Extending conceptual DFT to include external variables: the influence of magnetic fields

An extension of conceptual DFT to include the influence of an external magnetic field is proposed in the context of a program set up to cope with the ever increasing variability of reaction conditions and concomitant reactivity. The two simplest global reactivity descriptors, the electronic chemical potential (μ) and the hardness (η), are considered for the main group atoms H–Kr using current density-functional theory. The magnetic field strength, |B|, is varied between 0.0 and 1.0 B0 = ħe−1a0−2 ≈ 2.3505 × 105 T, encompassing the Coulomb and intermediate regimes. The carbon atom is studied as an exemplar system to gain insight into the behaviour of the neutral, cationic and anionic species under these conditions. Their electronic configurations change with increasing |B|, leading to a piecewise behaviour of the ionization energy (I) and electron affinity (A) values as a function of |B|. This results in complex behaviour of properties such as the electronegativity χ = −1/2(I + A) = −μ and hardness η = 1/2(I − A). This raises an interesting question: to what extent are atomic properties periodic in the presence of a magnetic field? In the Coulomb regime, close to |B| = 0, we find the familiar periodicity of the atomic properties, and make the connections to response functions central to conceptual DFT. However, as the field increases in the intermediate regime configurational changes of the atomic species lead to discontinuous changes in their properties; fundamentally changing their behaviour, which is illustrated by constructing a periodic table of χ and η values at |B| = 0.5 B0. These values tend to increase for groups 1–2 and decrease for groups 16–18, leading to a narrower range overall and suggesting substantial changes in the chemistry of the main group elements. Changes within each group are also examined as a function of |B|. These are more complex to interpret due to the larger number of configurations accessible to heavier elements at high field. This is illustrated for group 17 where Cl and Br have qualitatively different configurations to their lighter cogener at |B| = 0.5 B0. The insight into periodic trends in strong magnetic fields may provide a crucial starting point for predicting chemical reactivity under these exotic conditions.


Introduction
The behavior of atoms and molecules under the inuence of external elds has long been of interest to experimental and theoretical physicists and chemists. In particular, the inuence of oriented external electric elds on structure and reactivity of molecules was recently examined in detail by both theoretical and experimental chemists. 1,2 Evidence was presented, in the pioneering work by Shaik and coworkers, 2 that oriented external electric elds could potentially be used to exert unprecedented control over chemical reactivity, offering a plethora of new synthetic tools for organic, metallo-organic and bioorganic chemists to explore chemical space. 3 In short, "they are expected to be novel effectors of chemical change". 2 The inuence of (strong) magnetic elds on the other hand has received less attention. Theoretical studies on atomic systems have been motivated by the astrophysical discovery of strong magnetic elds on white dwarf and neutron stars, with elds of the order of 10 2 -10 5 T and 10 7 -10 9 T, respectively. 4 The energies of the most important low-lying states of a number of light atoms have been studied as a function of magnetic eld strength. [5][6][7][8][9][10][11][12][13] These studies have focused on determining how the electronic conguration of the ground state changes as the magnetic eld increases, and how the eld distorts the electron density. Calculations on molecular systems are technically more challenging since lower symmetries make it more difficult to apply accurate mesh-based approaches and nite basis set techniques must be adapted to allow for complex orbitals, whilst ensuring that the calculation of energies and physical observables remain independent of the gauge-origin associated with the vector potential describing the magnetic eld. This challenge was addressed in 2008 by Tellgren et al. 14 with the development of non-perturbative calculations using London atomic orbitals 15 for general molecular systems. In 2012, Lange et al. extended this approach to full-conguration interaction theory, 16 revealing a new bonding mechanism ("perpendicular paramagnetic bonding") occurring in magnetic elds of the order of 10 5 T. This leads to exotic new chemistry, for example the bb-component of the triplet state of the hydrogen molecule, unstable to dissociation under normal conditions, not only becomes bound but also becomes the ground state at high eld strengths. 16 Recently, the inuence of an external mechanical force 17,18 has also been considered. The eld of mechanochemistry refers to unusual chemical reactions induced by mechanical energy. It is the molecular analogue of grinding on the macroscopic scale. A prominent example of this type of reaction is the circumvention of the Woodward-Hoffmann rules for the electrocyclic ring opening of cyclobutene. 19 In the context of this ever increasing variability of reaction conditions and concomitant reactivity, theories that aim to provide qualitative or quantitative insight into aspects of reactivity should be broadened to account for the effect of reaction conditions. This inspired some of the present authors to embark on a program to extend conceptual density functional theory (DFT). [20][21][22][23] Central to the original conceptual DFT approach, 20,21,24 developed originally in the 1980s by Parr and coworkers, is the functional E ¼ E(N, v) for a given atom, molecule or solid state system, and its variation under perturbations of the system with respect to its number of electrons N and/or external potential v (i.e. the potential felt by the electrons due to the nuclei). These are precisely the perturbations experienced by a given atom or molecule at the onset of a chemical reaction. The various derivatives of the energy E with respect to N and/or v can be readily identied as response functions, quantifying the response of a system to the respective perturbation at the onset of a chemical reaction, hence their collective name of "reactivity descriptors".  20 and 28) are well documented and their signicance has been highlighted extensively in the literature on fundamental and applied aspects of conceptual DFT. Higher order derivatives have also been addressed. [29][30][31] The number of variables considered in the energy functional has been extended to include, for example, spin 32,33 and temperature. 34,35 External mechanical forces were also introduced recently by some of the present authors, [36][37][38] followed by the inclusion of external electric elds 39 aer pioneering work by Chattaraj and Pal. [40][41][42] These additional variables not only increase the scope of reaction conditions that conceptual DFT can be applied to, but also increase the number of relevant response functions that can be calculated. This signicantly extends the "response function tree" (see for example ref. 22), since the new descriptors are intertwined with the more conventional electric eld related atomic and molecular response functions, such as the (permanent) dipole moment, the polarizability and the rst hyperpolarizability. Variations of the electronic chemical potential, hardness, electron density and Fukui function with external elds in different orientations were recently calculated and analyzed for simple diatomic molecules (the dihalogens F 2 , Cl 2 , Br 2 , I 2 ) and H 2 CO in ref. 39.
A natural extension of this work on electric elds is the inclusion of a magnetic eld and the reactivity descriptors that arise. Some early work in this direction has been done by Chattaraj and coworkers. They concentrated on the secondorder derivative of E with respect to B, the magnetizability, and identied the magnetizability as a measure of soness (the inverse of the hardness), 43 proposing a minimum magnetizability principle 44 in analogy to the well-known maximum hardness 45 and the related minimum polarizability 46 principles.
In this work we will provide a systematic study of the two simplest global (i.e. r-independent) response functions, the electronic chemical potential or (minus) the electronegativity and the hardness, in the presence of an external magnetic eld, for atoms belonging to the main group elements of the rst four rows of the periodic table (H-Kr). In view of the basic formulas discussed in Section 2, when studying the eld dependence of electronegativity and hardness, the computation of atomic ionization energies and electron affinities as a function of the magnetic eld strength is a central requirement. A systematic study of these quantities for different eld strengths over the periodic table has, to the best of our knowledge, never been presented in the literature. Early work on the calculation of atomic energies in the presence of magnetic elds was focused on hydrogen and helium (for reviews, see ref. 5 and 11). Much less work has been undertaken to study atoms with more than two electrons in magnetic elds. An important early contribution to this eld was made in 1996 by Jones, Ortiz and Ceperley, 6 who published a study at the Hartree-Fock level for a series of light atoms/ions encompassing H, H À , He, Li and C, pinpointing the dependence of the ground state conguration on the eld strength. Similar studies, with increasing levels of theory were published in later years for Li, Be, and B by Ivanov, Schmelcher and coworkers. [7][8][9][10]12,13 In some of these studies, both the neutral atom and its cation (with possibly different evolution of their electronic congurations with eld strength) were examined, allowing the study of the ionization energy as a function of magnetic eld strength for Li, 8,12 Be, 10,13 and B. 9 A particularly interesting study in this series from a chemist's point-of-view is that by Ivanov and Schmelcher, 47 who published a very detailed Hartree-Fock level investigation of the carbon atom, in magnetic elds ranging from 0 to 2.23 Â 10 9 T, clearly revealing the appearance of chemically counter-intuitive ground state congurations that gradually maximise the number of b electrons and the total angular momentum, which becomes increasingly favourable with increasing magnetic eld strength due to the spin Zeeman interaction, up to fully spin-polarized congurations in the very high eld regime. These studies highlight how the ground state electronic conguration is very sensitive to the eld strength. As a result, different chemical behavior would be expected in various eld strength domains driven by these conguration changes.
In this work, we investigate the extent to which periodicity of chemical properties is preserved in the presence of strong magnetic elds. First, we evaluate c and h over a wide range of eld strengths varying from 0 to 1 B 0 (B 0 ¼ ħe À1 a 0 À2 z 2.3505 Â In Section 2 we give the essentials of current-densityfunctional theory and conceptual DFT relevant to the present work and in Section 3 the computational details for the calculations are given. In Section 4, we commence with a case study on the carbon atom in Section 4.1 to highlight how congurational changes as a function of jBj give rise to extra complexity in studying the electronegativity and hardness as a function of the magnetic eld strength. In Section 4.2, a periodic table of electronegativity and hardness is constructed and discussed for one jBj value (0.5 B 0 ), the central point of the range of magnetic eld strengths 0-1 B 0 studied in this work. Finally in Section 4.3, the response functions vm vB and vh vB at jBj ¼ 0 together with their periodicity are addressed.

Current-density-functional theory
To study the descriptors m and h in the presence of magnetic elds in the range 0-1 B 0 and their response to the eld at jBj ¼ 0, we require the ground state energies and electronic congurations of the neutral, cation and anion species in the presence of magnetic eld. The non-relativistic electronic Hamiltonian for an N-electron system in a magnetic eld B can be written as, where the rst term is the unperturbed zero-eld electronic Hamiltonian. The linear Zeeman terms are associated with the spin (ŝ i ) and orbital angular momentum (l i ¼ Àir i Â V i ) operators, describing the interaction of the electron i with the magnetic eld B. These terms split the energy levels and may raise or lower the energy relative to that in the absence of a eld. The remaining term, which is quadratic in B, is purely diamagnetic and raises the energy of the system relative to zero-eld. At sufficiently high eld strengths, the diamagnetic term will always dominate since it is quadratic in eld strength. For atomic systems with the eld oriented along the z-axis, the Hamiltonian of eqn (1) can be written as (see for example ref. 13), The range of eld strengths from jBj ¼ 0.0 to 1.0 B 0 spans both the Coulomb and intermediate regimes. At low eld (jBj ( 0.1 B 0 ), the Coulomb interactions present inĤ 0 are much more signicant than the magnetic interactions, which may be treated perturbatively, whereas at higher elds in the intermediate regime, both interactions are of comparable strengthpreventing the treatment of either by perturbative approaches. For very strong elds, typically much higher than 1.0 B 0 for the atomic systems in this work, the Landau regime is entered. This regime is not considered in the present work however an interesting study of many-electron systems in this context may be found in the work by Wunner et al. on the series He to Si for elds extending to 5 Â 10 8 T. 48 In the present work, we use a non-perturbative implementation of current-density-functional theory, suitable for systems in external magnetic elds of strength in the range 0.0-1.0 B 0 considered in this study. Implementations of nonpertubative calculations have been developed for general atomic and molecular systems at the Hartree-Fock, 14,49 conguration interaction, 16 complete active space self-consistent eld, Møller-Plesset, coupled-cluster and current-density-functional theory (CDFT) 50 levels in recent years. In the presence of a magnetic eld, density-functional theory (DFT) must be extended as shown by Vignale and Rasolt 51 since the energy is no longer dependent only on the charge density r but also on the paramagnetic current density j p . As shown in ref. 52 and 53, a formulation of CDFT analogous to Lieb's convex-conjugate formulation of DFT can be constructed by re-writing the energy functional E(v, A) depending on the external scalar potential v and the vector potential A associated with the magnetic eld in terms of the modied scalar potential This leads to the concave energy functional where (ujr) ¼ Ð u(r)r(r)dr, (Ajj p ) ¼ Ð A(r)$j p (r)dr and the convexconjugate universal density functional is which can be identied as the Vignale-Rasolt functional. 51 Adopting the Kohn-Sham (KS) ansatz, the functional in eqn (4) can be decomposed as in which the rst term is the non-interacting kinetic energy, J(r) the classical Coulomb electron-electron repulsion and E xc (r, j p ) the exchange-correlation energy, which now depends on both r and j p . The KS-CDFT equations take the form wherep ¼ ÀiV is the canonical momentum operator,ŝ is the spin operator, 3 p are the orbital energies and 4 p are the molecular orbitals. The charge and paramagnetic current densities can be expressed in terms of the molecular orbitals as and respectively, where i denotes occupied orbitals and s their spin.
The KS scalar and vector potentials are where (v ext , A ext ) are the physical external potentials, v J is the Coulomb potential, and the exchange-correlation potentials have scalar and vector components given respectively by In the present work the KS-CDFT equations are implemented in an unrestricted manner.
To ensure gauge-origin independence of the calculated energies, the molecular orbitals 4 p are expanded in a set of London atomic orbitals (LAOs). 15 These have the form u a ðrÞ ¼ f a ðrÞe ÀıA ðRaÞ$r (11) where A ðR a Þ ¼ 1 2 B Â ðR a À R o Þ is the vector potential at the position of the center of the LAO R a , relative to the gauge-origin R o . This denes a complex phase factor, which multiplies a standard Gaussian basis function f a (r). We note that in the present work on atoms the gauge-origin may always be chosen to coincide with the atomic centre, reducing the LAOs to a set of complex Gaussians. However, the framework presented here is general and the quantities analysed may be calculated for atoms and molecules alike.
The remaining challenge is then to select an appropriate form for the exchange-correlation energy in practical calculations. We have previously shown that the meta-GGA level cTPSS functional 54 provides good accuracy in the presence of strong magnetic elds. 55 At the meta-GGA level the modied kinetic energy densitỹ is used in place of the usual form s s (r) to ensure that E xc (r, j p ) is independently gauge invariant, as suggested by Dobson 56 and employed previously by Becke. 57 Utilising this approach, a family of cTPSS functionals have been applied in strong magnetic elds 58-60 including hybrid and range-separated hybrid variants. In the present work we will make use of the cTPSS meta-GGA and compare our results with those obtained at the Hartree-Fock level.

Conceptual density-functional-theory
The conceptual DFT descriptors m, c and h at a given eld strength, m(B), c(B) and h(B) were calculated using the wellknown nite difference approach 20,22 as where I(B) and A(B) are the ionization energy and electron affinity at a given eld strength jBj. It is well known that these quantities correspond to the le and right side derivatives of the energy with respect to the particle number N, as described by Perdew et al. 61 The expressions for m and h correspond to the averaged sum and difference of these derivatives, respectively. The factor of 1/2 present in eqn (14) is consistent with the original denition of Parr and Pearson 26 however is oen omitted from the denition of the hardness in more recent work; here we keep this factor for reasons of symmetry between the formulae for the electronegativity and hardness. As outlined previously, the central theme of this study is the extension of the E ¼ E(N, v) functional with an external magnetic eld, leading among others to reactivity descriptors of the type vm vB . We also note that, upon including spin in the E functional, a spin polarized version of conceptual DFT with functional E ¼ E(N, N S , v, B) was put forward 32,33 with N S being the difference between the number of a and b electrons. It is tempting to see if the reactivity descriptors as evaluated in our study, e.g. vm vB , could be considered in the context of this functional. It should however be noted here that upon adding or subtracting a complete electron, either a or b, as is the case when evaluating I or A, and seeking the lowest energy for the given value of M s , N S is not constant: this could only occur by adding or subtracting 0.5 a and 0.5 b electrons, which would not lead to the ground state conguration. In summary, the condition of constant N S in the partial derivatives with respect to B, in the context of the E ¼ E(N, N S , v, B) functional, is not compatible with obtaining the ground state energy.

Computational details
For the Hartree-Fock calculations in Subsection 4.1, which allow for comparison with literature data, a q-aug-cc-pVQZ basis was used. [62][63][64][65][66][67] All CDFT calculations were carried out using the cTPSS exchange-correlation functional, 54,68 a modied version of the TPSS functional 69 belonging to the meta GGA class of functionals, the prex c indicating that a modied version of the functional was used including the current density (through the modied form of the kinetic energy density given in eqn (12)), suitable for calculations in the presence of strong magnetic elds. 55 This functional has been shown to perform well relative to high-level ab initio calculations in strong magnetic elds in ref. 55. Preliminary studies for a subset of atomic systems also indicated that the use of a hybrid or rangeseparated hybrid variants of this functional (see ref. 58) does not signicantly alter the results in the present work. For these calculations, the d-aug-cc-pV5Z Gaussian basis set was employed. [62][63][64][65][66][67] As the aim of our study was to cover the main group elements up to and including the fourth row elements, K and Ca had to be excluded as this basis was not available for these elements.The magnetic eld was oriented along the z-axis in all calculations.  (13) and (14) respectively. This approach was not always straightforward since, for some species, a discontinuity appears in the E vs. jBj curve approaching zero eld, with the energy at jBj ¼ 0 not being that of the ground state. This is a manifestation of the multiplet problem associated with single-determinant methods such as HF and KS DFT, in which congurations which should be degenerate such as those in the 3 P zero-eld ground state of the carbon atom do not have the same energy with such methods. 56,70 However, in the presence of a magnetic eld this degeneracy is lied and also the different components of the multiplet exhibit a different variation in energy with respect to eld strength. Therefore, to ensure the E vs. jBj curve is continuous approaching zero eld, calculations were rst undertaken at a higher eld strength (typically 0.1 B 0 ) and the converged orbitals used as the initial guess for the lower and zero-eld calculations. In this way, a selected electronic conguration could be traced back to zero eld. In each case we selected the component of the multiplet that becomes the ground state for very low (but non-zero) magnetic elds as the conguration to study. For the calculations at higher eld strengths, the region jBj in the range 0.1 to 1 B 0 was scanned with intervals of 0.1 B 0 . For each jBj value, a range of M s values of the neutral atom was considered. For each M s value, the lowest energy and associated conguration was selected (since sometimes more than one conguration corresponds to a given M s ). Comparison of the lowest energy values for each M s then yields the ground state energy and corresponding conguration at a given jBj value. A similar procedure was followed for the cation and the anion, allowing the calculation of I(B) and A(B) and, from these, m(B) and h(B) (eqn (13) and (14)). This procedure is exemplied in Subsection 4.1 for the case of the carbon atom. Since our calculations only specify M s it is possible that the self-consistent eld calculations could converge to a solution that is not the ground state. To mitigate against this, continuity of the solutions as a function of jBj was carefully examined, both in terms of the energy and the nature of the orbitals involved. In practice, it was found that such issues were only problematic in a few cases where congurations were nearly degenerate close to jBj ¼ 0 and lower energy solutions could be readily obtained by using orbitals from higher elds and an initial guess.

Results and discussion
4.1 Electronegativity and hardness of the C atom As described previously, to compute the electronegativity and hardness values at a given jBj, according to eqn (13) and (14) respectively, the basic ingredients are the ionization energy and the electron affinity, which should be calculated at that value of jBj. As the electronic congurations for the neutral system, anion and cation may (and in most cases will) differ from their zero-eld counterparts, the search for an optimal conguration at a given jBj for the neutral, the cationic and anionic system should be conducted at that jBj value.
In order to illustrate this complexity and as a proof of concept, this analysis is undertaken for the carbon atom, for which the energy and associated congurations were scanned from jBj ¼ 0 to 1.0 B 0 as described in Section 3 for the neutral system, the cation and the anion. To the best of our knowledge this is the most extensive exercise of this type since the pioneering studies by Schmelcher on the neutral system, 47 permitting a direct comparison for the neutral C atom, and the more intricate combined studies on the neutral atom/cation combinations for Li, Be, B by the same group. 8-10,12 Table 1 shows the evolution of the ground state energy and conguration of the carbon atom, its cation and anion where, for each system, a change in conguration is indicated by a change in text color.
For the neutral C atom at zero-eld, the lowest energy component of the 3 P ground-state is that with the 2p 0 and 2p À1 orbitals singly-occupied by electrons of the same spin, here taken to be b. This conguration remains the ground-state in a very small eld, since it maximises the number of b electrons and minimises the M L value thus has the greatest decrease in energy with increasing eld strength. At jBj ¼ 0.2 B 0 the two 2s electrons decouple, giving rise to two extra unpaired b electrons with concomitant stabilization due to the spin-Zeeman effect, despite an electron now occupying the higher energy 2p +1 orbital (blue region). The M s value thereby jumps from À1 to À2. At higher eld, the conguration with an occupied 3d À2 orbital eventually becomes the ground state, since the energy of the 3d À2 orbital decreases with eld strength due to the orbital paramagnetic effect and eventually falls below that with the 2p +1 orbital, resulting in the ground-state conguration changing from .2p À1 2p 0 2p +1 to a 2p À1 2p 0 3d À2 (green region) but the value of M s remaining À2. This behavior is in line with that reported by Schmelcher. 47 Values are not identical due to differences in computational approach: whereas a numerical HF approach was followed by Schmelcher, the present calculations were carried out using a nite basis set expansion. The most important difference is that, whereas the M s ¼ À1 to À2 transition occurs at a similar eld strength (0.19 B 0 vs. 0.20 B 0 ), the change between the two M s ¼ À2 congurations occurs at a lower eld strength in Schmelcher's study compared to this work (0.49 B 0 vs. 0.60 B 0 ). It should be noted that a smaller size of interval in jBj could be used to rene the eld strength at which these transitions occur in the present work. Important is that the energy differences are on average 0.0006 E h for eld strengths below jBj ¼ 0.05 B 0 and 0.006 E h for eld strengths between 0.5 and 1.0 B 0 .
Overall, the comparison with Schmelcher's work suggests that the computational methods employed here are reliable and can be applied to analyse the cationic and anionic states of carbon. At low magnetic eld strengths, the 1s 2 2s 2 2p À1 conguration of the cation is lowest in energy, having values of M L ¼ À1 and M s ¼ À 1 2 , indicated by purple text in Table 1. At a magnetic eld of jBj ¼ 0.10 B 0 , 1s 2 2s2p À1 2p 0 becomes the ground-state electronic conguration (red text in Table 1) as a result of the unpairing of the 2s electrons, with a resulting change in M s to À 3 2 .
As for the cation, only two congurations are observed for the anion over this range of eld strengths: the 1s 2 2s 2 2p À1 2p 0 2p +1 conguration from jBj ¼ 0.0 to 0.1 B 0 with M s ¼ À 3 2 and, by the unpairing the 2s electrons at jBj > 0.1 B 0 , the 1s 2 2s2p À1 2p 0 2p +1 3d À2 with M s ¼ À 5 2 . It is interesting to note that the eld strengths at which the ground state congurations of the carbon anion, cation and atom change do not generally coincide, indicating that care should be taken when interpreting variations in the ionisation energy and election affinity as a function of the magnetic eld strength. Finally, it can be seen that from jBj ¼ 0.6 to 1.0 B 0 , the energy of the anion is higher than that of the atom, resulting in a negative electron affinity; for a more detailed discussion see Subsections 4.2 and 4.3. The behaviour of I and A as a function of magnetic eld strength are summarised in Table 2. Different colours are again used to indicate the regions in jBj where either of the two species involved in the calculation of these quantities changes its ground-state electronic conguration and through which I and/or A are expected to vary smoothly with jBj. As a result of this analysis, the ionisation energy can be split into four 'segments': from jBj ¼ 0.0-0.05 B 0 , 0.1 B 0 , 0.2-0.5 B 0 and 0.6-1.0 B 0 , indicated in Table 2. Though in each segment the electron is removed from a different orbital in the atom to form the cation (2p 0 , 2s, 2p +1 , 3d À2 respectively), remarkably the same cation is formed in segments 2, 3 and 4, but each time from a different neutral system conguration.
A similar observation can be made for the electron affinity, for which three regions are discerned where the attached electron occupies the 2p +1 , 3d À2 and then 2p +1 orbitals, but leading in the last two cases to the same conguration; in the rst case from jBj ¼ 0.2-0.5 B 0 the LUMO of the atom is the 3d À2 orbital which becomes occupied in the formation of the anion, whilst in the second case jBj > 0.5 the 3d À2 orbital is the HOMO of the atom and its LUMO is the 2p +1 orbital which becomes occupied on formation of the anion. Again, considering the electron affinity as a function of magnetic eld strength, discontinuities between the three or four segments can be discerned, which can be seen more clearly in Fig. 1 where we depict plots of I and A as a function of the eld strength. Fig. 1 clearly shows how different the behavior of both I and A can be in different regions of jBj: differences in slope (both in Table 1 Evolution of the ground state energies of C, C + and C À and their respective electronic configurations as a function of the magnetic field strength between 0 and 1.0 B 0 . Changes in configuration are indicated by changes in color. Singly occupied orbitals are always carrying a b electron (arbitrarily at jBj ¼ 0). All values are calculated at the HF level with the q-aug-cc-pVQZ basis magnitude but even in sign) show up, making a discussion on the behavior for the two quantities from jBj ¼ 0.0 to 1.0 B 0 perhaps unexpectedly complicated from a more chemical point of view. Their non-uniform behavior indeed hampers typical chemical intuition based thinking/reasoning: when a system is perturbed, a given quantity is either unchanged or it increases or decreases with, in the last two cases, a greater effect when the perturbation is larger. The value of response functions at jBj ¼ 0.0 thereby clearly emerges. This type of behavior was also found by Schmelcher and coworkers for the ionization energy of Li, Be and B, where for Li and Be the same piecewise behaviour of the I(B) curve was found in the range from jBj ¼ 0 to the fully decoupled states at very large eld strength 8,10 and where also for B a difference in the piecewise behaviour of the E vs. jBj curve between the neutral system and the cation was highlighted in that same range. 9 The analogous behavior of the electron affinity is not unexpected in view of the fundamental role which is played by changes in congurations of any atomic system as a function of an external magnetic eld.
Combining ionization energy and electron affinity into Mulliken's electronegativity in eqn (13) and Pearson's hardness expression in eqn (14) leads to further complications since then two quantities with their own, oen different "segments" in their jBj variation are to be combined, possibly leading to a further segmentation in the jBj variation of the electronegativity and/or hardness. In the case of carbon, by chance, there is no such further complication since the segmentation pattern of I and A in jBj are identical, leading again to four segments for both the electronegativity and the hardness ( Table 2 and Fig. 2).
Concentrating initially on the electronegativity, a rst regime is observed in segment one where c increases as a function of jBj quasi linearly (from an initial value of 0.207 to 0.230 E h ) with a slope of 0.46. The second regime is the single point at a higher value (0.242 E h ), followed by a third regime with an almost linearly decreasing electronegativity between jBj ¼ 0.2 and 0.5 B 0 with a slope of À0.10. The nal regime, from jBj ¼ 0.6 to 1.0 B 0 , also shows a decreasing electronegativity, again nearly linear, but with an even more pronounced decrease in c (slope value À0.16). For the hardness a similar structure in the plot may be expected, and Fig. 2 indeed shows again four regions corresponding with the segments in Table 2 with different slopes for the linear variation of h with respect to jBj in segments one, three and four. Table 2 Evolution of ionization energy, electron affinity, electronegativity and hardness as a function of field strength jBj. Different segments (see text) are indicated by different colors. Orbitals from which an electron is taken (I) or to which an electron is added (A) are indicated  Table 2.
It can be seen from Fig. 2 that the behaviour of electronegativity and hardness as a function of jBj for different atoms is extremely subtle such that predictions are more difficult to make: from the knowledge of the behavior of an atom of a given element, it is far from clear what the behavior of an atom from a different element in a magnetic eld will be. This issue will nevertheless be addressed in Subsection 4.2 where we try to discern patterns throughout the periodic table at a given jBj and in Subsection 4.3 where response functions at zero eld are discussed.
As stated previously, extensive calculations (encompassing all main group elements from H to Ar, except for K and Ca) were carried out with CDFT using the cTPSS exchange-correlation functional on account of its higher accuracy compared to Hartree-Fock (HF), although the results are qualitatively similar when considering the transitions in M s . In the case of the carbon atom, the transition from M s ¼ À1 to À2 occurs at 0.3 B 0 with cTPSS compared to 0.2 B 0 with HF, whilst for its cation the transition from M s ¼ À In these CDFT studies, as mentioned in Subsection 4.1, special care was taken in the case of negative electron affinity values for all elements and all eld strengths. In case of an unstable anion, the corresponding electron affinity was set to zero, as is oen done in conceptual DFT approaches. Please note that this strategy is different from that in the previous part of the paper where negative electron affinities were kept as such.  Table 2.  The data in the ESI † also adopts this strategy. Both strategies have been advocated in the literature. 71 Fig. 3 retrieves for jBj ¼ 0 the well known trends of increasing electronegativity from le to right (neglecting the noble gases for the moment) and from bottom to top in the periodic table. The lowest value is found at the bottom le (Na) and the highest value at the top right (F), as expected and in line with literature data. 20 The high electronegativity value for the noble gases, not reported in all tables and for example absent in the Pauling scale, arises due to their extremely high ionization energies (see the discussion in ref. 72 by some of the present authors) and negative electron affinities, which are taken to be zero as described above.
Generally positive values of A result in an increased value c, but since the ratio of electron affinity (when positive) and ionization energy is typically only of the order of 1/8, the nal result is that the very high ionization energies of the noble gases is sufficient to yield electronegativity values of the order of their halogen congeners, even resulting in He having the highest electronegativity. In the more detailed study in ref. 72, it was concluded that the combination of their high electronegativity with their extreme hardness determines the chemistry of the noble gases.
Considering now the case at jBj ¼ 0.5 B 0 , it rst should be noted that all anions at this jBj value were found to be stable since the addition of a b electron further stabilises the system due to the spin-Zeeman effect. This fundamental difference with the jBj ¼ 0 case has only three exceptions: He with a negative electron affinity of À0.009 E h and, remarkably O and F with electron affinities of À0.0007 E h and À0.0175 E h respectively. For He and O, the electron affinities become positive for jBj ¼ 0.5 B 0 to 1.0 B 0 whereas for F the electron affinity is positive for jBj ¼ 0.5 B 0 to 0.7 B 0 , turning negative for jBj ¼ 0.8 B 0 to 1.0 B 0 . At higher elds intricacies with the congurations of neutral and anionic system result in an unexpected behavior. Again, in cases were A was negative c was evaluated as I/2.
When considering the overall trends in electronegativity, it can be seen that the pattern at jBj ¼ 0.5 B 0 is signicantly different to that at zero eld; the rst four columns except for carbon strongly increase their electronegativity, the elements of columns 5 and 6 show a slight decrease, whereas the halogens and noble gases exhibit large decreases. This will be compared to the behaviour of the initial response of the electronegativity at zero eld in Subsection 4.3.
Overall the impression is that the electronegativity values show a tendency to be compressed in a smaller range in a strong magnetic eld. The result is that, in these conditions, the le hand side of the periodic table generally shows higher electronegativity values than the right hand side, leading to fundamental changes in chemistry (e.g. the polarity of bonds).
Whilst it can be difficult to nd trends in behaviour across a period due to the many discontinuities arising from changes in ground-state conguration of the elements at different eld strengths, it can be interesting to compare the behaviour of elements in the same group with respect to magnetic eld strength since the pattern of changes in ground-state conguration may be similar. This analysis is presented in Fig. 5 for certain columns of the periodic table. In Fig. 5(a) the variation in electronegativity with magnetic eld for H and Li are shown; they both have a roughly linear variation with eld strength, most likely because both atoms have M s ¼ À 1 2 and cations have M s ¼ 0 from jBj ¼ 0.0 to 1.0 B 0 , whilst both anions have M s ¼ À1 from at least jBj ¼ 0.1 B 0 to 1.0 B 0 .
The situation is different for the halogens, where the wellknown zero-eld situation of c F > c Cl > c Br evolves in a different way between F and its two heavier congeners, as can be seen in Fig. 5(b). Concentrating, for the sake of simplicity on the conguration of the neutral atoms, F only undergoes one transition from M s ¼ À 1 2 to À 3 2 at jBj ¼ 0.6 B 0 whereas Cl and Br undergo three changes from M s ¼ À 1 2 to À 3 2 then to À 5 2 and À 7 2 at jBj ¼ 0.3 B 0 , 0.5 B 0 and 0.7 B 0 respectively. This is reected in   5(b), where it can be seen that the slope of electronegativity with respect to magnetic eld changes sign only once for F whereas it changes at least three times for Cl and Br. This difference leads to the remarkable effect that in the jBj ¼ 0.4-0.6 B 0 region c F is still decreasing whereas c Cl and c Br increase and lie well above c F . At jBj > 0.6 B 0 , c F steadily increases whereas Cl and Br show an oscillatory behavior due to changes in the ground-state conguration, resulting in a F, Cl, Br sequence which changes several times. Fig. 5(c) shows the variation of electronegativity with eld strength for the halogens from jBj ¼ 0.0-0.1 B 0 ; in this region, the three curves behave in a similar way and no changes in M s occur in that region, indicating furthermore that taking response functions at jBj ¼ 0 would be useful to investigate.
The hardness values in Fig. 4 at zero eld exhibit the wellknown behavior of alternating values along a period 10,20,73 with maxima for the second, h and eighth groups with unoccupied, half-lled and fully occupied 2p shells respectively. 72 In each column a uniform decrease is observed, corresponding with increasing soness/polarizability. At jBj ¼ 0.5 B 0 nearly all atoms up to the h group show an increase in hardness relative to zero eld, but at this and subsequent groups the hardness is decreased relative to zero eld (with some minor exceptions), being most pronounced for uorine and especially the noble gases. The result is that values at the le and at the right of the periodic table become closer to each other and, just as in the case of the electronegativity, an overall compression of the hardness range is observed. Note however that, with exception of Si and Ne, in a given column the tendency of decreasing hardness moving down in the periodic table is preserved.
Combining the observations from the changes in electronegativity and hardness, relevant changes in the chemistry of main group elements compared to zero eld could be predicted, for example using Huheey's expressions for bond polarity and bond energy based on the electronegativity equalization principle. 22,74 Both quantities indeed reduce to a difference in electronegativity (squared in the case of bond energy) modulated by a sum of hardness values.

Response functions at jBj ¼ 0
One way in which the magnetic eld-dependence of these conceptual DFT quantities can be examined without the complications of the changes in ground-state conguration that occur as the magnetic eld strength increases is by considering the derivative of these quantities with respect to eld strength, evaluated at zero eld. These response quantities should lend themselves best to an overall comparison between the behavior of the main group elements. In Fig. 6 we depict the initial response of the ionization energy, with a periodic table representation showing both the numerical values and categorising them with a color code in which blue indicates a positive derivative, red a negative derivative and yellow a derivative that is zero or close to zero.
From these response properties, a periodicity can be discerned which is different to that for the zero eld electronegativity. In this case, the derivative is positive for the rst column of the s-block elements but negative for the second column. A similar pattern emerges for the p-block elements: starting from positive values in columns 3 and 4, they pass to nearly zero in columns 5 and 6 and end with highly negative values for the halogens and the noble gases. Within a given column where no changes in ground-state conguration are expected at the very low eld values used to obtain the derivative, neither for the neutral system nor for the cation, the values turn out to be of the same order of magnitude, be it with a few exceptions, without displaying a particular pattern.
For the electron affinity, a similar presentation is given in Fig. 7 but in which cases with negative A values at zero eld were eliminated. Again, a periodicity can be observed in which the derivatives within a given column have the same sign and are of a similar order of magnitude; the derivative is positive in the rst column of the p-block but decreases to near zero in the second and third columns, becoming increasingly negative in the h and sixth columns of the p-block. In the halogen group, bromine substantially deviates from chlorine and uorine; this apparent anomalous behavior was investigated in more detail and can be ascribed to the fact that the 5p orbitals are sufficiently low in energy that even at very low elds of the order of 0.0001 B 0 a change in ground-state conguration occurs with the 5p À1 orbital becoming occupied. Fig. 6 Initial responses of the ionization energy in a magnetic field as a periodic table representation showing both the numerical values and categorising them with a color code in which blue indicates a positive derivative, red a negative derivative and yellow a derivative that is zero or close to zero. Fig. 7 Initial responses of the electron affinity in a magnetic field as a periodic table representation showing both the numerical values and categorising them with a color code in which blue indicates a positive derivative, red a negative derivative, yellow a derivative that is zero or close to zero and grey those cases where electron affinity was negative and its derivative not computed (see text).
When evaluating the response of the electronegativity to the magnetic eld at zero eld, these derivatives were calculated as the average of the derivatives of ionization energy and electron affinity, where in case of negative A values (indicated in Fig. 7) only the ionization energy contribution was taken into account.
The overall picture in Fig. 8 looks similar to that for the ionization energy in Fig. 6, the main differences being that the derivatives for the rst column are slightly negative here due to the strongly negative electron affinity derivatives (the decrease in c at very low jBj for H and Li, not visible in Fig. 5, can be seen in Tables I and III respectively of the ESI †), whilst the derivatives in the sixth column are signicantly negative rather than near zero as in the case for the ionisation energy derivatives, again due to strongly negative electron affinity derivatives. For the pblock, the overall picture is similar to that for the response of I and A, with large positive values in the rst column decreasing to near zero in the third column and becoming large and negative in the h column. In summary, the electronegativities of the elements at either end of the p-block are most sensitive to perturbation by an external magnetic eld, leading to a large increase in electronegativity for elements on the le side of this block and a large decrease in electronegativity for elements on the right side. Since there is much less variation between values within a column, the average value of the response of the electronegativity to the magnetic eld is evaluated for each column and presented in Fig. 9.
Although this analysis is not directly analogous to the comparison of c at jBj ¼ 0 and 0.5 B 0 seen in Fig. 3 (due to the changes in ground-state conguration that occur over this range of eld strengths), some similarity in trends can be observed for the p-block elements. The increase in c in Fig. 3 for the third and fourth groups aligns with the positive derivative for c with respect to eld strength seen in Fig. 8, little change in c for the h group in Fig. 3 corresponds to a near zero derivative and a decrease in c for the last three groups in Fig. 3 corresponding to a negative derivative of c with respect to eld strength seen in Fig. 8.
Considering nally the response of hardness to an external magnetic eld, it would not necessarily be expected to show the same pattern as that for the electronegativity. This is because, whilst values of A are generally much smaller than values of I for a given element resulting in a relatively small difference between c and h, this is not necessarily the case for the derivatives of I and A with respect to the magnetic eld. The magnitude of the response of I and A with respect to the eld can be of the same order of magnitude; as such, the response of c (calculated from the sum of the responses of I and A) can be very different to that of h (calculated from the difference of the responses of I and A). Fig. 10 shows that, for elements with a fully occupied valence sub-shell the initial response of the hardness with respect to the eld is negative, for elements with a half-lled valence sub-shell the initial response of the hardness is near-zero and for the other elements, with partially occupied valence sub-shells, the initial response of the hardness is positive (with the exception of Br). Elements in the same column thus show a similar behaviour, although the magnitude of the response does not always follow a clear trend within a group. As for the electronegativity, there is not a direct correspondence between this analysis and the comparison of h at jBj ¼ 0 and 0.5 B 0 seen in Fig. 4, however for the p-block the sign of the change in value of h between jBj ¼ 0 and 0.5 B 0 generally aligns with that of the derivative of h at zero eld.
Several general observations can be made from this analysis: the rst is the similarity of the behaviour of atoms in a given group, secondly is that atoms belonging to a group with the highest value of the hardness in a period at zero eld have a tendency to lower their hardness or at least keep it unchanged, whereas all other main group elements increase in hardness in Fig. 8 Initial responses of the electronegativity in a magnetic field as a periodic table representation showing both the numerical values and categorising them with a color code in which blue indicates a positive derivative, red a negative derivative and yellow a derivative that is zero or close to zero. Fig. 9 Evolution of the average electronegativity response at jBj ¼ 0 for a given column in the periodic table for the main group elements. Fig. 10 Initial responses of the hardness in a magnetic field as a periodic table representation showing both the numerical values and categorising them with a color code in which blue indicates a positive derivative, red a negative derivative and yellow a derivative that is zero or close to zero. a magnetic eld. The same pattern of behaviour and similarity within a group is observed for the electronegativity. In a broader context, at least for the p-block elements, the trend of increasing c from bottom le to top right becomes attenuated by a decrease in c from le to right on application of an external magnetic eld, whereas for the s-block elements the situation is less clear-cut.
From the present study on atoms, some insight may already be gained into the changes in chemistry that molecules may undergo in the presence of magnetic elds. The compression of the electronegativity and hardness scales that is observed at jBj ¼ 0.5 B 0 and the signs of their derivatives with respect to eld strength at jBj ¼ 0 indicate that the polarity of the bonds may be signicantly affected by the magnetic eld, to the point of reversing altogether, as may be deduced from the Huheey equation for charge transfer between two species. 74 The effects predicted in this way may be compared with the effect of the magnetic eld on the electron density and dipole moment of diatomic molecules.

Conclusions
The extension of the energy functional E ¼ E(N, v) in conceptual DFT by inclusion of an external magnetic eld has been investigated by studying the two most important global properties in conceptual DFTthe electronic chemical potential (electronegativity) and the hardness of atoms belonging to the main group elements up to Kr. Compared to previous work on the inclusion of electric elds and given the range of magnetic eld strength considered (0.0-1.0 B 0 ) the evolution of these quantities, and the quantities on which they depend in their Mulliken and Pearson denitions respectively, the ionisation energy and electron affinity, transpires to be much more complex. The reason for this is that the ground-state conguration of the atom, its anion and cation, changes as the magnetic eld strength changes leading to discontinuities in their energies as functions of eld strength. Since the changes in ground-state conguration of the three species for each element can occur at different eld strengths, it can create a complicated piecewise structure for the electronegativity or hardness as a function of eld strength. This is demonstrated for carbon at the Hartree-Fock level.
To compare with trends across the periodic table known at zero eld for the hardness and electronegativity, a periodic table of c and h values for the main group elements evaluated using current density functional theory at jBj ¼ 0.5 B 0 is presented. Both for the electronegativity and hardness, an overall increase in values is observed on the le side of the periodic table, whereas a decrease is seen on the right side, with a similar behavior of elements within the same column. The overall picture is a compression of the electronegativity and hardness range across each period, which would lead to important changes to be expected for bond polarity following Huheey's electronegativity equalization based approach. 74 The derivatives of the electronegativity and hardness with respect to magnetic eld strength at zero eld, response functions in the conventional sense for conceptual DFT, present a simpler picture of the behaviour in a magnetic eld since the changes in ground-state conguration do not need to be considered given the denition of the properties at zero eld, thus more easily allowing a comparison across the periodic table. The behaviour of atoms within a group is seen to be similar both for c and h, having derivatives with the same sign and order of magnitude for the electronegativity. For the pblock, the general picture is that the increase in c from the bottom le to top right of the periodic table known at zero eld becomes attenuated by the tendency of c to decrease by an increasing magnitude from le to right across the periodic table in a magnetic eld. For the hardness, the atoms in the group with the highest h in a period at zero eld have a decreasing or unchanged hardness in the presence of a magnetic eld, whilst all other elements have an increasing hardness in a magnetic eld.
The present work focuses on chemical properties in static magnetic elds however further studies of the polarization of the density under non-uniform and oscillating magnetic elds (see, for example, ref. 75) may yield additional insight into this behavior.

Data availability
The datasets supporting this article have been uploaded as part of the ESI. †

Author contributions
R. F., T. J. P. I. and A. M. T. carried out all the calculations. A. M. T., F. D. P. and P. G. conceptualized the study, the study was supervised by A. M. T., T. J. P. I., F. D. P. and P. G. P. G. wrote the rst dra of the paper, all authors contributed to analyzing the results and the writing, reviewing and editing towards the nal version of the paper.

Conflicts of interest
There are no conicts to declare.