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.
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-- 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 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