Quantum chemical modelling of molecular properties - parameters of EPR spectra

Quantum chemical modelling of molecular properties - parameters of EPR spectra

EPR ( electric paramagnetic resonance) spectra can be obtained only for open-shell systems, since they rely on transitions between spin energy levels of an unpaired electron (or electrons) in external magnetic field. The phenomenon resembles NMR and it is described theoretically in a similar fashion (the employed effective Hamiltonian has a similar form). However, calculations of EPR spectra are different, since they require the use of quantum chemical methods for open-shell systems.

Description of open shell systems Methods based on spin-unrestricted Hartree-Fock (UHF) where it is assumed that spacial one-electron functions (orbitals) can be different for alpha and beta electrons. The method is simple conceptually and flexible, but involved spin contamination problems (total spin ceases to be a good quantum number). Methods based on spin-restricted Hartree-Fock (RHF), where the total wave function is a assumed to be a spin-adapted linear combination of Slater orbitals (so-called configuration state functions, CSF). The above also applies to Kohn-Sham DFT.

Effective EPR Hamiltonian H ESR = H ex +H sz +H ZFS +H hf +H nz +H Nq The meaning of the terms is as follows: 1 H ex = 2Js 1 s 2 is a so-called Heisenberg Hamiltonian, responsible for energy differences between different spin states. The differences are usually too big to be observed in EPR spectrum. J is so-called electron spin-spin coupling constant (related to exchange integral). 2 H sz = µ B S T gb is a term responsible for spin energy levels of an electron in external magnetic field g is a so-called g factor. For a free electron it is reduced to a number g e 2.0023193. In a molecule g is modified by the

presence of other electrons (similarly as the NMR shielding constant) g = g e 1+ g 3 H ZFS = S T DS so-called zero-field splitting, term present only in the case of more than one unpaired electron (biradicals etc). The D tensor is traceless (and symmetric). 4 H hf = K ( A fc K S T I K +S T A sd KI K ) The term describing hyperfine coupling, thats is interaction of electron magnetic moment with magnetic moments of the nuclei.

It s an important contribution, reponsible for splitting of the EPR sygnal. It consists of a Fermi contact contribution (purely isotropic) and a spin-dipole contribution (purely anisotropic). 5 Nuclear Zeeman term H nz = µ N g K B T I K This terms is the same as in the effective NMR Hamiltonian and its consequences are usually difficult to observe. The important terms (and therefore calculated by means of quantum chemical methods) are electronic g tensor (contribution 2 to the effective Hamiltonian) and the hyperfine coupling constant (contribution 4 to the effective Hamiltonian). K

Electronic g tensor In the case of radicals with one unpaired electron or high-spin states in the limit of high magnetic field, the energy of interaction with magnetic field is: E = µ B BT GB where G (symmetric G tensor, a quantity which can be obtained experimentally) is connected with the previously introduced g tensor in the following way:w G = g g T

Electronic g tensor, c.d. In the nonrelativistic theory there are two contributions to the g tensor which can be expressed as expectation values: spin Zeeman Ĥ s,b = 1 and magnetic dipole moment Ĥ B = e 2m e i l io = µ B l io = iµ B (r io i ) = m i i (l io = r io p i = i r io i ) The first term gives the value for an isolated electron (2 with no quantum electrodynamics corrections), the second is zero. All differences in g factor result therefore from relativistic and radiational effects.

Electronic g tensor, continued Taking into account relativistic correction to the αfs 2 order, the differences in the value of the g tensor ( g) are evaluated as: g = g SO + g RMC + g GC +O ( ) αfs 4 The most important contribution is usually g SO, calculated as a linear response with operators of spin-orbit coupling and orbital magnetic moment: g SO αβ = 2 ĤB α;ĥso β S z (max) The other two are calculated as expectation values:

g RMC αβ = g GC αβ = 2 S z (max) 2 S z (max) 0 0 ĤSZ/KE αβ ĤGC αβ 0 0 Ĥ SZ/KE (spin Zeeman/kinetic energy) and ĤGC (magnetic field/spin-orbit coupling) have complicated forms.

Computational requirements of the g tensor g exbibits some gauge dependence (although weak in comparison with other magnetic properties) - London orbitals are adviceable. The calculations are usually carried out using u-dft (spin-unrestricted DFT), sometimes spin-restricted DFT, and for small molecules MCSCF.

Hyperfine coupling constant The hyperfine coupling constant is composed (in non-relativistic approximation) of two terms: A K = A fc K + A sd K A fc K contains the Fermi contact opetator and is purely isotropic Afc K = A fc K 1. A sd K contains the spin-dipole coupling operator and is purely anisotropic - thus for freely rotating molecules in gas and liquid phase only A fc K remains.

Hyperfine coupling constant, continued Both these contributions can be calculated as expectation values: Ĥ K(FC) = α 2 fs 8π 3 δ(r ik )1 i and Ĥ K(SD) = α 2 fs i 3r ik r T ik r2 ik 1 r 5 ik but using spin density matrix D α β (difference between density matrices for electrons α i β).

A fc K = 1 2 g Kg e µ B µ N S Z 1 µν A sd K = 1 2 g Kg e µ B µ N S Z 1 µν φ µ Ĥ K(FC) φ ν φ µ Ĥ K(SD) φν D α β µν D α β µν where S Z is the total spin operator. The FC contribution can be thus interpreted as spin density on the nucleus. The calculated hyperfine couplings need to be multiplied by g K g e µ B µ N to convert them to MHz and compare with experiment.

Computational requirements of hyperfine coupling constant The FC contribution requires specially designed basis sets with tight functions (eg. EPR-III). The SD contribution requires wel described valence shell and polarization funcitons. U-DFT (spin-unrestricted DFT) usually performs well, and the differences between results obtained with various funcionals are usually small. However, the spin-contamination problem occurs and can be significant.

Programs for calculations of EPR properties ZFS in Dalton (HF, MCSCF). Hyperfine coupling constants in Dalton (HF, DFT, MCSCF), Gaussian (HF, DFT), ReSpect (DFT) and ADF (DFT). g factor in Dalton (HF, DFT, MCSCF), Gaussian (HF, DFT) [NMR keyword?], ReSpect (DFT) and ADF (DFT).