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Keywords:

  • difference dedicated configuration interaction;
  • magnetic coupling;
  • photochemistry

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Early Years–Semiempirical Calculations
  5. Moving to Ab Initio Calculations
  6. Difference Dedicated Configuration Interaction
  7. Orbital Manipulations
  8. Analysis of the Magnetic Coupling
  9. Other Applications of DDCI
  10. Perspective: Limits of the Methods and How to Overcome These
  11. Biography

On the occasion of the celebration of Prof. Rosa Caballol's 65's birthday, a symposium was organized by the Quantum Chemistry Group of the Universitat Rovira i Virgili (Tarragona, Spain) under the title “Theoretical Chemistry in Spain told by women.” The symposium gave a representative illustration of the research in theoretical chemistry in Spain, which is not only very diverse, but also in many areas at the forefront of the field. In this article, we summarize the contributions made by Rosa Caballol and coworkers. Emphasis lies on the developments of wave function based methods to accurately describe electron correlation effects in systems with unpaired electrons. Besides discussing some illustrative applications, we will also explore the limits of the existing methods and outline the perspectives of how one can possibly overcome them. © 2014 Wiley Periodicals, Inc.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Early Years–Semiempirical Calculations
  5. Moving to Ab Initio Calculations
  6. Difference Dedicated Configuration Interaction
  7. Orbital Manipulations
  8. Analysis of the Magnetic Coupling
  9. Other Applications of DDCI
  10. Perspective: Limits of the Methods and How to Overcome These
  11. Biography

The accurate treatment of electron correlation effects in medium- to large-sized systems remains an important challenge in nowadays theoretical and computational chemistry. Although many systems can be described with approaches based on density functional theory (DFT), there are several cases in which DFT does not provide a fully satisfactorily account of the electronic structure. Among others, important examples are the description of the potential energy surface of excited states, essential to photochemistry, and the transition metal compounds with unpaired electrons when studying the multiplet structure and coupling of spin angular momenta. In general, it can be stated that one should be careful with the application of DFT to systems with a marked multiconfigurational character of the electronic structure. In these cases, wave function based methods that use the Hartree–Fock determinant as reference point may suffer from the same shortcomings as DFT, and the coupled cluster singles-doubles with perturbative triples [CCSD(T)] approach—considered the golden standard of quantum chemistry—is not necessarily better than DFT. The key to an accurate and conceptually satisfactory description of the electronic structure is found behind the single-determinantal description, and taking into account the multideterminantal or multiconfigurational character from the start.[1-4]

In almost all cases, such descriptions of the electronic structure can be divided in two steps. First a reference wave function is constructed that accounts for the multideterminantal or multiconfigurational character of the wave function. This can be done in many ways, but most conveniently comprises a complete active space self-consistent field (CASSCF) wave function, in which a limited number of valence electrons is distributed in all possible ways over a set of valence orbitals. In this way, a wave function is built that (i) can treat near-degeneracies, (ii) is a spin eigenfunction, and (iii) also provides a reasonable starting point when spin-orbit coupling becomes important.[5] However, this description does not always give accurate relative energies due to the lack of dynamic correlation effects in the wave function. This is very hard to repair within the CASSCF approach, so other approaches need to be considered as the second step of the multiconfigurational description of the electronic structure.

The multiconfigurational or multideterminantal reference [multireference (MR) for short] wave function can be improved by perturbation theory (MR-PT), configuration interaction (MR-CI), and MR-CC approaches. The recent article of Szalay and coworkers[4] gives an excellent overview of the different variants of MR methods to treat dynamic electron correlation, and we refer the reader to this review for more details. The highest precision is in principle obtained with MR-CC but among the many different implementations of the theory, there does not seem to emerge a generally applicable scheme capable to tackle medium-sized systems. MR-CI has a long-standing history is rather precise but the largest drawbacks are the high-computational cost and the problems to rigorously correct for the size-consistency error. The PT approach is by far the most applied strategy and especially the CASPT2 implementation has become a popular working horse of wave function based quantum chemistry.

The state-of-the-art of computational and theoretical chemistry is developing at a surprising speed. This is due to the combination of ever growing computational power of the computers, important efforts in method development and the pressure of the scientific community demanding for better and more accurate calculations on bigger, more realistic systems. The progress can often be recognized in the evolution of the research topics of scientists that have been working in the field over the last decades, as the sum of such individual contributions makes science advance.

In this contribution, we want to illustrate some aspects of the evolution of quantum chemistry following the trail of the scientific career of Prof. Rosa Caballol. The description of the landscape starts in the early 1970s, when the semiempirical methods were in plain development and reaches present days with the latest developments in the difference dedicated CI (DDCI) method. Along the trail, we will see how the molecules that are studied gradually increase in complexity throughout the years, as illustrated in Figure 1.[6-10]

image

Figure 1. Molecules of increasing complexity. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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The Early Years–Semiempirical Calculations

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Early Years–Semiempirical Calculations
  5. Moving to Ab Initio Calculations
  6. Difference Dedicated Configuration Interaction
  7. Orbital Manipulations
  8. Analysis of the Magnetic Coupling
  9. Other Applications of DDCI
  10. Perspective: Limits of the Methods and How to Overcome These
  11. Biography

Rosa Caballol's research career begun in the early 1970s with the development of her PhD work in the group of professor Carbó. At that time, the study of the electronic structure of open shell systems within a spin-restricted setting took a large step forward. Many articles were published in these years that proposed energy expressions for molecules with more than one unpaired electron. These energy expressions opened the way to self-consistently optimize the orbital expansion coefficients, and hence, obtain a generalized open shell SCF theory.[6, 11-13] Computer power was still modest and typical applications of restricted open-shell Hartree–Fock theory were focussed on what are nowadays considered very small model systems. During the PhD of Rosa Caballol coupling operators were developed in the group of Carbó and applied to the excited singlet state of the He atom.[6] Simultaneously, the more practical semiempirical schemes were applied to study the conformational degrees of freedom of ethylamines,[14] the interaction of molecular fragments such as inline image, and ground state reactivity (intermolecular proton transfer in glycine). Excited states in ketene and diazomethane were studied and excited state reactivity in small molecules like formaldehyde, formic acid, methyl cyanide, and methylacethylene was also considered.[7] Most calculations were performed with the complete neglect of differential overlap or intermediate neglect of differential overlap approximations.

In the next stage along the trail, we are following, the spin-restricted formalism was (temporarily) set aside and organic (bi)radicals were intensively studied within the unrestricted Hartree–Fock scheme applying the MINDO/3 parametrization.[15] In collaboration with Poblet, Sarasa, Olivella, Canadell, and others, Rosa Caballol performed a long series of studies aimed at the understanding of organic radical reactivity.[8, 15-19] In the study of the conformational preference of the bicyclobutyl radical one can already recognize the fundamental interest that drives Rosa Caballol's research, that is, the rationalization of the results obtained. The relative stability of the different confomers was analyzed through an energy partition scheme and by a fragment molecular orbital analysis.[8]

A second example of this interest is the study of isomerization and fragmentation of the C3H6O2+ radical cation.[19] The calculation of a large collection of (local) minima and transition states on the complex potential energy surface permitted the authors to conclude that fragmentation is preceded by internal rearrangements. Based on the rate constants of the reactions calculated with the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and due to the complexity of the surface, a warning was issued concerning the interpretation of experiments with deuterium labeling, illustrating the role that theory/computation can play in this field.

Moving to Ab Initio Calculations

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Early Years–Semiempirical Calculations
  5. Moving to Ab Initio Calculations
  6. Difference Dedicated Configuration Interaction
  7. Orbital Manipulations
  8. Analysis of the Magnetic Coupling
  9. Other Applications of DDCI
  10. Perspective: Limits of the Methods and How to Overcome These
  11. Biography

The increasing computer power and the development of efficient algorithms brought the ab initio treatment of relevant molecular systems within reach. In the second half of the 1980s, we can find the first applications of the CIPSI (configuration interaction by perturbation with multiconfigurational zeroth-order wave function selected by iterative process) algorithm along our trail. This combined variational/perturbational scheme to account for electron correlation was first applied to XH4+ (X = Si, Ge, Sn) molecules,[20] but soon the step was made to transition metal chemistry. After the study of coordination complexes of carbon dioxide and copper or molybdenum,[21, 22] Rosa Caballol moved to binuclear complexes and the magnetic coupling research line emerged.[23-25]

Simultaneously, to these more applied investigations, she got involved in the development of new algorithms and accurate quantum chemical methods. Important contributions were made to the development of efficient truncated or full CI programs[26, 27] and the correction for size-consistency errors in of truncated MR-CI approaches.[28, 29] Another relevant contribution of this period was the obtention of improved virtual orbitals[30] to increase the efficiency of CI calculations and the development of dressing schemes of the singles and doubles CI matrix.[31] Many of these methodological developments were implemented in the SCIEL code,[32] which also conceived one of the first implementations of the DDCI method, which will be discussed in the next section.

However, let us first look with more detail to the mentioned methodological developments in the field of the (MR) CI designed to correct the size consistence error intrinsic to this method, and that establishes a bridge between CI and CC approaches. The size-consistency problem is one of the most important factors that explains why CI is relatively unpopular in the quantum chemical community. Several simple schemes have been proposed to correct the CI energy for the size-consistency error, among which the Davidson correction is one of the most applied corrections. These schemes are, however, rather approximate and may not give the desired accuracy, especially in multiconfigurational situations. In the early 1990s, a dressing scheme was proposed by Daudey et al.[33] that rigorously cancels the contribution of the unlinked contributions and obtain a strict size-consistent CI for any truncation level. For states that are dominated by a single Slater determinant, the dressing is defined as

  • display math(1)

where inline image is an excited determinant outside the model space and C0 and CJ are the CI coefficients of the respective determinants. As can be seen this dressing scheme is iterative as it depends on the coefficients of the final CI vector. Hence, it is generally recognized as the self-consistent-size-consistent CI, (SC)2-CI method.[34, 35]

In its original formulation, the scheme was set up for a wave function that is dominated by a single determinant inline image, but 1 year later, Malrieu et al. generalized the method for the multiconfigurational case.[29] The dressing is slightly more complicated in this case, but the essential feature of the single-determinantal situation is maintained: The effect of the triples and quadruples on the effective energies of the singles and doubles is the same ( inline image) as the effect of the singles and the doubles on the reference energy.[29] Any determinant outside the multideterminantal reference space can interact with various determinants in this reference space. Therefore, a weighting factor was introduced to convert the dressing into a weighted sum of different dressings that ensures the separability of the energy.

In line with this multiconfigurational dressing method, it was shown that the CI matrix can also be dressed in such a way that it becomes equivalent to a state-specific MR-CC calculation.[31] This type of dressing has the advantage that there is no need of going through the solution of a set of nonlinear equations, but one only needs to diagonalize the dressed CI matrix, which is often a computationally more simple and numerically more stable procedure. In this way, a bridge was made between the (MR) CI and CC techniques.

Difference Dedicated Configuration Interaction

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Early Years–Semiempirical Calculations
  5. Moving to Ab Initio Calculations
  6. Difference Dedicated Configuration Interaction
  7. Orbital Manipulations
  8. Analysis of the Magnetic Coupling
  9. Other Applications of DDCI
  10. Perspective: Limits of the Methods and How to Overcome These
  11. Biography

The fundaments of DDCI were laid by Malrieu in the 1960s.[36] He established that many terms in the nth-order correction to the energy are strictly the same for the ground state and any of the vertically excited states when a common orbital set is used. This means that while the calculation of the total energy requires the full summation over all excited Slater determinants, the excitation energies can be obtained from a smaller subset. This idea is one of the ingredients of the CIPSI method,[37] one of the first successful computational schemes to treat electron correlation effects in a multiconfigurational setting, and the predecessor of the spectroscopy-oriented CI (SORCI) of Neese[38] and the complementary space perturbative approach (CSPA) of Barone and et al.[39, 40]

The direct perturbative calculation of the energy difference was applied in the landmark article by de Loth et al.[41] to analyze the origin of the singlet-triplet energy difference in the copper acetate dimer Cu2(CH3COO)4(H2O)2. Using a reference wave function containing the determinants with up-down and down-up coupling of the unpaired electrons of Cu2+, it was demonstrated that the only differential contributions to the second-order correction to the energy arise from the determinants with not more than two changes in the orbitals that are doubly occupied (inactive) or empty (virtual) in the reference wave function. The second-order contribution of these determinants was analyzed class-by-class. The additivity of the perturbative contributions allowed the authors to give a clear-cut analysis of the mechanisms that determine the singlet-triplet splitting in this system.

The variational treatment of the list of determinants that differentially contribute to the second-order correction, that is, a CI calculation, further improves the precision of the procedure, although it must be kept in mind that the contributions of the different classes of excitations are no longer additive and that the selection criterion is strictly speaking only valid for PT. Broer and Maaskant performed such variational calculation when they analyzed the magnetic coupling in [Cu2Cl6]2−,[42] and performed in this way one of the first DDCI calculations, but the method was put on a firm basis a couple of years later by Miralles et al.[23, 24] In the first article, the degenerate reference space was analyzed and in the second one, the method was generalized to the nondegenerate case.

In the calculations of de Loth et al., the determinants of the reference wave functions are strictly degenerate. In such cases, only determinants with not more than two differences in the orbital occupations of inactive and virtual orbitals contribute to the energy difference, as stated earlier. In the general cases of nondegenerate reference spaces, there are more determinants that should be considered. When inline image and inline image belong to the reference space and inline image represents a determinant obtained from a single or double electron replacement with respect to any of the reference space, quasi-degenerate PT establishes the effect of the external determinants on the coupling between inline image and inline image with

  • display math(2)

In principle, the sum runs over all single and double excited determinants. Once the matrix elements of the effective Hamiltonian are calculated, the matrix can be diagonalized to obtain the energies of the states spanned by the determinants inline image, inline image,  inline image, and so forth, belonging to the reference space. However, not all determinants are relevant for the energy differences between the states. The double excitations from inactive to virtual orbitals give zero contribution to the nondiagonal elements. With inline image ( inline image and inline image are annihilation and creation operators, and the indices inline image and inline image refer to inactive and virtual orbitals, respectively) it is easily shown that the contribution of these determinants to the nondiagonal elements is strictly zero:

  • display math(3)

Because the number of different columns in inline image and inline image is larger than two when inline image. inline image does contribute to the diagonal matrix elements of the effective Hamiltonian, but this contribution is equal for all elements as it involves integrals that depend solely on the inactive and virtual orbitals, and hence, the same for all determinants of the reference space

  • display math(4)

Based on these considerations, the list of the multideterminantal reference CI of singles and doubles (MR-CISD) can be restricted to the determinants of the reference space and the excited determinants that involve at most three changes in the occupations of the inactive and virtual orbitals when one is only interested in energy differences. This subset of the full MR-CISD is known as the difference dedicated CI (DDCI) or sometimes also referred to as DDCI3.

The determinants generated by double replacements from inactive to virtual orbitals constitute normally the most numerous class of determinants in the MR-CISD expansion. The elimination of these determinants implies a very important computational saving and extended the applicability range of MRCI techniques far beyond the level of small model systems. Moreover, the DDCI wave function is less susceptible to size-consistency errors.

The variant used in the work of de Loth et al. and Broer and Maaskant is nowadays known as DDCI2. The further reduction of the CI space by eliminating the determinants with two electrons in the virtual orbitals or two holes in the occupied orbitals leads to a CAS + singles expansion (CAS+S) when a CAS is used as reference space (as is done in almost all cases). Restricting the external determinants to those with at most one change in the occupation of the occupied and virtual orbitals gives rise to DDCI1. Note, however, that the nomenclature of CAS+S and DDCI1 are not always consistently used and that one should be careful in comparing the results obtained with different computer implementations and data reported in the literature.

Orbital Manipulations

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Early Years–Semiempirical Calculations
  5. Moving to Ab Initio Calculations
  6. Difference Dedicated Configuration Interaction
  7. Orbital Manipulations
  8. Analysis of the Magnetic Coupling
  9. Other Applications of DDCI
  10. Perspective: Limits of the Methods and How to Overcome These
  11. Biography

The need for a common set of orbitals puts some conditions on the DDCI calculations. It is not always wise to use the CASSCF orbitals optimized for the ground state as this can lead to a bias toward the ground state and a poor description of the excited states. Whereas this is less critical when energy differences are calculated between states with the same orbital occupations (as in magnetic coupling), it is of utmost importance in studies of electronic absorption spectroscopy when electrons are excited to higher lying orbitals.

Natural orbitals

In these cases, it is highly recommended to use an average molecular orbital set. These orbitals can be obtained by summing the CASSCF density matrices of the different states and subsequently diagonalize the matrix to obtain average natural CASSCF orbitals. An alternative procedure is the construction of the density matrices after the DDCI step and diagonalize the matrix to obtain average DDCI natural orbitals. These average orbitals form the molecular orbital basis set for a new DDCI calculation and the procedure is repeated until the energy difference between the states is converged. In this way, the results are no longer dependent on the initial set of orbitals. This procedure is known as iterative DDCI (IDDCI) and was first applied on the excitation energies of CH2, CH2+, and SiH2, and the avoided crossing of LiF.[43] The comparison of DDCI and IDDCI results to full-CI benchmarks showed that IDDCI significantly improved the results.

Dedicated orbitals

The aforementioned canonical and natural orbitals are the most commonly used representations of the molecular orbital set, but not the only ones. Any unitary transformation leaves the wave function invariant and leads to an equally valid set of orbitals that can serve specific purposes. Over the years, an important part of the research of Rosa Caballol has been dedicated to design unitary orbital transformations that can help to reduce the computational cost of the DDCI calculation or to make easier the interpretation of the wave functions. A good example of how orbital manipulations can speed-up calculations is given by the dedicated orbital transformation, first reported in 1991[44] and further worked out in 2000.[45] In these studies, a strategy was developed to reduce the length of the CI expansion by eliminating the MOs that are less relevant for the energy difference between two states. A standard way proceeds through the ordering of the MOs by energy and elimination of the highest lying ones, but this is not always the optimal choice. A better strategy is to construct the density matrices for both states at a lower level of calculation (second-order PT, CAS+S, DDCI2) and then diagonalize the difference density matrix inline image. The eigenvalues of inline image give a measure to what extent the corresponding eigenvector contributes to the changes in the density, when the system is excited from state A to state B, and hence, to what extent the eigenvector is relevant to the energy difference. The eigenvalues are normally referred to as participation numbers and the eigenvectors received the label of dedicated orbitals. After ordering the orbitals by the participation numbers and eliminating the dedicated orbitals with the lowest participation numbers, a full DDCI calculation can be performed in a much smaller subspace of the full MO space without practically any loss of precision.

Projected orbitals

Many DDCI applications focus on the singlet-triplet splitting in systems with two spatially separated unpaired electrons, that is, the coupling of two inline image spin angular momenta. The generally accepted model is that this coupling is intermediated by the closed shell bridge that separates the two magnetic centers. DDCI gives a correct description of the coupling provided that a CASSCF wave function is used as reference, where the active space comprises the unpaired electrons and the magnetic orbitals. This recipe is still applicable when systems are considered with three or four unpaired electrons, but become cumbersome for systems with more unpaired electrons. As an alternative for the full DDCI calculation, Malrieu and Calzado suggested to extend the active space with ligand orbitals and to restrict the subsequent treatment of the dynamic electron correlation to the CAS+S level.[46, 47] The question now is what orbitals should be added? Again the canonical orbitals are not the best choice, as they are usually delocalized over several centers and it may be difficult to identify the orbital(s) that are most effective in the transmission of the coupling of the spin angular momenta. Different selection criteria have been published by Gellé et al.[48] and Calzado and Malrieu,[46, 47] but the projected orbital solution of Caballol and coworkers seems to be the most physically grounded criterion.[49]

The projection strategy is based on the understanding that the tails of the magnetic orbitals onto the bridging ligands provide the ideal shape to construct a model vector for projection. Hence, for each magnetic orbital, a model vector is defined using the tails of that specific magnetic orbital. These model vectors are orthonormalized to each other and then projected onto the MO space of the inactive orbitals. Among the original inactive orbitals, the one with the largest overlap with the projected vectors are eliminated and the other inactive orbitals are orthogonalized. This unitary transformation of the inactive MO space leads to a set of projected orbitals that are not only strongly localized on the bridging ligand but also have the optimal shape to interact with the unpaired electrons.

CAS+S based on the extended CAS (magnetic + projected ligand orbitals) significantly improves the singlet-triplet splitting with respect to a standard CAS+S treatment and is computationally much cheaper than the full DDCI calculation. The method gave excellent results for a series of binuclear inline image complexes[50] and a hetero binuclear complex with inline image and inline image,[51] but is not as general applicable as foreseen.[52]

Localized orbitals

Accurate calculations are a necessary but not sufficient condition to obtain useful results from a quantum chemical study. Equally important is the capacity to boil down the accurate numbers to simple concepts that describe the essential physics of the problem. The very long DDCI wave function—any medium-sized application leads to CI expansions with inline image determinants—can give very accurate results, but it is obviously not the ideal object to analyze. Effective Hamiltonian theory provides an important link between accuracy and simple models. This theory maps the DDCI results (or any other accurate multiconfigurational wave function) onto a smaller space in such a way that the diagonalization of the smaller space leads to exactly the same energies as obtained in the DDCI calculation and that the corresponding eigenvectors are the projections of the DDCI wave functions on the model space. In this way, accurate results are combined with compact, easy to analyze wave functions.[53]

The interpretation of the physics in the model space is usually done with orbitals that are localized on atoms or fragments and not with those that are delocalized over all the centers of the systems as happens with the canonical MOs. The projection method discussed in the previous subsection is a powerful technique to localize the active orbitals anywhere on the system through the definition of the different model vectors. Once the desired form of the active orbitals is obtained, the wave function is re-expressed in the new set of localized orbitals permitting an orthogonal Valence Bond reading of the results. Apart from transition metal binuclear complexes, these unitary transformations of the active orbitals have also been applied in the analysis of the CASSCF and DDCI wave functions of small polyoxometalates, organic biradicals, and biomimetic systems.[10, 54, 55] This projection technique is of course not the only way (and probably not the most generally applicable) to obtain localized orbitals, classical schemes such as Boys, Pipek-Mezey, or Ruedenberg schemes should also be mentioned. Closely related to the analysis of the wave function with projected orbitals is the study by Pierloot of the bonding of NO to metalo-porphyrines performed with orbitals obtained through the Cholesky localization scheme.[56, 57]

Analysis of the Magnetic Coupling

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Early Years–Semiempirical Calculations
  5. Moving to Ab Initio Calculations
  6. Difference Dedicated Configuration Interaction
  7. Orbital Manipulations
  8. Analysis of the Magnetic Coupling
  9. Other Applications of DDCI
  10. Perspective: Limits of the Methods and How to Overcome These
  11. Biography

Inspired by the work of de Loth and coworkers[41] and triggered by the accurate magnetic couplings provided by DDCI wave functions,[58] Malrieu, Caballol, and coworkers analyzed the magnetic coupling in great detail in a series of articles published between 2002 and 2011.[52, 59-63] These studies have been reviewed recently,[53] and here, we will only illustrate the main conclusions with an example to show the power of the applied methodology for analysis. First, the main mechanisms that contribute to the coupling between two inline image spins will be exposed and next it will be shown how a simpler description can be recovered through an effective Hamiltonian.

Mechanisms

Experimentally, it has been established that the singlet-triplet splitting in [N,N′,N″-trimethyl-1,4,7-triazacyclononane2Cu2(μ-1,3-N3)2]2+ (Cu-azido) is around 800 cm−1 with the open-shell singlet as lowest state. To a good approximation, the replacement of the bulky external ligands with NH3 groups does not alter the coupling of the inline image spin moments of the Cu2+ ions and this model will be used to analyze the origin of the 800 cm−1 energy gap. The orthonormal orbitals a and b, given in Figure 2, define a subspace of the full valence space that can be used to obtain the basic wave functions of the singlet (S) and triplet (T) states:

  • display math(5)
image

Figure 2. Gerade and ungerade magnetic orbitals of Cu-azido. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Taking the energy of the singlet as reference, the energy difference between the singlet and triplet ( inline image) is equal to inline image, twice the exchange integral involving orbitals a and b, inline image. This always leads to a triplet ground state and cannot be the final answer. In addition to inline image and inline image—normally labeled as the neutral determinants (no change of the oxidation state of the magnetic centers with respect to the formal one)—the subspace also contains the ionic inline image and inline image determinants. The diagonalization of the 4 × 4 matrix

  • display math(6)

leads to the following eigenvectors for the triplet and the lowest singlet:

  • display math(7)

and EST now reads

  • display math(8)

In the inline image matrix and the subsequent equations, U is the on-site repulsion parameter defined as relative energy of ionic determinants with respect to the neutral ones, and inline image is the hopping parameter. Under the assumption of inline image, the second term in the energy difference can be simplified 1 to

  • display math(9)

The second term is known as the kinetic exchange and favors the antiparallel alignment of the spin moments, contrary to the first term, the direct exchange, which stabilizes the triplet state.

The CI calculation in the space spanned by the neutral and ionic determinants gives direct access to the numerical estimates of these two antagonist interactions. The direct exchange contributes +12 cm−1 to the energy difference, whereas the kinetic exchange is much larger and of opposite sign, −94 cm−1. The sum of the two terms is approximately −80 cm−1,that is, just 10% of the total splitting and hence there must be other important contributions involving determinants with holes or electrons outside the small valence subspace considered up to now.

The first class of external determinants that play an important role are those included in the CAS+S calculation and comprise spin and charge polarization effects. 2 The charge polarization can be defined as the response of the electrons in the closed shells to the electron transfer from one magnetic site to the other. The minimal space description includes the charge transfer process through the presence of the ionic determinants in the wave function, but the energy of these determinants is largely overestimated and the process is not as effective as it should be. The determinants obtained from a single excitation from inactive to virtual orbitals do not interact with the neutral determinants, but do lower the effective energy of the ionic determinants, which gain weight in the wave functions. As the ionic determinants can only contribute to the singlet wave function [cf. Eq. (7)], the charge polarization favors the antiparallel coupling of the spin moments. For the example molecule the magnetic coupling is enhanced by −221 cm−1.

The spin polarization introduces extra spin density around the magnetic center through determinants with one hole in the inactive and one electron in the virtual orbitals (see Fig. 3). In these determinants, the unpaired electrons external to the reference space are coupled to a triplet, whereas the two electrons in the reference space are also parallel aligned. These two triplet coupled electron pairs can be coupled to singlet or to triplet. 3 This means that the effect of the spin polarization is not easily anticipated, it can both contribute in a ferro- and an antiferromagnetic manner. For the Cu-azido complex considered here, the contribution of the spin polarization is antiferromagnetic, −59 cm−1.

image

Figure 3. Schematic representation of the different types of determinants that contribute to the coupling of two localized magnetic spin angular moments. a and b refer to the active orbitals. The two determinants depicted in the reference space are examples of neutral (left) and ionic (right) determinants. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Adding the two contributions included in CAS+S calculation to the estimate obtained with the reference wave function, the magnetic coupling becomes −362 cm−1. This is already half way to the experimental estimate, but some effects are still missing. In the first place, one may add the double ligand-to-metal charge transfer (LMCT) and the double metal-to-ligand CT (MLCT) determinants to the wave function. 4 The contribution of these determinants is in general antiferromagnetic, but rather modest in size given the high energy of the determinants. In Cu-azido the contribution is just −13 cm−1.

Much more important are the determinants that combine a single excitation from inactive to virtual orbitals with either an inactive to magnetic or a magnetic to virtual electron replacement. In many applications of the DDCI method to magnetic coupling problems, only the inclusion of this class of determinants provides (nearly) quantitative agreement with experiment. As can be seen in Figure 3, these classes of determinants combine LMCT and MLCT excitations with single excitations from inactive to virtual orbitals. The latter are connected to the orbital optimization process. The coupling of the single inactive-to-virtual excitations to the LMCT and MLCT excitations provide relaxed CT excitations. The direct interaction of the determinants with those of the reference space is small, but coupling between external determinants in the CI makes that the MLCT and LMCT gain important weight in the wave function.[52] The nature of the contribution is variable although the relaxation of the LMCT tends to increase the antiferromagnetic character of the coupling and the relaxed MLCT usually acts in the opposite direction. The addition of these two classes of determinants to the CI expansion strengthens the magnetic coupling by −427 cm−1 and brings the theoretical estimate (−802 cm−1) in good agreement with experimental data that indicate that J is at least −800 cm−1. The disentanglement of the two types of determinants is rather complicated because the full contribution only appears when the determinants are coupled to all other external determinants, that is, the contributions are severely underestimated when attempting to separately include the relaxed MLCT or relaxed LMCT determinants.

The results of the earlier discussion of the mechanisms are summarized in Figure 4. The graph shows the importance of including the different mechanism in the wave function. Because of the nonadditivity of the different contributions, it is not possible to rigorously separate them and assign an absolute importance to each mechanism individually. The graph clearly shows how the first two contributions due to the reference space give at most a qualitative description of the coupling and that the quantitative correct picture only emerges, when all determinants that span the DDCI space are considered.

image

Figure 4. Increase of the magnetic coupling in Cu-azido by subsequently including the different mechanisms to the wave function. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Valence-only model

The earlier described procedure provides us with a detailed decomposition of the magnetic coupling, but this may not be the most appealing representation of the mechanism. To make the communication more fluid with the areas of chemistry outside the circle of quantum chemists, it may be preferable to avoid the extensive use of determinants and other jargon and try to come back to the intuitive picture provided by the minimal valence space. This consists of two rather simple antagonist contributions, namely the direct exchange integral and the kinetic or super exchange. It was shown that the bare description in the reference space gives a reasonable first approximation but does not correctly reproduce the magnitude of the coupling.

To get a more quantitative agreement, one can project the full DDCI result onto the reference space with effective Hamiltonian theory.[61] The most rigorous way consists of selecting the DDCI wave functions inline image, whose projections on the reference space has the highest norms. The energies and wave functions of these roots are then used to construct an effective Hamiltonian of the dimension of the reference space, but with the same energy eigenvalues as the selected DDCI roots. The corresponding eigenvectors coincide with the projections of the DDCI wave functions on the reference space. The effective Hamiltonian is written in the basis of the determinants inline image that are contained in the reference space (two neutral and two ionic determinants) and the matrix elements are calculated with the following formula

  • display math(10)

where Ek are the DDCI energies and inline image the orthogonalized projections of the DDCI wave function on the reference space.

The comparison of the numerical matrix elements of the effective Hamiltonian with the matrix given in Eq. (6) allows us to identify the effective values of Kab, tab, and U. The diagonalization of the effective Hamiltonian gives the same energy splitting between singlet and triplet as in the full DDCI calculation, but now we can directly relate the magnetic coupling of the correct strength with concepts as simple as direct and kinetic exchange using Eq. (9) with the parameters derived from the effective Hamiltonian procedure. These parameters are called effective parameters because they are no longer the bare parameters and they no longer have the clear-cut physical meaning they had in the bare reference space calculation, but are screened with the other effects due to determinants outside this minimal valence space.

This screening may lead to unexpected, or maybe better to say, counterintuitive values of the parameters. The most striking one is that the effective direct exchange integral may become negative favoring antiferromagnetic alignment of spins. The general tendency is that inline image is smaller than the bare U of the minimal space calculation and that the effective hopping parameter normally increases: inline image.

Other Applications of DDCI

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Early Years–Semiempirical Calculations
  5. Moving to Ab Initio Calculations
  6. Difference Dedicated Configuration Interaction
  7. Orbital Manipulations
  8. Analysis of the Magnetic Coupling
  9. Other Applications of DDCI
  10. Perspective: Limits of the Methods and How to Overcome These
  11. Biography

Although the study of the magnetic coupling has long been (and probably still is) the major area of application, DDCI has also been used to study vertical excitations spectra,[64, 65] ionization energies, and weakly avoided crossings,[66] potential energy surfaces in photochemistry,[67, 68] relative stability of different biomimetic adducts,[54] hopping probabilities in doped systems or mixed valence compounds,[69-72] conduction properties of 1D organic chains,[73, 74] and the electronic structure of systems with magnetoresistance effects,[75-77] among others.

In most cases, the application of DDCI follows the same line as for the magnetic coupling outlined earlier. That is, the orbitals with unpaired electrons in initial or final state should be contained in the active space of the CASSCF calculation that supplies the multiconfigurational reference wave function. When initial and final states possess different electronic configurations, one should be careful with the choice of the MO set to express the Slater determinants in the CI function to avoid a bias toward one of the states. The best solution is to use average (natural) orbitals.[43]

The fact that DDCI does not consider the Slater determinants generated by the double excitations from inactive to virtual orbitals limits the method to vertical excitation energies only. The total energies calculated at different geometries can, in principle, not be compared and the application of DDCI to study weakly avoided crossings or potential energy surfaces needs some additional action. The effect of the double inactive-to-virtual excitations can be incorporated in different ways. For instance, one can calculate the energy correction of these determinants with (second-order) PT and add it to the DDCI energy for each state. Alternatively, a reference potential energy curve for one of the electronic states can be calculated with a high-level computational scheme such as CCSDT(T). The curves for the other electronic states are then obtained by adding the DDCI energy difference to the CCSD(T) reference energy. This procedure can only be applied when one of the states of interest can be considered as monodeterminantal to a high extent, enabling a correct description of the reference curve with a single determinantal reference method. For instance, the quartet state of a system with three unpaired electrons is basically monodeterminantal and can accurately be calculated with CCSD(T). However, to describe the two doublet states necessarily a reference wave function with two or three determinants has to be applied. Figure 5 illustrates how the potential energy surface for one the doublets can be obtained by the application of

  • display math(11)
image

Figure 5. Construction of an excited state potential energy surface with DDCI. The blue curve represents the energy of a basically monodeterminantal electronic state calculated with (for instance) CCSD(T). The red curve represents the potential energy curve of the excited state obtained by adding the DDCI energy difference between the blue and red states to the CCSD(T) energy, see Eq. (11). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Perspective: Limits of the Methods and How to Overcome These

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Early Years–Semiempirical Calculations
  5. Moving to Ab Initio Calculations
  6. Difference Dedicated Configuration Interaction
  7. Orbital Manipulations
  8. Analysis of the Magnetic Coupling
  9. Other Applications of DDCI
  10. Perspective: Limits of the Methods and How to Overcome These
  11. Biography

Despite the fact that the computational cost is heavily reduced by eliminating the double inactive-to-virtual excitations, the application of DDCI is still limited to small or at most medium sized systems. There are basically two bottlenecks in the method that makes the application to larger systems not as easy as one would desire. The first one concerns the number of inactive and virtual molecular orbitals. The DDCI method scales as the third power in the number of molecular orbitals, and hence, either the increase of the basis set quality or the increase of the size of molecule severely conditions the applicability of DDCI to larger systems. The second bottleneck concerns the active space. Although DDCI scales only linearly on the size of the CAS, the dimension of the CAS itself grows exponentially with the number of active orbitals. Therefore, the large majority of the applications of DDCI reported in the literature are limited to systems with at most four electrons in four active orbitals.

In the light of these limitations, many initiatives are being taken to stretch the limits of applicability of the DDCI approach. One of the earliest ones has already been mentioned in Section Dedicated orbitals, and is based on ordering the MOs as function of their importance for the property under study, normally the energy difference between two states. This dedicated orbital transformation is a rather simple yet efficient way to eliminate the less important inactive and virtual molecular orbitals and hence, to reduce the computational cost of the DDCI calculation.

More elaborate schemes have been developed by Neese and by Barone and coworkers. Both are based on a combination of variational and PAs to estimate the effect of the determinants external to the CAS. The SORCI[38] of Neese makes a preselection of the determinants that only weakly interact with the determinants of the CAS and calculates the effects of these with second-order PT, whereas the more important ones are included in the CI. The CSPA of Barone et al.[39, 40] combines orbital transformations with the mixed variational/perturbative electron correlation treatment. The method first localizes the orbitals on different fragments of the system (typically, magnetic centers, bridge, and external ligands), and then orders the molecular orbitals within each fragment by increasing importance for the property under study. The DDCI is performed in a small subset of the whole MO space and the remainder is estimated with PT.

The cost of the DDCI calculation can also be reduced by selecting the determinants based on topological arguments for the orbitals and on numerical values of integrals as is done in the excitation selected CI (EXSCI) due to Maynau, Calzado, and coworkers.[78-81] As in the CSPA method, the EXSCI approach relies on a set of localized occupied and virtual orbitals. The transformation is made in such a way that all the molecular orbitals can be labeled with well-established chemically relevant labels, such as core, lone-pair, σ(C[BOND]H) or σ(C[BOND]C), and so forth. Orbitals are then excluded from the CI on topological arguments; for example, σ(C[BOND]C) orbitals located far away from the magnetic centers can be safely disregarded. On top of this selection, the method also applies a filter based on the numerical value of exchange integrals. Determinants external to the CAS are only included when the electron replacement involves interacting orbitals. Orbitals m and n are considered to interact, when the corresponding exchange integral Kmn is larger than a certain threshold. The combination of these two criteria provides a very powerful scheme and illustrative applications of the method can be found in the Refs. [81-83].

In summary, we can see that the DDCI method is an accurate and versatile strategy to calculate vertical excitation energies. The computational cost prevents the direct application to large, real-world molecules, but the developments based on localized orbitals and smart selection criteria makes it possible to address questions that are certainly relevant to nowadays (computational) chemistry. When one of the states of interest has a basically monodeterminantal character, the DDCI method can also be applied to study energy differences at different geometries as in photochemistry.

The methods proposed in the 1990s concerning a dressed (multiconfigurational) CI and its bridge to CC based techniques for treating open-shell systems have been recognized by the scientific community,[4, 84] but unfortunately not explored in an extensive way. It would be highly desirable to see how these methods perform in cases that go beyond the strict model systems[85] and to what extent they can provide an alternative to the DDCI method, which—although much less than standard MR-SDCI—suffers from the size-consistency problem and is only applicable to vertical excitation energies.

  • 1

    Taylor expansion: inline image

  • 2

    CAS+S also includes the determinants resulting from pure single excitations. The contribution of these is very small.[53]

  • 3

    Quintet coupling is of course also possible. However, this is not relevant here, as the electronic states have either singlet or triplet spin symmetry.

  • 4

    In organic biradical systems, these excitations correspond to double electron transfers from the inactive into the magnetic orbitals, and from the magnetic into the virtual orbitals.

Biography

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Early Years–Semiempirical Calculations
  5. Moving to Ab Initio Calculations
  6. Difference Dedicated Configuration Interaction
  7. Orbital Manipulations
  8. Analysis of the Magnetic Coupling
  9. Other Applications of DDCI
  10. Perspective: Limits of the Methods and How to Overcome These
  11. Biography
  • Image of creator

    Rosa Caballol received the degree in Chemical Engineering in 1972 from the Institut Químic de Sarrià (IQS, Institute of Chemistry of Sarrià) in Barcelona (Spain). Four years later, she finished her Ph. D. work at the IQS. Simultaneously, she graduated in Chemical Sciences at the University of Barcelona in 1976 and received the Ph. D. degree from the same University in 1983. During her Ph. D., supervised by Prof. Ramón Carbó, coupling operators were developed for the energy expression of open shell systems in a spin restricted setting. In 1977, she became a lecturer of the Chemistry faculty in Tarragona (Spain) and joined the Quantum Chemistry group of Prof. Enric Canadell. After being appointed as assistant professor in 1983, she directed the Quantum Chemistry group for approximately twenty years. In the 1980s, contacts with the research group of Malrieu, Daudey and collaborators were established and rendered permanent during various postdoctoral stays in Toulouse. In 1993, she was appointed as a full professor in Tarragona. The list of academic responsibilities includes the positions of vice-dean (1994–1995) and dean (1995–1998) of the Faculty of Chemistry, director of the Institut d'Estudis Avançats (Institute of Advanced Studies) (1999–2002), delegate of the rector for international relations (2002–2005) and secretary-general of the University Rovira i Virgili, Tarragona (2005–2006). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]