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.
- Groundstate.Quanty
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:
- Groundstate.out
# <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
