Helmut Eschrig Leibniz-Institut für Festkörper- und Werkstoffforschung Dresden Leibniz-Institute for Solid State and Materials Research Dresden ORBITAL POLARIZATION IN THE KOHN-SHAM-DIRAC THEORY with M. Sargolzaei and M. Richter The Kohn-Sham-Dirac (KSD Variational Principle The KSD Equation The Orbital Correlation Term The Orbital Current and L(r of an Atom The Functional Derivatives The Final KSD Equation Results for the 3d Series Results for the 4f Series
Kohn-Sham-Dirac (KSD Variational Principle KSD bispinor orbitals ψ k (r, ψ k = ψ γ 0 and orbital occupation numbers n k. Four-current density: J µ = c k n k ψk γ µ ψ k, ψ k ψ k = δ kk, Q = e k n k. Q is the total electron charge, if all ψ k are in the electron sector. The current density functional H[J µ ] of the Hohenberg-Kohn-Rajagopal variational principle is split into H[J µ ] = K[J µ ] + L[J µ ], { K[J µ ] = min k[ψ k, n k ] c ψ k,n k k n k ψk γ µ ψ k = J µ}.
The KSD variational principle for the ground state energy E[A, Q] in the external four-potential A µ (r reads { [ E[A, Q] = min k[ψ k, n k ] + L c n k ψk γ µ ψ k ] ψ k,n k k ec n k ψ k γ 0 γ µ A µ ψ k ψ k ψ k = δ kk, e k k } n k = Q,
Take the Hartree-Fock expression for k, k[ψ k, n k ] = k n k ψ k icα + βc 2 ψ k + + e2 2 kk n k n k e2 2 d 3 rd 3 r ψ k (rψ k(rψ k (r ψ k (r r r kk n k n k d 3 rd 3 r ψ k (rψ k (rψ k (r ψ k (r r r, and the local spin density approximation (LSDA E C [J µ ] for L.
The KSD Equation [ ] icα + βc 2 ecβγ µ (A µ + δ µâhf 0 + A C µ ψ k = ψ k ε k where ec (  HF ψ k (r = e k n k + e k n k d 3 r ψ k (r ψ k (r r r d 3 r ψ k (r ψ k (r r r ψ k (r + ψ k (r = ( ˆV HF ψ k (r is the Hartree-Fock potential operator, and is the correlation four potential. ea C µ(r = δec [J] δj µ (r
To get rid of the nasty vector potential: Gordon s decomposition of the three-current density J figuring in J µ = (cn, J: J = I + S, S = 1 2 n k ψk Σψ k, Σ = k J = 0 I = 0 I = (1/2 L : ( σ 0 0 σ. ej = 1 µ 0 M = e 2 (L + 2S. ea C µ(r = δec [J] δj µ (r = H C (r = δec δm(r. H C (r is local: it is simply a function of the position r. However, it depends also functionally on M, possibly in a most non-local way.
Alternative KSD equation: [ icα + βc 2 + V (r + ˆV ] HF + V C (r ψ k (r β d 3 r ( H(r + H C (r δm(r n k δ ψ k (r = ψ k(rε k with ( δm(r δ ψ k (r = µ δl(r Bohr δ ψ k (r + n kδ(r rσψ k (r. Here, we have to cope with the HF operator and with the non-local term δl(r /δ ψ k (r.
The HF operator can in most cases be replaced by the approved LSDA approximation, ˆV HF V H (r + Vss X (r, where V H is the Coulomb potential of the electron charge density (which includes the self-interaction of orbitals and V X is the LSDA exchange potential spin matrix: V X ss (r = V X (r + µ Bohr H X (r βσ With the above approximations, the scalar V X may be again combined with V C into the LSDA expression for V XC, and the action of H X may be combined with the action of H C on the spin part of M into the LSDA expression µ Bohr H XC (r βσψ k (r. This is known to work well except for the cases of so-called strong local correlation, and except for the semiconductor gap problem in which corrections to the scalar exchange potential V X must be introduced. In the rest of the H C expression, M is to be replaced with µ Bohr L, the remaining term which is not contained in the LSDA expression.
The Orbital Correlation Term After the previous discussion, the remaining orbital polarization term in the KSD equation is: µ Bohr β n k d 3 r ( H(r + H C (r δl(r δ ψ k (r. The first part (external field is the diamagnetic term, and the second is the orbital correlation term. In the following we skip the diamagnetic term. The spin only correlation field in collinear LSDA has the form (within 2 p.c. accuracy H C,S z (r = n(rf (n(rζ(r, ζ(r = m S(r n(r = H C,S (r = F (n(r2s(r, where ζ is the spin polarization degree, assumed in z-direction.
In the spirit of CDFT, where M = µ Bohr (L + 2S, we put and get the orbital correlation term µ Bohr β n k H C (r = H C,S (r + H C,L (r d 3 r ( H C,S (r + H C,L (r δl(r δ ψ k (r. We have calculated both contributions for atoms and find them of the same order of magnitude, implying that the first contribution, which is a correlation correction to spin-orbit coupling, is for scandium more than one order of magnitude to large. Since H C by nature is a curl (of A C and hence cannot be collinear, this defect must be attributed to the LSDA.
Hence we assume for the spin-orbit correction of the full non-local theory d 3 r H C,S (r L(r = 0. Since H C,S LSDA = F (n2s, we define (S L = d 3 r F (n(r S(r L(r = 0, and H C,L = H C,L LSDA HC,S LSDA (S L (S S. This yields the final orbital correlation term µ 0 eβ 2n k d 3 r H C,L (r δl(r δ ψ k (r.
The Orbital Current and L(r of an Atom I(r = k n k ( R( ψ k (ri ψ k (r + A(r ψ k (rψ k (r = 1 2 L(r Collinear approximation: H = e z H, L = e z L Symmetric gauge: A(ρ, φ, z = e φ A(ρ, z: ρ ρa(ρ, z = µ 0 dρ ρ H(ρ, z. Take scalar basis functions ϕ m e imφ, then the φ-averaged current is e φ ρ ( n k k m 0 m ϕ m (r 2 ψ k ϕ m β ϕ m ψ k + ρa(r ψ k (rψ k (r = e φ 2 L ρ.
The φ-averaging removes the off-diagonal terms e i(m mφ. The matrix element ψ k ϕ m is still a bispinor since ϕ m is a scalar. Integration over ρ yields L(r = 2 k n k ρ dρ ( m ρ ϕ m(r 2 ψ k ϕ m β ϕ m ψ k + m + A(r ψ k (r ψ k (r. This expression should also be useful for unfilled inner shells in a solid while for outer valence electron states the orbital currents from neighboring atoms to a far extent cancel like in Peierls argument on diamagnetism of the homogeneous electron gas.
The Functional Derivatives With spherical coordinates (ρ, z, φ, dρ = 1 ρ 2π d 3 r δ(z z θ(ρ ρ ρ and A = A C = A C,S + A C,L one finds δl(r δ ψ k (r =n k d 3 r δ(z z θ(ρ ρ π ρ ( m ρ ϕ m(r 2 ϕ m ψ k ϕ m (r + n k ψk (r ψ k (r δac (r n k k δ ψ k (r m + n k π δ(z z θ(ρ ρ A C (rψ k (r ρ +
The orbital correlation term becomes µ 0 eβ d 3 r H C,L (r δl(r 2n k δ ψ k (r = { = µ Bohr β d 3 r A C,L (r ( m ρ ϕ m(r 2 ϕ m ψ k ϕ m (r + n k ψk (r ψ k (r δac (r n m k k δ ψ k (r } + A C,L (ra C (rψ k (r +
and, with δa C (r ρ δ ψ k (r = µ 0 dρ ρ δh C (ρ, z 0 ρ δ ψ, k (r ζ(r = 1 ( n k ψk (rβσ z ψ k (r + L z (r, n(r k δζ L (r δ ψ k (r n(r = δl z(r δ ψ k (r, δh C (r ( δh C δ ψ k (r = βn (r k δn(r + βn k ζ(r δh C (r n(r δζ(r d 3 r 1 δh C (r n(r δζ(r + 1 δh C (r n(r δζ(r Σ z ψ k (r + δl z (r δ ψ k (r
The Final KSD Equation [ icα + βc 2 + V (r + V H (r + V XC (r + C V (r + + ( ( µ Bohr H(r + H XC (r ] + C Σ (r βσ z ψ k (r + + β m C m ϕ m ψ k ϕ m (r = ψ k (rɛ k.
{ β C V (r = 2µ Bohr A C,L (ra C (r + µ 0 C m = e ρ 0 ( dρ ρ δh C (ρ, z ρ δn(r d 3 r A C,L (r m ( ρ ρ ϕ m(r 2 1+µ 0 C Σ (r = 2µ Bohr d 3 r A C,L (r ñ(r d 3 r A C,L (r ñ(r dρ ρ 0 ρ 0 ζ(r n(r 1 ρ n(r δh C (ρ, z } δζ(r δh C (ρ, z δζ(r +O((H C 3, dρ ρ 1 δh C (ρ, z ρ n(r δζ(r., ñ = k n k ψk ψ k.
The leading term turns out to be the first contribution to C m, all other terms being 1.5 to 3 orders of magnitude smaller: The final KSD equation reads [ icα + βc 2 + V (r + V H (r + V XC (r + ] + µ Bohr (H(r + H XC (r βσ z ψ k (r + + β m C m ϕ m ψ k ϕ m (r = ψ k (rɛ k. C m = e d 3 r A C,L (r m ρ ϕ m(r 2.
Results for the 3d Series <φ m C V φ m > for 3d electrons [ionic state(+2] 4 2 0 (mev 2 4 m=1 m=2 6 8 10 0 1 2 3 4 5 6 7 8 9 10 Number of 3d electrons
<φ m C Σ φ m > for 3d electrons [ionic state(+2] 20 15 (mev 10 m=1 m=2 5 0 0 1 2 3 4 5 6 7 8 9 10 Number of 3d electrons
C m for 3d electrons [ionic state (+2] 0 100 (mev 200 m=1 m=2 300 400 0 1 2 3 4 5 6 7 8 9 10 Number of 3d electrons
Results for the 4f Series <φ m C V φ m > for 4f electrons [ionic state(+3] 0.1 0 (mev 0.1 0.2 m=2 m=1 m=3 0.3 0.4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Number of 4f electrons
<φ m C Σ φ m > for 4f electron [ionic state(+3] 4 m=1 3 m=2 m=3 (mev 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Number of 4f electrons
C m for 4f electrons [ionic state(+3] 0 50 100 (mev 150 200 250 m=1 m=2 m=3 300 350 400 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Number of 4f electrons