Ab initio analytic polarizability, first and second hyperpolarizabilities of large conjugated organic molecules: Applications to polyenes C4H6 to C22H24 Graham J. B. Hurst, Michel Dupuis, and Enrico Clementi Citation: J. Chem. Phys. 89, 385 (1988); doi: 10.1063/1.455480 View online: http://dx.doi.org/10.1063/1.455480 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v89/i1 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 22 Sep 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Transcript
Ab initio analytic polarizability, first and second hyperpolarizabilities of largeconjugated organic molecules: Applications to polyenes C4H6 to C22H24Graham J. B. Hurst, Michel Dupuis, and Enrico Clementi Citation: J. Chem. Phys. 89, 385 (1988); doi: 10.1063/1.455480 View online: http://dx.doi.org/10.1063/1.455480 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v89/i1 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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Ab initio analytic polarizability, first and second hyperpolarizabilities of large conjugated organic molecules: Applications to polyenes C4Ha to C22H24
Graham J. B. Hurst,a) Michel Dupuis, and Enrico Clementi IBM Corporation. Data Systems Division. Dept. 48B/MS 428. Kingston. New York 12401
(Received 28 December 1987; accepted 9 March 1988)
The static dipole polarizability and second hyperpolarizability tensors are calculated for polyene systems via ab initio coupled-perturbed Hartree-Fock theory. The effect of basis set augmentation on the calculated properties is explored for C4H6 and example basis sets are used to calculate the polarizability and second hyperpolarizability for the longer polyenes: C6Hs, CsHlo, C IOH I2, C 12H 14, CI4HI6,CI6HIS' C1sH 20, C2oH 22, C22H24. Results for the finite polyenes are extrapolated to predict the unit-cell polarizability and second hyperpolarizability of infinite polyacetylene. The working equations which take advantage of the 2n + 1 theorem of perturbation theory for calculating up to the second hyperpolarizability are given, and their implementation is briefly discussed. In particular it is shown that the implementation is readily amenable to parallel processing.
I. INTRODUCTION
Materials which exhibit strong nonlinear optical properties are of great importance. The expanding range of applications oflaser technology, from communications and medicine to the prospect of optical computers, has fostered a need for materials with specific optical properties. Materials with suitable nonlinear responses to incident light can be exploited to alter characteristics, such as the frequency, amplitude or phase, of the transmitted electromagnetic radiation. Thus these materials provide the means to tailor the nature oflight to suit particular applications.
The interaction of light, or other electromagnetic fields, with a molecule may polarize the charge distribution and alter the propagated field. The linear response to the field is described by the molecular polarizability, which determines the refractive index of a system. The nonlinear responses, which may be exploited in nonlinear optical devices, are described by the molecular hyperpolarizabilities. It is these molecular properties which we are interested in calculating here.
Conjugated organic molecules exhibit some of the largest nonlinear properties and have attracted much interest. 1-3 The anomalous second-order nonlinear responses are attributed to the motility of electrons in extended 1T-orbital systems, with longer conjugation lengths giving more pronounced nonlinear responses. The prototype extended 1T system is the infinite polyacetylene polymer (CH) 00 or its finite analogs called polyenes, H (CH) n H. Recent experiments4-6 have found that the second-order nonlinear response of polyacetylene is among the largest known.
In this report we consider a method for the ab initio calculation of molecular dipole polarizability and hyperpolarizabilities at the Hartree-Fock level, and apply it to polyene molecules in the series C4H6 to C22H24" The properties are evaluated by analytic expressions for electric field deriva-
tives7 of the self-consistent-field (SCF) energy.s The method follows the coupled-perturbed Hartree-Fock (CPHF) scheme adopted by many authors.9-IS In our implementation we obtain the 2n + 1 th energy derivatives from the nth wave function derivatives,I9-21 a strategy which results in improved accuracy and lessened computational efforts.
Applications of the method for the study of polarizability and hyperpolarizabilities of polyenes, and extrapolations to polyacetylene, are presented and analyzed. The choice of basis set is an important consideration for ab initio calculations of these properties.22 In general, energy -optimized basis sets need to be augmented with diffuse functions, for a better description of the wave function away from the nuclei, and with polarization functions, to describe distortions of the wave function. There is now a wide body of evidence that basis set selection is very crucial for small molecules.23-27
For large molecules it is believed that functions on other centers may lead to an adequate description of polarization effects associated with a given center, although computational evidence for this behavior has yet to be presented, owing to the numerically intense nature of such calculations for large systems. In the present work we consider several augmentations to a standard basis set for the short C4H6. Four of these basis sets have been used in calculations for longer polyenes. The properties show a marked dependence on conjugation length, and lessened sensitivity to basis with longer length. The smooth dependence of the polyene results on chain length has been fitted to a simple function which is then used to extrapolate the properties of infinite polyacetylene. Comparisons with experiment, where available, are given.
It is concluded that nonlinear optical properties may be successfully calculated by ab initio techniques, even for quite large molecular systems. We note however that in what follows we ignored the electron-electron correlation effects. There are indications that these effects may be somewhat larger for the higher order polarizabilities, as in the case of C4H6, e.g., below. Further studies of the magnitude of these effects are clearly in order.
J. Chern. Phys. 89 (1). 1 July 1988 0021-9606/88/130385-11$02.10 @ 1988 American Institute of Physics 385
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386 Hurst, Dupuis, and Clementi: Ab initio analytic polarizability
II. THEORETICAL CONSIDERATIONS
The energy of a molecule perturbed by a static, uniform electric field can be expanded7 as
E = E(O) - f.l}0) - ! aijFiFj - !{3ijkFiFjFk
- 14 r ijklFiFjFkF - "', (1)
where the subscripts, which identify the tensor components, are summed over the Cartesian axes x, y, and z. In the above equation E (0) is the unperturbed energy, Fi is the component of the field in the i direction, f.l(0) is the permanent dipole moment of the molecule, a is the static dipole polarizability tensor, and {3 and r are the first and second dipole hyperpolarizability tensors. These tensors describe the distortion of the molecule induced by the electric field. The elements of these tensors can thus be determined from derivatives of the energy with respect to electric field, either by finite-difference estimates of the partial derivatives, or from solving analytic derivative expressions. In this work we adopt the latter strategy.
A. Energy derivatives
In this section we summarize our working equations for completeness. The expression for the fourth derivative of the energy is the most complex one, and to the best of our knowledge has not been published previously in this form. For simplicity, our attention is confined to closed-shell systems described by the restricted Hartree-Fock formalism. The molecular SCF energy is given by
E = Vnue + 2I D.t (h., + F st ), st
(2)
where Vnue is the nuclear repulsion energy, sand t run over the basis functions, the density matrix elements D., are given by
occ
Dst = I C:Cti , (3) i
where i runs over the occupied molecular orbitals, h is the matrix of one-electron Hamiltonian integrals
h.t=(slh It) (4)
and Fis the Fock matrix, given by
F., = h.t + I Duv [2(st luv) - (sultv)], (5) uv
where Mulliken charge-cloud notation is used for the twoelectron integrals. The molecular orbital coefficients Csi are obtained by solving iteratively the Hartree-Fock equation
FC= see, (6)
subject to the usual orthonormality condition
CtSC= 1,
where S is the basis-function overlap matrix
Sst = (sit)
and € is the diagonal matrix of the eigenValues of F.
(7)
(8)
We now need expressions for the derivatives of the SCF energy with respect to electric field, up to the fourth derivative (to evaluate r). Pulay l4 has derived expressions for partial derivatives ofEq. (2), with respect to arbitrary param-
eters, up to the third derivative of the energy. For derivatives with respect to static electric field (or field derivatives), and assuming field-independent basis functions, several simplifications occur: the quantities Vnue , (st I uv) and S are constant with respect to electric field, and h does not have terms higher than first order with respect to electric field. 20 Using superscript letters to denote partial derivatives with respect to electric field, as in
EO=~E aF ' o
the analytic derivative expressions to fourth order are
In the above equations, 9t projects the real part of the expression, and the curly braces are used to abbreviate expressions that differ only by permutations of superscripts.
Equations (9)-(11) and (13) are equivalent to those used by Dykstra and Jasien l5 and Sekino and Bartlett. 16
With these equations, the nth derivative of the variational parameters is required for the (n + l)th energy derivative. However the situation for derivatives is analogous to that for standard perturbation theory: the nth wave function derivative can yield up to the (2n + l)th energy derivativel9
-21 as
in Eqs. (12) and (14). Equations (9), (10), and (12) are equivalent to those given by Lazzeretti and Zanasi. 27 The general equations of Pulay l4 and Gaw et al. 17 reduce to the above equations for derivatives with respect to electric fields.
To evaluate the above expressions, we need derivatives of D, F, C, and E. Differentiating Eqs. (3) and (5) with respect to electric field gives
occ
D:t = I (C:i*Cti + C:C~i)' (15) i
oce
D:tb = I (C:i~Cti + C:i*C~ + Ct;!c~ + C:C~ib), i
F:, = h:t + I D~v [st luv], uv
F:;= ID~:[st luv). uv
(16)
(17)
(18)
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Hurst, Dupuis, and Clementi: Ab initio analytic polarizability 387
Using the definitions
ca=cua,
C ab = cuab,
(19)
(20)
the wave function and eigenvalue derivatives are obtained through the CPHF equations
Uij = _1_ [ - €if + I C:F~tCtj] , Ej - E; st
(21)
Uijb = __ 1_ [ - €ifb + I C:F~:Ctj Ej - E; st
+ ~ ~ (C:F~tCtkUt + c:F~tCtkUkj)] (22)
resulting from differentiation of Eq. (6). A noncanonical treatment is adopted, where only the off-diagonal occupiedvirtual blocks of ~ and ~b are zero valued. The above expressions may thus be used for the occupied-virtual blocks of Ua
and Uab. The diagonal occupied--occupied and virtual-virtual blocks must satisfy derivatives of Eq. (7), so we have chosen
Uij=O, (23)
Uijb= -~ I (UkrU~j + U~U'tg), i,jeocc (24) 2 k
for the diagonal blocks. Substituting these into Eqs. (21)
and (22), respectively, determines the diagonal blocks of ~ and~b.
B. Implementation
The ab initio quantum chemistry program HOND028,29 has been extended to implement the above equations to calculate molecular dipole polarizability and first and second hyperpolarizabilities for closed-shell restricted Hartree-Fock wave functions. The program executes as follows:
E: The zeroth-order equations are solved in the usual manner.
p,: The dipole moment operator integrals are calculated and the components of the permanent dipole moment are evaluated via Eq. (9) where p,~0) = - Ea.
a: The first derivative of the wave function is solved iteratively (with optional damping), using the first-order CPHF equation (21). The elements of the dipole polarizability tensor are calculated via Eq. ( 10) where aab = - E abo
{3: The components of the dipole hyperpolarizability tensor are computed through Eq. (12), where Pabe = - E abe, using the first-derivative wave function from
above. r: The second-derivative wave function is solved iterati
vely (with optional damping), through the second-order CPHF equation (22). The dipole second hyperpolarizability tensor is evaluated with Eq. (14), where r abed = - E abed.
Iterative solutions for the CPHF equations were adopted so that large systems, up to several hundred basis functions, could be considered.
This implementation satisfies the 2n + 1 relation. For both the first and second hyperpolarizabilities, the order of
the required wave function derivatives is lower by one, compared to n + 1 schemes. This should give improved accuracy, because iterative solutions of the CPHF equations are generally not exact; thus additional sources of error are present in an n + 1 scheme. The magnitude of these errors will be reduced with CPHF solutions that are more tightly converged, but it seems preferable to avoid them.
The present scheme is also less computationally demanding than the n + 1 schemes. The iterative processes dominate the computation time required, and these are minimized in the current method; the hyperpolarizability is obtained for insignificant cost and the second hyperpolarizability is obtained for the same cost as for the first hyperpolarizability in an n + 1 scheme. This difference becomes even more pronounced with larger systems: for Nbasis functions, the time taken in the iterative processes is proportional to N\ while the tensor evaluations scale as N 3 (and take insignificant processor time) .
Another aspect of the computational savings is in the reduced dimensions, compared to the n + 1 schemes. For all the Cartesian components of the nth derivative of the wave function (n + 2) (n + 1) /2 iterative calculations are required. Thus the n + 1 schemes require three times as many iterative calculations (or three times as much storage if solved simultaneously) for the hyperpolarizability tensor and more than twice as many for the second hyperpolarizability tensor, compared to the current implementation.
The present implementation has also been incorporated into the parallel version of HONDO,29 which runs on the ICAP parallel computer systems developed by Clementi and co-workers.30 The parallelization strategy follows that used for the SCF calculation in HONDO.29 In a parallel HONDO job, the two-electron integral file is already partitioned among the attached processors (APs), so partial Fock matrices are constructed on each AP, using Eq. (5), and these are combined to form the full Fock matrix on the host processor. For parallelization of the CPHF equations, the derivative Fock matrices are formed in the same distributed manner, via Eqs. (17) and (18). In the program, the same parallel subroutine is used for the two-electron parts ofF, P, and Fob; all that differs is the density matrix passed to that subroutine. The other operations required in the iterative parts of the calculation, and in the evaluation of the tensor components, require insignificant time, so they are performed on the host processor.
III. APPLICATIONS
The static dipole polarizability and second hyperpolarizability tensors have been calculated for all-trans polyenes in the series C4H 6 to C22H24. This extends earlier theoretical work on these systems, undertaken in this laboratory, to study the structure, vibrational spectra, and electrical conduction of polyenes.31-34 The polyene geometries are those optimized31 with the 6-31G basis set,35 in C2h symmetry. The geometries change little when reoptimized with the polarized 6-31G** basis set,35 and they are in agreement with experimental structures. 31 Tests showed that using 6-31 G* * geometries did not significantly alter the calculated proper-
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388 Hurst, Dupuis, and Clementi: Ab initio analytic polarizability
ties so the 6-31 G geometries were used without reoptimization for all the basis sets employed here.
A right-handed inertial axis system is used to identify the tensor components. The inertial axes, in the order of increasing moments of inertia, are used for the Cartesian axes x, y, z. The x axis is in the molecular plane, in the chain direction (corresponding to the translation axis in polyacety lene ), the y axis is also in the U h plane (perpendicular to x), and the z axis is the twofold rotation axis. The origin is the center of symmetry for each polyene; the dipole and first hyperpolarizability tensors are thus zero and the polarizability and second hyperpolarizability tensors, respectively,have four and nine independent components,2<>t,ecause of the C2h
symmetry. All the basis sets used were based on the standard 6-31 G
basis, except for the STO-3G basis,36 but they differ in the extra functions added. Basis set comparisons were carried out for the C4H6 1,3-butadiene molecule, with the aim of determining a relatively compact basis, for the longer polyenes, that can yield quantitatively accurate second hyperpolarizabilities for at least the component in the chain direction. The first stage was to evaluate the improvement in the calculated properties with the addition of diffuse sand p shells on carbon and diffuse s shells on hydrogen. Once effective combinations of diffuse functions were established, the most promising (in terms of quality vs size) were augemented with higher angular momentum functions, in a systematic search for a compendious basis set for these property calculations.
TABLE I. Gaussian functions used to augment 6-31 G basis set for C4H6 •
Details of the augmentations to the 6-31G basis for C4H6 are given in Table I. Diffuse functions were generated in the usual way from the two most diffuse 6-31 G exponents: their ratio was used to make a geometric series. The polarization functions for K and L also form a geometric series, with emphasis on the diffuse polarization space; the largest exponents are taken from standard values for single polarization functions. The exponent for the single polarization function on each carbon in basis J was varied to determine which value gave the best agreement with the results obtained in the more saturated basis sets J and K, and with the experimental second hyperpolarizability.37 It was hoped that a small, augmented version of the 6-31 G basis could provide a good description of y, or at least y xxxx , and the basis chosen is labeled 6-31G + PD.
The polarizability and second hyperpolarizability tensors for series of polyenes were evaluated in four example basis sets. Our interest is primarily in the properties along the chain, so we desired to see how basis set effects varied with chain length. The STO-3G basis was not expected to give satisfactory results but was included for the sake of comparison. The standard 6-31 G and 6-31 G* bases were also used. The fourth basis set employed was the 6-31G + PD basis (in Table I); this basis has a diffuse p shell and a diffuse six-component Cartesian d shell on each carbon, in addition to the 6-31 G basis.
Ab initio calculations of polarizabilities and second hyperpolarizabilities have not previously been attempted for systems of the size considered here. The largest molecule
Exponents
Extra Carbon Hydrogen functions
Label carbon;hydrogen ;s ;p ;d ;s ;p
6-31G* Id; 0.8 A Is; 0.05 B Ip; 0.05 C Islp; 0.05 0.05 D Islp;ls 0.05 0.05 0.04 E 2s1p; 0.05 0.05
"Values of;d tried in basis J were: 0.8, 0.14, 0.1, 0.08, 0.06, 0.055, 0.05, 0.045, 0.04, and 0.02.
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Hurst, Dupuis, and Clementi: Ab initio analytic polarizability 389
here is C22H24, where all 46 atoms are described by a doublezeta-valence basis set for a total 246 contracted basis functions in 114 shells. The largest calculation, C I6H I8 in the 6-31 G + PO basis, involved 324 contracted basis functions in 116 shells.
The calculations were performed on the ICAP-l and ICAP-2 parallel computer systems.30 Integrals less than 10-9 a.u. were discarded and integral compression38 was used for the larger calculations (affecting the properties by less than 0.1 %) to reduce disc space and elapsed times. To avoid linear dependences, orbitals with overlap eigenvalues less than 10-5 were discarded. The SCF calculations were considered to be converged when no element in the density matrix changed by more than 10-6 a.u. between iterations. The CPHF iterations continued until the elements of the U matrices changed by less than 10-4 a.u. between iterations. Using seven FPS-I64 nodes of the ICAP-l computer, the calculation for C I6H I8 in the 6-31 G + PO basis required approximately 10 h elapsed time for the integrals and SCF, 16 h for the polarizability tensor, and 25 h for the second hyperpolarizability tensor. Symmetry was not exploited, so the calculation used nearly 300 million integrals (which the packing routines compressed by a factor of 0.4).
IV. RESULTS AND DISCUSSION
A. Basis augmentation
The SCF energy, polarizability tensor, and second hyperpolarizability tensor for C4H6, with different basis sets, are given in Tables II and III.
Table II gives the results for augmenting the 6-31 G basis with diffuse functions (and no higher angular momentum, or polarization, functions). For the components of a, the diffuse functions only have a pronounced effect for au, the out-of-plane component. The largest changes occur for the
addition of a single diffuse p shell on each carbon (basis B); the value of azz doubles while the other components increase by 5%-15%. The first diffuse s shell on each carbon also significantly increases au, but the effect with basis C is less than additive compared to the change observed with the diffuse s or p shells alone. The addition of further diffuse functions to the sand p space of carbon, or the s space of hydrogen, has little effect.
The experimental value39 ora has been determined to be 58.3 e2a~E h- 1 (where 7i is the scalar part of a, given in Table II). The 6-31G value is 27% less than this, but the addition of a diffuse sp shell on each carbon raises 7i to within 12% of the experimental value. This agreement is rather good, considering the lack of polarization functions. Further diffuse functions do not significantly improve the agreement.
A good description of r is much more dependent on the presence of diffuse functions than a. The results in Table II show that the greatest changes in r occur when a diffuse p shell is added to each carbon. Again the out-of-plane component, r zzzz in this case, shows the largest relative change (two orders of magnitude), though r ==' r xxzz' and ryyyy are also altered significantly. The component in the chain direction (r xxxx) where our ultimate interest lies, is trebled upon inclusion of a diffuse p shell on each carbon; this augmentation to the 6-31 G basis proved more important than any other for r xxxx' Although the relative size of the components of r is very sensitive to basis, r xxxx is consistently the largest.
The diffuse s shells are also significant. Augmentation of the 6-31 G basis by a diffuse s shell on each carbon, as in basis A, increases rxxu by an order of magnitude, but in concert with the diffuse p shell, in basis C, r zzzz is increased much more than the sum of the separate increases observed with bases A and B. The further addition of a diffuse s shell on each hydrogen, to give basis 0, increases ryyyy ' rzzzz' and
rxxzz'
TABLE II. Results for C4H6 with 6-31G basis augmented by ditruse functions. Basis labels as in Table I. Atomic units are used. The energies, in E., are subtracted from 154 E •. For a I a.u.=e2cloE hi:::: + 1.6488 X 1O-41 C2 m2 J- 1
, for y 1 a.u.=e4a~E.- 3::::6.2360X 1O-6s~ m4 J-3::::5.0366X 10-40 esu.
Orientationaily averaged values are a = 1/3(axx + ayy + au) and r = 1/5(y== + yyyyy + yuzz + 2yxxyy + 2yxxu + 2yyyu )'
A B C D E F G H I 6-31G IS; Ip; Islp; Is1p;ls 2s1p; Is2p; Is2p;ls 2s2p;ls 2s2p;2s
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390 Hurst, Dupuis, and Clementi: Ab initio analytic polarizability
TABLE III. Results for C4H6 with 6-310 basis augmented by polarization and diffuse functions. Basis labels as in Table I, except that 6-310 + PD is abbreviated to + PD. For basis sets with a single d shell on each carbon the exponent is indicated. Values are in atomic units, as in Table II.
The second set of diffuse (nonpolarization) functions is not as important as the first. The most significant change in r is with the second diffuse p shell on each carbon; comparing results with bases D and G, the out-of-plane component increases by 10% while rxxxx and rxxzz both increase by roughly 5%.
Basis sets D and G thus offer the best compromises between quality and size for augmentation with diffuse functions, so we decided to test the effect of higher angular momentum functions with these two supplemented 6-31 G basis sets. Basis D, with single diffuse shells on each atom, was augmented with a single d shell to form basis J, and basis G was supplemented with several polarization functions to form bases K and L. The polarization functions for K and L were chosen to approach saturation of the diffuse polarization space; their results are used here to assess basis set convergence of the properties. The exponent of the d shell in basis J was varied to see which gave the best agreement with the results in bases J and K and with experiment.
Selected results with the polarized basis sets are given in Table III. Values forbasisJ withtd = 0.1, 0.08, 0.055, 0.045 are omitted for brevity; these values are consistent with those listed in the table.
The standard 6-31G* d shell is not very effective for improving the calculated properties. Comparing the 6-31G and 6-31 G* results, a is only increased by 1 % and :y is decreased by 4%. The same exponent is also ineffective in improving basis D; both a and :y are increased by less than 1 % on going from basis D to basis J with td = 0.8.
Other values for the d-shell exponent have a greater effect on the polarizability and second hyperpolarizability with basis J. The polarizability changes most on going from td = 0.8 to td = 0.14; a xx increases by 2% and a yy and azz
increase by 8%. Further lowering of td slightly reverses these changes, but the choice of exponent is not critical for a.
For td = 0.14, a is 6% less than the experimental value of 58.3 e-a~E h- 1,39 for td = 0.05 itis 8% less. The extra polarization functions in bases K and L improve agreement to 5.2% and 4.4% less than the experimental a; further improvement probably requires inclusion of electron correlation effects. For all basis sets, axx is the largest component of a.
The components of r are very dependent on the value of the d-shell exponent in basis J. The values of:Y, :y xxxx' r xxyy'
r xxzz' ryyyy , and r J!J!ZZ are largest for td = 0.05. The only other component of r, r zzzz' is highest for the lowest exponent used: td = 0.02. For the components which are zero upon rotational averaging, r xxxy varies smoothly with exponent (with its greatest value at td = 0.04), but both rxyyy and rxyzz are small and vary erratically with td' The latter two components are thus quite sensitive to basis; a better description of the polarization space, including higher angular momentum functions, is probably required. Since r xxxy' r xyyy'
and r xyzz have no contribution to experimetnal :y (or r xxxx )
values, they will be given little consideration here. For all basis sets, rxxxx is the largest component of r.
Comparison with experiment is complicated by the fact that the properties are measured at the macroscopic level with oscillating fields. The measured property is thus the frequency-dependent third-order electric susceptibility tensorx(3)( -W;W I,W2,W3), wherew = WI + W2 + W3. Thetemperature-independent part is related to the frequency-dependent second hyperpolarizability r( - W;W I'W2'W3) by40
where N is the number of molecules per unit volume for volume susceptibilities (or N = 1 for molecular susceptibilities), andj(w), j(wl ), j(W2)' and j(w3), describe local
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Hurst, Dupuis, and Clementi: Ab initio analytic polarizability 391
field effects. The local field effects can be modeled41 but they are often approximated by unity.42,43 For isotropic media the Lorentz correction isf(llJ) = (1/; + 2)/3 where 1/", is the refractive index. In the limit of zero frequency, the frequency-dependent properties collapse to the static properties, but with optical frequencies they are larger than the static values in the order44.45
y(static) <r(Kerr) <y(dcSHG) <y(FWM) <r(THG)
due to frequency dispersion and vibrational effects. The above abbreviations, respectively, indicate the average sound hyperpolarizabilities for: static fields, y=y(O;O,O,O); Kerr effect, y( - llJ;llJ,O,O); dc field-induced second harmonic generation, y( - 2llJ;llJ,llJ,O); four wave mixing, y( - llJ3;llJ 1,llJ1, - llJ2) where llJ3 = 2llJ 1 - llJ2; and third harmonic generation, y( - 3llJ;llJ,llJ,llJ). These factors must be kept in mind when comparing the values of Y calculated here with experimental values.
An approximate comparison with experiment for C4H6
is possible. The experimental value of the temperature-independent, gas-phase molecular third-order susceptibility is i(3) = 2.30 ± 0.13 X 10-36 esu for dcSHG with an optical field of wavelength 694 nm.37 This corresponds to an average molecular second hyperpolarizability of y(dcSHG) = 27400 ± 1550 e2a~E;; I, ignoring local field effects. The
value of y calculated for basis J with td = 0.05 recovers 56% ± 3% of this dcSHG value, and with bases K and L 51 % ± 3% is recovered.
The basis chosen for the longer polyenes, denoted 6-31G + PD, is similar to basis J with td = 0.05, except that the diffuse s shells on carbon and hydrogen are omitted. For the calculations with basis J having td = 0.055, 0.05, 0.045, some orbitals were discarded because of near-linear dependencies between the diffuse carbon s shell and the spherical part of the six-component d shell. The carbon s shell thus seemed unimportant with td = 0.05; its omission from basis J affected y by less than 0.1 % and its components by less than 0.7% (except for a 1.9% change for yyyzz)' The further omission of the diffuse hydrogen s shells, to give the 6-31G + PD basis, decreases Yxxzz by 6% and Yyyzz by 16%, but the other components of y change less than 1 %. Thus the 6-31 G + PD provides a good compromise between quality and size,so it was chosen for the longer polyene calculations. However, y with the 6-31 G + PD basis is larger than with the more complete K and L bases, so the better agreement with experiment suggests some cancellation of errors. Thus the second hyperpolarizabilities calculated within the 6-31 G + PD basis may overestimate results with better basis sets, but they will be superior to results obtained with 6-31 G or 6-31G* basis sets.
B. Polyene series
Tables IV and V, respectively, present the scalar polarizability and second hyperpolarizability values for the polyene molecules in the STO-3G, 6-31G, 6-31G*, and 6-31 G + PD basis sets. Comparable experimental values are also included.
The basis set effects for a and y with the longer polyenes are less pronounced than those observed with C4H6 • The
TABLE IV. Values ofa for polyene molecules. Values are in atomic units, as in Table II.
STO-3G basis gives poor values and the 6-31G + PD basis yields significantly better results than the 6-31 G or 6-31 G* basis sets. The relative differences between a values with the four basis sets diminish somewhat with chain length; for C4H6 the ratio STO-3G:6-31G:6-31G*:6-31G + PD is 0.60:1.00:1.01:1.25, for CI6H 1g it is 0.64:1.00:1.00:1.12. For y, the change of basis effects with chain length is more significant than for a; the ratios between y values with the four bases are 0.46:1.00:0.96:13.97 for C4H6 and 0.48:1.00:0.91:1.21 for C16H 1g. Thus the basis augmentations in the 6-31 G + PD basis are much less important with increasing chain length, while the diminution of y from the 6-31G* d shell is slightly increased.
Comparison with experiment is limited by the lack of experimental data. The data for C4H6 was compared in Sec. IV A, where the agreement with experiment for a was better than for y; the 6-31G + PD basis recovers 91% and 54%, respectively. For y of C6Hg, 39% is recovered with the 6-31 G + PD basis, though there is more uncertainty in the experimental value; for C6Hg the experimental error estimate is ± 10% compared to ± 6% for C4H6• The C6Hg sample also contained from 10%-40% of the cis form. In the absence of more experimental data it is difficult to assess the accuracy of the present method, though the lessening ofbasis effects with chain length suggests that the theoretical results are more reliable for the longer systems. The results with the STO-3G and 6-31 G basis sets for C4H6, C6Hg, and
TABLE V. Values of rfor polyene molecules. Values are in atomic units, as in Table II.
C.H6 C6HS
CsHIO C,oH'2 C'2H,. C,.H'6 C,~,s C,sH20 C2oH22
C22H24
·Reference 37.
ST0-30
SOO S444
22500 61784
132416 240044 392074 S76363 804 877
1 O4S 129
6-310 6-310* 6-310 + PD
1098 9878
4077S 114624 253843 476398 808879
1230311 1780479 2380428
lOSS 9196
37637 lOS 242 231979 433 S15 732489
14846 3S 118 82212
178443 34S 721 603 S37 976279
Expt.
27400" 89700"
J. Chern. Phys., Vol. 89, No.1, 1 July 1988
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392 Hurst. Dupuis, and Clementi: Ab initio analytic polarizability
CsHIO are similar to those reported by Andre et al.,4b from finite-field and sum-over-states ab initio calculations of energy derivatives with respect to electric field. In ab initio calculations for other molecules, similar proportions of experimental r values are recovered 14,2,,4,,47; the reader is referred elsewhere for a detailed discussion.47
The scalar and longitudinal components of a and rare plotted in Fig. 1 for different chain lengths. The values have been divided by the number of repeating units N (more precisely the number of double bonds) so that they may be related to the unit-cell properties of polyacetylene as N approaches infinity.
Again we see that the basis-set effects are more important for the scalar parts than for the longitudinal components. The polarizability results in the STO-30 basis, although far too small, do seem to exhibit the same N dependence as a and axx with the better bases, as pointed out by Andre et al.46 However, for rand r xxxx' the STO-30 basis values do not exhibit the same N dependence at larger N. The ratio between r results with the 6-310 and 6-310 + PO basis sets decreases with increasing N; the extra diffuse functions which were crucial in the description of r for C4H6
quickly lose relative importance with N. Thus basis augmentation is not critical for r in the longer polyenes, but basis quality is still important; the values and trends obtained within a minimal basis set are not quantitatively reliable.
1.7
----,----~ 1.5
10 .... X
~ ]' 1.3
1.1
2_0
I.f
(a.)
""",-----------
N
(b)
N
---
6-31G+PD 6-31G 6-31G. STO-3G
10
6-31G+PD 6-31G 6-31G. STO-3G
10
The N dependence of the properties is of interest. For conjugated carbon chains without bond alternation, at infinitely large N Huckel theory4S predicts a xx a:N 3
, and rxxxx a: N' is predicted by the free electron model.49 Recent Pariser-Parr-Pople results show an average dependence of r xxxx a: N 4.25 for the polyenes C4H6 to C2oH22' so The limiting relation r xxxx ex: N is obtained from Huckel theory for chains with bond alternation where the delocalization length is less than the chain lengthS 1 and for finite, idealized polyene chains.s2 Thus therelationsaxx ex:N 3 and rxxxx a:N s maybe exhibited for some regions of N, but axxlN and rxxxxlN should asymptotically approach finite values at large N.
The N dependencies ofaxx and r xxxx in the present results do vary with N and with basis set. For the best basis set (6-310 + PO), a xx a:N1.S1 between N = 2 and 3, increasing to N 1.61 for N - 6, then varying with decreasing exponent. The proportionality with the 6-310 basis is similar for large N, going to a xx ex: N 1.40 for N - 11. The N dependence of r xxxx increases from N 3.67, at N - 2, to N 3.98 at N - 6, then decreases, approaching the 6-310 dependence with greater N. With the 6-310 basis r xxxx the proportionality exponent drops sharply with increasing N; between N = 2 and 3 r xxxx
a: N 5.00 but this reduces to N 3.04 from N = 10 to 11. Thus the decreasing N dependence at the largest N values suggests that axxlN and rxxxxlN are approaching asymptotic values.
1.$
~
~u I/"- ....
X
.~
1 u
2.5
1.0
:iu " .. : .... ~x
~ .2" f.O
3.0
(e)
N
(d)
N
---
6-31G+PD 6-31G 6-31G. STO-3G
10
6-31G+PD 6-31G 6-31G. STO-3G
10
FIG. L Plots ofbase 10 logarithms ofa. a"". r. and y"""" (over Nand 1 a.u.) vs N. N is the number of double bonds (approximately the number of repeating units); (a) logal(N Xl a.u_) vs N; (b) log a""/(N X 1 a.u.) vs N; (c) log r/(N X 1 a.u.) vs N; (d) log Y""""/(N X 1 a.u.) vs N.
J. Chem. Phys., Vol. 89, No. 1,1 July 1988
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Hurst, Dupuis, and Clementi: Ab initio analytic polarizability 393
The recent Pariser-Parr-Pople results of de Melo and Silbey50 show a similar N dependence. Their log r xxxx vs log N plot shows curvature, indicating that the N dependence of r xxxx is decreasing with larger N. Their fitted slope of 4.25 falls between the present 6-31 G and 6-31 G + PD results fitted over the same range. However their results for the other components of r are at odds with the ab initio values calculated here. We do not find that r xxxy changes sign with larger chain length or that the other components are negative, rather all components of yare positive, except r xxyy' and the signs of r xxxy' r xyyy' and r xyyz depend on the coordinate system and, for the latter two, the basis set. The component r xxyy is negative with the smaller basis sets, but with the 6-31 G + PD basis it is positive for the shorter polyenes, and it becomes negative with increasing N.
The limiting values for the polyene properties have been estimated by extrapolation. Assuming asymptotic limits as N approaches infinity, the values ofalN, axxlN, YiN, and axxxxlN have been fitted (by least squares) to
b c 10gA(N) = a + N + N 2 ' (26)
where A (N) is the function and a, b, c are the fitting parameters. The extrapolated values for infinite polyacetylene are thus
lim A(N) = lOa. (27) N-eo
The fitting parameters and extrapolated values for the different basis sets are listed in Table VI.
The extrapolated polyacetylene second hyperpolarizabilities may be converted to third-order electric susceptibilities via Eq. (25). The resulting susceptibilities, ignoring local field effects and using the polyacetylene mass density of 1.2 g cm- 3 53 to determine the number of C2H2 units per volume, are given in Table VII, with experimental values from THG experiments. The values are not strictly comparable, because these experimental values contain temperature-dependent contributions, and the local field effects are
TABLE VII. Estimated values of static i 31, using Eq. (25) and experimen
tal values offrequency dependent i 31• Values are in esu cm- 3
likely to be important for the condensed phases considered here. Also, frequency effects for TH G are large (particularly at resonant frequencies) so the discrepancy with experiment should be greater than was noted for the earlier comparison with temperature-independent gas-phase dcSHG molecular susceptibilities. Sources of error in the theoretical values include the absence of electron correlation, basis set deficiencies, the extrapolation procedure, and the conversion to volume susceptibilities. Of these, the latter two probably introduce the most uncertainty in the theoretical values, and the experimental values have been estimated to be accurate within a factor of2.4 The present estimates of X~ are within an order of magnitude of the experimental nonresonant value, which seems reasonable given the numerous approximations involved. The relative consistency of the extrapolated theoretical values suggests that trends in i 3
) may be accurately predicted by the present approach, even though the values are underestimated.
v. CONCLUSIONS
In this report we have outlined a method for the ab initio calculation of molecular dipole polarizability and hyperpo-
TABLE VI. Fitting parameters, for Eq. (26), and extrapolated values for N approaching infinity. a, b, and care dimensionless and the function values are in atomic units, as in Table II.
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394 Hurst, Dupuis, and Clementi: Ab initio analytic polarizability
larizabilities which we have successfully applied to polyene molecules in the series C4H6 to C22H24, and the results have been extrapolated to give predictions for polyacetylene.
The computation scheme is efficient and, with the ICAP parallel supercomputer, large molecular systems may be considered. The equations satisfy the relation that the nth derivative of the variational parameters is sufficient to determine the 2n + lth derivative of the energy. This scheme has been implemented into HONDO and may run in parallel or sequential mode. The principle advantages of this program, compared to programs which only exploit the n + 1 relation, are reduced sources of error and significantly lessened computational effort. The program was used to calculate polarizabilities and second hyperpolarizabilities for polyene systems with up to fifty atoms and several hundred basis functions, and larger calculations are possible.
The basis set dependence of the calculated properties has been investigated. For short polyenes, such as C4H6, extra diffuse functions and diffuse polarization functions are crucial for describing the second hyperpolarizability; their inclusion can increase y by an order of magnitude. As the polyene length increases, basis augmentation becomes less important, particularly for the component of y in the chain direction.
The calculated values ofa are in good agreement with experiment with the better basis sets, and the agreement of the static y values with dynamic values from experiment is in line with other ab initio calculations; the theoretical values are smaller, by up to a factor of 3. Theoretical predictions of a and y have been given for polyenes for which experimental results are not available, up to C22HW
The longitudinal polarizability and second hyperpolarizability increase with chain length. The N dependence varies withN. For the shorterpolyenes a xx cx:Nl.6 and Yxxxx cx:N
4.0
,
but these proportionalities taper off with larger N, presumably to linear relationships. The longitudinal components are largest in both a and y, reflecting the strong response of the delocalized 11" electrons to electric fields. Functions of N have been fitted to a, a xx , y, and Yxxxx' and these may be used to predict the values for longer polyenes.
The finite polyene results have been extrapolated to predict properties of poly acetylene. For the second hyperpolarizability, an approximate conversion to third-order electric susceptibilities (ignoring bulk effects, temperature, and frequency dependencies) gives values within an order of magnitude of experimental results. Although these predictions are quantitatively inaccurate for predicting high-temperature, frequency-dependent electric susceptibilities of condensed phases, they may prove useful in predicting relative nonlinear properties for different systems.
This work represents a first step in the ab initio prediction of nonlinear optical properties for organic systems with extended conjugation of 11" electrons. Results for polydiacetylene have also been obtained and work is proceeding in this laboratory on other promising organic molecules.
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Hurst, Dupuis, and Clementi: Ab initio analytic polarizability 395
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