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WIREs Comput Mol Sci
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# Calculations of magnetically induced current densities: theory and applications

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A review of computational studies of magnetically induced current density susceptibilities in molecules and their relation to experiments is presented. The history of the investigation of magnetically induced current densities and ring currents in molecules is briefly covered. The theoretical development of relativistic and nonrelativistic computational approaches for computing current densities in closed‐shell and open‐shell molecules is discussed and different state of the art methods to interpret calculated current densities are reviewed. Numerical integration approaches to assess global, semilocal, and local aromatic properties of multiring molecules are presented and demonstrated on free‐base trans‐porphyrin. We show that numerical integration of the current density combined with guiding visualization techniques of the current flow is a powerful tool for studies of the aromatic character of complicated molecular structures such as annelated aromatic and antiaromatic rings. Representative applications are reported illustrating the importance of careful current density studies for organic and inorganic chemistry. The applications include calculations of current densities and current strengths for aromatic, antiaromatic, and nonaromatic molecules of different kind. Current densities in spherical, cylindrical, tetrahedral, toroidal, and Möbius‐twisted molecules are discussed. The aromatic character, current pathways, and current strengths of porphyrins are briefly highlighted. Aromatic properties of inorganic molecules are assessed based on current density calculations. Current strengths as a noninvasive tool to determine strengths of hydrogen bonds are discussed. WIREs Comput Mol Sci 2016, 6:639–678. doi: 10.1002/wcms.1270

Schematic illustration of the induced magnetic field lines for benzene when approximating the molecular carbon ring as a coil.
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The current density of the adenine‐thymine (AT) DNA base pair calculated in a plane placed 1 bohr above the plane of the DNA pair. The external magnetic field is directed perpendicularly to the plane of the molecules, whereas in the calculation of the strength of the hydrogen bond, we recommend that the magnetic field is parallel to the aromatic molecules. The placement of the two integration planes, which are perpendicular to the molecular plane and the hydrogen bond, are indicated in yellow and light blue. The planes begin and end at vortices as shown in the picture.
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Current scheme for a pyrrolic ring of trans‐porphyrin illustrating on how local currents can be calculated using the integrated current strength analysis. Integration planes are displayed in green.
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The calculated current density for trans‐porphyrin in a plane placed 1 bohr above the molecular plane. The current density is represented in the same plane using (a) a streamline plot, (b) a vector plot, and (c) a line integral convolution (LIC) plot. The LIC streamlines of the stronger current streams are blue colored.
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Streamline plot of the current density of d Si6R6 in a plane containing the elongated Si⋯Si bond with the magnetic field set perpendicular to the plane. The silicon atoms are denoted with black dots.
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The molecular structure of d Si6R6, (R = 2,4,6‐iPr3C6H2).
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LIC streamline representation of the calculated current density for BH. The dominating vortex in the middle is paratropic sustaining a strongcurrent of −16.1 nA/T that passes a half‐plane perpendicular to BH beginning at the bond center.
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The two chiral structures of the toroidal C144H144 spiral. The molecular graphs have been made with MOLDEN and GIMP.
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The topology of the magnetically induced ring current in toroidal molecular rings. (Reprinted with permission from Ref . Copyright 2012 Verlag der Zeitschrift für Naturforschung)
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The two components of the magnetically induced ring current and the corresponding induced magnetic field of toroidal‐shaped molecular rings. (Reprinted with permission from Ref . Copyright 2012 der Zeitschrift für Naturforschung)
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The cycloparaphenylene dianion consisting of seven phenyls in the ring and the magnetically induced current density.
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The integrated current strength around $N H 4 −$ as a function of the distance from the center of the molecule. The external magnetic field is applied along one of the NH bonds. The picture has been made with MOLDEN, GIMP, and GNUPLOT.
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Ring‐current strengths were calculated for the open and closed rings. The gap of the open ring is 702 pm. The figure has been made with XMAKEMOL and GIMP.
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The integrated current strength around triptycene as a function of the distance from the center of the molecule. The magnetic field is applied along the C3 axis. The picture has been made with MOLDEN, GIMP and GNUPLOT.
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The degree of aromaticity of arsole (AsH = 1), phosphole (PH = 2) and cyclopentadiene (CH2 = 3) are compared to the degree of aromaticity for pyrrole. The degree of aromaticity has been estimated by using aromatic stabilization energies (ASE), magnetic susceptibility exaltations (Λ), nucleus‐independent chemical shifts in the molecular plane (NICS(0)), nucleus‐independent chemical shifts 1 Å above the molecular plane (NICS(1)), harmonic oscillator model of aromaticity (HOMA), and ring‐current strength (GIMIC). All data except the ring current strengths are taken from Ref . The ring current strengths are taken from Ref .
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Schematic overview of the different types of information being accessible through a current density analysis (black bold arrows). Some eventually indirectly accessible information is indicated by the gray arrows, which show how current density studies can be used in chemistry.
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Schematic illustration of the possible routes the ring current might take in a multiring compound with annelated or connected molecular rings. The current circling around the macroring is the net ring current (green). Local currents or semilocal currents are those circling around a particular part of the molecule, e.g., around one or several subrings as shown in blue. The current in the upper branch of the local current is stronger than the global one due to the local current circulation. The ring current can also bifurcate at subrings as illustrated in red with the current strengths fulfilling Kirchhoff's law.
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Illustration of the use of the terms local, semilocal, and global ring currents.
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The integrated current profile of bispentalene annelated benzene (a). The profile of the integrated current density that passes through a plane along the common CC bond between the annelated cyclopentanyl rings of one of the pentalene moieties and the benzene ring is shown in (b).
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Calculated profile of the current strength of naphthalene passing the plane shown in the left part of the figure. The integration plane begins in the center of the left ring and passes across the second ring. The placement and extension of the integration plane is formally indicated by the red arrow.
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Illustration of the stepwise scan (Δx) over the integration plane used for calculating the current strength profile of naphthalene. Steps of 0.05–0.1 bohr are typically applied.
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(a) Illustration of the placement of the integration plane across a particular bond and perpendicular to the molecular plane for benzene. (b) Illustration of the same integration plane as in (a) together with the signed modulus of the current density. Diatropic currents are displayed in blue and the paratropic ones are shown in red.
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The ACID isosurface of free‐base trans‐porphyrin obtained with isosurface cutoff values of (a) 0.003 and (b) 0.005.
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Signed modulus of the current density for naphthalene. Diatropic currents are illustrated in blue, paratropic ones in red.
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(a) Illustration of the placement of a visualization plane parallel to the molecular plane. (b) Vector plot visualization of the current density of naphthalene in a plane placed 1 bohr above the molecular plane. Diatropic currents are assumed to circle clockwise. (c) Streamline representation of the current density in the same plane. (d) LIC representation of the current density in the same plane.
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Schematic overview over different options for performing current density analyses. Note that the different approaches are complementary.
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Schematic illustration of the distinction between the classical (diatropic, clockwise) and nonclassical (paratropic, anticlockwise) directions of the induced current flow relative to the external magnetic field (green) using benzene as an example.
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