Electron Paramagnetic Resonance parameters from Wave Function Theory calculations. Laboratoire de Chimie et de Physique Quantiques Toulouse

Electron Paramagnetic Resonance parameters from Wave Function Theory calculations Hélène Bolvin Laboratoire de Chimie et de Physique Quantiques Toulouse

Outline Preliminaries EPR spectroscopy spin Hamiltonians time reversal operator and symmetry properties WFT theory for open shell systems transition metals, lanthanides and actinides 2 / 67

Outline Preliminaries EPR spectroscopy spin Hamiltonians time reversal operator and symmetry properties WFT theory for open shell systems transition metals, lanthanides and actinides EPR parameters from WFT theory NpCl 2 6 a fourfold degenerate ground state Ni 2+ in pseudo octahedric environment : an almost theefold degenerate ground state NpO 2+ 2 a twofold degenerate ground state : effect of the environment 2 / 67

Preliminaries 3 / 67

EPR spectroscopy Origin first observed by Soviet physicist Zavoisky in 1944 developed independently at the same time by Bleaney in Oxford. transitions between electronic magnetic states analogous to those of nuclear magnetic resonance (NMR), but the transitions occur between magnetic electronic states instead of magnetic nuclear states. concerns open-shell molecules 4 / 67

EPR spectroscopy Applications used in physics, chemistry, biology and geology interaction of the electronic spin with its microscopic neighborhood study of structure and dynamics of paramagnetic centers free radicals in solids, liquids and gaseous states biradicals transition metal and rare earth ions excited triplet states point defects in solids (F, VL centers) 5 / 67

Magnetoactive components of a molecule magnetic moment m L owing to the orbital angular momentum L = i l i 6 / 67

Magnetoactive components of a molecule magnetic moment m L owing to the orbital angular momentum L = i l i magnetic moment m S owing to the electron spin angular momentum S = i s i 6 / 67

Magnetoactive components of a molecule magnetic moment m L owing to the orbital angular momentum L = i l i magnetic moment m S owing to the electron spin angular momentum S = i s i magnetic moments of the nuclei m N owing to the nuclear spin I N 6 / 67

Magnetoactive components of a molecule For open shell molecules magnetic moment m L owing to the orbital angular momentum L = i l i magnetic moment m S owing to the electron spin angular momentum S = i s i 7 / 67

EPR : transitions between electronic Zeeman states interaction of a magnetic moment m with an external magnetic field B H Ze = m B for a pure spin doublet S = 1/2 with magnetic moment m S = µ B g e S S=1/2 M S =1/2 B M S =-1/2 hν = g e µ B B 0 8 / 67

EPR : transitions between electronic Zeeman states for technical reasons, the frequency is kept fixed, variation of the magnetic field microwaves region for the X band, ν = 9388 MHz for a spin doublet B = hν/ge µ B =0.33 T and hν=0.076 cm 1 for non pure spin systems, m µ B g e S the magnetic moment depends on the direction anisotropy 9 / 67

EPR : hyperfine coupling coupling with the spins I of the nuclei hyperfine structure m I = µ N g e I, since µ N µ B, each electron Zeeman level is split into 2I + 1 lines 10 / 67

EPR : triplet state D S=1 M S =1 M S =0 B M S =-1 when S 1, splitting of the M S components in absence of magnetic field zero-field splitting fine structure there are two transitions M S = 1 M S = 0 and M S = 0 M S = 1 High Field High Frequency EPR one can deduce the ZFS energies and the Zeeman components 11 / 67

g-factors from experiment energy gap between Zeeman states in a magnetic field B = B n E = g µ B B the g-factor depends on the direction of the magnetic field, this anisotropy is modelled by a spin Hamiltonian Experiments give access to the tensor H S = µ B B g S g = ± [ n gg n ] 1/2 G = gg One defines the principal g-factors g i from G i the principal values of G g i = ± G i 12 / 67

Determination of the sign of g-factors by use of circularly polarized radiation, the relative intensities of a given transition using right- and left-handed senses give information about the sign of g x g y g z in the case of hyperfine coupling, when the sign of the hyperfine constant is known 13 / 67

Semiclassical approach: precession around a magnetic field The anisotropy of the g-factors affects both the shape and the pulsation of the precession y gyµ0 x µ(t) gegzbzt g x µ0 B z the direction of the precession is defined by the sign of the product g x g y g z magnetic field to a principal axis z 14 / 67

Spin Hamiltonian Model Hamiltonian H S = S D S + µ B B g S it is a phenomenological relation expressed with spin operators S is a pseudo spin operator acting in the model space. With spin-orbit coupling, the true spin operator S is not a good quantum number. In the case of a small spin-orbit coupling, S S (transition metal complexes) otherwise, the value of S is chosen to suit the size of the model space the size of the model space is 2 S + 1 generated by the M S functions 15 / 67

Spin Hamiltonins spin algebra in the model space, algebra of spins in other directions S z M S z = M S M S z rotation in the spin space R: rotation z x y z R : rotation z y x z S + M S z = M S ( S+ M S +1)( S M S ) M S +1 z S M S z = M S ( S M S +1)( S+ M S ) M S 1 z S x M S x = M S M S x S y M S y = M S M S y S z M S z = M S M S z R M S z = M S x R M S z = M S y 16 / 67

Spin Hamiltonian EPR spin Hamiltonian H S = S D S + µ B B g S D is a two-rank tensor, usually traceless : the ZFS tensor g is in general not a tensor but G = gg is the principal axis of D and G are not the same in general 17 / 67

Spin space and real space the spin space is an ideal world in which the algebra is well defined : all is for the best in the best of all possible worlds (Candide Voltaire) the phenomenological parameters can be extracted from experimental data by the fitting of physical observables. the model is presupposed quantum chemical calculations 1 validation of the model 2 calculation of the parameters 18 / 67

Open shell systems 3d, 4d, 5d, 4f, 5f elements 19 / 67

Open shell systems ε 9 ε 8 ε 7 ε 6 ε 9 ε 8 ε 7 ε 6 S=1 one electron terms maximal filling of the orbitals = favors closed shell two electron terms ε 5 ε 4 ε 3 ε 2 ε 5 ε 4 ε 3 ε 2 coulombic integral = favors open shell exchange integral stabilizes parallel spins open shells systems are obtained with degenerate or quasi degenerate orbitals ε 1 ε 1 in the free ion : nd : 5-fold degeneracy, 7f :7-fold degeneracy ground state : maximal S, maximal L (Hund s rules) interaction with the environment : splitting of these orbitals, quenching of the orbital moment open shell molecules when the splitting is relatively small 20 / 67

WFT for open shell systems the wave function of an open shell system is multideterminantal for example 2 electrons in 2 orbitals a and b a triplet S = 1 1, 1 = ab 1, 0 = 1 ( ) a b + āb 2 1, 1 = ā b a singlet S = 0 0,0 = 1 2 ( a b āb ) 21 / 67

the SO-CASPT2 method 1st step spin-free calculation CAS CASSCF active orbitals : at the least the open shell orbitals CASPT2 dynamical correlation scalar relativistic effect are introduced at this stage, the wave functions belong to an irrep of the simple group and have a well defined spin S 2S+1 Γ 22 / 67

the SO-CASPT2 method 2nd step : spin-orbit coupling the spin-orbit matrix is written in the basis of the CASSCF { Ψ I } wave functions and diagonalized wave functions are no more eigenfunctions of the spin wave functions belong now to the irreps of the double group 23 / 67

Response to a magnetic field Hamiltonian of a molecule in a permanent magnetic field B (to 1st order) H = H ZF M B with ( ) M = µ B r i j i i=el with r i and j i position and current density of electron i in the non-relativistic limit ( ) M = µ B l i + g e s i i=el }{{ ) } = µ B (ˆ L + g e ˆ S + 1 4 ( ) r i A + diamag 24 / 67

Magnetic moment matrix 1- Solution of the zero field (ZF) equation H ZF Ψ I = E I Ψ I 2- Calculation of the magnetic moment matrix elements m IJ = µ B Ψ I L + g e S Ψ J = µ B Ψ I M Ψ J H = H ZF Ψ 0 Ψ 1 Ψ N Ψ 0 E 0 0 0 Ψ 1 0 E 1 0..... + M Ψ 0 Ψ 1 Ψ N Ψ 0 m 00 m 01 m 0N Ψ 1 m 10 m 11 m 1N..... B Ψ N 0 0 EN Ψ N m N0 m N1 m NN 25 / 67

Time reversal symmetry Electric field vs magnetic field Schrödinger equation of a particle with charge q in an electric field i h ψ(r,t) t ψ (r, t) is a solution of (1) in a magnetic field i h ψ(r,t) t ] = [ h2 2m 2 + qφ(r) ψ(r,t) (1) = 1 [ i h q ] 2 2m c A(r) ψ(r,t) (2) ψ (r, t) is not a solution of (2) but of the equation with reversed magnetic field 26 / 67

Time reversal operator Θ Θ is a antilinear and antiunitary operator time reversal operator in r representation, for systems with spatial and spin degrees of freedom where K is the conjugation operator. Θ = Ke iπs y / h = e iπs y / h K 27 / 67

Symmetry of the Hamiltonian under time reversal Zero field Hamiltonian Zero-field Hamiltonian H = p 2 2m + V ( r) + f (r) L S commutes with time reversal The angular momentum operators Θ MΘ = M M = L, S, J anticommutes with time reversal 28 / 67

Kramers degeneracy Θ 2 = e 2πiS y / h total spin rotation of angle 2π about the y-axis for bosons Θ 2 = 1 for fermions Θ 2 = 1 for H invariant under time reversal, [Θ,H] = 0 ; if H Ψ = E Ψ H(Θ Ψ ) = ΘH Ψ = E(Θ Ψ ) Ψ and Θ Ψ are degenerate if Ψ is non degenerate Θ Ψ = e iα Ψ Θ 2 Ψ = Θe iα Ψ = e iα Θ Ψ = Ψ not possible for fermions 29 / 67

Kramers degeneracy since for fermions Θ 2 = 1 In time-reversal invariant systems with an odd number of fermions, the energy levels have a degeneracy that is even. Kramers degeneracy is lifted by any effect that breaks the time-reversal invariance, for example external magnetic fields. 30 / 67

Magnetic moment of a non-degenerate state The magnetic moment is antisymmetrical under time reversal Θ Ψ M Ψ = Ψ M Ψ if Ψ is non degenerate Ψ M Ψ = Ψ M Ψ = 0 The magnetic moment of a non-degenerate state vanishes 31 / 67

Time reversal : conclusion 1 the magnetic moment operator is antisymmetrical towards time inversion Θ MΘ = M 2 the zero-field energy of a system with an odd number of fermions have a degeneracy that is even. This degeneracy is lifted by external magnetic field. 3 The magnetic moment of a non-degenerate state vanishes 32 / 67

EPR Parameters from WFT calculations 33 / 67

NpCl 2 6 a Kramers quartet NpCl 2 6, configuration (5f )3 cubic symmetry the ground state is of symmetry F 3/2u The highly symmetric lattice of Cs 2 ZrCl 6 preserves the symmetry of the Γ 8 state EPR spectrum (Bray, Bernstein, Dennis 1978) 34 / 67

8 NpCl 2 6 5f orbital in an octahedric environment E 1/2 2 Τ 1u G 3/2 Θ E 1/2 G 3/2 7/2;E 5/2 > J=7/2 5λ/2 2 Τ 2u G 3/2 G 3/2 2 Τ 2u;E 5/2 > 5/2;E 5/2 > J=5/2 2 Α u 2 Α u;e 5/2 > pure CF SOC 1st order SOC 2nd order pure SO λ=0 λlzsz λ λl.s O * h =0 O * 3 35 / 67

NpCl 2 6 : a Kramers quartet the ground state is 4-fold degenerate S = 3 2 S=3/2 M S=3/2 M S=3/2 gµ B B M S=1/2 M S=1/2 gµ B B MS=-1/2 gµ B B Γ 8 MS=-1/2 MS=-3/2 MS=-3/2 pure spin orbital contribution Spin Hamiltonian H S = µ B g B S + µ B G [ ] B x S x 3 + B y S y 3 + B z S z 3 x y and z are equivalent : g is isotropic One adds cubic terms to take into account the contribution of angular momentum 36 / 67

NpCl 2 : a Kramers quartet 6 model matrices basis set 3 2 1 2 1 2 3 2 M x = M y = 0 3( 1 2 g + 7 8 G) 0 3 4 G 3( 1 2 g + 7 8 G) 0 (g + 5 2 G) 0 0 (g + 5 2 G) 0 3( 1 2 g + 7 8 G) 3 4 G 0 3( 1 2 g + 7 8 G) 0 0 i 3( 1 2 g + 7 8 G) 0 i 3 4 G i 3( 1 2 g + 7 8 G) 0 i (g + 5 2 G) 0 0 i (g + 5 2 G) 0 i 3( 1 2 g + 7 8 G) i 3 4 G 0 i 3( 1 2 g + 7 8 G) 0 M z = 3 2 g 27 8 G 0 0 0 0 1 2 g 1 8 G 0 0 0 0 1 2 g + 1 8 G 0 0 0 0 3 2 g + 27 8 G 37 / 67

NpCl 2 : a Kramers quartet 6 SO-CASPT2 matrices basis set Ψ 1 Ψ 2 Ψ 3 Ψ 4 M x = 0.174 0.088 + 0.10i 0.572 + 0.003i 0.254 + 0.044i 0.088 0.101i 0.174 0.258 + 0.005i 0.561 + 0.114i 0.572 0.003i 0.258 0.005i 0.348 1.033 1.447i 0.254 0.044i 0.561 0.114i 1.033 + 1.447i 0.348 eigenvalues -2.028 2.028 0.096-0.096 M y = 0.886 0.194 0.827i 0.950 0.079i 0.112 + 0.139i 0.194 + 0.827i 0.886 0.083 0.158i 0.947i + 0.107i 0.950 + 0.079i 0.083 + 0.158i 0.413 0.131 + 0.737i 0.112 0.139i 0.947 0.107i 0.131 0.737i 0.413 eigenvalues -2.028 2.028 0.096-0.096 M z = 0.331 0.986i 1.251i 0.430 0.134i 0.236 + 0.614i 0.986 + 1.251i 0.331 0.11321 0.64851I 0.396 0.215i 0.430 + 0.134i 0.113 + 0.648i 0.369 0.1235 0.197i 0.236 0.614i 0.396 + 0.215i 0.1235 + 0.197i 0.369 eigenvalues -2.028 2.028 0.096-0.096 38 / 67

NpCl 2 : a Kramers quartet 6 SO-CASPT2 matrices in M z eigenvectors basis set R Ψ 1 R Ψ 2 R Ψ 3 R Ψ 4 M x = M y = M z = 0.000 0.799 0.248i 0.000 0.440 0.378i 0.799 + 0.248i 0.000 1.025 1.155i 0.000 0.000 1.025 + 1.155i 0.000 0.260 + 0.795i 0.440 + 0.378i 0.000 0.260 0.795i 0.000 0.000 0.247 0.799i 0.000 0.378 + 0.440i 0.247 + 0.799i 0.000 1.155 + 1.026i 0.000 0.000 1.155 1.026i 0.000 0.795 + 0.261i 0.378 0.440i 0.000 0.795 0.261i 0.000 2.026 0.000 0.000 0.000 0.000 0.096 0.000 0.000 0.000 0.000 0.096 0.000 0.000 0.000 0.000 2.026 39 / 67

NpCl 2 : a Kramers quartet 6 SO-CASPT2 matrices in M z eigenvectors, M x real basis set R Ψ 1 R Ψ 2 R Ψ 3 R Ψ 4 M x = M y = M z = 0.000 0.837 0.000 0.588 0.837 0.000 1.556 0.000 0.000 1.556 0.000 0.837 0.588 0.000 0.837 0.000 0.000 0.837i 0.000 0.588i 0.837i 0.000 1.556i 0.000 0.000 1.556i 0.000 0.837i 0.588i 0.000 0.837i 0.000 2.042 0.000 0.000 0.000 0.000 0.104 0.000 0.000 0.000 0.000 0.104 0.000 0.000 0.000 0.000 2.042 these three matrices match the three model matrices with g=-0.406 G=0.785 EPR experimental values g= -0.516 G= 0.882 40 / 67

NpCl 2 : a Kramers quartet 6 conclusion validation of the Spin Hamiltonian We have found the correspondance between the model space and the real space 3 Ψ3/2 = R Ψ 1 2 1 Ψ 2 1/2 = R Ψ 2 1 Ψ 1/2 = R Ψ 3 2 3 Ψ 2 3/2 = R Ψ 4 extraction of g and G in reasonable agreement with experimental values 41 / 67

NpCl 2 : a Kramers quartet 6 conclusion from the spin and orbital moment matrices one gets spin contribution gs = 0.027 G s = -0.250 orbital contribution gl = -0.460 G S =1.285 the orbital and spin contributions are opposite as they are in the free ion : J = L S the magnetic moments are very different from the free ion one S S Ψ±3/2 M z Ψ ±3/2 Ψ±1/2 M z Ψ ±1/2 = 2.04 3 2 g e = ±1.05 1 2 g e 42 / 67

Pseudooctahedric Ni(II) complex : S = 1 1 Γ 3 Γ 3 T 3 3 T 2 3 T 1 11017 11076 10987 10533 10631 10307 7769 7761 7619 Ni(II) 3d 8 without spin-orbit : pure spin, isotropic 3 A no SOC spin-orbit : some orbital moment of the excited states = anisotropic model space S = 1 SOC 18 2 43 / 67

S = 1 Spin Hamiltonian H S = S D S + µ B B g S In the frame of the principal axis of the ZFS tensor D HS ZF = 1 ( ) D x S x 2 + D y S y 2 + D z S z 2 2 basis set for the spin space 0 x, 0 y, 0 z 0 x = R 0 z = 1 2 ( 1 z + 1 z ) 0 y = R 0 z = i 2 ( 1 z + 1 z ) H ZF S 0 x = 1 2 (D y + D z ) = E 0 x H ZF S 0 y = 1 2 (D x + D z ) = E 0 y H ZF S 0 z = 1 2 (D x + D y ) = E 0 z 44 / 67

Pseudo-spin S = 1 Isotropic Zeeman interaction H S = S D S + µ B B g S Sx 0 x 0 y 0 z 0 x 0 0 0 0 y 0 0 i 0 z 0 i 0 S y 0 x 0 y 0 z 0 x 0 0 i 0 y 0 0 0 0 z i 0 0 S z 0 x 0 y 0 z 0 x 0 i 0 0 y i 0 0 0 z 0 0 0 with a magnetic field B = B x e x + B y e y + B z e z H S 0 x 0 y 0 z 0 x E 0 x iµ B gb z iµ B gb y 0 y iµ B gb z E 0 y iµ B gb x 0 z iµ B gb y iµ B gb x E 0 z the states do not have any magnetic moment since they are non degenerate 45 / 67

Pseudo-spin S = 1 Isotropic Zeeman interaction a magnetic field in direction x induces a coupling between 0 y and 0 z and induces a magnetic moment, even more that E z E y is small. H S 0 x 0 y 0 z 0 x E x 0 0 0 y 0 E y iµ B gb x 0 z 0 iµ B gb x E z 46 / 67

Pseudo-spin S = 1 Pseudooctahedric Ni(II) complex Spin Hamiltonian energies with E 0 x =0 E 0 y =2.5 E 0 z =18.0 and g=2.3 z is the axis of easy magnetization 47 / 67

Ni(II) complex : D tensor without spin-orbit coupling the three components 3 A,1 3 A,0 and 3 A, 1 are degenerate with spin-orbit Ψ 1 = c 11 3 A,1 + c 12 3 A,0 + c 13 3 A, 1 + c 1IMS 3 Γ I,M S + c 1J 1 Γ J,0 I,M S IJ Ψ 2 = c 21 3 A,1 + c 22 3 A,0 + c 23 3 A, 1 + c 2IMS 3 Γ I,M S + c 2J 1 Γ J,0 I,M S IJ Ψ 3 = c 31 3 A,1 + c 32 3 A,0 + c 33 3 A, 1 + c 2IMS 3 Γ I,M S + c 3J 1 Γ J,0 I,M S IJ with energies E 1, E 2 and E 3. Since the spin-orbit is a perturbation, ci1 2 + c2 i2 + c2 i3 close to 1. 48 / 67

Ni(II) complex : D tensor effective Hamiltonian technique effective Hamiltonian matrix in the 3 A,1 3 A,0 3 A, 1 target space C ij = c ij i,j = 1,3 E ij = E i δ ij properties of effective Hamiltonian H eff = C 1 EC H eff P Ψ 1 = E 1 P Ψ 1 H eff P Ψ 2 = E 2 P Ψ 2 H eff P Ψ 3 = E 3 P Ψ 3 P = MS 3 A,M S 3 A,M S projector on the target space 49 / 67

Ni(II) complex : D tensor effective Hamiltonian basis set 3 A,1 3 A,0 3 A, 1 H eff = 0.256 0.339 + 0.427i 2.882 + 1.175i 0.339 0.427i 0.512 0.339 0.427i 2.882 1.175i 0.339 + 0.427i 0.256 basis set 1 0 1 H S = 1 4 (D xx + D yy + 4D zz ) 1 2 (D zx id zy ) 1 4 (D xx D yy 2iD xy ) 1 2 (D zx + id zy ) 1 2 (D xx + D yy ) 1 2 ( D zx + id zy ) 1 4 (D 1 1 xx D yy + 2iD xy ) 2 ( D zx id zy ) 4 (D xx + D yy + 4D zz ) comparison of the two matrices all the elements of the D tensor the diagonalization of D lead to the magnetic axis Maurice, R et al JCTC 11 (2009) 2977. 50 / 67

Ni(II) complex : g tensor model matrices basis set 0 x 0 y 0 z M x = M y = M z = 0 ig xz ig xy ig xz 0 ig xx ig xy ig xx 0 0 ig yz ig yy ig yz 0 ig yx ig yy ig yx 0 0 ig zz ig zy ig zz 0 ig zx ig zy ig zx 0 51 / 67

Ni(II) complex : g tensor calculated matrices basis set Ψ 1 Ψ 2 Ψ 3 0 0.015 + 0.014i 0.004 + 0.332i M x = 0.015 0.014i 0 1.617 + 1.545i 0.004 0.332i 1.617 1.545i 0 M y = 0 0.045 0.044i 0.031 + 2.252i 0.045 + 0.044i 0 0.237 0.227i 0.031 2.252i 0.237 + 0.227i 0 M z = 0 1.692 + 1.663i 0.000 + 0.054i 1.692 1.663i 0 0.019 0.018i 0.000 0.054i 0.019 + 0.018i 0 52 / 67

Ni(II) complex : g tensor calculated matrices after rotation basis set Ψ 1 e α Ψ 2 e β Ψ 3 M x = 0 0.021i 0.333i 0.021i 0 2.237i 0.333i 2.237i 0 basis set 0 x 0 y 0 z M S x = 0 ig xz ig xy ig xz 0 ig xx ig xy ig xx 0 M y = 0 0.064i 2.252i 0.064i 0 0.328i 2.252i 0.328i 0 M S y = 0 ig yz ig yy ig yz 0 ig yx ig yy ig yx 0 M z = 0 2.373i 0.055i 2.373i 0 0.027i 0.055i 0.027i 0 M S z = 0 ig zz ig zy ig zz 0 ig zx ig zy ig zx 0 53 / 67

Ni(II) complex : g tensor one builds g one deduces G = gg one gets the principal axis of g and the g factors g 1 = 2.374 g 2 = 2.277 g 3 = 2.261 principal axis of G close to those of D 54 / 67

Pseudo-spin S = 1 Pseudooctahedric Ni(II) complex Spin Hamiltonian versus ab initio 55 / 67

Pseudo-spin S = 1 Pseudooctahedric Ni(II) complex 1 2 3 4 5 Ni(bipy) 3 Ni(bipy) 2 (NCS) 2 Ni(bipy) 2 ox Ni(bipy) 2 NO 3 NiL 2 NO 3 X-Ni-X 79 91 80 60 60 exp E 1 0.40 0.82 0.12 0.46 0.34 (cm 1) ) E 2 1.38 2.15 1.50 6.32 10.70 SO-CASPT2 E 1 1.046 3.471 1.972 2.969 0.963 E 2 3.330 5.292 4.525 7.108 12.011 56 / 67

Pseudo-spin S = 1 Conclusions for transition metal complexes with a quenched orbital momentum and S = 1 the anisotropic magnetic properties are due to the zero field splitting since the Zeeman interaction is almost isotropic the anisotropy is larger with a scheme E x E y E z 57 / 67

Kramers doublet S = 1/2 the model space is generated by Ψ and Ψ = Θ Ψ H = Ĥ ZF Ψ Ψ Ψ E 0 0 + Ψ 0 E 0 M Ψ Ψ Ψ m 0 m 1 B Ψ m 1 m 0 modelled by the Spin Hamiltonian Ĥ S = µ B B.g. S the spin space is generated by 1 2, 1 2 S x 12 12 12 0 1 2 1 1 2 2 0 S 12 12 y 12 0 1 2 i 1 1 2 2 i 0 S 12 12 z 12 1 2 0 1 2 0 1 2 58 / 67

Kramers doublet S = 1/2 By mapping the two matrices of Ĥ in the Ψ, Ψ basis set and ĤS in the 1 2, 1 2 basis set, one gets Equations defining the g matrix of a Kramers doublet g kx = 2Re ( Ψ L k + g e S k Ψ ) g ky = 2Im ( Ψ Lk + g e S k Ψ ) g kz = 2 ( Ψ L k + g e S k Ψ ) G = gg is a tensor. Its diagonalization leads to the magnetic axis. 59 / 67

[NpO 2 (NO 3 ) 3 ] and [NpO 2 Cl 4 ] 2 orbitals σ * u π * u φu(a 1 ) 5f φu δu 2p δu(e ) Γ φu(a 2 ) σg, σu, πg, πu NpO 2 (NO 3 ) 3 ] - Np NpO 2+ 2 2 O Np 6+ 5f 1 configuration NpO 2+ 2 φ and δ non bonding with 3 NO 3 symmetry D 3h splitting of the φ orbitals with 4 Cl symmetry D 4h splitting of the δ orbitals 60 / 67

NpO2 in Cs2 [UO2 (NO3 )3 ] 61 / 67

[NpO 2 (NO 3 ) 3 ] sym SO-CASPT2 exp E 1/2 E 0 g /g -3.554/±0.236-3.405 d ; 3.36 e /±0.205 d ; 0.20 e %φ/%δ 80/20 E 3/2 E (cm 1 ) 435 g /g 2.051/±0.042 %δ/%π 96/4 E 1/2 E (cm 1 ) 6997 6459.0 a,b g /g -3.792/±2.208 3.7 a /±2.1 b %φ/%δ 26/74 E 1/2 E (cm 1 ) 10583 9420.2 g /g 5.375/±2.441 ±4.9 a ± 5.29 b / ±2.5 a ± 2.17 b %φ/%δ 94/6 E 5/2 E (cm 1 ) 15167 17841 a 17483.6 b g /g 0.016/0.041 0.9 a ±0.12 b /±0.2 b E 3/2 E (cm 1 ) 17882 20816.3 a,b g /g 3.909/0.040 3.2 ± 0.4 a ±4.11 ± 0.09 b /0 a ;±0.2 b absorption a: Leung Phys.Rev. 187, 2 (1969) 504.b: Denning Mol.Phys. 46 (1982) 287. EPR d: Pryce Phys.Rev.Lett. 3 (1959) 375. e: Leung Phys.Rev. 180, 2 (1969) 380. 62 / 67

NpO 2 in Cs 2 [UO 2 Cl 4 ] 63 / 67

[NpO 2 Cl 4 ] sym free ion embedded exp E 1/2 E 0 0 0 g /g 3.274/±1.271-0.615/±1.430 ±1.32±0.02 a ±1.38 ± 0.07 c /1.3±0.3 a %φ/%δ 67/33 30/69 E 3/2 E (cm 1 ) 934 782 900-1050 g /g -1.100/±1.426-1.734/±0.748 %δ/%π 13/83 46/52 E 1/2 E (cm 1 ) 7751 7481 6880.4 b,c g /g 5.512/±0.027 5.397/±0.054 %φ/%δ 20/80 24/76 E 1/2 E (cm 1 ) 8600 9083 ca 7890 c g /g 7.719/0.003 7.675/±0.056 E 5/2 E (cm 1 ) 17442 16911 17241.4 g /g 0.085/0.000-0.108/±0.224 0.6 ± 0.45 c / 0.2 ± 0.3 c E 3/2 E (cm 1 ) 20252 19737 20080.8 g /g -3.884/±0.1281 3.883/0.033 3.8 ± 0.4 c /g = 0.1 ± 0.5 c absorption b: Stefsund Phys. Rev.. 180 (1969) 339. c: Denning Mol.Phys. 46, 2 (1982) 287 EPR a: Leung Phys.Rev. 180, 2 (1969) 380. 64 / 67

NpO 2+ 2 + equatorial ligands calculation of the g factors for the ground state and the excited states assignment of the signs good accordance with the experimental data for the nitrate system the chloride system is much more difficult to describe correctly since the ground state is close to an avoiding crossing. The effect of the quality of the description is very important the equatorial ligand have a dramatic effect on the magnetic behavior of the neptunyle ion 65 / 67

Conclusions in the case with S = 1/2, S = 1 and S = 3/2, calculation of the g and D tensor from first principles in each case, the full exploitation of the wave functions lead to more information than just taking the eigenvalues SO-CASPT2 values compare correctly to the experimental ones but numerical accuracy can be improved most of the presented results can be rationalized using crystal field theory. 66 / 67

Acknowledgments Liviu Chibotaru, Liviu Ungur (Leuven) Nathalie Guihéry, Jean-Paul Malrieu (Toulouse) Jochen Autschbach, Frédéric Gendron (Buffalo) previously in Toulouse Rémi Maurice (Nantes), Elena Malkin (Tromsø), Dayán Páez Hernández (Santiago) 67 / 67