Verbosity(0) -- This tutorial calculates the ground-state of NiO within the Ligand-field theory approximation -- in Ligand field theory we approximate the solid by a single transition metal atom d-shell -- interacting with a non-interacting Ligand shell. (Nowadays in the literature often called -- a bath) For transition metal oxides one can think of the ligand orbitals as the O-2p -- orbitals. For NiO there would be six O-2p orbitals and one might expect a cluster of -- 1 Ni-d shell and 6 O-2p shells (10+36=46 spin-orbitals in total). For a theory where -- calculation times scale roughly exponential with respect to number of orbitals going from -- 10 spin-orbitals (crystal-field theory) to 46 spin-orbitals slows thing down a lot. -- There is however a simple optimization one can make to speed up the calculations, without -- changing the final answer. One can make linear combinations of the O-2p orbitals to form -- ligand orbitals. Out-off the 36 O-2p orbitals only 10 interact with the Ni-d orbital. -- (see PRB 85, 165113 (2012) for nice pictures of the ligand orbitals in cubic symmetry or -- PRL 107, 107402 (2011) and J. Phys. Condens. Matter 24, 255602 (2012) for an example in -- lower symmetry (TiOCl)) -- In ligand field theory we thus have 20 spin-orbitals. 10 representing the Ni-3d shell and -- 10 representing the Ligand-d shell. -- we again take the ordering to be dn even and up odd NF=20 NB=0 IndexDn_3d={ 0, 2, 4, 6, 8} IndexUp_3d={ 1, 3, 5, 7, 9} IndexDn_Ld={10,12,14,16,18} IndexUp_Ld={11,13,15,17,19} -- we can define the angular momentum operators for the d-shell as we did in crystal-field -- theory OppSx_3d =NewOperator("Sx" ,NF, IndexUp_3d, IndexDn_3d) OppSy_3d =NewOperator("Sy" ,NF, IndexUp_3d, IndexDn_3d) OppSz_3d =NewOperator("Sz" ,NF, IndexUp_3d, IndexDn_3d) OppSsqr_3d =NewOperator("Ssqr" ,NF, IndexUp_3d, IndexDn_3d) OppSplus_3d=NewOperator("Splus",NF, IndexUp_3d, IndexDn_3d) OppSmin_3d =NewOperator("Smin" ,NF, IndexUp_3d, IndexDn_3d) OppLx_3d =NewOperator("Lx" ,NF, IndexUp_3d, IndexDn_3d) OppLy_3d =NewOperator("Ly" ,NF, IndexUp_3d, IndexDn_3d) OppLz_3d =NewOperator("Lz" ,NF, IndexUp_3d, IndexDn_3d) OppLsqr_3d =NewOperator("Lsqr" ,NF, IndexUp_3d, IndexDn_3d) OppLplus_3d=NewOperator("Lplus",NF, IndexUp_3d, IndexDn_3d) OppLmin_3d =NewOperator("Lmin" ,NF, IndexUp_3d, IndexDn_3d) OppJx_3d =NewOperator("Jx" ,NF, IndexUp_3d, IndexDn_3d) OppJy_3d =NewOperator("Jy" ,NF, IndexUp_3d, IndexDn_3d) OppJz_3d =NewOperator("Jz" ,NF, IndexUp_3d, IndexDn_3d) OppJsqr_3d =NewOperator("Jsqr" ,NF, IndexUp_3d, IndexDn_3d) OppJplus_3d=NewOperator("Jplus",NF, IndexUp_3d, IndexDn_3d) OppJmin_3d =NewOperator("Jmin" ,NF, IndexUp_3d, IndexDn_3d) Oppldots_3d=NewOperator("ldots",NF, IndexUp_3d, IndexDn_3d) -- And similar we can define the angular momentum operators for the ligand d-shell OppSx_Ld =NewOperator("Sx" ,NF, IndexUp_Ld, IndexDn_Ld) OppSy_Ld =NewOperator("Sy" ,NF, IndexUp_Ld, IndexDn_Ld) OppSz_Ld =NewOperator("Sz" ,NF, IndexUp_Ld, IndexDn_Ld) OppSsqr_Ld =NewOperator("Ssqr" ,NF, IndexUp_Ld, IndexDn_Ld) OppSplus_Ld=NewOperator("Splus",NF, IndexUp_Ld, IndexDn_Ld) OppSmin_Ld =NewOperator("Smin" ,NF, IndexUp_Ld, IndexDn_Ld) OppLx_Ld =NewOperator("Lx" ,NF, IndexUp_Ld, IndexDn_Ld) OppLy_Ld =NewOperator("Ly" ,NF, IndexUp_Ld, IndexDn_Ld) OppLz_Ld =NewOperator("Lz" ,NF, IndexUp_Ld, IndexDn_Ld) OppLsqr_Ld =NewOperator("Lsqr" ,NF, IndexUp_Ld, IndexDn_Ld) OppLplus_Ld=NewOperator("Lplus",NF, IndexUp_Ld, IndexDn_Ld) OppLmin_Ld =NewOperator("Lmin" ,NF, IndexUp_Ld, IndexDn_Ld) OppJx_Ld =NewOperator("Jx" ,NF, IndexUp_Ld, IndexDn_Ld) OppJy_Ld =NewOperator("Jy" ,NF, IndexUp_Ld, IndexDn_Ld) OppJz_Ld =NewOperator("Jz" ,NF, IndexUp_Ld, IndexDn_Ld) OppJsqr_Ld =NewOperator("Jsqr" ,NF, IndexUp_Ld, IndexDn_Ld) OppJplus_Ld=NewOperator("Jplus",NF, IndexUp_Ld, IndexDn_Ld) OppJmin_Ld =NewOperator("Jmin" ,NF, IndexUp_Ld, IndexDn_Ld) -- In order to calculate the total angular momentum in the cluster we can sum the operators OppSx = OppSx_3d + OppSx_Ld OppSy = OppSy_3d + OppSy_Ld OppSz = OppSz_3d + OppSz_Ld OppSsqr = OppSx * OppSx + OppSy * OppSy + OppSz * OppSz OppLx = OppLx_3d + OppLx_Ld OppLy = OppLy_3d + OppLy_Ld OppLz = OppLz_3d + OppLz_Ld OppLsqr = OppLx * OppLx + OppLy * OppLy + OppLz * OppLz OppJx = OppJx_3d + OppJx_Ld OppJy = OppJy_3d + OppJy_Ld OppJz = OppJz_3d + OppJz_Ld OppJsqr = OppJx * OppJx + OppJy * OppJy + OppJz * OppJz -- Just like in crystal-field theory we have Coulomb interaction on the Ni-d shell. -- We again expand the Coulomb interaction on spherical harmonics, where the angular part -- is solved analytical and the radial part gives three parameters F0, F2 and F4. -- We here define three operators separately and only later provide parameters OppF0_3d =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {1,0,0}) OppF2_3d =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {0,1,0}) OppF4_3d =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {0,0,1}) -- In ligand-field theory the ligand-field interaction is given by three different terms. -- There is an onsite splitting on the Transition metal d-shell -- There is an onsite splitting on the Ligand d-shell -- There is a hopping between the ligand d-shell and the Transition metal d-shell -- These interactions can be seen as effective potentials responsible for the splitting -- In order to enter these potentials we expand them on renormalized spherical harmonics -- and add the expansion coefficients to the function NewOperator("CF", ...) -- We thus need to know the potential expanded on spherical harmonics: -- Akm = {{k1,m1,Akm1},{k2,m2,Akm2}, ... } -- For specific symmetries we can use the function "PotentialExpandedOnClm" Which for cubic -- symmetry needs the energy of the eg and t2g orbitals. We here take the potential to be -- such that we have a 1 eV splitting and later multiply the operator with the actual size -- In crystal-field theory there is only an interaction on the transition metal d-shell -- In ligand field theory there is an interaction on the transition metal d-shell as well -- as on the ligand d-shell Akm = PotentialExpandedOnClm("Oh", 2, {0.6,-0.4}) OpptenDq_3d = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm) OpptenDq_Ld = NewOperator("CF", NF, IndexUp_Ld, IndexDn_Ld, Akm) -- We want to be able to calculate the occupation of the eg and t2g orbitals, we here use -- the same operators with potentials of 1 for the eg or 1 for the t2g orbitals to create -- number operators. (Note that there are many other options to do this) Akm = PotentialExpandedOnClm("Oh", 2, {1,0}) OppNeg_3d = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm) OppNeg_Ld = NewOperator("CF", NF, IndexUp_Ld, IndexDn_Ld, Akm) Akm = PotentialExpandedOnClm("Oh", 2, {0,1}) OppNt2g_3d = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm) OppNt2g_Ld = NewOperator("CF", NF, IndexUp_Ld, IndexDn_Ld, Akm) -- We also want to know hom many electrons are in the Ni-d and how many are in the Ligand-d -- shell. Here the number operators that count them. OppNUp_3d = NewOperator("Number", NF, IndexUp_3d,IndexUp_3d,{1,1,1,1,1}) OppNDn_3d = NewOperator("Number", NF, IndexDn_3d,IndexDn_3d,{1,1,1,1,1}) OppN_3d = OppNUp_3d + OppNDn_3d OppNUp_Ld = NewOperator("Number", NF, IndexUp_Ld,IndexUp_Ld,{1,1,1,1,1}) OppNDn_Ld = NewOperator("Number", NF, IndexDn_Ld,IndexDn_Ld,{1,1,1,1,1}) OppN_Ld = OppNUp_Ld + OppNDn_Ld -- Besides the onsite energy of the ligand and transition metal d-shell we need to define the -- hopping between them. We can use the same crystal-field operator, but now acting between -- two different shells. Akm = PotentialExpandedOnClm("Oh", 2, {1,0}) OppVeg = NewOperator("CF", NF, IndexUp_3d,IndexDn_3d, IndexUp_Ld,IndexDn_Ld,Akm) + NewOperator("CF", NF, IndexUp_Ld,IndexDn_Ld, IndexUp_3d,IndexDn_3d,Akm) Akm = PotentialExpandedOnClm("Oh", 2, {0,1}) OppVt2g = NewOperator("CF", NF, IndexUp_3d,IndexDn_3d, IndexUp_Ld,IndexDn_Ld,Akm) + NewOperator("CF", NF, IndexUp_Ld,IndexDn_Ld, IndexUp_3d,IndexDn_3d,Akm) -- Once all operators are defined we need to set parameters -- We follow the energy definitions as introduced in the group of G.A. Sawatzky (Groningen) -- J. Zaanen, G.A. Sawatzky, and J.W. Allen PRL 55, 418 (1985) -- for parameters of specific materials see -- A.E. Bockquet et al. PRB 55, 1161 (1996) -- After some initial discussion (some older papers use different definitions) the -- energies U and Delta refer to the center of a configuration -- The L^10 d^n configuration has an energy 0 -- The L^9 d^n+1 configuration has an energy Delta -- The L^8 d^n+2 configuration has an energy 2*Delta+U -- -- If we relate this to the onsite energy of the p and d orbitals we find -- 10 eL + n ed + n(n-1) U/2 == 0 -- 9 eL + (n+1) ed + (n+1)n U/2 == Delta -- 8 eL + (n+2) ed + (n+1)(n+2) U/2 == 2*Delta+U -- 3 equations with 2 unknowns, but with interdependence yield: -- ed = (10*Delta-nd*(19+nd)*U/2)/(10+nd) -- ep = nd*((1+nd)*U/2-Delta)/(10+nd) -- -- note that ed-ep = Delta - nd * U and not Delta -- note furthermore that ep and ed here are defined for the onsite energy if the system had -- locally nd electrons in the d-shell. In DFT or Hartree Fock the d occupation is in the end not -- nd and thus the onsite energy of the Kohn-Sham orbitals is not equal to ep and ed in model -- calculations. -- -- note furthermore that ep and eL actually should be different for most systems. We happily ignore this fact -- -- We normally take U and Delta as experimentally determined parameters. Especially -- core level photo-emission is sensitive to these parameters and can be used to determine -- the starting point of these models. (see the work of Bockquet et al as referenced above) -- number of electrons (formal valence) nd = 8 -- parameters from experiment (core level PES) U = 7.3 Delta = 4.7 -- parameters obtained from DFT (PRB 85, 165113 (2012)) F2dd = 11.142 F4dd = 6.874 tenDq = 0.56 tenDqL = 1.44 Veg = 2.06 Vt2g = 1.21 zeta_3d = 0.081 Bz = 0.000001 -- turning U and Delta to onsite energies (Including the transformation from U to F0) ed =(10*Delta-nd*(19+nd)*U/2)/(10+nd) eL = nd*((1+nd)*U/2-Delta)/(10+nd) F0dd = U+(F2dd+F4dd)*2/63 -- and our Hamiltonian is the sum over several operators Hamiltonian = F0dd*OppF0_3d + F2dd*OppF2_3d + F4dd*OppF4_3d + zeta_3d*Oppldots_3d + Bz*(2*OppSz_3d + OppLz_3d) + tenDq*OpptenDq_3d + tenDqL*OpptenDq_Ld + Veg * OppVeg + Vt2g * OppVt2g + ed * OppN_3d + eL * OppN_Ld Hamiltonian_CT = tenDqL*OpptenDq_Ld + Veg * OppVeg + Vt2g * OppVt2g + ed * OppN_3d + eL * OppN_Ld -- we now can create the lowest Npsi eigenstates: Npsi=3 -- in order to make sure we have a filling of 8 electrons we need to define some restrictions StartRestrictions = {NF, NB, {"1111111111 0000000000",nd,nd}, {"0000000000 1111111111",10,10}} psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi) oppList={Hamiltonian, OppSsqr, OppLsqr, OppJsqr, OppSz_3d, OppLz_3d, Oppldots_3d, OppF2_3d, OppF4_3d, OppNeg_3d, OppNt2g_3d, OppNeg_Ld, OppNt2g_Ld, OppN_3d} -- print of some expectation values print(" # "); for i = 1,#psiList do io.write(string.format("%3i ",i)) for j = 1,#oppList do expectationvalue = Chop(psiList[i]*oppList[j]*psiList[i]) io.write(string.format("%8.3f ",expectationvalue)) end io.write("\n") end -- Alternative way to get the wavefunctions. psiList1 = Eigensystem(Hamiltonian, StartRestrictions, Npsi, {{"Restrictions", StartRestrictions}}) psiList1 = Hamiltonian_CT * psiList1 Eigensystem(Hamiltonian, psiList1, {{'ExpandBasis', false}}) io.write("\n") -- print of some expectation values print(" # "); for i = 1,#psiList1 do io.write(string.format("%3i ",i)) for j = 1,#oppList do expectationvalue = Chop(psiList1[i]*oppList[j]*psiList1[i]) io.write(string.format("%8.3f ",expectationvalue)) end io.write("\n") end