It was shown that the application of the additivity rules [Chem.
Phys. Letters 13 (1972) 571] may be very helpfull for the determination
of the sign of the g tensor

Z. B. Maksić

English

Hrčak - Portal of scientific journals of Croatia

CR 0 AT IC A CHEMIC A ACT A 45 (1973) 431 CCA-780 539.19:538.214 Note On the Additivity of the Molecular Second Moments of Charge and Diamagnetic Susceptibility z. B. Maksic Institute »Ruder Boskovic, 41001 Zagreb, Croatia, Yugoslavia Received February 5, 1973 It was shown that the application of the additivity rules [Chem. Phys. Letters 13 (1972) 571] may be very helpfull for the determi-nation of the sign of the g tensor. Recently, a large number of experimental molecular quadrupoles, dia-magnetic and paramagnetic susceptibilities and second moments of charge became available1• This immence pool of experimental data gave strong impetus for the theoretical treatment of these quantities on the semiempirical level2-4 • The additivity of second moments was put forward first by Flygare et al.5 who have shown that the second moment related to the out of the heavy atoms plane coordinate c is easily obtained by using simple empirical additivity rules. These rules were subsequently rationalised within the framework of the semiempirical CND0/2 method where < c2 > second moments were cal-culated in the one-center approximation2• Flygare et aI.6 established recently an empirical scheme based on atomic dipoles which enables highly accurate estimates of second moments. This method takes into account the anisotropy of the electronic charge distribution in the vicinity of nuclei and also employs different parameters for the same atom in different moieties. We have shown7 that the molecular second moments of electronic charge can be easily calcu-lated, albeit not so accurate, by an additivity equation which employs only few parameters A toms < o I v 2 ! o > = ~ Z 11 v n 2 + ~ kp n p (1) n P where v denotes coordinates x, y and z of the molecular system of coordinates*, I 0 > is the ground state Slater determinant, Z 11 is atomic number of the nucleus n, nP is a number of atoms belonging to the p-th period of the Men-deleev system of elements and kv is a constant characterising p-th period. It is seen from the formula (1) that the atomic contribution to the molecular second moment is not only isotropic but i.t is also constant for all a toms be-longing to the same period. The values kP found earlier7 read as follows (in 10-16 cm2): * v refe.rs frequently to the principal inertial axes coordinates a, b and c. M:>- w TABLE I ~ Comparison between the Calculated and Experimental Second Moments and Diamagnetic Susceptibilities Molecule Calcd. Exp. I Calcd. Exp. ::E Z a2 = 22.1 < a2> = 25.1 25.2 X~a --43.7 -42.0 /o°""-n n ::EZnb; = 4.3 <b2> = 7.3 6.9 xgb = -49.2 -119.7 F F ::E Zn c; < c2> = 3.0 3.0 X~c = -137.4 -136.2 -0 0 ::E Z a2 = 19.0 < a2> = 22.0 25.7 ± 1.2 X~a = -35.6 -36.8 ± 2.6 o/ ""-o n n N ::EZnb; = 2.4 <b2> = 5.4 5.4 ± 1.2 x%b = -106.1 -122.6 ± 2.6 tJj ::E Znc; 0 < c2> = 3.0 3.2 ± 1.2 X~c = -116.2 -132.0 ± 4.6 ~ x!/F ;i:,. ::EZ x2 X~x = -68.7 ~ = 20.9 <x2> = 23.9 24.0 -66.711 rn n n ...... N~z ::E Zny~ <y2> = d -157.4 0 = 0 3.0 2.6 Xyy = -157.4 0 ::E Z z2 = 10.2 < z2> = 13.2 13.1 X~z = -114.l -112.6 n n xi H ::EZnx; = 0.8 <x2> = 4.0 3.8 X~x = -74.7 -77.712 N-N-NLz ::EZny; = 0 <y2> = 3.2 3.0 X~y =-78.1 -80.9 ::E Z z2 = 11.2 < z2> = 14.4 15.2 X~z --30.5 29.2 n n xt H I ::EZ x2 = 0.6 <x2> = 3.8 3.6b X~x = -130.7 -132.012 n n ::E Zny; = 0 <y2> = 3.2 3.1 X~y = -133.2 -134.5 . N/ o-c--z ::E Z z2 = 24.4 < z2> = 27.6 28.1 X~z = -29.7 -28.4 n n TABLE I (contd.) Molecule H,,, /H""" /H efB"' /B~ H H H s n...c/ """ -.; co3 H H ""N-C/ H/ ""O "'\°Ye 0 0F N ~ Znx; = 4.3 ~ Zny; = 2.0 ~ Znz; = 16.8 ~ Zna; = 40.1 ~ Znb; = 15.4 ~ Znc; = 3.2 ~ Zna; = 25.2** ~ Znb; -5.5 ~ Znc; -0 ~ Z0a; = 32.6 ~ Znb; = 31.4 ~Znc; = 0 ~ Z0a; = 78.2 ~ Z11b; = 48.3 ~ Znc; = 0 ~ Z0a; = 98.7 ~ Znb; = 49.9 ~ z11c; = 0 Calcd. < x2 > = 7.5 < y2 > = 5.2 < z2 > = 20.0 < a2 > = 44.8 < b2 > = 20.1 < c2 > = 7.9 < a2 > = 28.8 < b2 > = 9.1 < c2 > = 3.6 < a2 > = 38.4 < b2 > = 37.2 < c2 > = 5.8 < a2 > = 86.5 < b2 > = 48.6 < c2 > = 8.3 < a2 > = 106.5 <b2>= 57.7 < c2 > = 7.8 Exp. 45.0 ± 0.8 21.2 ± 0.8 9.0 ± 0.8 28.3 ± 0.9 8.4 ± 0.9 4.0 ± 0.9 37.4 ± 0.7' 37.7 ± 0.7 6.9 ± 0.7 86.5 ± 3.4" 47.9 ± 3.4 9.8 ± 3.4 105.7" 57.2 ± 5.1 8.7 Calcd. :X:~x = -106.2 d :X:yy = -115.6 d :X:zz = -53.8 :X:~a = -118.8 :x;gb = -223.6 :X:~c = -275.3 d :X:aa = -53.9 :x:gb = -137.4 :X:~c = -160.8 :X:~a = -182.4 :x;gb = -187.5 :X:~c = -320.7 Xd -aa = -241.4 :x;gb = -402.2 :X:~c = -573.1 d :X:aa = -277.9 :x;gb = -484.9 d x cc = --696.6 Exp. -110.813 -122.7 -56.0 -128.1 ± 1.814 -229.3 ± 2.2 -280.8 ± 2.4 -52.8 ± 1.115 -137.3 ± 2.9 -156.0 ± 3.1 -189.1 ± 1.816 -187.7 ± 1.9 -318.7 ± 2.0 -244.8 ± 8.217 -408.5 ± 9.5 -570.5 ± 10.0 -279.5 ± 10.08 -485.5 ± 15.1 -691.3 ± 15.1 ~ 0 t"' M () c:: t"' ;t> :i:J Ul M () 0 z t:l ~ 0 ~ M ~ ..., Ul >l=>- w • The experimental data are taken from the ref. (1) if it is not stated otherwise; •• A planar structure of formamide was assumed. w 434 Z. B. MAKSIC p 0 1 2 3 4 - --- ··----kn I I 0.2 1 2.5 3.5 5.5 I Here we give some more evidence for the additivity of the second moments in the sense of the equation (1) and for the additivity of the closely related diamagnetic susceptibility. We shall discuss in particular the potential use of the eqn. (1) in determining the sign of the g tensor. The earlier approaches used additivity of < c2 > second moments5 or semiempirical CND0/2 method4. The second moments obtained by the eqn. (1) and the diagonal elements of the diamagnetic susceptibility tensor X~a = (Le2/4mc02) < o I b2 + c2 Jo>* are compared with some ab initio results and the most recent experimental. data in the Table I. The agreement with experiment and ab initio calculations is very good, the exception being ozone. This discrepancy indicates that either some special bonding feautres are present in this molecule or perhaps that the experimental results should be reinvestigated. The second moments in 2-fluoropyridine are not unambiguously determined by the experiment due to the indeterminacy in sign of the g values8. The set of second moments presented in Table I corresponds to the diagonal g components - 0.088, - 0.041 and 0.023 for K1a, g0b and gee respectively. The alternative choice of sign in g would lead to the negative < c2 > = -10.7 value and could be rejected as physically unacceptable since < v2 > is always positive. Nevertheless the fine agreement obtained for the former set of second moments illustrates the potential capability of the eqn. (1) to make a proper choice of sign, what could be very useful in cases where both sets of second moments are physically acceptable. If the bulk magnetic susceptibility is not known then only the anisotropies in the second moments could be experimentally obtained. The anisotropies for dimethylketene, propynal and SF., are listed in the Table II. The agreement with experiment for propynal and SF, is satisfactory. There are two possible sets of second moments for dimethylketene9. The correspond-ing values are close to each other and it is hard to make a choice. Despite of that it seems that the set g < 0 is slightly preferable. To conclude, one can say that the present results provide further support of the simple additivity formula (1). It gives also a very simple criterion for the choice of the sign of g values. Therefore it is not n ecessary to perform semiempirical calculation of second moments for this purpose as we reco-mended earlier4 since the straightforward application of the formula (1) will suffice if the two possible sets are not too close to each other. It should be mentioned that the second moments calculated by the eqn. (1) could be easily transformed to the gauge invariant values10 simply by transforming the terms ~ Znv~ to the center of the positive charge of a molecule. n * The remaining two expressions are easily obtained by the cyclic permutation of the coordinates while the constants have their usual meaning. TABLE II Comparison between the Calculated and Experimental Anisotropies in Second Moments Molecule HF""-c=c~o 11:3c/ H / H-C=C~ 0 F /___.F s·· ~F F ~ Z11a~ = 70.3 ~ z b2 = 36.9 n n ~ Z11c~ = 3.2 ~ Z 11a~ = 56.9 ~ Z11b~ = 5.0 ~ Z11c~ = 0 ~ Z a2 = 48.6 n n ~ Znb~ = 25.8 ~ Znc~ = 10.1 Calcd. g<O Exp. < a2 > -< b2 > = 33..4 33.8 ± 1.0" < b2 > -< c2 > = 33.7 34.4 ± 0.9 < c2 > -< a2 > = -67.1 -68.2 ± 1.1 < a2 > -< b2 > = 51.9 51.6 ± 0.518 < b2 > -< c2 > = 5.0 4.3 ± 0.7 < c2 > -< a2 > = -56.9 -55.9 ± 0.4 < a2 > -< b2 > = 22.8 24.3 ± 0.919 < b2 > -< c2 > = 15.7 17.0 ± 1.3 < c2 > -< a2 > = -38.5 -41.3 ± 1.1 g>O 31.3 ± 1.29 34.1 ± 0.9 -65.9 ± 1.1 ~ 0 t"' tr:! (') c: t-< ;:.. ::d r.n tr:! (') 0 z ti ~ 0 ~ tr:! z ..., r.n ~ C;.) (JI 436 Z. B . MAKSIC Acknowledgment. I would like to thank to Dr. W. Hilttner for critical reading of the manuscript and useful comments. REFERENCES 1. W. H. F 1 y gar e and R. C. Benson, Mol. Phys. 20 (1971) 225, and the refe-rences cited therein. 2. J.E. Bloor and z. B. Maksic, Mol. Phys. 22 (1971) 351; Z. B. Maksic and J. E. B 1 o or, Croat. Chem. Acta 44 (1972) 435. 3. J . R. Rabinowitz and R. Rein, Int. J . Quant. Chem. 6 (1972) 669; R. Rein, G. R. Pack and J. R. Rabinowitz, J. Magn. Res. 6 (1972) 360. 4. J . E . B 1 o or and Z. B. Maks i c, J. Chem. Phys. 57 (1972) 3572. 5. R. P. B 1 i ck ens de r fer, Y. H. S. Wang, and W. H. F 1 y gar e, J. Chem. Phys. 51 (1971) 3196. 6. T. D. Gierke, H . L. Ti g e 1 a a r, and W. H. F 1 y gar e, J. Am. Chem. Soc. 94 (1972) 331. 7. Z. B. Maksic and J.E. Bloor, Chem. Phys. Lett.13 (1972) 571; Z. B. Maksic and J. E. B 1 o or, J. Phys. Chem. (in press). 8. D. Sutter, Z. Naturf. 26a (1971) 1644. 9. D. Sutter, L. Ch a r pent i er, and H. Dre i z 1 er, Z. Naturf. 27a (1972) 597. 10. L. Weise nth a 1 and A. M. de Graf, Phys. Rev. Lett. 27 (1971) 470. 11. Ab initio results: C. Petrongolo, E. Scrocco, and J. Tomasi, J. Chem. Phys. 48 (1969) 407. 12. Ab initio results: R. Bonacconsi, C. Petrongolo, E Scrocco, and J. Tom as i, J. Chem. Phys. 48 (1968) 1500. 13. Ab initio results: E. A. Laws, R. M. Stevens, and W. N. Lipscomb, J. Am. Chem. Soc. 94 (1972) 4461. 14. E . Hamer, D. H . Sutter, and H. Dre i z 1 er, Z . Naturf, 27a (1972) 1159. 15. H. L. Ti g e 1 a a r and W. H . F 1 y gar e, J. Am. Chem. Soc. 94 (1972) 343. 16. B. Bak, E. Hamer, D. H . Sutter, and H. Dre i z 1 er, Z. Naturj. 27a (1972) 705. 17. W. Czieslik, D. Sutter, H. Dreizler, C. L. Norris, S. L. Rock, and W. H . F 1 y gar e, Z. Naturf, 27a (1972) 1691. 18. R. C. B ens on, R. S. S c o t t, and W. H. F 1 y g a re, J. Phys. Chem. 73 (1969) 4359. 19. R. G. Stone, H. L. Ti g e 1 a a r , and W. H. F 1 y gar e, J. Chem. Phys. 53 (1970) 3947. IZVOD 0 aditivnosti drugih momena ta elektronskog naboja molekula i dijamagnetske susceptibilnosti z. B. Maksic Pokazano je da primjena piravila aditivnosti [Chem. Phys. Letters 13 (1972) 571] moze pomoci kod odredivanja predznaka g-tenzora. INSTITUT »RUDER BOSKOVIC« 41000 ZAGREB Primljeno 5. veljace 1973. 

On the Additivity of the Molecular Second Moments of Charge and Diamagnetic Susceptibility

Abstract

Similar works

Full text

Available Versions

Hrčak - Portal of scientific journals of Croatia