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