Analysis of the magnetic coupling in binuclear complexes. I. Physics of the coupling

Analysis of the magnetic coupling in binuclear complexes. I. Physics of the coupling Carmen J. Calzado Laboratoire de Physique Quantique, IRSAMC, Universite ´Paul Sabatier, 118, Route de Narbonne, 31062 Toulouse, France Jesu ´s Cabrero Departament de Quı ´mica Fı ´sica i Inorga `nica and Institut d’Estudis Avanc¸ats, Universitat Rovira i Virgili, Pl. Imperial Tarraco, 1. 43005 Tarragona, Spain Jean Paul Malrieu Laboratoire de Physique Quantique, IRSAMC, Universite ´Paul Sabatier, 118, Route de Narbonne, 31062 Toulouse, France Rosa Caballol Departament de Quı ´mica Fı ´sica i Inorga `nica and Institut d’Estudis Avanc¸ats, Universitat Rovira i Virgili, Pl. Imperial Tarraco, 1. 43005 Tarragona, Spain 共Received 4 September 2001; accepted 6 November 2001兲 Accurate estimates of the magnetic coupling in binuclear complexes can be obtained from ab initio configuration interaction 共CI兲calculations using the difference dedicated CI technique. The present paper shows that the same technique also provides a way to analyze the various physical contributions to the coupling and performs numerical analysis of their respective roles on four binuclear complexes of Cu (d9) ions. The bare valence-only description 共including direct and kinetic exchange兲does not result in meaningful values. The spin-polarization phenomenon cannot be neglected, its sign and amplitude depend on the system. The two leading dynamical correlation effects have an antiferromagnetic character. The first one goes through the dynamical polarization of the environment in the ionic valence bond forms 共i.e., the M⫹ ¯M⫺structures兲. The second one is due to the double excitations involving simultaneously single excitations between the bridging ligand and the magnetic orbitals and single excitations of the environment. This dispersive effect results in an increase of the effective hopping integral between the magnetic orbitals. Moreover, it is demonstrated to be responsible for the previously observed larger metal-ligand delocalization occurring in natural orbitals with respect to the Hartree–Fock ones. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1430740兴 I. INTRODUCTION The field of molecular and solid state magnetism arouses increasing interest in organic and organometallic chemistry as well as in material science.1–3 These systems are characterized by the existence of localized unpaired electrons, usually located on metallic ions. The properties of the material are governed by the interaction between their unpaired electrons on neighbor centers, which may be viewed as an effective interaction between site-centered spins, and mapped onto a Heisenberg–Dirac–Van Vleck Hamiltonian,4 H ˆ⫽⫺兺 i,jJijS ˆiS ˆj.共1兲 For a bicentric system with ms⫽⫾1/2 on each center, the coupling constant J⫽1E⫺3Eis negative for a singlet ground state 共antiferromagnetism兲and positive in the opposite case 共ferromagnetism兲.Jis usually very small, ranging from a few hundreds to a few cm⫺1, and its calculation is a challenge for theoreticians. Quantum chemistry may be successfully applied for such evaluations. Numerous studies employ the density functional theory approach, mostly with the popular B3LYP exchange correlation potential,5–10 using Noodleman’s approximation,11 in which the magnetic coupling is evaluated from the difference between the Unrestricted 共spin polarized兲highest multiplet and the brokensymmetry 共BS兲determinant. Although this procedure may be useful, in particular to predict or to analyze trends in magneto-structural correlations, it suffers from two weaknesses: the dependence on the Vxc multiparametric exchange correlation potential and the spin contamination problems, especially severe for the broken-symmetry solution. Properly using Noodleman’s formula, i.e., taking into account the very small overlap between the magnetic orbitals in the BS solution, the B3LYP functional usually gives values of Jabout twice the experimental ones.7,8,12 Moreover, this procedure does not allow to analyze the various contributions to Jand to check the validity of the semiempirical rationales historically proposed to interpret the sign and the magnitude of Jand its structural dependencies that will be briefly recalled. Ab initio techniques that work with the exact Hamiltonian overcome these difficulties and limits provided that all the meaningful physical effects are incorporated in the treatment, resulting in a good agreement with experiment. JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 7 15 FEBRUARY 2002 27280021-9606/2002/116(7)/2728/20/$19.00 © 2002 American Institute of Physics 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 09:45:28 Among the successful methods one may first quote the nonorthogonal configuration interaction 共NOCI兲,13 which introduces a very limited number of valence bond configurations, including various ligand to metal charge transfer states.14,15 The list suffers some arbitrariness. The method requires specific optimization of the orbitals of each VB configuration and has not received numerous applications. More recently, the CASPT2 共Refs. 16 and 17兲method has been used in this field. The second order expansion gives rather satisfactory results provided that the variational complete active space is large enough and properly chosen.18–21 In the recent past, an alternative configuration interaction 共CI兲technique has been designed for a direct evaluation of vertical energy differences, the difference dedicated configuration interaction 共DDCI兲,22 which has been successfully employed in a wide series of studies, concerning molecular23–26 as well as solid state magnetic materials.27–30 An error smaller than 10 cm⫺1is typical for a value of J⬃100 cm⫺1. The definition of the DDCI space is based on second order quasidegenerate perturbation theory arguments as explained in our early papers in this domain.22,23 A secondorder perturbative approach has the advantage of providing a partition of the final Jvalue into a sum of different physical contributions31 such as direct exchange, Anderson’s antiferromagnetic 共or kinetic exchange兲contribution, spin polarization, etc. However the second order perturbative expansion is not numerically reliable because of convergence problems and arbitrariness in the choice of the zero order Hamiltonian. The DDCI method goes through an exact diagonalization and hence gains in accuracy, but the analysis of the various physical mechanisms involved in the coupling is not straightforward any more. However, the usefulness of such a rigorous analysis is indubitable since it would allow to check the validity of the qualitative models, frequently used in an a posteriori rationalization and in the molecular design of new materials.1 The target of the present paper is to show that it is possible to combine numerical accuracy 共going through large basis sets and long CI expansions兲and analysis in terms of physical contributions. Four different antiferromagnetic systems have been considered in the present work, all of them involving two electrons on two Cu (d9) sites: the 关Cu2Cl6兴2⫺, the 关Cu2( ␮ -N3)2(NH3)6兴2⫹, and the Cu2( ␮ -CH3COO)4(H2O)2complexes and the Cu2O7cluster, a model of the La2CuO4perovskite. Section II describes the computational details. After recalling the valence only treatments in Sec. III, the theoretical development and the strategy of analysis are presented in Sec. IV and applied on these four binuclear systems. Finally, Sec. V reanalyzes the role of the different contributions when using natural orbitals instead of the triplet Hartree–Fock orbitals used in Secs. III and IV. II. DESCRIPTION OF THE MODELS AND COMPUTATIONAL DETAILS The four systems considered here, represented in Fig. 1, contain two Cu(d9) centers, bridged by different ligands and with different relative positions. Figure 1共a兲depicts our first model, the 关Cu2Cl6兴2⫺, which magnetostructural dependence has been extensively studied in the recent past.9,23,32,33 We have concentrated on the planar structure, where each Cu atom has a square planar coordination and bears its unpaired electron in a dxy-type orbital 关Fig. 2共a兲兴. The Cu–Cu distance in the real complex is 3.44 Å. Experimentally, the planar structure of this complex belongs to the Cipoint group but only very slightly distorted from D2h, considered in the present work. A similar situation is found in the 关Cu2( ␮ -N3)2(NH3)6兴2⫹complex 关Fig. 1共b兲兴 with a square pyramidal coordination of the Cu centers 关Fig. 2共b兲兴. The Cu–Cu distance is 5.19 Å, larger than in the 关Cu2Cl6兴2⫺ complex, but the magnetic coupling constant in the former is one order of magnitude larger than in the latter, showing that the end-to-end bridged (N3)⫺groups play a crucial role in the exchange phenomenon.34,35 As in the precedent case, the experimental Cigeometry has been symmetrized to C2h. Regarding the Cu2( ␮ -CH3COO)4(H2O)2molecule 关Fig. 1共c兲兴, the Cu atoms have also a square pyramidal coordination, but are placed in parallel planes, bridged by four bidentate acetato-ligands and separated by 2.64 Å only. Each Cu atom bears an unpaired electron in a dx2–y2orbital 关Fig. 2共c兲兴. The experimental Cigeometry36 has been used in the calculations. The nature of the magnetic interaction in this system was a matter of controversy during three decades: Do the two dx2–y2unpaired electrons interact directly via a ␦ -bond or indirectly through the acetato-ligands?37 The experimental works of Gu ¨del et al.38 and Ewald and Sinn39 together with the ab initio calculation of de Loth et al.31 ruled out the ␦ -bond hypothesis and showed the role of the acetato-ligand in the Cu–Cu exchange. The last considered system is a model of the antiferromagnetic perovskite La2CuO4. It consists of a Cu2O7cluster 关Fig. 1共d兲兴, embedded in a set of point charges to mimic the Madelung field of the La2CuO4lattice. The unpaired electron is placed in a dx2–y2orbital, but unlike the precedent models, the Cu atoms are bridged by just one oxygen ligand 关Fig. 2共d兲兴. The magnetic coupling in this compound is quite large FIG. 1. Schematic representation of the four models considered: 共a兲the 关Cu2Cl6兴2⫺complex, in a planar geometry; 共b兲the 关Cu2( ␮ -N3)2(NH3)6兴2⫹ complex with end-to-end bridging azido ligands; 共c兲the Cu2( ␮ -CH3COO)4(H2O)2complex; 共d兲the Cu2O7cluster. 2729J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Magnetic coupling 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 09:45:28 共around ⫺1000 cm⫺1兲, well established from Raman40 and neutron diffraction41 experiments. Different theoretical approaches have been employed to evaluate J关for instance, finite8and periodic42 DFT calculations, nonorthogonal CI 共Ref. 14兲and DDCI 共Refs. 27, 28兲calculations兴the CI procedures giving the most correct results. Since the four models presented here have been studied in previous independent works, different basis sets have been used in the HF–CI calculations. In the 关Cu2Cl6兴2⫺complex A, the azido-bridged complex B, and copper acetate C, an effective core potential of Barandiara ´n and Seijo has been employed for the Cu atoms, where the valence electrons are described by a (9s6p6d)/关3s3p4d兴basis set.43 ANO-type functions44 have been used for Cl, N, and H, with the following contractions: ANO-s.4s3p1dfor Cl, ANO-s.3s2p1dfor the bridging N atoms, ANO-s.3s2pfor the N atoms of the external NH3groups, and ANO-s.2sfor the H atoms. For the copper acetate C, due to the size of the system, the effective core potentials proposed by Huzinaga45 have been used for the C and O atoms, with basis sets (5s6p1d)/关2s2p1d兴for oxygen atoms and (5s5p1d)/ 关2s2p1d兴for the carbon atoms. In the CH3groups and H2O molecules, minimal basis sets have been used, (6s3p)/关2s1p兴for O and C, and (3s)/关1s兴for H.46 Finally, in the Cu2O7cluster D, the Hay–Wadt pseudopotential and basis functions have been used for Cu atoms.47 For the oxygen atoms an all electron basis set (10s5p) contracted to 关3s2p兴has been employed,48 enlarged with a dpolarization function ( ␣ ⫽0.85) in the bridging-oxygen atom. The ROHF molecular orbitals have been obtained by using the MOLCAS 4.1 package.49 The CASDI 共Ref. 50兲program has been used in the CI calculations and the NATURAL 共Ref. 51兲program in the determination of the natural MOs. III. VALENCE-ONLY DESCRIPTION A. Theory 1. One-band model From the early seventies, several qualitative models were proposed52–54 to interpret the physical factors governing the magnetic coupling. Early analysis pointed out the existence of two antagonist contributions, J⫽JF⫹JAF , where Fand AF indicate ferromagnetic and antiferromagnetic contributions, respectively. JFis attributed to the direct exchange between the magnetic orbitals 共always ferromagnetic兲, and JAF is interpreted through the delocalization effect that can only occur in the singlet state 共hence antiferromagnetic兲and is sometimes called kinetic exchange. These rationalizations may follow orthogonal valence bond 共VB兲,52 nonorthogonal VB 共Ref. 53兲or valence configuration interaction 共VCI兲共Ref. 54兲arguments, but in general they handle two electrons in two magnetic 共local or symmetry-adapted兲 orbitals. Let us consider the VCI and the orthogonal VB approaches for the simplest symmetrical A•–B•system with two electrons in two orbitals. The orbitals are supposed to be previously determined, giving orthogonal core orbitals 共closed shell兲and magnetic orbitals. These magnetic orbitals may be either symmetry-adapted, gand u, or their equivalent localized transforms, aand b, g⫽a⫹b &and u⫽a⫺b & or 共2兲 a⫽g⫹u &and b⫽g⫺u &. With two active electrons in the magnetic orbitals, the CI space is limited to four determinants. In the orthogonal VB approach, for the Ms⫽0 component, there are two neutral determinants, 兩 ab ¯ 典 ⫽ 兩 core ab ¯ 典 and 兩 ba ¯ 典 ⫽ 兩 core ba ¯ 典 ,共3兲 and two ionic determinants, 兩 aa ¯ 典 ⫽ 兩 core aa ¯ 典 and 兩 bb ¯ 典 ⫽ 兩 core bb ¯ 典 .共4兲 The CI matrix takes the form 兩 ab ¯ 典 兩 ba ¯ 典 兩 aa ¯ 典 兩 bb ¯ 典 冋 0Kab tab tab Kab 0tab tab tab tab UK ab tab tab Kab U 册 ,共5兲 where the energy of the neutral VB determinants is considered as the energy origin. The direct exchange, Kab ⫽ 具 ab ¯ 兩 1/r12 兩 ba ¯ 典 , is necessarily positive since it is the selfrepulsion of the ab overlap electronic distribution, as well as U, the energy difference between the ionic and the neutral forms. The hopping integral tab⫽ 具 ab ¯ 兩 H ˆ 兩 aa ¯ 典 ⫽ 具 a 兩 F ˆ 兩 b 典 , where F ˆis the Fock operator, is the coupling between the neutral and the ionic VB forms. As shown by Hay et al.54 this integral can also be expressed by the diagonal Fock elements of the symmetry adapted orbitals, tab⫽1 2(Fgg ⫺Fuu). The four solutions are: 共i兲in the usymmetry: FIG. 2. Relative orientation of the magnetic orbitals in 共a兲the 关Cu2Cl6兴2⫺ complex; 共b兲the 关Cu2( ␮ -N3)2(NH3)6兴2⫹complex; 共c兲the Cu2( ␮ -CH3COO)4(H2O)2molecule; and 共d兲the Cu2O7cluster. 2730 J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Calzado et al. 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 09:45:28 共a兲a purely neutral triplet state, 兩 Tu 典 ⫽1 &共 兩 ab ¯ 典 ⫺ 兩 ba ¯ 典 )共6兲 of energy 3Eu⫽⫺Kab .共7兲 We also refer to this Ms⫽0 component of the triplet configuration as Tab 0⫽1 &共ab ¯ ⫺ba ¯ 兲;共8兲 共b兲a purely ionic singlet state, 兩 Su 典 ⫽1 &共 兩 aa ¯ 典 ⫺ 兩 bb ¯ 典 )共9兲 of energy 1Eu⫽U⫺Kab ;共10兲 共ii兲in the gsymmetry, two singlet states which are mixtures of the neutral and the ionic components: 共a兲the lowest one is essentially neutral, 兩 Sg 典 ⫽ ␦ 共 兩 ab ¯ 典 ⫹ 兩 ba ¯ 典 )⫹ ␥ 共 兩 aa ¯ 典 ⫹ 兩 bb ¯ 典 ), 共 ␦ ⬎ ␥ ⬎0兲.共11兲 Its energy is 1Eg⫽Kab⫹U⫺ 冑 U2⫹16tab 2 2.共12兲 It is also useful to define the singlet configuration, Sab⫽1 &共ab ¯ ⫹ba ¯ 兲;共13兲 共b兲the second one is essentially ionic, 兩 Sg ⬘ 典 ⫽⫺ ␥ 共 兩 ab ¯ 典 ⫹ 兩 ba ¯ 典 )⫹ ␦ 共 兩 aa ¯ 典 ⫹ 兩 bb ¯ 典 ), 共14兲 and much higher in energy, 1Eg ⬘⫽Kab⫹U⫹ 冑 U2⫹16tab 2 2.共15兲 The singlet–triplet energy separation is given by J⫽⌬EST⫽1Eg⫺3Eu⫽2Kab⫹U⫺ 冑 U2⫹16tab 2 2.共16兲 When UⰇ 兩 tab 兩 , a power expansion, equivalent to a perturbation of the neutral singlet by the ionic one, is possible and the coupling constant tends to52 J⫽2Kab⫺4tab 2 U,共17兲 where the opposite signs of both contributions become evident. It may be worth to introducing a diagrammatic picture of the coupling. The two neutral VB determinants may be depicted as in Diagram 1: Diagram 1 where the double arrows indicate the magnetic orbitals. Hereafter, simple left-to-right arrows →symbolize holes, i.e., inactive doubly occupied orbitals, from which an electron is excited, and simple right-to-left arrows ←symbolize particles, i.e., virtual orbitals, into which electrons are excited. The direct exchange appears as a first order interaction, represented in Diagram 2: Diagram 2 while the kinetic exchange is a second order effect, as shown in Diagram 3: Diagram 3 The same result is obtained when working with symmetryadapted MOs, 兩 Sg 典 ⫽␭ 兩 gg ¯ 典 ⫺ ␮ 兩 uu ¯ 典 , 兩 Tu 典 ⫽1 &共 兩 gu ¯ 典 ⫺ 兩 ug ¯ 典 ), 共18兲 兩 Sg ⬘ 典 ⫽ ␮ 兩 gg ¯ 典 ⫹␭ 兩 uu ¯ 典 , 兩 Su 典 ⫽1 &共 兩 gu ¯ 典 ⫹ 兩 ug ¯ 典 ). The coefficients in both approaches are related by 2 ␦ ⫽␭⫹ ␮ and 2 ␥ ⫽␭⫺ ␮ .共19兲 When 兩 t/U 兩 is small, ␭and ␮ are both close to 1/&and ␥ is small. The relations that give the values of Kab ,tab , and U from the solutions of the (2e⫺/2MO) CI are easily established, Kab⫽ 1Eg⫹1Eg ⬘⫺3Eu⫺1Eu 4,共20兲 2731J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Magnetic coupling 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 09:45:28 U⫽1Eu⫺3Eu,共21兲 tab⫽⫺ 冑 共1Eg ⬘⫺1Eg兲2⫺共1Eu⫺3Eu兲2 4.共22兲 Hence this variational CI calculation enables us to evaluate the direct exchange, Kab , as well as the hopping integral, tab , and the on-site repulsion, U. Analogously, effective Kab ,tab , and Uvalues that include the effects of the dynamic correlation can be extracted in an exact way from large CI expansions, as will be shown in part II of this work.58 2. Two-band model Analogous elementary pictures may explicitly introduce ligand orbitals, when the exchange between the metal centers is supposed to proceed through a bridging-ligand in a A•–L–B•structure. For simplicity, the ligand orbitals will be limited to a unique doubly occupied orbital, l. The two neutral VB determinants may be depicted as in Scheme 1: Scheme 1 In addition to the previously discussed direct coupling, a fourth-order indirect coupling between them is possible that proceeds through a ligand to metal charge transfer 共LMCT兲 process, according to Scheme 2: Scheme 2 where ⌬ECT is the excitation energy 共hence, positive兲to the ligand to metal charge transfer states (A⫺L⫹B•). Obviously, there is an equivalent contribution involving (A•L⫹B⫺) intermediate. The diagrammatic representation of one of them is shown in Diagram 4: Diagram 4 The corresponding contribution to Jis antiferromagnetic, J←⫺4tla 2tlb 2 ⌬ECT 2U共23兲 and is the usual interpretation of superexchange. The contribution of the consecutive charge transfer processes through the ligand may be included in an effective hopping integral, tab eff⫽tab⫹tlatlb ⌬ECT ,共24兲 and when tab is small, J⫽2Kab⫺4共tab eff兲2 U,共25兲 which returns to the one-band model, Eq. 共17兲. In principle, other mechanisms proceeding through the double charge transfer (A⫺L⫹⫹B⫺) intermediates 共of relative energy ⌬E2CT兲may also be considered giving a contribution, J←⫺8tla 2tlb 2 ⌬ECT 2⌬E2CT.共26兲 A typical representation is Diagram 5: Diagram 5 This double ligand to metal transfer is known in solid state physics as the Goodenough mechanism.55,56 Since the relative energy of the doubly ionic structure ⌬E2CT is certainly larger than Uthis mechanism may be supposed to bring very small contributions, although it has been invoked sometimes.29,56 B. Numerical results The present section reports calculations of the valenceonly CASCI 共2e⫺in 2MOs兲for the four model problems. It 2732 J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Calzado et al. 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 09:45:28 has been shown57 that the orbitals of the ROHF triplet and of the (2e/2MO) CASSCF singlet are almost identical. The results presented in Table I are obtained from the triplet state MOs and show that: 共1兲The Kab ferromagnetic direct exchange is small, without a clear correlation with the Cu–Cu distance, since the orientation of the magnetic orbitals 共Fig. 2兲is not the same and the tails on the bridging-ligands depend on their chemical nature. 共2兲The tab hopping integral contribution is quite different from one system to another. Comparing the chloride and the azido complexes, both with dxy magnetic orbitals 共Fig. 2兲, makes evident the role of the delocalization tails on the bridging ligand since tab is larger in the second one despite the larger Cu–Cu distance. The energy difference between the neutral and the ionic VB components, U, is very large 共around 2⫻105cm⫺1,⬃24–25 eV兲, much larger than usually assumed in model Hamiltonians. One may notice that this quantity is almost invariant of the ligand and of the metal-metal distance. This near-invariance is in better agreement with the onsite repulsion of the simple Hubbard model,52 than with more elaborated methods54 in which U⫽Jaa⫺Jab ,共27兲 Jaa and Jab being the one-center and two-centers Coulomb repulsions, respectively. 共3兲The overall Jvalue is very far from the experimental one. It is even of the wrong sign for the chloride complex. It shows that a lot of physics is lacking at this stage which nevertheless contains the two leading factors usually involved for interpretation. It is clear that the antiferromagnetic contribution is grossly underestimated. IV. BEYOND THE VALENCE-ONLY DESCRIPTION: DYNAMICAL CORRELATION EFFECTS A. The Brillouin’s theorem and its consequences At this stage it is important to discuss the effects of the orbitals used in the calculations. In Secs. III and IV, selfconsistent orbitals obtained from a variational restricted open-shell Hartree–Fock calculation of the triplet state are used. These orbitals satisfy the Brillouin’s theorem, which consequences can be summarized as follows. When acting on the Tutriplet state, the single excitations ai ⫹aj共where a and a⫹are annhilation and creation operators, respectively兲 lead to ai ⫹ajTudeterminants that do not interact with Tu, 具 ai ⫹ajTu 兩 H ˆ 兩 Tu 典 ⫽0. 共28兲 Three types of single excitations are to be distinguished: 共1兲The 1hdeterminants correspond to the single excitations that send one electron from an occupied horbital to the amagnetic orbital, aa ⫹ahor aa ¯ ⫹ah ¯ , 兩 hh ¯ (ab ¯ ⫺ba ¯ ) 典 ⇒ 兩 ha ¯ ab ¯ ⫺ah ¯ ba ¯ 典 . The Brillouin’s theorem imposes 具 a 兩 h ˆc⫹J ˆa⫹J ˆb 兩 h 典 ⫽0, 共29兲 where h ˆcrepresents the Fock operator for the closed shells of the system, h ˆc⫽h ˆ⫹兺 k苸core 2J ˆk⫺K ˆk,共30兲 J ˆand K ˆbeing the usual Coulomb and exchange operators. 共2兲The 1pdeterminant set contains all the single excitations from the magnetic orbitals to the virtual set, ap ⫹aaor ap ¯ ⫹aa ¯ , 兩 hh ¯ (ab ¯ ⫺ba ¯ ) 典 ⇒ 兩 hh ¯ pb ¯ ⫺hh ¯ bp ¯ 典 . The Brillouin’s theorem imposes in this case, 具 p 兩 h ˆc⫹J ˆa⫺K ˆa⫹J ˆb⫺K ˆb 兩 a 典 ⫽0. 共31兲 共3兲For the 1h⫹1pexcitations, ap ⫹ah, the Brillouin’s theorem gives 具 p 兩 h ˆc⫹J ˆa⫺K ˆa 2⫹J ˆb⫺K ˆb 2 兩 h 典 ⫽0. 共32兲 This condition implies that the state obtained under the action of the singlet ap ⫹ah⫹ap ¯ ⫹ah ¯ excitations, Shp⫽1/&(hp ¯ ⫹ph ¯ ) on the triplet state, 兩 Shp•Tu 典 ⫽ 兩 Shp•Tab 0 典 ⫽1 2 兩 共hp ¯ ⫹ph ¯ 兲共ab ¯ ⫺ba ¯ 兲 典 共33兲 does not interact with it, 具 Shp•Tu 兩 H ˆ 兩 Tu 典 ⫽0. 共34兲 TABLE I. Valence-only description with ROHF molecular orbitals. Direct exchange, Kab , hopping integral, tab , on-site repulsion, U, coupling constant, J. All results in cm⫺1. 关Cu2Cl6兴2⫺关Cu2(N3)2(NH3)6兴2⫹Cu2(CH3COO)4(H2O)2Cu2O7 dCu–Cu(Å) 3.44 5.19 2.64 3.78 2Kab 27 12 4 67 2tab ⫺1756 ⫺4436 ⫺2085 ⫺7917 U(⫻10⫺4) 19.0 20.8 19.1 19.4 J⫹11 ⫺82 ⫺19 ⫺255 Jexp 0,⫺40a⬍⫺800b⫺286c,⫺294⫾4d⫺1032⫾48e ⫺1081⫾40f aReference 62. dReference 38. bReference 63. eReference 40. cReference 64. fReference 41. 2733J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Magnetic coupling 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 09:45:28 It is easy to show that the same excitation acting on the singlet state, Shp•Sab⫽1 2 兩 共hp ¯ ⫹ph ¯ 兲共ab ¯ ⫹ba ¯ 兲 典 ,共35兲 satisfies a similar equation, 具 Shp•Sab 兩 H ˆ 兩 Sab 典 ⫽0, 共36兲 where Sab⫽1/&(ab ¯ ⫹ba ¯ ) was defined in Eq. 共13兲. In summary, single excitations from the core to the virtual space do not interact with any of both states. Single excitations from the core to the active orbitals, Eq. 共29兲,or from the active orbitals to the virtual ones, Eq. 共31兲, do not interact with the SCF triplet state but they can give small interactions with the singlet that have to be carefully analyzed. As a consequence, the ligand-to-metal charge transfer excitations responsible for the delocalization tails on the ligand do not interact with the triplet state. It is shown in Sec. IVB1a that their interaction with the singlet is also extremely small. The off-diagonal elements tla and tlb of the Fock operator are null and consequently the superexchange mechanisms, Eq. 共23兲and Eq. 共26兲, do not contribute. The variational optimization of the orbitals in the SCF procedure defines an optimal delocalization between the ligand and the magnetic centers in such a way that these effects are incorporated in the valence-only CI. B. Taking only the neutral determinants as model space 1. Theory Following the logics of a previous work,31 the quasidegenerated perturbation theory may be used as a guideline to understand the effective coupling between the neutral forms, 兩 ab ¯ 典 and 兩 ba ¯ 典 . The perturbative analysis is essentially conceptual, since the numerical analysis presented here is based on variational calculations. These two determinants may be considered as defining a physically meaningful twodimensional model space, upon which quasidegenerate perturbation theory allows to build an effective Hamiltonian, H ˆeff . Its off-diagonal matrix element gives J ⫽2 具 ab ¯ 兩 H ˆeff 兩 ba ¯ 典 . Up to the second order, 具 ab ¯ 兩 H ˆeff 共2兲 兩 ba ¯ 典 ⫽Kab⫹兺 兩 ␣ 典 ⫽ 兩 ab ¯ 典 , 兩 ba ¯ 典 具 ab ¯ 兩 H ˆ 兩 ␣ 典具 ␣ 兩 H ˆ 兩 ba ¯ 典 E0 共0兲⫺E ␣ 共0兲, 共37兲 where E0 (0)⫽E 兩 ab ¯ 典 . The second order contributions may be schematized as in Diagram 6: Diagram 6 In the preceding section, the role of 兩 ␣ 典 ⫽ 兩 aa ¯ 典 , 兩 bb ¯ 典 has been analyzed. Other outer-space determinants can be obtained by excitations involving nonvalence 共inactive兲orbitals, either core-orbitals or virtual orbitals. The list of determinants contributing up to the second order as well as the corresponding physical contributions have been discussed by de Loth et al.31 The determinants of this list will be labeled hereafter by the number nof the excited electrons from the core or holes, nh, and the number mof promoted electrons to the virtual space or particles, mp. Up to the second order, 兩 ␣ 典involve five types of determinants: 1h,1p,1h⫹1p,2h, and 2p,as represented in Fig. 3. a. The 1h and 1p determinants. Among the previously defined 1hdeterminants, the most important are the ligand to metal (L→M) charge transfer 共LMCT兲states, which do not interact with the triplet due to Brillouin’s theorem, but lead to a small interaction with the singlet state, 冓 1 &共hb ¯ ⫹bh ¯ 兲aa ¯ 冏 H ˆ 冏 1 &hh ¯ 共ab ¯ ⫹ba ¯ 兲 冔 ⫽2 具 h 兩 K ˆb 兩 a 典 ⫽2共hb,ab兲,共38兲 where the conventional notation of bielectronic integrals is used. The integral, 具 i共1兲j共2兲 兩 1 rl2 兩 k共1兲l共2兲 典 ⫽ 具 ij 兩 kl 典 ⫽共ik,jl兲共39兲 represents the interaction between the ik and jl overlap distributions. Since the Aand Bmagnetic centers are usually well separated, the ab distribution is very small. This results in a small integral and consequently small coefficients of the LMCT states in the singlet wave function, typically of the order of 10⫺2. FIG. 3. Schematic representation of the various excitation operators leading to the relevant CI spaces. 2734 J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Calzado et al. 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 09:45:28 Similarly, the 1pdeterminants, which involve metal to ligand charge transfer excitations, do not interact with the triplet state, but their interaction with the singlet state is 冓 1 &hh ¯ 共ap ¯ ⫹pa ¯ 兲 冏 H ˆ 冏 1 &hh ¯ 共ab ¯ ⫹ba ¯ 兲 冔 ⫽2 具 p 兩 K ˆa 兩 b 典 ⫽2共pa,ab兲.共40兲 As in the precedent case, for two distant magnetic centers the magnitude of this integral is small and therefore the coefficients of the 1pdeterminants on the singlet wave function are small as well. b. The 1h ⫹ 1p determinants. The 1h⫹1pdeterminants belong to three different classes. 共i兲The pure singlet excitations in the inactive part, Shp , represented in Scheme 3: Scheme 3 act on the Sab and Tab 0model space states giving the singlet, Shp•Sab⫽1 2 兩 (hp ¯ ⫹ph ¯ )(ab ¯ ⫹ba ¯ ) 典 , and the triplet, Shp•Tab 0 ⫽1 2 兩 (hp ¯ ⫹ph ¯ )(ab ¯ ⫺ba ¯ ) 典 , defined in Eqs. 共35兲and 共33兲, respectively. The interaction of these single-excited states with the singlet and triplet states does not bring any contribution to the S–Tenergy difference, due to the Brillouin’s theorem, as shown in Sec. IVA. 共ii兲The triplet excitations in the inactive part, as shown in Scheme 4: Scheme 4 introduce the spin polarization effect. The inactive part becomes Thp 0⫽1 &共hp ¯ ⫺ph ¯ 兲,Thp ⫹⫽hp,Thp ⫺⫽h ¯ p ¯ ,共41兲 which acting on the model space gives 共1兲two triplet states: Thp 0•Sab⫽1 2 兩 共hp ¯ ⫺ph ¯ 兲共ab ¯ ⫹ba ¯ 兲 典 , 1 &共Thp ⫹ •Tab ⫺⫺Thp ⫺ •Tab ⫹兲⫽1 & 兩 共hpa ¯ b ¯ ⫺h ¯ p ¯ ab兲 典 ;共42兲 one singlet state: 1 )共Thp ⫹ •Tab ⫺⫹Thp ⫺ •Tab ⫹⫺Thp 0•Tab 0兲.共43兲 The interaction of the triplet states with Tab 0brings the following contributions: 具 Thp 0•Sab 兩 H ˆ 兩 Tab 0 典 ⫽1 & 具 p 兩 K ˆa⫺K ˆb 兩 h 典 , 共44兲 1 & 具 Thp ⫹ •Tab ⫺⫺Thp ⫺ •Tab ⫹ 兩 H ˆ 兩 Tab 0 典 ⫽ 具 p 兩 K ˆa⫹K ˆb 兩 h 典 . For the singlet state, 1 ) 具 Thp ⫹ •Tab ⫺⫹Thp ⫺ •Tab ⫹⫺Thp 0•Tab 0 兩 H ˆ 兩 Sab 典 ⫽⫺ 3 & 具 p 兩 K ˆa⫺K ˆb 兩 h 典 .共45兲 The local K ˆaand K ˆbexchange operators introduce spin polarization of the inactive orbitals 共or closed shells兲. The total spin polarization energy in the triplet state is 3Ehp 共2兲⫽⫺ 1/2 具 p 兩 K ˆa⫺K ˆb 兩 h 典 2⫹ 具 p 兩 K ˆa⫺K ˆb 兩 h 典 2 ⌬Eh→p,共46兲 where ⌬Eh→pis a positive quantity representing the excitation energy of the promoted inactive electrons, E 兩 hp ¯ ab ¯ 典 ⫺E 兩 hh ¯ ab ¯ 典 . The total spin polarization energy of the singlet state is 1Ehp 共2兲⫽⫺ 3/2 具 p 兩 K ˆa⫺K ˆb 兩 h 典 2 ⌬Eh→p.共47兲 The contributions to both states are significant. The terms 具 p 兩 K ˆa 兩 h 典 2/⌬Eh→p共resp. b兲, which introduce the spin polarization of the electrons around each magnetic center, contribute equally to the singlet and the triplet states and cancel in the transition. Hence, the contribution to the S–Tenergy splitting comes only from the crossed terms, J←⫹4 具 h 兩 K ˆa 兩 p 典具 p 兩 K ˆb 兩 h 典 ⌬Eh→p.共48兲 This contribution, labeled ␧DSP 2in de Loth et al.,31 is nonnegligible only when the hp distribution is important near A and B, i.e., for hand pbeing bridging-ligand orbitals. Its sign cannot be predicted since it depends on the signs of the hp distribution near the magnetic centers, which depends on the ligand. The corresponding second-order diagrams to the coupling are of Diagram 7 type. The contribution of the spin polarization to Jis easily identified by extending the CI space, spanned by the valence CAS, with the determinants of the Thp ⫹ •Tab ⫺and Thp ⫺ •Tab ⫹type, which are responsible for the differential effect, 2735J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Magnetic coupling 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 09:45:28 Diagram 7 共iii兲The inactive single excitations with a spatial change in the active part, shown in Scheme 5, namely, the single excitations on the ionic VB forms 共singlet by nature兲give Shp•aa ¯ and Shp•bb ¯ singlets and Thp 0•aa ¯ and Thp 0•bb ¯ triplets:Scheme 5 The interaction of Shp•aa ¯ with the neutral singlet is 具 Shp•aa ¯ 兩 H ˆ 兩 Sab 典 ⫽2共ph,ab兲⫺共pb,ah兲,共49兲 while the interaction of Thp 0•bb ¯ with the triplet state is 具 Thp 0•bb ¯ 兩 H ˆ 兩 Tab 0 典 ⫽⫺共pb,ah兲.共50兲 The final second-order contribution to the singlet–triplet splitting is J←⫺8共ph,ab兲2⫺4共ph,ab兲关共pa,bh兲⫹共pb,ah兲兴 共U⫹⌬Eh→p兲. 共51兲 The ab distribution has weak amplitude everywhere and therefore this second order contribution, labeled ␧SE⫹P 2in the work of de Loth et al.,31 is expected to be small. The corresponding processes are depicted in Diagram 8: Diagram 8 However these outer space 1h⫹1pdeterminants have larger interactions with the ionic VB determinants 兩 aa ¯ 典 and 兩 bb ¯ 典 which are parts of the valence CI singlet state, 兩 Sg 典 关Eq. 共11兲兴. Actually, 1 & 具 共hp ¯ ⫹ph ¯ 兲aa ¯ 兩 H ˆ 兩 hh ¯ aa ¯ 典 ⫽& 具 p 兩 h ˆc⫹2J ˆa⫺K ˆa 兩 h 典 . 共52兲 The Brillouin’s theorem implies that 关Eq. 共32兲兴, 具 p 兩 h ˆc⫹J ˆa⫹J ˆb⫺K ˆa 2⫺K ˆb 2 兩 h 典 ⫽0, 共53兲 which gives 1 & 具 共hp ¯ ⫹ph ¯ 兲aa ¯ 兩 H ˆ 兩 hh ¯ aa ¯ 典 ⫽& 具 p 兩 J ˆa⫺K ˆa 2⫺J ˆb⫹K ˆb 2 兩 h 典 . 共54兲 The 具 p 兩 J ˆa⫺J ˆb 兩 h 典 integral is large since it represents the coupling between the hp transition distribution with the A⫺ ¯A⫹valence dipole, resulting from the modification of the field in the ionic VB structure with respect to the mean field. The corresponding energy correction represents the effect of the dynamical polarization of the ligands under the effect of the valence charge fluctuation in the singlet state. The matrix elements may be visualized as in Diagram 9: Diagram 9 2736 J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Calzado et al. 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 09:45:28 form as Eq. 共63兲, such as represented in Diagrams 21 and 22, respectively: Diagram 21 Diagram 22 These contributions of 1h⫹2pand 2h⫹1pdeterminants are nonadditive contributions since mixed fourth-order terms also exists, such as represented in Diagram 23: Diagram 23 Adding the 2h⫹1pand 1h⫹2pdeterminants to the DDCI2 list leads to the full DDCI space and has been shown in the recent past26–28,60,61 to provide a systematic agreement with experiment, in all the systems studied. The effect of the 1h ⫹2pdeterminants goes through a modification of tab involving products of two bielectronic integrals, each of them corresponding to the interaction of two transition dipoles. The bielectronic integral (ap⬘,hp)关resp. (bp⬘,hp)兴is involved in the dispersion energy between the electrons in a共resp. b兲 and in h, which is proportional to (ap⬘,hp)2关resp. (bp⬘,hp)2兴. The contribution to the effective hopping integrals no longer involves the quadratic terms, but a crossed product (ap⬘,hp)(bp⬘,hp). This contribution may therefore be considered of dispersive origin. A similar physical content can be attributed to the 2h⫹1pprocesses, now involving al transition dipoles. The effect of 2h⫹1pand 1h⫹2pcan therefore be considered as a dispersive contribution to the kinetic exchange. In other terms, the polarizability of the spectator electrons of the ligand helps to transfer the active electrons from one magnetic site to the other. It is expected that the more polarizable the bridging-ligand, the larger this effect will be. 2. Numerical results When comparing DDCI2 and DDCI results in Table II it appears that the results are significantly improved over DDCI2. The 关Cu2Cl6兴2⫺complex is now weakly antiferromagnetic, in accordance with some experimental data.62 Moreover, in all cases a reasonable quantitative agreement with experiment is obtained now. When looking at the isolated contribution of 2h⫹1pand 1h⫹2pdeterminants, reported in Table III, several features can be commented. Table III shows that the overall effect of these determinants in the systems studied is systematically antiferromagnetic. Additional calculations adding only the 2h⫹1por the 1h⫹2pto DDCI2 have been performed on the CuO2lattice fragment and on the copper acetate. They show that: 共i兲the 2h⫹1pcontribution is large and antiferromagnetic 共leading to an exaggerated value of J⫽⫺1532 cm⫺1in the cuprate and ⫺284 cm⫺1in the acetate兲; 共ii兲the 1h⫹2pcontribution is ferromagnetic 共reducing J to ⫺563 cm⫺1in the cuprate and to ⫺62 cm⫺1in the acetate兲; 共iii兲they interact at fourth order, as previously noticed 共cf. Diagram 23兲since the final contribution is not the sum of the separate contributions. The role of the bridging-ligand polarizability is manifest. The relative effect of the dispersive contribution to the total value of Jis much larger for the azido and the acetato ligands 共where they multiply Jby a factor close to 3兲than for the oxo bridge 共30% increase of J兲. The chloride case is exceptional since the effect changes the sign of an overall very small J. V. USE OF NATURAL ORBITALS Natural orbitals obtained from diagonalization of the one-electron density matrix, calculated from the DDCI most 2743J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Magnetic coupling 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 09:45:28 exact wave function can be used to reanalyze the problem. To apply the difference dedicated technique, a common set of MOs is to be used for both states, and therefore mean natural orbitals have been calculated from the average of the singlet and triplet states density matrices. As rationalized in Sec. IV of the present work, the delocalization tails between the ligand and the metallic centers become larger in the natural orbitals than in the ROHF ones, and therefore the ab overlap distribution is expected to have larger amplitudes, especially on the bridging-ligand. The remaining parameters should be modified in consequence: the direct exchange, Kab , is expected to be enlarged as well as the hopping integral, 兩 tab 兩 , and the on-center effective repulsion, U, should be slightly reduced. The results in Table V by comparing with Table I show that 共i兲Kab is drastically increased, by a factor 5 in 关Cu2Cl6兴2⫺and Cu2O7, by 20 in the acetate, by 60 in the azido complex; 共ii兲 兩 tab 兩 is larger, multiplied by 2 in 关Cu2Cl6兴2⫺and in the acetate, by 1.4 in Cu2O7, and by 3 in the azido complex; 共iii兲Uis reduced by 20%–25% in all the systems. The resulting valence-only CI remains unreliable. It is wrong in 关Cu2Cl6兴2⫺,⫹78 cm⫺1共incorrect sign兲, and weakly improved in Cu2O7,⫺452 cm⫺1共instead ⫺255 cm⫺1兲, more significantly in the azido complex, ⫺253 instead of ⫺82 cm⫺1, and ⫺33 instead of ⫺19 cm⫺1in the acetate. There is no reason to hope that the complex dynamical correlation effects can be kept through a mere revision of the orbitals. It is interesting to repeat the DDCI1, DDCI2, and DDCI calculations with these orbitals. Comparing Tables II and VI one sees that the DDCI1 and DDCI2 results are significantly improved, i.e., closer to the final value. The final 共DDCI兲results are quite comparable to those obtained from ROHF orbitals, but in slightly better agreement with experiment. The 2h⫹1pand 1p⫹2hintroduced in DDCI have now a much weaker effect, as shown in Table VII 共⫹6cm ⫺1 instead ⫺30 cm⫺1for ROHF orbitals in 关Cu2Cl6兴2⫺,⫺185 cm⫺1instead of ⫺333 cm⫺1in the Cu2O7fragment, ⫺310 cm⫺1instead of ⫺427 cm⫺1in the azido complex and practically invariant in the acetate兲. In the azido complex these effects now increase Jby 38% instead of 114% when using ROHF orbitals. Their effect still doubles the value of Jfor the acetate, while it multiplies it by a factor 3 when using ROHF orbitals. These two last cases show that it is not possible to rely on the use of quasinatural orbitals to rest on the DDCI2 level of calculation, omitting the 2h⫹1pand 1h ⫹2pexcitations. One may actually observe that the coefficients of the ligand to metal charge transfer configurations are no longer zero at the 1h⫹1por the DDCI2 level, cf. Table IV, since the Brillouin’s theorem is not satisfied anymore. However these coefficients are decreased when the 2h⫹1pand 1h⫹2pconfigurations are involved. This is opposite to the phenomenon observed when starting from ROHF orbitals. VI. CONCLUSIONS Recent works have shown the possibility to obtain accurate values of the magnetic coupling constant through ab initio CI calculations, especially when using the difference TABLE V. Contributions to the coupling constant, Jat the CASCI level, using quasinatural orbitals. Direct exchange, Kab , hopping integral, tab , on-site repulsion, U. All results in cm⫺1. 关Cu2Cl6兴2⫺关Cu2(N3)2(NH3)6兴2⫹Cu2(CH3COO)4(H2O)2Cu2O7 2Kab 162 720 66 334 2tab ⫺3528 ⫺12200 ⫺3992 ⫺11179 U(⫻10⫺4) 14.8 15.2 16.0 15.8 J⫹78 ⫺253 ⫺33 ⫺452 Jexp 0,⫺40a⬍⫺800b⫺286c,⫺294⫾4d⫺1032⫾48e ⫺1081⫾40f aReference 62. dReference 38. bReference 63. eReference 40. cReference 64. fReference 41. TABLE VI. Coupling constant, J共in cm⫺1兲, at various CI levels using quasinatural orbitals. 关Cu2Cl6兴2⫺关Cu2(N3)2(NH3)6兴2⫹Cu2(CH3COO)4(H2O)2Cu2O7 CASCI ⫹78 ⫺253 ⫺33 ⫺452 DDCI1 ⫺9⫺745 ⫺114 ⫺903 DDCI2 ⫺21 ⫺815 ⫺120 ⫺952 DDCI ⫺15 ⫺1125 ⫺238 ⫺1137 Jexp 0,⫺40a⬍⫺800b⫺286c,⫺294⫾4d⫺1032⫾48e ⫺1081⫾40f aReference 62. dReference 38. bReference 63. eReference 40. cReference 64. fReference 41. 2744 J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Calzado et al. 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 09:45:28 dedicated CI technique. However the result essentially consisted in a number, and any physical analysis of the factors leading to the final value of Jwas lacking. This prevented any confrontation with the qualitative models, popular among the specialists of the domain, and frequently used as a tool in the design of new magnetic architectures. The present work is an attempt to overcome this difficulty and to show that it is possible from accurate calculations to analyze the physical effects contributing to the observable. While perturbative approaches provide a natural way for such analysis 共and we actually have followed this way for analytic derivation and qualitative considerations兲, they are not quantitatively reliable, since many partial series are too slowly convergent 共for instance the dynamical polarization of ionic VB structures兲. We have combined localization of the magnetic orbitals into atom centered magnetic orbitals and appropriate partitions of the CI space, in order to reach the desired information. The most relevant conclusions are the following: 共1兲The physics cannot be kept restricted to the valence space, with the simple balance between the direct 共K兲 and kinetic (⫺4t2/U) exchanges, whatever the definition of the valence space 共mean field variational or natural magnetic orbitals兲, the action of the exact Hamiltonian in this restricted space leads to values of Jwhich are one order of magnitude smaller than experiment 共and sometimes even of incorrect sign兲. 共2兲The spin polarization is non-negligible, although is not the main effect beyond the valence space. Its nature 共ferro or antiferro兲is system-dependent. 共3兲The main effect beyond the CAS and belonging to the DDCI2 list, 共built from second order arguments兲comes from 1h⫹1pouter space determinants and is a fourth 共and higher兲order correction. It consists in the dynamical repolarization of the ionic VB structures. 共4兲The processes involving two inactive holes or two inactive particles have a much smaller effect. 共5兲Beyond the DDCI2 space, which is not sufficient to reach a quantitative agreement with experiment, one must involve 2h⫹1pand 1h⫹2pexcitations. Their effect is large. It is not a dynamical polarization of the ligand to metal charge transfer states, but it proceeds through a dynamical coupling of the ligand-metal transitions dipole with transition dipoles of the surrounding electrons, and an increase of the effective hopping integral of dispersive origin. At this stage, it may be interesting to see whether and how these different physical effects are included in alternative ab initio techniques. The NOCI method certainly does not include the spin polarization effects since it works with restricted open-shell SCF descriptions of each VB form. Since it only introduces a limited number of l→acharge transfer states 共with their specific h→prelaxations兲, selected on rather intuitive considerations, it certainly lacks part of the 2h⫹1pand 1h⫹2peffects, which have opposite trends. The importance of the lacking contributions may depend on the nature of the bridging-ligands. The CASPT2 approach, when starting from the valence CI space 共minimal CAS兲, introduces the effects of all DDCI perturbers and in principle does not miss any of the above considered effects. However, it is a contracted scheme and does not revise the composition of the valence part of the wave function, namely the ionic/neutral ratio in the singlet state. As shown in paper II,58 the dynamical correlation effects dramatically increase this ratio 共multiplied by a factor between 2 and 5兲through higher order effects, which are incorporated in a variational treatment. Using contracted schemes results in an underestimation of the perturbation, especially when polarizable ligands are present in the molecular structure. In these cases, the recipe consists in enlarging the CAS to include in this way part of the higher order effects. A detailed discussion comparing CASPT2 and DDCI methods will be given elsewhere. This paper has performed the above analysis with two rather different sets of orbitals, the Hartree–Fock ones, and quasinatural MOs. The magnetic orbitals are significantly more delocalized on the ligands in the second set, as phenomenologically observed,57 and rationalized here. This effect leads to larger zero-order values of the direct exchange, K, and the hopping integral t. The DDCI2 values become more accurate when natural orbitals are used, but the 2h ⫹1pand 1h⫹2pexcitations still have a non-negligible effect on the final value of J. The present analysis of the physical effects that contribute to the magnetic coupling may seem quite complex. At TABLE VII. Contributions to the coupling constant, J共in cm⫺1兲, using natural orbitals. 关Cu2Cl6兴2⫺关Cu2(N3)2(NH3)6兴2⫹Cu2(CH3COO)4(H2O)2Cu2O7 Direct exchange 162 720 66 334 Kinetic exchange ⫺84 ⫺973 ⫺99 ⫺786 1h⫹1p⫺87 ⫺492 ⫺81 ⫺451 2h,2p⫺12 ⫺70 ⫺6⫺49 2h⫹1p,1h⫹2p6⫺310 ⫺118 ⫺185 Total ⫺15 ⫺1125 ⫺238 ⫺1137 Jexp 0,⫺40a⬍⫺800b⫺286c,⫺294⫾4d⫺1032⫾48e ⫺1081⫾40f aReference 62. dReference 38. bReference 63. eReference 40. cReference 64. fReference 41. 2745J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Magnetic coupling 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 09:45:28 this stage it seems to invalidate the models that essentially stay within the valence-only space. However the next paper58 shows that the application of the effective Hamiltonian theory, makes it possible to return to a simplified picture, i.e., to the valence model space. The use of effective energies and interactions modified in order to incorporate the complex and massive effects of the outer space excitations allows us to return closer to qualitative pictures. ACKNOWLEDGMENTS These two works have been stimulated by the lectures given 共in particular by M. Verdaguer兲in the ESF Tutorial on Theoretical Aspects of Molecular Magnetism, Vienna, November 2000. The authors are indebted to C. de Graaf for many comments and suggestions. They thank the SpanishFrench Scientific Cooperation 共Integrated Action HF20000030兲. J.C. and R.C. want to thank the DGICYT of the Ministerio de Educacio ´n y Cultura of Spain 共Project No. PB981216-CO2-02兲and the CIRIT of the Generalitat de Catalunya 共Grant No. SGR99-182兲for their financial support. 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