclaw Centre for

Networking and Supercomputing (WCSS) is greatly appreciated.

Authors declare no competing financial interest.

Keywords: electric dipole polarizability � first electric dipole

hyperpolarizability � second electric dipole hyperpolarizabil-

ity � long–range–corrected density functional theory � LPol-n

basis sets

How to cite this article: A. Baranowska–Ła� czkowska, W. Bartko-

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Received: 11 July 2012Revised: 27 October 2012Accepted: 30 October 2012Published online on 28 December 2012



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On the performance of long-range-corrected density functional theory and reduced-size polarized...

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