d- and s-orbital populations in the d block: unbound atoms in physical vacuum versus chemical elements in condensed matter. A Dronskowski-population analysis

1.1 Free atoms

A century ago, Bohr initiated the description of atomic states in terms of atomic electron configurations [1, 2]. After the invention of quantum mechanics in 1925, the consistent theory of atomic structure emerged on this sound basis, and was reviewed in 1935 [3]. Model theories of periodic solids and of individual molecules were set off, too [4], [5], [6], [7]. A configuration is the specification of spin orbital functions occupied by single electrons, the anti-symmetrized product of which often is a reasonable approximation for the true electronic matter-field, represented by a many-electron wavefunction. The single-electron orbitals are usually assumed to be of the ‘canonical’ (orthogonal and delocal) type, approximating many single-electronic ionization and excitation processes. Thereby, these orbitals can be connected to observable data [8]. This is an important aspect, since the orbital model is the best approach toward a deeper understanding of chemical bonding and reactions [9], [10], [11], [12], [13], [14], [15], [16], [17].

After World War II, Moore published her reviews of the theoretical analyses of a century of atomic vacuum Vis–UV spectroscopy [18, 19]. At the beginning of each section, she displayed the dominant electron configuration, from which the observed atomic orbital (AO) and spin-coupled ^S L _J ^π (p) ground level is approximately derived (quantum numbers for total spin S with multiplicity 2S+1, total spatial angular momentum L, total angular momentum J, parity π, and parentage p). In the following years, in particular after the first centenary of the Periodic Table in 1969, textbooks of general chemistry (e.g., refs. [20], [21], [22], [23]) began selling the dominant electron configurations of chemically unbound, angular-momentum-coupled ground states of the neutral atoms in physical vacuum as paradigmatic for “the chemical elements”, i.e., the bound, partially charged, spatially perturbed atoms confined in chemical compounds. Combined with an oversimplified Aufbau rule (i.e., neglecting the two-electron repulsion energies), the energetic order of AOs is often assumed as independent of the element number Z, the atomic charge q and the bonding state, following always the so-called (n+ℓ,n) rule, see e.g., refs. [24], [25], [26], [27], [28]. A section of the (n+ℓ,n) AO energy pattern (namely the one relevant for the 3d elements in period n = 4) is displayed as rule (1).

the (n + ℓ, n) textbook rule 1, fitting: the group 2 atoms and their chemistry

But in reality, the (n+ℓ,n) energy rule, i.e., ns < (n−1)d  < np, simulates the observable one-electron energy level pattern only for group 2 valence shells (i.e., neither for groups 0–1 nor for groups 3–18, nor for the inner atomic core (X-ray) levels, see e.g., refs. [18, 19, 29], [30], [31], [32]). For the alkali metal atoms of group 1, spectroscopic data shows that even ns < np < (n−1)d  ≷ (n+1)s, while for groups 12–18 (n−1)d < ns (rule 3). Already in 1923, Bohr had documented the strong variation of d and f orbital levels versus the more regular pair of s and p levels [2].

Bohr’s century-old diagrams of single-electron ionization energies ε (from the smallest to the largest ones in terms of √|ε| values, in a single graphic) versus the nuclear charge or element numbers (up to 92) show a rather smooth behavior for the pairs of s and p (p½, p ³/₂) levels, while the d and the f levels vary in a stepwise manner, crossing with the s,p pairs up and down. After WWII, the deductions from the spectroscopic observations were supported by many quantum-mechanical atomic structure calculations at the single-configuration density-functional or Hartree–Fock (HF) or Dirac–Fock (e.g., refs. [29], [30], [31], [32]) and the various multi-configurational levels. In particular, the involved variation of the order of the AO energies in the small-energy valence range was eventually unraveled and is depicted here in Figure 1 for the first four periods of elements. The insight, notably that the rules (1)–(3) are specifically relevant for the s, d, and p elements, respectively, entered only into a few chemistry textbooks (e.g., refs. [33], [34], [35], [36], [37], [38], [39]).

Figure 1:

Free atomic configuration-averaged 2nd ionization energies |ε| (A⁺ → A²⁺) plotted as √|ε| (in √eV) versus element numbers Z, for A = ₂He through ₃₆Kr; derived from the NIST data [19]. A⁺ yields representative orbitals for chemically bonded atoms (usually slightly contracted in comparison to those of the free neutral atoms). |ε| corresponds to the negative of the one-electron Kohn–Sham (KS) atomic-orbital energy. For increasing Z, the pairs of ns and np energy levels (in blue) descent together rather smoothly, while the d (red; and f, lilac) levels vary in steps. With regard to the energy, for the lightest elements: 3d « 4s (hydrogenic rule 3); from ₁₂Mg to ₂₀Ca: 4s < 3d (the (n+ℓ,n) textbook rule 1); for Ar and K: 3d ≈ 5s; for 3d elements ₂₁Sc to ₂₉Cu: 3d ≲ 4s (the ‘true’ transition element rule 2), and from Zn onward: 3d « 4s (for the core shells of heavy elements again the hydrogenic rule 3).

1.2 Bonded atoms

One origin of these problems is that the dominant atomic electron configuration may be different for the four situations: (i) atoms delocally bound in metallic substances at high coordination numbers (CNs), (ii) atoms with two-center bonds in common chemical coordination compounds at intermediate CN, (iii) atoms in small molecules with low CN (e.g., diatomics), or (iv) unbound atoms in vacuum. Further, chemists (and physicists) often apply the same word ‘element’ to three different objects and concepts: (1) the free atoms of element number Z, (2) all the various pure allotropes and phases of bound atoms Z, (3) the ‘abstract’ elements of atoms Z in various chemical compounds, conserved in chemical transmutations, and sometimes even for (4) a sole nucleus less or more dressed by electronic shells. IUPAC’s Gold book [40] explicitly defines the ‘chemical element’ as either a blend of the 1st and 3rd concept, “A species of atoms Z”, bound or unbound, or as the 2nd concept, “A pure chemical substance Z”.

When chemistry of the elements flourished again after WWII, a successful comprehensive model for d-metal complexes was developed (ligand field theory), based on rule (2) [33], [34], [35], [36]. However, the main drawback of the (n+ℓ,n) rule (1) is not the ‘improvable’ (n−1)d > ns energy pattern but the missing indication of the chemically most important large orbital energy gaps. In rules (2) and (3), the symbol “≪” indicates those large gaps (i.e., large in comparison to chemical bond energies). The large hydrogen-like (n−1)sp ≪ n(sp) gap distinguishes the closed (n−1)(sp)⁸ noble gas shells, i.e., chemically inert-core shells from the chemically active n(sp) ^g valence shells of elements in group g. This gap is the physical reason and origin of the chemical periodicity of the post-noble gas elements under ambient chemical conditions [41], [42], [43]. A large gap of type (n−1)spd < n(sp) also develops after the d ¹⁰-shell filling (in periods n ≥ 4), from group 12 onward, causing the double periodicity of main and transition groups.

Actually, there are many examples for different dominant electron configurations of an ‘element’ in different environments (compounds), corresponding to the different concepts of an ‘element’. For instance, the dominant configuration of the free Ni atom in the ground state is 3d ⁸ 4s ² (just 2½ kJ mol⁻¹ below a 3d ⁹ 4s ¹ state), of Ni in the metallic phase it is 3d ⁹ 4s ¹ ; in many small (diatomic) molecules it is also 3d ⁹ 4s ¹ , but in the tetra-carbonyl nickel Ni(CO)₄ complex is 3d ¹⁰ 4s ⁰ , and for oxidized Ni^II in common nickel salts such as [Ni(H₂O)₆]Cl₂ it is 3d ⁸ 4s ⁰ [44], [45], [46], [47], [48], [49], [50], [51], [52], [53]. In his early plots of AO energies versus the element numbers, Bohr was not sure at which element number the valence (n–1)d level would fall below the valence ns level [1, 2]. Löwdin concluded his famous article on the Periodic Table’s centenary of 1969 [54] with the ‘most important’ question, at which degree of oxidation the electron configuration of an element would change. Yet, discourses on the Periodic Table usually cite another note of Löwdin, namely that a generally valid (n+ℓ,n) rule has not been derived from the Schrödinger equation. Remarkably this fault is not assigned to the inadequacy of the (n+ℓ,n) rule, but often to the incompleteness of the Schrödinger equation in the chemical realm. It is often not considered that the total state energies sensitively depend on both the one-electron orbital energies (negative due to the dominating nuclear attraction) and the repulsive Coulomb interactions (positive) between all the electrons in spin-orbital pairs coupled in various manners among each other.

In the 1930s–1950s it was not yet clear how many electrons were to be accommodated in the inner and the outer s, p, and d AOs of the transition metal atoms in their compounds [55, 56]. Pauling was influential, advertising 3d-4s-4p and even 4d AO populations of 3d transition elements, despite the spatial diffuseness of the 4s,p,d orbitals. This narrative is rather slowly disappearing from the textbooks. Chemists tended to assume that the, somewhat irregular, dominant electronic configurations of the ground states of the free atoms have some relevance for the chemically bound atoms. There are 27 (3 × 9), 3d, 4d, and 5d elements between the d ⁰ s ² and d ¹⁰ s ² groups 2 and 12. The free atomic ground states of 17 of these d elements are derived from (n−1)d ^g−2 ns ² configurations, following the (n+ℓ,n) rule. The 10 ‘exceptions’ from the rule of the free atomic states (Cr, Cu; Nb, Mo, Ru to Ag; Pt, Au) are mentioned in most elementary textbooks, e.g., refs. [20], [21], [22], [23].

During the 1950s, transition metal chemists found that the dominant electron configurations of the metal ions M ^q+ (q = formal oxidation state) in common transition metal complex species, where the metal atom is surrounded by a set of closed-shell ligands (Lewis bases), are 3d ^g−q 4s ⁰ [33], [34], [35], [36], [37], [38], [39, 41], [42], [43, 50], [51], [52], [53]. The situation is different in solid metallic phases, where the metal atoms are coordinated by more ‘ligands’, though of the same ‘metallic character’ with small occupied core shells, with normal and diffuse valence shells, and without dative electron pairs [57], [58], [59], [60], [61], [62], [63].

We here investigate the dominant configurations of the elements of groups g = 2–12. We compare the free atoms M₁, the atomic clusters M _m (m = 1–13), a few complex molecules ML₆, and the metallic phases [M_∞]. Professor Dronskowski’s LOBSTER AO-Population tool [64, 65] is particularly helpful to answer our conceptual questions concerning the dominant electron configurations of the transition metals in different chemical environments.

2.1 Free atoms

We discuss some experimental ‘orbital’ energies ε _exp, as displayed in Figure 1. Figure 2 presents the values of free transition metal atoms (early and late ones, light and heavy ones). ε _exp is defined as the configuration-averaged single-electronic detachment energy from one orbital without changing the population of the other orbitals (ionization energy, IE; corresponding to the work function of solids). Four points must be noted.

Figure 2:

Configuration-averaged one-electron ionization energies (IE in eV) of neutral transition metal atoms (Sc, La and Ni, Pt) derived from NIST atomic spectral data [19]. The d levels (red) vary more than the s levels (blue), in particular for the 3d series. The orbital energy patterns strongly depend on the d ^g−x s ^x orbital population distributions.

First, the valence shell contains energetically near-degenerate orbitals of rather different spatial size, the compact (n−1)d and the diffuse ns orbitals. As a rule of thumb, the radius of ns is ca. three times larger than that of (n−1)d [17, 32]. The orbital energies of the compact (n−1)d orbitals change significantly for different d-s population schemes, they change more than the diffuse ns orbitals, in particular for the 3d elements, and for the late d elements. The (n−1)d ≷ ns orbital energy order may also change for different population schemes such as d ^g s ⁰, d ^g−1 s ¹, d ^g−2 s ². Hence, there is no general fixed orbital energy pattern for the (n−1)d-ns valence shell of the d-block elements!

Second, to derive sensible ε _exp values, ionization processes need to be selected where no significant orbital rearrangement happens. That is not necessarily the lowest energy ionization, which is usually displayed in the textbooks. For instance, the 1st ionization of several transition metal atoms, such as V, Co, or Ni, occurs with a d-s population change. As an example, the “first” ionization of Ni is Ni⁰-³F₄ ^e(3d ⁸4s ² ) → Ni⁺-⁵D_5/2 ^e(3d ⁹4s ⁰ ). However, for the ε _exp values of Ni-3d, and Ni-4s, respectively, the ionization processes Ni⁰-³F₄ ^e(3d ⁸4s ²) → Ni⁺-⁴F_4.5 ^e(3d ⁷4s ², and 3d ⁸4s ¹) are the appropriate ones.

Third, the experimentally derived orbital energies (and the exact KS density-theoretical ones [66]) cover also the relaxation of the final cation, what reduces the ε _exp value, while the self-consist calculations of orbital energies refer to the orbital charge distributions in the initial state and do not cover the ionic orbital relaxation effect. Note that common density-functional approximations usually yield reasonable molecular structures and energies, while the orbital energies are much smaller in value than the ionization energies, mainly due to the incompletely corrected ‘self-interaction error’ [66].

Forth, atoms, clusters, and molecules each have their specific spatial symmetries, which lead to specific orbital level splittings and specific orbital populations. For instance, the most stable state of the half-filled d shell of a free atom in a vacuum (of O₃ symmetry) is ⁶S_5/2 ^e- d ⁵ , while for the atom in a strong ligand field (of O_h symmetry), the particularly stable states are ⁴A_1g(3/2)- d(t _2g ) ³ and ¹A_1g(0)- d(t _2g ) ⁶ . Hence, which AO population numbers are preferred depends on the atomic environment. Accordingly, the orbital coupling of the free atomic ground state is of less relevance for chemistry in general. In any case it is better to average over the different states of a given configuration. Hence, we display the configuration-averaged one-electron ionization energies; in particular we average over the spin-orbit split levels.

The staircase-like variation of the d levels with increasing Z versus the more smooth hydrogen-like variation of the pair of s and p levels is shown in Figure 1. We deduce the following fuzzy empirical rule for the free atoms of d-block elements: “energetically (n–1)d ≈ ns at the beginning of the d series, with the tendency toward (n–1)d < ns at the end”. Pilar even claimed “4s is always above 3d”, in contrast to the textbook rule ([67], however [68]).

We here note that ε _exp[(n−1)d] < ε _exp[ns] in particular for the cases where the lowest energy configuration is (n−1)d ^g−2 ns ². Small d population means small d-d repulsion and low d-orbital energy. Apparently, caution is recommended when comparing experimental ionization energies (the configuration averaged ones, and even more so referring to the more ‘directly observed’ ones between individual states) with the orbital energies of orbital models.

There are no apparent violations of the correct Aufbau rules concerning the orbital populations in single-configuration models of the ab-initio (HF or Dirac–Fock) or density-functional (KS) type. For the ab-initio orbital model “the lower orbital levels are fully occupied at first, provided the orbital energy gap is bigger than the two-electron interaction correction”. We stress that the one-electron energy values have different meanings concerning: the experimental ionization; the average-configuration derived values; the ab-initio HF approach; the exact density-functional KS approach; the common KS approximations [66]. For instance, the HF orbital energies do not account for Coulomb correlation nor for cationic relaxation, which often partially cancel each other. The common KS approximations suffer from incomplete self-interaction corrections. If (n−1)d ^g−2 ns ² is the lowest energy configuration (as in many free d atoms), yet ε _exp [(n−1)d] < ε _exp [ns] and ε _HF [(n−1)d] < ε _HF [ns] may hold (violation of the simple Aufbau rule), but ε _KS,appox [ns] < ε _KS,approx [(n−1)d] usually holds (Janak’s rule). Indeed, we find with the PBE density functional for Sc-3d ¹4s ²: 4sα (−4.3 eV) < 4sβ (−4.0 eV) < 3d ₁α (−3.5 eV) (see also Supplementary Material available online, Sect. S3).

2.2 Clusters of transition metal atoms

For transition metals M, we compare the atom M₁ and the close-packed solid [M_∞] with cluster molecules M _m , m = 5, 9, and 13 (the latter has the ‘metallic’ CN = 12 for the central atom). The geometric structures are displayed in Figure 3, the calculated bond lengths and orbital populations are shown in Table 1. We have chosen two 3d elements with atomic ground configurations 3d ^g−24s ², namely Mn (g = 7) and Ni (g = 10).

Figure 3:

Ball-and-stick models of clusters of transition metal atoms (M) for M = Mn or Ni. In the M₁₃ (D _3d) cluster, the red (thin-blue) lines represent shorter (longer) interatomic distances.

From the single atom to the metallic phase, the 3d population increases by ca. one unit, while the 4s population decreases, 4s partially hybridized with 4p. These trends may be rationalized as follows. The slight overlap of the rather compact 3d orbitals causes bonding by populating the ‘band’ of d orbital levels. The 4s-4p population is more complex. These diffuse AOs strongly overlap with each other, yielding bonding and strongly antibonding levels. In addition there is an overall Pauli repulsion by the occupied core shells of the adjacent metal atoms. Only the bottom of the s-p band of molecular orbitals is bonding and low in energy and will be occupied. Due to the large overlap values, in particular for high CNs, the atomic reference basis becomes nearly degenerate, and the AO population values sometimes become numerically unstable, i.e., inaccurate.

In summary, while the single atoms in vacuum have a tendency toward high population of the diffuse ns orbital (paradigmatic 3d ^g−24s ²), the clustering of the metal atoms toward the metallic phase tends to medium sp population (paradigmatic 3d ^g−14(sp)¹) with filling the lower, bonding half of the s-type band.

2.3 Solid transition metals

Three different low-energy crystal structures were investigated for the solid metals Ca to Zn of groups g = 2–12 in period 4 (and for the heavier homologs, yielding similar trends, see the Supplementary Material, Sect. S1): the hexagonal close-packed (hcp) one, the cubic close-packed (ccp = fcc, face centered cubic) one, both with (CN = 12), and the body-centered cubic (bcc) one (CN = 8); for technical details see the Methodology (Sect. 4, below).

Figure 4 displays the quantum-chemically optimized interatomic nearest neighbor distances (at T = 0 K and p = 0 bar). The experimentally determined values (for STP) of the 4th, and also of the 5th and 6th periods are very similar (see Sect. 4 of the Supplementary Material) (For the calculations of Mn at 0 K, we chose the high-temperature fcc structure instead of the unusually complex, highly coordinated, spatially expanded structure at STP). Four points are not new, yet noteworthy: (i) The quantum chemical calculations correctly reproduce the empirical trends. (ii) The inter-atomic distances vary smoothly with group number, the shortest ones occur around groups 8 and 9. (iii) The irregularity of the electronic configurations of the ground levels of the free atoms ((n−1)d ^g−2 ns ² or (n−1)d ^g−1 ns ¹ or (n−1)d ^g ns ²) appears not to affect the bond distances, and also not the melting and boiling temperatures of the solid phases (Supplementary Material, Sects. S1.3 and S1.4). (iv) The secondary periodicity is obvious, meaning that period four differs from the mutually rather similar periods 5 and 6.

Figure 4:

Solid transition metals: Calculated (for T = 0 K) interatomic distances (in pm) of the most stable hcp, ccp, or bcc structure (but for Mn, the close-packed high-T structure at T = 0 K, instead of the complex, less-dense low-T structure). Fourth period: black squares ■; 5th period: red circles ; 6th period: blue triangles ; all with eye-guiding connections (no relation to the d ^g−2 s ² vs. d ^g−1 s ¹ configurational differences of the ground states of the free atoms is visible).

The orbital populations of the atoms in the metallic phases of the elements in the 4th period are shown in Figure 5 (for more details and for the heavier homologs, see the Supplementary Material, Sect. S1.2). Again, the variation is rather smooth. If the occupied valence band structure is projected onto the 3d and 4s AOs, one obtains for the metals Sc to Cu approximately (within ±0.2) 3d ^g−0.94s ^0.9, and for the projection onto 3d, 4s, and 4p, approximately 3d ^g−1.44(sp)^1.4. Calcium and Zink from groups 2 and 12 differ a little. The Ca metal with ca. 3d ^0.6 has slightly more 4s population, but significantly less than the 3d ⁰4s ² of the free atom. Zn metal has a filled 3d ¹⁰ core shell with only a little additional 4d polarization of the 4sp valence band. Our AO populations agree within ca. ±0.2 with the ones in the review book of Papaconstantopoulos, giving on the average 3d ^g−1.05 4s ^0.65 4p ^0.4 [69, 70].

Figure 5:

Atomic orbital (AO) Mulliken populations (3d, 4s, 4s, and 4p) of face-centered cubic phases of 4th row metals Ca to Zn (From density functional calculations with the SCAN+rVV10 meta-gradient-approximation with dispersion). The full (open) circles connected by full (dotted) lines refer to calculations with (without) 4p orbitals for the population analysis, respectively. Other density functionals, other cubic and hexagonal phases, yield similar trends (see the Supplementary Material, Sect. S1.3).

The s,p,d populations in the transition metals may be derived from various spectroscopic data or from band structure computations. The literature data scatter quite a bit (e.g., refs. [47], [48], [49]). In some cases small or even negative ns populations were reported, and in other cases also higher populations of ns ^1.5 to ns ^>2. This (non-systematic) scattering of deduced parameters does not really prove that the atoms at high CN = 12 in hexagonal or cubic solid environment follow the d,s population behavior of free atoms in vacuum with CN = 0.

Figure S5 in the Supplementary Material displays good correlations between the calculated cohesive energies and the boiling and melting temperatures of the metallic elements. Figure S6 displays the variation of the melting and boiling temperatures along the series of transition metals. Both Figures indicate an increasing bond strength where the s(p) bonding of the metals of groups 2 or 12 is supported by d bonding, which is strongest for group 6 metallic phases with dominant (n−1)d ⁵ ns ¹ configuration and with half-filled valence bands. There is no apparent correlation with the d ^g s ⁰, d ^g−1 s ¹, d ^g−2 s ¹ configurations of the free, chemically isolated atoms.

2.4 Some transition metal complexes

Since the development of the crystal and ligand field models before and after WWII, transition metal chemists describe the lower electronic states of complexes of metal cations M ^q+ by the dominant configuration (n−1)d ^g−q ns ⁰. We have reproduced this concept for three complexes, formally with 5, 6, and 7 electrons in the valence shell of the metal atom, using the same computational approaches as applied so far.

For T_d symmetric Na₄[MnCl₆], we obtain a spin sextet and populations Mn-3d ^5.494s ^0.124p ^0.66, meaning Na⁺ ₄[Mn²⁺ (3d ⁵4s ⁰)Cl⁻ ₆] where the Cl⁻ ligands donate ca. 6 × 0.5 into the half-populated 3d shell, and ca. 0.1 and 0.7 e into the diffuse 4s and 4p shells, respectively. For the geometric structure, effective charges and spin populations of this high-spin and the respective excited low-spin complexes, see the Supplementary Material, Sect. S2.

For O_h symmetric [Cr(CO)₆], we obtain a spin singlet and populations Cr-3d ^5.334s ^0.004p ^0.77, meaning [Cr⁰ (3d ⁶4s ⁰) (CO⁰)₆]⁰, where the six CO ligands donate ca. 2e from their C-2pσ lone-pairs into the empty 3d(e_g) and 4p(t_1u) orbitals, while ca. 2e from the filled 3d(t_2g) orbitals are back-donated into the empty CO-π* orbitals.

Adding 1 e, we obtain the O_h symmetric spin doublet complex anion [Cr(CO)₆]⁻ with populations Cr-3d ^5.294s ^1.04 4p ^0.76, meaning [Cr⁻ (3d ⁶4s ¹) (CO⁰)₆]⁻ (see also the Supplementary Material, Sect. S2). The attached electron populates the diffuse Cr-4s orbital, a typical phenomenon for formally anionic transition metal atoms. In contrast, cationic metal complexes with formally seven valence electrons are d ⁷ s ⁰ systems. Examples are the Rh²⁺ or Co²⁺ complexes, e.g., the spin quartet [Co²⁺ (3d ⁷4s ⁰) (H₂O,NH₃)⁰ ₆]²⁺ [44], [45], [46].