mean polarizabilities of organic molecules. a comparison of restricted hartree fock, density...

Mean polarizabilities of organic molecules. A comparison of RestrictedHartree Fock, Density Functional Theory and Direct Reaction Field

results1

Marcel Swarta,* , Piet Th. van Duijnenb, Jaap G. Snijdersa

aMaterials Science Centre and Laboratory for Chemical Physics, Rijksuniversiteit Groningen, Nijenborgh 4, 9747 AG Groningen,Netherlands

bStratingh Institute (OMAC), Rijksuniversiteit Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlands

Received 15 January 1998; accepted 10 April 1998

Abstract

The polarizabilities of 15 organic molecules are calculated using the Restricted Hartree Fock (RHF) method, the TimeDependent Density Functional Theory (TD-DFT) and the Direct Reaction Field (DRF) approach. The RHF method gives ratherpoor results, while the other two give average deviations comparable to the experimental uncertainty. The DRF approach is veryfast (, 1 s) and transferable to other (classes of) molecules, but underestimates the anisotropy of molecules containingp-bonds.Three DFT methods were used (Local Density Approximation, Becke–Perdew and LB94 potential) which need more time (9–80 h) but give a better overall accuracy, which improves towards the basis set limit. The LB94 potential improves the Becke–Perdew potential, which in turn gives better results than the Local Density Approximation. The DFT methods providepolarizability components which show a deviation from experimental values which is less than the experimental uncertainty,while the frequency dependency of the polarizabilities is also calculated properly by the TD-DFT method.q 1999 ElsevierScience B.V. All rights reserved.

Keywords:Polarizabilities; Molecular modeling; Density functional theory

1. Introduction

There is a growing interest in atomic and molecularpolarizabilities within the scope of the development ofaccurate force fields to be used in QM/MM methods[1–6]. Dispersion and induction forces are more andmore being recognized as essential parts in the

description of intermolecular interactions in chemicalenvironments [4,7–13].

It has also been noticed that using a molecularpolarizability located in the center of the moleculeoften leads to improper behavior [4,14]. This can beovercome by using effective atomic polarizabilitiesthat represent the molecular value. A set of theseatomic parameters has been constructed within theframework of the Direct Reaction Field (DRF)method, which predicts the molecular polarizabilitieswith a deviation comparable to the experimentaluncertainty (2–4%) [15,16]. In this paper, we present

Journal of Molecular Structure (Theochem) 458 (1999) 11–17

0166-1280/99/$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.PII: S0166-1280(98)00350-9

1Dedicated to Professor Paul S. Bagus on the occasion of his 60thbirthday.* Corresponding author: Tel.: +31 50 3634377; fax: +31 50

3634441; e-mail: [email protected]

DRF results for a representative series of organicmolecules. For comparison, we have also performedRestricted Hartree Fock (RHF) and Time DependentDensity Functional Theory (TD-DFT) calculations onthe same set of molecules, the latter also at a non-zerofrequency.

2. Methods

The calculation of a polarizabilitya is based on aTaylor expansion of the total energy around the elec-tric field strengthE:

U =U (0) −m(0)i Ei −

12!

aij EiEj −13!

bijk EiEjEk

−14!

gijkl EiEjEkEl…

with U (0) the unperturbed total energy,m (0) the perma-nent dipole moment,a the polarizability, andb andg

the first and second hyperpolarizabilities.The RHF results are obtained with Coupled

Perturbative Hartree Fock (CPHF) equations [15] (asimplemented in the HONDRF program [5]), whichtake the response of a molecule to an applied fieldanalytically into account. It is also possible to use afinite field procedure (with the same results), but sincethis is much slower, it was not used. Dunning’s triplezeta valence plus polarization functions basis set(TZP) [16] was used for all molecules, which is thelargest standard (Gaussian) basis set in the HONDRFprogram and has proven to give an underestimation ofthe polarizabilities by,22% [17].

A lot of effort has been put into constructing basissets specially designed for accurate calculations of thepolarizability [18–22]. However, they are based onGaussian-type orbitals, which makes a comparisonwith the ADF results (using Slater-type orbitals)impossible. Moreover, it is not the purpose of thisarticle to comparebasis setsor to construct them forthe ADF program, but rather to make an honest com-parison between the threemethodsusing the programsand basis sets that are readily available.

The DFT results were all obtained by using theRESPONSE code [23–25] in the Amsterdam DensityFunctional (ADF) program [26–29]. This TimeDependent DFT (TD-DFT) method at zero frequencyis similar to a CPHF procedure, but then applied to

DFT SCF equations which gives the same polarizabil-ities as when using a finite field procedure.

The ADF program uses basis sets of Slater func-tions, of which two were used: a triple zeta valenceplus polarization (TZP, basis set IV in ADF; of equalquality as Dunning’s TZP basis) and one with triplezeta valence plus double polarization functions and dif-fuse s, p, d functions (TZ2P+ + , basis VII) which hasbeen reported to give accurate polarizabilities [25].Several potentials were used: LDA (VWN [30],GGA’s (Becke88 exchange [31]–Perdew86 correla-tion [32] potential, Van Leeuwen-Baerends LB94[33]).

Within the DRF approach [5], a molecular polariz-ability is being constructed from interacting atomicpolarizabilitiesap:

mp =ap E0 + ∑N

qÞpTpqmq

� �with the modifieddipole field tensorT:

Tpq

ÿ �ij = − f E

pq·dij=r3 +3f T

pq· ri rj=r5ÿ �

The modification is present in the screening factorsf Epq

and f Tpq, which represents the damping due to over-

lapping charge densities. Several functions havebeen tested for this damping [34], of which twohave survived: a conical and an exponential function.The exponential function has been used in this paper.

Exponential functions:

r(v) =a3

8pe−av f E

pq =1−12v2 +

12v+1

� �e−v

v=ar

(apaq)1=6f Tpq =1−

16v3 +

12v2 +

12v+1

� �e−v

In these equations thea-factor and the atomic polar-izabilities are adjustable parameters obtained in the fitprocedure [17].

In matrix notation:

M=AE; A= a −1 −Tÿ �−1

This A-matrix maps the linear response of themolecule to an applied electric field, i.e. its polariz-ability in a 3N × 3N matrix representation. It can bereduced to a ‘normal’ 3× 3 polarizability tensor:

amn= ∑N

i, j =1Aij

ÿ �mn; m,n [ { x,y,z}

12 M. Swart et al. / Journal of Molecular Structure (Theochem) 458 (1999) 11–17

3. Results

The calculated mean polarizabilities of the 15molecules are given in Table 1, together with experi-mental data. The experimental mean values areobtained mostly from the refractive indexn (at5893 A; sodium D-line) and the Lorentz–Lorentzequation (withM molecular weight,r macroscopicdensity,Nav Avogadro’s number):

n2 −1n2 +2

Mr

=43pNava

These values should be extrapolated to infinite wave-length (or zero frequency) to obtain the static polariz-ability. This scales the values down by 1–4% [35],and gives an estimate of the experimental uncertaintywhen using the uncorrected values.

The experimental data can also be obtained fromthe Kerr constants [35], but this introduces, apartfrom the wavelength dependency, an additionaluncertainty. One has to make an assumption of thegeometry, thereby limiting this method to (small)symmetric molecules. The additional uncertainty hasbeen estimated to be 5–10% [35], giving a total uncer-tainty of at least 6–14%.

In Table 2, some experimental polarizability com-ponents are given, which were taken from Refs [36]and [35]. Some other components were left out,because of spurious assumptions that increase theuncertainty even more. If necessary, the componentswere uniformly scaled to reproduce the average meanpolarizability. It should be noted also, that someexperimentalcomponents reported by Applequist etal. [36] are extrapolated empirical values; i.e.experimental Kerr constants were obtained for somemolecules which were then extrapolated to similar(larger) molecules.

The RHF results give an average deviation of24.9%, which has been reported before [17], andcan be attributed to two factors. The first is the(small) basis set; it is known that specially constructed[18–22] or large (including very diffuse functions)[25] basis sets are needed to obtain accurate results.This is especially apparent from the results of planarand linear molecules, where the out-of-plane polariz-ability is very small in comparison to the experimentaland DFT results.

The second factor is the absence of electron corre-lation in the RHF method. Inclusion of this correlation(by, for instance, Configuration Interaction, MultiConfiguration SCF, Coupled Cluster or Møller–Plesset methods) increases the work neededenormously, and can be applied only to relativelysmall molecules.

Much more promising and accurate are the DFTresults. The results from all three potentials in theTZP basis set give a much better accuracy (deviation6–8%) for the mean values. When using the TZ2P++basis set, which has been reported as the basis set limitfor (other) organic molecules [25], the deviationsbecome even smaller (LDA 5.8%, Becke–Perdew3.6%, LB94 potential 3.0%). In both basis sets, theLB94 results are smaller than either the LDA or theBecke–Perdew results.

The DFT polarizability components show, like themean value, a much smaller deviation from theexperimental data than the RHF results, far less thanthe experimental uncertainty of these values.

However, we are comparing with theuncorrectedmean values. A better comparison can then be madeby calculating the (frequency dependent) polarizabil-ities with the frequency corresponding to the wave-length (5893 A˚ ) of the sodium D-line. The frequencydependent polarizabilities from the LB94 potential inthe TZ2P+ + basis set are on average 2.90% largerthan the static values, which is in good agreement withthe estimated 1–4% from experimental data [35].They are also on average 4.7% larger than the experi-mental values, which is a deviation comparable to theexperimental uncertainty.

In principle, both CPHF and TD-DFT should scaleasN3, whereN is the number of basis functions. Whilethe CPHF method (taking about 25% of the total CPUtime) indeed practically scales asN3, the TD-DFTmethod more or less scales linearly and takes approxi-mately 40% of the CPU time. Apparently the currentsystems are too small for theN3 behavior of TD-DFTto become dominant.

The DRF approach gives polarizabilities with anaccuracy of 3.6%, at a very low computational cost(, 1 s) and with high transferability to othermolecules [17]. It should be noticed that, like theRHF method, some problems arise with linear andplanar molecules, where the anisotropy of themolecular polarizability (i.e.p-bonds) cannot be

13M Swart et al. / Journal of Molecular Structure (Theochem) 458 (1999) 11–17

obtained with interaction tensors based on the atomicpositions. Adding extra fit points increases theaccuracy of the polarizabilities [14], but leads alsoto bad interaction energies in classical DRF energycalculations [4].

4. Computational details

The DRF and RHF calculations were carried out on theCray J932 supercomputer in Groninigen. All ADF resultswere calculated at a SGI Power Challenge workstation.

Table 1Molecular polarizabilities (Bohr3)

aexpv aDRF aRHF

TZP aLDAIV aBP

IV aLB94IV aLDA

VII aBPVII aLB94

VII aLB94VII,v

Acetamidep 40.5R1 38.57 31.08 40.46 39.54 39.22 43.65 42.63 41.18 43.13Acetylenet 22.5Kg 21.90 18.01 19.56 19.27 19.18 23.80 23.48 22.39 23.09Benzenep 70.11R1 61.87 61.87 66.66 65.66 65.86 70.75 69.56 69.70 72.64Chlorinep 31.1Kg 31.20 18.73 25.27 24.92 25.41 31.96 31.31 29.88 31.67Cyclohexanolt 79.9R1 77.97 69.23 79.15 76.97 77.03 82.11 79.93 80.72 82.68Dimethyletherp 35.00Rg 35.36 27.62 34.79 33.78 33.43 37.22 35.99 35.55 36.27Formaldehydep 16.5Rg 18.25 13.52 16.58 16.25 16.05 19.08 18.62 17.83 18.25Hydrogenp 5.33Kg 4.90 2.62 4.79 4.57 4.74 5.87 5.51 5.62 5.76Methylcyanidep 29.7R1 29.84 24.59 28.36 27.77 28.20 31.17 30.52 30.76 31.06Neopentanet 68.97R1 65.49 58.96 67.79 65.67 65.66 69.52 67.41 68.56 70.29Propanep 42.40R1 42.19 35.87 42.08 40.77 41.45 43.67 42.27 42.12 44.08TCFM#t 57.47R1 56.45 40.00 53.18 52.83 52.65 60.89 60.04 59.29 60.87TCMC#t 70.5R1 68.26 54.00 66.10 64.96 66.08 74.57 73.29 73.66 75.17TFM#p 19.0Kg 18.96 13.09 17.87 17.53 16.77 20.68 20.30 18.62 18.85Waterp 9.94R1 10.06 5.56 8.40 8.25 7.97 10.55 10.29 9.20 9.38

Time , 1 s 57 h 9 h 14 h 23 h 32 h 53 h 80 h 80 h

Average deviation − 1.78 − 24.94 − 6.01 − 8.12 − 8.16 5.84 3.17 0.89 3.79(%) 6 4.80 6 11.50 6 5.79 6 5.46 6 5.52 6 3.70 6 3.55 6 3.73 6 3.69

Average absolute 3.60 24.94 6.07 8.12 8.16 5.84 3.62 2.98 4.65deviation (%) 6 3.64 6 11.50 6 5.72 6 5.46 6 5.52 6 3.70 6 3.09 6 2.41 6 2.52

Relative to LB94 + 2.35 + 0.07 + 4.98 + 2.35 + 2.90potential (%) 6 1.92 6 2.03 6 3.75 6 3.91 6 1.35

#: TCFM = trichlorofluoromethane: TCMC= trichloromethylcyanide; TFM= trifluoromethane.p: Molecules used to obtain DRF-parameters [17].t: Molecules used to test DRF-parameters.K: Obtained from Kerr constants (uncertainty ca. 5%).R: Obtained from refractive index (uncertainty ca. 0.2%).g: Gas phase.I: Liquid phase.DRF: Direct Reaction Field approach result [17].TZP: CPHF value with TZP basis set.LDA: Local Density Approximation result (Vosko–Wilk–Nusair potential [30]).BP: Becke88 exchange [31] with Perdew correlation [32] potential.LB94: Van Leeuwen–Baerends (LB94) potential with correct asymptotic behavior [33].IV: DFT value with basis set IV (TZP).VII: DFT value with basis set VII (TZ2P+ +).v: Polarizabilities at 5893 A˚ .


Table 2Polarizability components (Bohr3)

(a)axx,exp ayy,exp azz,exp a xx,DRF ayy,DRF azz,DRF a xx,RHF ayy,RHF azz,RHF

Acetylene 30.5 18.5 18.5 35.56 15.07 15.07 31.00 11.51 11.51Benzene 82.7 82.7 45.0 74.90 74.90 35.83 74.79 74.79 36.03Chlorine 44.5 24.4 24.4 39.02 27.28 27.28 34.16 11.01 11.01Dimethylether 42.6 29.3 33.0 39.41 32.61 34.07 31.24 25.69 25.94Hydrogen 6.3 4.9 4.9 6.51 4.10 4.10 6.52 0.67 0.67Methylcyanide 38.1 25.5 25.5 39.73 24.90 24.90 35.37 19.20 19.20Neopentane 69.0 69.0 69.0 65.49 65.49 65.49 58.96 58.96 58.96TCFM# 59.7 59.7 53.0 59.97 59.97 49.42 44.30 44.30 31.42TCMC# 72.3 69.6 69.6 70.61 67.09 67.09 56.02 52.99 52.99TFM# 19.4 19.4 18.1 18.83 18.83 19.22 13.38 13.38 12.50

Average component − 3.38 − 27.71deviation (%) 6 9.35 6 20.52

Average absolute 8.01 28.05component devia-tion (%)

6 5.88 6 20.05

(b) DFT polarizability components (Bohr3)axx,LDA ayy,LDA azz,LDA a xx,BP ayy,BP azz,BP a xx,IV ayy,IV azz,IV axx,VII ayy,VII a zz,VII

Acetylene 31.15 13.76 13.76 30.72 13.54 13.54 31.72 12.92 12.92 31.77 17.71 17.71Benzene 80.26 80.26 39.50 79.15 79.15 38.68 80.18 80.18 37.23 83.71 83.71 41.69Chlorine 37.87 18.97 18.97 37.37 18.69 18.69 38.28 18.98 18.98 39.97 24.84 24.84Dimethylether 39.81 32.10 32.46 38.79 31.04 31.50 38.73 30.78 30.80 41.03 32.88 32.75Hydrogen 6.79 3.79 3.79 6.42 3.64 3.64 6.74 3.74 3.74 6.97 4.95 4.95Methylcyanide 40.92 22.08 22.08 40.14 21.58 21.58 41.03 21.79 21.79 42.36 24.96 24.96Neopentane 67.79 67.79 67.79 65.67 65.67 65.67 65.66 65.66 65.66 68.56 68.56 68.56TCFM# 43.65 57.94 57.94 42.79 56.85 56.84 42.45 57.76 57.76 51.26 63.31 63.31TCMC# 68.71 64.79 64.79 67.58 63.64 63.64 68.44 64.90 64.90 84.61 68.18 68.18TFM# 18.26 18.26 17.09 17.92 17.92 16.73 17.23 17.23 15.85 19.10 19.10 17.64

Average deviation − 8.14 − 10.25 − 10.06 0.94(%) 6 10.64 6 10.24 6 10.99 6 7.15

Average absolute 10.55 11.67 12.24 4.98deviation (%) 6 8.25 6 8.59 6 8.49 6 5.21

#: TCFM = trichlorofluoromethane; TCMC= trichloromethylcyanide; TFM= trifluoromethane.exp: Experimental polarizability components [35,36] (accuracy 6–14%).DRF: Direct Reaction Field approach result [17].TZP: CPHF value with TZP basis set.LDA: Local Density Approximation result (Vosko–Wilk–Nusair potential [30]) in basis set IV (TZP).BP: Becke88 exchange [31] with Perdew correlation [32] potential in basis set IV (TZP).IV: Van Leeuwen–Baerends (LB94) potential with correct asymptotic behavior [33] in basis set IV (TZP).VII: Van Leeuwen–Baerends (LB94) potential with correct asymptotic behavior [33] in basis set VII (TZ2P+ +).


5. Geometry

The geometries of the molecules were taken asmuch as possible from experimental data [37]. Theremaining (internal) coordinates were optimized bythe PM3 method in MOPAC93 [38].

6. Conclusions

The DRF approach provides mean polarizabilityvalues at low computational cost with an accuracy(3–4%) equal to experimental uncertainty [17]. Itsonly setback is the underestimation of the anisotropyof linear and planar molecules, but that is more thancompensated for with a high transferability to othermolecules without the need to reparameterize.

The RHF method gives rather poor mean polariz-abilities (deviation 25% in TZP basis), with high costfor improvement upon these results. Even with thespecially constructed (polarized) basis sets, oneneeds correlated wavefunctions, thereby limiting theapplicability to small molecules.

TheDFT methods give good mean values with theTZP basis set (6–8% deviation), and accuratevalues (deviation 3–6%) with the TZ2P+ + basisset. The latter needs only 3–4 times more CPUtime, so it is rather easy to improve the TZP results.It is also evident that the LDA gives larger valuesthan the gradient corrected potentials (+2–5%),while the Becke–Perdew results are substantially(2.4%) improved by the Van Leeuwen–Baerendspotential.

The impact of the frequency dependency on thepolarizabilities is reflected properly by the TD-DFTmethod which was shown with the Van Leeuwen–Baerends potential. The frequency dependent polariz-abilities are on average 2.9% larger than the staticvalues, which is in perfect agreement with theextrapolation estimate of 1–4%.

The polarizability anisotropy is very well repro-duced by allDFT methods, and theDRF method,giving deviations (5–12%) from experimentalvalues within the experimental uncertainty (6–14%).The RHF method again shows deviations (28%)which are large in comparison to experimentaldeviations.

References

[1] A. Warshel, M. Levitt, J. Mol. Biol. 103 (1976) 227–249.[2] J. Gao, J. Phys. Chem. 99 (1995) 16460.[3] M.A. Thompson, J. Phys. Chem. 100 (1996) 14492–14507.[4] P.T. van Duijnen, A.H. de Vries, Int. J. Quant. Chem. 60

(1966) 1111–1132.[5] A.H. de Vries, P.T. van Duijnen, A.H. Juffer, J.A.C.

Rullmann, J.P. Dijkman, H. Merenga, B.T. Thole, J. Comp.Chem. 16 (1995) 37–55 and 1445–1446.

[6] A.H. de Vries, P.T. van Duijnen, R.W.J. Zijlstra, M. Swart, J.Elec. Spectr. 86 (1997) 49–56.

[7] A.H. de Vries, P.T. van Duijnen, Int. J. Quant. Chem. 57(1996) 1067–1076.

[8] F. Grozema, P.T. van Duijnen, J. Phys. Chem. A, in press.[9] R.W.J. Zijlstra, P.T. van Duijnen, A.H. de Vries, Chem. Phys.

204 (1996) 439–446.[10] J. Cioslowski, J. Sauer, J. Hetzenegger, T. Karcher, T.

Hierstetter, J. Am. Chem. Soc. 115 (1993) 1353–1359.[11] T.R. Furlani, J. Gao, J. Org. Chem. 61 (1996) 5492–5497.[12] S.J. Stuart, B.J. Berne, J. Phys. Chem. 100 (1996) 11934–

11943.[13] B.D. Bursulaya, D.A. Zichi, H.J. Kim, J. Phys. Chem. 99

(1995) 10069–10074.[14] A.H. de Vries, Modelling condensed-phase systems.

From quantum chemistry to molecular models. Ph.D. Thesis,1995.

[15] M. Dupuis, A. Farazdel, S.P. Karma, S.A. Maluendes, in:MOTECC-90, ESCOM, Leiden, 1990, pp. 277–342.

[16] T.H. Dunning Jr, P.J. Hay, in: Methods of Electronic StructureTheory, 1977, pp. 1–27.

[17] P.T. van Duijnen, M. Swart, (1998) J. Phys. Chem. A 102(1998) 2399–2407.

[18] H.-J. Werner, W. Meyer, Mol. Phys. 31 (1976) 855–872.[19] A.J. Sadlej, Theor. Chim. Acta 81 (1991) 329.[20] A.J. Sadlej, Theor. Chim. Acta 81 (1991) 45.[21] A.J. Sadlej, M. Urban, J. Mol. Struct. (THEOCHEM) 80

(1991) 234.[22] A.J. Sadlej, Theor. Chim. Acta 79 (1991) 123.[23] S.J.A. van Gisbergen, J.G. Snijders, E.J. Baerends, Chem.

Phys. Lett. 259 (1996) 599.[24] S.J.A. van Gisbergen, J.G. Snijders, E.J. Baerends, J. Chem.

Phys. 103 (1995) 9347.[25] S.J.A. van Gisbergen, V.P. Osinga, O.V. Gritsenko, R. van

Leeuwen, J.G. Snijders, E.J. Baerends, J. Chem. Phys. 105(1996) 3142–3151.

[26] ADF 2.3.3, Theoretical Chemistry, Vrije Universiteit,Amsterdam, 1997.

[27] C. Fonseca Guerra, O. Visser, J.G. Snijders, G. te Velde,E.J. Baerends, in: Methods and Techniques in ComputationalChemistry: METECC-95, STEF, Cagliari, 1995, pp. 305–395.

[28] E.J. Baerends, D.E. Ellis, P. Ros, Chem. Phys. 2 (1973) 41.[29] G. te Velde, E.J. Baerends, J. Comp. Phys. 99 (1992) 84.[30] S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200.[31] A.D. Becke, Phys. Rev. A38 (1998) 3098.[32] J.P. Perdew, Phys. Rev. B33 (1986) 8822. Erratum: Phys. Rev.

B 34 (1986) 7406.


[33] R. van Leeuwen, E.J. Baerends, Phys. Rev. A49 (1994)2421–2431.

[34] B.T. Thole, Chem. Phys. 59 (1981) 341–350.[35] C.J.F. Bottcher, P. Bordewijk, Theory of Electric Polarization,

Elsevier, Amsterdam, 1978.

[36] J. Applequist, J.R. Carl, J.K. Fung, J. Am. Chem. Soc. 94(1972) 2952.

[37] J.J.P. Stewart, MOPAC 93.00 Manual, Fujitsu Ltd., Tokyo, 1993.[38] J.J.P. Stewart, MOPAC 93, Tokyo, Japan, 1993.


mean polarizabilities of organic molecules. a comparison of restricted hartree fock, density...

Documents