Fixed Moment Calculations Research Articles

Structure and properties of metal-exchanged zeolites studied using gradient-corrected and hybrid functionals. I. Structure and energetics

The structural and energetic properties of purely siliceous, proton-, and Cu- and Co-exchanged chabazite have been studied using periodic density-functional (DFT) calculations with both conventional gradient-corrected exchange-correlation functionals and hybrid functionals mixing exact (i.e., Hartree-Fock) and DFT exchange. Spin-polarized and fixed-moment calculations have been performed to determine the equilibrium and excited spin-configurations of the metal-exchanged chabazites. For the purely siliceous chabazite, hybrid functionals predict a slightly more accurate cell volume and lattice geometry. For isolated Al/Si substitution sites, gradient-corrected functionals predict that the lattice distortion induced by the substitution preserves the local tetrahedral symmetry, whereas hybrid functionals lead to a distorted Al coordination with two short and two long Al-O bonds. Hybrid functionals yield a stronger cation-framework binding that conventional functionals in metal-exchanged zeolites, they favor shorter cation-oxygen bonds and eventually also a higher coordination of the cation. Both types of functionals predict the same spin in the ground-state. The structural optimization of the excited spin-states shows that the formation of a high-spin configuration leads to a strong lattice relaxation and a weaker cation-framework bonding. For both Cu- and Co-exchanged chabazite, the prediction of a preferred location of the cation in a six-membered ring of the zeolite agrees with experiment, but the energy differences between possible cation locations and the lattice distortion induced by the Al/Si substitution and the bonding of the cation depends quite significantly on the choice of the functional. All functionals predict similar energy differences for excited spin states. Spin-excitations are shown to be accompanied by significant changes in the cation coordination, which are more pronounced with hybrid functionals. The consequences of electronic spectra and chemical reactivity are analyzed in the following papers.

Possible magnetism in small palladium clusters

Possible magnetism of small palladium clusters having cubic geometry has been studied using the Gaussian orbital basis and the local spin-density approximation. Extensive calculations have been done on ${\mathrm{Pd}}_{19}$ clusters with face-centered-cubic symmetry by varying atomic spacing parameters. Possible magnetism of the ${\mathrm{Pd}}_{15}$ cluster with body-centered-cubic symmetry has also been studied. Fixed moment calculations were done to find all possible metastable states of the cluster. In addition to a nonmagnetic state, the ${\mathrm{Pd}}_{19}$ cluster is found to have a magnetic state with a moment of $0.32{\ensuremath{\mu}}_{B}.$ The two states are very close in total energy but the magnetic state has slightly lower total energy. The ${\mathrm{Pd}}_{15}$ cluster is found to have only one magnetic state with a moment of $0.53{\ensuremath{\mu}}_{B}.$ From our calculations, we conclude that palladium clusters are very likely to be magnetic although the magnetic moment is very small.

THE POSSIBILITY OF TWO MAGNETIC STATES IN Co19 CLUSTER

The possibility of more than one magnetic state has been explored for the face-centered-cubic Co19 cluster using the local spin-density linear combination of the Gaussian orbital method. Fixed moment calculations were done and the converged states were used as an input potential to obtain self-consistent solutions. Two different magnetic states with the magnetic moment per atom of 1.00 μB and 2.05 μB were obtained. The high-spin moment state has lower energy and is considered the ground state, and the low spin moment a metastable state. The local magnetic moment of the center, first- and second-shell atoms are found to be 1.90, 1.64, and −0.72 μB respectively for the low spin state, and 1.87, 1.92, and 2.01 μB respectively for the high spin state. Considering that the center-site atom has shown slow convergence to bulk moment for other transition-metal clusters in previous studies, Co clusters are found to be exceptional as far as this property is concerned. Density of states (DOS) of the cluster and bulk have been compared after the cluster levels were Gaussian-broadened. The DOS similarity between cluster and bulk is also found to be reasonably good.

Possibility of many magnetic states in cluster systems: V and Cr clusters.

The possibility of more than one magnetic state in cluster systems has been explored using the local spin density linear combination of Gaussian orbitals method. Fixed moment calculations were done and their converged states were used to determine input potentials to find as many stable states as possible for the V and Cr clusters having body-centered-cubic (bcc) symmetry. Several calculations were made at different atomic spacings for ${\mathrm{V}}_{9}$ and ${\mathrm{Cr}}_{9}$ clusters. It was found that for some atomic spacings, as many as four or five magnetic states exist for a cluster. In addition to the states reported previously with small average magnetic moments of 0.33${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ and 0.67${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ per atom, states with moments as large as 2.78${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ and 3.78${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ per atom are also found to be possible for ${\mathrm{V}}_{9}$ and ${\mathrm{Cr}}_{9}$ clusters, respectively, as well as intermediate-moment states in some cases. The ${\mathrm{Cr}}_{15}$ cluster is found to have a state with 0.80${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ per atom in addition to the 0.40${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ state reported previously. The local moments for the 0.80${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ state in ${\mathrm{Cr}}_{15}$ clusters are found to be very close to that which has been reported by other groups.

Theoretical study of the ferromagnetism in UNi2.

Electronic-structure calculations for the ferromagnet ${\mathrm{UNi}}_{2}$ in the hexagonal ${\mathrm{MgZn}}_{2}$ structure (C14) are reported. In this treatment the uranium 5f electrons are considered as itinerant and are included in the self-consistent local-density calculations on the same footing as the other valence electrons. By adding the spin-orbit interaction and orbital polarization to the Hamiltonian, a substantial cancellation between the spin and orbital moments on the uranium atom is obtained. This is in good qualitative agreement with neutron-scattering experiments and can only be achieved by inclusion of the orbital polarization. From fixed-moment calculations, where the total magnetic moment is chosen to agree with data, a good agreement with the experimental uranium form factor is obtained. We emphasize the contribution of the 6d electrons to the uranium magnetic form factor for small Q vectors, which in fact gives rise to a substantial improvement of the agreement with data when compared to an unconstrained fit where only 5f contributions to the uranium magnetic density are taken into account. The magnetism in ${\mathrm{UNi}}_{2}$ is found to be driven by the uranium atoms, in contrast to the related system ${\mathrm{UFe}}_{2}$ in which it is caused by the iron atoms.