Abstract
We develop a methodology for calculating, analyzing, and visualizing nuclear magnetic shielding densities which are calculated from the current density via the Biot–Savart relation. Atomic contributions to nuclear magnetic shielding constants can be estimated within our framework with a Becke partitioning scheme. The new features have been implemented in the GIMIC program and are applied in this work to the study of the 1H and 13C nuclear magnetic shieldings in benzene (C6H6) and cyclobutadiene (C4H4). The new methodology allows a visual inspection of the spatial origins of the positive (shielding) and negative (deshielding) contributions to the nuclear magnetic shielding constant of a single nucleus, something which has not been hitherto easily accomplished. Analysis of the shielding densities shows that diatropic and paratropic current-density fluxes yield both shielding and deshielding contributions, as the shielding or deshielding is determined by the direction of the current-density flux with respect to the studied nucleus instead of the tropicity. Becke partitioning of the magnetic shieldings shows that the magnetic shielding contributions mainly originate from the studied atom and its nearest neighbors, confirming the localized character of nuclear magnetic shieldings.
Highlights
IntroductionSecond-order magnetic properties such as nuclear magnetic shieldings, indirect spin−spin coupling constants, and magnetizabilities are usually calculated by using the gradient theory of electronic structure calculations as the second derivative of the electronic energy with respect to the external magnetic perturbation(s) in the limit of vanishing perturbation(s).[1−4] the elements of the nuclear magnetic shielding and magnetizability tensors can be obtained as second derivatives of the magnetic interaction energy, which can be written as an integral over the scalar product of a current density caused by a magnetic perturbation and the vector potential of the second magnetic perturbation.[5−7] The current density JB(r) induced by an external magnetic field B or the current density JmI (r) induced by the nuclear magnetic moment mI of nucleus I is formally defined as the real part (9) of the mechanical momentum density
As will be discussed in later in this work, the magnetic properties evaluated within this scheme have no reference to the magnetic gauge origin if the current density is gauge origin independent, as is the case in our GIMIC approach[8−11] as well as in the ipsocentric approach.[12−16]
While the end results of the gradient-theory and the integration approaches are the same, the method based on integration can be used for providing additional information about orbital and spatial contributions to a given magnetic property
Summary
Second-order magnetic properties such as nuclear magnetic shieldings, indirect spin−spin coupling constants, and magnetizabilities are usually calculated by using the gradient theory of electronic structure calculations as the second derivative of the electronic energy with respect to the external magnetic perturbation(s) in the limit of vanishing perturbation(s).[1−4] the elements of the nuclear magnetic shielding and magnetizability tensors can be obtained as second derivatives of the magnetic interaction energy, which can be written as an integral over the scalar product of a current density caused by a magnetic perturbation and the vector potential of the second magnetic perturbation.[5−7] The current density JB(r) induced by an external magnetic field B or the current density JmI (r) induced by the nuclear magnetic moment mI of nucleus I is formally defined as the real part (9) of the mechanical momentum density.
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