Analysis of the magnetic coupling in binuclear complexes. II. Derivation of valence effective Hamiltonians from ab initio Cl and DFT calculations

Analysis of the magnetic coupling in binuclear complexes. II. Derivation of valence effective Hamiltonians from ab initio CI and DFT calculations 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 4 December 2001兲 Most interpretations of the magnetic coupling Jbetween two unpaired electrons rest upon simple valence models that involve essentially the ferromagnetic direct exchange contribution, Kab , and the antiferromagnetic effect of the delocalization resulting from the interaction between neutral and ionic determinants, tab , whose energy difference is U.Ab initio valence-only calculations give very poor estimates of J, whatever the definition of the magnetic orbitals, and large CI expansions are required to evaluate it properly. It is, however, possible to define valence effective Hamiltonians from the knowledge of the eigenenergies and the eigenvectors of these accurate CI calculations. When applied to four different complexes, this strategy shows that spin polarization may change the sign of the direct exchange interaction, Kab , and that dynamical correlation results in a dramatic reduction of the effective repulsion U. The present article also shows how Kab ,tab , and Ueffective parameters can be extracted from density functional theory 共DFT兲calculations and that the typical overestimation of Jin DFT can be attributed to an excessive lowering of the effective on-site repulsion. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1446024兴 I. INTRODUCTION The magnetic properties of molecular biradicals, intermolecular complexes, transition metal binuclear, or polynuclear architectures are the subject of an intense research effort. In material science as well, magnetic lattices receive an increasing attention.1–4 The basic characteristics of these systems, which all involve localized unpaired electrons, are the sign and the amplitude of the coupling Jbetween the unpaired electrons on neighbor sites. This information may be introduced in a Heisenberg–Dirac–Van Vleck spin-only Hamiltonian:5 H ˆ⫽⫺兺 i,jJijS ˆiS ˆj,共1兲 where S ˆiand S ˆjare the spin operators on sites iand j. The experimental values of Jare obtained by fitting the results coming from measurements of magnetic susceptibility, neutron scattering, or Raman spectroscopy to those obtained by assuming the Heisenberg microscopic Hamiltonian. The magnetic coupling constant Jmay be negative 共antiferromagnetism, AF兲or positive 共ferromagnetism, F兲.6This coupling is essentially local.7–9 The interaction between the nearest neighbors usually prevails, although in some architectures the second neighbor interaction has been assumed to be similar to the nearest one as in CuGeO310 and Li2CuO2.11 From the very beginning of this domain, models have been proposed which essentially rest upon a very limited set of electrons and orbitals. Most of them invoke the magnetic orbitals and the unpaired electrons only. Hence for a binuclear problem with ms⫽⫾1 2on each magnetic center, only two unpaired electrons in two local orbitals aand bare considered. The models may be formulated in a nonorthogonal valence bond 共VB兲model,12,13 in an orthogonal valence bond description14 if aand bhave been orthogonalized, i.e., 具 a 兩 b 典 ⫽0, or in a valence configuration interaction 共VCI兲 picture.15 All these models stay on a minimal valence description of the problem. They have led to some qualitative conclusions: 共i兲the direct exchange Kab between the magnetic orbitals is a ferromagnetic 共triplet favoring兲contribution; 共ii兲the other contribution is antiferromagnetic and comes from the specific electronic delocalization occurring in the singlet, through the mixing of the neutral dominant singlet VB configurations (1/&)(ab ¯ ⫹ba ¯ ) with JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 10 8 MARCH 2002 39850021-9606/2002/116(10)/3985/16/$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 10:08:14 the ionic VB determinants 兩 aa ¯ 典 and 兩 bb ¯ 典 . Its amplitude is governed by the effective hopping integral tab : 2tab⫽ 冓 1 &共ab ¯ ⫹ba ¯ 兲 冏 H ˆ 冏 1 &共aa ¯ ⫹bb ¯ 兲 冔 共2兲 and by the energy difference between the neutral and ionic VB structures: U⫽1 2关 具 共aa ¯ ⫹bb ¯ 兲 兩 H ˆ 兩 共aa ¯ ⫹bb ¯ 兲 典 ⫺ 具 共ab ¯ ⫹ba ¯ 兲 兩 H ˆ 兩 共ab ¯ ⫹ba ¯ 兲 典 兴.共3兲 Up to the second order, the antiferromagnetic contribution14 to Jis ⫺4tab 2/Uand the quantity 2tab can be related to the energy difference between the symmetry-adapted gand u molecular orbitals:15 2tab⫽␧g⫺␧u. These models have been widely used not only as a posteriori rationalization of the experiment, for instance to interpret the structural dependence of Jin a series of complexes,16–18 but also as an intellectual guide, for instance, to build ferroor ferrimagnetic lattices.19 However, ab initio calculations apparently fail to support the validity of these elementary pictures. It is actually possible to define accurate magnetic orbitals from self-consistent field 共SCF兲calculations on the upper or lower multiplets of a binuclear system. These calculations minimize the energy of a simple description of these states 共in terms of two determinants for a two electron in two orbital problem兲in a mean field approximation. Natural magnetic orbitals may as well be defined from very accurate descriptions, involving extensive configuration interaction 共CI兲expansions. But in both cases the valence-only description, i.e., the interaction between 兩 ab ¯ 典 , 兩 ba ¯ 典 , 兩 aa ¯ 典 , and 兩 bb ¯ 典 determinants, gives very poor results, the values of Jbeing frequently of incorrect sign and, when not, one order of magnitude too small.20–25 Accurate values of Jcan be obtained by ab initio CI techniques.8,9,22–30 A perturbative second-order analysis was performed by Malrieu et al.31–33 showing the importance of processes which involve other orbitals and electrons. Since the perturbation expansion is not very reliable due to convergence problems,32 a selected CI scheme has been defined from perturbative arguments. The energies and wave functions of the desired states are obtained by diagonalizing different selected spaces. The dynamical correlation effect is obtained through excitations involving either occupied 共holes, h兲or virtual 共particles, p兲inactive orbitals. Up to the second order in a perturbative expansion, the excitation can concern at most two holes and two particles. These (2h ⫹2p) excitations do not contribute to the energy difference and can be omitted in the CI expansion, leading to the variational so-called difference dedicated CI method.34 This approach has led to very accurate values of the magnetic coupling in a wide series of systems. A preceding paper35 has shown that it is possible to analyze the role of the various types of processes which go beyond the valence-only description 共spin-polarization, dynamical repolarization of ionic VB structures, etc.兲using different CI spaces. The first aim of this work is to analyze whether it is possible to return from this complex picture to a simple valence space description, in which the interactions are no longer the bare ones imposed by the direct action of the Hamiltonian, but effective interactions incorporating the effects of external correlation. The theory of effective Hamiltonians is a tool for a rigorous concentration of the information.36,37 Its principle is recalled in Sec. III. When applied to a two-electron in two-orbital problem it gives a dressed valence-only Hamiltonian. The comparison between the bare and the dressed valence-only Hamiltonians, obtained for a series of four binuclear complexes of Cu 共II兲ions 共described in Sec. II兲, shows the action of the dynamical correlation, i.e., the modification of the Kab ,tab , and U parameters. Three strategies are employed for this concentration of information, namely, 共i兲the original theory of Bloch,38 which uses information coming from four exact eigenstates, when available; 共ii兲an alternative formalism, which defines a Hermitian effective Hamiltonian, by using the Gram–Schmidt orthogonalization;39 and 共iii兲the theory of intermediate effective Hamiltonians, proposed by Malrieu et al.,40 which handles the two lowest eigenstates only. All these methods lead to a consistent conclusion concerning a dramatic reduction of the effective on-site repulsion Uas an effect of the dynamical polarization. The second prospect of the present work concerns DFT calculations on magnetic complexes 共Sec. IV兲. It is shown that it is possible to derive DFT evaluations of the integrals Kab ,tab, and U, from various solutions of the Kohn–Sham equations 共closed shell singlet, restricted open shell triplet, and broken-symmetry singlet solutions兲. These calculations lead to an overestimation of the 兩 tab /U 兩 ratio, compared to the best ab initio 兩 tab eff/Ueff 兩 value, resulting in a too large ionic VB component in the broken-symmetry solution. This explains the generally observed overestimation of Jin these approaches. The key role of the exchange potential will be illustrated and discussed. Finally, Sec. V contains the conclusions of both papers. II. DESCRIPTION OF THE SYSTEMS AND COMPUTATIONAL DETAILS Four binuclear systems involving two d9Cu共II兲centers i.e., two active electrons in two magnetic orbitals have been considered. The first one is a fragment of the CuO2square lattice of the La2CuO4perovskite, which presents superconducting behavior after hole doping, while it is a twodimensional antiferromagnetic lattice with J⬃⫺1000 cm⫺1 before doping (Jexp⫽⫺1032⫾48 cm⫺1,⫺1081⫾40 cm⫺1).41,42 ACu 2O7cluster properly embedded in a set of pseudopotentials and point charges 关Fig. 1共a兲兴 has been used to represent this system. The remaining three examples come from chemistry, namely, 共i兲the 关Cu2Cl6兴2⫺complex in a planar geometry, Fig. 1共b兲, with a weak antiferromagnetic character (Jexp ⫽0,⫺40 cm⫺1);43 3986 J. Chem. Phys., Vol. 116, No. 10, 8 March 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 10:08:14 共ii兲关Cu2(N3)2(NH3)6兴2⫹with end-to-end bridging-azido groups, Fig. 1共c兲, leading to a very strong antiferromagnetic behavior (Jexp ⬍⫺800 cm⫺1);44 共iii兲Cu2(CH3COO)4(H2O)2with four acetato bridges, Fig. 1共d兲, presenting an intermediate antiferromagnetism (Jexp ⫽⫺286,⫺294⫾4cm ⫺1).45,46 The relative orientation of the magnetic orbitals in these four systems is presented in the Scheme: for 共a兲the Cu2O7cluster, 共b兲the 关Cu2Cl6兴2⫺complex, 共c兲 the 关Cu2(N3)2(NH3)6兴2⫹complex, and 共d兲the Cu2(CH3COO)4(H2O)2molecule, respectively. A more detailed description of these systems has been reported in paper I,35 together with the computational details for the CI calculations. The MOLCAS 4.1 package47 has been used to obtain the ROHF molecular orbitals. The CASDI program48 has been used in the CI calculations and the NATURAL program49 in the determination of the natural MOs. All DFT calculations have been performed by means of the GAUSSIAN 98 code.50 The B3LYP51 parametrization has been used in the DFT calculations, with the following basis sets: for the 关Cu2Cl6兴2⫺and 关Cu2( ␮ -N3)2(NH3)6兴2⫹complexes, the Hay and Wadt core potential and basis functions have been used for copper atoms.52 All electron basis sets have been used for the remaining atoms, the 6-311G basis set for chlorine53 and the 6-31G one for hydrogen and nitrogen atoms.54 For the Cu2( ␮ -CH3COO)4(H2O)2molecule, the effective core pseudopotential and basis functions proposed by Stevens, Basch, and Krauss55 have been employed for the Cu, O, and C atoms. For the cuprate, the same basis functions as in the HF–CI calculations have been used for Cu and O atoms. In order to analyze various aspects of the dynamical correlation, four different CI spaces are used, namely, 共i兲the bare valence CAS 共CASCI兲; 共ii兲the DDCI1 space containing all the configurations reached by single excitations on the top of the CAS. Three types of excitations can be distinguished: •1h, where an electron moves from an inactive occupied orbital to an active orbital, with a possible simultaneous single excitation inside the active space; •1p, where an active electron is moved to a virtual orbital, and as in the preceding case, this movement can be coupled with single excitations inside the active space; •1h⫹1p, where an electron moves from an occupied to a virtual orbital, with possible simultaneous single excitations inside the active space. The spin polarization contributions, i.e., a simultaneous excitation of a core electron to the active space and of an active electron of different spin to a virtual orbital, are included. 共iii兲the DDCI2 space, which adds to the previous space the 2hdeterminants, i.e., two core electrons moving to the active orbitals, and 2pexcitations, i.e., two electrons from the active orbitals to the virtual ones; 共iv兲the DDCI space, containing also the 2h⫹1pdeterminants 共two core electrons moving to an active orbital and to a virtual one, respectively兲and 1h⫹2pdeterminants 共one core and one active electrons moving to two virtual orbitals兲. Regarding the DFT calculations, Noodleman’s brokensymmetry approach56 has been used to establish the value of the magnetic coupling. Since the overlap between the magnetic orbitals is rather small in all the studied systems 共see Sec. IVB兲, the limit of strong orthogonality has been considered, Jbeing calculated from J⫽2共EBS⫺ET兲,共4兲 where EBS and ETare the unrestricted broken symmetry determinent and triplet state energies, respectively. III. STRICT DETERMINATION OF VALENCE EFFECTIVE HAMILTONIANS FROM ACCURATE AB INITIO CI CALCULATIONS A. The bare valence-only Hamiltonian Let us recall briefly the nature of the model space, built from two orthogonal magnetic local orbitals, aand b.Itis composed of four determinants, two neutral, 兩 ab ¯ 典 and 兩 ba ¯ 典 , and two ionic ones, 兩 aa ¯ 典 and 兩 bb ¯ 典 . The Hamiltonian expressed in this reduced complete active space takes the form FIG. 1. Schematic representation of the four models considered: 共a兲the Cu2O7cluster; 共b兲the 关Cu2Cl6兴2⫺complex, in a planar geometry; 共c兲the 关Cu2( ␮ -N3)2(NH3)6兴2⫹complex with end-to-end bridging azido ligands; and 共d兲the Cu2( ␮ -CH3COO)4(H2O)2molecule. 3987J. Chem. Phys., Vol. 116, No. 10, 8 March 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 10:08:14 兩 ab ¯ 典 兩 ba ¯ 典 兩 aa ¯ 典 兩 bb ¯ 典 冋 HNN Kab tab tab Kab HNN tab tab tab tab HNN⫹UK ab tab tab Kab HNN⫹U 册 ,共5兲 where HNN is the energy of the neutral determinants. This space generates 共i兲a purely neutral triplet state 兩 Tu 典 ⫽(1/&)( 兩 ab ¯ 典 ⫺ 兩 ba ¯ 典 ), 共ii兲a purely ionic singlet state of usymmetry 兩 Su 典 ⫽(1/&)( 兩 aa ¯ 典 ⫺ 兩 bb ¯ 典 ), 共iii兲two singlet states of gsymmetry, 兩 Sg 1 典 and 兩 Sg 2 典 , that may be expressed as linear combinations of a purely neutral singlet 兩 Sg N 典 ⫽(1/&)( 兩 ab ¯ 典 ⫹ 兩 ba ¯ 典 ), also denoted Sab in paper I,35 and a purely ionic singlet 兩 Sg I 典 ⫽(1/&)( 兩 aa ¯ 典 ⫹ 兩 bb ¯ 典 ). The Hamiltonian can be expressed in the basis of S ˆ2eigenvector configurations. Taking the energy of the triplet 3Eu⫽HNN⫺Kab 共6兲 as the energy origin, the Hamiltonian can be written 兩 Sg N 典 兩 Sg I 典 兩 Tu 典 兩 Su 典 冋 2Kab 2tab 00 2tab 2Kab⫹U00 0000 000U 册 .共7兲 The 兩 Su 典 state lies at the energy Uabove the triplet state. In the gsymmetry the lowest singlet state 兩 Sg 1 典 ⫽cN 兩 Sg N 典 ⫹cI 兩 Sg I 典 共with cN⬎cI⬎0兲is essentially neutral. Its energy is 1Eg 1⫽Kab⫹U⫺ 冑 U2⫹16tab 2 2.共8兲 The second root 兩 Sg 2 典 ⫽⫺cI 兩 Sg N 典 ⫹cN 兩 Sg I 典 is essentially ionic and much higher in energy: 1Eg 2⫽Kab⫹U⫹ 冑 U2⫹16tab 2 2,共9兲 lying close to Uwhen 兩 tab 兩 ⰆU. Paper I of this series35 has shown how to obtain the values of the Kab ,tab , and Uintegrals from the CASCI solution. A second-order perturbative expansion leads to the well-known expression of the magnetic coupling:14 J⫽2Kab⫺4tab 2 U.共10兲 Using the relations 2tab⫽␧g⫺␧uand U⫽EI⫺EN⫽Jaa ⫺Jab 共Jaa and Jab being the one and two-center Coulomb repulsions, respectively兲, Eq. 共10兲can be written as15 J⫽2Kab⫺共␧g⫺␧u兲2 Jaa⫺Jab ,共11兲 exploited in most of the qualitative rationalizations of the magnetic coupling and of its structural dependence.16–18 B. The effective Hamiltonian approach Whatever the definition of the valence space 共Hartree Fock or natural orbitals兲, the physics of the magnetic coupling cannot be contained in the two elementary features, namely direct exchange and kinetic exchange. It would be impossible as well to reduce it to an enlarged valence space including one or a few occupied MOs of the bridging ligand, as suggested by the two-band model, popular in solid state physics, since dynamical correlation phenomena involving virtual MOs appear to be crucial in the determination of the amplitude of the magnetic coupling. But it is possible to project the exact information obtained from the large CI calculations into the valence space, considered as a model space, using the rigorous theory of effective Hamiltonians. This is the scope of the present section. 1. Quasi-degenerate perturbation theory A way to produce an effective Hamiltonian, spanned by a given model space S, consists in using the well-known quasidegenerate perturbation theory.38,57–59 This is at least a conceptual guide, as it was in the preceding paper,35 to which we shall refer for the identification of the various effects. The second-order expression of the matrix elements of H ˆeff are given by 具 I 兩 H ˆeff 共2兲 兩 L 典 ⫽ 具 I 兩 H ˆ 兩 L 典 ⫹兺 兩 ␣ 典 苸S 具 I 兩 H ˆ 兩 ␣ 典具 ␣ 兩 H ˆ 兩 L 典 EL 共0兲⫺E ␣ 共0兲, ᭙I,L苸S,共12兲 where Sis the model space and the denominator is the zeroth-order energy difference between the right component 兩L典of the matrix element and the outer space determinant 兩 ␣ 典. From this formulation it is possible to identify which matrix elements are affected by the various perturbers 兩 ␣ 典. It should be pointed out that the formalism is not Hermitian, as shown by Eq. 共12兲.If兩I典and 兩L典have different zeroth-order energies, as occurs in magnetic systems when one of them is a neutral determinant and the other an ionic determinant, then 具 I 兩 H ˆeff (2) 兩 L 典 ⫽ 具 L 兩 H ˆeff (2) 兩 I 典 . In particular, the consequence to be expected is the effect of the perturbers to be larger for the 具 neutral 兩 H ˆeff (2) 兩 ionic 典 than for the 具 ionic 兩 H ˆeff (2) 兩 neutral 典 effective hopping, since the energy denominators are larger in absolute value for the latter. Returning to the developments of paper I,35 some predictions can be made: 共i兲The spin polarization correction directly affects Kab since it couples 兩 I 典 ⫽ 兩 core ab ¯ 典 and 兩 L 典 ⫽ 兩 core ba ¯ 典 through the spin polarized aa ¯ ⫹ah ¯ ap ⫹aa 兩 core ab ¯ 典 determinant. 共ii兲The 1h⫹1pdeterminants giving the ap ⫹ahaa ¯ ⫹ab ¯ 兩 core ab ¯ 典 intermediate states change the active part of the wave function and lead to • a small second-order modification of Kab due to the interaction between the two neutral VB determinants, • a modification of tab due to the interaction between the neutral and ionic determinants, and • a more important modification of Udue to the polarization of the ionic forms, as shown in the Diagram: 3988 J. Chem. Phys., Vol. 116, No. 10, 8 March 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 10:08:14 which may be seen as the internal part of Diagram 11 in paper I.35 It gives Ueff⫽U⫹⌬U⫽U⫺兺 h兺 p 具 h 兩 J ˜ a⫺J ˜ b 兩 p 典 2 U⫹⌬Eh→p,共13兲 where J ˜ a⫽Ja⫺Ka/2 and ⌬Eh→pis the excitation energy to the h→ppromotion. The corrected kinetic exchange is obtained accordingly: ⫺tab 2 Ueff ⫽⫺ tab 2 U 冉 1⫹兺 h兺 p 具 h 兩 J ˜ a⫺J ˜ b 兩 p 典 2 U共U⫹⌬Eh→p兲⫹higher orders 冊 . 共14兲 The fourth-order term was derived in the previous paper 关Eq. 共56兲of Ref. 35兴. The higher orders introduced by the change from Uto Ueff are physical contributions to Jthrough infinite summations of the diagrams. 共iii兲The 2hand 2pexcitations essentially consist in modifications of the exchange integral through the intermediate states aa ¯ ⫹ah ¯ ⬘ab ⫹ah 兩 coreab ¯ 典 and ap ¯ ⬘ ⫹ab ¯ ap ⫹aa 兩 coreab ¯ 典 ; 共iv兲the 2h⫹1pand 1h⫹2pdeterminants essentially touch the hopping integral up through the intermediate states ap ⫹ah⬘aa ¯ ⫹ah ¯ 兩 core ab ¯ 典 and ap⬘ ⫹ahap ¯ ⫹ab ¯ 兩 core ab ¯ 典 . To summarize this section, the expected results are 共a兲the spin polarization should modify the Kab value, the sign of the correction being system dependent; and 共b兲the 1h⫹1pdynamical polarization of the ionic components should reduce the energy difference between the neutral and the ionic parts, Ueff. As shown in the Appendix, it may be predicted that at the third order the same perturbers should lower the 兩 具 aa ¯ 兩 ⌬H ˆ(3) 兩 ab ¯ 典 兩 , i.e., the 兩 具 ionic 兩 ⌬H ˆ(3) 兩 neutral 典 兩 ⫽ 兩 tIN 兩 value, while the 兩 具 ab ¯ 兩 ⌬H ˆ(3) 兩 aa ¯ 典 兩 element, that is the 兩 具 neutral 兩 ⌬H ˆ(3) 兩 ionic 典 兩 ⫽ 兩 tNI 兩 counterpart, should be left practically unchanged, contributing to the nonHermitian character of the effective Hamiltonian. The 2h⫹1pand 1h⫹2pdeterminants should affect essentially the hopping integrals, and paper I35 has shown that the effect is an increase of the tab magnitude. Of course the present discussion is based on low-order considerations and the construction of H ˆeff from the variational CI calculations may somewhat differ from second-order developments. Moreover, the QDPT expansion has no chance to converge when working with this four-dimensional space. Actually, some ligand to metal charge transfer 共LMCT兲 states lie below the ionic M⫹M⫺ 兩 Su 典 and 兩 Sg 2 典 states. These LMCT states act as intruders, resulting in small positive energy denominators and inducing the divergence of the series. Hence the QDPT arguments are purely qualitative.60 We now go on to nonperturbative approaches using our variational calculations in order to build effective Hamiltonians. 2. Effective Hamiltonian from the exact spectrum The theory developed by Bloch38 establishes a procedure to define effective Hamiltonians, expanded in a lowdimensional model space, from the knowledge of the eigenenergies and eigenstates of the exact Hamiltonian. Let us consider a model space, S, of small dimension, i.e., spanned by nvectors 兩 ⌽I 典 . Its projector is P ˆS⫽兺 I苸S 兩 ⌽I 典具 ⌽I 兩 .共15兲 Here we are interested in a valence CAS, spanned by the four VB determinants S⫽ 兵 兩 ab ¯ 典 , 兩 ba ¯ 典 , 兩 aa ¯ 典 , 兩 bb ¯ 典 其 , or their four combinations: S⫽ 兵 兩 Sg N 典 , 兩 Sg I 典 , 兩 Su 典 , 兩 Tu 典 其 . Suppose we know a large number of eigensolutions of the exact Hamiltonian: H ˆ 兩 ⌿k 典 ⫽Ek 兩 ⌿k 典 .共16兲 Consider now the neigenvectors of H ˆhaving the largest components 共or projections兲in the model space S. These eigenvectors define a target space, ST, isodimensional to S: P ˆST⫽兺 k苸ST 兩 ⌿k 典具 ⌿k 兩 .共17兲 The wave operator ⍀ ˆsends Sto ST,P ˆST⫽⍀ ˆP ˆS. We would like to define an effective Hamiltonian in S, the eigensolutions of which are the most exact and informative. This means that we want an effective Hamiltonian: H ˆeff⫽P ˆSH ˆeffP ˆS,共18兲 such that its neigenvalues are exact, and that its eigenvectors are projections of the corresponding exact eigenvectors in the model space: H ˆeff Bloch 兩 P ˆS⌿k 典 ⫽Ek 兩 P ˆS⌿k 典 .共19兲 This is the definition of the Bloch effective Hamiltonian.38 Since the projections of the 共necessarily orthogonal兲eigenvectors may be nonorthogonal, Skl⫽ 具 P ˆS⌿k 兩 P ˆS⌿l 典 ⫽0 for k⫽l,共20兲 the Bloch effective Hamiltonian may be non-Hermitian.61 It is more convenient to orthogonalize the 兩 P ˆS⌿k 典 vectors by a procedure that modifies them as little as possible. The S⫺1/2 transformation, 3989J. Chem. Phys., Vol. 116, No. 10, 8 March 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 10:08:14 兩 ⌿k ⬘ 典 ⫽S⫺1/2 兩 P ˆS⌿k 典 .共21兲 presents such a property and leads to the des Cloizeaux effective Hamiltonian:57 H ˆeff dC 兩 ⌿k ⬘ 典 ⫽Ek 兩 ⌿k ⬘ 典 .共22兲 Other orthogonalizations are possible, for instance the Gram–Schmidt one39 that is used hereafter. Of course the previous developments are applicable when using an approximate spectrum of H ˆ, resulting, for instance, from truncated CI calculations. Let us specify this technique in our two-electron/two-orbital problem. The model space being split into three subspaces of different spin and space symmetry, as shown previously, the effective Bloch Hamiltonian can only take the following form, taking as zero of energy the triplet state one: 兩 Sg N 典 兩 Sg I 典 兩 Tu 典 兩 Su 典 冏 2Kab B2tNI B00 2tIN BUB⫹2Kab B00 000 0 000UB⫹2共Kab B⫺Kab ⬘B兲 冏 ⫽H ˆeff Bloch , 共23兲 where Kab B⫽ 具 ab ¯ 兩 H ˆeff Bloch 兩 ba ¯ 典 ,共24兲 Kab ⬘B⫽ 具 aa ¯ 兩 H ˆeff Bloch 兩 bb ¯ 典 ,共25兲 tNI B⫽ 具 ab ¯ 兩 H ˆeff Bloch 兩 aa ¯ 典 ,共26兲 tIN B⫽ 具 aa ¯ 兩 H ˆeff Bloch 兩 ab ¯ 典 ,共27兲 and UB⫽ 具 aa ¯ 兩 H ˆeff Bloch 兩 aa ¯ 典 ⫺ 具 ab ¯ 兩 H ˆeff Bloch 兩 ab ¯ 典 ⫹Kab ⬘B⫺Kab B. 共28兲 This Hamiltonian is non-Hermitian, tNI B⫽tIN B,共29兲 and introduces five integrals: tNI B,tIN B,UB,Kab B, and Kab ⬘B. The projections of the eigenvectors 兩 3⌿u 典 and 兩 1⌿u 典 onto the model space are fixed by symmetry. It is not the case for the singlet states 兩 1⌿g 1 典 and 兩 1⌿g 2 典 , whose projections in the model space have a degree of freedom, namely the ratio of the coefficients on 兩 Sg N 典 and 兩 Sg I 典 : 兩 P ˆS1⌿g 1 典 ⫽cN 兩 Sg N 典 ⫹cI 兩 Sg I 典 ,cN⬎cI⬎0, 共30兲 兩 P ˆS1⌿g 2 典 ⫽⫺cN ⬘ 兩 Sg N 典 ⫹cI ⬘ 兩 Sg ⬘ 典 ,cI ⬘⬎cN ⬘⬎0. Hence the knowledge of the energy of the four states, 3Eu, 1Eu,1Eg 1, and 1Eg 2, and of the two cI/cNand cI ⬘/cN ⬘ratios fixes univocally the values of the five parameters of the Bloch effective Hamiltonian. Let us define the overlap between the projections of the two singlet states as s⫽ 具 P ˆS1⌿g 1 兩 P ˆS1⌿g 2 典 ⫽⫺cNcN ⬘⫹cIcI ⬘.共31兲 The overlap matrix takes the form S⫽ 冉 1s s1 冊 .共32兲 The biorthogonal vectors are defined by 兩 P ˆS⌿k † 典 ⫽S⫺1 兩 P ˆS⌿k 典 ,共33兲 which for the case of the projections 兩 P ˆS1⌿g 1 典 and 兩 P ˆS1⌿g 2 典 become 兩 P ˆS1⌿g 1† 典 ⫽1 1⫺s2共 兩 P ˆS1⌿g 1 典 ⫺s 兩 P ˆS1⌿g 2 典 ), 共34兲 兩 P ˆS1⌿g 2† 典 ⫽1 1⫺s2共⫺s 兩 P ˆS1⌿g 1 典 ⫹ 兩 P ˆS1⌿g 2 典 ). Using the spectral representation of the effective Hamiltonian, Heff Bloch⫽兺 i⫽1,4 兩 P ˆS⌿i 典 Ei 具 P ˆS⌿i † 兩 ,共35兲 it is possible to extract the expression for the five effective parameters tNI B,tIN B,UB,Kab B, and Kab ⬘B: 2tNI B⫽1 1⫺s2关1Eg 1共cNcI⫺scNcI ⬘兲⫹1Eg 2共scN ⬘cI⫺cN ⬘cI ⬘兲兴, 2tIN B⫽1 1⫺s2关1Eg 1共cNcI⫹scN ⬘cI兲⫺1Eg 2共scNcI ⬘⫹cN ⬘cI ⬘兲兴, UB⫽1 1⫺s2关1Eg 1„cI 2⫺cN 2⫺s共cI ⬘cI⫹cNcN ⬘兲… ⫹1Eg 2„cI ⬘2⫺cN ⬘2⫺s共cI ⬘cI⫹cNcN ⬘兲…兴,共36兲 2Kab B⫽1 1⫺s2关1Eg 1共cN 2⫹scNcN ⬘兲⫹1Eg 2共cN ⬘2⫹scN ⬘cN兲兴⫺3Eu, 2Kab ⬘B⫽1 1⫺s2关1Eg 1共cI 2⫺scIcI ⬘兲⫹1Eg 2共cI ⬘2⫺scI ⬘cI兲兴⫺1Eu. The exchange Kab ⬘Bis different from Kab B, but for simplicity, its value will not be reported nor discussed hereafter. Let us consider now the procedure to obtain a Hermitian effective Hamiltonian. The S⫺1/2 orthogonalization of the eigenvectors projector onto the model space, leading to the des Cloizeaux effective Hamiltonian, symmetrically affects the essentially neutral state 兩 1⌿g 1 典 , and the mostly ionic state 兩 1⌿g 2 典 . Since we are mainly interested in the lowest 共neutral兲singlet, it is preferable to leave 兩 P ˆS1⌿g 1 典 unchanged, and to orthogonalize 兩 P ˆS1⌿g 2 典 共the projections of the ionic eigenstate兲to 兩 P ˆS1⌿g 1 典 . This is a Gram–Schmidt orthogonalization. The second eigenvector has to satisfy 具 P ˆS1⌿g 2† 兩 P ˆS1⌿g 1 典 ⫽0, and thus cN ⬘⫽cI,cI ⬘⫽cN, 共37兲 兩 P ˆS1⌿g 2† 典 ⫽⫺cI 兩 Sg N 典 ⫹cN 兩 Sg I 典 3990 J. Chem. Phys., Vol. 116, No. 10, 8 March 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 10:08:14 is the second eigenvector of the Gram–Schmidt effective Hamiltonian H ˆeff GS , with 1Eg 2as eigenenergy. This effective Hamiltonian takes the form 兩 Sg N 典 兩 Sg I 典 兩 Tu 典 兩 Su 典 冋 2Kab GS 2tab GS 00 2tab GS UGS⫹2Kab GS 00 000 0 000UGS⫹共2Kab GS⫺2Kab GS兲 册 ⫽H ˆeff GS .共38兲 This Hermitic Hamiltonian only introduces four parameters. The knowledge of the three energy differences and of the cI/cNratio is sufficient to determine the values of these parameters. Since in this case the overlap between 兩 P ˆS1⌿g 1 典 and 兩 P ˆS1⌿g 2† 典 is zero, as follows from Eqs. 共30兲and 共37兲, we obtain the relations 2tab GS⫽cIcN共1Eg 1⫺1Eg 2兲, UGS⫽共cI 2⫺cN 2兲共1Eg 1⫺1Eg 2兲,共39兲 2Kab GS⫽cN 21Eg 1⫹cI 21Eg 2⫺3Eu. In practice it is more convenient to write the projection of the ground singlet state 兩 P ˆS1⌿g 1 典 in terms of the 兩 gg ¯ 典 and 兩 uu ¯ 典 symmetry-adapted valence determinants: 兩 P ˆS1⌿g 1 典 ⫽␭ 兩 gg ¯ 典 ⫺ ␮ 兩 uu ¯ 典 ,␭⬎ ␮ ⬎0, 共40兲 where cN⫽␭⫹ ␮ &and cI⫽␭⫺ ␮ &.共41兲 Equation 共39兲can be written as 2tab GS⫽1 2共␭2⫺ ␮ 2兲共1Eg 1⫺1Eg 2兲, UGS⫽⫺2␭ ␮ 共1Eg 1⫺1Eg 2兲,共42兲 2Kab GS⫽␭ ␮ 共1Eg 1⫺1Eg 2兲⫹1 2共1Eg 1⫹1Eg 2兲⫺3Eu. 3. Numerical results a. The Bloch effective Hamiltonian. Table I contains the Bloch effective Hamiltonian obtained from either ROHF orbitals 共ROHF MOs兲or natural orbitals 共NOs兲using the DDCI wave functions for the Cu2O7fragment of the perovskite lattice. Figure 2 shows the effective parameters tNI B, tIN B,UB, and Kab Band the cI/cNratio obtained for this system by using increasingly correlated CI wave functions. When using ROHF MOs, several features are relevant: 共i兲The bare Kab Bvalue is small and important changes appear at the DDCI level. 共ii兲The non-Hermiticity is important: (tNI B⫺tIN B)/(tNI B ⫹tIN B)⬇20%–30% and 兩 tNI B 兩 ⬎ 兩 tIN B 兩 , as expected, at all post-CASCI level. Figure 2 shows that 兩 tIN B 兩 begins to decrease due to third-order effects related with the 1h⫹1pexcitations 共DDCI1兲as explained in the Appendix. As shown from QDPT 共already in paper I35兲 the 2h⫹1pperturbers appearing at the DDCI level enhance the effective 兩 tab B 兩 value. 共iii兲The dominant effect is the drastic decrease of UB, reduced to 30% of its bare value by the effect of the dynamical polarization 共1h⫹1p, DDCI1兲. Table I and Fig. 2 共bottom兲show that NOs significantly change the zeroth-order values 共larger Kab B, larger 兩 tab B 兩 , smaller UBat the CASCI level, due to larger delocalization tails in the magnetic orbitals62兲, but the trends are similar and the final DDCI effective interactions are very close 共same value of UB, same value of the product tNI B•tIN B兲. b. The Gram–Schmidt effective Hamiltonian. In order to avoid the uncomfortable non-Hermiticity problem, the Gram–Schmidt effective Hamiltonians are given for three systems: the cuprate cluster 共Table I, Fig. 2兲, the chloride complex 共Table II, Fig. 3兲, and the azido complex 共Table III, Fig. 4兲. The case of copper acetate is discussed at the end of the section. As in Table I, Tables II and III contain the final DDCI results for the choride and azido systems, while as in Fig. 2, Figs. 3 and 4 also report values obtained from shorter CI expansions. The conclusions are quite similar to the preceding ones: 共i兲The Kab GS effective exchange may become negative. This change of sign may be correlated with the sign of the spin polarization contribution, which is antiferromagnetic in the cuprate and the azido complex 共cf. paper I35兲, but the changes in Kab GS are larger than the spin polarization contribution to J. Hence, the spin polarization is only a part of the effects contributing to Kab . 共ii兲When starting from ROHF MOs 共top of figures兲, the 兩 tab GS 兩 value decreases under the effect of the 1h⫹1p determinants 共compare CASCI and DDCI1 in Figs. 2–4兲and raises close to the original CASCI value at TABLE I. Magnetic coupling 共J兲, ionic/neutral ratio of the ground state (cI/cN), and effective parameters 共tab ,Kab , and U兲for the Cu2O7fragment of the La2CuO4lattice from CI and B3LYP calculations. All the values in cm⫺1, except Uwhich is in eV. For the CI calculations, the CASCI and the DDCI lists of determinants have been employed, with two sets of orbitals 共ROHF and natural orbitals兲. The CI parameters have been obtained by using the three kinds of effective Hamiltonians: Bloch, Gram–Schmidt 共GS兲, and intermediate 共Int兲. MOs Level JacI/cNHeff U2Kab tab ROHF CASCI ⫺255 0.041 24.12 67 ⫺3960 DDCI ⫺1077 0.118 Bloch 7.24 581 tIN ⫺3541 tNI ⫺7041 GS 7.44 ⫺229 ⫺3589 Int 8.21 ⫺139 ⫺3960 Natural CASCI ⫺451 0.070 19.65 334 ⫺5589 DDCI ⫺1136 0.136 Bloch 7.21 339 tIN ⫺4057 tNI ⫺5412 GS 7.31 ⫺22 ⫺4089 Int 9.98 386 ⫺5589 B3LYP ⫺1689 0.230 4.20 108 ⫺3903 aJexp :⫺1032⫾48 cm⫺1,⫺1081⫾40 cm⫺1, from Refs. 41 and 42. 3991J. Chem. Phys., Vol. 116, No. 10, 8 March 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 10:08:14 the DDCI level under the effect of the 2h⫹1pand 1h⫹2pdeterminants. When starting from NOs 共bottom of figures兲, the CASCI value of 兩 tab GS 兩 is much larger. It is again reduced by the 1h⫹1peffects and slightly raises under the effects of the 2h⫹1pand 1h⫹2pexcitations. The DDCI value with NOs is intermediate between the zeroth-order values 共CASCI兲 of 兩 tab 兩 from ROHF MOs and NOs. 共iii兲From both sets of orbitals, the main effect is the dramatic decrease of UGS to5eVin关Cu2Cl6兴2⫺and 7.5 eV in the cuprate. This decrease is essentially due to the 1h⫹1pexcitations 共dynamical repolarization of ionic VB structures兲. For the azido complex the results have to be considered with caution since the identification of the essentially ionic valence 兩 1⌿g 2 典 state was quite difficult and ambiguous. For instance, at the DDCI2 level with NOs, two states, which are the 14th and 24th of their symmetry, have practically equal weights on the 兩 Sg I 典 configuration. At the DDCI level, the identification of the pertinent roots was impossible when using ROHF MOs. The rather arbitrary choice between them leads to very different effective integrals. There are a large number of ligand to metal or metal to ligand charge transfer states that can have lower energy than the metal to metal charge transfer, i.e., the ionic valence-bond state, which explains the high rank of the 兩 1⌿g 2 典 state. The same is true for the 兩 1⌿u 典 ionic state. A strong mixing between LMCT and ionic states can occur, which results in the difficult assignment of the predominantly valence ionic states. For the acetato complex the identification of these states happened to FIG. 2. Effective parameters tab ,Kab , and U共in eV兲and ionic/neutral ratio (cI/cN) in the ground state wave function obtained for the Cu2O7cluster from increasingly correlated CI wave function. Three types of effective Hamiltonians: Bloch, Gram–Schmidt 共GS兲, and intermediate 共Int兲have been used and two different sets of molecular orbitals, ROHF 共on the top兲and natural orbitals 共on the bottom兲, have been considered. TABLE II. Magnetic coupling 共J兲, ionic/neutral ratio of the ground state (cI/cN), and effective parameters 共tab ,Kab , and U兲for the 关Cu2Cl6兴⫺2 complex. For the CI calculations, the CASCI and the DDCI lists of determinants have been employed, with two sets of orbitals 共ROHF and natural orbitals兲. The CI parameters have been obtained by using the Gram– Schmidt 共GS兲and the intermediate 共Int兲effective Hamiltonians. All values in cm⫺1, except for U, which is in eV. MOs Level JacI/cNHeff U2Kab tab ROHF CASCI 11 0.009 23.61 27 ⫺878 DDCI ⫺22 0.042 GS 5.07 49 ⫺853 Int 5.22 52 ⫺878 Natural CASCI 78 0.024 18.32 162 ⫺1764 DDCI ⫺15 0.056 GS 5.07 113 ⫺1145 Int 7.81 182 ⫺1764 B3LYP ⫺99 0.122 2.15 161 ⫺1062 aJexp :0,⫺40 cm⫺1, from Ref. 43. 3992 J. Chem. Phys., Vol. 116, No. 10, 8 March 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 10:08:14 be technically difficult, the ionic states being out of the first 25 lowest roots of the corresponding symmetry. These remarks illustrate the conceptual limit of the strict effective Hamiltonian approach when the model space generates a set of eigenstates with a broad energy spectrum. In such a case intruder states appear. Their impact is not simply the divergent behavior of the QDPT, as usually believed. The intruder states result in an impossible or arbitrary definition of the target space, i.e., of the set of the exact eigenvectors which are supposed to be generated from the model space. This comment supports the idea that one has to define less ambitious effective Hamiltonians, which, in this case, no longer try to generate the ionic excited states and concentrate on the two low-lying magnetic states. Such effective Hamiltonians may be defined using the concept and theory of the intermediate effective Hamiltonians. C. Valence intermediate effective Hamiltonian 1. Theory The intermediate Hamiltonian,40 built on an n-dimensional model space, is only asked to reproduce m(m⬍n) exact eigenvalues and the projections of the corresponding m eigenstates onto the model space: H ˆint 兩 P ˆS⌿k 典 ⫽Ek 兩 P ˆS⌿k 典 ,k⫽1, m⬍n.共43兲 This imposes m(m⫺1) conditions and there is an intrinsic flexibility in the definition of H ˆint. When only the kstate is looked for, it is possible to impose H ˆint⫽P ˆSH ˆP ˆS⫹P ˆS⌬ ˆkP ˆS,共44兲 where ⌬ ˆkis a state specific diagonal operator. For the particular determinant 兩I典of the model space, the eigenequations for the exact Hamiltonian lead to FIG. 3. Effective parameters tab ,Kab , and U共in eV兲and ionic/neutral ratio (cI/cN) in the ground state wave function obtained for the 关Cu2Cl6兴2⫺complex from increasingly correlated CI wave function. The Gram–Schmidt and intermediate effective Hamiltonians have been used and two different sets of molecular orbitals, ROHF 共on the top兲and natural orbitals 共on the bottom兲, have been considered. TABLE III. Magnetic coupling 共J兲, ionic/neutral ratio of the ground state (cI/cN), and effective parameters 共tab ,Kab , and U兲for the 关Cu2( ␮ -N3)2(NH3)6兴2⫹complex. For the CI calculations, the CASCI and DDCI lists of determinants have been employed, with two sets of orbitals 共ROHF and natural orbitals兲. The CI parameters have been obtained by using the Gram–Schmidt 共GS兲and the intermediate 共Int兲effective Hamiltonians. All values in cm⫺1, except for U, which is in eV. MOs Level JacI/cNHeff U2Kab tab ROHF CASCI ⫺82 0.021 25.77 12 ⫺2218 DDCI ⫺802 0.095 Int 5.73 ⫺380 ⫺2218 Natural CASCI ⫺253 0.080 18.85 720 ⫺6100 DDCI ⫺1103 0.159 GS 4.57 ⫺147 ⫺3007 Int 9.27 837 ⫺6100 B3LYP ⫺3026 0.341 3.26 ⫺144 ⫺4482 aJexp :⬍⫺800 cm⫺1, from Ref. 44. 3993J. Chem. Phys., Vol. 116, No. 10, 8 March 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 10:08:14 lations in Exchange Coupled Systems, edited by R. D. Willett, D. Gatteschi, and O. Khan, NATO Advanced Studies Series, C 共Reidel, Dordrecht, 1985兲, Vol. 140, p. 87. 34J. Miralles, O. Castell, R. Caballol, and J. P. Malrieu, Chem. Phys. 172,33 共1993兲. 35C. J. Calzado, J. Cabrero, J. P. Malrieu, and R. Caballol, J. Chem. Phys. 116, 2728 共2002兲. 36P. Durand, J. Mol. Struct.: THEOCHEM 120, 443 共1985兲. 37P. Durand and J. P. Malrieu, in Ab initio Methods in Quantum Chemistry I, edited by K. P. Lawley 共Wiley, New York, 1987兲, p. 320. 38C. Bloch, Nucl. Phys. 6, 329 共1958兲. 39See, for instance, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes 共Cambridge University Press, Cambridge, 1986兲. 40J. P. Malrieu, P. Durand, and J. P. Daudey, J. Phys. A 18, 809 共1985兲. 41P. E. Sulewski, P.A. Fleury, K. B. Lyons, S. W. Cheong, and Z. Fisk, Phys. Rev. B 41,225共1990兲; R. P. Singh, P. A. Fleury, K. B. Lyons, and P. C. Sulewski, Phys. Rev. Lett. 62, 2736 共1989兲. 42G. Aeppli, S. M. Hayden, H. A. Mook, Z. Fisk, S. W. Cheong, D. Rytz, J. P. Remeika, G. P. Espinosa, and A. S. Cooper, Phys. Rev. Lett. 62, 2052 共1989兲; Y. Endoh, K. Yamada, R. J. Birgeneau et al., Phys. Rev. B 37, 7443 共1988兲; S. M. Hayden, G. Aeppli, R. Osborn, A. D. Taylon, T. G. Perring, S. W. Cheong, and Z. Fisk, Phys. Rev. Lett. 67, 3622 共1991兲. 43R. D. Willett, in Magneto-Structural Correlation in Exchange Coupled Systems, edited by R. D. Willett, D. Gatteschi, and O. Khan, NATO Advanced Studies Series C 共Reidel, Dordrecht, 1985兲, Vol. 140. p. 389; G. Maass, B. Gerstein, and R. D. Willett, J. Chem. Phys. 46, 401 共1967兲;G. O’Bannon and R. D. Willett, Inorg. Chim. Acta 53, 6131 共1983兲. 44P. Chaudhuri, K. Oder, K. Wieghardt, B. Nuber, and J. Weiss, Inorg. Chem. 25, 2818 共1986兲. 45B. N. Figgis and R. L. Martin, J. Chem. Soc. 3837 共1956兲. 46H. U. Gu ¨del, A. Stebler, and A. Furer, Inorg. Chem. 18, 1021 共1979兲. 47K. Andersson, M. R. A. Blomberg, M. P. Fu ¨lscher et al.,MOLCAS version 4, Lund University, Sweden 共1997兲. 48CASDI program, N. Ben Amor and D. Maynau, Chem. Phys. Lett. 286,211 共1998兲. 49NATURAL program, V. M. Garcı ´a and O. Castell 共1997兲. 50M. J. Frisch, G. W. Trucks, H. B. Schlege et al.,GAUSSIAN 98, Revision A.3, Gaussian, Inc., Pittsburgh, PA, 1998. 51A. D. Becke, J. Chem. Phys. 98,5648共1993兲. 52P. J. Hay and W. R. Wadt, J. Chem. Phys. 82,299共1985兲. 53A. D. McLean and G. S. Chandler, J. Chem. Phys. 72, 5639 共1980兲;R. Krishnan, J. S. Binkley, R. Seeger, and J. A. Pople, ibid. 72,650共1980兲. 54R. Ditchfield, W. J. Hehre, and J. A. Pople, J. Chem. Phys. 54,724共1971兲; W. J. Hehre, R. Ditchfield, and J. A. Pople, ibid. 56,2257共1972兲. 55W. Stevens, H. Basch, and M. Krauss, J. Chem. Phys. 81, 6026 共1984兲;W. Stevens, M. Krauss, H. Basch, and P. G. Jasieu, Can. J. Chem. 70,612 共1992兲. 56L. Noodleman and J. G. Norman, Jr. J. Chem. Phys. 70,4903共1979兲;L. Noodleman, ibid. 74, 5737 共1981兲; L. Noodleman and E. R. Davidson, Chem. Phys. 109, 131 共1986兲; L. Noodleman, C. Y. Peng, D. A. Case, and J. M. Mouesca, Coord. Chem. Rev. 144,199共1995兲. 57J. des Cloizeaux, Nucl. Phys. 20,321共1960兲. 58J. H. Van Vleck, Phys. Rev. 33, 467 共1929兲. 59B. H. Brandow, Int. J. Quantum Chem. 15, 207 共1979兲. 60This remark only concerns the four-dimensional model space, but it does not invalidate the expansion from the neutral VB determinants 共twodimensional strictly degenerate model space兲which has been successfully employed in the past 关Ref. 32; P. De Loth, P. Karafiloglou, J. P. Daudey, and O. Kahn, J. Am. Chem. Soc. 110, 5676 共1988兲; P. De Loth, J. P. Daudey, H. Astheimer, L. Walz, and W. Haase, J. Chem. Phys. 82,5048 共1985兲兴. 61The usual QDPT expansion when converged leads to the Bloch effective Hamiltonian. 62J. Cabrero, C. J. Calzado, D. Maynau, R. Caballol, and J. P. Malrieu, submitted. 63R. Caballol, O. Castell, F. Illas, I. de P. R. Moreira, and J. P. Malrieu, J. Phys. Chem. A 101, 7860 共1997兲. 64C. Adamo, V. Barone, A. Bencini, F. Totti, and I. Ciofini, Inorg. Chem. 38, 1996 共1999兲. 65J. F. Sanz, C. J. Calzado, and A. Ma ´rquez, Int. J. Quantum Chem. 76,458 共2000兲. 66C. J. Calzado and J. P. Malrieu, Chem. Phys. Lett. 317,404共2000兲. 67R. L. Martin and F. Illas, Phys. Rev. Lett. 79, 1539 共1997兲; F. Illas and R. L. Martin, J. Chem. Phys. 108, 2519 共1998兲. 4000 J. Chem. Phys., Vol. 116, No. 10, 8 March 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 10:08:14