Calculation of EPR parameters by WFT H´ el` ene Bolvin Laboratoire de Chimie et de Physique Quantiques Toulouse pNMR Training Course Mariapfarr Feb 22-24 2014
Outline Generalities EPR spectroscopy spin Hamiltonians SO-CASSCF based methods time reversal operator and symmetry properties (Toulouse) 2 / 68
Outline Generalities EPR spectroscopy spin Hamiltonians SO-CASSCF based methods time reversal operator and symmetry properties Examples NpCl 2 − a fourfold degenerate ground state 6 Ni 2 + in pseudo octahedric environment : an almost threefold degenerate ground state (Toulouse) 2 / 68
Generalities (Toulouse) 3 / 68
Magnetoactive components of a molecule m L owing to the orbital angular momentum � L = ∑ i � magnetic moment � l i (Toulouse) 4 / 68
Magnetoactive components of a molecule m L owing to the orbital angular momentum � L = ∑ i � magnetic moment � l i magnetic moment � m S owing to the electron spin angular momentum � S = ∑ i � s i (Toulouse) 4 / 68
Magnetoactive components of a molecule m L owing to the orbital angular momentum � L = ∑ i � magnetic moment � 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 (Toulouse) 4 / 68
Magnetoactive components of a molecule m L owing to the orbital angular momentum � L = ∑ i � magnetic moment � 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 rotational magnetic moment � m R owing to the rotational angular momentum of the molecule (Toulouse) 4 / 68
Magnetoactive components of a molecule For open shell molecules m L owing to the orbital angular momentum � L = ∑ i � magnetic moment � l i magnetic moment � m S owing to the electron spin angular momentum � S = ∑ i � s i (Toulouse) 5 / 68
EPR : transitions between electronic Zeeman states m with an external magnetic field � interaction of a magnetic moment � B m � � H Ze = − � B m S = − µ B g e � for a pure spin doublet S = 1 / 2 with magnetic moment � S M S =1/2 S=1/2 M S =-1/2 B h ν = g e µ B B 0 (Toulouse) 6 / 68
EPR : transitions between electronic Zeeman states microwaves region ◮ for the X band, ν = 9388 MHz ◮ for a spin doublet B = h ν / g e µ B =0.33 T and h ν =0.076 cm − 1 m � = µ B g e � for non pure spin systems, � S the magnetic moment depends on the direction ⇒ anisotropy for a magnetic field of 1 T = 10000 G µ B B = 0 . 46 cm − 1 (Toulouse) 7 / 68
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 2 I +1 lines (Toulouse) 8 / 68
EPR : triplet state M S =1 S=1 D M S =0 M S =-1 B 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 (Toulouse) 9 / 68
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 B † � g � � H S = µ B � ˜ S � � 1 / 2 n † � gg † � � g = ± � n Experiments give access to the tensor G = gg † One defines the principal g-factors g i from G i the principal values of G � g i = ± G i (Toulouse) 10 / 68
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 (Toulouse) 11 / 68
Semiclassical approach: precession around a magnetic field The anisotropy of the g-factors affects both the shape and the pulsation of the precession y gy µ0 x µ (t) the direction of the precession is gx µ0 defined by the sign of the B gegzBzt z product g x g y g z magnetic field � to a principal axis z (Toulouse) 12 / 68
Spin Hamiltonian Model Hamiltonian H S = � S † � D � � B † � g � � ˜ S + µ B � ˜ ˜ 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. → − S ≈ � ˜ In the case of a small spin-orbit coupling, 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 (Toulouse) 13 / 68
Spin Hamiltonian spin algebra in the model space, algebra of spins ˜ S z | ˜ M S z � M S | ˜ M S z � = √ S + | ˜ ˜ M S z � (˜ S + ˜ M S + 1 )(˜ S − ˜ M S ) | ˜ M S + 1 z � = M S √ S − | ˜ ˜ M S z � (˜ S − ˜ M S + 1 )(˜ S + ˜ M S ) | ˜ M S − 1 z � = M S in other directions S x | ˜ ˜ M S | ˜ M S x � M S x � = ˜ S y | ˜ M S | ˜ M S y � M S y � = S z | ˜ ˜ M S | ˜ M S z � M S z � = rotation in the spin space R : rotation z → x → y → z R ′ : rotation z → y → x → z R | ˜ M S z � | ˜ M S x � = R ′ | ˜ M S z � | ˜ M S y � = (Toulouse) 14 / 68
Spin Hamiltonian EPR spin Hamiltonian H S = � S † � D � � B † � g � � ˜ S + µ B � ˜ ˜ 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 (Toulouse) 15 / 68
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 validation of the model 1 calculation of the parameters 2 (Toulouse) 16 / 68
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 � a ¯ � + | ¯ | 1 , 0 � = √ b ab | 2 � � � ¯ a ¯ � | 1 , − 1 � = b ◮ a singlet S = 0 �� � � | 0 , 0 � = 1 � a ¯ � −| ¯ √ b ab | 2 in the case of no ZFS, | 1 , 1 � | 1 , 0 � and | 1 , − 1 � are degenerate, but the response to a magnetic field is different is different for the three components (Toulouse) 17 / 68
CASSCF method multireference wave function of the Ith state Ψ I = ∑ C κ I Φ κ κ ∈ CAS CAS Φ κ = | φ i ··· φ l | Slater determinant variational SCF procedure optimization of the molecular orbitals at the same time as the C κ I (Toulouse) 18 / 68
CASSCF method multireference wave function of the Ith state Ψ I = ∑ C κ I Φ κ κ ∈ CAS CAS Φ κ = | φ i ··· φ l | Slater determinant variational SCF procedure optimization of the molecular orbitals at the same time as the C κ I scalar relativistic effects included the wave functions belong to an irrep of the simple group and have a well defined spin S 2 S + 1 Γ active orbitals ◮ at the least the open shell orbitals : non dynamical correlation ◮ increase of the active in order to include some of the dynamical correlation variationnaly (Toulouse) 18 / 68
dynamical correlation second order perturbation theory ◮ CASPT2 ◮ NEVPT2 Interaction Configuration ◮ CAS-SDCI ◮ DDCI (Toulouse) 19 / 68
Spin-Orbit Coupling the spin-orbit matrix is written in the basis of the CASSCF {| Ψ I , M S �} wave functions and diagonalized ··· ··· ··· | Ψ 1 , S 1 � | Ψ 1 , S 1 − 1 � | Ψ 1 , − S 1 � | Ψ 2 , S 2 � | Ψ 2 , − S 2 � | Ψ 3 , S 3 � E 1 � Ψ 1 , S 1 | E 1 � Ψ 1 , S 1 − 1 | H SO E 1 IJ E 1 � Ψ 1 , − S 1 | E 2 � Ψ 2 , − S 2 | E 2 H SO E 2 � Ψ2 , − S 2 | JI E 3 � Ψ3 , S 3 | E 3 E I SF energies (CASSCF, CASPT2, NEVPT2, CAS+SDCI ...) � � H SO � � � � ′ H SO is the spin-orbit operator obtained � ˆ ˆ H SOC = Ψ K , M S � Ψ L , M IJ S from a 4c to 2 c transformation (Toulouse) 20 / 68
Spin-Orbit Coupling the spin-orbit matrix is written in the basis of the CASSCF {| Ψ I , M S �} wave functions and diagonalized ··· ··· ··· | Ψ 1 , S 1 � | Ψ 1 , S 1 − 1 � | Ψ 1 , − S 1 � | Ψ 2 , S 2 � | Ψ 2 , − S 2 � | Ψ 3 , S 3 � E 1 � Ψ 1 , S 1 | E 1 � Ψ 1 , S 1 − 1 | H SO E 1 IJ E 1 � Ψ 1 , − S 1 | E 2 � Ψ 2 , − S 2 | E 2 H SO E 2 � Ψ2 , − S 2 | JI E 3 � Ψ3 , S 3 | E 3 E I SF energies (CASSCF, CASPT2, NEVPT2, CAS+SDCI ...) � � H SO � � � � ′ H SO is the spin-orbit operator obtained � ˆ ˆ H SOC = Ψ K , M S � Ψ L , M IJ S from a 4c to 2 c transformation the solutions belong to the irreps of the double group ◮ wave functions are no more eigenfunctions of the spin one improves the quality of the calculations by increasing the number of SF roots (Toulouse) 20 / 68
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