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Applications of Topological Methods in Molecular Chemistry pp 151–226Cite as

Topology of Quantum Mechanical Current Density Vector Fields Induced in a Molecule by Static Magnetic Perturbations

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Part of the book series: Challenges and Advances in Computational Chemistry and Physics ((COCH,volume 22))

Abstract

It is shown that the quantum mechanical theory of static magnetic properties can be reformulated in terms of electronic current densities induced by an external magnetic field and permanent magnetic dipole moments at the nuclei. Theoretical relationships are reported to evaluate magnetizability, nuclear magnetic shielding and nuclear spin-spin coupling via the equations of classical electromagnetism, assuming that the current density is evaluated by quantum mechanical methods. Emphasis is placed on the advantage of the proposed formulation, as an alternative to procedures based on perturbation theory, as regards interpretation of response allowing for the ideas of current density tensor and current susceptibility vector. Visualisation of the electronic interaction with a magnetic field and intramolecular perturbations, e.g., nuclear magnetic dipoles, is made possible via current density maps, nuclear shielding density maps and plots of nuclear spin-spin coupling density. Topological analysis of the quantum mechanical current density in terms of Gomes stagnation graphs is shown to yield fundamental information for understanding magnetic response. Examples are given for a few archetypal molecules. A topological definition of delocalized electron currents is proposed.

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Notes

  1. 1.

    See Eq. (8a) of Ref. [34]. By combining the first two addenda on the l.h.s. of Eq. (2) of Ref. [36], a quantum-mechanical relationship of the same form as the Newton’s second law is obtained for a particle acted upon by the Lorentz force and by the nonlocal Bohm potential, see Eq. (94) of Ref. [15].

  2. 2.

    The f function is fully arbitrary, provided it is continuous and has the physical dimensions of a magnetic flux density times the square of length. It is well-behaved for \( {\kern 1pt} r \to \infty \) and satisfies the condition \( \nabla^{2} f = 0 \).

  3. 3.

    The topological index \( \iota \) counts the number of times that the current density vector \( \varvec{J}^{{\mathbf{B}}} \) rotates completely while one walks counterclockwise around a circle of radius \( \epsilon \), so small that \( \varvec{J}^{{\mathbf{B}}} \) has no zeroes inside except the SP at its centre. The topological index \( \iota \) of a saddle (vortex) line is −1 (+1). Both SPs have \( (r,s) = (2,0) \).

  4. 4.

    If the eigenvalues are complex one defines the signature as the difference between the number of eigenvalues having a positive real part and the number of eigenvalues having a negative real part.

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  1. Dipartimento di Scienze chimiche e geologiche, Università degli Studi di Modena e Reggio Emilia, Via Campi 213/b, 41125, Modena, Italy

    P. Lazzeretti

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Editors and Affiliations

  1. LCC-CNRS, Toulouse, France

    Remi Chauvin

  2. LCC-CNRS, Toulouse, France

    Christine Lepetit

  3. Théorique, UMR 761, Laboratoire de Chimie, Paris, France

    Bernard Silvi

  4. CNRS/UPM, MONARIS UMR 8233 (LADIR + LM2N), Paris, France

    Esmail Alikhani

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Lazzeretti, P. (2016). Topology of Quantum Mechanical Current Density Vector Fields Induced in a Molecule by Static Magnetic Perturbations. In: Chauvin, R., Lepetit, C., Silvi, B., Alikhani, E. (eds) Applications of Topological Methods in Molecular Chemistry. Challenges and Advances in Computational Chemistry and Physics, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-29022-5_7

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