Closed-form ground state expression for the chemical shift tensor

Volume 142, number 6

CHEMICAL PHYSICS LETTERS

25 December I987

CLOSED-FORM GROUND STATE EXPRESSION FOR THE CHEMICAL SHIFT TENSOR J. HERZFELD Department of Chemistry, Brandeis University, Waltham. MA 02254, USA Received I October 1987

Steepestdescent perturbation theory is used to derive an equation for the chemical shift tensor elements that requires neither any knowledge of the excited states of the molecule nor an explicit calculation of the molecular wavefunction in the presence of the applied magnetic field. This result is compared to the ground state expression derived earlier from the average energy approximation.

The chemical shift tensor reflects the electronic environment of a nucleus and, as such, it provides useful information about local molecular structure. Ideally, measured tensors could be compared with ones calculated for specific molecular structures in order to test the plausibility of the latter. However, because the calculation of shift tensors is quite difficult, the experimental results are more often interpreted empirically, on the basis of model compound data. The main difficulty in calculating chemical shift tensors arises from the fact that the perturbation of the electronic wavefunction by the magnetic field makes an important contribution [ 11. This may be taken into account, using finite perturbation theory (PPT), by explicitly calculating the ground state wavefunctions in the presence of small magnetic fields in three mutually orthogonal orientations, a laborious procedure [ 2,3]. Another approach is to use Rayleigh-Schriidinger perturbation theory (RSPT), which requires evaluation of infinite sums over excited states [ 1,4]. Recently, Cioslowski has described a steepest-descent perturbation theory (SDPT) that requires knowledge of only the unperturbed ground state wavefunction and performs well compared with RSPT [ 51. Application of this theory, as follows, provides a rigorous closed-form ground state expression for the chemical shift tensor elements. Since the derivation depends only OF infinitesimal fields, considerable confidence can be placed in the perturbation treatment. The individual elements of the chemical shielding tensor for nucleus A are given by #I

(1) Here y, is the normalized ground state electronic wavefunction in the presence of an applied field of magnitude i; rnAis the position of the nth electron relative to nucleus A; v, is the velocity of the nth electron; go is the normalized ground state electronic wavefunction in the absence of an applied field;

Bj in direction j; Ei is the unit vector in direction

s, = (dlmc)

c I m* I -3(E,‘L) n

,

(2)

” For details see e.g. refs. [ 6,7].

0 009-2614/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Volume 142, number 6

CHEMICAL PHYSICS LETTERS

25 December 1987

where LnAis the angular momentum of the nth electron relative to nucleus A; and

S~,=(e2/2mC2) C Ir~*I-'{Ei.[m*X(EjXr,)]} n =(e2/2mC2) C l~n~l-3[(r~~‘~,)Bij-(Ei’r,)(~ji.r~~)1 n

,

(3)

where r, is the position of the nth electron relative to an arbitrary origin. The last two terms in eq. (1) are usually evaluated using FPT or RSPT. Alternatively, SDPT provides that to first order IWj>=(I~o>-~j~Bj~jlI~))lCj

(4)

*

Here & 1 - I q&o>(q&,I is the projector onto the Hilbert space complementary to the zero-field ground state; I?j= (&2TYZC) C (Ej’L,) n

7

(5)

is the perturbing electronic Hamiltonian for a field of unit magnitude in direction j, where L, is the orbital angular momentum of the nth electron relative to an arbitrary origin; Cj=(1+IUj12B/z(~~lIj~~jI~~})1’2 is the appropriate normalization constant, and a, is the variational parameter chosen to minimize the perturbed energyE(aj)=(WjlA,+B~jIw,), where A,, is the zero-field electronic Hamiltonian [ 51. For small values of Bj, when fiO and Aj do not commute, ~~~~~~Ol~j~~lI~>~~~OIIl~Oo>z~~~~~~l~ijA~~

l~~>-<~Ol~~l~~o>(~Oltij~jl~~O))

7

(6)

independent of Bj, and ~A,,~~~~l~ijlI~~~aj<~~~l~i~~I~~~++(~~IfiiP~~

~I$O))

*

(7)

This result permits calculation of the shift tensor elements based only on the zero-field ground state wavefunction $o. It is not necessary to determine the field perturbed wavefunctions wj explicitly. The restriction of these results to systems for which Z& and Z?jdo not commute is not a serious limitation. In non-linear molecules, i?,., A, and & do not commute with all the nuclear interaction terms in the electronic Hamiltonian. In linear molecules (with the molecular axis designated as the z direction), I& and fiY also do not commute with all the nuclear interaction terms in A,. However, if spin-orbit coupling is neglected, fiZ does commute with fro for a linear molecule. In this case, the eigenfunction of no may be chosen to also be eigenfunctions of fiZ so that i%= Ir$,,) = 0 and, according to eq. (4), )y,) = I &o). This is the same result as obtained by RSPT, and it implies that, for linear molecules, there is no paramagnetic contribution to cAlrwhen spin-orbit coupling is neglected. For systems with only one nucleus, fix, fly and fi, all commute with Z?,, unless spin-orbit coupling is included, It follows then, that eqs. (6) and (7) cannot be used for isolated atoms if spin-orbit coupling is neglected. Iqio) = 0 for all j [ 81, Thus eqs. (6) and (7) will usually It should be noted that for most molecules ( $. II?, simplify to

~*,,~~~Ol~~jI~OO~“j~~*l~iii~+~j~ilI~>

*

(9)

It is interesting to directly compare the SDPT derived expression in eq. (4) with the RSPT derived lirstorder expansion in the orthonormal basis set of zero-field eigenfunctions 470

Volume142,number6

Iv,>=

CHEMICAL PHYSICSLETrERS

25December1987

(l&lo)--4 Jo wk1-l Ik)(~klmo))/e,.

(10)

Here @kand A.+ ( ek 1fro 1$,J - (q&I fi,, I &,) are, respectively, the wavefunction and the excitation energy of the kth excited state in the absence of a magnetic field, and Cj is again the appropriate normalization constant. For many years it was common practice to circumvent this sum over excited states by using the average energy approximation and the closure relation to give

(11) and

Ivj) X[ I~o>-~~(ll~)~fj,I~~,)llcj.

(12)

Apart from the questionable accuracy of the approximation, the difficulty with this approach was in determining the appropriate value for (l/m). However, the resulting expression for v, (eq. (12)) is the same as that obtained above from SDPT (eq. (4)) if we take (l/U)=Uj

a

(13)

It is apparent, therefore, that (i) the average energy approximation corresponds to adjustment of the wavefunction in the direction of steepest descent of the expectation value of the Hamiltonian, (ii) the best value of m is obtained by variation to minimize the expectation value of the Hamiltonian, and (iii) in light of eq. (13), fi, I q&o>in eq. (8) may be interpreted as the “average excited state” in the presence of a magnetic field of unit magnitude in direction i. The average energy approximation is thus seen to be more legitimate than has been thought. However, it remains that most previous applications of it have been flawed, not only by the lack of an expression for (l/m), but also by the further assumption that (IlhE) is independent of the orientation of the applied field. This work was supported by NIH grant GM-36810.

[ I] N.F.Ramsey,Phys. Rev. 78 (1950) 699. [2] R. Ditchfield, Mol. Phys. 27 (1974) 789. [ 31 E. Vauthier, S. Odiot and F. Tonard, Can. J. Chem. 60 (1982) 957. [ 41 N.F. Ramsey, Phys. Rev. 86 (1952) 243. [5] J. Cioslowski, J. Chem. Phys. 86 (1987) 2105. [ 61 A. Carrington and A.D. McLachlan, Introduction to magnetic resonance (Harper and Row, New York, 1967) pp. 54-57. [ 71 L. Ando and G.A. Webb, Theory of NMR parameters (Academic Press, New York, 1983) pp. 47-53. [ 81 J.H. van Vleck, Theory of electric and magnetic susceptibilities (Oxford Univ. Press, Oxford, 1932) pp. 273,274.

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