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From Hartree-Fock to Electron Correlation: Application to Magnetic Systems

Vincent Robert, Mikaël Képénékian, Jean-Baptiste Rota, Marie-Laure Bonnet, and Boris Le Guennic

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-Tc 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:

which is to be solved, defining the N-electron eigenfunction and eigenvalue E of the Hamiltonian H. The nonrelativistic Hamiltonian is written as a sum of different kinetic and potential contributions arising from interacting electrons and nuclei:

Since the nuclei are much heavier than the electrons, their kinetic energy is much smaller and, consequently, can be considered as motionless. In the study of the electronic problem, the nuclei positions are parameters for the motion of the electrons, and the problem is solved by considering only the electronic part of the Hamiltonian (so-called the Born-Oppenheimer approximation [33]). Thus, the electronic Hamiltonian using atomic units reads

While the first two terms are monoelectronic in nature, the third one is the electron-electron repulsion which excludes any analytical resolution of the manybody problem.

Traditionally, one looks for a step-by-step procedure to incorporate the important physical contributions in a hierarchical way. A reasonable zeroth-order wavefunction is accessible within the Hartree-Fock scheme. Such treatment relies on a meanfield approximation where each electron moves in the field generated by the nuclei and the average electronic distribution arising from the N - 1 other electrons (see Section 1.2.2.1). It was rapidly understood that such single determinantal strategy fails to properly describe bond breakings. As a matter of fact, as a bond is stretched, the independent electron approximation breaks down as the electrons tend to localize in a concerted way one on each nuclei. To overcome this failure and incorporate the so-called static correlation, the CASSCF procedure has been proposed [1,2]. Along this procedure, the wavefunction becomes intrinsically multireference (see Section 1.2.3.1). Finally, contributions which tend to reduce the electron-electron repulsion account for the dynamical correlation. Its main effect is the digging of the Coulomb hole to increase the probability of finding two electrons in different regions of space, distinguishing radial and angular correlations. This concept has been widely used in the understanding of DFT approaches.

As both static and dynamical correlations are turned on top of a Hartree-Fock solution, electrons are allowed to occupy arbitrarily (respecting spin and space symmetries!) all the MOs, introducing other electronic configurations which may be necessary to describe the physical state of interest. In a sense, the expansion of the wavefunction as a linear combination of Slater determinants (configuration interaction, CI) tends to recover the physical effects absent in the initial orbital approximation.

1.2.2 Finding a Solution

Let us start from an infinite set of MOs, ?i, and a zeroth-order approximation to the N-electron problem. The MOs are split into two sets, either doubly occupied or empty (referenced as (a, b, c, .) and (r, s, t, .), respectively), defining |0> as The wavefunction can be developed upon |0> and the electronic configurations built from |0> by successive excitations (see Figure 1.1),

Figure 1.1 (a) |0>, (b) single, and (c) double excited determinants.

where represents single excited determinants, double excited, and so on.

Solving the electronic problem consists in the determination of (i) the MOs, and (ii) the amplitudes of different electronic configurations The first task is achieved along the Hartree-Fock procedure, while the second calls for numerical demanding methods which are constantly under intense investigations.

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 H and

where Ja and Ka 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. h is the sum of...