Magnetic Shielding of Nuclei in Molecules

TLDR

The magnetic field at a nucleus differs from the externally applied field because electron motion generates a Lamb‑type diamagnetic term and a second‑order paramagnetic term analogous to high‑frequency matrix elements in molecular diamagnetism. The authors develop an expression for the nuclear magnetic field induced by an external field in polyatomic molecules that possess no net electron orbital or spin angular momentum. The expression separates the electron contribution into a simple diamagnetic correction and a complex second‑order paramagnetic component. The two‑term formula shows that the paramagnetic term can be large, varies with molecular environment, explains observed chemical shifts in nuclear moments, and for H₂ yields a –0.56×10⁻⁵ correction that results in a total shielding constant of 2.68×10⁻⁵.

Abstract

An expression is developed for the magnetic field at a nucleus resulting from the application of an external magnetic field to a polyatomic molecule which has no resultant electron orbital or spin angular momenta in the absence of the external field. The field at the nucleus is not the same as the externally applied field because of the field arising from the motion of the electrons in the molecule. The expression for the electron contribution to the magnetic field is shown to consist of two parts. The first is a simple term that is similar to the diamagnetic correction developed by Lamb for atoms. The second is a complicated one arising from second-order paramagnetism and is analogous to the term dependent on the high frequency matrix elements in the theory of molecular diamagnetism. Under certain circumstances the second-order paramagnetic term can become quite large. Since both of these terms are altered when the same nucleus is in different molecules, they at least partially and perhaps completely explain the chemical effect that has been reported by various observers in measurements of nuclear moments. For linear molecules, the second-order paramagnetic term is shown to be directly related to the experimentally measurable spin-rotational magnetic interaction constant of the molecule. This relation is particularly valuable in the important case of molecular hydrogen where it is shown that the correction for second-order paramagnetism is -0.56\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$. When this is added to the Lamb-type term as calculated by Anderson, the total magnetic shielding constant for molecular ${\mathrm{H}}_{2}$ becomes 2.68\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$.