Analysis of the magnetic coupling in binuclear systems. III. The role of the ligand to metal charge transfer excitations revisited

Analysis of the magnetic coupling in binuclear systems. III. The role of the ligand to metal charge transfer excitations revisited Carmen J. Calzado,1,a兲Celestino Angeli,2David Taratiel,3Rosa Caballol,3and Jean-Paul Malrieu4 1Departamento de Química Física, Universidad de Sevilla, Profesor García González s/n, E-41012 Sevilla, Spain 2Dipartimento di Chimica, Università di Ferrara, via Borsari 46, I-44100 Ferrara, Italy 3Departament de Química Física i Inorgànica, Universitat Rovira i Virgili, Marcel lí Domingo s/n, E-43007 Tarragona, Spain 4Laboratoire de Physique et Chimie Quantiques, Université Paul Sabatier, 118 Rte de Narbonne, 31062 Toulouse Cedex, France 共Received 26 March 2009; accepted 4 July 2009; published online 31 July 2009兲 In magnetic coordination compounds and solids the magnetic orbitals are essentially located on metallic centers but present some delocalization tails on adjacent ligands. Mean field variational calculations optimize this mixing and validate a single band modelization of the intersite magnetic exchange. In this approach, due to the Brillouin’s theorem, the ligand to metal charge transfer 共LMCT兲excitations play a minor role. On the other hand the extensive configuration interaction calculations show that the determinants obtained by a single excitation on the top of the LMCT configurations bring an important antiferromagnetic contribution to the magnetic coupling. Perturbative and truncated variational calculations show that contrary to the interpretation given in a previous article 关C. J. Calzado et al., J. Chem. Phys. 116, 2728 共2002兲兴 the contribution of these determinants to the magnetic coupling constant is not a second-order one. An analytic development enables one to establish that they contribute at higher order as a correlation induced increase in the LMCT components of the wave function, i.e., of the mixing between the ligand and the magnetic orbitals. This larger delocalization of the magnetic orbitals results in an increase in both the ferro- and antiferromagnetic contributions to the coupling constant. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3185506兴 I. INTRODUCTION An enormous effort has been devoted in the past to the understanding of the factors controlling the interactions between the spin moments on magnetic systems. This effort has been driven by the aim to rationalize the macroscopic magnetic behavior and to help in the design of new materials with particular properties. Since the seminal work of Anderson,1,2many models have been proposed to explain magnetic exchange interactions. These models intend to express the magnetic coupling constant Jon the basis of a reduced number of parameters, which take into account the main physical effects of the coupling. Most of them only deal with the magnetic orbitals and the unpaired electrons in their simplest version with only one electron per metallic center. For S=1 2binuclear systems, these one-band models reduce to two electrons on two magnetic orbitals aand b, and Jcan be expressed in the Anderson mechanism as a function of only three parameters: the direct exchange between the active orbitals Kab, the one-electron transfer between metal centers tab, and the on-site Coulomb repulsion U, J=2Kab −4tab 2/U=JF+JAF,共1兲 where the former term represents the ferromagnetic 共F兲direct exchange contribution to the coupling and the latter takes into account the delocalization effects favoring the antiferromagnetic 共AF兲kinetic exchange contribution. This AF component is driven by the coupling between the valence bond 共VB兲neutral N=兵兩ab ¯ 兩,兩ba ¯ 兩其 determinants and the VB ionic I=兵兩aa ¯ 兩,兩bb ¯ 兩其 determinants. These configurations define a minimal valence complete active space 共CAS兲. The other models introduce also orbitals located on the ligands bridging the metallic centers, that is, a two-band model, recently revisited by Van den Heuvel and Chibotaru.3 The simplest version adds only one ligand orbital, resulting in a model with four electrons in three orbitals. In this case, the ligand to metal charge transfer 共LMCT兲states play a crucial role, mediating the coupling between the neutral and ionic VB determinants. Therefore, additional parameters need to be included, namely, the one-electron hopping integral between metal and ligand orbitals tal or tbl, the on-site ligand Coulomb repulsion Ul, and the ligand-to-metal charge transfer excitation energy ⌬ECT. Simultaneous to the development of these models, ab initio calculations have provided more and more accurate values of Jon systems of increasing complexity and size. Both density functional theory based approaches 共DFT兲and a兲Electronic mail: [email protected]. THE JOURNAL OF CHEMICAL PHYSICS 131, 044327 共2009兲 0021-9606/2009/131共4兲/044327/14/$25.00 © 2009 American Institute of Physics131, 044327-1 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.230.47 On: Fri, 28 Oct 2016 09:02:18 wave function-based methods have extensively been used and their performances and limitations discussed by several authors.4–6Bencini7recently gave an excellent review on the use of these computational tools. Together with numerical accuracy, the importance of these calculations lies in the fact that they have clarified some aspects of the factors governing the coupling. Wave function-based methods are especially useful for these interpretation purposes. In particular they have put in evidence that the Jvalues resulting from the action of the exact Hamiltonian on the model space are at least one order of magnitude smaller than the experiment or even of incorrect sign. Consequently, the physics of the coupling can neither be restricted to the interaction of the neutral and ionic VB determinants, as in one-band models, nor to their coupling with the LMCT states, as in two-band ones. Different approaches have been used to analyze the physical contributions to the magnetic coupling beyond the active-electron only approximation. Particularly remarkable is the pioneering work carried out by de Loth et al.8in the early 1980s and all the applications reported during the following decade.9–12 They have provided expressions for the perturbative evaluation of Jfrom the energy gap between the singlet and triplet states of binuclear Cu 共II兲complexes. They followed an idea developed by Malrieu13 in the 1960s who showed that it was not necessary to calculate the total correlation energy for both singlet and triplet states when using perturbation theory. Inspired by this work, Broer and Maaskant14 performed early variational calculations including the same configurations on copper dinuclear complexes. Following the same ideas, Malrieu and co-workers15,16 developed the difference dedicated configuration interaction 共DDCI兲approach where the CI matrix only contains those configurations playing a role on the energy difference of the states involved in the coupling. Based on the same idea, Neese and co-workers17,18 recently developed and coded the spectroscopy oriented CI method focused to the study of transition metal complexes. Recently, Barone et al.19 also coded a modified DDCI2 version of the method and applied to organic biradicals. From a computational point of view the DDCI procedure is much more expensive than the perturbative evaluations of de Loth et al.,8but, in contrast, it is free of convergence problems and of the implicit arbitrariness related to the definition of the zeroth-order Hamiltonian. So far, the DDCI approach has been extensively employed in the evaluation of Jin molecular and solid state magnetic materials with a remarkable good agreement with experiment.20–32 Some years ago we took benefit of this methodology to analyze in depth the physical contributions to the magnetic coupling on a series of binuclear Cu 共II兲complexes.33 The use of the DDCI strategy allows not only to obtain quite accurate Jvalues but also to analyze the various physical effects by generating CI spaces of increasing lengths that include different types of determinants. We also reported a procedure to define valence effective Hamiltonians starting from the eigenenergies and eigenvectors of these CI calculations.34 Indeed, the procedure provides a scheme to return to the qualitative models with effective parameters, which incorporate not only the effects brought by the model space but also those beyond the active-electron approximation, which in most of the cases have their origin in the electronic correlation. After classifying the excited determinants according to the number of doubly occupied orbitals 共h兲and virtual orbitals 共p兲implied in the excitation processes on the CAS determinants, our results indicate that the main contributions to the coupling come from the following: 共i兲The excitations involving one inactive occupied orbital hand one inactive virtual orbital p. This 1h-1p class of excitations introduces spin polarization effects as well as the polarization of the ionic VB forms, whose main effect consists of lowering the effective energy of ionic configurations. Both effects are particularly important when they involve orbitals centered in the bridging ligands. 共ii兲The 2h-1pand 1h-2pclasses of excitations. Some of them can be considered as single excitations on the LMCT and metal to ligand charge transfer 共MLCT兲 configurations, respectively. Their impact on the J value is at least 20%–30% of the global value, even more than 50% in some cases. When this type of excitations is included in the CI expansions computed J values are closer to the experiment. Previous interpretation of the mechanism governing the effect of these later excitations on the magnetic coupling was essentially formulated in terms of an enhancement of the effective hopping integral tab. The present work shows the irrelevance of this second-order based interpretation and identifies alternative mechanisms. Section II describes a series of Cu 共II兲dimers, either fragments of magnetic lattices or molecular systems, on which calculations are performed. Sections III and IV situate the problem and recall the previous second-order based interpretations. In Sec. V truncated CI calculations are employed to show that the second-order corrections brought by the 2h-1pand 1h-2pexcited configurations are in general much smaller than their contribution in the DDCI calculations. To get insight in the role of these excitations, a series of class-partitioned variational calculations is performed by adding selected classes of excited configurations to the valence CAS. The set of results confirms that the effect of 2h-1pand 1h-2pexcitations is not a second-order one. In Sec. VI we try to identify which classes of excitations interact with them giving significant higherorder contributions to the magnetic coupling constant. Some general conclusions are given in Sec. VII. II. CHOICE OF TEST SYSTEMS AND COMPUTATIONAL DETAILS The systems considered here, represented in Fig. 1, contain two Cu d9centers and differ by the number and type of bridging ligands and by the amplitude and sign of the magnetic constant. Geometries from x-ray crystal structures have been used for all the systems considered. A variety of complexes is reported with the following: 共a兲one single bridging ligand, as in La2CuO4共Refs. 35–38兲, and Sr2CuO2Cl2共Ref. 39兲cuprates, structures 1and 2, respectively where a Cu2O7fragment embed- 044327-2 Calzado et al. J. Chem. Phys. 131, 044327 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.230.47 On: Fri, 28 Oct 2016 09:02:18 ded in a set of point charges and total ion potentials simulating the Madelung potential of the infinite crystal has been chosen to evaluate the magnetic coupling; 共b兲two OH−bridging ligands in the 关LCu–共OH兲2–CuL兴 complexes, giving AF coupling in system 3共Refs. 40 and 41兲with L=N,N,N⬘,N⬘-tetramethylethylene- diamine and F coupling in 4共Refs. 42 and 43兲with L=2⬘-bipyridine; 共c兲two Cl−bridging ligands, giving also F coupling in the nonplanar 关Cu–共Cl兲2–Cu兴unit in 关As共C6H5兲4兴 ⫻关CuCl3兴共Refs. 44–46兲and AF coupling for the planar geometry in KCuCl3, structures 5and 6, respectively;44,47,48 共d兲two bidentated end-to-end azido groups in 关LCu–共NNN兲2–CuL兴, where L=N,N⬘,N⬙-tri- methyl-1,4,7-triazacyclononane25,49 in structure 7and 共e兲one bisbidentated bridging ligand such as oxalato group in 关LCu–共C2O4兲–CuL兴with a large range of AF coupling constants for L=1,1,4,7,7-pentaethyldie- thylene-triamine in 8.50 9is a model geometry of 8 used in previous works.51 In structures 7and 8, external ligands were modeled with NH3groups with the coordinated N at the experimental position. The counterions were not included in structures 3 and 5–8, since their distance to the anions was large. In structures 1–7, core electrons of Cu atoms 共up to 3s兲 were replaced with effective core potentials, and the 共9s6p6d兲/关3s3p4d兴basis set was used for the valence electrons of Cu.52 For systems 8and 9, Cu atoms are represented by means of atomic natural orbital ANO-type basis functions with contractions 关5s4p3d1f兴on the basis of our previous experience with these systems.21 For systems 1and 2,O atoms basis set is of DZP 共double zeta with polarization兲 quality with contractions 共10s5p1d兲/关3s2p1d兴as in previous works.23 For the remaining systems, ANO-type basis functions are used for the ligands of different quality depending on the specific position. Contractions 关3s2p1d兴are employed for C, N, and O in bridging positions and 关4s3p1d兴for bridging Cl. For the external ligands, contractions 关3s2p兴are used for atoms directly bonded to the metal centers and 关2s1p兴for the remaining positions. H atoms are represented by means of ANO-type functions with a contraction 关2s1p兴when placed on the bridging ligands, and 2sor 1sfor the rest.53–55 At the variational level, the DDCI method has been used in its CASDI implementation.56 To analyze the effect of the different types of excitations, different truncated CI calculations have been performed including different subsets of excitation classes. Regarding the perturbative approaches, CASPT2 calculations have been performed from the minimal 共2,2兲valence CAS by using the standard H0Hamiltonian implemented in MOLCAS 6.4 code.57 These calculations use a different molecular orbital 共MO兲set for every state and are state specific 共SS兲. The NEVPT2 method which uses a partially bielectronic H ˆ0that avoids intruder states problem has been used in the partially contracted version.58–61 A direct evaluation of the second-order effect of 2h-1pand 1h-2phas also been coded independently, as will be commented in Sec. V A. III. IMPLICATIONS OF THE VARIATIONAL DEFINITION OF THE ORBITALS An initial crucial point is to precise how the magnetic orbitals are defined, since the use of mean-field variational calculations has specific consequences linked to Brillouin’s theorem.62,63 Let us consider the basic problem of magnetic coupling between two S=1 2spins located on two metallic sites Aand B, such as two Cu d9ions. In a zeroth-order picture two unpaired electrons occupy two metal-centered orthogonal magnetic orbitals aand b. These two electrons and two orbitals lead to one triplet and three singlet valence states. The magnetic orbitals may be uniquely defined through a restricted open-shell self-consistent field 共SCF兲calculation of the single determinant Tutriplet state. The core closed-shell orbital his optimized as well. The Ms=0 component of this Tustate is Tu=1 冑2共兩hh ¯ gu ¯ 兩−兩hh ¯ ug ¯ 兩兲 =1 冑2共兩hh ¯ ab ¯ 兩−兩hh ¯ ba ¯ 兩兲,共2兲 where gand uare symmetry adapted magnetic orbitals, and the localized aand borbitals are obtained by rotation a,b=1/冑2共g⫾u兲. Local orthogonal valence orbitals are the basic stones of a fruitful mode of analysis of the wave function and of the physics that it obeys, which has been a usual tool of interpretation of magnetic coupling8,33 and electron transfer processes64 共for recent contributions on orthogonal VB analysis of CAS wave functions, see Refs. 65 and 66兲. The lowest singlet state is dominated by the neutral configuration SN, FIG. 1. Structures of binuclear Cu共II兲compounds described in Sec. II. 044327-3 Magnetic coupling in binuclear systems J. Chem. Phys. 131, 044327 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.230.47 On: Fri, 28 Oct 2016 09:02:18 Sg= ␣ 兩gg ¯ 兩− ␤ 兩uu ¯ 兩 =␭ 冉 兩ab ¯ 兩+兩ba ¯ 兩 冑2 冊 + ␮ 冉 兩aa ¯ 兩+兩bb ¯ 兩 冑2 冊 =␭SN+ ␮ SI, 共3兲 ␣ ⬎ ␤ ⬎0, ␭= ␣ + ␤ 冑2Ⰷ ␮ = ␣ − ␤ 冑2. As previously discussed,33 Brillouin’s theorem that cancels the interaction between the triplet configuration and those obtained by single excitations also gives a negligible 共for 1h and 1p兲or null 共for 1h-1p兲interaction between the neutral singlet configuration SNand the singly excited ones. Regarding the ionic components of the singlet state I=兵兩hh ¯ aa ¯ 兩,兩hh ¯ bb ¯ 兩其, the interaction with 1h共LMCT兲configu- rations gives 具hh ¯ bb ¯ 兩H ˆ冏1 冑2共a ˆa +a ˆh+a ˆa ¯ +a ˆh ¯ 兲hh ¯ bb ¯ 冔 =冑2具h兩J ˆb−J ˆa−K ˆb兩a典. 共4兲 This interaction has a non-negligible AF effect due to 具h兩J ˆa兩a典term. Similar considerations apply for 1p共MLCT兲 configurations. These non-negligible interactions will play a role in the forthcoming discussion. The singlet 1h-1pexcitations on the Iconfigurations lead to a strong interaction, reflecting the change in the electrostatic field with respect to the mean one, 具hh ¯ bb ¯ 兩H ˆ冏1 冑2共a ˆp +a ˆh+a ˆp ¯ +a ˆh ¯ 兲hh ¯ bb ¯ 冔 =冑2具h兩J ˆb−J ˆa+1 2共K ˆa−K ˆb兲兩p典.共5兲 If alternatively CASSCF singlet MOs are used, where the singlet state Sg=␭SN+ ␮ SIhas a small contribution of ionic VB components 关see Eq. 共3兲兴, the magnetic orbitals are not strictly equivalent to those resulting from a SCF calculation of the triplet state, since they have larger tails on the ligands, making larger their interaction. The same argument holds for 1pexcitations. A third possibility consists of using a different MO set for each state 共SS calculations兲. This is the procedure, for instance, in CASPT2 calculations. The use of SS orbitals in DDCI calculations makes null the differential contributions of the 1hand 1pexcitation classes for both states because of Brillouin’s theorem. Although DDCI calculations are almost independent on the choice of the MO set and the difference between SCF triplet and CASSCF singlet MOs is not appreciable in the drawing of their amplitudes, the small differences have nevertheless consequences on the effect of the different types of excitations on J, as will be discussed in Sec. V A. Whatever the SCF procedure for obtaining the MO set, the energy optimization produces an optimal 共or nearly optimal兲mixing of the metal and ligand orbitals. This has the crucial consequence of drastically reducing the weight of both the LMCT and the MLCT configurations, which belong to the 1hand 1pclasses, respectively. Consequently, with a self-consistent MO set, the Anderson two-band model2can- not be the reference to understand the different contributions to the coupling, since in this case the hopping integrals between magnetic and ligand MOs, tal and tbl, are null. Variational optimization of the MOs compels one to stay in the simple one-band model. IV. SECOND-ORDER EVALUATION OF THE MAGNETIC COUPLING At all stages of calculation the wave functions of the lowest triplet and singlet states are largely dominated by the neutral VB configurations Tuand SN, respectively, which may be considered as reference zeroth-order functions. Their energy difference is positive, 具SN兩H ˆ兩SN典−具Tu兩H ˆ兩Tu典=2Kab.共6兲 The interaction between the ionic and neutral determinants contributes to the second order to an energy stabilization of the singlet state ␧g 共2兲=−4tab 2/U, as derived in the one-band Anderson mechanism 共1兲. Numerical results have shown that in general the CASCI evaluation of Jgives at most 20%–30% of the experimental AF coupling constants, as illustrated by CAS entry in Table I, which reports the magnetic coupling values obtained with different CI spaces for the binuclear Cu 共II兲complexes described in Sec II. It is therefore necessary to go beyond va- TABLE I. Calculated magnetic coupling constants 共cm−1兲for different partially truncated post-CASCI levels for systems 1–9. La2CuO4 1 Sr2CuO2Cl2 2 关Cu2共OH兲2兴 3 关Cu2共OH兲2兴 4 关Cu2Cl2兴 5 关Cu2Cl2兴 6 关Cu2共N3兲2兴 7 关Cu2共C2O4兲兴 8 关Cu2共C2O4兲兴 9 CAS ⫺255 ⫺160 ⫺35 33 13 11 ⫺82 ⫺7⫺13 CAS+S ⫺706 ⫺464 ⫺159 73 28 17 ⫺362 ⫺22 ⫺39 DDCI2 ⫺744 ⫺482 ⫺184 63 24 8 ⫺375 ⫺23 ⫺41 DDCI2+2h-1p⫺1462 ⫺1256 ⫺750 195 80 ⫺61 ⫺1470 ⫺158 ⫺344 DDCI2+1h-2p⫺569 ⫺386 ⫺117 57 20 16 ⫺324 ⫺18 ⫺32 DDCI ⫺1077 ⫺952 ⫺500 157 54 ⫺21 ⫺802 ⫺78 ⫺160 Expt. 关⫺1030, ⫺1096兴a⫺1008b⫺509c172d46e0, ⫺40e⬍−800 f⫺75g¯ aReferences 35–38. bReference 39. cReference 41. dReference 43. eReference 44. fReference 49. gReference 50. 044327-4 Calzado et al. J. Chem. Phys. 131, 044327 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.230.47 On: Fri, 28 Oct 2016 09:02:18 lence configurations and to consider nonvalence configurations to reproduce the experimental values. As shown in the 1960s 共Ref. 13兲and numerically exploited in the 1980s on copper acetate8and other S=1 2dimers,9–12 only those determinants contributing at second order to the singlet-triplet energy difference must in principle be taken into account. When considering neutral determinants as zeroth-order space this differential list involves the valence ionic configurations as well as 1h,1p,2h,2p, and 1h-1pclasses of excited determinants. This is the so-called DDCI2 space. The commonly used CAS+Singles CI 共CAS+S兲only includes 1h, 1p, and 1h-1pclasses. The CI restricted to this space usually gives 50%–60% of the experimental AF coupling, as shown in Table I共entries CAS+S and DDCI2兲. As discussed in Sec. III, the 1hand 1pconfigurations interact with the ionic VB component of the CAS wave function. Since the coefficient of this ionic configuration is small at this stage, the direct contributions of 1hand 1pon the singlet remain small. At these CI levels, the 1h-1pconfigurations play the leading role. Their effect may be split into the following: 共a兲The spin polarization correction resulting from interaction of the neutral Tuand SNconfigurations with the product of a triplet single excitation of the core electrons by a triplet configuration of the magnetic electrons of the type a ˆp ¯ +a ˆb +a ˆb ¯ a ˆh兩hh ¯ ab ¯ 兩=兩h ¯ p ¯ ab兩.共7兲 This contribution is a second-order effect on the neutral zeroth-order functions Tuand SNand may be either F or AF. 共b兲The dynamical polarization of the ionic VB configurations resulting from the product of a singlet single excitation of the core electrons by a singlet single excitation of the magnetic electrons which can be written as a single excitation on the top of the ionic forms as: 1 冑2共a ˆp +a ˆh+a ˆp ¯ +a ˆh ¯ 兲兩hh ¯ aa ¯ 兩=1 冑2共兩ph ¯ aa ¯ 兩+兩hp ¯ aa ¯ 兩兲.共8兲 The direct interaction of these configurations with the ionic valence ones is large and may be considered as lowering of the effective energy of ionic configurations Ueff =U+兺 h,p 具hh ¯ aa ¯ 兩H ˆ兩ph ¯ aa ¯ 典具ph ¯ aa ¯ 兩H ˆ兩hh ¯ aa ¯ 典 共−U+␧h−␧p兲,共9兲 where the matrix elements are given in Eq. 共5兲. As previously indicated, DDCI2 calculations include all the differential contributions to the energy difference between singlet and triplet states up to second order from a neutral reference space. The ionic valence configurations belong to these differential contributions but in view of their important role, it is preferable to include them in the reference space and to consider the CAS as a better reference space than the neutral VB determinants only. It increases the list of excitations, which have a differential effect on the energies of the valence states up to the second order, and the truncated CI space has to be enlarged as well. At this level, the purely inactive double excitations a ˆp⬘ +a ˆp +a ˆh⬘a ˆhinduce an equal energy shift on the valence states and can be discarded. The so-obtained expansion is called the DDCI space, where 2h-1pand 1h-2pconfigurations are added to the DDCI2 one. Many calculations have shown the performance of DDCI estimates of the magnetic coupling.20–32 The results reported in Table Ishow that the Jvalues obtained at CAS+S and DDCI2 levels are far from experimental ones, which suggest that an important part of the effects is missed. It is therefore concluded that 2h-1pand 1h-2pconfigurations bring from 30% to 50% of magnetic coupling. In the past these effects have been explained by suggesting that these excitations strengthen the coupling between the neutral and ionic forms 共N↔2h−1p↔I兲, which produces an increase in tab and consequently of the AF contribution of the magnetic coupling. The second-order correction to the effective hopping integral between neutral and ionic states mediated by the 2h-1pde- terminants is tab eff =tab +具SN兩H ˆ兩2h−1p典具2h−1p兩H ˆ兩I典 ⌬E2h1p =tab +⌬tab 2h1p, 共10兲 where the main contribution to ⌬tab 2h1pis ⌬tab 2h1p←−兺 h兺 h⬘兺 p 2共hp,ah⬘兲共bh⬘,hp兲 ⌬E2h1p .共11兲 The sign of tab depends on the type of magnetic orbitals and is found negative in this type of systems. The sign of ⌬tab 2h1p depends on the symmetry of the inactive occupied h⬘orbital. It has been shown that if the contribution is dominated by single excitations on the LMCT configuration from antisymmetric orbital of the bridging ligand, as bridging oxygen 2pz orbital in cuprates, this contribution is negative and therefore increases its absolute value. The 1h-2p excitations also participate in the coupling between the ionic and neutral forms 共N↔1h−2p↔I兲, modulating the effective hopping integral tab. In this case, the dominant contribution is ⌬tab 1h2p←兺 h兺 p兺 p⬘ 2共hp,ap⬘兲共bp⬘,hp兲 ⌬E1h2p .共12兲 Similarly, the sign of these contributions also depends on the symmetry of p⬘. In the past, it has been assumed that the sum of these mechanisms results in an overall increase in 兩tab eff兩and therefore of the kinetic exchange. Regarding the singlet and triplet wave functions, going from DDCI2 to DDCI produces a remarkable increase in the weights of the LMCT configurations, which has been related to the impact of the 2h-1p excitations.23 In fact, single excitations on the LMCT configurations belong to the 2h-1pclass of excitations and the effect of this excitation class was first interpreted as a lowering of the effective energy of the LMCT configurations. This would be relevant in a two-band model but since the LMCT configurations have a negligible interaction with the neutral determinants, this interpretation was discarded. On 044327-5 Magnetic coupling in binuclear systems J. Chem. Phys. 131, 044327 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.230.47 On: Fri, 28 Oct 2016 09:02:18 the basis of this remark, two pathways were proposed to explain the enhancement of AF character as follows: 共1兲A third-order mechanism, N↔I↔共2h-1p兲↔N, which involves ⌬tab. 共2兲A fourth-order one, N↔共2h-1p兲↔I↔共2h-1p兲↔N, which involves 兩⌬tab兩2. V. IRRELEVANCE OF THE SECOND-ORDER INTERPRETATION OF THE ROLE OF THE 2h-1p AND 1h-2pEXCITED CONFIGURATIONS On the basis of the perturbative interpretation one may be tempted to rely on CAS-based second-order calculations or to combine DDCI2 variational calculations with perturbative estimates of the effect of 2h-1pand 1h-2pconfigura- tions, which are the largest subspaces in DDCI method, thus saving most of the computational effort. Class-partitioned variational calculations which add separately classes of excited configurations to the CAS may alternatively be considered in order to check whether the effect of the various classes is essentially additive, as they should be if dominated by second-order effects. For the sake of comparison with the perturbative methods 共by default SS兲and to check the behavior of DDCI results with respect to the MO set, coupling constants for systems 1–4 have been evaluated from different sets of MOs: SCF triplet MOs, 共2,2兲CASSCF singlet MOs, as well as the SS MOs. Table II shows that DDCI results are remarkably independent of the set of MOs used. Hereafter, we will concentrate on compounds 1–4 to perform a detailed analysis. A. Second-order CAS-based perturbations For consistency of comparison with class-partitioned variational calculations, although CASPT2 calculations are in general based on enlarged CAS,24 only 共2,2兲minimal CAS will be considered here. The same strategy is applied in the NEVPT2 calculations. In both methods, the energy corrections to singlet and triplet states are class additive and so is the energy difference. Tables III and IV report the details of the contributions of different classes for compounds 1–4. The perturbative calculations suffer for some arbitrariness in the definition of the zeroth-order Hamiltonian. In order to avoid this source of uncertainty and to check whether the class contributions are additive, variational calculations have been performed by adding specific subspaces of the outer space determinants to the CAS. The CI spaces are thus composed of the CAS and the indicated set of determinants. To evaluate the independent effect of each contribution on J, the CAS Jvalue is subtracted. This makes the comparison with perturbative contributions possible. From this series of class-partitioned calculations, some relevant observations can be made as follows: 共a兲The 2h-2pinactive double excitations bring a negligible contribution, despite the fact that the MOs are different from the singlet and the triplet states. This validates their elimination in DDCI method. 共b兲There is a good agreement between the CASPT2 and CI calculations from 1hto 2pdifferential contributions. When the CI space includes all these contributions 共DDCI2 level兲, the total result of Jis similar, which indicates the approximately additive character of all these excitation subsets. 共c兲1hand 1pcontributions, which are obviously null in SS calculations, have a small AF effect when using a unique set of MOs in the variational calculations, as discussed in Sec. III. 共d兲The overall CASPT2 Jvalue is very good in La2CuO4 1but significantly underestimated for Sr2CuO2Cl22 for the AF complex 关Cu2共OH兲2兴3, as well as for the F complex 关Cu2共OH兲2兴4共66%, 50%, and 50% of the experimental value, respectively兲. The NEVPT2 values are even smaller than the CASPT2 ones with the same distribution between the various classes. The relative failure of the second-order approaches already reported in previous works28,67 might possibly be attributed to the internally contracted character of these formalisms. The zeroth-order wave function severely underestimates the coefficient of the ionic VB component of the singlet state. The dynamical polarization effects included in 1h-1pexcitations increase this coefficient, as recalled in Sec. IV. We have therefore tested the performance of a recently proposed procedure which allows a decontraction of the CAS composition of the wave function.68 This procedure actually leads to an enlargement of the ionic component resulting in an increase in the AF character. The results are excellent for system 1since this procedure gives J=−1106 cm−1 TABLE II. Effect of the MO set 共T: SCF triplet MOs, S: CASSCF singlet MOs, SS: state specific MOs兲on the magnetic coupling constant value J共in cm−1兲 for systems 1–4 at different CI levels. Jexpt La2CuO4 1 关⫺1030, ⫺1096兴a Sr2CuO2Cl2 2 ⫺1008b 关Cu2共OH兲2兴 3 ⫺509c 关Cu2共OH兲2兴 4 +172 d CI space T S SS T S SS T S SS T S SS CAS ⫺255 ⫺387 ⫺315 ⫺160 ⫺237 ⫺195 ⫺35 ⫺72 ⫺52 33 31 32 CAS+S ⫺706 ⫺780 ⫺783 ⫺464 ⫺505 ⫺507 ⫺159 ⫺178 ⫺179737374 DDCI2 ⫺744 ⫺825 ⫺828 ⫺482 ⫺527 ⫺530 ⫺184 ⫺205 ⫺206636264 DDCI ⫺1077 ⫺1149 ⫺1174 ⫺952 ⫺1002 ⫺1026 ⫺500 ⫺523 ⫺526 157 156 178 aReferences 35–38. bReference 39. cReference 41. dReference 43. 044327-6 Calzado et al. J. Chem. Phys. 131, 044327 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.230.47 On: Fri, 28 Oct 2016 09:02:18 with the NEVPT2 externally partially contracted version, instead of −795 cm−1 before the internal decontraction, as reported in Table III. But they remain unsatisfactory for system 2,J=−690 cm−1, after the decontraction instead of −515 cm−1, far from the experiment, −1008 cm−1, and DDCI, −952 cm−1, and similarly for system 3, where the internal decontraction gives J=−284 cm−1, a significant improvement when compared to −173 cm−1 共see Table IV兲, but still insufficient since the experimental value is −509 cm−1. Therefore the internal contraction cannot be considered the main responsible for the failure of the second-order treatments. 共e兲The effect of 1h-2pexcitations is similar in perturbative and class-partitioned CI calculations with a moderate F contribution to the Jvalue in AF systems 1–3, and an almost negligible AF contribution in F system 4. 共f兲The direct effect of 2h-1pconfigurations both in perturbative and class-partitioned CI calculations deserves a more accurate analysis. This class of excitations does not bring a significant contribution to Jwhen a common set of MOs is used in the variational CAS +2h-1pcalculations, as shown in the rightward columns of Tables III and IV 共⫺19, ⫺7, and ⫺3 in compounds 1–3兲. This result questions the interpretation of their role given in Ref. 33 and recalled in Sec. IV. NEVPT2 calculations performed with triplet state MOs for both states give the confirmation: the contribution to Jof this class of excitations is small: ⫺45, ⫺31, ⫺20, and 3 cm−1 for 1–4, respectively. As a further verification, the second-order correction to tab given in expressions 共11兲and 共12兲has been directly computed, and a negligible contribution has also been found. The second-order contribution observed in CASPT2 and NEVPT2 calculations in Tables III and IV is linked to the state specificity, as confirmed by SS variational calculations 共column 3 in both tables兲. This effect can be understood as due to larger metal-ligand mixing of the magnetic orbitals in the singlet state, as discussed in Sec. III. The 共hp,ah⬘兲,共hp,bh⬘兲integrals in expression 共11兲have larger amplitude for singlet state orbitals, since they are proportional to this mixing. It results in a larger stabilization of the singlet state and therefore in an enhancement of the AF character. An independent TABLE III. Independent contributions of various classes of excitations in perturbative and variational approaches for systems 1and 2共in cm−1兲. The truncated CI spaces include the corresponding excitation class on top of the CAS. La2CuO4 1 Jexpt=关−1030,−1096兴a CASPT2 SS MOs NEVPT2 SS MOs Truncated CI SS MOs Truncated CI T MOs CAS共2,2兲⫺315 ⫺315 ⫺315 ⫺255 1h00 0 ⫺21 1p00 0 ⫺30 1h-1p⫺434 ⫺301 ⫺418 ⫺268 2h⫺16 ⫺5 2p⫺7⫺6 1h-2p77 38 50 66 2h-1p⫺287 ⫺221 ⫺181 ⫺19 2h-2p16 16 Total ⫺966 ⫺795 ⫺1174b⫺1077b Sr2CuO2Cl2 2 Jexpt=−1008 c CASPT2 SS MOs NEVPT2 SS MOs Truncated CI SS MOs Truncated CI T MOs CAS共2,2兲⫺195 ⫺195 ⫺195 ⫺160 1h00 0 ⫺9 1p00 0 ⫺20 1h-1p⫺294 ⫺190 ⫺269 ⫺176 2h⫺9⫺2 2p⫺3⫺3 1h-2p40 16 22 37 2h-1p⫺230 ⫺164 ⫺137 ⫺7 2h-2p25 23 Total ⫺667 ⫺515 ⫺1008b⫺952b aReferences 35–38. bThis value is not the sum of independent class contributions but that of the DDCI calculation. cReference 39. 044327-7 Magnetic coupling in binuclear systems J. Chem. Phys. 131, 044327 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.230.47 On: Fri, 28 Oct 2016 09:02:18 calculation of ⌬tab directly coded and evaluated confirms the SS character of the role of 2h-1pand 1h-2p classes in perturbative calculations. In any case, it must be noticed that the 2h-1psecond-order correction in SS approaches is in general not sufficient and let the second-order calculation far from experiment. 共g兲The most important evidence of this set of classpartitioned results is the very different impact of the 2h-1pand 1h-2pclasses of excitations at the CI level when acting on the bare CAS or when added to the DDCI2 space. For compound 1, for instance, as reported in Table V,Jat CAS+2h-1plevel with triplet MOs is −273 cm−1, i.e., the 2h-1pclass gives an AF contribution of −19 cm−1, while at DDCI2+2h-1pJis −1462 cm−1, i.e., an AF enhancing of −718 cm−1 with respect to DDCI2 value. This value is largely overestimated when compared to experiment but when adding 1h-2pexcitations 共DDCI space兲, a correct value of −1077 cm−1 is obtained due to the opposite effect of this last class of excitations. Similar conclusions can be extracted for systems 2–4. From these observations it may be concluded that the large overall effect of these two classes, as it appears in the variational calculations going from DDCI2 to DDCI, is not a second-order effect. It proceeds through their interaction with other classes of nonvalence determinants. To elucidate the mechanism of their indirect impact on the Jvalues, truncated CI gathering various classes of excitations have been performed. B. Tracing the indirect mechanism of action of 2h-1p and 1h-2pexcitations Table Vreports the CI results on systems 1–4 when adding 2h-1pand/or 1h-2pexcitations to different truncated CI spaces. In AF systems 1–3, the 2h-1pexcitations only bring an important AF contribution once 1hLMCT configurations have been added to the CAS. Considering 1as an example, while the addition of 2h-1pto the CAS only brings a correction of −19 cm−1 to J, an effect of −240 cm−1 is obtained when the 1hconfigurations are present in the CI space. The AF effect of this class of excitations is dramatically amplified when the 1h-1pconfigurations are included in the CI, this contribution being −672 cm−1 in system 1共CAS+S, 2h-1pentry in Table V兲. On the other hand, the 1pexcita- tions have a negligible effect. TABLE IV. Independent contributions of various classes of excitations in perturbative and variational approaches for systems 3and 4共in cm−1兲. The truncated CI spaces include the corresponding excitation class on top of the CAS. 关Cu2共OH兲2兴 3 Jexpt=−509 a CASPT2 SS MOs NEVPT2 SS MOs Truncated CI SS MOs Truncated CI T MOs CAS共2,2兲⫺52 ⫺52 ⫺52 ⫺35 1h00 0 ⫺2 1p00 0 ⫺14 1h-1p⫺120 ⫺72 ⫺106 ⫺67 2h⫺22 ⫺3 2p⫺4⫺8 1h-2p43 26 30 25 2h-1p⫺94 ⫺61 ⫺42 ⫺3 2h-2p⫺4⫺4 Total ⫺253 ⫺173 ⫺526b⫺500b 关Cu2共OH兲2兴 4 Jexpt=+172 c CASPT2 SS MOs NEVPT2 SS MOs Truncated CI SS MOs Truncated CI T MOs CAS共2,2兲32 32 32 33 1h00 0 0 1p00 0 1 1h-1p22 21 36 28 2h⫺13 ⫺2 2p⫺3⫺5 1h-2p15 18 22 3 2h-1p62 50 60 7 2h-2p⫺24 ⫺20 Total 95 94 178b157b aReference 41. bThis value is not the sum of independent class contributions but that of the DDCI calculation. cReference 43. 044327-8 Calzado et al. J. Chem. Phys. 131, 044327 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.230.47 On: Fri, 28 Oct 2016 09:02:18 The effect is similar but of opposite sign in the F system 4, since 2h-1pexcitations bring an important F contribution only when the 1hLMCT configurations are present in the CI space, 149 cm−1. The dominant mechanisms of action of 2h-1pconfigurations must therefore be different in AF and F systems. The effect of 1h-2pconfigurations is F in AF systems, while it is negligible in the F system 4. In systems 1–3 this F effect increases when 1h-2pexcitations are added to CI space from CAS to DDCI2: for 1,⌬J=66 cm−1 for CAS+1h-2pspace, 164 cm−1 for CAS+S+1h-2p, and 175 cm−1 for DDCI2+1h-2p. Since the same trend but larger in absolute value is observed for 2h-1pexcitations, this F effect seems to be a damping of the AF mechanisms. In Sec. VI, some rationalizations of these effects will be proposed. VI. RATIONALIZATION OF THE ROLE OF 2h-1p AND 1h-2pEXCITATIONS A. AF effect of 2h-1pconfigurations in AF systems It has previously been shown33 that the 2h-1pclass of excitations is responsible for the occurrence of large coefficients of LMCT 共1h兲configurations. In that work the increase in the LMCT coefficients was not supposed to have an impact on the magnetic coupling constant, since the direct interaction between LMCT and the neutral configurations is negligible according to Brillouin’s theorem, as discussed in Sec. III. However, there is an important interaction between LMCT and ionic configurations ruled by the magnitude of the 具h兩J ˆa兩a典term. One may therefore suspect that the interaction between LMCT and ionic configurations has a signifi- TABLE V. Effect of adding 2h-1por 1h-2pclasses of excitations to CAS, CAS+S, and DDCI2 spaces 共in cm−1兲for systems 1–4 共T MOs兲.⌬Jis referred to CAS, CAS+S, or DDCI2 values, respectively. Jexpt La2CuO4 1 关⫺1030, ⫺1096兴a Sr2CuO2Cl2 2 ⫺1008b 关Cu2共OH兲2兴 3 ⫺509c 关Cu2共OH兲2兴 4 +172 d J⌬JJ⌬JJ⌬JJ⌬J CAS共2,2兲⫺255 ¯⫺160 ¯⫺35 ¯33 ¯ +2h-1p⫺273 ⫺18 ⫺167 ⫺7⫺38 ⫺340 7 +1h+2h-1p⫺494 ⫺240 ⫺381 ⫺221 ⫺100 ⫺65 182 149 +1h-2p⫺195 66 ⫺123 37 ⫺10 25 36 3 +1p+1h-2p⫺221 33 ⫺140 20 ⫺14 21 41 8 CAS+S ⫺706 ¯⫺464 ¯⫺159 ¯73 ¯ +2h-1p⫺1378 ⫺672 ⫺1199 ⫺735 ⫺650 ⫺523 248 175 +1h-2p⫺542 164 ⫺372 92 ⫺94 65 71 ⫺2 +2h-1p+1h-2p⫺1023 ⫺317 ⫺916 ⫺452 ⫺428 ⫺269 203 130 DDCI2 ⫺744 ⫺482 ⫺184 63 +2h-1p⫺1462 ⫺718 ⫺1256 ⫺774 ⫺750 ⫺566 195 132 +1h-2p⫺569 175 ⫺386 96 ⫺117 67 57 ⫺6 DDCI ⫺1077 ⫺952 ⫺500 157 DDCI-DDCI2 ⫺333 ⫺470 ⫺316 94 aReferences 35–38. bReference 39. cReference 41. dReference 43. TABLE VI. Some representative coefficients of the singlet wave function at different CI levels for systems 1,3, and 4.CI/CN, ionic/neutral valence configurations ratio. LMCT, the most significant LMCT excitation in the singlet state. The left column indicates the specific excitation class共es兲added in each case to CAS, CAS+S, or DDCI2 spaces. La2CuO41关Cu2共OH兲2兴3关Cu2共OH兲2兴4 JC I/CNLMCT JC I/CNLMCT JC I/CNLMCT CAS ⫺255 0.0406 ¯⫺35 0.0218 ¯33 0.0009 ¯ 1h+2h-1p⫺494 0.0685 0.1312 ⫺100 0.0495 0.1158 182 0.0060 0.1022 1p+1h-2p⫺221 0.0406 ⬍0.001 ⫺14 0.0215 ⬍0.001 41 0.0010 ⬍0.001 CAS+S ⫺706 0.0910 0.0292 ⫺159 0.0544 0.0197 73 0.0005 0.0098 2h-1p⫺1378 0.1385 0.1615 ⫺650 0.1159 0.1483 248 0.0173 0.1225 1h-2p⫺542 0.0789 0.0222 ⫺94 0.0459 0.0158 71 0.0017 0.0087 2h-1p+1h-2p⫺1023 0.1137 0.1239 ⫺428 0.0926 0.1197 203 0.0005 0.1017 DDCI2 ⫺744 0.0955 0.0300 ⫺184 0.0571 0.0208 63 0.0004 0.0104 2h-1p⫺1462 0.1452 0.1632 ⫺750 0.1218 0.1514 195 0.0181 0.1244 1h-2p⫺569 0.0818 0.0227 ⫺117 0.0478 0.0166 57 0.0017 0.0092 DDCI ⫺1077 0.1180 0.1250 ⫺500 0.0966 0.1218 157 0.0132 0.1031 044327-9 Magnetic coupling in binuclear systems J. Chem. Phys. 131, 044327 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.230.47 On: Fri, 28 Oct 2016 09:02:18