410 ¹ 2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/cphc.200300979 ChemPhysChem 2004,5,410±414 Rovibrational Corrections to Transition Metal NMR Shielding Constants Michael B¸hl,* [a] Petra Imhof, [a, b] and Michal Repisky [c] Dedicated to Prof. Dr. W. Kutzelnigg on the occasion of his 70th birthday. Introduction The development of theoretical tools to calculate NMR proper- ties continues unabated. [1] A current focal point is to go beyond the treatment of molecules as vibrationless entities at 0 K and to account for the effect of vibrational averaging on chemical shifts or spin ± spin coupling constants. Procedures have been devised to include rovibrational effects in the NMR calculation, for instance, by suitable perturbational treatments [2] or by approx- imating full or partial solutions of the nuclear Schrˆdinger equation. [3] The modeling of these effects by averaging chemical shifts over trajectories from classical molecular dynamics (MD) simulations is enjoying some popularity, [4, 5, 6] partly because these simulations can readily be extended to explicitly include a solvent. We have used the latter approach to calculate thermal and solvent effects on the chemical shifts of transition metals. [6] Since the corresponding magnetic shielding constants can be quite sensitive to geometrical parameters, [7] pronounced rovibrational effects may be expected. In fact, large such effects, amounting to several hundreds of parts per million, were modeled for s( 57 Fe) of some iron complexes, but smaller variations, on the order of a few dozen parts per million, were found for s( 51 V) and s( 55 Mn) in vanadate complexes and permanganate ion, respectively. [6] Quantities averaged over classical MD trajectories do not include zero-point effects, which are purely quantum-mechan- ical in nature. For magnetic shieldings of lighter nuclei, these zero-point effects can be much larger than the thermal effects superimposed on them. [2, 3] We argued [6c,d] that thermal effects on transition-metal magnetic shieldings, evaluated from classical MD, should at least show the correct qualitative trend, since both classical and quantum-mechanical averaging would tend to increase the metal ± ligand bond lengths with respect to their equilibrium values, and it is typically these distances that dominate the metal shieldings. [7b] To test this reasoning, we now report rigorous rovibrational corrections to shieldings of some typical transition metal complexes, evaluated perturba- tionally according to the procedure from ref. [2] Results and Discussion For our purpose, we adapted the corresponding implementation from the Dalton program package [8] such that energies, energy derivatives, and properties produced with another quantum chemistry code can be processed (specifically, from Gaussi- an98 [9] ). This was necessary since at the beginning of this project the latest version of Dalton did not allow for DFT-based electronic-structure calculations, the method of choice for structures and properties of transition metal complexes. [10] Briefly, the procedure consists of two parts: First, an effective geometry r eff is constructed from the equilibrium geometry r e , the harmonic frequencies w e , and the cubic force field V (3) [Eq.(1)]. [11] r eff,j  r e,j  1 4 w 2 e;j X m V 3 e;jmm w e;m (1) Second, the magnetic shielding hypersurface s(r) is expanded around this effective geometry, and the vibrationally averaged s 0 is calculated from s eff  s(r eff ), the second derivative of the shielding surface s (2) , and the harmonic frequencies w, all evaluated at r eff [Eq.(2)], [2] s 0  s eff  1 4 X i s 2 eff;ii w eff;i (2) [8] H. Qian, S. Saffarian, E. L. Elson, Proc. Natl. Acad. Sci. USA 2002, 99,10376± 10381. [9] A.Pikovsky, A. Zaikin, M. A. de la Casa, Phys. Rev. Lett. 2002, 88, 050601. [10] G. Schmid, I. Goychuk, P. H‰nggi, Europhys. Lett. 2001, 56, 22±28. [11] P. Jung, J. Shuai, Europhys. Lett. 2001, 56, 29±35. [12] P. Gaspard, J. Chem. Phys. 2002, 117 , 8905±8916. [13] D. T. Gillespie, J. Phys. Chem. 1977 , 81, 2340 ± 2361. [14] D. T. Gillespie, J. Chem. Phys. 2001, 115, 1716 ± 1733; D. T. Gillespie, and L. R. Petzold, J. Chem. Phys. 2003, 119, 8229 ± 8234. [15] W.H. Press, S.A.Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, 2nd Ed., Cambridge University Press, New York, 1992. [16] H. Gang, T. Ditzinger, C. Z. Ning, H. Haken, Phys. Rev. Lett. 1993, 71, 807 ± 810. [17] D. T. Gillespie, J. Chem. Phys. 2000, 113, 297 ± 306. [18] L. Gammaitoni, P. H‰nggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 1998, 70, 223±285; P. H‰nggi, ChemPhysChem 2002, 3, 285±290. [19] Z. H. Hou, H. W. Xin, Phys. Rev. E 1999, 60, 6329 ± 6332. [20] C. V. Rao, D. M. Wolf, A. P. Arkin, Nature 2002, 420, 231±237. [21] C. S. Zhou, J. Kurths, B. B.Hu, Phys. Rev. Lett. 2001, 87 , 098101, 1 ± 4; A. Neiman, L. Schimansky-Geier, A. Cornell-Bell, F. Moss, Phys. Rev. Lett. 1999, 83, 4896 ± 4899; J. F. Lindner, S. Chandramouli, A. R. Bulsara, M. Lˆcher, W. L. Ditto, Phys. Rev. Lett. 1998, 81, 5048 ± 5051. Received: September 12, 2003 [Z969] [a] Priv.-Doz.Dr. M. B¸hl, Dr. P. Imhof Max-Planck-Institut f¸r Kohlenforschung Kaiser-Wilhelm-Platz 1 45470 M¸lheim an der Ruhr (Germany) Fax: ( 49) 208-306 2996 E-mail: buehl@mpi-muelheim.mpg.de [b] Dr. P. Imhof Curent address: Interdisziplin‰res Zentrum f¸r Wissenschaftliches Rechnen Universit‰t Heidelberg Im Neuenheimer Feld 368, 69120 Heidelberg (Germany) [c] M. Repisky Institute of Inorganic Chemistry, Slovak Academy of Sciences Dubravska cesta 9, 84236 Bratislava (Slovak Republic) Supporting information for this article is available on the WWW under http:// www.chemphyschem.org or from the author.