Electron localisation function in current-density-functional theory

ABSTRACT We present a generalisation of the electron localisation function (ELF) to current-density-functional theory as a descriptor for the properties of molecules in the presence of magnetic fields. The resulting current ELF (cELF) is examined for a range of small molecular systems in field strengths up to B0 = 235 kT (one atomic unit). The cELF clearly depicts the compression of the molecular electronic structure in the directions perpendicular to the applied field and exhibits a structure similar to that of the physical current densities. A topological analysis is performed to examine the changes in chemical bonding upon application of a magnetic field.

Recently, we have studied exchange-correlation functionals in current DFT (CDFT) for calculations of molecules in the presence of strong uniform magnetic fields [22][23][24][25]. In particular, in [25], we showed that the CONTACT Andrew M. Teale andrew.teale@nottingham.ac.uk current-dependent generalisation of the Tao-Perdew-Staroverov-Scuseria (TPSS) meta-generalised-gradientapproximation (meta-GGA) functional [26], denoted as cTPSS (see also [27] for the use of cTPSS in response theory), provides a reasonable description of molecules in strong magnetic fields. In such fields, a new form of bonding, perpendicular paramagnetic bonding, has recently been identified [28]. The nature of this bonding interaction was accurately captured and analysed in terms of molecular-orbital energies and electron-density differences at the CDFT level in [25] using the cTPSS functional.
In Section 2, we propose a generalisation of the ELF for molecules in a magnetic field. This generalisation is applied to study chemical bonding in Section 3.1, whereas a topological analysis is performed in Section 3.2. Finally, in Section 3.3, the physical current density induced by the field is visualised alongside the generalised ELF, illustrating the striking similarity in the topologies of these quantities.

Theory
After reviewing the ELF as introduced in Hartree-Fock theory in Section 2.1, we consider its adaption for Kohn-Sham theory in Section 2.2. Finally, we discuss the generalisation of the ELF to systems in magnetic fields in Section 2.3.

ELF in Hartree-Fock theory
The definition of the ELF by Becke and Edgecombe [1] focused on Hartree-Fock theory and the same-spin pair density, Expansion of the spherically averaged same-spin conditional pair density to leading order gives where s is the radius of a spherical shell around r. The term in brackets is related to the Fermi-hole curvature derived by Becke [29] and generalised to non-zero current densities by Dobson [30]. Becke and Edgecombe [1] proposed to use this term as a measure of electron localisation, introducing the relative ELF measure Here, D σ (r) is defined in Equation (3), while D 0 σ (r) is the corresponding quantity for a homogeneous electron gas Defined in this manner, the ELF is a dimensionless quantity between 0 and 1. The upper limit f ELF,σ (r) = 1 corresponds to perfect localisation, whereas f ELF,σ (r) = 0.5 indicates behaviour close to that of a uniform gas with the same density.

ELF in Kohn-Sham theory
The ELF as proposed for Hartree-Fock theory relies on the definition of the conditional pair-density, P σ σ cond (r, s), for the interacting system. This quantity is not accessible in the same manner in Kohn-Sham DFT, since the determinantal wave function is then used to describe a system of non-interacting electrons rather than to approximate the physical system. On the other hand, Savin et al. [14] noted that the leading term in Equation (3) is accessible in Kohn-Sham theory, being related in a simple manner to the Pauli kinetic-energy density: where The term τ σ (r) is the σ -spin contribution to the noninteracting kinetic-energy density in the everywherepositive gauge. Integration of this quantity over all space yields the σ -spin component of the non-interacting kinetic energy T s used in Kohn-Sham theory. This form reflects the fact that, although non-interacting, the Kohn-Sham wave function is a single Slater determinant, preserving the fermionic characteristics of the electrons. The second term in Equation (6), τ vW σ (r), is the von Weizsäcker (vW) kinetic-energy density, which can be interpreted as the kinetic-energy density for a system of bosonic particles with orbitals proportional to √ ρ σ . The difference in Equation (6) can then be interpreted as the change in the kinetic-energy density induced by the Pauli principle. Since the von Weizsäcker kinetic-energy density provides a lower bound on the non-interacting kinetic energy density, the Pauli kinetic-energy density is everywhere nonnegative. Given that τ Pauli σ (r) contains the same information as does D σ (r), we may consider whether the ratio D σ (r)/D 0 σ (r) required for f ELF,σ (r) may be derived from it. Savin et al. [14] noted that this can indeed be achieved by using the kinetic-energy density of a uniform electron gas yielding The ELF may then be interpreted as a measure of localisation that reflects changes arising from the fermionic nature of the electrons and the satisfaction of the Pauli principle. This observation goes a long way towards rationalising the practical utility of f ELF,σ (r) for chemical interpretation, bearing in mind that the Pauli principle and its consequences govern many aspects of how chemical phenomena may be interpreted. The Pauli principle leads to Fermi-correlation (exchange) effects between electrons of the same-spin, which are typically an order of magnitude larger than those between opposite-spin electrons (dynamical correlation). As a result, the interpretation of same-spin interactions, as in ELF, can serve as a useful qualitative tool for interpretation. The ELF formula discussed above involves two separate functions, one for each spin. Kohout and Savin [3] proposed instead the spin-polarised form We consider closed-shell systems, for which the values of f ELF (r) from Equations (4) and (11) are identical.

Extension of ELF to magnetic fields
In a magnetic field, f ELF (r) becomes gauge-dependent through its dependence on the kinetic energy density τ , which itself is a gauge variant quantity (see [25,27,[30][31][32][33][34] for further discussion). This unphysical dependence can be removed by introducing terms dependent on the paramagnetic current density, such as those that arise in the discussion of the spherical average of the exchange hole by Dobson [30]. This can be accomplished either in the expansion of Equation (3) or by replacing the first term of Equation (6) according to where j pσ (r) is the paramagnetic current density, This generalisation has previously been used to compute time-dependent ELFs [35,36] and in the generalisation of meta-GGA functionals to calculate response properties perturbatively [27] and magnetic properties nonperturbatively [25]. The latter implementation allows for the self-consistent determination of molecular energies and orbitals in a magnetic field using London atomic orbitals [37][38][39] (also known as gauge-including atomic orbitals (GIAOs)). We note that the gauge independent kinetic energy density is not unique, and several other forms have been proposed [32,34]. However, the choice Equation (12) has the advantage that the kinetic energy density is independent of the external magnetic field, as well as satisfying Equation (8). These issues have previously been examined in the context of meta-GGA functionals in [34]. Here, we use the cTPSS functional to perform calculations at a finite magnetic field, computing the generalised ELF as a function of field strength. Following the notation cTPSS for the TPSS functional with the substitution in (12), we use the acronym cELF for the similarly modified ELF.

Results
All cELF calculations use the London program [40,41] with the CDFT implementation of cTPSS described in [25] via the XCFun library [42] to determine the required quantities for ELF at different field strengths. All calculations have been carried out in an uncontracted Cartesian aug-cc-pCVTZ basis set [43,44] of London atomic orbitals [37]. Unless otherwise noted, zero-field TPSS geometries are used.

Electron confinement
For a magnetic field of strength B along the z-axis, the molecular electronic Hamiltonian takes the form (in atomic units) where H 0 is the unperturbed electronic Hamiltonian, 1 2 BL z is the orbital Zeeman operator expressed in terms of the orbital angular momentum operator L z in the field direction, BS z is the spin Zeeman operator expressed in terms of the spin angular momentum operator S z in the field direction, and 1 While the Zeeman terms may raise or lower the energy, the diamagnetic term always raises the energy, becoming dominant in a field stronger than one atomic unit B 0 235 kT. From the form of the diamagnetic term, we see that the field confines the system in the directions perpendicular to the field vector. As a result, the field exerts a significant influence over the electronic structure and, in particular, the localisation of the electrons.

... Paramagnetic bonding
We first consider the recently discovered phenomenon of perpendicular paramagnetic bonding [28]. In Figure 1, we present iso-volumes for He 2 in a zero field (blue) and in a uniform field of strength B 0 perpendicular to the internuclear axis (red). The bond distance of 2.864a 0 has been optimised at the cTPSS/u-aug-cc-pCVTZ level in the same field. In [28], the paramagnetic bonding in He 2 was rationalised in terms of induced electron rotation. The bonding was further analysed at the Kohn-Sham level in [25], in terms of molecular-orbital energies and density distortions relative to non-bonded atoms in the same field.
The confinement induced by the field, shown in red in Figure 1, is clearly captured by the cELF. In the field, the iso-surfaces contract around the atoms, becoming much more compact in the perpendicular directions than in the zero field (blue). Relaxation to the equilibrium geometry at zero field (not presented) gives similarly diffuse but near-spherical zero-field ELF iso-surfaces.

... Covalent bonding
To illustrate the utility of the cELF in understanding the features of covalent bonding in magnetic fields, we consider methane, ethane, ethene, and ethyne as prototypical systems containing single, double, and triple bonds. In Figure 2, the methane results are shown. In the absence of a field, the blue iso-surfaces are consistent with those obtained in earlier studies; in the presence of field, the iso-surfaces around the atoms contract. The structure of the cELF is similar for ethane (see Figure 3). An additional region of electron localisation is seen at the centre of the C-C bond both at zero field and at B = 0.4B 0 . As the field increases, the localisation becomes less pronounced, eventually splitting into two separate regions.  The zero-field ELF plots for ethene in Figure 4 are consistent with those presented elsewhere [2], with a characteristic dumb-bell shaped structure at the centre of the C=C bond. With increasing field strength, the dumbbell structure splits into two, developing eventually into two new structures with localisation above and below the C=C bond. Around the hydrogen atoms, the cELF surfaces of ethene contract in a manner similar to that observed for methane and ethane.
Finally, in Figure 5, we present plots for ethyne. At zero field, the ELF structure is similar to that reported elsewhere [2], with a characteristic ring structure about the triple bond. The ring is distorted at 0.1B 0 , splits into two at 0.5B 0 , and eventually becomes reminiscent of that for ethene at 1.0B 0 , with localisation above and below the bond.
The features of the cELF observed here can help to rationalise the success of the cTPSS meta-GGA functional in strong fields relative to conventional GGA functionals and also the local-density approximation (LDA) (see [25]). In the cTPSS functional, the same modification of Equation (12) is employed, entering the functional via α(r) = [τ σ (r) − τ vW σ (r)]/τ UEG σ (r) and τ vW σ (r)/τ σ (r). Note that α(r) contains the same information as the cELF, to which it is related as f ELF,σ (r) = 1/(1 + α 2 σ (r)). Hence, α(r) also characterises the nature of different bonding regions, vanishing in one-orbital regions. However, it is not restricted to values between 0 and 1. The ratio τ vW σ (r)/τ σ (r) also serves as a useful indicator of one-orbital regions.
Our visualisations suggest that, as the field increases, regions of large f ELF (r) and low α(r) values distort considerably. Since meta-GGAs are designed to minimise one-electron self-interaction errors in these regions, the good performance of the cTPSS functional in strong magnetic fields observed in [25] may reflect an increased importance of these regions.

Topological analysis
The function f ELF is a continuous scalar field and can therefore be subjected to a topological analysis similar to that used for the electron density by Bader [16]. This approach has been extensively used to highlight the features of the ELF related to the chemical bonding in systems at zero field. In particular, the positions of attractors, which correspond to maxima of f ELF , highlight the regions of high localisation and are known to coincide with traditional chemical notions of bonding [2]. Changes in the number and arrangement of attractors in a magnetic field can therefore be indicative of deeper changes in the electronic structure induced by the field.
The attractors for ethane (determined on a uniform grid with spacing 0.05a 0 ) are shown in Figure 3. At zero field and 0.4B 0 , a single attractor is present at the centre of the C-C bond. As the field increases, two attractors appear along the C-C bond axis, reflecting two separate maxima in the cELF. In Figure 4, the central dumbbell shape of ethene encompasses two attractors. With increasing field strength, new features evolve, with two pairs of attractors appearing above and below the C=C bond.
In Figure 5, the attractors for ethyne are shown, the high symmetry of this system leading to a ring attractor about the bond. A magnetic field perpendicular to the bond axis destroys the linear symmetry, breaking the ring attractor into two point attractors. At 0.5B 0 , two further attractors appear above and below the CࣕC bond axis. Finally, at 1.0B 0 , the cELF resembles that for ethene in the same field, with attractors above and below the CࣕC bond axis.

Physical current densities
The total physical current density j is the physically observable electron current induced by the magnetic field. It is related to the total paramagnetic current density that entersτ =τ α +τ β (Equations (11) and (12)) and hence the cELF by where the second term is the diamagnetic current, dependent on the magnetic vector potential related to the physical field B as B = ∇ × A.
Visualisation of this quantity can aid with the understanding of the magnetic field's influence on a system. Figure 6 shows the physical current vector field superimposed over a cELF contour plot for ethyne at 1.0B 0 in the molecular plane (left) and 1.15a 0 above it (right). The current streamlines follow the structure of the cELF, circulating around the attractors and following contour regions of high localisation. In this sense, the current plots are complementary to the cELF, highlighting localised and bonding regions in a field. However, unlike the cELF, the current density vanishes in field-free DFT and is only of interest in a magnetic field.

Conclusions
We have presented the current-ELF (cELF), a generalisation of the ELF to systems in magnetic fields. The cELF constitutes a useful tool for understanding the nature of changes in chemical bonding upon application of a magnetic field. In systems bound by the perpendicular paramagnetic bonding mechanism [28], such as He 2 , the effects of the field in confining the electronic structure in directions perpendicular to the field are particularly evident. For covalently bound systems, the effects are more subtle but the cELF provides a clear representation of field-induced changes in electronic structure, as illustrated by application to methane, ethane, ethene and ethyne. We expect cELF to become a useful tool for the interpretation of changes in chemical bonding and reactivity in magnetic fields.