Density functional calculations of molecular

JOURNAL OF CHEMICAL PHYSICS

VOLUME 109, NUMBER 18

8 NOVEMBER 1998

Density functional calculations of molecular polarizabilities and hyperpolarizabilities P. Calaminici, K. Jug, and A. M. Ko¨ster Theoretische Chemie, Universita¨t Hannover, Am Kleinen Felde 30, 30167 Hannover, Germany

~Received 12 May 1998, accepted 22 July 1998! This paper presents dipole moments, static polarizabilities, first hyperpolarizabilities and second hyperpolarizabilities calculated in the framework of density functional theory. All calculations have been performed using a finite field approach implemented in our new density functional theory program ALLCHEM. The calculations were of all-electron type. Both local and gradient-corrected functionals have been used. The influence of first- and second-order field-induced polarization functions, the external field strength, the numerical integration technique and the exchange-correlation functionals on the calculation of polarizabilities and hyperpolarizabilities is discussed in detail. A systematic study including 23 small and medium size molecules demonstrates that the obtained polarizabilities as well as the first and second hyperpolarizabilities are in good qualitative agreement with experimental data. The described density functional method provides polarizabilities and hyperpolarizabilities considerably better than the Hartree–Fock method and almost as accurate as much more expensive correlation treatments. This work demonstrates that reliable predictions of electro-optical properties for molecules with 20 and more atoms are possible using an efficient implementation of density functional theory. © 1998 American Institute of Physics. @S0021-9606~98!30340-2# I. INTRODUCTION

larizabilities, polarizability anisotropies and mean first and second hyperpolarizabilities of 23 molecules obtained with the finite field approach implemented in ALLCHEM. The calculated results are compared with available experimental data and other theoretical results.

In the last 40 years there has been a significant development of experimental methods used for the determination of the polarizability a, first hyperpolarizability b and second hyperpolarizability g such as molecular beam methods,1–4 collision-induced scattering of light,5–8 Kerr-effect experiments,9,10 electric field-induced second harmonic generation experiments ~ESHG, dcSHG, EFISH!11–18 and thirdharmonic generation techniques ~THG!.19–23 At the same time a large theoretical knowledge of atomic and molecular electrical properties by means of different quantum chemical methods has been accumulated and it would be beyond the scope of this paper to review all of them here ~see Refs. 24–26, and references therein!. This variety of theoretical work has led to a better understanding of the role of these properties in certain chemical and physical phenomena such as laser technology where the development of materials with specified optical properties is required. For the computational chemist, this means the determination of polarizabilities and hyperpolarizabilities of molecules, initially in the gas phase. In the last few years, density functional theory ~DFT! programs have been used in order to calculate static polarizabilities and hyperpolarizabilities.27–36 The static polarizabilities and hyperpolarizabilities can be studied either with the finite field approach27–34 or with the coupled KohnSham method.35,36 Both approaches yield equivalent results. Although the calculated static hyperpolarizabilities cannot be directly compared with experimental data, they are highly valuable to understanding the influence of the molecular and electronic structure on hyperpolarizabilities. Recently we have implemented the finite field approach in our new density functional program ALLCHEM.37,38 In this article we present static properties such as dipole moments, mean po0021-9606/98/109(18)/7756/8/$15.00

II. THEORETICAL AND COMPUTATIONAL METHOD A. Theory

The static response properties of a molecule can be defined by expanding the field-dependent energy E(F) in a series, E ~ F! 5E ~ 0! 2 2

1 24

a i j F i F j 2 ( b i jk F i F j F k (i m i F i 2 ( i, j i, j,k

(

1 2

i, j,k,l

1 6

g i jkl F i F j F k F l 2•••,

~2.1!

where E(0) is the total energy of the molecular system in the absence of the electric field and the quantities, m i , a i j , b i jk and g i jkl are components of the static dipole moment, polarizability, first hyperpolarizability and second hyperpolarizability tensors, respectively. In an alternative way the static response properties of a molecule can be defined by expanding the field-dependent dipole moment, calculated from the field-induced charge distribution, as a series of the external electric field

m i ~ F! 5 m i ~ 0! 1 ( a i j F j 1 12 ( b i jk F j F k j

1 61 7756

(

j,k,l

j,k

g i jkl F j F k F l 1••• .

~2.2!

© 1998 American Institute of Physics

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 148.247.92.8 On: Wed, 06 May 2015 23:50:53

Calaminici, Jug, and Ko¨ster

J. Chem. Phys., Vol. 109, No. 18, 8 November 1998

The equivalence of these two definitions for fieldindependent basis sets follows from the Hellmann–Feynman theorem.39,40 We have used the dipole moment expansion in our finite field calculations because in this expression the field occurs only in the third power for the calculation of g and not in the fourth as in the energy expansion. The used finite field method was developed by Kurtz et al.41 for semiempirical methods. In this approach a uniform electric field is aligned along the axes and the 45° line between the axes of the molecule. By using various field strengths the following equations for the tensor components of a, b and g can be derived

a ii F i 5 @ m i ~ F i ! 2 m i ~ 2F i !#

order to accelerate the convergence the unperturbed density matrix is used as starting density for the perturbed SCF runs. Using this implementation more than 80% of the CPU time is spent in the numerical integration of the exchangecorrelation potential which scales linearly with the size of the system. From the calculated polarizability components the mean polarizability, ¯a 5 31 ~ a xx 1 a y y 1 a zz ! ,

uD au25

~ principal axes! ,

~2.4!

b iii F 2i 5 13 @ m i ~ 2F i ! 1 m i ~ 22F i !# 2 13 @ m i ~ F i ! 1 m i ~ 2F i !# ,

~2.5!

b i j j F 2j 5 13 @ m i ~ 2F j ! 1 m i ~ 22F j !# 2 31 @ m i ~ F j ! 1 m i ~ 2F j !# ,

~2.6!

g iiii F 3i 5 21 @ m i ~ 2F i ! 2 m i ~ 22F i !# 2 @ m i ~ F i ! 2 m i ~ 2F i !# ,

~2.7!

g ii j j F i F 2j 5 12 @ m i ~ F i ,F j ! 2 m i ~ 2F i ,F j ! 1 m i ~ F i ,2F j ! 2 m i ~ 2F i ,2F j !# 2 @ m i ~ F i ! 2 m i ~ 2F i !# . ~2.8! For the calculation of the a and b components 12 selfconsistent field ~SCF! runs are necessary with the field strengths 6F i and 62F i (i5x,y,z) along the molecular axes. For the calculation of the g ii j j components another 12 runs are necessary with the field strengths (F i ,F j ), (2F i ,F j ), (F i ,2F j ) and (2F i ,2F j ) along the 45° lines between the molecular axes. The independent calculation of the symmetric off-diagonal elements of g, i.e., g xxy y and g y yxx , can be used to control the numerical stability of the finite field approach. For the calculation of the necessary induced dipole moments only the core Hamiltonian has to be modified in the different SCF runs, H ~m1n! 5H ~m0n! 1F i ^ m u r i u n & . H m(0)n

3tr a2 2 ~ tr a! 2 2

~2.9!

is an element of the unperturbed core HamilHere tonian, F i is a component of the applied field and ^ m u r i u n & is a dipole moment integral. When the perturbed core Hamiltonian H„1… is constructed and the perturbed density matrix P„1… is obtained self-consistently, the induced electric dipole moments can be calculated from P„1… and the corresponding dipole moment integral matrix. Therefore the Coulomb integrals are not modified during these calculations. Since the evaluation of four-center Coulomb integrals is avoided in ALLCHEM, a very efficient implementation of this approach is possible by storing the Coulomb integrals in memory. In

~ general axes!

~2.11!

5 12 @~ a xx 2 a y y ! 2 1 ~ a xx 2 a zz ! 2 1 ~ a y y 2 a zz ! 2 #

~2.3!

a i j F j 5 23 @ m i ~ F j ! 2 m i ~ 2F j !# 2 121 @ m i ~ 2F j ! 2 m i ~ 22F j !# ,

~2.10!

and the polarizability anisotropy,

2 3

2 121 @ m i ~ 2F i ! 2 m i ~ 22F i !# ,

7757

~2.12!

can be calculated. The calculated b components permit the calculation of the mean first hyperpolarizability, which may be defined as nine-fifths of the partial derivative of the mean polarizability with respect to a field F z oriented along the direction of the permanent dipole moment, ¯b 5

9 ] ¯a 3 5 5]Fz 5

(i b iiz .

~2.13!

For the second hyperpolarizability, the quantity of interest is the mean value given by ¯g 5

1 5

g ii j j . ( i, j

~2.14!

These are the usual, experimentally determined polarizability and hyperpolarizability invariants. B. Computational details

All presented calculations have been carried out using our new DFT program ALLCHEM.37,38 ALLCHEM works with Gaussian basis sets and Gaussian-type auxiliary functions in order to avoid the calculation of four-center Coulomb integrals. A new efficient integral algorithm for the calculation of three-center Coulomb integrals was applied37 which exploits some special features of the used auxiliary function sets. Recently, an adaptive numerical integrator was successfully implemented,38 which allows a very efficient automatic optimization of the grid for individual molecular systems. For a given tolerance of the numerical integration this adaptive grid generator automatically adjusts the grid to the basis set and the molecular structure. This feature increases the reliability of the numerical integration considerably and the number of expensive reference grid calculations can be reduced. The DFT polarizabilities and hyperpolarizabilities were calculated via the finite field method described above. In the DFT code ALLCHEM both the linear combination of Gaussian-type orbital local spin density ~LCGTO-LSD! and the corresponding generalized gradient approximation ~LCGTO-GGA! methods are implemented. In the case of


7758



TABLE I. Exponents of field-induced polarization functions.

a, b Atom

s

H C N O F S

0.027 94 0.034 83 0.042 16 0.049 93 0.02917

p

g d

0.116 00

d

f

0.071 11 0.079 20 0.098 74 0.119 53 0.141 55 0.082 68

0.052 96 0.066 03 0.079 93 0.094 65 0.055 29

local density functional calculations the exchange-correlation contributions proposed by Vosko, Wilk and Nusair ~VWN!42 have been used, while in the case of generalized gradient approximated spin density functional calculations the functional of Becke43 for the exchange and of Lee, Yang and Parr44 for the correlation contribution ~BLYP! was used. The exchange-correlation potential was numerically integrated on the adaptive grid described above. Details concerning the choice of the grid accuracy for the calculation of the polarizabilities and hyperpolarizabilities will be given below. The Coulomb energy was calculated by the variational fitting procedure proposed by Dunlap, Connolly and Sabin.45,46 It is well known that a general characteristic required for basis sets to perform well for polarizability calculations is that they should contain diffuse functions.47 An economical strategy for constructing these kinds of basis sets is to augment valence basis sets of reasonably good quality with additional polarization functions.48–51 We have chosen as valence basis a triple zeta basis set ~TZVP! which was optimized for local DFT calculations.52 The basis set was then augmented with field-induced polarization ~FIP! functions due to Zeiss et al.49 They derived the FIP function exponents from an analytic analysis of the fieldinduced charges in hydrogen orbitals. The exponents of the first- and second-order Gaussian-type FIP functions53 are listed in Table I. The first-order FIP functions are used for the calculation of a and b. For the calculation of g secondorder FIP functions, which are f -type Gaussians for mainrow elements, are additionally added to the basis set. In order to avoid the contamination of the valence basis set with the diffuse s- and p-type Gaussians of the FIP functions, spherical basis functions are used in all calculations. We have named the resulting basis sets TZVP-FIP1 and TZVP-FIP2. In order to judge the performance of the TZVP-FIP1 basis set we have compared, for a selected set of molecules, polarizability calculations using the TZVP-FIP1 basis and the basis set due to Sadlej48 which is more frequently used in Hartree–Fock based a calculations. It should be mentioned that both basis sets are of similar size. To study the importance of second-order FIP functions for the calculation of g we have compared local density functional calculations of g with the TZVP-FIP1 and TZVP-FIP2 basis sets. For the fitting of the density the auxiliary function set A252 was used in all calculations. The SCF energy convergence criterion54 was set to 10 29 a.u. in all calculations. For the optimization of molecular structures the VWN functional42 and a local density functional optimized double

zeta basis set ~DZVP! were used. For the numerical integration during the structure optimization the grid accuracy was set to 1024 . The optimization was performed with a quasiNewton update method55 using analytic energy gradients. The structure optimization convergence was based on the gradient and displacement vectors with a threshold of 1024 and 1023 a.u., respectively. For the calculation of a, b and g polar molecules were oriented with their permanent dipole moment along the z axis pointing in the positive z direction. Nonpolar molecules were oriented according to the standard orientation of the corresponding symmetry group.

C. Influence of the the electric field strength, grid accuracy, and basis set

One of the most severe problems of the described finite field procedure is the choice of an appropriate field strength. Two different and opposite requirements govern the choice of the field and both of them should be satisfied at the same time. First, the field must be large enough so that the imprecision in the calculated dipole moments is small compared to the contribution to the dipole moment expansion ~2.2! of the term whose coefficient is desired. At the same time the field must be small enough so that the error incurred by the truncation of the expansion is acceptably small. The choice of field strengths could be even more problematic for the calculation of first and second hyperpolarizabilities with DFT methods, as was discussed in the literature.29 Such problems are usually attributed to numerical inaccuracies resulting from the numerical integration of the exchange-correlation contribution in DFT calculations. The use of an adaptive grid generator that automatically adjusts the grid to the basis set and the molecular structure, as implemented in ALLCHEM, represents a very satisfactory solution to these numerical problems, as will be shown below. Our work strategy was first to use a very high grid accuracy and to vary the field in order to find an appropriate field strength that can be used in all calculations. Therefore, the tedious work to adjust the field strength for each molecule can be avoided. With this aim in mind, test calculations on a set of small molecules ~H2O, NH3 , HF, H2S and CO!, for which reliable experimental and theoretical dipole moments, polarizabilities and hyperpolarizabilities are available,56–69 have been performed in order to study the influence of the electric field and grid accuracy on the calculation of ¯a , u D a u , ¯b and ¯g . The experimental gas-phase structures70–72 of these molecules were used in the presented calculations. The influence of the electric field was studied using a grid accuracy of 1027 and calculating a, b and g tensors with the described finite field method using the VWN functional and applying different field strengths. In Fig. 1 we present representative results for H2 O. As expected, the second hyperpolarizability shows the strongest dependency from the applied field whereas a is almost independent of the field strength. The most sensitive components of the first @Fig. 1~a!# and second @Fig. 1~b!# hyperpolarizability tensors of water are plotted against the electric field strengths. As Fig. 1 shows, a field strength of 1022.5 a.u. is most appropriate for a and b calculations. For g a field strength of 1022




7759

FIG. 1. Influence of the electric field strength on the most sensitive components of the first ~a! and second ~b! hyperpolarizability tensor of H2 O.

FIG. 3. Influence of the basis set for the calculations of ~a! mean polarizabilities ~a.u.! and ~b! polarizability anisotropies ~a.u.! for TZVP-FIP ~1! and Sadlej (3) basis.

should be used. The same results were found for the other molecules in the above series. The influence of the grid accuracy was studied in a similar way. Test calculations have been performed using the above derived field strengths for the calculation of a, b and g varying the grid accuracy for the numerical integration from 1022 to 1026 . In Fig. 2 we present results for NH3 that are representative for the above set of molecules. The most sensible tensor components of b @Fig. 2~a!# and of g @Fig. 2~b!# are plotted against the grid accuracy. As can be seen from Fig. 2, the tensor components are converged for a grid accuracy of 1024 . Therefore a grid accuracy of 1024 and a field strength of 22.5 10 for the calculation of a and b and a field strength of 1022 for the calculation of g was chosen for all the following calculations. After the definition of the electric field strengths and the grid accuracy the influence of the basis set for the calculation of the polarizability tensor and of the polarizability anisotropy was investigated. For this study the experimental gasphase structures70–72 of NH3 , H2 O, HF, CO, NO and CO2

were used. For this series of molecules the polarizability and the polarizability anisotropy calculations using the TZVPFIP1 basis set and the Sadlej basis set were compared with experimental data. For all calculations the VWN functional42 was used. As already mentioned in Sec. II B, the underlying valence basis of the TZVP-FIP1 basis set was optimized for this level of theory. In Fig. 3 the calculated molecular polarizabilities @Fig. 3~a!# and polarizability anisotropies @Fig. 3~b!#, obtained with these two basis sets, are plotted against the experimental ones. The maximum error of the calculated average polarizabilities ¯a is 12.7% in the case of the Sadlej basis set and 7% in the case of the TZVP-FIP1 basis set. The corresponding average error is 5.5% for the Sadlej basis set and 2.6% for the TZVP-FIP1 basis set. This is an indication of the importance of a DFT optimized valence basis set for polarizability calculations. The comparison between experimental and calculated values of u D a u shows also that the average error is larger, if the Sadlej basis set is used. However the differences of the two basis sets are less pronounced in the calculation of u D a u . III. RESULTS AND DISCUSSION

The calculated dipole moments, average polarizabilities and polarizability anisotropies for the above-defined set of small molecules ~H2 O, NH3 , HF, H2 S and CO! are listed in Table II and compared with available experimental data and other theoretical results. For the reported DFT finite field calculations the parameters derived in Sec. II were used in combination with the TZVP-FIP1 basis. In all reported calculations the experimental gas-phase structures70–72 of these molecules were used. The dipole moments obtained with the correlated methods are in good agreement with the experimental data. The Hartree–Fock SCF dipole moments show serious errors, e.g., inversion of the sign for CO, with respect to the experimental ones. The comparison of the theoretical polarizabilities shows that the inclusion of electron correlation increases the calculated ¯a . The DFT ¯a values are usually slightly larger than the coupled-cluster single double triple @CCSD~T!# ones. The average deviation of the DFT

FIG. 2. Influence of the grid accuracy on the most sensitive components of the first ~a! and second ~b! hyperpolarizability tensor of NH3 . This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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7760



TABLE II. Dipole moments ~Debye!, polarizability components, average polarizabilities and polarizability anisotropies ~a.u.! of small molecules. Molecule Method H2 O

NH3

HF

ma

TABLE III. Components and average values of the first hyperpolarizabilities ~a.u.! of small molecules.

a xx

ayy

a zz

¯a

uDau

Molecule H2 O

VWN BLYP SCFb CCSD~T!b Expt.

1.88 1.83 1.98 1.83 1.85

10.32 10.44 9.16 10.02 10.31c

10.08 10.06 7.96 9.64 9.55c

10.21 10.27 8.52 9.73 9.91c

10.20 10.26 8.55 9.79 9.92c

0.20 0.32 1.05 0.34 0.66c


1.57 1.52 1.62 1.50 1.47

14.16 14.17 12.77 13.71 14.00d

14.16 14.17 12.77 13.71 14.00d

16.56 16.48 13.35 15.71 15.60d

14.96 14.94 12.96 14.38 14.53d

2.40 2.31 0.58 2.01 1.94e


1.81 1.76 1.92 1.78 1.82

5.58 5.61 4.51 5.34 5.03f

5.58 5.61 4.51 5.34 5.03f

6.65 6.78 5.76 6.44 6.51f

5.94 6.00 4.93 5.71 5.52f

1.07 1.16 1.25 1.10 1.62g


1.15 1.07 1.11 0.98 0.97

25.12 24.90 23.84 25.36 25.92h

24.70 24.53 23.76 24.35 25.20h

25.01 24.73 23.43 24.40 25.29h

24.95 24.72 23.68 24.70 25.47h

0.37 0.32 0.37 0.99 0.66i


0.24 0.19 20.25 0.15 0.11

12.25 12.25 11.11 11.73 12.21j

12.25 12.25 11.11 11.73 12.21j

15.65 15.79 14.47 15.65 15.70j

13.39 13.43 12.23 13.04 13.37j

3.40 3.54 3.37 3.92 3.59e

NH3

HF

H2 S

CO H2 S

CO

Experimental dipole moments ~Ref. 57!. Calculations ~Ref. 58!. c Measurement of the depolarization ratio for Rayleigh scattering from water vapor at 514.5 nm ~Ref. 59!. d Reference 60. e Depolarized light scattering (l 5 6328 Å! ~Ref. 61!. f Dielectric constant measurement on the liquid using the Lorenz-Lorentz relation ~Ref. 62!. g Molecular-beam electric-resonance spectrometry ~Ref. 63!. h Rotational Raman spectra 1 refractive index measurement 1 depolarized light scattering 1 Kerr effect measurement ~Ref. 64!. i Depolarized light scattering ~Ref. 65!. j Refractive index measurement ~Ref. 66!.

a

b

calculated mean polarizabilities from the experimental data is less than 5% and in the same range as for the CCSD~T! results. The Hartree–Fock SCF results show larger deviations from the experimental data as can be seen from Table II. The differences between local ~VWN! and gradientcorrected ~BLYP! results are small and unimportant for the prediction of qualitative trends. The calculated DFT anisotropies u D a u differ much more from the experimental data as the average polarizabilities. The situation is similar for the CCSD~T! calculations using considerably larger basis sets including lone-pair functions. Therefore the relative small size of the TZVP-FIP1 basis used in the DFT calculations is not the source of this disagreement. We believe that this discrepancy between theory and experiment is due to vibrational contributions that change u D a u considerably, but have only little effect on ¯a .73 In Table III the calculated hyperpolarizabilities for the same set of molecules are listed. For the calculation of b the z axis of the molecule was oriented along the permanent

Method

b xxz

b y yz

b zzz

¯b

¯b exp

VWN 26.12 213.42 217.80 222.41 222.060.9b BLYP 25.88 214.31 216.98 222.30 SCFa 21.40 29.40 27.10 210.80 CCSD~T!a 26.20 210.20 213.70 218.00 VWN 212.06 212.06 232.70 234.10 248.961.2b BLYP 213.07 213.07 232.00 234.87 SCFa 27.00 27.00 211.10 215.10 CCSD~T!a 28.80 28.80 239.60 234.30 VWN 22.12 22.12 211.72 29.57 211.061.0c BLYP 22.14 22.14 212.20 29.89 SCFa 20.28 20.28 28.40 25.38 CCSD~T!a 21.27 21.27 29.62 27.30 VWN 4.88 216.35 23.65 29.08 210.162.1b BLYP 3.30 216.76 22.85 29.79 SCFa 5.60 29.90 7.90 2.20 CCSD~T!a 24.40 29.50 1.10 27.70 VWN 5.27 5.27 27.59 22.88 30.263.2b BLYP 5.11 5.11 26.43 21.99 SCFa 3.50 3.50 28.20 21.10 CCSD~T!a 6.60 6.60 26.00 23.50

Calculations ~Ref. 58!. ESHG measurement ~l5694.3 nm! ~Ref. 68!. c ESHG measurement ~l5694.3 nm! ~Ref. 69!. a

b

dipole moment of the system. Therefore the calculated ¯b values can be directly compared with static electric fieldinduced second-harmonic generation measurements. However, the reported experimental data were not measured in the static limit which makes the quantitative comparison between the calculated and experimental data impossible. Nevertheless the qualitative trend of the experimental data is reproduced by the static DFT calculations. The quantitative comparison of the different theoretical methods shows a very satisfactory agreement between the DFT and CCSD~T! calculations. In contrast the Hartree–Fock SCF results show qualitative differences, e.g., inversion of the sign of ¯b for H2 S. This underlines the importance of the correlation contribution in the calculation of hyperpolarizabilities. Again gradient corrections have only minor effects on ¯b as can be seen from the comparison of the VWN and BLYP results in Table III. In Table IV we have listed the calculated second hyperpolarizabilities g for the above set of molecules. For the reported DFT calculations a field strength of 1022 a.u. was used as discussed in Sec. II C. In order to study the influence of the second-order FIP functions on ¯g we have performed local DFT calculations with the TZVP-FIP1 ~VWN/FIP1! and TZVP-FIP2 ~VWN/FIP2! basis sets. The comparison of these calculations shows that the second-order FIP functions increase the calculated ¯g values. In general the inclusion of these functions improves the correlation between the DFT results and the CCSD~T! ones considerably resulting in a satisfactory agreement of these two correlation methods. An exception is H2 S where the DFT ¯g is seriously underestimated compared to the CCSD~T! one. The reason for this underestimation is the large difference of the g xxxx component in the DFT and CCSD~T! calculation. At this point it



J. Chem. Phys., Vol. 109, No. 18, 8 November 1998 TABLE IV. Diagonal components and average values of the second hyperpolarizabilities ~a.u.! of small molecules. Molecule H2 O

NH3

HF

H2 S

CO

Method

g xxxx

gyyyy

VWN/FIP1 1 216.5 811.3 VWN/FIP2 3 118.6 1130.5 BLYP 3 222.5 1179.2 CCSD~T!a 2 900.0 820.0 VWN/FIP1 1 722.3 1722.3 VWN/FIP2 2 350.0 2350.0 BLYP 2 414.7 2414.7 CCSD~T!a 1 800.0 1800.0 VWN/FIP1 362.9 362.9 VWN/FIP2 756.4 756.4 BLYP 786.7 786.7 CCSD~T!a 650.0 650.0 VWN/FIP1 375.1 2001.6 VWN/FIP2 4 981.4 3254.9 BLYP 5 154.1 3274.3 CCSD~T!a 14 400.0 3400.0 VWN/FIP1 1 500.1 1500.1 VWN/FIP2 1 679.5 1 679.5 BLYP 1634.3 1634.3 CCSD~T!a 1 470.0 1470.0

g zzzz

¯g

¯g exp

1472.1 1790.2 1793.7 1500.0 3357.7 7666.0 7788.7 8200.0 445.8 484.3 498.0 390.0 2875.9 4397.2 4329.8 5100.0 2389.5 2474.4 2692.3 1880.0

1312.9 2118.4 2179.3 1800.0 3032.1 4594.6 4742.5 4200.0 396.9 665.5 689.6 560.0 2370.3 4884.7 4943.3 7900.0 1747.1 1880.8 1960.7 1590.0

2311.06120.0b

7761

TABLE V. Average polarizabilities, polarizability anisotropies and average second hyperpolarizabilities ~a.u.! of selected molecules using experimental structures. Calculation Molecule O2 NO N2 HCN CO2 CH4 CH3 F N2 O C2 H4 n-C3 H8 C6 H6

6147.06110.0b

842.06120.0c

10300.06260.0b

¯a 10.59 11.64 11.84 17.19 17.42 17.62 17.84 19.45 28.00 42.50 69.45

uDau

6.70 5.44 5.02 8.81 13.47 0.00 1.34 18.88 11.75 7.10 37.58

Experiment ¯g

908.4 1 908.3 1 018.8 2 840.2 1 425.8 3 538.5 3272.8 1 892.3 8 253.8 13 894.5 21 520.4

uDau

¯a a

10.66 11.74a 11.74e 17.48f 17.48h 17.54i 17.60j 19.77h 28.48k 42.09m 71.53a

¯g b

7.42 5.69b 4.70b 13.50g 14.17b 0.00 1.30j 19.97b 12.21l 37.92b

962.066.0c 2830.0684.0d 917.065.0c 1368.067.0d 3 257.0616.0d 2875.06230.0d 9120.06205.0d 10740.0654.0d 23900.06480.0d

Dielectric constant measurement ~Ref. 74!. Depolarized light scattering (l 5 6328 Å! ~Ref. 61!. c Static g ~Ref. 75!. d ESHG measurement (l 5 694.3 nm! ~Ref. 25!. e Reference 60. f Refractive index measurement ~Ref. 76!. g Kerr Effect ~Ref. 76!. h Reference 77. i Reference 65. j Reference 78. k Depolarized light scattering (l 5 5145 Å! ~Ref. 79!. l Depolarized light scattering (l 5 6328 Å! ~Ref. 66!. m Reference 80.

a

b

1720.0648.0

b

Calculations ~Ref. 58!. ESHG measurement (l5694.3 nm! ~Ref. 68!. c ESHG measurement (l5694.3 nm! ~Ref. 69!. a

b

must be noted that in the CCSD~T! calculation lone pair functions were used for the sulfur and that the reported CCSD~T! ¯g is too large.58 Nevertheless there is an underestimation of g xxxx in the DFT calculation that seems to be due to the description of the sulfur lone pairs in H2 S. This is supported by the large shift in g xxxx going from the TZVPFIP1 to the TZVP-FIP2 basis. The inclusion of f functions at the sulfur considerably improves the description of the fieldinduced charges at the sulfur and thus the g xxxx component. Therefore the true static ¯g of H2 S will be between the reported DFT TZVP-FIP2 and CCSD~T! values, considerably different from the reported experimental value at 694.3 nm. Even the DFT calculated ¯g values are static and no attempt was made to include rotational and vibrational contributions. The basic trend of the experimental data is very well reproduced with the TZVP-FIP2 basis considering that experimental ¯g values from different nonlinear optical processes can differ as much as 40%. The influence of gradient corrections is negligible for qualitative predictions. In conclusion the above comparison of a, b and g DFT calculations with CCSD~T! and experimental data shows that the obtained results with the described DFT finite field method are generally accurate within 10% for a, 20% for b and 30% for g. With this accuracy qualitative trends of the hyperpolarizabilities due to the change of the molecular and electronic structure of systems can be reliably predicted. In order to test the reliability for larger and more complicated structures we have calculated polarizabilities and second hyperpolarizabilities for 11 molecules with different electronic structure features. The calculations were performed as described in Sec. II using the TZVP-FIP1 basis for a and b and the TZVP-FIP2 basis for g calculations. The experimental gas-phase structures70–72 were employed for all

molecules. Because we already observed that gradient corrections are of minor influence for DFT a, b and g estimations the local VWN functional was used in all calculations. In Table V we compare our calculated results with available experimental data. The agreement between the calculated and measured mean polarizabilities is very good. This can be seen also from Fig. 4~a!, which graphically displays the correlation between experiment and theory. The analysis of the polarizability anisotropies shows one serious discrepancy between experiment and theory. We suggest for HCN a u D a u value around 8.8 a.u. that is quite independent from the theoretical method used. The difference to the experimental

FIG. 4. Correlation between calculated and experimental ~a! mean polarizabilities ~a.u.! and ~b! second hyperpolarizabilities ~103 a.u.! for the molecules listed in Table II, IV and V. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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TABLE VI. Average polarizabilities, polarizability anisotropies and average second hyperpolarizabilities ~a.u.! of aliphatic and aromatic hydrocarbons using optimized structures. Calculation Molecule n-C3 H8 n-C4 H10 n-C5 H12 n-C6 H14 n-C7 H16 C6 H5 NH2 C10H8 C14H10

¯a 43.69 56.34 69.49 82.77 96.36 83.77 122.24 171.68

uDau

6.73 13.09 18.75 26.08 33.79 51.98 93.59 86.47

Experiment ¯g 14 538.2 20 482.6

47 349.9 44 681.1

¯a

¯g a

42.09 54.07a 66.07a 79.51c 91.89c 77.81d 111.35d 171.42d

10 740.0640b 13 730.0670b

32 599.6d 61 941.5d

a

Reference 80. ESHG measurement (l5694.3 nm! ~Ref. 25!. c Reference 81. d Reference 24. b

value of 13.5 a.u. that we found to the best of our knowledge in the literature76 is one order of magnitude larger than for all other systems. We suggest therefore another experimental measurement of u D a u for HCN. The agreement between the calculated static and measured dynamic mean second hyperpolarizability is very satisfactory. Here one should keep in mind that ¯g values from different nonlinear optical process can differ by as much as 40% from each other. As Table V and Fig. 4~b! show the basic qualitative trend of the experimental ¯g values can be predicted with the static DFT calculations. It should be noted that also for molecules with more complicated electronic structures, such as O2 , NO, N2 O or C6 H6 , the calculated polarizabilities and hyperpolarizabilities are in the same error range as for simple molecules. This demonstrates the reliability of the presented DFT finite field method. In order to use this method for the design of molecules with large nonlinear optical properties reliable predictions of a, b and g for optimized molecular structures are necessary. In Table VI we compare our calculated mean polarizabilities and hyperpolarizabilities of optimized aliphatic and aromatic hydrocarbons with experimental data. The optimizations were performed as described in Sec. II B. The agreement between theory and experiment is for the optimized structured almost as good as for the experimental ones. This comparison demonstrates that also for optimized structures a reliable prediction of polarizabilities and hyperpolarizabilities is possible using the described DFT method. IV. CONCLUSIONS

In this paper we have reported calculations of dipole moments, molecular static polarizabilities, polarizability anisotropy, first hyperpolarizabilities and second hyperpolarizabilities calculated in the framework of density functional theory. The calculations have been performed applying the density functional method both at local and GGA level of theory. A representative set of 23 small and medium size molecules was investigated. We have demonstrated that the frequently discussed problems of numerical instabilities in


the calculation of a, b and g in density functional methods can be avoided using an adaptive grid procedure. The use of an adaptive grid generator that automatically adjusts the grid to the used basis sets and to the molecular structure increases the numerical stability of the finite field procedure substantially. A grid size of only 1024 used for the numerical integration was sufficient to produce numerically stable polarizabilities and hyperpolarizabilities. Because the evaluation of four-center Coulomb integrals is avoided in ALLCHEM a very efficient implementation of the finite field method originally described by Kurtz et al.41 is possible in this DFT program. Using this implementation more than 80% of the CPU time is spend in the numerical integration of the exchange-correlation potential that scales linearly with the size of the system. Therefore much larger systems than the examples presented here can be calculated with the described method. We have already calculated hyperpolarizabilities for molecules with more than 40 atoms on a small workstation. The presented studies show that DFT optimized basis sets, e.g., TZVP, augmented with first-order FIP functions are most appropriate for the determination of a and b with density functional methods. For the calculation of g with DFT methods second order FIP functions improve the results considerably. The influence of the auxiliary functions is less important and the relatively small A2 auxiliary function set is most adequate for polarizability and hyperpolarizability calculations as the presented results show. The influence of gradient-corrected functionals for the calculations of a, b and g is not important for qualitative trends. The calculated values of m , ¯a and u D a u are in good agreement with the available experimental data. The agreement between theory and experiment for ¯b and ¯g is satisfactory considering that the hyperpolarizabilities were calculated in the static limit without inclusion of rotational and vibrational contributions. The comparison with other theoretical methods shows that the presented DFT finite field method provides dipole moments, polarizabilities and hyperpolarizabilities considerably better than Hartree–Fock and almost as good as expensive correlation treatments. The only qualitative difference between calculated and experimental data was observed for u D a u of the HCN molecule. For the polarizability anisotropy of HCN we suggest a value around 8.8 a.u. compared with the experimental value of 13.5 a.u. in the literature.76 Another measurement is needed. In conclusion, the very encouraging results obtained in this work show that DFT methods can be used in order to predict fairly accurate values of a, b and g. The presented method opens the way for the design of new nonlinear optical materials using density functional methods. The small grid size requests for the numerical integration in combination with the efficient implementation in ALLCHEM allows the calculation of large systems. Therefore the presented method represents an attractive reliable alternative to the commonly used semiempirical programs for the prediction of molecular polarizabilities and hyperpolarizabilities.




ACKNOWLEDGMENTS

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