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BAND 27 b ZEITSCHRIFT FÜR NATURFORSCHUNG HEFT 1

The Nephelauxetic Effect — Calculation and Accuracy of the Interelectronic Repulsion Parameters

II. Application to d3 and d7 Single Crystal Spectra at Cryogenic Temperatures *

E . K Ö N I G

Institut für Physikalische Chemie II, Universität Erlangen-Nürnberg, 8520 Erlangen, Germany

(Z. Naturforsch. 27 b, 1—5 [1972] ; received September 28, 1971)

Expressions are reviewed which may be used to determine 10 Dq and B from the spin-allowed bands in the optical spectra of d3 and d7 electron systems within octahedral and tetrahedral sym metry. Application to low-temperature single crystal spectra demonstrates that (i) the semi-empiri-cal ligand field theory reproduces transition energies with sufficient accuracy; (ii) differences in the values of 10 Dq and B observed with different fitting methods may be attributed to the in-accuracy of experimental data; (iii) there are generally valid values of B35 and /?33 for each com-plex ion.

The semi-empirical ligand field theory provides means to completely determine the electronic d — d spectra of transition metal ions of octahedral sym-metry in terms of three parameters: the octahedral splitting parameter 10 Dq ( = A) and the inter-electronic repulsion parameters ( = Racah para-meters) B and C which are linear combinations of the C o n d o n - S h o r t l e y parameters F2 and 1. An analogous statement applies to tetrahedral sym-metry and one or two additional parameters (e. g. Ds and Dt in tetragonal environment) are required if the symmetry is lower than cubic. In general, the numerical values of the parameters B and C, as determined from d — d spectra, are lower than the values in a free transition metal ion. This observa-tion is well known as the nephelauxetic effect 2' 3.

In the first part of this study 4, the author has re-cently reviewed and tested methods which may be used to determine 10 Dq, B, and C from electronic spectra. To focus the attention on the value of B, these methods were applied to the spin-allowed d — d bands in high-spin d2, d3, d7, and d8 complexes of octahedral and tetrahedral microsymmetry. In the expressions of the corresponding transition energies, the parameter C does not occur5. In addition, 10 Dq may always be fixed by a suitable choice of the calculation method. A convenient check on the accuracy of the employed numerical procedure is provided by calculating the extra band energy, if

Requests for reprints should be sent to Doz. Dr. E. KÖNIG, Institut für Phys. Chem. II der Universität Erlangen-Nürn-berg, D-8520 Erlangen, Fahrstr. 17.

those complex ions are considered where all three spin-allowed d — d bands are observed. Room tem-perature solution and single crystal spectra of al-most fifty complexes and impurity ions of the transition metals were subject of the analysis. The results 4 may be summarized as follows: (i) The accuracy of B and 10 Dq depends on the

method adopted to their calculation. Conse-quently, certain methods may be selected which provide the "best" possible fit to the experi-mental data;

(ii) Given a specific method of calculation, an un-systematic variation in the deviations between calculated and observed transition energies is often encountered. It was suggested that this is due to insufficient accuracy of the experi-mental room temperature (solution) data.

In the present contribution, the same methods as used previously4 will be applied to single crystal spectra measured at cryogenic temperatures. It will be demonstrated that results somewhat different from those of room temperature spectra are ob-tained.

I. Ligand Field Theory of d3 and d7 Ions in Cubic Fields

The general treatment of ligand field theory is adequately covered in several textbooks 7 - 9 to which

* For the first part of this study refer to E. KÖNIG, Struct. Bonding 9, 175 [1971].

G ZT-I'H)^

E. KÖNIG

reference is made here. With respect to the single crystal data available at present, we will concentrate primarily on the theory of d3 and d7 ions in cubic fields. The relevant energy expressions have been derived previously4. For convenience, we will briefly introduce those quantities and list explicitly those expressions which will be needed in the sub-sequent numerical calculations.

Thus in the octahedral d3 configuration three spin-allowed transitions from the 4A2g ground state to the excited states 4T 2 g , a 4 T l g , and b 4T i g are expected. Within the approximation considered here, the energy of the lowest transition is always determined as vx (4A2g —> 4T2g) = 10 Dq. The ener-gies of the two higher transitions follow from

>'2,3= 2 (15B + 3O0<7)+-2 t ( 1 5 ß - 1 0 ^ ) 2

+ 12 B • 10 Dq]1'1. (1)

The parameter B may then be obtained according to four different methods: (a) fitting the second band,

5 = (2v12 + v22-3vlv2)/ (15 v2 — 21 vt), (2)

(b) fitting the third band, B = (2 Vi2 + f32 - 3 n r3) / (15 v3 - 27 vt), (3)

(c) fitting the sum of the second and third band, B = ( v 2 + r 3 - 3 v 1 ) / 1 5 , (4)

(d) fitting the difference between the second and third band,

B = ^ [3 f ! ± (25 0>3 - v2)2 - 1 6 V ) I / 2 ] • (5)

The expressions (1) to (5) apply to tetrahedral d7

ions as well. In the octahedral d7 configuration, the ground

state is a4T lg and the excited quartest states are, in the order of increasing energy, 4T2 g , 4 A 2 g , and b 4T l g . The energy of the three spin-allowed transi-tions is determined according to

"i (a4Tig 4T2g) = I (10 Dq - 1 5 B) + ,

v2 (a4Tig -> 4A2g) = vx +10 Dq , (6) r3(a4T l g-^b4Tig) = [(10Z)q + 15 5) 2

— 12 5-10 Dq]1/!.

There are again four different methods which may be employed to obtain the parameters 10 Dq and B: (a) fitting the first and second band, 10 Dq — v2 — vl,

B= (2v12-v1v2)(Uv2-27 vt), (7)

(b) fitting the first and third band, 10 Dq = 2 ?>! — j>3 + 15 B, (8)

B = jo[-{2Vl~ ± ^ ~ + *»* + "1 >'3>,/!] ,

(c) fitting the second and third band,

10 Dq= I (2 v2 — v3) + 5 5 ,

ß = 510 2 ) ± 3 { 8 1 r s 2 - 1 6 ^ ( r 2 - r a ) } ' ' • ] .

(9)

(d) fitting the difference between the first and second band,

10 Z)g = — J'i, B=(v2 + V3-3V1)/15. (10)

II. Application to Single Crystal Spectra

In order to asses the accuracy of the parameter values of 10 Dq and B, the equations listed in sec-tion I will be applied below to some recent low tem-perature single crystal spectra. Following the first part of this study 4 it will be assumed, for the sake of argument, that the three-parameter (10 Dq, B, C) theory is valid exactly. The question then arises about the significance of the calculated transition energies. In ligand field theory, all energy dif-ferences are calculated at a constant value of 10 Dq (viz. "vertical" transitions in the T a n a b e - S u -g a no diagram). Since the relation 10 Dq ~ R~5

holds to a reasonable approximation 10, this is equi-valent to a fixed metal-ligand distance, R. In the spin-allowed d — d transitions considered here, the states involved originate in different strong field configurations g eg and, consequently, the potential minima of the excited state and the ground state do not coincide. The calculated transition energy cor-responds, therefore, to the energy of a transition from the zero-point vibrational level of the elec-tronic ground state to an excited vibrational level of the excited state (cf. "vertical" transition ac-cording to the F r a n c k - C o n d o n principle).

As far as the comparison between theoretical and experimental energies is concerned, two limiting conditions may be distinguished:

(i) In centrosymmetric (e.g. octahedral) com-plexes, all d — d transitions are rigorously forbidden on the basis of parity. The forbidden electronic transitions may gain intensity through coupling to odd vibrations (vibronic mechanism n ) . At low

THE NEPHELAUXETIC EFFECT 3

temperatures, each band thus consists of a progres-sion in one or more even vibrations superimposed upon one quantum of the odd ("permitting") vibra-tion. The no-phonon (0" —> 0') band is absent or of very weak intensity 12. Therefore, within reason-able approximation, the calculated vertical transition energy should be associated with the maximum of the vibronic band determined, in principle, at 0 °K.

(ii) In non-centrosymmetric (e. g. tetrahedral) complexes, the d — d transitions become partly al-lowed on account of mixing with odd-parity states of the central ion (e.g. p states11). One observes, at low temperatures, a progression in the totally symmetric vibrational mode originating in the no-phonon (0"—>-0') band. Normally, the highest intensity would be expected in one of the higher vibrational sub-bands 0 " - > n / e v e i l . However, in the example studied at present13'14, the most pro-minent band is associated with the no-phonon transition. This situation is encountered if ground and excited state potential minima occur at the same internuclear distances12. It has been suggested13, therefore, that the geometry of the Co2® ion in the relevant excited states is not greatly different from that in the ground state. Here it may be more ap-propriate to approximate the calculated vertical transition energy by the average of the energies of the no-phonon bands in the most intense spin-orbit components determined again, in principle, at 0 °K.

If the temperature is increased, in both cases higher vibrational levels of the electronic ground state become successively populated and the cor-responding band in the electronic spectrum is pro-gressively shifted to lower energy 15. The magnitude of the shift is dependent on the distribution of the vibrational levels in the electronic ground state and on the intensity of the resulting hot bands and will differ from compound to compound. Therefore, in general, spectra measured at higher than cryogenic temperatures cannot compare favorably with theory.

III. Results and Discussion

Results of the present analysis are compiled in Tables 1 to 3. For each compound, experimental transition energies determined according to sec-tion II are listed in line 1. Subsequent lines contain the calculated transition energies, their deviation from the corresponding experimental value, dv — ''calc (in cm 1 and in percent), and the values of the parameters B3- and ßS5 . In Table 2, values of 10 Dq and of the deviation, (5(10Z)g), from the value of v2 — vt are listed in addition. Each line ap-plies to a different method marked with reference to section I.

In the octahedral d3 configuration, the only well evidenced low-temperature single crystal spectrum where the 4A2g —> b4Tig transition has been unequi-vocally assigned is that of VC12 16 '17 . If the quasi-molecular model is assumed type (i) behaviour is expected. No fine structure has been observed in the 4A2g —> 4T2g and 4A2g a4Tig bands, and thus the band maxima listed in Table 1 are associated with the vertical transition energies. In the 4 A 2 g -> b4T lg band where a vibrational progression in a mode of ~ 234 c m - 1 is encountered, the first pro-minent vibrational component (viz. 22,244 c m - 1 ) seems to well approximate the centroid of the band (viz. Fig. 1 of 1. c . 1 8 ) . The parameter values calcu-lated by KIM et al.16 are incorrect, since applying eq (1) the fit was based on the estimated energy of the no-phonon transition. No suitable data on the d3 configuration in tetrahedral environment are known.

Recently, low temperature single crystal spectra of RbCoCl3 and of the Co2® ion in several chloride lattices became available 19. The a4Tig ground state of the octahedral d7 configuration is split by spin-orbit interaction into the -T6, r 8 a , .T8b and .T7

levels in the order of increasing energy20. Since and are K r a m e r s doublets, the a4Tig term

Compound Method vi, c m - 1 j>2, c m - 1 c m - 1 öv B35, c m - 1 ^35 4A2g 4 T 2 g 4 A 2 g ^ a 4 T l g 4 A 2 g —> b 4 T i g [ c m - 1 ] [ % ]

VC12 (22°K) exp 9300 14,220 22,244 a 10 Dq flitted 22,231 - 7 0.03 570.1 0.74 b 10 Dq 14,226 fitted + 6 0.04 571.3 0.75 c 10 Dq 14,224 22,240 ± 4 0.02 570.9 0.75 d 10 Dq 14,231 22,255 + 11 0.06 572.3 0.75

Table 1. Observed and Calculated Transition Energies of Octahedral Vanadium(II) (B^*® = 766 c m - 1 ) .

4 E. KÖNIG

Compound Me- vi, c m - 1 i>2, e m - 1 j'3, c m - 1 6 V £35 ßäö 10 Dq d (10 Dq) thod a 4 T l g ^ 4 T 2 g a 4 T l g - > 4 A 2 g a 4 T i g - > b 4 T i g [ c m " 1 ] [%] [ c m - 1 ] [%]

RbCoCls exp. 6450 13,500 17,210 b fitted 13,855 fitted + 355 2.56 781.0 0.80 7405 + 355 4.79 c 6278 fitted fitted - 172 2.74 791.8 0.82 7223 + 173 2.4.3 d 6134 13,184 fitted - 316 3.77 757.3 0.78 7050 0 0

K M g C l 3 : Co2© exp. 5740 12,000 16,475 b fitted 12,367 fitted + 367 2.97 774.8 0.80 6627 + 367 5.54 c 5563 fitted fitted - 177 3.18 785.8 0.81 6437 + 177 2.7.3 d 5415 11,675 fitted - 325 4.39 750.3 0.77 6260 0 0

CsMgCls: Co2© exp. 6450 13,300 17,190 b fitted 13,854 fitted + 554 3.99 779.6 0.80 7404 + 554 7.48 c 6181 fitted fitted - 269 4.35 796.5 0.82 7119 + 269 3.78 d 5957 12,807 fitted - 493 6.06 742.7 0.77 6850 0 0

MgClo : Co2© exp. 6450 13,570 17,025 b fitted 13,849 fitted + 279 2.01 768.2 0.79 7399 + 279 3.77 c 6314 fitted fitted - 136 2.15 776.8 0.80 7256 + 136 1.87 d 6202 13,322 fitted - 248 2.93 749.7 0.77 7120 0 0

CdCl 2 : Co2© exp. 6040 12,500 16,930 b fitted 13,001 fitted + 501 3.85 787.4 0.81 6961 + 501 7.20 c 5797 fitted fitted - 243 4.19 802.5 0.83 6702 + 242 3.61 d 5596 12,056 fitted - 444 5.81 754.0 0.78 6460 0 0

CsCdCl3 : Co2© exp. 5263 11,236 16,400 b fitted 11,376 fitted + 140 1.23 799.1 0.82 6113 + 140 2.29 c 5196 fitted fitted - 67 1.29 803.2 0.83 6040 + 67 1.11 d 5140 11,113 fitted - 123 1.75 789.8 0.81 5973 0 0

L iCl : Co2© exp. 6386 13,570 17,300 b fitted 13,727 fitted + 157 1,14 791.2 0.82 7341 + 157 2.14 c 6310 fitted fitted - 76 1.20 796.0 0.82 7260 + 76 1.06 d 6247 13,431 fitted - 139 1.63 780.8 0.80 7184 0 0

Table 2. Observed and Calculated Transition Energies of Octahedral Cobalt (II) in Chloride Lattices at 5 °K Co2®

( ß f r e e = 9 7 1 c m - ) .

Compound Method v\, c m - 1 i>2, c m - 1 j% c m - 1 ö v B35, c m - 1 ^35 4 A 2 - ^ 4 T 2 4 A 2 a 4 T i 4 A 2 - > b 4 T i [ c m " 1 ] [ % ]

Z n O : Co2© exp. 4140 7186 (4.2°K) a 10 Dq fitted

b 10 Dq 7106 c 10Dq 7109 d 10 Dq 7103

Z n A l 2 0 4 : Co2© exp. 4015 6914 (1.8°K) a 10 Dq fitted

b 10 Dq 6940 c 10 Dq 6939 d 10 Dq 6941

15,384 17,108 + 1724 10.08 790.4 0.81 fitted - 62 0.87 671.4 0.69 15,443 ± 59 0.61 675.5 0.70 15,319 - 65 0.67 666.9 0.69 16,210 15,467 - 743 4.80 689.1 0.71 fitted + 26 0.38 740.3 0.76 16,185 ± 25 0.26 738.6 0.76 16,237 + 27 0.28 742.2 0.76

Table 3. Observed and Calculated Transition Energies of Tetrahedral Cobalt (II) in Oxidic Lattices ( # f r e e — 971 cm *).

has been effectively stabilized against the action of the J a h n - T e l l e r effect. Thus, assuming type (i) conditions, the experimental energies included in Table 2 were obtained directly from the reported spectra. If a vibrational structure was observed (viz. the a 4 T l g -> 4 T 2 g band in CsMgCl3 : Co2® and in LiCl : Co2®), the centroid of the vibronic band was

estimated. It should be observed that, due to its low intensity ("two-electron jump"), a high experimen-tal uncertainty should be assumed for the a4T lg (t|g e2 ) 4A2g ( 4 e4) band.

In the single crystal spectra of tetrahedrally co-ordinated Co2 0 ions, type (ii) conditions are pre-valent and the experimental energies are determined

THE NEPHELAUXETIC EFFECT 5

accordingly. Thus, in ZnO: Co 2 0 , it is the energies of the highest intensity (no-phonon) vibrational bands which are listed in Table 3, since individual vibrational components at higher energies are con-siderably weaker in intensity14. In ZnAl204 : Co2®, the transition energies employed in Table 3 are mean values of the most intense no-phonon bands in each of the electronic transitions13. It should be kept in mind that, due to the complicated structure of the bands, the totality of the weak vibrational transitions may still contribute a significant fraction to the overall transition energy 21.

Inspection of the results which have been col-lected in Tables 1, 2, and 3 reveals considerable differences to the results from room temperature (solution) spectra 4. If those results are disregarded which were obtained by the most unfavorable methods [viz. (a) in Table 3], the agreement be-tween calculated and observed transition energies is surprisingly good. On the other hand, there is not much that could be gained by choosing between methods (b), (c), and (d) in octahedral d3 and tetrahedral d7 spectra. Most important, the deviation between the values of B35 calculated by different methods is negligible and that of ß35 is practically non-existent. In Table 2, differences between results obtained by methods (b), (c) , and (d) are some-what larger but still considerably smaller than in the room temperature spectra 4. In addition, the smallest differences are encountered in those spectra where the best resolution in the a4Tl!? 4A2g. band has been achieved, cf. CsCdCl3 : Co2® and LiCl : Co2®. Thus, if the complications inherent in the data are

1 E . U . C O N D O N a n d G . H . SHORTLEY, T h e T h e o r y o f A t o -mic Spectra, Cambridge University Press, Cambridge 1959.

2 C. K. JORGENSEN, Progr. inorg. Chem. 4, 73 [1962]. 3 C . E . SCHÄFFER a n d C . K . JORGENSEN, J . i n o r g . n u c l e a r

Chem. 8, 143 [1958]. 4 E. KÖNIG, Struct. Bonding 9, 175 [1971]. 5 In this way certain problems associated with the treatment

of C are eliminated. Thus there is evidence that the para-meter C is not susceptible to the nephelauxetic effect to the same extent as B 6.

8 H. WITZKE, Theor. chim. Acta 20, 171 [1971]. 7 C. J. BALLHAUSEN, Introduction to Ligand Field Theory,

McGraw-Hill, New York 1962. 8 J. S. GRIFFITH, The Theory of Transition Metal Ions,

University Press, Cambridge 1961. 9 H . L . SCHLÄFER U. G . GLIEMANN E i n f ü h r u n g i n d i e L i g a n -

denfeldtheorie, Akademische Verlagsgesellschaft, Frank-furt a. M. 1967.

10 E. KÖNIG and K. J. WATSON, Chem. physic. Letters 6, 457 [1970].

taken into account, the results of octahedral Co2® ions are similar to those discussed above.

IV. Summary and Conclusions

We have reviewed expressions which were derived previously, on the basis of the semi-empirical ligand field theory, to determine 10 Dq and B from the spin-allowed bands of d3 and d7 electron systems of octahedral and tetrahedral stereochemistry. These equations were applied to low-temperature single crystal spectra of suitable compounds and the extra band energy was calculated.

Provided the spectral data employed are repre-sentative for most systems of the studied electron configuration, the conclusions arrived at are as fol-lows :

(i) the semi-empirical ligand field theory repro-duces quite accurately the transition energies, at least in spin-allowed bands of d3 and d7

systems of cubic symmetry; (ii) the differences in the parameter values 10 Dq

and B resulting from the application of dif-ferent fitting methods are due essentially to in-accuracy of the experimental data;

(iii) there exist generally valid values of the inter-electronic repulsion and nephelauxetic para-meters B35 and ß35 for each complex ion which, however, may be determined only if sufficient-ly accurate experimental data are available.

This study has been sponsored in part by research grants of the Deutsche Forschungsgemeinschaft, the Stiftung Volkswagenwerk, and the Fonds der Chemi-schen Industrie whose support is gratefully appreciated.

11 C. J. BALLHAUSEN, Progr. inorg. Chem. 2, 251 [I960] , 12 G. HERZBERG, Molecular Spectra and Molecular Structure,

Vol. 3, Van Nostrand, New York 1966. 1 3 J . FERGUSON, D . L . W O O D , a n d L . G . V A N UITERT, J . c h e m .

P h y s i c s 5 1 , 2 9 0 4 [ 1 9 6 9 ] . 1 4 R . PAPPALARDO, D . L . W O O D , a n d R . C . LINARES, J . c h e m .

P h y s i c s 3 5 , 2 0 4 1 [ 1 9 6 1 ] . 15 J. LEE and A. B. P. LEVER, J. molecular Spectroscopy 26,

1 8 9 [ 1 9 6 8 ] . 16 S. S. KIM, S. A. REED, and J. W. STOUT, Inorg. Chem. 9,

1 5 8 4 [ 1 9 7 0 ] . 17 Recently, SMITH 18 likewise studied single crystal spectra

of VCl, and of the V2® ion in several chloride lattices at 6 °K. However, since quantitative energies were not re-ported, these results could not be included into the present study. SMITH 18 claims that ligand field theory fits his re-sults rather well. This would support and extend our pre-sent conclusions.

1 8 W . E . SMITH, J . d i e m . S o c . [ L o n d o n ] A 1 9 6 9 , 2 6 7 7 .

6 H. BREUER UND H.-H. PERKAMPUS

19 F. C. GILMORE, US At. Energy Comm. ORNL-TM-2507 [ 1 9 6 9 ] ,

20 A. D. LIEHR, J. physic. Chem. 67, 314 [1963].

21 Apparently, the centers of gravity reported 14 or estimated 4

in these two tetrahedral cobalt (II) systems are not ap-propriate to the present analysis.

Auswertung nmr-spektroskopischer Messungen der Mischassoziation yon Nucleosid-Derivaten bei vergleichbaren

Konzentrationsverhältnissen Evaluation of NMR-spectroscopical Results about Co-association of Nucleoside

Derivatives in Comparable Concentrations H . BREUER * u n d H . - H . PERKAMPUS

Institut für Physikalische Chemie der Universität Düsseldorf (Z. Naturforsch. 27 b, 6—12 [1972] ; eingegangen am 31. August 1971)

In this paper we described a method how to calculate from NMR the association — caused by H-bonds — of two components. These components may be present in equal concentration and may be partially self-associated.

The may act as H-donors and acceptors. The method is demonstrated using some nucleoside derivatives as examples.

In einer vorangehenden Arbeit haben wir über NMR-spektroskopische Untersuchungen zur Misch-assoziation von vier Nucleosid-Derivaten berichtet1. Die Auswertung erfolgte mit Hilfe einer modifizier-ten B e n e s i - H i l d e b r a n d - Auftragung:

i . i i + Cma V b ~ Vmb Vxb — Vmb ( t>xb~i>mb) ' ^ x

Hierin bedeuten: v^ gemessene chemische Verschie-bung der Komponente b, i>mb extrapolierte chemische Verschiebung des Monomeren b, fxb berechnete che-mische Verschiebung der Mischassoziate, Kx Misch-assoziations-Konstante in 1/Mol, cma Monomeren-konzentration der Komponente a.

cma läßt sich leicht berechnen, wenn die zugehö-rige Eigenassoziations-Konstante bekannt ist.

Um dieses Verfahren anwenden zu können, müs-sen einige Voraussetzungen erfüllt sein. Für die Substanz im Überschuß muß gelten: gute Löslichkeit und ideales Verhalten sowie genaue Kenntnis der Parameter der Eigenassoziation. Die Substanz im Unterschuß muß ein H-Brücken-Protonensignal lie-fern, das möglichst weit von Signalen ihrer Mi-schungspartners entfernt liegt. Die Verbindungen sollen ferner eine geringe Eigen-, aber eine starke Mischassoziation zeigen.

Herleitung der Auswertemehode In den beiden Molekülen eines Mischassoziats be-

finden sich die sauren Protonen in chemisch ver-

Sonderdruckanforderungen an Prof. Dr. H.-H. PERKAMPUS, Institut für Physikal. Chemie der Univ. Düsseldorf, D-4000 Düsseldorf, Fa. Henkel, Gebäude Z 10.

schiedener Umgebung, es müssen also auch zwei verschiedene chemische Verschiebungen für das „reine" Mischassoziat existieren, die jeweils die Lage der zwei beobachteten Signale beeinflussen. Sie gehen dabei mit dem einfachen Gewicht der Misch-assoziations-Konzentration ein, wenn ein 1 : 1 -Mischassoziat vorliegt. Die folgende Abb. 1 veran-schaulicht dies. Dabei bedeuten die gestrichelten Linien die hypothetischen Signale der einzelnen Spezies, und die ausgezogenen Linien entsprechen

~ca = cma+2cna + cab

~cb=cmb+2cnb + cab

~cab ~2cn

J2cnb

'Cab 'cmb

vxa vna va vnb vma vxb vb vmb chemische Verschiebung

Abb. 1. Schematisches Spektrum der H-Brücken-Protonen-signale für die Mischassoziation, v, vn , i>m gemessene Ver-schiebung, Eigenassoziat-Verschiebung, Monomerenverschie-bung, c, c n , c m Einwaage-, Eigenassoziat- und Monomeren-

Konzentration.

* Jetzige Anschrift: Dr. H. BREUER, 34 Göttingen, Theodor-Heuss-Str. 18.