Basis set and electron correlation effects on initial convergence for vibrational nonlinear optical properties of conjugated organic molecules

Using three typicalp-conjugated molecules~1,3,5-hexatriene, 1-formyl-6-hydroxyhexa-1,3,5triene, and 1,1-diamino-6,6-dinitrohexa-1,3,5-triene ! we investigate the level of ab initio theory necessary to produce reliable values for linear and nonlinear optical properties, with emphasis on the vibrational contributions that are known to be important or potentially important. These calculations are made feasible by employing field-induced coordinates in combination with a finite field procedure. For many, but not all, purposes the MP2/6-31 1G(d) level is adequate. Based on our results the convergence of the usual perturbation treatment for vibrational anharmonicity was examined. Although this treatment is initially convergent in most circumstances, a problematic situation has been identified. ©2004 American Institute of Physics. @DOI: 10.1063/1.1667465 #


I. INTRODUCTION
Materials that exhibit strong nonlinear optical ͑NLO͒ properties are of great importance due to potential applications in communications, medicine, optical computers, and holography.At the molecular level the NLO properties are determined by first and second hyperpolarizabilities.Organic oligomers formed from quasi-one-dimensional -conjugated organic chains, which are capped with a donor group on one end and an acceptor group on the other, often have large hyperpolarizabilities and are of practical interest for that reason. 1,2sing a clamped nucleus approximation 3 the hyperpolarizability can be decomposed into electronic and vibrational contributions.The latter are often as large, or even larger, than their electronic counterparts. 4 -102][13] Within the Bishop-Kirtman ͑BK͒ method the vibrational hyperpolarizabilities may be separated into pure vibrational ( P v ) and zero-point vibrational average ( P zpva ) components.The P zpva component is defined, as usual, to be the difference between the averaged electronic property in the vibrational ground state and the electronic property evaluated at the equilibrium geometry, whereas P v includes all other vibrational effects.Based on their power series expansion Bishop and Kirtman organized the expressions for P v and P zpva into ''square bracket''-type terms, which denote the electrical properties involved and the total order of electrical and mechanical anharmonicity.Successively higher-order anharmonic terms involve successively higher-order derivatives ͑or combinations of derivatives͒ of the electrical properties and the pure vibrational potential energy with respect to normal coordinates.
The bottleneck in computing BK vibrational hyperpolarizabilities is the calculation of the higher-order derivatives.In fact, it is computationally prohibitive for medium/large organic molecules to go beyond the lowest-order square bracket terms using flexible basis sets and/or accurate ab initio techniques.Fortunately, there exists an alternative approach that avoids explicit determination of the derivatives. 14 -17It can be shown that the pure vibrational contribution is given exactly by the sum of a so-called nuclear relaxation ͑NR͒ term P nr and a so-called C-ZPVA term, P c-zpva : 15 P v ϭ P nr ϩ P c-zpva , ͑1͒ where P nr contains the lowest order nonvanishing anharmonicity contributions of each square bracket type and P c-zpva includes all higher-order anharmonicity terms ͑hence all the higher-order derivatives͒.The alternative methodology, known as the finite field ͑FF͒ method, involves carrying out a geometry optimization for the molecule in the presence of a static electric field.By numerical differentiation of the field-dependent change in the electronic properties due to the shift in equilibrium geometry one can obtain an accurate approximation ͑see below͒ for P nr .In an analogous manner the change in P zpva yields an accurate approximation for P c-zpva .It is of interest at this point to describe the relationship between the various molecular vibrational contributions defined above and the corresponding quantities that occur in the modern solid-state physics literature of nonlinear static field and dynamic electro-optical properties.In the latter discipline it is usual to invoke a classical approximation, which neglects the zero-point vibrational averaging term P zpva .Then the remaining pure vibrational contribution is factored a͒ Author to whom correspondence should be addressed.Electronic mail: josepm.luis@udg.esinto an ionic term and a piezoelectric term. 18The piezoelectric term is associated with strain and accounts for the acoustic modes that have zero frequency in the infinite periodic solid.In a finite molecule, the analogous modes have nonvanishing frequencies and are treated in the same manner as all other vibrations ͑a detailed analysis of the role of acoustic modes in molecular calculations has been given elsewhere 19 ͒.Although, in principle, the total pure vibrational contribution in periodic solids is included in the sum of the piezoelectric and ionic terms, practical calculations are currently limited to the nuclear relaxation component.This is primarily because of the difficulties in obtaining the higherorder derivatives. 20t can be foreseen that advances made in molecular calculations, particularly with respect to FF treatments, will be transferred to solid-state physics in the not-too-distant future.2][23] Furthermore, the nonclassical ZPVA computation can be made practical by taking advantage of the methodology described in Sec.II of this paper ͓see, particularly, Eq. ͑5͔͒.Although the C-ZPVA contribution may be obtained from a finite field treatment of the ZPVA term, it is important to bear in mind that the former ͑i.e., C-ZPVA) can be quite significant even when the latter is not as we will see in the current paper.For both ZPVA and C-ZPVA the leading term depends inversely on the square root of the mass, which tends to make the contribution smaller in solidstate oxides, for example, than it is in -conjugated organic molecules that contain hydrogen.However, anharmonicity parameters can easily play a more decisive role than the mass effect and thus the importance of the ZPVA and C-ZPVA contributions needs to be investigated for both types of systems.Finally, there is an important distinction between quasilinear organic molecules and three-dimensional ͑3D͒ solids regarding the electronic structure treatment that is employed.Whereas density-functional theory ͑DFT͒ is the method of choice for 3D solids, it has been shown that conventional functionals lead to a dramatic overshoot of the ͑hyper͒polarizabilitities calculated for quasilinear organic molecules. 24,25here have recently been two new formulations of DFT that report a major improvement in the case of the linear polarizability, 26,27 which leads to some optimism that a reasonably successful treatment of hyperpolarizabilities might not be too far in the future.
Focusing again on medium size organic molecules, we note that the FF method demands substantially less computational resources than are required to calculate the higherorder derivatives that appear in the analytical expressions for P nr and, especially, P c-zpva . 28 -30There is a price to pay in accuracy, although it is ordinarily a small price.This is due to the fact that the FF values pertain to the case where all optical frequencies are infinite ͑i.e., the infinite frequency approximation͒ or only a static field is present.For typical laser optical frequencies it has been shown [31][32][33] that the infinite frequency approximation does not lead to a significant loss of accuracy.However, the error can increase rapidly when the optical frequencies approach the IR region. 31t is well known that the perturbation series for the en-ergy of a harmonic oscillator perturbed by cubic ͑quartic,...͒ terms will eventually begin to diverge at some order.Similarly it can be expected that the BK treatment will also fail since it is based on a power series expansion in the normal coordinates.For weakly bound systems, such as HF 34,35 or H 2 O 34 dimers, calculations reveal that the perturbation series diverges immediately, which means that this treatment cannot be used in its current form.On the other hand, for typical -conjugated NLO molecules, it is usually found or assumed that the perturbation series is, at the least, initially convergent. 2 For a few -conjugated molecules it turns out that the first-and second-order anharmonic terms make larger contributions to P nr than the harmonic terms. 10,29,36owever, P nr terminates at second order.In order to monitor the convergence of P v therefore it was suggested that the behavior of two distinct sequences-which add up to the total-should be separately examined: ͑ A͒ P e , ͓ P zpva ͔ I , ͓ P zpva ͔ III ,..., ͑ B͒ P nr , P ͑ c-zpva͒͑I͒ , P ͑ c-zpva͒͑III͒ ,... .
In the ͑A͒ series the Roman superscript refers to the total order in anharmonicity and the absence of even order reflects the fact that such terms vanish exactly.P (c-zpva͒͑I) is derived from ͓ P zpva ͔ I and contains the second lowest-order anharmonicity terms of each square bracket type in P v ; P (c-zpva͒͑III) is derived from ͓ P zpva ͔ III and contains the third lowest-order BK terms of each square bracket type of P v , etc.For typical -conjugated NLO molecules, ͓ P zpva ͔ I is usually small in comparison with P e .In some instances it is over two-thirds as large 9,29 but the sequence ͑A͒ has always been found thus far to be initially convergent.However, in a previous investigation of seven typical medium size NLO molecules this was not always true for sequence ͑B͒. 29For infinitefrequency approximation dynamic hyperpolarizabilities the ratio P (c-zpva͒͑I) / P nr was never greater than 0.68, but for static hyperpolarizabilities it was sometimes larger than unity.In the study just described we obtained very substantial differences between Hartree-Fock ͑HF͒/6-31G and secondorder Møller-Plesset perturbation theory MP2/6-31G results, not only for the individual values of the electronic and vibrational hyperpolarizabilities, but also for their relative values.The most pronounced differences were found for ͓ P zpva ͔ I and P (c-zpva͒͑I) .This made us doubt the reliability of our MP2/6-31G results and motivated us to investigate the effect of basis set and level of correlation treatment on the calculated P e and P nr for three representative -conjugated organic molecules. 30It was found that the 6-31G basis set does not systematically provide even qualitative results.In a few cases even the sign of the contribution changes.However, semiquantitative accuracy was achieved using the 6-31 ϩG(d) basis set, ͑maximum error 17%͒.Compared with quadratic configuration interaction with single and double excitations ͑QCISD͒, MP2 gives a significant fraction of the correlation correction, but does not provide semiquantitative accuracy for P e or P nr ͑errors up to 94%͒.The accuracy of MP2 is far better for the ratio P nr / P e , in which case the maximum error is 32% for the largest basis set considered.Differences between QCISD and coupled cluster with single and double excitations ͑CCSD͒ values, calculated at the QCISD geometry, were always less than 8%.
From the results of Ref. 30 we know what level of ab initio theory we need to use to compute reliable values for P e and P nr .In order to assess the initial convergence of the two sequences above we also need reliable values of the first-order ZPVA and C-ZPVA contributions.Hence the goal of the present paper is to carry out a study of a similar type for ͓ P zpva ͔ I and P (c-zpva͒͑I) .The latter properties are far more computationally demanding to determine than P e and P nr .Nevertheless, by using field induced coordinates 37 ͑FIC's͒ to obtain P zpva and the FF method of Kirtman, Luis, and Bishop 14 to calculate P (c-zpva͒͑I) we were able to do these calculations.The FIC's constitute a minimum set of coordinates that allow the property to be obtained exactly. 31,37In contrast with normal coordinates their number does not increase with the number of the atoms of the system and there are far less than 3NϪ6 of them for the molecules considered here.Consequently, this is the first time that the ZPVA and C-ZPVA͑I) contributions to the hyperpolarizability have been calculated at an ab initio level beyond MP2/6-31G for a medium size organic molecule.
Our presentation is organized as follows: Section II summarizes the methodological and computational considerations.It is followed in Sec.III by a discussion of the results for P e , ͓ P zpva ͔ I , P nr , and P (c-zpva͒͑I) for three representative -conjugated organic molecules.Finally, our conclusions are given in Sec.IV.

II. METHODOLOGICAL AND COMPUTATIONAL CONSIDERATIONS
The usual expression for ͓ P zpva ͔ I is given by where i is the harmonic vibrational frequency of mode i, Q i is the corresponding mass-weighted normal coordinate, and F i j j is the cubic vibrational force constant defined by F i j j ϭ(‫ץ‬ 3 V/‫ץ‬Q i ‫ץ‬Q j 2 ) Qϭ0 .The first term on the right-hand side of Eq. ͑2͒ is first order in electrical anharmonicity, since it involves second ͑as opposed to first͒ derivatives of the electrical property with respect to nuclear coordinates, but zeroth-order in mechanical anharmonicity.In the second term the orders are reversed.This term is second-order in mechanical anharmonicity because it contains third derivatives with respect to the vibrational potential energy, which is one order higher than the harmonic force constants ͑e.g., F j j ϭ(‫ץ‬ 2 V/‫ץ‬Q j 2 ) Qϭ0 ).In either case anharmonicity introduces higher-order derivatives that become increasingly more tedious to determine as the number of normal modes of the chemical system increases.
If we regard the harmonic vibrational force constants as a function of the normal coordinates and the electric field, then Eq. ͑2͒ can be also be written as 37 [ where n is 1 for the dipole moment, 2 for the linear polarizability, 3 for the first hyperpolarizability, and 4 for the second hyperpolarizability.For convenience, the designation of the components of the electric field F has been suppressed.As shown in Eq. ͑3͒ the electrical anharmonicity terms can be computed by taking derivatives of F j j with respect to the finite field.For large molecules this is computationally cheaper than taking second derivatives of the property with respect to the normal coordinates.Furthermore, if one defines the harmonic FIC associated with the property P by the relation ͑again, nϭ1 corresponds to the dipole moment, n ϭ2 to the linear polarizability, etc.͒ 36 then Eq. ͑3͒ can be simplified to 20 [ where n,har 2 ϭ(‫ץ‬ 2 V/‫ץ‬ n,har 2 ) Qϭ0 .The calculation of ͓ P zpva ͔ I using Eq.͑5͒ requires evaluation of a single derivative of F j j with respect to the FIC instead of 3NϪ6 derivatives with respect to the entire set of normal coordinates as in Eq. ͑2͒.

III. RESULTS AND DISCUSSION
In order to complement our previous investigation 30 concerning the effect of basis sets and electron correlation on the calculated P e and P nr we chose the same three representative molecules ͑see Fig. 1͒: 1,3,5-hexatriene ͑I͒, 1-formyl-6hydroxyhexa-1,3,5-triene ͑II͒, and 1,1-diamino-6,6dinitrohexa-1,3,5-triene ͑III͒.I is nonpolar; II is polar with a dominant valence bond ground state; and III is polar with a ground state that has mixed valence bond-charge transfer character. 47Tables I-VI summarize the results obtained for the longitudinal component of P e , ͓ P zpva ͔ I , P nr , and P (c-zpva͒͑I) in the case of molecules I and II, respectively.Tables VII and VIII show corresponding quantities for molecule III except that P (c-zpva͒͑I) has been omitted because the large basis set calculations proved too time consuming.Even without P (c-zpva͒͑I) some useful conclusions can be drawn from the remaining data.A fairly systematic study of basis set and electron correlation effects on P (c-zpva͒͑I) was conducted for molecules I and II.The 6-31G results are taken from Ref. 29 while some of the smaller basis sets results for P e and P nr were also calculated in Ref. 30 but were obtained using the FF method as noted in the tables.For P nr the difference between previous numerical FF calculations and the present analytical treatment using harmonic FIC's is small.Small deviations arise because of numerical errors in calculating the derivatives.
Let us begin by examining ␣ zz (0;0).Much of the following analysis for this property and all the others will be based on the relative magnitude of the various contributions compared to the total value indicated by a superscript t.For example, in the case of ␣ zz (0;0) the ratio ͉␣ e /␣ t ͉ is always greater than 0.94 for molecule I regardless of the basis set or level of calculation.We conclude that the electronic contribution is dominant for this molecule.While the electronic contribution remains the most important for molecules II and III, the NR contribution is also significant.The ratio ͉␣ nr /␣ t ͉ obtained at the highest level of calculation for II ͓MP2/6-311ϩϩG(d, p)] and III ͓MP2/6-31ϩG(d)] is 0.13 and 0.38, respectively.On the contrary, ͉͓␣ zpvd ͔ I /␣ t ͉ and ͉͓␣ zpva ͔ I /␣ t ͉ for molecule II are both less than about 0.03 at the highest level of calculation ͓MP2/6-31ϩG(d)͔.Clearly, the initial convergence is quite satisfactory for ␣ zz (0;0).
There are a couple of additional respects in which the results for ␣ zz (0;0) are worthy of notice.For any given basis set the HF and MP2 values may differ substantially with regard to the magnitude of the individual contributions to a total property value.They are similar to one another, however, as far as the relative importance of the various contributions, i.e., the fraction of the total property value, is concerned.The maximum difference in this fraction is 0.02, 0.07, and 0.15 for molecules I, II, and III, respectively.For molecule I we can also compare MP2/6-31G with QCISD/6-31G.While the maximum difference between the individual MP2 and QCISD contributions is 0.05, the difference in the relative contributions is always less than 0.01.Second, it is interesting to observe that the 6-31G and 6-31ϩG(d) basis sets give similar results for the relative importance of the various contributions at both the Hartree-Fock and MP2 levels.In fact, the largest absolute difference ͑in all cases at the MP2 level͒ is 0.01 for molecule I, 0.03 for molecule II, and 0.04 for molecule III.
Due to symmetry, the longitudinal first hyperpolarizability vanishes for molecule I.For molecules II and III we first discuss the NR results.At the HF level ␤ zzz nr (0;0,0) yields the major contribution to ␤ zzz t (0;0,0) ͓except for molecule II with the 6-311ϩϩG(d, p) basis which will be discussed later͔.In addition, although the dynamic NR hyperpolarizability, ␤ zzz nr (Ϫ;,0) →ϱ , is smaller than the static value ͑with the exception of molecule III in the 6-31G basis͒ it is still larger than ␤ zzz e (0;0,0) for molecule III and about 2 3 as large as ␤ zzz e (0;0,0) for molecule II.There are major changes in the NR and/or electronic terms upon going from HF to MP2.Most notably, the relative value of ␤ zzz nr with respect to the static ␤ zzz e is reduced considerably.However, the qualitative picture is similar.Thus for molecule III at the MP2 level ␤ zzz nr (0;0,0) remains substantially larger than ␤ zzz e (0;0,0), while ␤ zzz nr (Ϫ;,0) →ϱ now becomes comparable in size.In the case of molecule II at the MP2 level, ␤ zzz nr (0;0,0) and ␤ zzz nr (Ϫ;,0) →ϱ become, respectively, about 2 3 and 1 4 as large as the static electronic term.Turning to the ZPVA and, particularly, C-ZPVA results we can see that, in general, they are much more sensitive to basis set than the NR values.Nonetheless, it is clear that ␤ zzz zpva (0;0,0) is very small compared to ␤ zzz t (0;0,0) and ␤ zzz e (0;0,0) regardless of the method or basis set.We conclude that the initial convergence for sequence ͑A͒ is rapid.
The story for sequence ͑B͒ is not as simple.In the case of molecule II ␤ zzz (c-zpva͒͑I) (0;0,0) is smaller in magnitude than ␤ zzz nr (0;0,0) in most of the calculations.This is not true, however, at the MP2/6-31ϩG level.We believe that this egregious 6-31ϩG result is unreliable because the basis set contains relatively too many diffuse functions.In calculating properties other than the energy with a limited basis, the importance of using a ''balanced'' basis set is well known.For the same reason the HF/6-311ϩG(d) and 6-311ϩ ϩG(d,p) values for ␤ zzz (c-zpva͒͑I) (0;0,0) may be considered as spurious.Although these values are correct for the given basis set, we consider the results spurious because they depart so much from the values obtained with the largest basis sets.In this connection we note the utility of the 6-31ϩG(d) basis which appears to be well balanced ͑by virtue of the generally good agreement with the largest basis sets͒ and, at the same time, is fairly compact.Using the latter basis the MP2 ratio ͉␤ zzz (c-zpva͒͑I) (0;0,0)/␤ zzz nr (0;0,0)͉ is 0.59.This is considerably larger than the HF result ͑0.38͒ because correlation increases the magnitude of the C-ZPVA contribution while keeping the NR contribution essentially the same.With or without correlation, however, an initially convergent ͑B͒ sequence is obtained.As we have seen in the past, for dynamic processes such as the Pockels effect ͓i.e., ␤͑Ϫ;,0͔͒ the initial convergence of sequence ͑B͒ is much improved over the static case.Indeed, for molecule II the ratio ͉␤ zzz (c-zpva͒͑I) (Ϫ;,0) →ϱ /␤ zzz nr (Ϫ;,0) →ϱ ͉ is 0.12 at the MP2/6-31ϩG(d) level and even smaller at the HF/6-31 ϩG(d) level since correlation affects the dynamic properties in much the same way as the static properties.In connection with the MP2 calculations it is apparent that the smaller ͓than 6-31ϩG(d)] basis set results are not reliable.
The vibrational contributions are also of key importance in determining the second hyperpolarizability of the molecules in this study.In order to compute ␥ zzzz nr (0;0,0,0) both harmonic and anharmonic FIC's are needed. 36As a result the FF method 17 utilizes less computational resources ͑just as it does in the C-ZPVA calculation͒ and thus it has been used to calculate ␥ zzzz nr (0;0,0,0).Our calculated results ͑see Tables VII and VIII͒ show that for molecule III ␥ zzzz nr (0;0,0,0) is two orders of magnitude larger than ␥ zzzz e (0;0,0,0).The dynamic ␥ zzzz nr ͑infinite frequency approximation͒ values for the optical Kerr effect ␥ zzzz nr (Ϫ;,0,0) →ϱ ͑OKE͒ and the intensity dependent refractive index ␥ zzzz nr (Ϫ;,Ϫ,) →ϱ ͑IDRI͒ are both larger than ␥ zzzz e (0;0,0,0) as well, while the dc-second harmonic generation ␥ zzzz nr (Ϫ2;,,0) →ϱ ͑dc-SHG͒ is about the same magnitude, even though all three properties are much reduced from the static NR value.For molecules I and II the situation is not as extreme, although NR is quite important in a number of instances.Thus, at the MP2/6-31ϩG(d) level, ͉␥ zzzz nr (0;0,0,0)/␥ zzzz e (0;0,0,0)͉ is 1.52 for molecule II and 0.34 for molecule I, replacing the static NR with its dynamic counterpart reduces this ratio to less than 0.55 in all cases.
Although the static ZPVA contribution appears to be about 1 3 as large as the static electronic term for molecule III, it is less important for the other two molecules and we can certainly say that sequence ͑A͒ is initially convergent for static ␥.Sequence ͑B͒ is also well behaved, for both the static and dynamic processes, with one important exception: the ratio ͉␥ zzzz (c-zpva͒͑I) (0;0,0,0)/␥ zzzz nr (0;0,0,0)͉ for molecule II is 1.6 at the HF/6-31ϩG(d) level and 2.2 at the MP2/6-31 ϩG(d) level.For larger basis sets the HF ratio remains either about the same or increases, but the corresponding MP2 calculations were not done.Thus the effect of correlation on the ratio is a bit uncertain.Nevertheless, we see that the initial convergence of sequence ͑B͒ can be problematic for the static second hyperpolarizability.This situation is not signaled by either the covalent versus ionic character of the ground state or by the relative magnitude of the NR hyperpolarizability in comparison with the electronic term.
Correlation always increases ␥ zzzz e (0;0,0,0) while ␥ zzzz nr (0;0,0,0) either decreases or remains essentially the same.The effect on the relative importance of these two contributions is substantial.For instance, in the case of molecule I, the HF/6-31ϩG(d) NR term is about 4 3 the electronic term whereas, at the MP2 level the same ratio becomes about 1 3 .For the dynamic ␥ zzzz nr vs ␥ zzzz e (0;0,0,0) the general trend is the same, though not as pronounced.At this point it is worthwhile to remark on the comparison between MP2 and QCISD based on the calculations done for molecule I using the 6-31G basis.Except for a couple of small terms ͑Ͻ10% of the total property͒ QCISD is less than 50% different from MP2.The QCISD values are generally smaller with the result that the relative importance of the various contributions is more accurately maintained.Thus, for example, the ratio ͉␥ zzzz nr (0;0,0,0)/␥ zzzz e (0;0,0,0)͉ changes from 0.25 for MP2 to 0.32 for QCISD whereas ␥ zzzz e (0;0,0,0) itself changes from 1.56ϫ10 5 to 1.05ϫ10 5 .We tentatively conclude that MP2 is adequate for investigating the relative importance of the different contributions and for obtaining a reasonable estimate of the total.
It is important to observe that, for molecule III at the HF level, the 6-31G and 6-31ϩG values often have the wrong sign.This is true for all contributions to the static second hyperpolarizability as well as the NR contribution to the OKE.In contrast, as already noted the 6-31ϩG(d) basis set appears to be fairly well balanced, and in almost all cases it gives values that are similar to those obtained for the much larger 6-311ϩϩG(2d,2p) basis.Although there are some glaring exceptions, overall the ͑hyper͒polarizability values

FIG. 1 .
FIG. 1. Structural formula of molecules studied in this paper.
also calculated in Ref. 30 but using the FF method.

TABLE I .
Electronic and vibrational polarizabilities and second hyperpolarizabilities of molecule I calculated at the HF level.All quantities are in a.u.

TABLE II .
Electronic and vibrational polarizabilities and second hyperpolarizabilities of molecule I calculated at the MP2 and QCISD levels.All quantities are in a.u.

TABLE III .
Electronic and vibrational polarizabilities and first hyperpolarizabilities of molecule II calculated at the HF level.All quantities are in a.u.

TABLE IV .
Electronic and vibrational second hyperpolarizabilities of molecule II calculated at the HF level.All quantities are in a.u.

TABLE V .
Electronic and vibrational polarizabilities and first hyperpolarizabilities of molecule II calculated at the MP2 level.All quantities are in a.u.
a Results taken from Ref.29except for ␥ zzzz nr (Ϫ;,,Ϫ) →ϱ .b All but 6-311ϩϩG(d,p) results were also calculated in Ref. 30 but using the FF method.c Value not calculated.

TABLE VI .
Electronic and vibrational second hyperpolarizabilities of molecule II calculated at the MP2 level.All quantities are in a.u.

TABLE VII .
Electronic and vibrational polarizabilities, first and second hyperpolarizabilities of molecule III calculated at the HF level.All quantities are in a.u.

TABLE VIII .
Electronic and vibrational polarizabilities, first and second hyperpolarizabilities of molecule III calculated at the MP2 level.All quantities are in a.u.Results taken from Ref. 29 except for ␥ zzzz nr (Ϫ;,,Ϫ) →ϱ .Results were also calculated in Ref. 30 but using the FF method. b