Density functional restricted-unrestricted approach for nonlinear properties: application to electron paramagnetic resonance parameters of square planar copper complexes

The density functional restricted-unrestricted approach for treatments of spin polarization effects in molecular properties using spin restricted Kohn-Sham theory has been extended from linear to nonlinear properties. It is shown that the spin polarization contribution to a nonlinear property has the form of a quadratic response function that includes the zero-order Kohn-Sham operator, in analogy to the lower order case where the spin polarization correction to an expectation value has the form of a linear response function. The developed approach is used to formulate new schemes for computation of electronic g-tensors and hyperfine coupling constants, which include spin polarization effects within the framework of spin restricted Kohn-Sham theory. The proposed computational schemes are in the present work employed to study the spin polarization effects on electron paramagnetic resonance spin Hamiltonian parameters of square planar copper complexes. The obtained results indicate that spin polarization gives rise to sizable contributions to the hyperfine coupling tensor of copper in all investigated complexes, while the electronic g-tensors of these complexes are only marginally affected by spin polarization and other factors, such as choice of exchange-correlation functional or molecular structures, will have more pronounced impact on the accuracy of the results.     

Spin-unrestricted character of Kohn-Sham orbitals for open-shell systems

It is argued that Kohn-Sham calculations on open-shell systems should be spin unrestricted in character. An application to the proton hyperfine constant in the methyl radical is presented. © 1995 John Wiley & Sons, Inc.     

Mechanisms of spin transmission: Isotropic hyperfine interactions

The contributions of spin polarization and spin delocalization mechanisms to the proton hyperfine coupling constant is investigated. It is shown that these non-observables are not uniquely defined in most calculations. Arguments are presented which suggest that these non-observables may be profitably defined if both unrestricted Hartree-Fock and restricted Hartree-Fock calculations are made.     

Spin-spin contributions to the zero-field splitting tensor in organic triplets, carbenes and biradicals-a density functional and ab initio study

An evaluation study for the direct dipolar electron spin-spin (SS) contribution to the zero-field splitting (ZFS) tensor in electron paramagnetic resonance (EPR) spectroscopy is presented. Calculations were performed on a wide variety of organic systems where the SS contribution to the ZFS dominates over the second-order spin-orbit coupling (SOC) contribution. Calculations were performed using (hybrid) density functional theory (DFT), as well as complete active space self-consistent field (CASSCF) wave functions. In the former case, our implementation is an approximation, because we use the two-particle reduced spin-density matrix of the noninteracting reference system. In the latter case, the SS contribution is approximated by a mean-field method which, nevertheless, gives accurate results, compared to the approximation free computation of the SS part in a CASSCF framework. For the case of the triplet dioxygen molecule, it was shown that restricted open-shell density functional theory (RODFT), as well as CASSCF, can provide accurate spin-spin couplings while spin-unrestricted DFT leads to much larger errors. Furthermore, 15 organic radicals, including several 1,3 and 1,5 diradicals, dinitroxide biradicals, and even a chlorophyll a model system, were examined as test cases to demonstrate the accuracy and efficiency of our approach within a DFT framework. Accurate D values with root-mean-square deviations of 0.0035 cm(-1) were obtained. Furthermore, all trends, including those due to substituent effects, were correctly reproduced. In a different set of calculations, the polyacenes benzene, naphthalene, anthracene, and tetracene were studied. Applying DFT, the absolute D values were noticeably underestimated, but it was possible to correctly reproduce the trend to smaller D values with larger size of the systems. Finally, it was demonstrated that our approach is also well-suited for the study of carbenes. The smaller organic radicals of this work were also studied, through the use of CASSCF wave functions. This was a special advantage in the case of the triplet polyacenes, where the CASSCF approach gave better results than the DFT method. In comparing spin-restricted and spin-unrestricted results, it was shown through a natural orbital analysis and comparison to high-level ab initio calculations that even small amounts of spin polarization introduced by the unrestricted calculations lead to large deviations between the unrestricted Kohn-Sham (UKS) and restricted open-shell Kohn-Sham (ROKS) approaches. It is challenging to understand why the ROKS results show much better correlation with the experimental data.     

Dipolar-induced dephasing of triplet electron spins

In this paper, magnetic dipolar-induced spin dephasing is considered for localized electronic triplet spin states in solids. Using a projection operator formalism, expressions are derived to describe the Hahn-echo decay behavior for an ensemble of triplet spins at zero- and low-magnetic field strengths. For triplet states localized on non-axially symmetric molecules (or defects) it is shown that, at zero field, cross-relaxation with rapidly relaxing spins is essential in the dipolar-induced dephasing process; secular spin-spin interactions become important only in the presence of a static magnetic field or hyperfine couplings. The results are used to relate experimental dephasing data previously obtained for photoexcited triplet states of axially- and non-axially symmetric defects in CaO.     

Implementation of a density functional theory-based method for the calculation of the hyperfine A-tensor in periodic systems with the use of numerical and Slater type atomic orbitals: application to paramagnetic defects

The A-tensor parameterizes the "hyperfine" interaction of an "effective" electronic spin with the magnetic field due to the nuclear spin as monitored in an electron paramagnetic resonance (EPR) experiment. In this account, we describe an implementation for the calculation of the A-tensor in systems with translational invariance based on the Kohn-Sham form of density functional theory (KS DFT). The method is implemented in the periodic program BAND, where the Bloch states are expanded in the basis of numerical and Slater-type atomic orbitals (NAOs/STOs). This basis is well-suited for the accurate representation of the electron density near the nuclei, a prerequisite for the calculation of highly accurate hyperfine parameters. Our implementation does not rely on the frozen core approximation tacitly assumed in the pseudopotential schemes. The implementation is validated by performing calculations on the A-tensor for small atoms and molecules within the supercell approach as well as for paramagnetic defects in solids. In particular, we consider the A-tensor of "normal" and "anomalous" muonium defects in diamond and of the hydrogen cyanide anion radical HCN(-) in a KCl host crystal lattice.     

Magnetic field effects due to spin—orbit coupling in transient intermediates

The effects of anisotropic spin—orbit coupling on the radical yield and CIDEP in chemical reactions of short-lived triplet intermediate are treated in the density matrix formalism. Analytical expressions for the magnetic field effect (MFE) on the lifetime of the triplet precursor and electron spin polarization are obtained in terms of the molecular parameters, reorientational correlation time, and magnetic field strength.     

Spin-multiplet energies from time-dependent density functional theory

Starting from a formally exact density-functional representation of the frequency-dependent linear density response and exploiting the fact that the latter has poles at the true excitation energies, we develop a density-functional method for the calculation of excitation energies. Simple additive corrections to the Kohn-Sham single-particle transition energies are derived whose actual computation only requires the ordinary static Kohn-Sham orbitals and the corresponding eigenvalues. Numerical results are presented for spin-singlet and triplet energies. © 1996 John Wiley & Sons, Inc.     

Generation of generalized branching diagram spin functions by Schmidt orthogonalization of spin-paired functions

The generalized branching diagram (GBD ) spin representation is defined as the method of sequentially coupling together a number of subsystem spin eigenfunctions using the general rules of angular momentum coupling. It is shown that any GBD representation may also be obtained by Schmidt orthogonalizing a set of cannonical spin–paired (SP ) functions, provided the SP basis is suitably ordered. The ordering procedure used is well suited to computer implementation. This is a generalization of results known in the literature for the Yamanouchi–Kotani and for the Serber spin representations.     

Multireference spin-adapted variant of density functional theory

A new Kohn-Sham formalism is developed for studying the lowest molecular electronic states of given space and spin symmetry whose densities are represented by weighted sums of several reference configurations. Unlike standard spin-density functional theory, the new formalism uses total spin conserving spin-density operators and spin-invariant density matrices so that the method is fully spin-adapted and solves the so-called spin-symmetry dilemma. The formalism permits the use of an arbitrary set of reference (noninteracting) configurations with any number of open shells. It is shown that the requirement of degeneracy of the total noninteracting energies of the reference configurations (or configuration state functions) is equivalent to the stationary condition of the exact energy relative to the weights of the configurations (or configuration state functions). Consequently, at any molecular geometry, the weights can be determined by minimization of the energy, and, for given reference weights, the Kohn-Sham orbitals can be determined. From this viewpoint, the developed theory can be interpreted as an analog of the multiconfiguration self-consistent field approach within density functional theory.     

Zero-point vibrational corrections to isotropic hyperfine coupling constants in polyatomic molecules

The present work addresses isotropic hyperfine coupling constants in polyatomic systems with a particular emphasis on a largely neglected, but a posteriori significant, effect, namely zero-point vibrational corrections. Using the density functional restricted-unrestricted approach, the zero-point vibrational corrections are evaluated for the allyl radical and four of its derivatives. In addition for establishing the numerical size of the zero-point vibrational corrections to the isotropic hyperfine coupling constants, we present simple guidelines useful for identifying hydrogens for which such corrections are significant. Based on our findings, we critically re-examine the computational procedures used for the determination of hyperfine coupling constants in general as well as the practice of using experimental hyperfine coupling constants as reference data when benchmarking and optimizing exchange-correlation functionals and basis sets for such calculations.     

Density-functional calculations of relativistic spin-orbit effects on nuclear magnetic shielding in paramagnetic molecules

Terms arising from the relativistic spin-orbit effect on both hyperfine and Zeeman interactions are introduced to density-functional theory calculation of nuclear magnetic shielding in paramagnetic molecules. The theory is a generalization of the former nonrelativistic formulation for doublet systems and is consistent to O(alpha4), the fourth power of the fine structure constant, for the spin-orbit terms. The new temperature-dependent terms arise from the deviation of the electronic g tensor from the free-electron g value as well as spin-orbit corrections to hyperfine coupling tensor A, the latter introduced in the present work. In particular, the new contributions include a redefined isotropic pseudocontact contribution that consists of effects due to both the g tensor and spin-orbit corrections to hyperfine coupling. The implementation of the spin-orbit terms makes use of all-electron atomic mean-field operators and/or spin-orbit pseudopotentials. Sample results are given for group-9 metallocenes and a nitroxide radical. The new O(alpha4) corrections are found significant for the metallocene systems while they obtain small values for the nitroxide radical. For the isotropic shifts, none of the three beyond-leading-order hyperfine contributions are negligible.     

Plane wave/pseudopotential implementation of excited state gradients in density functional linear response theory: a new route via implicit differentiation

This work presents the formalism and implementation of excited state nuclear forces within density functional linear response theory using a plane wave basis set. An implicit differentiation technique is developed for computing nonadiabatic coupling between Kohn-Sham molecular orbital wave functions as well as gradients of orbital energies which are then used to calculate excited state nuclear forces. The algorithm has been implemented in a plane wave/pseudopotential code taking into account only a reduced active subspace of molecular orbitals. It is demonstrated for the H(2) and N(2) molecules that the analytical gradients rapidly converge to the exact forces when the active subspace of molecular orbitals approaches completeness.     

The generator coordinate method in the unrestricted Hartree‐Fock formalism

The generator coordinate method was implemented in the unrestricted Hartree‐Fock formalism. Weight functions were built from Gaussian generator functions for 1s, 2s, and 2p orbitals of carbon and oxygen atoms. These weight functions show a similar behavior to those found in the generator coordinate restricted Hartree‐Fock method, i.e., they are smooth, continuous, and tend to zero in the limits of integration. Moreover, the weight functions obtained are different for spin‐up and spin‐down electrons what is a result from spin polarization. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

A spin-adapted coupled-cluster based linear response theory for double ionization potentials

We developed in this article a spin-adapted formulation of the coupled-cluster based linear response theory (CC-LRT) for computing double-ionization potentials (DIPs), which may be experimentally observed by Auger spectroscopy. CC-LRT is a multireference generalization of the CC theory where the energy differences have no disconnected vacuum (core) diagrams, signifying core-extensivity. For the spin-adaptation of the CC-LRT equations for the singlet and triplet manifolds, we used the Young-Yamanouchi orthogonal spin-eigenfunctions. The orbital version of the CC-LRT equations are then automatically generated by the conjugate projection operators of Young-Yamanouchi spin functions. We illustrated the working of our spin-adaptation procedure by confining our CC-LRT equations to the space of 2h and 1p–3h ionized determinants. As numerical application of our formalism, we computed the Auger kinetic energies of HF and H₂O. We also analyzed the nature of size-extensivity of the DIPs generated by CC-LRT and showed explicitly that when the molecule is composed of two noninteracting fragments the computed DIPs are either DIPs of fragment A or B or a composite DIP depending on both A and B, which are just not sum of ionization potentials (IPs) of A and B. This analysis is done to underscore the fact that DIPs from CC-LRT is only core-extensive and not fully extensive. © 1996 John Wiley & Sons, Inc.     

Grassmann coherent states representation of the path integral: Evaluation of the generating function for spin systems

An interacting spin system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a discrete time formalism are used in combination with a basis set involving Grassmann variables coherent states to get a many‐body analytic propagator. The generating function thus obtained leads, after an adequate tracing over Grassmann variables in the imaginary time domain, to the partition function. A spin 1/2 Hamiltonian involving the whole set of interactions is considered. Fermion operators satisfying the standard anticommutation relations are constructed from the raising and lowering spin operators via the Jordan–Wigner transformation. The partition function obtained is more general than the partition function of the traditional Ising model involving only first‐neighbor interactions. Computations were performed assuming that the coupling as a function of the distance can be reasonably well represented by an Airy function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Light-induced electron spin polarization in vanadyl octaethylporphyrin: I. Characterization of the excited quartet state

Laser flash induced spin-polarized transient electron paramagnetic resonance (TREPR) spectra for vanadyl octaethylporphyrin in isotropic and partially ordered frozen solutions are presented and compared with corresponding luminescence data. The TREPR spectra show well-resolved hyperfine couplings to the vanadium nucleus and a multiplet polarization pattern with features typical of zero-field splitting (ZFS). The principal values of the vanadium hyperfine coupling tensor evaluated from the spectra are 1/3 of the corresponding values found from steady-state EPR spectra of the ground state. On the basis of these characteristics and numerical simulations, the polarization patterns are assigned to the excited quartet state. The values of the ZFS parameters of the trip-quartet obtained from simulation of the spectra (D = 17.5 mT and E = 1.5 mT) are comparable to those of the triplet state of the zinc and free base octaethyl porphyrin. The lifetime of the spin polarization is found to be temperature dependent and is essentially the same as that of the optical emission. The temperature dependence is rationalized using a model in which the decay to the ground state occurs from both the trip-quartet and trip-doublet, which are in thermal equilibrium even at 15 K. A fit of the model to the observed spin polarization lifetimes yields an energy gap of 47 cm(-1) between the trip-quartet and trip-doublet. It is shown that the spin polarization evolves from a multiplet pattern at early times to a net absorptive pattern at late times following the laser flash. It is proposed that the establishment of thermal equilibrium leads to the evolution of the spin from multiplet to net polarization.     

Coordination complexes of 1-(4-[N-tert-butyl-N-aminoxyl]phenyl)-1H-1,2,4-triazole with paramagnetic transition metal dications

1-(4-(N-tert-Butyl-N-aminoxylphenyl))-1H-1,2,4-triazole (NIT-Ph-Triaz) forms isostructural cyclic 2:2 dimeric complexes with M(hfac)(2), M = Mn, Ni, Co, hfac = hexafluoroacetylacetonate. For M = Cu, only a sufficient sample for crystallographic analysis was isolated. For M = Mn, Ni, and Co, the M-NIT exchange is strongly antiferromagnetic. The intradimer exchange coupling between M-NIT units is J/k = +0.53 K for M = Mn, J/k = (-)3.5 K for M = Ni. For M = Co, J/k < 0 K, with the magnetic susceptibility tending toward zero at low temperatures. The exchange behavior is consistent with an intradimer spin polarization mechanism linking M-NIT units through the conjugated pi-system of the radical. Computational modeling of NIT-Ph-Triaz gives Mulliken spin populations in good accord with experimental electron spin resonance hyperfine coupling constants, and is consistent with the presumed radical spin density distribution in the complexes. The results provide useful guidelines to anticipate spin polarization effects in organic pi-radical building blocks in magnetic materials, particularly when qualitative connectivity-based analyses are clouded by nonalternant molecular connectivities.