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Validity of the Energy Band Picture B. Atomic Properties (Localized versus Itinerant Electrons) and Band Expectations; Comparison with the Rare-Earth Metals . . C. Localization of f Electrons and the Role of Coulomb Correlation in Determining Localized versus Itinerant Descriptions II. Energy Bands in Solids: The Relativistic Augmented-Plane-Wave Method . A. Description of the Method; Comparison between Nonrelativistic and Relativistic Formulations B. Limitations and Complications Expected (and Found) for Band Calculations on Heavy Metals III.

The former is lumped with Η and treated as un­ perturbed Hamiltonian. The latter is treated as perturbation. To perform the perturbation calculation, we introduce a projection operator Ρ that projects on the lower multiplets and its complementary operator Q that projects on Ht/kT Ht/kT 1. CRYSTAL FIELD THEORY 41 the higher. The properties of these operators are Ρ = P, Q = Q, PQ = QP = 0, and P + Q = 1. The Hamiltonian may be rewritten as 2 2 H = Η — βΗ · ( Ρ + β ) μ ( Ρ + β ) = Η + # „ t (43) where / ϊ = · ( Ρ μ Ρ + β μ β ) is the unperturbed Hamiltonian with eigenvalues ε and if = - βΗ · ( Ρ μ β + βμΡ) is the perturbation.

We have n n „ &g =g' -g n n „ = > n / -e /kT e _ -e„/kT\ m e > η Kn\PμQ + ~ m \ m e QμP\m)(m\0\ny J η ε JY^e- ' xl en /kT " ^ [ < Γ | 0 | Γ > < Γ | μ | Γ > + <Γ|μ|Γ><Γ|0|Γ>· Γ p'HlkT Γ' v. _ -E ,lkT T In this expression, g is the pseudonuclear g factor obtained from the E N D O R experiments, g is the nuclear g factor obtained from the atomic beam resonance experiments, Σ 3 Σ sums over the lower and upper multiplets, respectively, and Σ sums over both multiplets. The number of terms in the double sum Σ Σ depends on the number of nonzero matrix elements and hence on the symmetry of Γ and Γ'.

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