# Differences

This shows you the differences between two versions of the page.

 — documentation:tutorials:nio_ligand_field:energy_level_diagram [2016/10/10 09:41] (current) 2016/10/09 15:23 Maurits W. Haverkort created 2016/10/09 15:23 Maurits W. Haverkort created Line 1: Line 1: + {{indexmenu_n>​3}} + ====== Energy level diagram ====== + ### + In order to do temperature averaging it is important to understand the number of excited states that are important. One can learn a lot by looking at the energy level diagram. Here we plot one for Ni$^{2+}$. + ### + + ### + The input file is: + ​ + Verbosity(0) + + -- In order to understand the physics / chemistry of a system it is often good + -- to make energy level diagrams. i.e. plot the eigen-state energy as a function + -- of some parameter one varies. + + -- Here we create the energy level diagram of all 190 states of NiO in the ligand + -- field approximation as a function of the Ni onsite crystal-field strenght + + 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} + + -- angular momentum operators on the d-shell + + 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) + + -- Angular momentum operators on the Ligand 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) + + -- total angular momentum + + 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 + + -- define the coulomb operator + -- we here define the part depending on F0 seperately from the part depending on F2 + -- when summing we can put in the numerical values of the slater integrals + + 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}) + + -- define onsite energies - crystal field + -- Akm = {{k1,​m1,​Akm1},​{k2,​m2,​Akm2},​ ... } + + 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) + + 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) + + 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 + + -- define L-d interaction + 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) + + -- 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 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 + + -- 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 + + 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 + + Hamiltonian0 =  F0dd*OppF0_3d + F2dd*OppF2_3d + F4dd*OppF4_3d + zeta_3d*Oppldots_3d + Bz*(2*OppSz_3d + OppLz_3d) + + tenDqL*OpptenDq_Ld + Veg * OppVeg + Vt2g * OppVt2g + + ed * OppN_3d + eL * OppN_Ld + ​ + Npsi=190 + StartRestrictions = {NF, NB, {"​1111111111 0000000000",​8,​8},​ {"​0000000000 1111111111",​10,​10}} + psiList = Eigensystem(Hamiltonian0,​ StartRestrictions,​ Npsi) + + file = assert( io.open("​EnergyLevelDiagram",​ "​w"​)) + + for i=0, 30 do + tenDq = 0.1*i + file:​write(string.format("​%14.7E ",​tenDq)) + Hamiltonian=Hamiltonian0 + tenDq * OpptenDq_3d + Eigensystem(Hamiltonian,​ psiList) + for key,value in pairs(psiList) do + energy = value * Hamiltonian * value + file:​write(string.format("​%14.7E ",​energy)) + end + file:​write("​\n"​) + end + + file:​close() + + gnuplotInput = [[ + set autoscale ​ + set xtic auto + set ytic auto + set style line  1 lt 1 lw 1 lc rgb "#​000000"​ + + set xlabel "10Dq (eV)" font "​Times,​12"​ + set ylabel "​Energy (eV)" font "​Times,​12"​ + + set out '​EnergyLevelDiagram.ps'​ + set size 1.0, 1.0 + set terminal postscript portrait enhanced color  "​Times"​ 8 + + plot for [i=2:191] "​EnergyLevelDiagram"​ using 1:i notitle with lines ls  1 + ]] + + -- write the gnuplot script to a file + file = io.open("​EnergyLevelDiagram.gnuplot",​ "​w"​) + file:​write(gnuplotInput) + file:​close() + + -- call gnuplot to execute the script + os.execute("​gnuplot EnergyLevelDiagram.gnuplot"​) + -- change the postscript file to pdf or eps + os.execute("​ps2pdf EnergyLevelDiagram.ps ; ps2eps EnergyLevelDiagram.ps ; mv EnergyLevelDiagram.eps temp.eps ; eps2eps temp.eps EnergyLevelDiagram.eps ; rm temp.eps"​) + ​ + ### + + As in example 4 Quanty returns a nice plot. Note that one can add labeling. For this have a look at example 4. + | {{:​documentation:​tutorials:​nio_ligand_field:​energyleveldiagram.png?​nolink |}} | + ^Energy level diagram showing the energies of the different multiplets as a function of $10Dq$ ^ + + + ### + The script does not write to standard output. + ### + + + ===== Table of contents ===== + {{indexmenu>​.#​1|msort}}