E L S E V I E R Chemical Physics 216 (1997) 53-66
Chemical Physics
Cotton-Mouton effect and shielding polarizabilities of ethylene:
an MCSCF study
Sonia Coriani a, Antonio Rizzo
a , ,Kenneth Ruud b, Trygve Helgaker b
a lstituto di Chimica Quanti,s'tica ed Energetica Molecolare de! Consiglio Nazion.a!e delle Ricerche. Via Risor£imento 35. 1-56126 Pisa.
halv
b Department of Chemistry. University t~'Oslo. P.O.B. 1033 Blindern. N-0315 Oslo. Norway Received i 0 October 1996
A b s t r a c t
The static hypermagnetizabilities and nuclear shielding polarizabilities of the carbon and hydrogen atoms of ethylene have been computed using multiconfigurational linear-response theory and a finite-field method, in a mixed analytical- numerical approach, Extended sets of magnetic-field-dependent basis functions have been employed in large MCSCF calculations, involving active spaces giving rise to a few million configurations in the finite-field perturbed symmetry. The convergence of the observables with respect to the extension of the basis set a~ well as the effect of electron correlation have been investigated. Whereas for the shielding polarizabilities we can compare with other published SCF results, the ab initio estimates for the static hypermagnetizabilities and the observable to which they are related - the Cotton-Mouton constant, - are presented for the first time.
1. I n t r o d u c t i o n
The development of ever more sophi:,ticated com- putational methods and ~he increasing availability of powerful computers has led several researchers to the field of nonlinear optical - electric and magnetic - properties. In the last few years, in pa_rticu!ar, t.he interest in the effects of perturbing electric fields on magnetic properties has been growing steadily.
In our group hypermagnetizabilities (magnetic po- larizabilities) and the hypermagnetizability anisotro- py, which is directly related to the Cotton-Mouton effect (CME), have been computed for Ne [1] and Ar [2] and the electric-field dependence of" the magnetiz-
* Corresponding author.
abilities and nuclear magnetic shieldings have been studied for N 2, CEH 2, HCN, H 2 0 [3], CO and CH 4 [4]. Here we present restdts for the hypermagnetiz- abilities and nuclear magnetic shielding polarizabili- ties of ethylene. As in the previous cases, the ele- ments of the tensors are calculated in a mixed analyt- ical-numerical approach. SCF and MCSCF magnett- zabilities and nuclear magnetic shieldings are first evaluated in the presence of external electric fields, and the polarizabilities are then obtained by finite- field (FF) numerical differentiation. The use of atomic basis sets that depend explicitly on the exter- nal magnetic field (London Atomic Orbitals, LAOs or Gauge Including Atomic Orbitals, GIAOs) en- sures gauge-origin independent results and faster ba- sis-set convergence. The choice of the natural con- nection for the perturbation-dependent basis set elim- 030 !-0104/97/$ i 7.00 Copyright © ! 997 Elsevier Science B.V. All rights reserved.
Pli S030 ! -0 ! 04(97)000 ! 9-0
54 S. Coriani et a l . / Chemical Physics 216 t 1997) 53-66
inates numerical problems that would otherwise severely limit the accuracy of the higher-order prop- erties computed by numerical differentiation.
Other groups have also been active in this field recently. Hypermagnetizabilities for several atomic and molecular systems have been computed by Bishop and coworkers and by ,~,gren and his group.
Limiting ourselves to the most recent literature, we mention for instance the third-order Moller-Plesset (MP3) and linea_rized coupled-clusters doubles (LCCD) plus FF calculations on N 2, HF, CO and H 2 by Cybulski and Bishop [5], the SCF cubic response [6,7] and very recentl.y the MCSCF cubic response [8] calculations by Agren et al. In Ref. [9] the interested reader will find a thorough account of experiment and theory of the CME. The literature on the calculation of chemical shielding polarizabilities is vast and the recent review by Raynes [10] is an excellent introduction to the subject. See also Refs.
[3] and [4] for a list of pertinent references.
We present here the first theoretical calculations of the hypermagnetizabilities and the Cotton-Mou- ton constant of ethylene. Experimental CME data are also available [ I ! - 14]. The shielding polarizabilities of ethylene were calculated with a finite-field Cou- pled Hartree-Fock approach by Grayson and Raynes for both the carbon [15] and hydrogen [16] atoms.
Augspurger et al. [17] have published an estimate of the rotationally averaged shielding polarizability pa- rameter A,, as defined below.
Theory
d m ,
The theory and techniques used for computing hypennagnetizabilities and shielding polarizabilities with LAOs have been discussed in detail in Ref. [3].
Here we shall briefly review some definitions, with particular emphasis on the relationships with experi- ment for the system under study.
The molecular magnetizability, X, and the nuclear magnetic shielding of nucleus K, tr(K), are defined
as
o ,,n m, I (',
X = - 0 2 B a = o
O E(B,m) [
o ' ( K ) = l + (2)
OBbmr IB=m=O
where e(B, m) is the molecular energy in the pres- ence of the external magnetic field induction B and the nuclear magnetic moments m. m r represents the nuclear magnetic moment of the If th nucleus.
2.1. Hypermagnetizabilities and the Conon-Mouton Effect (CME)
In the presence of an external weak, homoge- neous electric-field perturbation E, the elements of the molecular magnetizability tensor A',~p may, in our case, be expanded as [18]
= + 1 + . . -
(3)
since the first-order contributions vanish because of symmetry. The hypermagnetizability tensor r/,~t3.~ is
- OE,,~Et3 (4)
In the first definition of r/,t~.~, the electric-dipole polarizability tensor ot,,a is introduced. In our ap- proach, r/,,tj.~a is obtained by numerical differentia- tion of analytically calculated magnetizabilities - that is, from the last expression in Eq. (4).
The experiment to which the hypermagnetizabili- ties are related is the CME [19] - the birefringence of light in gases in a constant magnetic field. Isotropic substances show a weak birefringence An = nil- n±
(anisotropy of the refractive index) when radiation passes through a sample in a direction orthogonal to a strong magnetic field. Experimentally, one mea- sures the eilipticity ~ , directly proportional to An, acquired by polarized light passing through the sam- ple in a strong, uniform magnetic field. The resulting magneto-optic effect is often extremely weak, and its accurate detection is consequently difficult [9].
For rigid diamagnetic molecules in the gas phase,
S. Coriani et al./ Chemical Physics 216 t 19971 53-66 55 Buckingham's classical treatment [9,1 l] leads to the
expression
27B 2 wBZNA
A n = 2Vm(4.n.E0 ) m C = Vm(4.n.Eo)
[art+Q(r)]
w B 2 p
(4"rre0) kT [art + Q(T)] (5)
l( , )
AT/= ~- r/,,a.~t~-
-~-'r/,,,~.m3
(6),( , )
Q( T) = "g'~ ot,~# X,~ - "~ tx,~,~ Xt~tj (7) where NA is Avogadro's number, V m the raolar volume, k the Boltzmann constant, T the tempera- ture and P the pressure. The Einstein summation convention has been adopted. The last expression in Eq. (5) is valid for an ideal gas only.
The anisotropy of the refractive index has thus two contributions: one contribution that depends on the inverse of the temperature and involves the hy- permagnetizability anisotropy, and a second contri- bution - often referred to as "Langevin's term' - that depends on the inverse of the square of the temperature and describes the molecular orienta- tional effect. In the case of ethylene, with the molecule in the xz plane and making use of the molecular symmetry [20], Eqs. (6) and (7) reduce to Ar~ = ]~" (2rh..,..~.,. + 2r/yy.yy + 2 r / : : . : : - I rl.,.~..yy
--'l~xx,ZZ- ~yy.x.l"- 'i~yy,z:- ~ZZ..t'x- 'OZZ,yy +6"O.,y.a.y + 67/.,-=..,-= + 6"qyz,yz) ( 8 ) Q ( T ) = I
lSkr [(
'~.,.,--",.,,) (
X.,.,- X,,)+(,.,.,-,::)(x,..,.-x::)
+( x,r- x:=)]
= ~ l y ' 15kT i<j
i,j=x,y,z
k ol ii,jjA xii.jj .
(9)
The frequency-dependent hypermagnetizability can be written as a combination of a quadratic (the
diamagnetic contribution) and a cubic (the paramag- netic contribution) response function [9]
rt,,t~.~ ( - ~o; o~,0,0)
p d
= r/g, t3.~,~ + "qdt~.~,~
1
- 4 ((r~;r~'L~'L~))'-~.°'.°.°
- - ( (
4 1 r~;rl3" r26"~ r, ra ( -)))
-o,.~,.o, , ° ,
"'-,"Here r denotes the position and L the angular momentum operator. Taking advantage of symmetry and reducing the expression for the paramagnetic contribution to a Cauchy-moment expansion, the fre- quency dependence of the hypermagnetizability ani- sotropy of the Ne and Ar atoms was studied using the multiconfigurational response method [1,2]. The same approach cannot straightforwardly be em- ployed for molecules. The finite electric field tech- nique employed here limits our study to static hyper- magnetizabilities.
2.2. Chemical shielding polarizabifities
The nuclear shielding polarizabilities of nucleus K can be defined from the perturbative expansion of the nuclear shielding with respect to the electric field
= o',,t3 (K) + o',~#.,( K)E~
+ ~tr~;.,a( K ) E , e I 6 + "'" ( l l ) t r )
cr~t~.~( K ) = aE v (12)
r )
o',~.~,,(K)
= OE,,,aE~(13)
and .are obtained in this investigatiGn by numerical differentiation of the analytically computed chemical shieldings.
A description of the effect of an uniform electric field on the nuclear shielding was given by Bucking- ham in 1960 [21]. The change in the mean shielding, after averaging over all molecular orientations in the NMR external magnetic field, keeping the electric field r, voa ,..q.~,;,,o ,,, Lhe ,-.,,,! .... !o ;o l i h ~ . , I , / l i l ~ i i l l g l l k I t U i l l ~ J l l ~ l h . , U i ~ . , , 1 0
-%e,- 8.e,e
(14)56 S. Coriani et al./ Chemical Physics 216 (1997) 53-66
where (Einstein summation implied)
Ay= - - - - 3 o;,,,,.~, 1 ' (15)
1
B~8 -" - 3(1 + ~,8 ) or~'~"Y~ (16)
The number of non-vanishing elements in the tensors or, or' and or" for a given nucleus depends on the local 'on-site' symmetry at the position of the nu- cleus [20]. For ethylene in a D2h (equilibrium) sym- metry, the carbon atoms are in a C2v site symmetry environment, whereas the protons are in a C s site symmetry. With the molecule in the xz plane, Eq.
(14) for the two atoms becomes
~r(C) ffi - A z E , - B x , ~ E 2 - B y y E 2 - B z z E 2 (17)
~r( H ) ffi - A x E x - A z E , - BxxE2 - ByyE 2
- Bzze2 - B x z e x e z (i8)
In ethylene, the number of non-vanishing compo- nents in the or, or' and or" tensors is 3, 7, and 15, respectively, for the carbon atoms, and 5, 14, and 28 for the protons [20]. In our tables we shall report all tensor elements for the carbon atoms. For the hydro- gen atoms, we shall report and discuss only those elements that contribute to the mean shieldings in
(18).
As pointed out by Raynes in Ref. [10], there are now several areas where the effect of the electric field on the chemical shielding is invoked to explain experimental observations such as intermolecular in- teractions in gases, effects of the solvent in liquids, intramolecular electric fields due to polar groups in molecules or the effect of electric fields in solids.
2.3. Conversion factors
The following conversion factors from atomic units to other units are useful:
• 1 au of ~ = e 2 a 2 E ~ I / _ _ 1.64878X 10 -41 C2m2j - 1 ____ 1.48185 x 10- 25 (4,treo) cm 3.
• I au of a ' - e2a2m~ ! --- 7.89104 x l0 -29 JT -2 _-- 7.89104 × 10- 3o erg G- 2.
• I au of r l - e4a~m~SE~ 2 --- 2.98425 x 10 -52 C 2 m 2 j - I T - 2 - - - 2.68211 × l0 -44 (41reo) cm3G -2.
• I au of o r ' - p p m ( a 2 / e ) - 1.94469)< 10 -is mV -t -- 5.83003 × 10-t4 cm statV-i (esu);
• 1 a u× of or"--ppm ( a 2 / e ) 2= 3.78182x 10 -30 m2V -2 -" 3.39892 × 10 -21 cm 2 statV -2.
3. Computational details
All calculations were performed with the DAL- TON quantum chemistry program [22].
3.1. Basis sets
Calculations were performed with four different (spherical) Gaussian basis sets [23,24]:
• Basis IV: a I l l s 7p 3d l f / 6 s 3p ld]/(Bs 7p 3d l f / 5 s 3p ld) set (178 functions);
• Basis IVa: a [12s 8p 4d 2f/7s 4p 2d]/(9s 8p 4d 2f/6s 4p 2d) set (248 functions);
• Basis IVb: a[13s 9p 5d 2f/Bs 5p 2d]/(10s 9p 5d 2f/7s 5p 2d) set (280 functions);
• Basis IVi: a [12s 8p 4d l f / 7 s 4p ld]/(9s 8p 4d l f / 6 s 4p ld) set (212 functions).
Basis IV is based on a compilation by Huzinaga [25] and is described in Ref. [26]. Basis sets IVa and IVb were first employed in Ref. [3] and are obtained by adding diffuse functions to basis IV. In basis IVi the spd space for the C atoms and the sp space for the H atoms are those of basis set IVa. A single f function (with exponent 0.8) and a single d function (with exponent equal to the average of the exponents of the d functions for H of set IVa) were added to the C and H atom basis sets, respectively.
Basis IVi was tested at the SCF level and then used for the correlated calculations. Because of the low symmetry arrangement imposed by the FF ap- proach, this basis represents the largest 'sensible' choice of a basis set that could be used in our correlated calculations.
3.2. Active spaces
For the correlated calculations, two complete ac- tive spaces (CAS) were employed: the Full Valence CAS (FV-CAS) including 3 ag, 2 b3u, 1 b2u, 3 bt.., 2 b2g and
1 b3g
active orbitals, and a larger CAS, here referred to as CAS2, which includes 4 ag, 3 b3u, 2 b2u, 3 b~u, 2 b2g and1 b3g
active orbitals. The two innermost orbitals - the first ag orbital and the first b~u orbital - were kept frozen and the active orbitalsS. Corivni ¢l a l . / Chemical Phy~i¢~ 2;6 (1997) 53-66 57
were selected on the basis of second-order M¢ller- Plesset (MP2) occupation numbers, including for CAS2 orbitals with an MP2 occupation number greater than 0.01. All CASSCF calculations were carried out in the IVi basis.
3.3. Geometry
For the hypermagnetizabilities and carbon atom shielding polarizabilities, the molecule was placed in the xz plane, with the z axis along the molecular axis, in a conventional D2h geometrical arrangement.
As noted in the previous section, all first derivatives of the magnetizabiiity are zero with this setup [20].
For the hydrogen shielding polarizabilities, we chose the H (0,0,z) atom in a coordinate frame centered on the C atom, with the z axis pointing along a C - H bond towards the proton and the x axis lying in the molecular plane, pointing towards the other nearest proton. This setup may be obtained by rotating the D2h frame around the y axis and was adopted for easy cornparison with the results of Ref. [16]. The geometr] was taken from experiment [27]: Rcc = 1.339 A; R c n = 1.085 A; < H C C = 121.09";
< HCH - 117.825.
3.4. Finite fieM
An electric field strength of 0.008 au was found to be adequate to obtain numerically stable hyper- magnetizabilities and shielding polarizabilities in the finite-field numerical differentiation.
In the presence of one perturbing field, the sym- metry of the system is reduced to C2v, yielding about six million variational parameters in our largest CASSCF calculations. F'or a C s set up, needed for instance with two crossed perturbing electric fields or to evaluate in the most economical way the correlated shielding polarizabilities at the hydrogen atoms, the number of variables increases to about twelve million. The CASSCF calculation then be- comes exceedingly expensive, both in terms of CPU as well as disk space requirements. The double-field calculations are needed, on the other hand, to deter- mine the mixed-index shielding polarizabilities of the carbon atom (o',,a.~t 3, o-~a.a~ a #/3 = x, y, z), which do not enter the rotational averages, and the mixed-index hypermagnetizabilities (r/,~a.,,o - v/,,a.t3~
a :~/3 - x, y, z). The latter enter the expression for the hypermagnetizability anisotropy, and thus the observable - the Cotton-Mouton constant.
To calculate the CME of ethylene, the hypermag- netizability, electric polarizability and magnetizabil- ity anisotropies were computed, both at the SCF and the CASSCF levels. The electric polarizability ani- sotropy was computed at different frequencies (at the SCF level only with basis IVb). Since the Langevin term dominates the CME in ethylene, this approach allows us to estimate the frequency-dependence be- havior of the anisotropy of the refractive index in the presence of a magnetic field.
4. Results and discussion 4.1. Magnetizability
The magnetizability and the hypermagnetizabili- ties of ethylene are reported in Table 1, together with the hypermagnetizability anisotropy. We list the re- sults obtained at the SCF level with the three basis sets IVi, IVa and IVb, and those obtained with basis IVi in the FV-CAS and CAS2 correlated calcula- tions.
According to the data in Table l, basis set conver- gence is achieved already with basis IVi for the average magnetizability and for the individual com- ponents. Our best result for the averaged SCF mag- netizability ( - 4.5008 au) improves by about 2% the SCF-GIAO result previously obtained in our group [28], and it is in magnitude about 6% smaller than the SC~'-IGLO result of Ref. [29], obtained with a much smaller basis. The authors of Ref. [32] give ab initio values of the individual components of the magnetizability tensor, cited as private communica- tion from van Wiillen and Kutzelnigg (see also Ref.
[33]). The largest disagreement is in the X~x compo- nent, where our value (-3.5280 au) is some 9%
smaller in absolute value than that of van W'tillen and Kutzelnigg.
To our knowledge, no con'elated results are avail- able in the literature for the magnetizability of ethy- lene. Correlation has apparently a negligible influ- ence on the average magnetizability. Our IVi CAS2 value (-4.4955 au) is only 0.12% smaller than the IVb SCF result. Actually, this remarkably small
58 S. Coriani et a l . / Chemical Physics 216 (1997) 53-66
correlation effect results from cancellation among the individual components: the X~ component varies by as much as 2.6%, whereas )try and )tz~ change by - 0 . 9 % and - 1 . 3 % respectively when introduc- ing correlation. A detailed comparison of experiment and SCF results was made in Ref. [28]. We refer the interested reader to that paper for an extended and critical review of experimental data. Since correla- tion leaves the SCF estimate practically unaffected, the conclusions drawn in Ref. [28] on possible cali- bration errors in Barter's experiments [31] remain valid. The experimental value of Ref. [31], once multiplied by the factor of 1.07 as suggested in [28], is still some 2% less negative than our current estimate, whereas the more recent experimental value of Oldenziel and Trappeniers [30] is about 8% smaller in magnitude. The authors of Ref. [32] were able to estimate the individual components of the magnetiz- ability tensor, and their values are listed in the footnote of Table 1. Our correlated values are in absolute value larger than theirs by 6% to 11%.
4.2. Hypermagnetizabilities
For the hypermagnetizabilities, convergence with respect to the basis set is much slower than for the magnetizability, as shown in Table I. The results obtained with Basis IV (not reported) demonstrate the inadequacy of that set, especially for 7/x~.~x ( + 30.13 au, cf. - 7 4 . 4 6 au in basis IVb) and ~yy.yy
(+61.47 au, cf. - 1 0 9 . 8 2 au in basis IVb). The stability of the SCF results going from basis IVi to IVb is, however, more satisfactory. With the remark- able exception of */zz.~- which is much smaller than the other components and whose effect on the anisot- ropy in Eq. (11) is extremely limited - the compo- nents of the hypermagnetizability tensor 7/ vary on the average by 7% to 8%, the maximum effect being felt by ¢/,,,,.zz (about 14%) as the basis set gets larger.
The hypermagnetizability anisotropy varies by less than 0.2%, but we note that AT/= 75.58 au whea basis IV is employed.
The SCF analysis indicates that the use of the IVi
Table I
Ethylene, m a g n e t i z a b ! l i t y and h y p e r m a g n e f i z a b i l i t i e s (au)
S C F F V - C A S C A S 2
Basis IVi Basis IVa Basis IVb Basis !Vi Basis IVi
Xav - 4.5022 - 4 . 5 0 1 9 - 4,5008 a - 4.4887 - 4.4955
X . b - 3.5303 - 3,5292 - 3.5280 - 3 . 7 0 3 6 - 3.6192 c
~ . - 5.6667 - 5.6679 - 5.6640 - 5 . 4 8 1 4 - 5.6117
Xu - 4.3095 - 4 . 3 0 8 6 - 4 . 3 1 0 4 - 4.2811 - 4.2555
vh,~.x, ~ - 73.45 - 74.62 - 74.46 - 51.67 - 65.71
~ . , y y 96.45 93.19 104.49 120.94 99.42
v/.,zz - 118.96 - 119.84 - 104.42 - 9 8 . 8 3 - 108.46
~ y y . . - 301.91 - 302.47 - 310.74 - 192.06 - 264.63
v/yy.yy - 102.61 - 102.97 - 109.82 - 67.06 - 94.69
Vlyy.zz - 196.22 - 195.63 - 2 1 2 . 8 1 - 140.09 - 173.72
~ z , . - 51.05 - 53.56 - 47.30 - 43.47 - 47.73
~'z,yy - - 68.96 - 66.50 - 64.37 - 63.67 - 80.6 I
~z.zz 2.99 4.61 4.30 - 13.1 ! - 22.01
~,ty.xy 62.05 63.31 62.45 36.38
~,~z.~z 12.83 12.82 13.56 10.95
v~yz.y z 42.46 43.69 44.28 29.03
A ~ 66.57 67.85 66.46 40.78 61.00 + 7.97 d
a S C F - G I A O - 4 . 4 1 5 au, Ref. [28]; S C F - I G L O - 4 . 7 5 2 au, Ref. [29]; e x p e r i m e n t : - 4 . 1 4 + 0.08 au, R e f . [30]; - 3 . 9 5 + O. 17 au, R e f . [31];
- 4 . 1 5 + 0.19 au, R e f . [32]; - 4 . 2 3 + 0 . I 0 au, R e f . [33].
b T h e o r y , I G L O : ; ~ , - 3 . 8 4 au; Xyy - 5 . 5 3 au; Xzz - 4 . 3 7 au. G i v e n as "private c o m m u n i c a t i o n f r o m Ch. v a n Wiillen a n d W. K u t z e l n i g g ' in Ref. [32], a n d e x t r a c t e d from Ref. [34].
c Experiment: X , - 3.33 ± 0.2 au; Xyy - 5.28 __. 0.2 au; Xzz - 3.83 _-I- 0 . 2 au. R e f . [32].
d E x p e r i m e n t - 22 ± 67 au, at A - 632.8 n m . R e f . [ 13].
S. Coriani et a l . / Chemical Ph):s'ic.s 216 (1997153-66 59
basis for CASSCF calculations is adequate. When correlation is introduced, the hypermagnetizability tensor behaves in a peculiar manner. Comparing the IVi SCF, FV-CAS and CAS2 results for the compo- nents not involving mixed indices - which we could evaluate in all three approaches - we find that FV-CAS in most cases strongly overestimates the effects of correlation. Again neglecting ~ . ~ , the FV-CAS on average modifies the other eight tensor components by about 24%, whereas the effect is reduced to about 10% with CAS2. However, a larger correlation effect is observed in the CAS2 calcula- tions for 'l~zz.yy and for 7h~.~ ~ than in the FV-CAS.
By analyzing the effect of correlation on the remaining hypermagnetizability tensor components, which - with the noticeable exception of the fully parallel component, r/z~.~- never exceeds 17%, we may assume that the effect of correlation on the remaining components ,also does not exceed 17% of the SCF value and we _are thus able to provide upper and lower limits to the anisotropy. Note that the uncertainty in the mixed-index hypermagnetizability is reduced to about 11% in the anisotropy, Eq. (11), and - due to the dominance of the Langevin term in the CME of ethylene - to about 0.5% in the Cotton-Mouton constant. Note also that the only available experimental number for the hypermagneti- zability anisotropy ( A t / = - 22 + 67 au at A = 632.8 nm, [13]) comes with a 300% uncertainty, allowing plenty of room for agreement.
It should be noted that the value given for )he experimental hypermagnetizability anisotropy in Table 1 of Ref. [13] is ten times smaller than the one reported in our tables. An analysis of the data in the figure of Ref. [13] is, on the other hand, more consistent with a value of A'q of - 2 2 _+ 67 au at A = 632.8 nm. Moreover, a similar analysis of the data for acetylene in the same figure - confirmed by other recent experimental results for the same molecule [35] - leads us to believe that there is an error in the headline of Table I of Ref. [13] (the hypermagnetizability anisotropies are actually given in units of l0 -42 cm3G -2 and not 10 -43 cm3G -2 as stated).
As far as rhz.~ ~ is concerned, it displays a correla- tion behavior similar to that observed for the rh~.~ ~ component in N 2 and HCN [3]. In all cases, the overall effect on the hypermagnetizability anisotropy
proved to be limited. Moreover, we have never obser,¢ed 'anomalous' correlation effects on the mixed-index tensor components in other similar stud- ies. In ethylene, the correlated At/ is on the average only about 8% lower than the corresponding SCF anisotropy.
4.3. Cotton-Mouton constant
The molar Cotton-Mouton c o n s t a n t m C and the corresponding anisotropy of the refractive index at a pressure of P = I atm and with an induction field of
B=I T - A n , , = A n ( P = I atm, B = I T),seeEq.
(8) and Ref. [9] - at room temperature ( T = 293.15 K) are repoiied in Table 2, where the results ob- tained with basis IVb at the SCF level and with basis IVi at the CAS2 level are displayed.
To be able to compare with experiment, which is always performed at a finite electric-field frequency, we have in Table 2 reported the results of calcula- tions performed both in the static regime and at three different electric-field frequencies. As mentioned above, our finite electric-field approach to the calcu- lation of the hypermagnetizabilities does not allow the study of the frequency-dependence of At/ for non-spherical systems. We therefore limit our fie- quency-dependence analysis to the study of the elec- tric polarizability anisotropy A a . The data in Table 2 show that the Langevin's term Q(T) contributes about 95% of the CME of ethylene at room tempera- ture. This behavior parallels that observed for most of the molecular systems investigated in the v,*~ ....
[3,4] - exceptions being H 2 0 [3] and CH4 [4] .... and justifies the neglect of the frequency-dependence of the hypermagnetizability anisotropy in the calcula- tion of the CME.
The frequency dependence of the electric-dipole polarizability a ( - t o ; to) is displayed in Table 2. In this Table, a comparison is also made w ~h experi- ment [14,37] and with the results of S~:kino and Bartlett [36], who obtained the individuat compo- nents of a, the average and the anisotropy, at A = 633.4 nm with TDHF and various correlated meth- ods - MP2, CCSD and CCSD(T). We also mention that a calculation of the static dipole polarizability of ethylene with a moderately-sized basis set at both the SCF and MP2 levels [40], followed by a TDHF analysis of the frequency-dependence of u obtained
60 s. Coriani et al. / Chemical Physics 216 (1997) 53-60 t h r o u g h a p o w e r s e r i e s e x p a n s i o n i n v o l v i n g the l o w -
est C a u c h y m o m e n t s w a s d o n e S p a c k m a n [41].
A s o b s e r v e d b y the a u t h o r s o f R e f . [36], the e f f e c t o f c o r r e l a t i o n o n the electric d i p o l e p o l a r i z a b i l i t y o f e t h y l e . e is q u i t e s m a l l , the c o r r e l a t e d p o l a r i z a b i l i t y b e i n g a b o u t 5 % s m a l l e r t h a n t h e S C F v a l u e . D i s p e r - sion a l s o m o d i f i e s the o b s e r v a b l e b y a f e w p e r c e n t . N o t e that the c o m p a r i s o n in T a b l e 2 is b e t w e e n S C F
a n d c o r r e l a t e d r e s u l t s o b t a i n e d w i t h d i f f e r e n t b a s i s sets.
M a g n e t i z a b i l i t y a n i s o t r o p i e s a r e a l s o r e p o r t e d in T a b l e 2. R e c e n t d a t a f o r e x p e r i m e n t a l a n i s o t r o p i e s e x i s t in t h e l i t e r a t u r e [32], a n d the a g r e e m e n t w i t h o u r r e s u l t s is a c c e p t a b l e , e s p e c i a l l y at t h e c o r r e l a t e d level.
T h e d i s p e r s i o n o f the e l e c t r i c d i p o l e p o l a r i z a b i l i t y Table 2
Electric polarizabilities, magnetizabilities and their averages and anisotropies at different frequencies (au)..,C in cm3G - 2mol-~(4,rreo) , Q(T) in au. An,, = An(P = 1 atm, B = ! T ) = 1.64518 X 10 7 T - i XmC[cm3G- 2mol(4~eeo)]. At~.,:j = t ~ . - ot~j; AX..i~ = X . - X1j
SCF - Basis IVb CAS2 - Basis IVi
7` (nm) ~ 632.8 5 i,~.5 455.6 ~ 632.8 514.5 455.6
0% 28.218 a 29.134 b 29.472 30.084 26.743 c,d 27.486 e.f 27.757 g 28.241
ot~ 24.787 25.240 25.401 25.683 24.499 24.944 h 25. I 01 25.377
a~y 22.977 23.682 23.943 24.413 22. ! 66 22.802 23.035 23.453
azz 36.890 38.479 39.073 40.156 33.563 34.712 35.134 35.892
Aotx~.y ~ !.81 1.558 i !.458 i.270 2.333 o 2.142 j 2.066 1.924
AOtyy,z z - 13.913 - 14.797 - 15.13 - 15.743 - ! 1.397 - 1 !.91 - 12.099 - 12.439
Aa~x,z z - 12.103 - 13.269 - 13.672 - 14.473 - 9 . 0 6 4 - 9 . 7 6 8 - 10.033 - 10.515
AA'~.yy k 2.136 !.9925
AXy~,zz - !.3536 - 1.3562
Aa'~,z z 0.7824 0.6363
A~ 66.461 61.00 ± 7.97
(273.15) i019,55 999.98 993.96 978.66 1104.97 1094.73 1089.82 1079.92
,,~C X 10 ~a (273.15) 4.08 4.01 3.99 3.93 4.38 + 0.02 4.34 ± 0.02 4.33 ± 0.02 4.29 ± 0.02
Anu(293.15)× 10 ~ 2.29 2,25 2.24 2,21 2.46__.0.01 2.44± 0.01 m 2.43±0.01 n 2.41 +0.01
a SCF c%, : 28.47 au; a~,: 24.655 au; oey r 22.965 au; of..: 36.769 au, Ref. [5].
b S C F - T D H F a,,: 28.02 au; oq~: 24.622 au; ayr: 23.064"au; a:.: 36.373 au. at 7` --- 644.3 rim, Ref. [36].
c Experiment: 28.47 au, Ref. [37].
d Experiment, vibrational correction included (electronic contribution in parentheses): a,," 28.70 au (27.80 au); a.L.,: 24.6 __ 0.7 au (24,5 4- 0.7 au); a.vv: 24.1 ± 0.7 au (21.7 + 0,7 au); ,x::: 37.3 ~+ 0.6 au (3'7.1 ± 0.6 au). But also, with a different analysis of data: oq..
26,1 4- 0,2 au (26.0 4. 0.2 au); ayy: 24.8.4- 0.2 au (22.4 + 0.2 au); a:.: 35.2 ± 0.I au (35.1 ± 0.I au), see Ref. [14]
e MBPT(2) 27.41 au; C C S D 26.93; CCSD(T) 27.08. at 7` ffi 644.3 nm, Ref. [36]. Experiment (including zero-point corrections) 28.70, Ref.
138].
t Experiment, vibrational correction included: aa,: 28.58 au; a ~ : 28.0 + 0.4 an; avv: 23.4 + 0.2 au; a : . : 34.3 ± 0.5 au. But also, with a different analysis of data: a~x: 26.7 ± 0.2 au; ofyy: 23.0 ± 0.2 au; a . . : 36.0 ± 0.1 au. at 7` = 632.8 nm, Ref. [14],
z Experiment, vibrational correction included: a , , : 29.02 au; a~.~: 26.4 + 0.2 au; Ofyy: 23.4 ± 0.2 au, ~.." 37.2 ± 0.1 au at 7, = 514.5 nm,
Ref. [I 4]. ""
h a ~ : MBPT(2) 25.431 au; CCSD 24.934 au; CCSD(T) 25.148 au. o~v~: MBPT(2) 22.853 au, CCSD 22.033 au:, CCSD(T) 22.239 au. a . . :
MBPT(2) 34.034 au; CCSD 33.834 au; CCSD(T) 33.854 au. Ref. [36]. ""
i A a ~ ( l / ~ " ) [(~xx -- ~yy)2 4. (~xx -- Olzz )2 "l" ( ~ y y -- Otzz)2]l/2 = 12.60 au, SCF-TDHF. at A - 644.3 nm, Ref. [36]. Cf. 14.08 au, this work, 7` = 632.8 nm.
o Afif= (i//~/~") [(ax.~--~yy)2 .l_ (~,~ z _ Or.z)2 " t ' ( ~ y y - G,z)2]I/2 = !1.46 au, exp., Ref. [39]. Cf. our 13.10 au, 5CF, and 10.43 au, CAS2.
J A ~ m ( 1 / ~ " ) [(Ot.~,x_ ~vv)2 + ( a . ~ _ ~ . , ) 2 "1- ( ~ v v - ~:.:)2]1/2 m !0.17 ~u, MBPT(2); 10.65 au, CCSD; 10.47, CCSD(T); at A = 644.3 nm. Ref. [36]. 10.98, experiment, Ref. [38]~'Cf. ! 1.00 au, ~is work, 7` = 632.8 nm.
k Experiment: (2X.. - A'~ - A'vv) ffi 0.962 4- 0.027 au; (2A'~ ~ - A'vv - X::) = 2.442 ± 0.057 a u ; ( 2 X v y - X : : - ,Yx~) = -3.403 +_ 0.053 au;
Ref. [32]. Cf. our ()~5712, 2.918,~, 3.4896 (SCF); 0.7199, 2.6288, 3.3487 (CASSCF).
, Experiment: A'¢ = - 2 2 4- 67 au, 7` ffi 632.8 nm. Ref. [13].
m Experiment: A n,,(293.15) ffi (2.96 ± 0.22) x I 0 - ~ 3 Rvf. [ ! 2], A = 632.8 nm. A n,,(293.15) = (3. i 3 ± 0.06) x ; 0 ~ 3 Ref. [ ! 3J, A -- 632.8 rim.
" Experiment: An,,(293.15) ffi (2.47 + 0.24) x 10-,3 Ref. [I !], A = 546.1 nm.
S. Coriani et a l . / Chemical Physics 216 (1997) 53-66 61
and its anisotropy is reflected in the Q(T) contribu- t|on to the Cotton-Mouton effect, as shown in Fig.
1. The moderate effects of the dispersion, which is slightly stronger at the SCF than the correlated level, and of correlation itself are evident.
We are aware of three experimental values for the Cotton-Mouton constant of gaseous ethylene [11- 13]. In terms of the direct observable - that is, the anisotropy of the refractive index at P - 1 atm and B = 1 T, A n , , - we compare our correlated value at A = 632.8 rim, 293.15 K, (2.44 + 0.01) × 10-~3 to Corfield's (2.96 + 0.22) × 10 -13 [12] and to the more recent value of (3.13 __+ 0.06) × 10 -~3 of Kling et al.
[13], and we compare our value at A = 514.5 nm, 293.15 K, (2.43 -I- 0.01) × 10- ] 3, to the 1967 esti- mate by Buckingham et al. (2.47 + 0 . 2 4 )× 10 -13 [11 ]. The agreement with Ref. [ 11 ] is remarkable, whereas a disagreement, in particular with Ref. [13], appears in the two other cases. This disagreement cannot be ascribed to the uncertainty in our hyper- magnetizability anisotropy (whose contribution to the CME of ethylene is quite small), or to an incom- plete description of correlation or dispe, rsion effects.
The anisotropies entering Eq. (9) should in principle be zero-point vibrationail7 averaged as well as in- clude pure vibrational c6ntributions in polar a n d / o r paramagnetic molecules [9]. It seems quite unlikely that the effect of the zero-point vibrational average, neglected in this paper, may be responsible for such a large deviation of our results from the e×perimental values. Vibrational averages apd especially pure vi- brational contributions to magnetizabilities, electric dipole polarizabilities and in pexticular hypermagne- tizabilities can be quite important, but it is unlikely that they can be invoked to explain a 22% disagree- ment for a system whose magnetic birefringence is strongly dominated by the L&agevin's orientational contribution.
4.4. Carbopi nuclear magnetic shielding polarizabili- ties
Table 3 summarizes our results for the nuclear magnetic shielding and its polarizabilities for the
~, 'il" "~
carbon atom of eth),lene. Tt~c ,olar~zao ~1l~s are compared with the only literature data available for
1120
~100
1000
1060 Q(T) (a.u.)
1040
1020
1000
980
960
~mmm'mmm,.m~mm,mmam,.~m,~
[
~ S C F I- - -
• • • , i g • e w •
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.00 0.09 0.1
Frequency (a.u.)
Fig. !. Frequency de~ndence of the temperature dependent Q(T) term, Eq. (9).
62 S. Coriani et al./ Chemicai Physics 216 (!997) 53-66
the individual tensor components: the SCF estimates of Grayson and Raynes [15]. Whereas the authors of Ref. [15] report results only for the shielding polariz- abilities that enter the rotational averages in Eqs.
0 4 ) to (17), we have in Tattle 3 listed all non- vanishing components of the tensors tr, or' and o'".
As for the magnetizabilities, the IVi basis set provides SCF results that, for the chemical shielding at the carbon atom, do not differ much from those obtained with the larger sets. The actual average value (57.476 ppm) is 4% smaller than Gauss' esti- mate in Ref. [42] and 5.5% smaller than that of Grayson and Raynes [15]. Again, FV-CAS overesti-
mates the correlation effects, in particular for tzxx,
~ and, consequently, for the average tra~, whereas Cryy is essentially unaffected. O,,r CAS2 result (Cra~
= 76.188 ppm) is apparently still too large - about 7% larger than Gauss' MBPT(2) or CCSD results [43]. Note that Gauss improves his agreement with experiment ( ~ =64.5 ppm, Ref. [44], about 18%
smaller than our best correlated estimate) by employ- ing an MBPT(4) approach [43].
A surprisingly good convergence with respect to the basis set size is observed for the majority of the chemical shielding polarizabilities. With the excep- tions of ~.yy (which increases in magnitude by
Table 3
Shieldings and shielding polafizabilities for C(0,0, z) (ppm and ppm au) S C F
This w o r k Ref. [ 15]
Basis IVi Basis IVa Basis IVb
F V - C A S C A S 2 This w o r k Tnis w o r k
Basis IVi Basis IVi
o'av 57.508 57.476 57.476 a 60.840 82.479 76.188 b
o'xx - 86.333 - 86.380 - 86.376 - 80.960 - 34.467 - 45.630
o'yy ! 76.599 176.6 ! 5 176.610 ! 78.861 178.101 i 79.930
o' u 82.256 82.192 82. ! 93 84.617 ! 03.804 94.264
o~x.z - 1714.5 - 1715.7 - 1716.0 - 1666.4 - 1223.7 - 1365.8
oy'y.z 170.4 ! 70.2 170.3 163.7 183. i 180.6
tr~.~ - 1959.3 - 1960.3 - 1960,3 - 1929.3 - 1453.0 - 1548.7
tz~'.,~ 385. I 385.5 385.7 329.5 358.0
¢~,x 753.6 753.5 753.3 586.6 640.7
%'z,y 98.8 99. I 99.2 95.8 95.6
O'z'y,y -- 299.9 -- 299. I -- 298.9 -- 217.4 -- 226.7
O'~'x ,~ ~ ! 417.7 1432,7 1429.7 1202.5 - 8.2 664.0
O'~,yy 6665.0 6649.3 6638.4 6796.5 350.2 2084.0
¢"xx,u 9211.8 9224.2 9221.8 8686.0 - I i 04.4 924.0
O'~,xx - 7389.5 - 7376.6 - 7357.4 - 7029.5 - 7320.4 - 7026.4
o'~.yy 82.9 87.2 93.0 157.5 - 92. ! - I 01.8
o'~,zz 438.9 421.5 4 ! 2.7 416.7 ! 98.6 353.3
O",~ ,xx 1801.8 ! 809.3 1849. 2 i 797.0 1065,5 1613.9
O',~.~,y - 697.8 - 749.8 - 832.9 - 848.0 - 596.6 - 1390.2
o'~,u - 10448. - 10493. - 10490. - 10399.3 - 8 2 3 0 . 0 - 8 7 9 2 . 8
o'~,xz i 973.9 1965.6 1967.5 1006.8
o'z~.xz 2154.1 2152.0 2151.9 1609.9
o'~.xy - 2188.8 - 2203.5 - 2211.5 - |398.4
o'~.xy - 3 1 7 3 - 303.3 - 300. ! - 27.4
o'~.y z I 1.6 4.6 - 0.4 - 283.3
u,~,yz - 1719.5 - 1708.2 - 1690.3 - 1382.7
Az ! 167.8 1168.6 ! 168.7 1144.0 c 831.2 91 !.3
Bxx 695.0 689. I 679.8 671.7 1043.8 79 i .4
Bye, - 1008.3 - 997.8 - 983. ! - 10 ! 7.7 56.4 - 98.7
Bn 139.5 141.1 142,6 217.1 1522.6 1252.6
a SCF 59,9, Ref. [42].
t, MBP'r(2) 71.20. Ref. [42]; MBPT(4) 67.90; SDQ-MBPT(4) 70.0; CCSD 71.30. Ref. [43]. Experiment 64.5. Ref. [44].
c Az _- 1144.5. Ref. [17].
S. Coriani et a l . / Chemical Physics 216 (1997) 53-66 6 3
about 16% going from basis IVi to IVb),
tTyy.yy
(which is 10% "~oo small in basis IVi), and O'y'~.y~
(which does not enter the rotational average and is much smaller than the other components), the per- centage variations are well below 10% along the SCF columns of Table 3. The basis set effect on
O';'y,yy
andO'zz,yy
is reflected inByy,
but to a much smaller extent (-- 2.5%).Comparison with Ref. [15] reveals an agreement to within 4% for the first polarizabilities, and agree- ments to within 6 % for most of the second polariz- abilities. Our tr~"~.~ is 16% smaller than that of Grayson and Raynes, whereas
O';'y,yy
is about 70%smaller. On the other hand, the rotational averaging reduces these disagreements for both B~ and Byy, differing by only l . l % and 3.5% respectively, whereas a cancellation between large polarizabilities of opposite sign gives rise to a 52% disagreement for Bz~, even though the differences in the individual components never exceed 6%. The reason for the strong disagreement between our results and those of Grayson and Raynes [15] for the two 'anomalous' second polarizab;dities, o'~.~ and O';y.yy is not easy to identify. We however note ',hat the authors of Ref.
[15] use a smaller double-zeta electric-field polarized basis set and that they discuss evidence hinting at the possibility of some basis set dependence.
Correlation has a strong effect on the polarizabili- ties ahd a dramatic influence on some of the aver- ages: the CAS2 Byy value is only one tenth of the SCF value and the CAS2 Bz~ value is alrr.ost ten times larger than the SCF value. For FV-CAS the effects are even more dramatic. The averages are once again affected both by the enormous variation of some tensor components (see for example o'~.~, which varies from +9221.8 ppm - SCF - to - 1 1 0 4 . 4 ppm - FV-CAS - to +924.0 ppm - CAS2) and by the cancellation in the combination of large contributions with opposite sign. As for the mixed-index hypermagnetizabilities, it proved im- possible to compute the mixed-index shielding polar- izabilities with the larger CAS2 wave function, and we had instead to resort to the FV-CAS results, which most certainly overestimate the e"fects of correlation.
All in all, some of the conclusions of our preced.
ing studies on other molecular systems concerning the shielding polarizabilities [3,4] can easily be ex-
T a b l e 4
S h i e l d i n g s a n d s h i e i a m ~ , p e l ' : ~ z a ~ ' d t ; . e ~ f o r H ( 0 , O , z ) ( p p m a n d p p m a u )
S C F F V - C A S C A S 2
T h i s w o r k R e f . [16] T h i s w o r k T h i s w o r k B a s i s I V b B a s i s I V i B a s i s I V i
O'av 2 6 . 0 9 2 6 . 6 9 2 6 . 9 2 2 6 . 4 8 a
O'xx 2 7 . 5 7 2 8 . 2 5 2 7 . 4 3 2 7 . 2 5
O'yy 2 5 . 7 5 2 6 . 5 5 2 5 . 9 4 2 5 . 8 0
trzz 2 4 . 9 5 2 5 . 2 7 2 7 . 4 0 2 6 . 3 8
O'xz 1.81 i .66 i .67 1.73
trzx 4 . 7 7 4 . 5 3 3 . 9 0 4.21
°'x'x.x - 9 . 9 - i 3.3 - 7.3
o-x'x, z - 6 0 . 6 - 6 5 . 3 - 5 6 . 7
tr~r,~x - 8 . 6 - 7.5 - 4 . 4
O'~y.z - 8 i .2 - 8 2 . 2 - 7 i .4
O'zz.x - 9 1 . 8 - 8 9 . 9 - 6 5 . 4
o-z'z, z - 5 0 . 2 -- 5 0 . 3 - 38~8
°'x'~.xx - 25 !. ! - 2 0 4 . 0 - ! 9 8 . 9
o-~t.yy - 5 8 1 . 2 - 5 1 4 . 5 - 4 2 3 . 1 - 5 2 5 . 0
o-,~'x .zz - 9 9 . 4 - 2 4 . 0 - 1 3 3 . 0
O'x'x.xz 1 9 0 . 0 122.7 1 ! 5 . 7
O-yy.x x - 2 4 9 . 2 - 2 8 0 . 3 - 2 6 5 . 5
try~.yy - 1 4 9 . 2 - 101.5 - 122.5 - 142.6
o'~.z z 2 0 . 7 3 3 . 3 - 8 4 . 8
tr~.xz - 73.1 - 6 8 . 2 - ! 8 8 . 7
o-z~.x,~ - 2 3 2 . 7 - 2 4 6 . 3 - 3 7 8 . 7
orz~.yy 1 4 3 . 4 156.5 - 5 i .3 - 2 7 . 6
crz'z.zz - ! 12.9 - ! 18.7 - 2 3 5 . 2
trz'z.xz - 56. ! - 8 7 . 7 - 142.8
A x 3 6 . 7 3 6 . 9 2 5 . 7
A z 6 4 . 0 6 6 . 7 5 5 . 6
Bxx 1 2 2 . 2 ! 2 1 . 8 1 4 0 . 5
Byy 9 7 . 6 7 6 . 6 9 9 . 5 I 15.9
B u 3 1 . 9 18.3 7 5 . 5
Bxz - 2 0 . 3 - ! 1.0 3 8 . 6
a E x p e r i m e n t : 2 5 . 4 3 . R e f . [34].
tended to ethylene. Correlation is extremely impor- tant and its effects are not easy to predict, and a limited CASSCF approach is often not sufficiently flexible to guarantee a proper description of the electric field effects on the chemical shieldings. A similar conclusion was drawn by Bishop and Cybul- ski with respect to MP2 [45].
4.5. Hydrogen nuclear magnetic shielding polariz- abilities
Table 4 shows the nuclear magnetic shieldings and the components of the shielding polarizability tensors that contribute to the rotational averages for
64 S. Coriani et a l . / Chemical Phy~'ics 2/6 (1997) 53-66
the hydrogen atom located in the (0,0, + z) position in the geometrical arrangement of Section 3. For comparison, we report the corresponding estimates of Grayson and Raynes [16], who performed the calculations with the same geometry and with the origin of the magnetic vector potential at the proton of interest.
The calculation of the shielding polarizabilities at the H atoms could in principle be performed by resorting to a C2v arrangement, but only at the cost of increasing noticeably the number of finite-field arrangements. Each calculation requires several hours of CPU time on a high-performance SGI computer and several gigabytes of storage memory. However, our experience with methane [4] indicates that, at least for that molecule, t_h.e effect of correlation on the shielding polarizabilities of hydrogen is small. As pointed out above, the low symmetry (C s) finally adopted is not suitable for CAS2 calculations. We therefore restricted our study to SCF calculations with basis IVb and FV-CAS calculations with basis IVi. In Table 4, we include only few CAS2 results - the ones that could be extracted at no extra cost from the (C2v) calculations of magnetizabilities and car- bon nuclear shieidings.
Again, with a few exceptions (o-~"~.~ for example), the agreement with Ref. [16] is quite good for all tensor components, whereas some averages are in disagreement, see for instance B,, and B~,. Correla- tion is quite unimportant for the chemical shielding (our best value for %v, 26.48 ppm in the CAS2 approximation, is about 4% larger than experiment [34]) but, in contrast to what was observed for methane [4], not negligible for the polarizabilities.
FV-CAS, whose limitations already have been stressed in the preceding discussion, has a strong effect on most of the components, and reverses the sign of o-y~.~ and tr~z.yy. The few CAS2 estimates in N
Table 4 show once again how this complete active space tends to p~aly cancel the correlation effects predicted by FV-CAS. Nonetheless, for instance, tr~"z.yy, which is equal to 143.4 ppm at SCF level, is computed at - 51.30 ppm by FV-CAS and at - 27.6 ppm by CAS2. According to FV-CAS, the rotational averages are also strongly affected by correlation and we conclude that SCF does not seem appropriate for describing the shielding polarizabilities of the hydro- gen atoms of ethylene.
5. Summary
We have presented the results of extended mag- netic gauge-origin independent SCF and MCSCF calculations of hypermagnetizabilities and shielding polarizabilities of ethylene. To our knowledge, these are the first correlated results for both properties.
Our study has been extended to obtain as by-prod- ucts correlated results for the magnetizability and for the frequency dispersion of the electric-dipole polar- izability. Comparisons with experiment have been made for the Cotton-Mouton effect and the hyper- magnetizability anisotropy. There is no experimental counterpart of the CME that can afford data for comparison of our results for the shielding polariz- abilities.
The use of different extended basis sets and of two complete active spaces enabled a detailed study
^¢ the basis set and correlation effects on the proper-
O l
ties. It is not unexpected in view of the results of our previous studies [ 1-4] and those of others ill the field (see Refs. in [9]), that we need 212 to 280 basis functions in order co achieve basis set convergence, and that correlation effects are in some case dra- matic, to the point that, especially for the shielding polarizability, it is difficult, if not impossible, to claim convergence with respect to the description of the correlation effects even with as many as six million variational parameters in the response calcu- lations.
We were not able to study the frequency depen- dence of the hypermagnetizability anisotropy, some- thing that would have been possible using a finite magnetic-field approach and complex algebra from the second derivatives of the electric dipole dynamic polarizability - as done by Bishop and coworkers [5,46] - or by resorting to Cubic Response, now available at the MCSCF level of approximation [8].
On the other hand, the dominance of the orienta- tional term in the CME of ethylene permits a quite accurate study of the dispersion of the observable itself - the anisotropy of the refractive index - through an analysis of the dispersion of the electric- dipole polarizability anisotropy. Also, in this study we have neglected zero-point vibrational averages and pure vibrational contributions to the observables.
These corrections can be large, especially for the ,,;hielding polarizabilities, but their calculation can be
S. Coriani et a l . / Chemical Physics 216 (1997) 53-66 65
quite challenging [9,47]. Some recent advances in this field appear to be promising [48].
The importance of ethylene as the first member in the series of polyenes, whose peculiar nonlinear optical properties are exploited in the design and development of optical polymeric materials, is note- worthy. This work is intended as a contribution to a wider understanding of the mechanisms governing this nonlinearity in a field - that of magnetic in- duced birefringence and electric field dependence of magnetic properties in general - that is rapidly ex- panding.
Acknowledgements
Most of the correlated calculations were per- formed on the SGI computer at the Department of Chemistry of the University of ,~a'hus, and many thanks are due to Poul Jorgensen and to the whole Theorctical Chemistry group there, for their assis- tance and patience. These studies were made possi- ble by the efforts over the years of several re- searchers. The invaluable help of Micha| Jaszufiski, who has been a constant source of advice and com- ments, must be acknowledged. This work has re- ceived support from the Norwegian Supercomputer Committee (TRU) through a grant of computing time.
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