The theory of chemical shifts in nuclear magnetic resonance I. Induced current densities By J. A. Pople Department of Theoretical Chemistry, University of Cambridge {Communicated by C. A. Coulson, —Received 16 October 1956) The chemical shifts of nuclear magnetic resonance frequencies are determined by the secondary magnetic field due to the electronic current induced by the applied field. This paper is concerned with the general problem of finding the current density vector field. An approximate orbital theory is developed which enables this to be divided up, under certain conditions, into local diamagnetic and paramagnetic circulations about individual atoms.
542 J. A. Pople where —e,m are the electronic charge and mass, A(r:/) is the vector potential of the external field at r;- and V' is the potential energy (representing the electron-nuclear and interelectronic interactions). p?- is the quantum-mechanical momentum which, in the Sehrodinger representation, is replaced by = (2-2) We wish to find the expectation values of the current density operator j(r) = “ 2m S {(Pi + (e/c) A(r^))<5(r - r,) + S(r -r,) + (e/c) A(r,))} (2-3) (where £(r — r3-) is the Dirac ^-function) to the first order in the applied magnetic field. The most direct approach would be to use ordinary perturbation theory, writing (2*1) in the form Jf = + (2-4) where is the unperturbed Hamiltonian, and then expressing the perturbed wave function for the ground state as a linear combination of the ground and excited wave functions of the unperturbed molecule. This method, although suitable for an atom or a simple molecule such as H2 has serious disadvantages when applied to a complex molecule. This is primarily due to the difficulty of choosing a single vector potential which will lead to equivalent results for equivalent atoms. Thus if the potential has the form A = -|rx H , (2*5) where is the applied uniform field, then an arbitrary choice has to be made for the origin of r. If this is taken to be at a particular nucleus, the interpretation for other atoms will be complicated.
Chemical shifts in nuclear magnetic resonance. I 543 expressions for localized currents on each atom. The property of the new atomic orbitals which leads to simplification is their behaviour under the operator p + (e/c) A which appears in the Hamiltonian and the current density operator j.
544 J. A. Pople If Jc = lthey give the occupation numbers of the various parts of the total electron density function in the state k. If we evaluate a similar integral for the modified functions and we obtain |V? S «(r - r,) T,dr = S (r) ;&,(!•). (3-4) J j stfiv The diamagnetic current density jdla- can now be written in terms of the coeffi cient J“Sr) = “ 4 ? J5(r - ff(pi + <e/c> A(r,)) ’f .+nt(P< + m A(r,)) Y0]} dr 2 {[(P + (e/«)A) X.J* & +>?,4[(P+(«/«) A)3i,]}. (3-5) “ -■35 Using (2*7), this becomes ~2nie jdla-(r) (A,-A|).r 2m StflV p + -(A-A s))& ^tv + 0S/1 P + ~ (A—A,)) (3-6) ([( If we now neglect terms in this expansion involving the product of atomic orbitals on different atoms (i.e. s=M), and further use p* = — p, then (3*6) reduces to jai“-(r) = psju a - K) &,<r> ^<r)- (3-7) W S /IV pr>(r) = «Ur) Since (3-8) represents the unperturbed electron density on atom $ (again neglecting inter atomic terms) and A -A s = - |( r - r s)xH, (3-9) (3*7) can be written e2 (3-10) jdl“'(r) = ^ ? P r[(r- r<)xH]- It clearly represents the sum of the effects of local Larmor precession with angular frequency eHj2mc in each atom.
Chemical shifts in nuclear magnetic resonance. I 545 where Ek — E0 is the excitation energy of state Jc in the unperturbed molecule.
546 J. A. Pople 5. Further simplification and examples The formulae given in the previous two sections are sufficient to calculate the local atomic currents provided that suitable wave functions in terms of atomic orbitals are available. If we are dealing with atoms of low atomic number, the l.c.a.o. treatment can be developed satisfactorily in terms of s andp functions only, enabling the formulae for paramagnetic currents to be put in more explicit form.
Chemical shifts in nuclear magnetic resonance. I 547 which have to be coupled to form a singlet. The transition density function (equation (3*1)) is well known for such wave functions. If the excitation is from to then if the excited state is symbolized by i- _ ^2 i/rf (5-4) The values of the numerical coefficients are then obtained by writing ifrt and ^ in l.c.a.o. form.
548 J. A. Pople Simple hydrides Returning to the general theory, equations (5-2) and (5-3) can be further sim plified for simple hydrides XHn for which there is only one atom about which para magnetic circulations can occur. For the paramagnetic moment in such systems we have p%h2H 1 .para. _ ° _ V_ (p(kO) _ pk0)\2 N ~ 8n2m2c2*;> o Ek — r vx) • If the excitation energies Ek — E0 are now replaced by some average A and l.c.a.o. molecular orbital functions are used, it becomes possible to carry out the summation over excited states. If we suppose that the molecular orbitals have the l.c.a.o. form (5-7) A then, for the excited state tfrt -> ftp from (5*4) p%ru&)= p x ixx jy, (5-8) so that, if the sum over excited states is restricted to a sum over such singly excited configurations, ooo. unocc.