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International audience; The Crystal program for quantum-mechanical simulations of materials has been bridging the realm of molecular quantum chemistry to the realm of solid state physics for many years, since its first public version released back in 1988. This peculiarity stems from the use of atom-centered basis functions within a linear combination of atomic orbitals (LCAO) approach and from the corresponding efficiency in the evaluation of the exact Fock exchange series. In particular, this has led to the implementation of a rich variety of hybrid density functional approximations since 1998. Nowadays, it is acknowledged by a broad community of solid state chemists and physicists that the inclusion of a fraction of Fock exchange in the exchange-correlation potential of the density functional theory is key to a better description of many properties of materials (electronic, magnetic, mechanical, spintronic, lattice-dynamical, etc.). Here, the main developments made to the program in the last five years (i.e., since the previous release, Crystal17) are presented and some of their most noteworthy applications reviewed.

A computational procedure is developed for the efficient calculation of derivatives of integrals over non-separable Gaussian-type basis functions, used for the evaluation of gradients of the total energy in quantum-mechanical simulations. The approach, based on symbolic computation with computer algebra systems and automated generation of optimized subroutines, takes full advantage of sparsity and is here applied to first energy derivatives with respect to nuclear displacements and lattice parameters of molecules and materials. The implementation in the Crystal code is presented, and the considerably improved computational efficiency over the previous implementation is illustrated. For this purpose, three different tasks involving the use of analytical forces are considered: (i) geometry optimization; (ii) harmonic frequency calculation; and (iii) elastic tensor calculation. Three test case materials are selected as representatives of different classes: (i) a metallic 2D model of the Cu(111) surface; (ii) a wide-gap semiconductor ZnO crystal, with a wurtzite-type structure; and (iii) a porous metal-organic crystal, namely the ZIF-8 zinc-imidazolate framework. Finally, it is argued that the present symbolic approach is particularly amenable to generalizations, and its potential application to other derivatives is sketched.

The nature of chemical bonding in actinide compounds (molecular complexes and materials) remains elusive in many respects. A thorough analysis of their electron charge distribution can prove decisive in elucidating bonding trends and oxidation states along the series. However, the accurate determination and robust analysis of the charge density of actinide compounds pose several challenges from both experimental and theoretical perspectives. Significant advances have recently been made on the experimental reconstruction and topological analysis of the charge density of actinide materials [Gianopoulos et al. IUCrJ, 2019, 6, 895 [PMC free article] [PubMed]]. Here, we discuss complementary advances on the theoretical side, which allow for the accurate determination of the charge density of actinide materials from quantum-mechanical simulations in the bulk. In particular, the extension of the Topond software implementing Bader’s quantum theory of atoms in molecules and crystals (QTAIMAC) to f- and g-type basis functions is introduced, which allows for an effective study of lanthanides and actinides in the bulk and in vacuo, on the same grounds. Chemical bonding of the tetraphenyl phosphate uranium hexafluoride cocrystal [PPh4+][UF6–] is investigated, whose experimental charge density is available for comparison. Crystal packing effects on the charge density and chemical bonding are quantified and discussed. The methodology presented here allows reproducing all subtle features of the topology of the Laplacian of the experimental charge density. Such a remarkable qualitative and quantitative agreement represents a strong mutual validation of both approaches—experimental and computational—for charge density analysis of actinide compounds.

In mineral exploration and geologic mapping of igneous and metamorphic terranes, the background is often dominantly resistive. The most important electromagnetic interaction is between a discrete conductor and an overlying sheet of conductive overburden (e.g., glacial clays or weathering products of the basement rocks). To enable the electromagnetic modeling of these common situations, here I provide closed-form expressions for the approximate electromagnetic response of a sphere embedded in highly resistive rocks and interacting with an overlying thin sheet. The sphere is assumed to be dipolar and excited by a locally uniform field. The expressions in the time and frequency domains are represented as sums of complete and incomplete cylindrical functions. New asymptotic approximations are provided for the efficient evaluation of the required incomplete cylindrical functions. The frequency-domain formulas are validated by numerical transformation to the time domain and comparison to the time-domain solution.

We study pressure-induced isostructural electronic phase transitions in the prototypical mixed valence and strongly correlated material EuO using the global-hybrid density functional theory. The simultaneous presence in the valence of highly localized $d$- and $f$-type bands and itinerant $s$- and $p$-type states, as well as the half-filled $f$-type orbital shell with seven unpaired electrons on each Eu atom, have made the description of the electronic features of this system a challenge. The electronic band structure, density of states, and atomic oxidation states of EuO are analyzed in the 0--50 GPa pressure range. An insulator-to-metal transition at about 12 GPa of pressure was identified. The second isostructural transition at approximately 30--35 GPa, previously believed to be driven by an oxidation from Eu(II) to Eu(III), is shown instead to be associated with a change in the occupation of the Eu $d$ orbitals, as can be determined from the analysis of the corresponding atomic orbital populations. The Eu $d$ band is confined by the surrounding oxygens and split by the crystal field, which results in orbitals of ${e}_{g}$ symmetry (i.e., ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and ${d}_{2{z}^{2}\ensuremath{-}{x}^{2}\ensuremath{-}{y}^{2}}$, pointing along the Eu-O direction) being abruptly depopulated at the transition as a means to alleviate electron-electron repulsion in the highly compressed structures.

The chemistry of f-electrons in lanthanide and actinide materials is yet to be fully rationalized. Quantum-mechanical simulations can provide useful complementary insight to that obtained from experiments. The quantum theory of atoms in molecules and crystals (QTAIMAC), through thorough topological analysis of the electron density (often complemented by that of its Laplacian) constitutes a general and robust theoretical framework to analyze chemical bonding features from a computed wave function. Here, we present the extension of the Topond module (previously limited to work in terms of s-, p- and d-type basis functions only) of the Crystal program to f- and g-type basis functions within the linear combination of atomic orbitals (LCAO) approach. This allows for an effective QTAIMAC analysis of chemical bonding of lanthanide and actinide materials. The new implemented algorithms are applied to the analysis of the spatial distribution of the electron density and its Laplacian of the cesium uranyl chloride, Cs2UO2Cl4, crystal. Discrepancies between the present theoretical description of chemical bonding and that obtained from a previously reconstructed electron density by experimental X-ray diffraction are illustrated and discussed.

International audience; We revise formal and numerical aspects of collinear and non-collinear density functional theories in the context of a two-component self-consistent treatment of spin–orbit coupling. Theoretical and numerical analyses of the non-collinear approaches confirm their ability to yield the proper collinear limit and provide rotational invariance of the total energy for functionals in the local-density or generalized-gradient approximations (GGAs). Calculations on simple molecules corroborate the formal considerations and highlight the importance of an effective screening algorithm to provide the sufficient level of numerical stability required for a rotationally invariant implementation of non-collinear GGA functionals. The illustrative calculations provide a first numerical comparison of both previously proposed non-collinear formulations for GGA functionals. The proposed screening procedure allows us to effectively deal with points of small magnetization, which would otherwise be problematic for the evaluation of the exchange–correlation energy and/or potential for non-collinear GGA functionals. Both previously suggested formulations for the non-collinear GGA are confirmed to be adequate for total energy calculations, provided that the screening is achieved on a sufficiently fine grid. All methods are implemented in the Crystal program.

International audience; A noncanonical coupled perturbed Kohn–Sham density functional theory (KS-DFT)/Hartree-Fock (HF) treatment of spin–orbit coupling (SOC) is provided. We take the scalar-relativistic KS-DFT/HF solution, obtained with a relativistic effective core potential, as the zeroth-order approximation. Explicit expressions are given for the total energy through the 4th order, which satisfy the 2n + 1 rule. Second-order expressions are provided for orbital energies and density variables of spin-current DFT. Test calculations are carried out on the halogen homonuclear diatomic and hydride molecules, including 6p and 7p elements, as well as open-shell negative ions. The computed properties through second or third order match well with those from reference two-component self-consistent field calculations for total and orbital energies as well as spin-current densities. In only one case (At2–) did a significant deviation occur for the remaining density variables. Our coupled perturbation theory approach provides an efficient way of adding the effect of SOC to a scalar-relativistic single-reference KS-DFT/HF treatment, in particular because it does not require diagonalization in the two-component spinor basis, leading to saving factors on the number of required floating-point operations that may exceed one order of magnitude.

The spin-current density functional theory (SCDFT), when formulated in a basis of Pauli spinors, provides a proper theoretical framework for the study of materials in an arbitrarily oriented external magnetic field and/or upon inclusion of spin-dependent relativistic effects, such as spin-orbit coupling. The SCDFT is formulated in terms of the particle-number density $n$, the Cartesian components of the magnetization ${m}_{x},$ ${m}_{y},$ and ${m}_{z}$, the orbital-current density $\mathbit{j}$, and the three spin-current densities ${\mathbit{J}}^{x},$ ${\mathbit{J}}^{y},$ and ${\mathbit{J}}^{z}$, where each of these density variables depends on specific blocks of the density matrix. Exchange-correlation (xc) functionals within the SCDFT should therefore depend on all of these eight fundamental density variables: ${F}_{xc}[n,{m}_{x},{m}_{y},{m}_{z},\mathbit{j},{\mathbit{J}}^{x},{\mathbit{J}}^{y},{\mathbit{J}}^{z}]$, which makes their parametrization a formidable task. Here, we formulate the adiabatic connection of the SCDFT for a treatment of exact Fock exchange in the theory. We show how the inclusion of a fraction of Fock exchange in standard functionals of the (spin) DFT (either in their collinear or noncollinear versions: ${F}_{xc}[n],$ ${F}_{xc}[n,{m}_{z}]$ and ${F}_{xc}[n,{m}_{x},{m}_{y},{m}_{z}]$) allows for the two-electron potential to depend on all those blocks of the density matrix that correspond to the eight density variables of the SCDFT, in a sensible and yet practical way. In particular, in the local-density and generalized-gradient approximations of the SCDFT, the treatment of the current densities solely from the Fock exchange term is formally justified by the short-range behavior of the exchange hole. We discuss that the adiabatic coupling strength parameter modulates the two-electron coupling of the orbital- and spin-current densities with the particle-number density and magnetization. Formal considerations are complemented by numerical tests on a periodic model system in the presence of spin-orbit coupling and in the absence of an external magnetic field.

The geometrical and electronic parameters for the interaction of two toxic heavy metals, namely: cadmium (Cd) and lead (Pb), on graphene and graphene oxide (GO) surfaces are investigated by using local Gaussian type basis sets and the hybrid PBE0 functional as implemented in the CRYSTAL code. The role of including long-range dispersion (D3) contribution as well as the basis set superposition error (BSSE) on the adsorption process is found to be crucial in the description of such interactions. Generally, Cd and Pb adatoms are found to be adsorbed more strongly on GO rather than pristine graphene due to the incorporation of oxygen functional groups (epoxy and/or hydroxyl) on the surface. Moreover, Pb has interestingly been found to play as an electron donor and to form covalent bonding with the GO surface. Such findings could have an impact in water treatment applications using graphene-based nanomaterials.

We perform a formal analysis of relativistic density functional theory for the treatment of spin-orbit coupling (SOC), noncollinear magnetization (NCM), and orbital current density (OCD). We identify specific components of the spinors (namely, those mapped onto imaginary diagonal spin-blocks of the density matrix) that arise from the SOC operator and define the OCD. We show that these pieces of the spinors only enter in the bielectronic part of the potential through the exact Fock exchange (FE) operator. The lack of FE therefore leads to a correspondingly incorrect physical description of SOC, NCM, and OCD. This analysis is complemented with an illustrative example, where we show that, while in the absence of FE, the theory fails even at reproducing the expected right-hand relationship between the NCM and OCD, its inclusion provides results that match those from a reference SOC configuration-interaction calculation.

We revise formal and numerical aspects of collinear and noncollinear density functional theory (DFT) in the context of a two-component self-consistent treatment of spin-orbit coupling (SOC). While the extension of the standard one-component theory to a noncollinear magnetization is formally well-defined within the local density approximation, and therefore results in a numerically stable theory, this is not the case within the generalized gradient approximation (GGA). Previously reported formulations of noncollinear DFT based on GGA exchange-correlation potentials have several limitations: (i) they fail at reducing (either formally or numerically) to the proper collinear limit (i.e., when the magnetization is parallel or antiparallel to the z axis everywhere in space); (ii) they fail at ensuring a quantitative rotational invariance of the total energy and even a qualitative rotational invariance of the spatial distribution of the magnetization when a SOC operator is included in the Hamiltonian; (iii) they are numerically very unstable in regions of small magnetization. All of the above-mentioned problems are here shown (both formally and through test examples) to be solved by using instead a new formulation of noncollinear DFT for GGA functionals, which we call the "signed canonical" theory, as combined with an effective screening algorithm for unstable terms of the exchange-correlation potential in regions of small magnetization. All methods are implemented in the CRYSTAL program and tests are performed on simple molecules to compare the different formulations of noncollinear DFT.

This paper is a comparative study of the Energy Envelope and the T -component response for interpreting airborne electromagnetic (AEM) data. The Energy Envelope is the square root of the sum of squares of three component AEM data along with their Hilbert transforms, while the T -component response is a similar quantity, except without the Hilbert transform terms. These quantities can be used to determine approximate geometrical parameters of compact anomalous targets. The approximate parameters are useful for constraining automatic interpretation algorithms. Synthetic examples are generated using a dipole conductor model. The synthetic models show that the Hilbert transform terms included in the Energy Envelope yield no additional benefits with regard to AEM data interpretation. Hence, the T -component response is a more efficient quantity for AEM modeling. The position of the peak of the T -component response can be used to estimate the position of a compact target that is consistent with the measured response. In particular for a MEGATEM configuration and when the target lies directly below the flight line and the line spacing of the survey is 200 m, the error in predicting the position of the target is under 200 m. This error is improved in situations where the conductor is at an offset to the flight line, or when the line spacing is decreased. The strike of the conductor can also be estimated, as a series of peaks will align along the strike direction. Once the position and strike of the conductor is known, look-up-tables are generated for these specific parameters. The look-up-tables can be used to determine the depth and dip of the target. The depth can be estimated from the full width at half magnitude of the T -component response. The dip can be estimated from the asymmetry of the T -component response. Tests over the Chibougamau field site yield results in reasonable agreement with previous work of the same authors.

Lanthanide sesquioxides are strongly correlated materials characterized by highly localized unpaired electrons in the $f$ band. Theoretical descriptions based on standard density functional theory (DFT) formulations are known to be unable to correctly describe their peculiar electronic and magnetic features. In this study, electronic and magnetic properties of the first four lanthanide sesquioxides in the series are characterized through a reliable description of spin localization as ensured by hybrid functionals of the DFT, which include a fraction of nonlocal Fock exchange. Because of the high localization of the $f$ electrons, multiple metastable electronic configurations are possible for their ground state depending on the specific partial occupation of the $f$ orbitals: the most stable configuration is here found and characterized for all systems. Magnetic ordering is explicitly investigated, and the higher stability of an antiferromagnetic configuration with respect to the ferromagnetic one is predicted. The critical role of the fraction of exchange on the description of their electronic properties (notably, on spin localization and on the electronic band gap) is addressed. In particular, a recently proposed theoretical approach based on a self-consistent definition---through the material dielectric response---of the optimal fraction of exchange in hybrid functionals is applied to these strongly correlated materials.

The electronic and vibrational features of the VHn (n = 1 to 4) family of defects in diamond (hydrogen atoms saturating the dangling bonds of the atoms surrounding a vacancy) are investigated at the quantum mechanical level by using the periodic supercell approach, an all electron Gaussian type basis set, hybrid functionals, and the CRYSTAL code. Most of the results have been collected for supercells containing 64 atoms; however, in order to explore the effect of the defect concentration on both the IR and Raman spectra, supercells containing 216, 512 and 1000 atoms have also been considered in the VH4 case. For each system, all the possible spin states are considered; their relative stability, band structure, charge and spin density distributions are thoroughly described. All the investigated systems present specific IR and Raman spectra, with vibrational spectroscopic features that can in principle be used as fingerprints for their characterization. This is particularly true for the C–H stretching, that ranges between 2500 and 4400 cm−1. The stretching modes are strongly affected by anharmonicity that has been evaluated in this work; it turns out to be extremely sensitive to the H load and spin state of the system, and ranges from −335 cm−1 for VH1 to +85 cm−1 for VH4. All of the investigated defects have very low C–H stretching IR intensity, so that they essentially appear as silent, the exception being VH1. The situation is different for the Raman spectra: the stretching modes of all defects do have similar large intensity; unfortunately here it is the experimental evidence that is lacking.

The Linear Combination of Atomic Orbitals approach implemented into the public Crystal program for quantum-chemical simulations of n-dimensional periodic systems ( $$n = 0,1,2,3$$ ) is here extended to g-type basis functions. A general algorithmic procedure is devised for the calculation of the coefficients needed for the analytical evaluation of one- and two-electron integrals for energy and forces within the recursive McMurchie–Davidson strategy, up to arbitrarily high quantum numbers. Explicit routines are generated for the calculation of all the coefficients needed for g-type functions, which ensure a very high computational efficiency. The code has been generalized in many respects so as to allow for the use of g-type functions for: (1) Hartree–Fock energy and forces; (2) density functional theory energy and forces (in either a local density, generalized gradient, meta-GGA or various hybrid approximations); (3) all-electron and pseudo-potential basis sets; (4) spin-restricted and unrestricted calculations; (5) coupled-perturbed Hartree–Fock/Kohn–Sham (hyper)-polarizability calculations; (6) projected density-of-states. The g-type basis functions are expected to play an important role in (1) the description of the electronic structure of heavy elements and in particular of lanthanides and actinides with occupied 4f and 5f bands, respectively, where they represent the first polarization, (2) those calculations requiring an accurate description of the electronic polarization.

Phosphogypsum formed from the production of phosphoric acid represents by far the biggest accumulation of gypsum-rich wastes in the world and commonly contains elevated radionuclides, including uranium, as well as other heavy metals and metalloids. Therefore, billions-of-tons of phosphogypsum stockpiled worldwide not only possess serious environmental problems but also represent a potential uranium resource. Gypsum is also a major solid constituent in many other types of radioactive mine tailings, which stems from the common usage of sulfuric acid in extraction processes. Therefore, management and remediation of radioactive mine tailings as well as future beneficiation of uranium from phosphogysum all require detailed knowledge about the nature and behavior of uranium in gypsum. However, little is known about the uptake mechanism or speciation of uranium in gypsum. In this study, synthesis experiments suggest an apparent pH control on the uptake of uranium in gypsum at ambient conditions: increase in U from 16 μg/g at pH = 6.5 to 339 μg/g at pH = 9.5. Uranium L 3 -edge synchrotron X-ray absorption spectroscopic analyses of synthetic gypsum show that uranyl (UO 2 ) 2+ at the Ca site is the dominant species. The EXAFS fitting results also indicate that uranyl in synthetic gypsum occurs most likely as carbonate complexes and yields an average U-O distance ∼0.25 A shorter than the average Ca-O distance, signifying a marked local structural distortion. Applications to phosphogypsum from the New Wales phosphoric acid plant (Florida, USA) and uranium mine tailings from the Key Lake mill (Saskatchewan, Canada) show that gypsum is an important carrier of uranium over a wide range of pH and controls the fate of this radionuclide in mine tailings. Also, development of new technologies for recovering U from phosphogypsum in the future must consider lattice-bound uranyl in gypsum.

A discrete conductor model was used to estimate the volume of influence of a dual transmitter loop ground time-domain electromagnetic system (the InfiniTEM system). A sphere model in locally uniform field was used to calculate the signal from a subsurface target where the currents are constrained to flow vertically. The noise was determined from two field surveys. The signal-to-noise ratio was determined at each subsurface target location and each receiver location. The sensitivity of the InfiniTEM system at each target location was defined as the maximum of the absolute value of the signal-to-noise ratio for the ensemble of receiver positions in the survey. The volume of influence is defined as the volume where all targets have a sensitivity greater than one. The manner in which volume of influence varies can be used to determine the optimal design parameters of an InfiniTEM survey. Our analysis reveals that the InfiniTEM system should be operated with a loop separation distance of 1.5 times the loop width (where width and separation are measured parallel to the traverse lines); and that there should be 4 traverse lines between the loops, corresponding to a traverse line spacing of 250 m for a loop width of 1000 m. For the purposes of delineating highly conductive targets, the optimal waveform parameters are a high duty cycle (in our case 0.75), a low base frequency (in our case 10 Hz), and measurements should be made in the B field domain. For the purposes of finding less conductive targets, the base frequency should be high (in our case 30 Hz), the duty cycle should be low (in our case 0.25), and measurements should be made in the ∂ B /∂ t domain. Our study confirms that the InfiniTEM system can detect a 100 m radius sphere at up to 925 m depth. We have determined that electromagnetic systems are most sensitive to bodies striking perpendicular to the traverse line. As well, we have confirmed that the InfiniTEM system is most effective at detecting vertical targets.

Electromagnetic exploration methods have important applications for geologic mapping and mineral exploration in igneous and metamorphic terranes. In such cases, the earth is often largely resistive and the most important interaction is between a conductor of interest and a shallow, thin, horizontal sheet representing glacial tills and clays or the conductive weathering products of the basement rocks (both of which are here termed the “conductive overburden”). To this end, we have developed a theory from which the step and impulse responses of a sphere interacting with conductive overburden can be quickly and efficiently approximated. The sphere model can also be extended to restrict the currents to flow in a specific orientation (termed the dipping-sphere model). The resulting expressions are called semianalytical because all relevant relations are developed analytically, with the exception of the time-convolution integrals. The overburden is assumed to not be touching the sphere, so there is no galvanic interactions between the bodies. We make use of the dipole sphere in a uniform field and thin sheet approximations; however, expressions could be obtained for a sphere in a dipolar (or nondipolar) field using a similar methodology. We have found that there is no term related to the first zero of the relevant Bessel function in the response of the sphere alone. However, there are terms for all other zeros. A test on a synthetic model shows that the combined sphere-overburden response can be reasonably approximated using the first-order perturbation of the overburden field. Minor discrepancies between the approximate and more elaborate numerical responses are believed to be the result of numerical errors. This means that in practice, the proposed approach consists of evaluating one convolution integral over a sum of exponentials multiplied by a polynomial function. This results in an extremely simple algorithmic implementation that is simple to program and easy to run. The proposed approach also provides a simple method that can be used to validate more complex algorithms. A test on field data obtained at the Reid Mahaffy site in Northern Ontario shows that our approximate method is useful for interpreting electromagnetic data even when the background is thick. We use our approach to obtain a better estimate of the geometry and physical properties of the conductor and evaluate the conductance of the overburden.

We show the advantages of using multi-component transmitter-receiver systems for determining the geometry of a compact planar target whose electromagnetic response can be approximated by a dipole. Our approach is based on a modified version of an algorithm that we previously published using a single-component (vertical) transmitter. Tests on synthetic models reveal that single transmitter systems are unable to resolve the orientation of a dipole conductor that approaches axial symmetry with respect to the traverse line. This occurs as a result of lack of a noticeable y-component anomaly, where the y-component is oriented transverse to the flight-line direction. For a plate-like conductor, axial symmetry equates to being at a small offset and having a strike parallel or perpendicular to the traverse line. Here, the term 'offset' is used to denote the lateral distance from the centre of the conductor to the flight line. The ambiguities can be resolved through measuring specific components of a multi-component transmitter-receiver system; namely, Rxz and Rzz with one of Rxy, Ryy, Rzy, Ryx and Ryz, where the first letter denotes the orientation of the transmitter and the second letter denotes the orientation of the receiver. However, for the case of a MEGATEM system geometry, measuring Rzx, Rzz and Ryx is most suitable for determining the geometry of conductors striking nearly perpendicular or parallel, and at small offset to the traverse line. The minimum system capable of determining the correct geometrical parameters of a dipole conductor for the small-offset symmetric case would therefore consist of a two-component (y- and z-directed) transmitter, as well as a two component (x- and z-directed) receiver. Tests on line 15701 of the MEGATEM survey in Chibougamau, Quebec, confirm the inability of single-transmitter systems to determine geometrical parameters of a dipole conductor for the case where y-component data is unavailable.

We have developed a method for displaying or imaging data from a ground-time-domain electromagnetic system and for extracting the geometric parameters of a small conductor. The parameters are determined directly from the data using combinations of the spatial components of the secondary fields and their Hilbert transforms. The position of the target coincides with the peaks of the energy envelope (EE) or the [Formula: see text]-component of the response. Here, the EE is the square root of the sum of the squares of the three spatial components and their Hilbert transforms, whereas the [Formula: see text]-component response is an analogous quantity that excludes the Hilbert transform terms. Studies on synthetic models indicate that the [Formula: see text]-component response is sharper than the EE in most possible target orientations. Once the position of a body has been determined using the peak of the [Formula: see text]-component response, the dip of the target can be quantified using the ratio of the full-width at half-magnitude (FWHM) of the [Formula: see text]-component response and the [Formula: see text]-component Hilbert transform response, which is analogous to the EE but excludes the untransformed quantities. Finally, once all other geometric parameters have been determined, the depth of the target can be evaluated using the FWHM of the [Formula: see text]-component response. The proposed modeling method was tested over an anomaly acquired at the Coulon field site during an InfiniTEM survey in the Abitibi greenstone belt of Quebec. The extracted geometric parameters were consistent with the available geologic information.