Experimental and theoretical evidence for substitutional molybdenum atoms in theTiO2(110)subsurface

2006

Molybdenum was deposited at room temperature on the ${\mathrm{TiO}}_{2}(110)$ surface in the 0--1.3 equivalent monolayer (eqML) range and was then annealed at $400\phantom{\rule{0.2em}{0ex}}\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$ in order to reach a kind of equilibrium state. A threshold was found in the behavior of the deposit: below 0.2 eqML, substitutional molybdenum occurs in titanium sites located under the bridging oxygen atoms of the ${\mathrm{TiO}}_{2}(110)$ surface. In this position, molybdenum atoms are in a structural and chemical ${\mathrm{MoO}}_{2}$-like environment. Density-functional theory calculations show that this molybdenum site is actually the most stable one in …

Multichannel multiple scattering calculation ofL2,3-edge spectra ofTiO2andSrTiO3: Importance of multiplet coupling and band structure

2010

We report a theoretical study on x-ray absorption spectroscopy at the Ti-${L}_{2,3}$-edge of rutile and anatase ${\text{TiO}}_{2}$ as well as ${\text{SrTiO}}_{3}$. Using the first-principles multichannel multiple-scattering method, we obtain good agreement with experiment in all cases. We show that both multiplet-type electron correlation effects and the long-range band structure strongly influence the spectra. The differences in line shape between the three compounds are essentially a long-range effect which reflects the different crystal structures on a length scale of 1 nm.

Defects at the TiO2(100) surface probed by resonant photoelectron diffraction.

2006

We report photoelectron diffraction (PED) experiments of weakly sub-stoichiometric TiO 2 (100) rutile surfaces. Apart from standard core-level PED from the Ti-2p3/2 line, we have studied valence band PED from the defect induced Ti-3d states in the insulating band gap. For maximum yield, the latter were resonantly excited at the Ti-2p absorption edge. The PED patterns have been analyzed within the forward scattering approximation as well as by comparison with simulated PED patterns obtained in multiple scattering calculations. The analysis shows that the defect induced Ti-3d charge is mainly located on the second layer Ti atoms. © 2007 Elsevier B.V. All rights reserved.

Molybdenum Clusters on a TiO2(110) Substrate Studied by Density Functional Theory.

2009

International audience; A theoretical study on molybdenum clusters adsorbed on a rutile TiO2(110) substrate is reported. Using density functional theory, equilibrium geometries, atomic charges, and total energies have been calculated for clusters containing up to five Mo atoms. Isolated Mo adatoms are strongly oxidized and repel each other. The Mo oxidation state is considerably lowered as soon as the first short Mo−Mo bond is formed. The relative stability of different cluster geometries can be understood from the competition between Mo−Mo and Mo−O bonding. Some low-energy structures for two and three Mo atoms involve large displacements of a substrate oxygen atom. The most stable five-ato…

Interaction of Mo(CO) 6 and its derivative fragments with the Cu(001) surface: Influence on the decomposition process

2014

A theoretical study on the adsorption and decomposition of molybdenum carbonyl on the copper (001) surface is reported. The adsorption structures and energies of Mo(CO)n molecules (n = 1 … 6) are computed systematically using density functional theory with Van der Waals corrections. By analyzing the energies of the various conformations, the main factors that determine the stable adsorption geometry are identified. Insight into the thermodynamics of decomposition is gained by calculating the reaction energy for dissociation of Mo(CO)n into Mo(CO)n−1 and CO. In the gas phase, this reaction is highly endothermic for all n. On the Cu surface, however, removal of the first CO group (n = 6) beco…

Defect States at theTiO2(110)Surface Probed by Resonant Photoelectron Diffraction

2008

The charge distribution of the defect states at the reduced ${\mathrm{TiO}}_{2}(110)$ surface is studied via a new method, the resonant photoelectron diffraction. The diffraction pattern from the defect state, excited at the $\mathrm{Ti}\mathrm{\text{\ensuremath{-}}}2p\mathrm{\text{\ensuremath{-}}}3d$ resonance, is analyzed in the forward scattering approach and on the basis of multiple scattering calculations. The defect charge is found to be shared by several surface and subsurface Ti sites with the dominant contribution on a specific subsurface site in agreement with density functional theory calculations.

Field-induced tip–sample oxygen transfer in scanning tunneling microscopy on TiO2(110) (1 1).

2008

International audience; A study on the field-induced tip–surface oxygen transfer at room temperature and its influence on the tunneling conditions for stable STM imaging of the TiO2(110) (1 1) surface is reported. A simple model of field-induced transfer is applied to tungsten and platinum–iridium tips. The oxygen transition rates from the sample to the tip or from the tip to the sample depend on the oxygen desorption barriers formed at tunneling distance. For stable imaging the applied bias voltage has to balance the oxygen transfer probabilities in both directions. In the case of Pt/Ir tips, the tunneling conditions for images with clear evidence of bridging oxygen point defects have been…

Giant lateral electrostriction in ferroelectric liquid-crystalline elastomers

2001

Mechanisms for converting electrical energy into mechanical energy are essential for the design of nanoscale transducers, sensors, actuators, motors, pumps, artificial muscles, and medical microrobots. Nanometre-scale actuation has to date been mainly achieved by using the (linear) piezoelectric effect in certain classes of crystals (for example, quartz), and 'smart' ceramics such as lead zirconate titanate. But the strains achievable in these materials are small--less than 0.1 per cent--so several alternative materials and approaches have been considered. These include grafted polyglutamates (which have a performance comparable to quartz), silicone elastomers (passive material--the constri…

Multiple scattering theory for non-local and multichannel potentials.

2012

International audience; Methodological advances in multiple scattering theory (MST) in both wave and Green's function versions are reported for the calculation of electronic ground and excited state properties of condensed matter systems with an emphasis on core-level photoemission and absorption spectra. Full-potential MST is reviewed and extended to non-local potentials. Multichannel MST is reformulated in terms of the multichannel density matrix whereby strong electron correlation of atomic multiplet type can be accounted for in both ground and excited states.

Resonant Photoelectron Diffraction

2013

A layout of a resonant photoelectron diffraction, RESPED, experiment is described from the theoretical basis to the data acquisition and analysis procedures. The theory of the resonance between the directly emitted photoelectron of a selected valence band and the electron emitted by autoionization (Auger) of the same valence band is presented within a formal frame. The critical issue of the angular symmetry and distribution of the resonating electron is discussed in connection with the current computational protocols for photoelectron diffraction, PED, analysis. A few representative applications are presented, where RESPED is shown to overcome some limitations of conventional PED thanks to …

Intrinsic Nature of the Excess Electron Distribution at theTiO2(110)Surface

2012

The gap state that appears upon reduction of TiO2 plays a key role in many of titania's interesting properties but its origin and spatial localization have remained unclear. In the present work, the TiO2(110) surface is reduced in a chemically controlled way by sodium adsorption. By means of resonant photoelectron diffraction, excess electrons are shown to be distributed mainly on subsurface Ti sites strikingly similar to the defective TiO2(110) surface, while any significant contribution from interstitial Ti ions is discarded. In agreement with first principles calculations, these findings demonstrate that the distribution of the band gap charge is an intrinsic property of TiO2(110), indep…

Influence ofsp−dhybridization on the electronic structure of Al-Mn alloys

2008

The influence of $sp\text{\ensuremath{-}}d$ hybridization on the electronic structure of different Al-Mn alloys has been studied by photoelectron spectroscopy. Experimental evidence of a pseudogap in a crystalline binary Hume-Rothery alloy is provided. The pseudogap varies systematically with Mn concentration. The $sp\text{\ensuremath{-}}d$ hybridization alone, even in the absence of Hume-Rothery mechanism, can produce the pseudogap. Existence of the pseudogap, suppression of the $\mathrm{Mn}\phantom{\rule{0.2em}{0ex}}2p$ satellite, and decrease in the Doniach-\ifmmode \check{S}\else \v{S}\fi{}unji\ifmmode \acute{c}\else \'{c}\fi{} asymmetry parameter are the consequences of the $sp\text{\e…

Cationic vacancies and anomalous spectral-weight transfer in Ti1−xTaxO2thin films studied via polarization-dependent near-edge x-ray absorption fine …

2013

We report the electronic structures of Ta-doped anatase TiO 2 thin films grown by pulsed laser deposition (PLD) with varying magnetization using a combination of first-principles calculations and near-edge x-ray absorption fine structure (NEXAFS) spectroscopy. The roles of Ta doping and Ti vacancies are clarified, and the observed room-temperature ferromagnetism is attributed to the localized magnetic moments at Ti vacancy sites ferromagnetically ordered by electron charge carriers. O K -edge spectra exhibit significant polarization dependence which is discussed and supported by first-principles calculations in relation to both the crystal symmetry and the formation of defects. In particula…

Monolayer Formation of Molybdenum Carbonyl on Cu(111) Revealed by Scanning Tunneling Microscopy and Density Functional Theory

2012

International audience; Molybdenum carbonyl Mo(CO)(6) was adsorbed on the Cu(111) surface at 160 K in the monolayer coverage range and studied by scanning tunneling microscopy. A well-ordered monolayer of hexacarbonyl molecules was observed experimentally for the first time. The monolayer has a hexagonal structure compatible with a (root 7 x root 7)R19 superlattice on the copper (111) plane. The arrangement and orientation of the molecules on the surface were determined by density functional theory calculations, including van der Waals interactions. The comparison of adsorption and cohesive energies reveals that the molecule-substrate interaction is stronger than the intermolecular one, whi…

Second-order Raman scattering in CuO

2013

Polarized second-order Raman scattering spectra of CuO single crystals are reported. It is shown that for some scattering geometries the second-order processes dominate the inelastic light scattering spectra. Group-theoretical symmetry analysis of the selection rules for the first- and second-order scattering processes is performed and phonon dispersion relations are calculated within density functional theory. The main spectral features of the two-phonon spectra are assigned to overtones of the vibrational branches at various special points across the Brillouin zone.

Dynamics of molybdenum nano structure formation on the TiO2(110) surface: A kinetic Monte Carlo approach

2006

Abstract The rutile TiO 2 (1 1 0) surface is a highly anisotropic surface exhibiting “channels” delimited by oxygen rows. In previous experimental and theoretical DFT works we could identify the molybdenum adsorption sites. They are located inside the channels. Moreover, experimental studies have shown that during subsequent annealing after deposition, special molybdenum nano structures can be formed, especially two monolayer high pyramidal chains of atoms. In order to better understand the dynamics of nano structure formation, we present a kinetic Monte Carlo study on diffusion and adsorption of molybdenum atoms on a TiO 2 (1 1 0) surface. A quasi one-dimensional lattice gas model has been…

Kirkwood-Buff integrals from molecular simulation

2019

The Kirkwood-Buff (KB) theory provides a rigorous framework to predict thermodynamic properties of isotropic liquids from the microscopic structure. Several thermodynamic quantities relate to KB integrals, such as partial molar volumes. KB integrals are expressed as integrals of RDFs over volume but can also be obtained from density fluctuations in the grand-canonical ensemble. Various methods have been proposed to estimate KB integrals from molecular simulation. In this work, we review the available methods to compute KB integrals from molecular simulations of finite systems, and particular attention is paid to finite-size effects. We also review various applications of KB integrals comput…

Mo(CO)6 dissociation on Cu(111) stimulated by a Scanning Tunneling Microscope

2013

Abstract The surface of Cu(111) was exposed to molybdenum hexacarbonyl Mo(CO)6 with monolayer coverage at temperature 160 K and studied by a Scanning Tunneling Microscope. The monolayer structure has a hexagonal arrangement and forms a (√7 × √7) R19 superlattice on the copper (111) plane. Upon repeated scanning the monolayer is transformed into a (1 × 2) superstructure with 3-fold oriented domains. The domains of (1 × 2) superstructure can change orientation under scanning according to 3-fold surface symmetry. From analysis of the domain mobility, it follows that CO groups of carbonyl fragments are organized in the (1 × 2) superstructure conditioning the domain reorientation. The observed s…

Real-space multiple scattering method for angle-resolved photoemission and valence-band photoelectron diffraction and its application to Cu(111)

2011

Abstract: A computational method is presented for angle-resolved photoemission spectra (ARPES) and photoelectron diffraction (PED) in the ultraviolet regime. The one-step model is employed and both initial valence and final continuum states are calculated using the finite-cluster, real-space multiple scattering method. Thereby the approach is versatile and provides a natural link to core-level PED. The method is applied to the Cu(111) valence band and good agreement with experiment is found for both ARPES spectra and PED patterns. When the PED patterns are integrated over a filled band of a single-orbital symmetry, such as Cu-3d, we show, both numerically and analytically, that the exact th…

Molybdenum thin film growth on a TiO2 (1 1 0) substrate.

2009

International audience; We report a first principles study on the structure and energetics of thin films of molybdenum on a (1 1 0) surface of rutile TiO2. Mo films with 1 × 1 epitaxy in the coverage range between 0.5 and 2 monolayer are investigated. The most stable structures are those which maximize the number of Mo–Mo bonds. This leads to two-dimensional structures with zigzag Mo–Mo coordination for 1 monolayer coverage and three-dimensional structures with approximately body-centered cubic coordination for higher coverage. For a coverage up to 1.5 monolayers, the interface Mo atoms preferentially occupy the so-called upper hollow adsorption site with three Mo–O bonds

Kirkwood-Buff Integrals for Finite Volumes.

2012

Exact expressions for finite-volume Kirkwood−Buff (KB) integrals are derived for hyperspheres in one, two, and three dimensions. These integrals scale linearly with inverse system size. From this, accurate estimates of KB integrals for infinite systems are obtained, and it is shown that they converge much better than the traditional expressions. We show that this approach is very suitable for the computation of KB integrals from molecular dynamics simulations, as we obtain KB integrals for open systems by simulating closed systems.

Electronic structure, lattice dynamics and thermodynamic stability of paramelaconite Cu4O3

2014

Abstract An ab initio study of the electronic structure, lattice dynamic and thermodynamic properties of paramelaconite Cu 4 O 3 is reported. The insulating, mixed-valence character of Cu 4 O 3 is elucidated by analyzing the band structure and the spin-orbital symmetry of the Cu-3 d hole states. Exchange coupling constants between Cu 2+ ions are computed which confirm the frustrated antiferromagnetism of the spin lattice. The lattice dynamics is studied from first principles and main features of the vibrational spectrum are assigned to the different chemical species Cu + , Cu 2+ and O. The thermodynamic stability of Cu 4 O 3 is investigated by calculating the free energy of the decompositio…

Optical and Acoustic Vibrations Confined in Anatase TiO2 Nanoparticles under High-Pressure

2014

International audience; The effect of an applied high pressure on the optical and acoustic vibrations of small anatase TiO2 nanoparticles is studied using Raman scattering. All the Raman peaks show a significant variation of their frequency with pressure, except for the low-frequency peak which is due to acoustic vibrations confined in the nanoparticles. These variations (or lack thereof) are compared to first-principles calculations of the stiffness tensor and phonons of bulk anatase TiO2 as a function of pressure. In particular, the variation of the shape of the low-frequency peak is explained by the increase of the elastic anisotropy of anatase TiO2 as pressure is increased.

A First Principles Study on Charge Dependent Diffusion of Point Defects in Rutile TiO2

2010

A first principles theoretical study on the diffusion mechanism of Ti interstitials and O vacancies in rutile TiO2 is reported. We find that the diffusion depends strongly on the defect charge. Wea...

Reactivity between molybdenum and TiO2(110) surfaces: evidence of a sub-monolayer mode and a multilayer mode

2005

Small amounts of molybdenum (from 0.03 to 1.3 eqML) were deposited on non-stoichiometric TiO 2 (1 1 0) surface. The deposits were investigated by means of LEED and X-ray/UV photoemission using synchrotron radiation. For the smallest coverage (<0.2 eqML), deposition leads to oxidation of molybdenum into species close to Mo 4+ .In such a case, states appearing in TiO 2 band gap are mainly due to reduced titanium. For higher coverages, metallic behaviour of molybdenum is observed. This phenomenon was explained, thanks to first principle calculations, as a decrease of the Mo-O interactions for the benefit of the Mo-Mo interactions as the surface molybdenum atom density increases.

X-ray absorption spectra at the CaL2,3edge calculated within multichannel multiple scattering theory

2004

We report a theoretical method for x-ray absorption spectroscopy (XAS) in condensed matter which is based on the multichannel multiple scattering theory of Natoli et al. and the eigen-channel $R$-matrix method. While the highly flexible real-space multiple scattering (RSMS) method guarantees a precise description of the single-electron part of the problem, multiplet-like electron correlation effects between the photoelectron and localized electrons can be taken account for in a configuration interaction scheme. For the case where correlation effects are limited to the absorber atom, a technique for the solution of the equations is devised, which requires only little more computation time th…

Theory of CaL2,3-edge XAS using a novel multichannel multiple-scattering method

2003

A new method for calculating X-ray absorption spectroscopy (XAS) at the L2,3 edges of Ca and transition metals is presented. It is based on the multichannel multiple-scattering theory by Natoli et al. [Phys. Rev. B, (1990), 42, 1944-1968] combined with the eigen-channel R-matrix formalism. Atomic multiplet-like effects, owing to the Coulomb interaction of photoelectrons and the 2p hole, are taken into account through a configuration interaction ansatz for the final-state wavefunction. The various multiplet states lead to a set of channels for the photoelectron wavefunction, which is calculated in multiple-scattering theory. The method is applied to Ca, an important element for biological ap…

Understanding the2pcore-level spectra of manganese: Photoelectron spectroscopy experiments and Anderson impurity model calculations

2007

Using high-resolution core-level photoelectron spectroscopy and modified Anderson impurity model calculations, we study the $\mathrm{Mn}\phantom{\rule{0.2em}{0ex}}2p$ spectrum of manganese metal and resolve the current debate about its spectral shape. An unusual satellite feature, $1\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ from the main peak, is observed in the $\mathrm{Mn}\phantom{\rule{0.2em}{0ex}}2{p}_{3∕2}$ spectrum of a thick Mn layer grown on Al. It originates from intra-atomic multiplet effect related to Mn atoms with large local moment. The satellite decreases in intensity for thin Mn layers and for Al deposition on bulklike Mn because of enhanced $\mathrm{Mn}\phantom{\rule{0.2em}{0e…

Scanning Probe Microscopy Study of the Metal-Rich Layered Chalcogenides TaM2Te2 (M = Co, Ni)

1998

The compounds TaNi2Te2 and TaCo2Te2 have been examined by scanning tunneling and atomic force microscopy. The title phases crystallize in layered structures with metal slabs sandwiched by tellurium atoms. Scanning probe microscope images of the surfaces of these materials arise from the surface tellurium atoms anddepending on the experimental conditionscan show very different features. The images have been simulated through surface charge densities calculated within the Extended Huckel and LMTO frameworks.

Electronic Structure Changes across the Metamagnetic Transition in FeRh via Hard X-Ray Photoemission

2012

International audience; Stoichiometric FeRh undergoes a temperature-induced antiferromagnetic (AFM) to ferromagnetic (FM) transition at similar to 350 K. In this Letter, changes in the electronic structure accompanying this transition are investigated in epitaxial FeRh thin films via bulk-sensitive valence-band and core-level hard x-ray photoelectron spectroscopy with a photon energy of 5.95 keV. Clear differences between the AFM and FM states are observed across the entire valence-band spectrum and these are well reproduced using density-functional theory. Changes in the 2p core levels of Fe are also observed and interpreted using Anderson impurity model calculations. These results indicat…

How to apply the Kirkwood–Buff theory to individual species in salt solutions

2013

It is generally assumed that the Kirkwood–Buff (KB) theory cannot be applied to anions and cations individually in a solution, as one cannot simulate this system in an open ensemble due to the electroneutrality constraint. By applying our recently derived KB theory for closed systems, we show that one does have access to single-ion properties in Molecular Dynamics. Our findings are supported by simulations for a model of a salt solution in which particles interact with WCA potentials, as well as for the NaCl/water system using the Particle Mesh Ewald technique for electrostatics.

Multiplet coupling and band structure in L2,3-edge XAS through multi-channel multiple scattering theory.

2009

International audience; Using the recently developed multi-channel multiple scattering (MCMS) method we have calculated the x-ray absorption spectra (XAS) at the L2,3-edge of transition metal compounds. The MCMS method is an ab initio scheme which combines an accurate description of the band structure of the material with a correlated many-electron wave function on the absorber atom. Thereby configuration interaction in the XAS final state, in particular multiplet effects, can be taken into account. In the present implementation, we use an electron-hole wave function and treat the interaction with all other electrons on a mean-field level. The calculated spectra agree well with experiment f…

First-Principles Calculations of Angle-Resolved and Spin-Resolved Photoemission Spectra of Cr(110) Surfaces at the2p−3dCr Resonance

2013

A first principles approach for spin- and angle-resolved resonant photoemission is developed within multiple scattering theory and applied to a Cr(110) surface at the 2p-3d resonance. The resonant photocurrent from this nonferromagnetic system is found to be strongly spin polarized by circularly polarized light, in agreement with experiments on antiferromagnetic and magnetically disordered systems. By comparing the antiferromagnetic and Pauli-paramagnetic phases of Cr, we explicitly show that the spin polarization of the photocurrent is independent of the existence of local magnetic moments, solving a long-standing debate on the origin of such polarization. New spin polarization effects are…

Adsorption and diffusion of a molybdenum atom on theTiO2(110)surface: A first-principles study

2007

A study on the structure and energetics of a single molybdenum atom adsorbed on the $\mathrm{Ti}{\mathrm{O}}_{2}(110)$ surface is reported. All possible adsorption sites have been determined. Moreover, it is found that incorporation of the Mo atom into the first surface layer leads to considerably more stable structures than on any adsorption site. Different channels for migration of the molybdenum atom have been identified. The diffusion barriers of these channels have been determined. The results on structure and energetics are discussed by analyzing the electronic properties of the $\mathrm{Mo}∕\mathrm{Ti}{\mathrm{O}}_{2}(110)$ systems.

Theoretical study of resonant x-ray emission spectroscopy of Mn films on Ag.

2006

We report a theoretical study on resonant x-ray emission spectra (RXES) in the whole energy region of the Mn $L_{2,3}$ white lines for three prototypical Mn/Ag(001) systems: (i) a Mn impurity in Ag, (ii) an adsorbed Mn monolayer on Ag, and (iii) a thick Mn film. The calculated RXES spectra depend strongly on the excitation energy. At $L_3$ excitation, the spectra of all three systems are dominated by the elastic peak. For excitation energies around $L_2$, and between $L_3$ and $L_2$, however, most of the spectral weight comes from inelastic x-ray scattering. The line shape of these inelastic ``satellite'' structures changes considerably between the three considered Mn/Ag systems, a fact tha…

Kirkwood–Buff Integrals Using Molecular Simulation: Estimation of Surface Effects

2020

Kirkwood&ndash

Kirkwood–Buff integrals of finite systems

2018

The Kirkwood–Buff (KB) theory provides an important connection between microscopic density fluctuations in liquids and macroscopic properties. Recently, Krüger et al. derived equations for KB integrals for finite subvolumes embedded in a reservoir. Using molecular simulation of finite systems, KB integrals can be computed either from density fluctuations inside such subvolumes, or from integrals of radial distribution functions (RDFs). Here, based on the second approach, we establish a framework to compute KB integrals for subvolumes with arbitrary convex shapes. This requires a geometric function w(x) which depends on the shape of the subvolume, and the relative position inside the subvolu…

Adsorption of 3d Transition Elements on a TiO2(110) Surface.

2008

International audience; A first-principles study on the adsorption of 3d transition metal atoms on a stoichiometric TiO2(110) surface is reported. For all 3d elements except Cu, the most stable on-surface adsorption site is a site where the adatom binds to two twofold and one threefold surface oxygen atoms. For Ti, V, and Cr, however, a subsurface site, where the adatom substitutes a sixfold Ti atom, is more stable. The adatoms are oxidized in all cases. The charge transfer to the substrate is larger for the substitutional site than for the on-surface adsorption sites and decreases with atomic number along the 3d series. The relative stabilities of the adsorption sites are discussed in term…

Tutorial : Modélisation de l'absorption X dans l'approche mono-électronique.

2008

Recent advances in multichannel multiple scattering theory for electron spectroscopies.

2006

Multichannel en absorption X.

2006

EELS modeling.

2009

International audience

Kirkwood–Buff integrals of finite systems: shape effects

2018

The Kirkwood–Buff (KB) theory provides an important connection between microscopic density fluctuations in liquids and macroscopic properties. Recently, Krüger et al. derived equations for KB integrals for finite subvolumes embedded in a reservoir. Using molecular simulation of finite systems, KB integrals can be computed either from density fluctuations inside such subvolumes, or from integrals of radial distribution functions (RDFs). Here, based on the second approach, we establish a framework to compute KB integrals for subvolumes with arbitrary convex shapes. This requires a geometric function w(x) which depends on the shape of the subvolume, and the relative position inside the subvolu…