1.1 Introduction

At the beginning of last century, quantum mechanics broke out and the famous Schrödinger’s and Dirac’s equations were derived and constituted tremendously important milestones. Even though they aimat describing the nanoscopic correlated world, it is known that the analytical solution is limited to the two-particle system, a prototype of which being the H atom. In particular, the description of a simple system as H₂ necessarily relies on approximations. One may first consider electrons as independent particles moving in the field of fixed nuclei. The appealing strategy of a mean field approximation was thus suggested along with the important picture of screened nuclei. How much the fluctuation with respect to this description dominates the physical properties has been a widely debated challenging issue.

This review will be organized as follows. First, the different methods traditionally used in quantum chemistry are briefly recalled starting from the Hartree–Fock description to the introduction of correlation effects. Since quantum chemistry aims at describing the interactions between atomic partners, the one-electron functions (so-called molecular orbitals, MOs) are derived from one-electron atomic basis sets localized on the atoms (atomic orbitals, AOs). However, it is known that a major drawback in this single determinantal description of the wavefunction is its inability to properly account for bond breaking. The H₂ case is used as a pedagogical example in Section 1.2.2.2 to exemplify the need for multireference SCF algorithms. For the study of homolitic breaking of such a single bond, it is recalled that both bonding and antibonding MOs must be introduced to incorporate the so-called nondynamical correlation effects. In this hierarchical construction of the wavefunction, the Complete Active Space Self-Consistent Field (CASSCF) [1, 2] procedure is described (see Section 1.2.3.1). Such methodology is particularly efficient since along bond stretching, two electrons become strongly correlated and the CASSCF treatment tends to localize one electron in each atom. The important dynamical correlation effects are then exemplified deriving the H₂–H₂ interactions, and the short distance behavior (1/R⁶) of the van der Waals potential is recovered (see Section 1.3.1).

In the last section, the machinery and efficiency of ab initio techniques are demonstrated over selected examples. A prime family is represented by magnetic systems which have attracted much attention over the last decades considering their intrinsic fundamental behaviors and possible applications in nanoscale devices. Chemists have put much effort to design and fully characterize new families of systems which may exhibit unusual and fascinating properties arising from the strongly correlated character of their electronic structures. From a fundamental point of view, high-T_c superconducting copper oxides [3–5], and colossal magnetoresistant manganite oxides [6–11] are such families which cannot be ignored in the field of two- and three-dimensional materials. One-dimensional chains [12–15] as well as molecular systems mimicking biological active centers [16,17] have more recently been considered as promising targets in the understanding of dominant electronic interactions. In such materials, a rather limited number of electrons are responsible for the observed intriguing properties. Reasonably satisfactory energetics description of such systems can be obtained by the elegant broken-symmetry (BS) method [18–21]. Let us mention that, in particular, BS density functional theory (DFT) calculations have turned out to be very efficient in the determination of magnetic coupling constants and EPR parameters (see [22–29] and references therein). On the other hand, the DFT methodology has been extensively used in surface science to follow at a microscopic level the reactant transformation leading to products. Nevertheless, this description has shown to suffer from an unrealistic description of physisorption [30]. Thus, a combined approach based on the periodic DFT method with MP₂ correction has been proposed to overcome this intrinsically methodological drawback [31, 32].

These examples aim at shedding light over a selected number of systems in materials science, catalysis, and enzymatic activity which may call for explicitly correlated calculations.

1.2 Methodological Aspects of the Electronic Problem

1.2.1 The Electronic Problem

Physical properties of molecules take their origin in electron assembly phenomena. To understand these properties, one has to investigate the electron distributions and interactions. This information is contained in the electronic wavefunction governed by Schrödinger’s equation:

(1.1)

1.2.2.1 Hartree–Fock Approximation

The goal is to find a set of MOs sustaining the reference determinant .These orbitals should form an orthonormal basis of one-electron functions. Under these constraints, the Hartree–Fock equations are easily derived by minimizing the expectation value of Ĥ and

(1.5)

where J_a and K_a represent the Coulomb and the exchange operators, respectively.

The eigenfunction problem(s) must be solvediteratively (self-consistent field procedure, SCF) since the Fock operator is constructed on the occupations of its own eigenvectors. ĥ is the sum of the kinetic energy and nuclei–electron interactions, while the sum defines the Hartree–Fock potential that averages the interelectronic repulsion so as to give a monoelectronic operator.

1.2.2.2 Example

In order to clarify the Hartree–Fock SCF framework, let us concentrate on the quantum chemist’s “swiss army knife” system, namely H₂ in a minimal AO basis set {a, b}. From symmetry consideration, one can build two MOs, symmetric (g)

(1.6)

and antisymmetric (u)

(1.7)

Evidently, the Hartree–Fock solution for the ground state is

(1.8)

or returning to the AOs,

are referred to as the ionic forms since the two electrons are localized on the same atomic center. This is to be contrasted with the combination which is the neutral singlet. Thus, consists of an equal weight of ionic and neutral forms. While the H₂ molecule should clearly dissociate into (see Figure 1.2), the Hartree–Fock procedure overestimates the weight of the ionic forms. As a matter of fact, the latter should physically become vanishingly small as the bond is stretched. This is a major pitfall of the Hartree–Fock theory which is being taken care of in a multireference approach.

1.2.2.3 Beyond Hartree–Fock Treatment: Electronic Correlation

As mentioned previously, the main effort is to calculate the coefficients in the wavefunction expansion. This calculation gives a correlated wavefunction, the energy of which defines the correlation energy as

Practically the expansion cannot be carried out upon an infinity of excitations and a selection of excited determinants must be made in the configuration interaction treatment. In practice, this procedure cannot lead to the exact solution of the many-electron problem for two main reasons:

• It is impossible to handle infinite one-electron basis sets. Therefore, the constructed determinants {|_i〉} cannot form a complete N-electron function basis set.
• Even for relatively small basis sets, the number of determinants to be considered may become extremely large. Thus, in practice, one will not take into account all determinants (full-CI procedure) but only a small part of these (truncated-CI), in single and/or double (CIS, CISD) calculations.

Traditionally, one distinguishes static and dynamical correlations in the CI approach. In the next section, we will clarify these notions using the H₂ example.

1.2.3.1 Static Correlation: CASSCF Approach

Let us concentrate on the problem of bond breaking of H₂. In the g orbital, the maximum of electronic density is in the middle of the bond. Conversely, the u function displays a nodal plane and the maximum of density is concentrated on the nuclei.

To overcome the major failure of the Hartree–Fock description, one may introduce in a multireference expansion other determinants. Clearly, g and u become quasidegenerate in the long-distance regime. Thus, may as well significantly participate in the two-electron wavefunction. By allowing the occupations of two MOs by two electrons, leaving all the other orbitals either doubly occupied (inactive) or vacant (virtual), one performs a Complete Active Space Self-Consistent Field CAS(2,2)SCF calculation (see Figure 1.3). From a physical point of view, this procedure consists in treating exactly the correlation in the active space and let the inactive orbitals react to the field generated by different configurations built in the active space. This point constitutes the major difficulty of the CASSCF calculation. Indeed, since the active space is the only part of the system where the correlation is treated with fine accuracy, it has to include the necessary configurations to describe the property of interest.

One defines the best set of MOs under this constraint. The inactive MOs respond to the occupations of the active MOs, treating democratically the configurations. The comparison between H₂ and F₂ systems is instructive since the latter holds such inactive shells. The CAS(2,2) inactive MOs of F₂ will be the best compromise between the double occupancy of g and u.

For H₂ in a minimal basis set, the correlation energy is analytical from the 2 × 2 matrix diagonalization (see Ref. [34] for derivations). Writing and

(1.9)

(1.10)

These results call for two important comments. First the correlation energy is negative as a result of the flexibility offered to the wavefunction. Then, the amplitude c of being negative reduces the weight of the ionic forms. Eventually, as it can be shown that c →−1 and the wavefunction reduces to . The electrons are no longer independent, they are said to be correlated. The reduction of the ionic forms stresses the demand of atoms to recover their neutral character. The nondynamical correlation strikes back again the delocalization preference arising from the Hartree–Fock scheme. Along the CASSCF procedure one introduces the leading physical contributions in a multireference wavefunction. This allows one to treat on the same footing quasidegenerated electronic configurations given in a predefined active space (so-called CAS). Typically, the dissociation of H₂ can be properly discussed using a CAS(2,2)SCF calculation (see Figure 1.2).

1.2.3.2 Dynamical Correlation

In the light of the previous considerations, let us again concentrate on H₂ close to the equilibrium distance. Consequently, the value is large whilst c is almost negligible. is almost monoreference. A statistical analysis of the wavefunction shows that the electrons spend much time in the g orbital and sometimes explore the u one. In this case, the correlation is a fluctuation of the electronic density around an average value. This is part of the origin for dynamical terminology. The dynamical correlation brings a correction to the energy and wavefunction, but the qualitative results of the Hartree–Fock approach are not deeply changed.

More generally, on top of the CASSCF wavefunction one traditionally performs either second-order perturbation theory treatment (CASPT2) [35,36] or variational CI such as the so-called first-order CI which incorporates in a variational way all the single excitations on the CAS determinants. These contributions account for the electronic relaxations which respond to the instantaneous field modifications or spin polarization in the active space.

In this respect, the Difference Dedicated CI (DDCI) methodology [37–39] has shown to provide impressive results in magnetically coupled systems [40–42]. The conceptual guideline is the quasidegenerated perturbation theory (QDPT) developed by Bloch [43]. For a two-electron/two-MO system one looks for the singlet–triplet energy difference 2J, J being the one-parameter model Heisenberg Hamiltonian The model space consists of two neutral forms upon which the QDPT defines an effective Hamiltonian . At the second order of perturbation theory, the off-diagonal element of Ĥ_eff is precisely J and reads

being outer-space determinants, including ionic forms If the sum is restricted to then J reads

with and One recovers the famous competition between ferromagnetic and antiferromagnetic contributions. For |α〉 to be simultaneously coupled to it should not defer by more than two spin orbitals (Slater’s rule). Thus, the determinants are traditionally listed according to the number of holes (h) and particles (p) generated on the model space. As soon as this space is enlarged to the full valence space (i.e., including the ionic forms), it can be shown that 1h, 1p, 1h + 1p, 2h, 2p, 2h + 1p, and 2p + 1h participate in the hierarchical organization of the singlet and triplet.

Indeed the purely inactive excitations 2h + 2p simply shift the diagonal matrix elements. As shown in Figure 1.4, the selection gives rise to DDCI-1, -2, and -3 levels of calculations. Being a truncated-CI methodology, DDCI suffers from intrinsic size-consistency issue which has been elegantly corrected in the so-called Size-Consistent Self-Consistent (SC)² framework [44]. The physical effects (spin polarization, dynamical correlation) have been clarified by considering different levels of calculations [45–47].

In order to remedy to this size-consistency problem, alternative approaches have been proposed and coupled pairs methodology turned out to be very efficient [48]. Unfortunately, the cost of such calculations does not allow one to handle even moderate size systems. Nevertheless, the CASPT2 method [35, 36] offers a remarkable compromise, introducing at second-order of perturbation theory the correlation effects. The corresponding atomic effects are properly incorporated in this contracted treatment of correlation effects. Such methodology has proven to be remarkably efficient in the inspection of magnetic properties of molecular and extended systems.

1.3.1 Dipoles Interactions: Example of (H₂)₂

Let us consider two H₂ molecules well separated in space (l L, see Figure 1.5).

If a, b, c, and d refer to the AOs, one can built the g and u MOs on each H₂ fragment (see Figure 1.6).

Thus, a zeroth-order wavefunction is given by

One can observe in the development of |₀〉 that the doubly ionic structures ‘‘H⁺H⁻H⁺H⁻’’ and ‘‘H⁺H⁻H⁻H⁺’’ hold equal weights, in disagreement with naive electrostatic argument. However, the double excitation g₁g₂ → u₁u₂ (see Figure 1.7) enhances the former and reduces the latter thanks to configurations interaction:

The bielectronic Coulomb integrals can be approximated as the inverse of interatomic distances,

Thus, a second-order development (l L) gives

Using second-order perturbation theory to evaluate the correlation energy, the L⁻⁶ dependence of the dispersion energy is recovered.

The origin of the dispersion energy is clear in this procedure. Indeed, the development of the doubly excited determinant on the atomic orbitals a, b, c, and d (see Figure 1.8) exhibits the role of correlation between the fluctuations of the positions of the electrons in the two bonds. When the electrons move from b to a, then the probability of a concerted displacement from d to c is larger than the one of a movement from c to d.

1.3.2 Open-Shell Ligands: Noninnocence Concept

Considering the possibility of generating high oxidation states ions (in iron chemistry for instance, let us mention notable examples of Fe(IV) [49, 50], Fe(V) [51–53] and Fe(VI) [54]), much synthetic effort has been devoted to the preparation of specific multidentate ligands. The use of such ligands, known as noninnocent, has opened up the route to original synthetic materials, involving open shells on both metal and ligands partners [55–61]). The spectacular excited-state coordination chemistry concept in which a ligand coordinates in an excited electronic state to a metal center has emerged from this class of compounds [62]. The generation of radical ligands in coordination compounds has given rise to a promising route to magnetic materials.

From the theoretical point of view, DFT as well as CI calculations have been undertaken to scrutinize the electronic structures of such noninnocent ligandbased systems [58–60,62,63]. In particular, the comparison between experimental and calculated exchange-coupling constants and the analysis of the magnetic interactions has been the subject of intense work. While DFT has sometimes failed to fully account for the low-energy spectroscopy, the wavefunction-based DDCI method has elucidated the unusual behavior of several complexes [58, 62]. Among those, a striking example is given by the Fe(gma)CN complex containing the glyoxalbis(mercaptoanil) (gma) ligand (see Figure 1.9) [22]. Even though the noninnocent character of the gma ligand was clearly demonstrated both experimentally and theoretically, DFT calculations were only partially successful in the description of the electronic structure of the full complex [62]. The magnetic susceptibility and zero-field Mössbauer measurements clearly favored a doublet ground state. Nevertheless, DFT calculations did not provide any clear evidence in that sense, the M_s = 1/2 solution exhibiting a low-spin Fe(III) (S_Fe = 1/2) coupled to a closed shell gma ligand (S_gma = 0). Clearly, for a good description of the electronic structure of such system, DFT and its monodeterminantal character is not appropriate and correlated ab initio calculations might be desirable.

Based on this statement, correlated ab initio calculations on this particular system by means of DDCI-2 calculations on the top of the CAS(5,5)SCF wavefunction were performed [22]. Interestingly, the active orbitals consist of three metal-centered and two ligand-centered MOs (see Figure 1.10) [62]. The calculations showed that the low-energy spectrum exhibits a 200 cm⁻¹ quartet–doublet gap, in agreement with different experiments, and that the observed strong antiferromagnetic is due to important ligand-to-metal charge transfer (LMCT). The resulting ground-state wavefunction which exhibits an intermediate magnetic/covalent character is rather strongly correlated and is dominated by local (S_Fe = 3/2 and S_gma = 1) electronic configuration. Finally, whereas the gma ligand is clearly a closed-shell singlet when considered alone, it is likely to be a triplet when coordinated to the iron center. The multiconfigurational nature of the wavefunction has been identified in this example and makes this class of compounds still very challenging for theoreticians. It has been recently suggested that the energetics of low-lying states of coordination complexes based on porphyrins and related entities may not be accessible by means of DFT methodology (see Ref. [23] and references therein). More troublesome is the dependence of the spin density maps on the functional choice.

1.3.3 Growing 1D Materials: Ni-Azido Chains

With the generation of magnetic properties goal in mind, experimentalists have prepared higher dimensionality materials. One of the main challenges in the synthesis of extended 1D systems is to prevent the local magnetic moments from canceling out. In the presence of most frequent antiferromagnetic interactions, pioneer approaches were devoted to regular heterospin ferrimagnetic chains [12] holding alternating spin carriers, coupled through a unique exchange constant. Another strategy consists in varying the magnetic exchange constants between homospin carriers [64, 65]. Finally, the use of strong anisotropic metal ions to reduce the magnetization relaxation has generated the promising field of the single-chain magnet (SCM) [13–15].

In this respect, the azido ligand turned out to be extremely appealing in linking metal ions and a remarkable magnetic coupler for propagating interactions between paramagnetic ions. The structural variety of the azido complexes ranges from molecular clusters to extended 1D to 3D materials [66–71]. An interesting prototype of such a system has been recently synthesized where a single azido unit bridges in an alternating End-On (EO) and End-to-End (EE) way the Ni(II) ions (see Figure 1.11) [72]. The system can be considered from the chemical point of view as a quasi-1D chain. However, based on magnetic susceptibility measurements, it was suggested that the system should be described from the magnetic point of view as isolated dimers. Indeed, the introduction of a second magnetic interaction was shown to be irrelevant. Therefore, the question of the nature and amplitude of the magnetic interactions between the nearest Ni(II) ions deserved special attention. The alternation of EO and EE units strongly suggested the presence of two magnetic exchange pathways which can be accessible through Ni₂ dimers spectroscopy analysis. Thus, CAS(5,6)SCF/DDCI-2 calculations were performed on the molecular EE and EO fragments extracted from the available crystal structure.

The active orbitals consist of the in-phase and out-of-phase linear combinations of the d_z2 and d_x2−y2 metallic AOs (see Figure 1.12) and the nonbonding MO of the N⁻₃ bridge.

Since the Ni(II) ion is formally d⁸, it is expected that exchange interactions between S = 1 ions should give rise to three spin states in the Ni₂ units, namely singlet (S), triplet (T), and quintet (Q) states. In a Heisenberg picture (S₁ = S₂ = 1), the energy separations are 6| J| and 4| J| between the quintet and singlet, quintet and triplet states, respectively (see Figure 1.13). Within the EE unit, a relatively large antiferromagnetic exchange constant (J_EE ∼−50 cm⁻¹)was calculated in good agreement with the unique value extracted from experiment (∼ −40 cm⁻¹). This is to be contrasted with the EO Ni₂ unit, which exhibits a negligibly small magnetic interaction (α = J_EO/|J_EE| ratio ∼0.02) (see Figure 1.14).

The correlated calculations not only confirmed the isolated dimers picture, but also associated the leading antiferromagnetic exchange pathway with the EE bridging mode. In the light of the calculated (E_Q − E_S)/(E_Q − E_T) ratio, let us mention that the deviation from a pure Heisenberg picture is negligible (less than 2%) ruling out the speculated participation of quadratic terms. The attempt to generate high-enough ferromagnetic interactions between S = 1 sites looked very promising since the antiferromagnetic coupling between the resulting S = 2 units through EE bridges might have resulted in a Haldane chain with vanishingly small spin gap [73, 74]. The versatility of the azido magnetic coupler should still be considered to generate synthetic models for theoretical physics analysis.