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 — documentation:tutorials:nio_ligand_field:groundstate [2016/10/10 09:41] (current) 2016/10/09 15:20 Maurits W. Haverkort created 2016/10/09 15:20 Maurits W. Haverkort created Line 1: Line 1: + {{indexmenu_n>​1}} + ====== Groundstate ====== + ### + The first example looks at the ground-state of NiO. Ni in NiO is $2+$ and thus has locally formal valence of $8$ electrons in the Ni $d$-shell. The lowest state has two holes in the $e_g$ orbitals with $S=1$ ($\langle S^2\rangle=1(2+1)=2$). Covalence allows the $d^8$ configuration to mix with a $d^9$ and $d^{10}$ configuration. In ligand field theory there is only a single shell of $d$ symmetry representing linear combinations of the ligand orbitals. There are 45 states in the $d^8$ configuration $10\times 10$ states in the $d^9L^9$ configuration and 45 states in the $d^{10}L^8$ configuration. The following example calculates these 190 eigen-states. + ### + + ### + ​ + 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 + + -- we now can create the lowest Npsi eigenstates:​ + Npsi=190 + -- 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(" ​ #    <​E> ​     <​S^2> ​   <​L^2> ​   <​J^2> ​   <​S_z^3d>​ <​L_z^3d>​ <​l.s> ​   <​F[2]> ​  <​F[4]> ​  <​Neg^3d>​ <​Nt2g^3d><​Neg^Ld>​ <​Nt2g^Ld><​N^3d>"​);​ + 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 + ​ + ### + + ### + The output is: + ​ + # ​   <​E> ​     <​S^2> ​   <​L^2> ​   <​J^2> ​   <​S_z^3d>​ <​L_z^3d>​ <​l.s> ​   <​F[2]> ​  <​F[4]> ​  <​Neg^3d>​ <​Nt2g^3d><​Neg^Ld>​ <​Nt2g^Ld><​N^3d>​ + 1   ​-3.395 ​   1.999   ​12.000 ​  ​15.147 ​  ​-0.905 ​  ​-0.280 ​  ​-0.319 ​  ​-1.043 ​  ​-0.925 ​   2.189    5.989    3.823    6.000    8.178 + 2   ​-3.395 ​   1.999   ​12.000 ​  ​15.147 ​  ​-0.000 ​  ​-0.000 ​  ​-0.319 ​  ​-1.043 ​  ​-0.925 ​   2.189    5.989    3.823    6.000    8.178 + 3   ​-3.395 ​   1.999   ​12.000 ​  ​15.147 ​   0.905    0.280   ​-0.319 ​  ​-1.043 ​  ​-0.925 ​   2.189    5.989    3.823    6.000    8.178 + 4   ​-2.435 ​   1.979   ​11.981 ​  ​16.757 ​  ​-0.000 ​  ​-0.000 ​  ​-0.876 ​  ​-1.035 ​  ​-0.910 ​   3.099    5.025    3.904    5.971    8.124 + 5   ​-2.435 ​   1.979   ​11.981 ​  ​16.757 ​  ​-0.000 ​   0.000   ​-0.876 ​  ​-1.035 ​  ​-0.910 ​   3.099    5.025    3.904    5.971    8.124 + 6   ​-2.416 ​   1.999   ​11.998 ​  ​15.842 ​  ​-0.457 ​  ​-0.053 ​  ​-0.476 ​  ​-1.036 ​  ​-0.910 ​   3.102    5.021    3.905    5.971    8.123 + 7   ​-2.416 ​   1.999   ​11.998 ​  ​15.842 ​   0.000    0.000   ​-0.476 ​  ​-1.036 ​  ​-0.910 ​   3.102    5.021    3.905    5.971    8.123 + 8   ​-2.416 ​   1.999   ​11.998 ​  ​15.842 ​   0.457    0.053   ​-0.476 ​  ​-1.036 ​  ​-0.910 ​   3.102    5.021    3.905    5.971    8.123 + 9   ​-2.367 ​   1.997   ​11.989 ​  ​12.563 ​  ​-0.465 ​  ​-0.146 ​   0.251   ​-1.036 ​  ​-0.911 ​   3.092    5.033    3.904    5.971    8.125 + ​10 ​  ​-2.367 ​   1.997   ​11.989 ​  ​12.563 ​  ​-0.000 ​   0.000    0.251   ​-1.036 ​  ​-0.911 ​   3.092    5.033    3.904    5.971    8.125 + ​11 ​  ​-2.367 ​   1.997   ​11.989 ​  ​12.563 ​   0.465    0.146    0.251   ​-1.036 ​  ​-0.911 ​   3.092    5.033    3.904    5.971    8.125 + ​12 ​  ​-2.348 ​   2.000   ​12.003 ​  ​12.000 ​   0.000   ​-0.000 ​   0.481   ​-1.036 ​  ​-0.911 ​   3.095    5.029    3.905    5.971    8.124 + ​13 ​  ​-1.812 ​   1.989   ​11.386 ​  ​18.624 ​  ​-0.000 ​  ​-0.000 ​  ​-1.397 ​  ​-1.016 ​  ​-0.911 ​   3.604    4.492    3.950    5.954    8.097 + ​14 ​  ​-1.756 ​   0.812   ​10.078 ​   9.840   ​-0.000 ​  ​-0.000 ​  ​-0.831 ​  ​-0.977 ​  ​-0.904 ​   2.829    5.383    3.807    5.981    8.212 + ​15 ​  ​-1.756 ​   0.812   ​10.078 ​   9.840   ​-0.000 ​   0.000   ​-0.831 ​  ​-0.977 ​  ​-0.904 ​   2.829    5.383    3.807    5.981    8.212 + ​16 ​  ​-1.749 ​   1.998   ​11.302 ​  ​15.074 ​  ​-0.465 ​   0.803   ​-0.461 ​  ​-1.016 ​  ​-0.913 ​   3.592    4.507    3.947    5.954    8.099 + ​17 ​  ​-1.749 ​   1.998   ​11.302 ​  ​15.074 ​   0.000    0.000   ​-0.461 ​  ​-1.016 ​  ​-0.913 ​   3.592    4.507    3.947    5.954    8.099 + ​18 ​  ​-1.749 ​   1.998   ​11.302 ​  ​15.074 ​   0.465   ​-0.803 ​  ​-0.461 ​  ​-1.016 ​  ​-0.913 ​   3.592    4.507    3.947    5.954    8.099 + ​19 ​  ​-1.651 ​   1.998   ​11.120 ​   9.922   ​-0.465 ​   0.420    0.723   ​-1.012 ​  ​-0.917 ​   3.567    4.537    3.942    5.954    8.104 + ​20 ​  ​-1.651 ​   1.998   ​11.120 ​   9.922   ​-0.000 ​   0.000    0.723   ​-1.012 ​  ​-0.917 ​   3.567    4.537    3.942    5.954    8.104 + ​21 ​  ​-1.651 ​   1.998   ​11.120 ​   9.922    0.465   ​-0.420 ​   0.723   ​-1.012 ​  ​-0.917 ​   3.567    4.537    3.942    5.954    8.104 + ​22 ​  ​-1.566 ​   1.210   ​11.020 ​   9.045    0.000   ​-0.000 ​   1.857   ​-0.993 ​  ​-0.909 ​   3.080    5.100    3.849    5.971    8.180 + ​23 ​  ​-1.566 ​   1.210   ​11.020 ​   9.045    0.000    0.000    1.857   ​-0.993 ​  ​-0.909 ​   3.080    5.100    3.849    5.971    8.180 + ​24 ​  ​-0.706 ​   0.018   ​17.151 ​  ​17.207 ​  ​-0.000 ​  ​-0.000 ​   0.071   ​-0.902 ​  ​-0.949 ​   2.598    5.771    3.640    5.991    8.369 + ​25 ​  ​-0.635 ​   0.122    8.089    8.316   ​-0.028 ​  ​-0.344 ​  ​-0.574 ​  ​-0.939 ​  ​-0.871 ​   3.250    4.952    3.850    5.948    8.202 + ​26 ​  ​-0.635 ​   0.122    8.089    8.316   ​-0.000 ​   0.000   ​-0.574 ​  ​-0.939 ​  ​-0.871 ​   3.250    4.952    3.850    5.948    8.202 + ​27 ​  ​-0.635 ​   0.122    8.089    8.316    0.028    0.344   ​-0.574 ​  ​-0.939 ​  ​-0.871 ​   3.250    4.952    3.850    5.948    8.202 + ​28 ​  ​-0.178 ​   1.995    3.249    6.671   ​-0.000 ​  ​-0.000 ​  ​-0.261 ​  ​-0.820 ​  ​-1.047 ​   3.584    4.591    3.898    5.927    8.175 + ​29 ​  ​-0.178 ​   1.995    3.249    6.671   ​-0.000 ​   0.000   ​-0.261 ​  ​-0.820 ​  ​-1.047 ​   3.584    4.591    3.898    5.927    8.175 + ​30 ​  ​-0.154 ​   1.881    3.675    6.908   ​-0.430 ​  ​-0.307 ​   0.306   ​-0.827 ​  ​-1.037 ​   3.568    4.610    3.894    5.928    8.178 + ​31 ​  ​-0.154 ​   1.881    3.675    6.908    0.000    0.000    0.306   ​-0.827 ​  ​-1.037 ​   3.568    4.610    3.894    5.928    8.178 + ​32 ​  ​-0.154 ​   1.881    3.675    6.908    0.429    0.307    0.306   ​-0.827 ​  ​-1.037 ​   3.568    4.610    3.894    5.928    8.178 + ​33 ​  ​-0.127 ​   1.968    3.289    3.679   ​-0.438 ​  ​-0.360 ​   0.301   ​-0.817 ​  ​-1.050 ​   3.549    4.634    3.889    5.928    8.183 + ​34 ​  ​-0.127 ​   1.968    3.289    3.679   ​-0.000 ​   0.000    0.301   ​-0.817 ​  ​-1.050 ​   3.549    4.634    3.889    5.928    8.183 + ​35 ​  ​-0.127 ​   1.968    3.289    3.679    0.438    0.360    0.301   ​-0.817 ​  ​-1.050 ​   3.549    4.634    3.889    5.928    8.183 + ​36 ​  ​-0.101 ​   1.989    3.035    1.789    0.000   ​-0.000 ​   0.601   ​-0.815 ​  ​-1.053 ​   3.543    4.641    3.887    5.928    8.185 + ​37 ​  ​-0.025 ​   0.035   ​19.364 ​  ​19.395 ​  ​-0.016 ​  ​-0.467 ​   0.091   ​-0.873 ​  ​-0.933 ​   3.217    5.048    3.794    5.941    8.265 + ​38 ​  ​-0.025 ​   0.035   ​19.364 ​  ​19.395 ​   0.000    0.000    0.091   ​-0.873 ​  ​-0.933 ​   3.217    5.048    3.794    5.941    8.265 + ​39 ​  ​-0.025 ​   0.035   ​19.364 ​  ​19.395 ​   0.016    0.467    0.091   ​-0.873 ​  ​-0.933 ​   3.217    5.048    3.794    5.941    8.265 + ​40 ​   0.895    0.004   ​15.844 ​  ​15.848 ​  ​-0.000 ​  ​-0.000 ​   0.096   ​-0.870 ​  ​-0.876 ​   3.961    4.189    3.978    5.872    8.150 + ​41 ​   0.895    0.004   ​15.844 ​  ​15.848 ​  ​-0.000 ​   0.000    0.096   ​-0.870 ​  ​-0.876 ​   3.961    4.189    3.978    5.872    8.150 + ​42 ​   0.960    0.005   ​17.378 ​  ​17.383 ​  ​-0.002 ​  ​-1.644 ​   0.078   ​-0.858 ​  ​-0.887 ​   3.944    4.214    3.974    5.869    8.157 + ​43 ​   0.960    0.005   ​17.378 ​  ​17.383 ​   0.000    0.000    0.078   ​-0.858 ​  ​-0.887 ​   3.944    4.214    3.974    5.869    8.157 + ​44 ​   0.960    0.005   ​17.378 ​  ​17.383 ​   0.002    1.644    0.078   ​-0.858 ​  ​-0.887 ​   3.944    4.214    3.974    5.869    8.157 + ​45 ​   3.052    0.003    4.466    4.464   ​-0.000 ​  ​-0.000 ​   0.118   ​-0.925 ​  ​-0.942 ​   3.751    4.798    3.727    5.724    8.549 + ​46 ​   3.492    0.014   ​11.999 ​  ​12.000 ​  ​-0.000 ​  ​-0.000 ​  ​-0.185 ​  ​-1.143 ​  ​-1.143 ​   3.006    5.993    3.001    6.000    8.999 + ​47 ​   3.493    1.999   ​12.009 ​  ​14.759 ​  ​-0.175 ​  ​-0.028 ​  ​-0.160 ​  ​-1.143 ​  ​-1.143 ​   3.005    5.995    3.001    6.000    8.999 + ​48 ​   3.493    1.999   ​12.009 ​  ​14.759 ​  ​-0.000 ​  ​-0.000 ​  ​-0.160 ​  ​-1.143 ​  ​-1.143 ​   3.005    5.995    3.001    6.000    8.999 + ​49 ​   3.493    1.999   ​12.009 ​  ​14.759 ​   0.172    0.027   ​-0.160 ​  ​-1.143 ​  ​-1.143 ​   3.005    5.995    3.001    6.000    8.999 + ​50 ​   3.494    1.997   ​12.015 ​  ​14.572 ​  ​-0.247 ​  ​-0.042 ​  ​-0.148 ​  ​-1.143 ​  ​-1.143 ​   3.004    5.995    3.002    5.999    8.999 + ​51 ​   3.494    1.997   ​12.015 ​  ​14.572 ​  ​-0.000 ​   0.000   ​-0.148 ​  ​-1.143 ​  ​-1.143 ​   3.004    5.995    3.002    5.999    8.999 + ​52 ​   3.494    1.997   ​12.015 ​  ​14.572 ​   0.250    0.043   ​-0.148 ​  ​-1.143 ​  ​-1.143 ​   3.004    5.995    3.002    5.999    8.999 + ​53 ​   3.494    1.995   ​11.997 ​  ​14.383 ​  ​-0.422 ​  ​-0.054 ​  ​-0.134 ​  ​-1.142 ​  ​-1.143 ​   3.003    5.996    3.002    5.999    8.999 + ​54 ​   3.494    1.995   ​11.997 ​  ​14.383 ​   0.000   ​-0.000 ​  ​-0.134 ​  ​-1.142 ​  ​-1.143 ​   3.003    5.996    3.002    5.999    8.999 + ​55 ​   3.494    1.995   ​11.997 ​  ​14.384 ​   0.423    0.054   ​-0.134 ​  ​-1.142 ​  ​-1.143 ​   3.003    5.996    3.002    5.999    8.999 + ​56 ​   4.081    1.987   ​12.000 ​  ​14.934 ​  ​-0.542 ​  ​-0.304 ​  ​-0.388 ​  ​-1.137 ​  ​-1.113 ​   2.912    5.972    3.118    5.998    8.884 + ​57 ​   4.081    1.987   ​12.000 ​  ​14.934 ​   0.000   ​-0.000 ​  ​-0.388 ​  ​-1.137 ​  ​-1.113 ​   2.912    5.972    3.118    5.998    8.884 + ​58 ​   4.081    1.987   ​12.000 ​  ​14.934 ​   0.542    0.304   ​-0.388 ​  ​-1.137 ​  ​-1.113 ​   2.912    5.972    3.118    5.998    8.884 + ​59 ​   4.466    1.875   ​12.986 ​  ​17.651 ​  ​-0.000 ​  ​-0.000 ​  ​-0.718 ​  ​-1.133 ​  ​-1.129 ​   3.650    5.300    3.291    5.759    8.950 + ​60 ​   4.466    1.875   ​12.986 ​  ​17.651 ​  ​-0.000 ​   0.000   ​-0.718 ​  ​-1.133 ​  ​-1.129 ​   3.650    5.300    3.291    5.759    8.950 + ​61 ​   4.478    1.782   ​11.873 ​  ​15.127 ​  ​-0.188 ​   0.079   ​-0.550 ​  ​-1.137 ​  ​-1.134 ​   3.673    5.293    3.328    5.706    8.966 + ​62 ​   4.478    1.782   ​11.873 ​  ​15.127 ​   0.000    0.000   ​-0.550 ​  ​-1.137 ​  ​-1.134 ​   3.673    5.293    3.328    5.706    8.966 + ​63 ​   4.478    1.782   ​11.873 ​  ​15.127 ​   0.188   ​-0.079 ​  ​-0.550 ​  ​-1.137 ​  ​-1.134 ​   3.673    5.293    3.328    5.706    8.966 + ​64 ​   4.499    1.569   ​14.522 ​  ​17.563 ​  ​-0.061 ​  ​-0.151 ​  ​-0.519 ​  ​-1.136 ​  ​-1.136 ​   3.592    5.379    3.408    5.622    8.971 + ​65 ​   4.499    1.569   ​14.522 ​  ​17.563 ​  ​-0.000 ​  ​-0.000 ​  ​-0.519 ​  ​-1.136 ​  ​-1.136 ​   3.592    5.379    3.408    5.622    8.971 + ​66 ​   4.499    1.569   ​14.522 ​  ​17.564 ​   0.061    0.151   ​-0.519 ​  ​-1.136 ​  ​-1.136 ​   3.592    5.379    3.408    5.622    8.971 + ​67 ​   4.526    1.554   ​17.593 ​  ​19.202 ​  ​-0.000 ​  ​-0.000 ​  ​-0.585 ​  ​-1.119 ​  ​-1.122 ​   3.362    5.564    3.422    5.652    8.926 + ​68 ​   4.526    1.554   ​17.593 ​  ​19.202 ​   0.000    0.000   ​-0.585 ​  ​-1.119 ​  ​-1.122 ​   3.362    5.564    3.422    5.652    8.926 + ​69 ​   4.541    1.989   ​11.221 ​  ​12.000 ​  ​-0.000 ​   0.000    0.386   ​-1.140 ​  ​-1.137 ​   3.626    5.351    3.372    5.651    8.977 + ​70 ​   4.556    1.887   ​14.623 ​  ​14.589 ​   0.045   ​-0.312 ​   0.342   ​-1.140 ​  ​-1.139 ​   3.535    5.447    3.453    5.565    8.982 + ​71 ​   4.556    1.887   ​14.623 ​  ​14.589 ​   0.000   ​-0.000 ​   0.342   ​-1.140 ​  ​-1.139 ​   3.535    5.447    3.453    5.565    8.982 + ​72 ​   4.556    1.887   ​14.623 ​  ​14.589 ​  ​-0.045 ​   0.312    0.342   ​-1.140 ​  ​-1.139 ​   3.535    5.447    3.453    5.565    8.982 + ​73 ​   4.568    1.697   ​16.245 ​  ​16.043 ​  ​-0.103 ​   0.043    0.177   ​-1.139 ​  ​-1.141 ​   3.462    5.527    3.540    5.471    8.989 + ​74 ​   4.568    1.697   ​16.245 ​  ​16.042 ​  ​-0.000 ​   0.000    0.177   ​-1.139 ​  ​-1.141 ​   3.462    5.527    3.540    5.471    8.989 + ​75 ​   4.568    1.697   ​16.244 ​  ​16.042 ​   0.103   ​-0.043 ​   0.177   ​-1.139 ​  ​-1.141 ​   3.462    5.527    3.540    5.471    8.989 + ​76 ​   4.569    1.980   ​19.639 ​  ​19.620 ​   0.000   ​-0.000 ​   0.042   ​-1.137 ​  ​-1.141 ​   3.447    5.538    3.551    5.464    8.986 + ​77 ​   4.580    0.568   ​14.051 ​  ​15.098 ​   0.054   ​-0.213 ​   0.224   ​-1.140 ​  ​-1.141 ​   3.417    5.575    3.579    5.429    8.992 + ​78 ​   4.580    0.568   ​14.051 ​  ​15.098 ​   0.000   ​-0.000 ​   0.224   ​-1.140 ​  ​-1.141 ​   3.417    5.575    3.579    5.429    8.992 + ​79 ​   4.580    0.568   ​14.051 ​  ​15.098 ​  ​-0.053 ​   0.211    0.224   ​-1.140 ​  ​-1.141 ​   3.417    5.575    3.579    5.429    8.992 + ​80 ​   4.580    0.534    8.335    8.968   ​-0.101 ​  ​-0.216 ​   0.129   ​-1.140 ​  ​-1.142 ​   3.402    5.592    3.601    5.405    8.994 + ​81 ​   4.580    0.534    8.335    8.968    0.000   ​-0.000 ​   0.129   ​-1.140 ​  ​-1.142 ​   3.402    5.592    3.601    5.405    8.994 + ​82 ​   4.580    0.534    8.336    8.969    0.101    0.218    0.129   ​-1.140 ​  ​-1.142 ​   3.402    5.592    3.601    5.405    8.994 + ​83 ​   4.652    0.638   ​12.301 ​  ​12.344 ​   0.000    0.000    0.461   ​-1.097 ​  ​-1.090 ​   2.944    5.900    3.396    5.760    8.843 + ​84 ​   4.652    0.638   ​12.301 ​  ​12.344 ​   0.000    0.000    0.461   ​-1.097 ​  ​-1.090 ​   2.944    5.900    3.396    5.760    8.843 + ​85 ​   4.883    1.976   ​12.167 ​  ​15.996 ​  ​-0.255 ​  ​-0.035 ​  ​-0.307 ​  ​-1.142 ​  ​-1.130 ​   3.256    5.692    3.758    5.294    8.948 + ​86 ​   4.883    1.976   ​12.167 ​  ​15.996 ​  ​-0.000 ​  ​-0.000 ​  ​-0.307 ​  ​-1.142 ​  ​-1.130 ​   3.256    5.692    3.758    5.294    8.948 + ​87 ​   4.883    1.976   ​12.167 ​  ​15.996 ​   0.255    0.035   ​-0.307 ​  ​-1.142 ​  ​-1.130 ​   3.256    5.692    3.758    5.294    8.948 + ​88 ​   4.887    1.963   ​12.161 ​  ​17.219 ​  ​-0.000 ​  ​-0.000 ​  ​-0.239 ​  ​-1.141 ​  ​-1.128 ​   3.252    5.691    3.744    5.313    8.943 + ​89 ​   4.887    1.963   ​12.161 ​  ​17.219 ​   0.000    0.000   ​-0.239 ​  ​-1.141 ​  ​-1.128 ​   3.252    5.691    3.744    5.313    8.943 + ​90 ​   4.899    1.980   ​11.769 ​  ​12.838 ​  ​-0.221 ​   0.004   ​-0.068 ​  ​-1.141 ​  ​-1.128 ​   3.290    5.651    3.717    5.342    8.941 + ​91 ​   4.899    1.980   ​11.769 ​  ​12.838 ​  ​-0.000 ​  ​-0.000 ​  ​-0.068 ​  ​-1.141 ​  ​-1.128 ​   3.290    5.651    3.717    5.342    8.941 + ​92 ​   4.899    1.980   ​11.769 ​  ​12.838 ​   0.221   ​-0.004 ​  ​-0.068 ​  ​-1.141 ​  ​-1.128 ​   3.290    5.651    3.717    5.342    8.941 + ​93 ​   4.905    1.997   ​11.743 ​  ​12.000 ​   0.000    0.000    0.014   ​-1.141 ​  ​-1.127 ​   3.319    5.619    3.694    5.368    8.939 + ​94 ​   5.123    1.903    5.329    7.412   ​-0.261 ​  ​-0.309 ​  ​-0.277 ​  ​-1.088 ​  ​-1.130 ​   3.403    5.458    3.629    5.510    8.861 + ​95 ​   5.123    1.903    5.329    7.411   ​-0.000 ​  ​-0.000 ​  ​-0.277 ​  ​-1.088 ​  ​-1.130 ​   3.403    5.458    3.629    5.510    8.861 + ​96 ​   5.123    1.903    5.329    7.411    0.261    0.309   ​-0.277 ​  ​-1.088 ​  ​-1.130 ​   3.403    5.458    3.629    5.510    8.861 + ​97 ​   5.133    1.957    6.440    8.653   ​-0.000 ​   0.000   ​-0.217 ​  ​-1.089 ​  ​-1.127 ​   3.443    5.419    3.614    5.524    8.862 + ​98 ​   5.135    1.973    6.077    7.516   ​-0.260 ​  ​-0.048 ​  ​-0.074 ​  ​-1.089 ​  ​-1.130 ​   3.463    5.399    3.599    5.539    8.862 + ​99 ​   5.135    1.973    6.077    7.516    0.000    0.000   ​-0.074 ​  ​-1.089 ​  ​-1.130 ​   3.463    5.399    3.599    5.539    8.862 + 100    5.135    1.973    6.077    7.516    0.260    0.048   ​-0.074 ​  ​-1.089 ​  ​-1.130 ​   3.463    5.399    3.599    5.539    8.862 + ​ + ### + + + + ===== Table of contents ===== + {{indexmenu>​.#​1|msort}}