Calculation of Dipole Hyperpolarizabilities of H20 NH3 CH,and CH,F BY G. P. ARRIGHINI M. MAESTRO AND R. MOCCIA Centro di Chimica Teorica del C.N.R. Istituto di Chimica Fisica Via Risorgimento 35 56100 Pisa Italy Received 29th August 1968 From the first-order corrections of the SCF MO induced by a static homogeneous electric field the third-order corrections to the SCF molecular energies have been evaluated. These SCF hyper- polarizabilities were determined by using basis sets of STF of various sizes to express the unperturbed and the perturbed orbitals for HzO NH3 CH4 and CH3F. In some cases the calculated value compare badly with the experimental estimates. The possible reasons for these disagreement are briefly discussed. INTRODUCTION The distorsion of the electronic charge of a molecule under the influence of an electric field is important in determining several phenomena of interest.The evalua- tion of this distorsion for general situations represents a formidable problem which cannot be solved accurately not even for molecules of moderate complexity. Fortu- nately the computations can be carried out with reasonable accuracy for a particular situation which is not so restrictive as to diminish their importance. This situation is realized when a molecule is subjected to a static electric field whose sources can be considered external. For this situation the theory leads to the definition of several quantities through which the interaction energy between the molecule and the field is expressed. While some of these quantities like the permanent dipole moment and the polarizability can be determined experimentally with ease most of them like the higher permanent multipole moments and the higher polarizabilities are much more difficult to measure '9 both for experimental reasons and because of the complexities involved in the interpretation of the quantities experimentally determined.2* It seems therefore of interest to take recourse to the theory in order to evaluate them although the actual computation must be of an approximate nature.In this paper we report the results for the dipole hyperpolarizabilities of H,O NH3 CH4 and CH3F calculated by the rigorous perturbed H-F scheme where both perturbed and unperturbed orbitals were expressed by linear combinations of STF. The theory of the molecular polarizabilities has been developed by Bucking- ham.lV 3 9 McLean and Yoshimine have published a complete review always referring their treatment to Cartesian coordinates.Here we will limit ourself to a brief sketch using spherical coordinates which present some advantages over the car- tesian ones. If the electric field is charge of the molecule static and its sources are completely external to the electronic the perturbing potential V(r) may be written as 48 G . P . ARRIGHINI M. MAESTRO AND R . MOCCIA 49 where VL,M are the parameters which characterize the field and Y L M are the usual spherical harmonics. The transformation properties under rotations of the quantities VL,M will be those of the irreducible h a n k tensors while there the transformation properties under translations can be easily obtained from those of the YL,M.6 In presence of the field (2.1) the molecular energy E can be expanded as (ll3!)C C (a3E/avL,MavLi,M~avL~,M~)OvL,MvL~,M~vL2,M~+* * * (2'2) L M L i M i L 2 3 2 The coefficients appearing in the successive summations in eqn.(2.2) may be ideiitifiedwith the permanent multipole moments the general polarizabilities and so on. The use of the spherical coordinates and of (2.1) automatically eliminates from the expansion of the energy E all terms which must be zero because of Y(r) must obey the Laplace equation AV = 0. The fact that the VL,M behave as irreducible tensors can be exploited to express their product as linear combinations of new irreducible tensors,'. i.e.? VL M vL. 1 M2 = Cc( L 1 9 L' ; M ) v,&:2 + M 1 (2.3) L' where the C(L,L1,L';M,M1) are Clebsch-Gordan coefficient^.^ By standard procedure 7,8 it is thus possible to bring eqn.(2.2) in a form which contain only contractions of irreducible tensors. At this stage it may be expedient to return to Cartesian axes where only real quantities appear. This can be readily accomplished by using suitable unitary transformations. Once this last form of eqn. (2.2) is obtained the trans- formation properties of the various coefficients appearing in it are evident and it will be easy to find a particularly convenient set of axes. A suitable arrangement will be that coincident with those used to represent the symmetry operations of the molecular point group. In this case the only tensors not vanishing will be those which transform according to the total symmetric representation. As an example we consider an homogeneous field characterized only by the vector 8 = VV(r).In this case only the VI,M coefficients of eqn. (2.1) will be different from zero. Going through the steps previously described it is possible to write E = E0+47cN1 x 3)-1CY,,m(')[Y,,m(tl)E]o +(I x 3 x 5)-1Cy,,,,(b)Cyz,,(tI)Elo+ in m (1 x 3 x 5 x 7)- 'Cy,,m('>[y3,m(q1)E10 + (1/6)~2~0,0(')[r12~0,0(~)EI~ + m (1/30>~~2~1,,('>[~z~~,m~~)EIo + * * * * (2.4) m where the vector operator q stands for ( d j d 8 ) and the cSPl,m(r) are real harmonic polynomials obtainable by a suitable unitary transformation from the usual solid harmonics gI,m(r) = r Z Yl,m(0,q5).8 In eq. (2.4) the quantities [cSPl,m(q)E]O are proportional to the permanent dipole moments the [Y2,m(q)E]0 and [y2cSPo,o(q)E]o to the polarizabilities and so on.The evaluation of the above quantities requires the computation of the derivatives (dE/o'iffg)O (d2E/i%'gd8'gr)o (a3E/a8'gaiffg,a8'gll) etc. (9 g' 9" = X y 2). Because of the impossibility of their rigorous evaluation several approximate methods have been p r o p ~ s e d . ~ - ' ~ It seems that among them the perturbed H-F method without further approximation the so-called coupled H-F scheme (CHF) is capable of giving reliable results for the atomic polarizabilities 1 4 9 as well as for the molecular polarizabilities 9* '* l6 and magnetic sus~eptibilities.~~ This method permits one to obtain the rigorous derivatives of the H-F energies and hyperpolarizabilities ' 50 CALCULATION OF HYPERPOLARIZABILITIES EHF with respect to a parameter which give the intensity of a one-electron perturbation.Thus the coupled H-F scheme has been employed here to calculate the third derivatives of the molecular energy with respect to the three components of 8. As it has been shown,18J9 it is necessary to calculate only the first-order corrections 4:) of the occupied orbitals 4 j ) in order to evaluate the energy derivatives up to the third order with respect to the parameter p of a one-electron perturbation phP. Here only closed shell cases will be considered. These corrections 4$) must satisfy the equation wheref" is the unperturbed H and F hamiltonian YP indicates the correction upon the H-F electronic potential 2o which depend upon the &)'s while the other symbols are self-evident. In the above equation the off-diagonal Lagrange multiplier corrections appear.Without lack of generality the 4;)'s can be projected upon the unoccupied orbitals only. Thus the following formula is obtained which can be solved iteratively.16* 2 o also the following formula for the energy derivatives are obtained. With a little manipulation by exploiting the higher order perturbation equations FIRST DERIVATIVE * O SECOND DERIVATIVE occup. EgF = C [($; I hY I 4;)+(4; I hP I4S)+C.C.]. (2.8) i THIRD DERIVATIVE occup. occup. j i . j EaBy = 2c [(44 I hP+ GP I 4:) + C.C.] - 2 c [~gi(@$ I (by) + C.C.] +(pya)+(y~p). (2-9) In the last equation E:~ = (4; I ha+ga I 4:) and (pya)+(yap) means that the contributions corresponding to the others aPy arrangements must be added to the first one. In the present computations the orbitals 4;) and +$) were approximated by the Roothaan method i.e.4:) = { x>C. ;) 4:) = ( x } C . $. Once a given basis set { x ) is employed it can be shown 21 that thefollowingexpres- sion is obtained in place of eqn. (2.6) unoccup. C.\+(WP + GP)CPj 0 &g-&E,O Ctj = c CP (2.6a) Subsequently it is easy to derive the analogues of the eqn. (2.7) (2.8) and (2.9). Thus (2.7a) G . P . ARRIGHINI M . MAESTRO A N D R . MOCCIA 51 occup. occup. j L j Igg. = 2 c [cajt(Hfl+Gfl)c:j+c*c.]+ -2 c [C:J(Wa+Ga) cp,x C!JSC’ii+C.C.]+ (PY4 +(Yap>. (2.94 where q l y = (x I hP I x,> q = (x I gfl I x,> and = (x 1 x,>. The symbol EsCF has been used to distinguish it from the true H-F quantity EHF. If the basis set employed ( x ] is complete they will be coincident. For a static homogeneous electric field the perturbation matrices Ha WP and HI’ are those of the dipole moment operator.RESULTS AND DISCUSSION As reported elsewhere the first-order corrections to the SCF MO have been evaluated with eqn. (2.6a) for H20,16 NH3,16 CH4 16 2 2 and CH3F 2 3 using several basis sets of STF some of them of such a size as to obtain an unperturbed energy EiCF which are estimated to be close to the H-F limit. These corrections were used to evaluate through eqn. (2.8a) the dipole polarizabilities 16* 22* 2 3 and have presently been used to evaluate by eqn. (2.9a) the dipole hyperpolarizabilities. In table 1 are reported the basis sets employed and the SCF molecular energies E& obtained by the standard Roothaan procedure. The calculations were performed for fixed molecular geometries coincident with the experimental ones.Only for CH3F due to reasons given el~ewhere,~ the geometry employed is negligibly different from the experimental one. The figures reported in table 1 indicate that the largest bases give energies close to the estimated H-F Table 2 reports the results of our calculation for the dipole hyperpolarizabilities of HzO. For a reference frame which has the z-axis coincident with the two-fold symmetry axis and the zx plane coincident with the molecular plane only the [q29’,,,(q)E], [93,2(q)E]0 and [9,,o(q)E]o tensor components are not vanishing therefore table 2 reports only three quantities which are proportional to them. These quantities are expressible by the p coefficients as defined by Buckingham.’ For these molecules with a reference frame which has the z axis coincident with the three-fold symmetry axis and the y z plane coincident with one of the reflection planes only the [q29,,,(q)E], [ 9 3 0 ( q ) E ] 0 and [93,-3(q)E]0 tensor components are different from zero.As for H20 table 3 reports three quantities proportional to them expressed as linear combination of the p coefficients. Finally table 4 reports the CH4 results for the only not vanishing tensor component proportional to [ 9 3 - 2 ( q ) E ] 0 when the orienta- tion of the reference frame is such to have the x y z-axes coincident with the three twofold symmetry axes. The experimental results listed in table 3 and 4 (there are no available data for H20) were taken from ref. (3). Values in parentheses are approximate only. Tables 2 3 and 4 show that with the exception of CH4 the calculated values compare poorly with the available experimental data.The disagreement is so great as to make apparently negligible the importance of the size of the basis sets employed. At first sight this would seriously undermine the accuracy of the computed values but it seems to us that it would be a hasty conclusion. We examine e.g. if the fault might lie in the computing scheme we adopted i.e. the CHF technique. That this seenis not the case may be conjectured by the following arguments. First we observe that the CHF technique has proved to be satisfactory for the atomic polarizabilities I4 and hyperpolarizabilities I 5 and certainly superior to other methods even for the 24 Table 3 shows the results obtained for NH and CH3F. 52 HzO I H20 I1 NH3 I NH3 I1 CH4 I CH4 I1 CHSF CALCULATION OF HYPERPOLARIZABILITIES TABLE 1 .-BASIS SETS EMPLOYED basis functions no.of basis functions 7 27 8 32 9 39 47 lSN(6.67457) ; 2S~(1.9426) ; 2P~(1*9426) ; lS~(1.19545) EicF = -56.0058 a.u. lSc(5.716) ; 2Sc(l.625) ; 2Pc(1*625) ; lS~(1.28) E;& = - 40.1 129 a.u. (a) The 2P components perpendicular to the molecular plane are omitted. (b) 0. Salvetti to be published. (c) Located at a distance of 0.7 a.u. from the nitrogen approximate position of the centroid of the ( d ) R. Moccia and C . Vergani to be published. ( e ) ref. (22). (f) ref. (23). lone pair (CLP). TABLE DIPOLE HYPERPOLARIZABILITY RESULTS FOR H20 t2.S.U.) (Pxxz + by y z f P z zz) (Pxxz - PYYZ) (282 z z - ~ P X X Z - ~ P Y Y Z ) H20 I + 0.1723 + 0.1087 -0.1983 H20 I1 + 0.1598 +0~1000 - 0.0778 expt. I - - TABLE 3.-DIPOLE HYPERPOLARIZABILITY RESULTS FOR NH3 AND CHjF ( t3.S.U.) ( P x x z - t P y y z - k P z z z ) (2Pz2z-313xxz-3317,Yz) U P X X Y - P Y Y Y ) NH3 I - 0.05 197 + 0.1 451 4 -0.1884 NH3 I1 -0.1363 + 0-2739 - 0.3051 expt.(- 11.7) I - expt. (-3.33) - - CH3F - 0.2667 - 0.021 1 + 0.0944 TABLE 4.-DIPOLE HYPERPOLARIZABILITY RESULTS FOR CH4 ( t3.S.U.) ( P X Y Z ) CH4 I + 0.0400 CH4 I1 +0.0191 expt. - f0*01 G . P. ARRIGHINI M. MAESTRO AND R. MOCCIA 53 molecular polari~abilities.~* l 1 On the other hand this result had to be expected because as it could be easily shown the CHF technique give the derivative of an approximate energy which is as much corrected with respect to the fluctuation potential 2 5 as the unperturbed H-F energy. It appears rather improbable that the higher corrections due to the neglected part of the electronic correlation should drastically change the values of these derivatives.Thus the CHF technique is expected to give results of satisfactory accuracy for quantities like the electrical polarizabjlity hyperpolarizability paramagnetic susceptibility etc. If this reasoning is correct one is led to conclude that the cause of the observed discrepancies between calculated and experimental hyperpolarizability is due to the fact that our calculations were performed even in the best cases with basis sets of large but limited extension. To this criticism a definite answer cannot be given at the moment ; however some consideration may be helpful. The differences among the values calculated by using basis sets of different sizes are percentage wise not negligible and of the right magnitude as it may be inferred from the differences encountered for other quantities like the dipole polarizabilities 6 22 and the paramagnetic su~ceptibilities.~~ As for the hyperpolarizabilities these quantities depend only on the first-order corrections of the orbitals.In addition the results obtained with basis sets of increasing size (which are not reported here) show a regular behaviour and their values tend to those corresponding to the largest bases. It is rather improbable then that by increasing the size of the bases which is tantamount to approaching the H and F limit the results would change drastically. It is our opinion therefore that the computed values cannot be that wrong and that the experimental estimates should be reviewed. This conclusion is also supported by the fact that the only reliable experimental result which refers to CH4 is in satisfactory agreement with that calculated by the largest basis.Accordingly it would establish a + sign for this quantity. A. D. Buckingham Quart. Rev. 1959 13 183. A. D. Buckingham R. L. Disch and D. A. Dunmer J . Amer. Chem. Soc. 1968,90,3104. A. D. Buckingham and B. J. Orr Quart. Rev. 1967 21 195. A. D. Buckingham Adv. Chem. Physics 1967 12 107. A. D. McLean and M. Yoshimine J. Chem. Physics 1967,47 1927. Y. N. Chiu J. Math. Physics 1964 5 283. ’ M. E. Rose Elementary Theory of Angular Momentum (J. Wiley and Sons 1957 New York). see e.g. A. R. Edmonds Angular Momentum in Quantum Mechanics (Princeton University Press 1960) p. 124. R. M. Stevens and W. N. Lipscomb J. Chem. Physics 1964 40,2238.R. E. Wyatt and R. G. Parr J. Chem. Physics 1964 41 514. l 1 H. J. Kolker and M. Karplus J. Chem. Physics 1964,41 1259 ; and ibid. 1964 41,2011. l 2 H. D. Cohen and C. C. J. 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