Groundstate
The first example looks at the ground-state of NiO. Ni in NiO is $2+$ and thus has locally $8$ electrons in the Ni $d$-shell. The lowest state has two holes in the $e_g$ orbitals with $S=1$ ($\langle S^2\rangle=1(2+1)=2$). The output of this example shows several expectation values of the $45$ multiplets available in the $d^8$ sector.
The input is:
- Groundstate.Quanty
-- This tutorial calculates the ground-state of NiO within the crystal-field theory approximation -- Within crystal-field theory the solid is approximated by a single atom in an effective -- potential. Although an extremely crude approximation it is useful for some cases. -- For correlated transition metal insulators it captures the right symmetry of the -- localized open d-shell. It is useful to determine magnetic g-factors, energies of d-d -- excitations or core level x-ray absorption. (2p to 3d excitations L23 edges) -- One should notice that the effective crystal-field potential is an effective potential -- it is there to mimic the interaction with neighboring ligand atoms. In real materials -- there do not exist such large electro static potentials. -- In order to do crystal-field theory for NiO we need to define a Ni d-shell. -- A d-shell has 10 elements and we label again the even spin-orbitals to be spin down -- and the odd spin-orbitals to be spin up. NF=10 NB=0 IndexDn_3d={0,2,4,6,8} IndexUp_3d={1,3,5,7,9} -- We define several operators acting on the Ni d-shell OppSx =NewOperator("Sx" ,NF, IndexUp_3d, IndexDn_3d) OppSy =NewOperator("Sy" ,NF, IndexUp_3d, IndexDn_3d) OppSz =NewOperator("Sz" ,NF, IndexUp_3d, IndexDn_3d) OppSsqr =NewOperator("Ssqr" ,NF, IndexUp_3d, IndexDn_3d) OppSplus=NewOperator("Splus",NF, IndexUp_3d, IndexDn_3d) OppSmin =NewOperator("Smin" ,NF, IndexUp_3d, IndexDn_3d) OppLx =NewOperator("Lx" ,NF, IndexUp_3d, IndexDn_3d) OppLy =NewOperator("Ly" ,NF, IndexUp_3d, IndexDn_3d) OppLz =NewOperator("Lz" ,NF, IndexUp_3d, IndexDn_3d) OppLsqr =NewOperator("Lsqr" ,NF, IndexUp_3d, IndexDn_3d) OppLplus=NewOperator("Lplus",NF, IndexUp_3d, IndexDn_3d) OppLmin =NewOperator("Lmin" ,NF, IndexUp_3d, IndexDn_3d) OppJx =NewOperator("Jx" ,NF, IndexUp_3d, IndexDn_3d) OppJy =NewOperator("Jy" ,NF, IndexUp_3d, IndexDn_3d) OppJz =NewOperator("Jz" ,NF, IndexUp_3d, IndexDn_3d) OppJsqr =NewOperator("Jsqr" ,NF, IndexUp_3d, IndexDn_3d) OppJplus=NewOperator("Jplus",NF, IndexUp_3d, IndexDn_3d) OppJmin =NewOperator("Jmin" ,NF, IndexUp_3d, IndexDn_3d) Oppldots=NewOperator("ldots",NF, IndexUp_3d, IndexDn_3d) -- Next we define the Coulomb operator. The Coulomb operator is given as: -- Sum_{i,j} e^2/|r_i-r_j| -- This is actually a nastier sum than one might think as r_i-r_j=0 for r_i=r_j. -- The solution is to expand this operator on spherical harmonics -- ~ sum_k,m Y(t_i,p_i) Y(t_j,p_j)^* f(r_i,r_j) -- the part depending on the angles (theta and phi) can be solved analytically. -- The radial part (f(r_i,r_j)) is calculated numerically and results in the Slater integrals -- The sum runs in principle over all possible values of k. The expansion coefficient k is -- related to the momentum transfer of two electrons when they scatter due to the Coulomb -- interaction. There is a maximal transfer of momentum that can happen between two electrons -- which is 2l. Furthermore k has to be even due to parity. -- There are thus l+1 = 3 Slater integrals for two electrons in the d-shell. -- F[0], F[2] and F[4]. F[0] is the spherical interaction between two d electrons. F[2] and -- F[4] represent interaction or scattering processes whereby the electrons exchange an angular -- momentum of 2 or 4. -- -- We here define the part depending on F0, F2 and F4 separately. -- when summing we can put in the numerical values -- of the slater integrals. OppF0 =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {1,0,0}) OppF2 =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {0,1,0}) OppF4 =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {0,0,1}) -- Next we need to define the crystal field. -- The crystal-field is a potential that lifts the degeneracy of the d electrons. -- There are several ways to add such an interaction. We here follow the classical -- notation as for example used in Ballhausen or other textbooks on crystal-field theory. -- We expand the crystal-field potential on (renormalized) spherical harmonics and -- enter the expansion coefficients of this potential. -- For specific symmetries there are relations between the different expansion coefficients. -- The function PotentialExpandedOnClm takes as an input the energies of the eigen-orbitals -- and creates the potential expanded on spherical harmonics. The order of the eigen-energies -- is the same as used in most point-group tables. (in cubic symmetry it is eg before t2g) -- We here create an operator that splits the d-shell into t2g orbitals at an energy of -0.4 -- and eg orbitals at an energy of 0.6, such that the total splitting is one and the center -- of gravity is not changed. One can change the total splitting by multiplying this operator -- Akm = {{k1,m1,Akm1},{k2,m2,Akm2}, ... } Akm = PotentialExpandedOnClm("Oh",2,{0.6,-0.4}); OpptenDq = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm); -- We do not only want to give the t2g and eg electrons a different energy, it is also nice -- to define operators that count the number of t2g and eg electrons. If we define an operator -- where the eg electrons have an onsite energy of 1 and the t2g electrons have an onsite -- energy of 0 then the expectation value of such an operator is proportional to the total -- number of eg electrons. This is what we do in the next 4 lines, once to get an operator -- that counts the number of eg electrons, once to create an operator that counts the number -- of t2g electrons Akm = PotentialExpandedOnClm("Oh",2,{1,0}); OppNeg = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm); Akm = PotentialExpandedOnClm("Oh",2,{0,1}); OppNt2g = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm); -- once all operators are defined we can set some parameter values. -- the value of U drops out of a crystal-field calculation as the total number of electrons -- is always the same U = 0.000 -- F2 and F4 are often referred to in the literature as J_{Hund}. They represent the energy -- differences between different multiplets. Numerical values can be found in the back of -- my PhD. thesis for example. http://arxiv.org/abs/cond-mat/0505214 F2dd = 11.142 F4dd = 6.874 -- F0 is not the same as U, although they are related. Unimportant in crystal-field theory -- the difference between U and F0 is so important that I do include it here. (U=0 so F0 is not) F0dd = U+(F2dd+F4dd)*2/63 -- tenDq in NiO is 1.1 eV as can be seen in optics or using IXS to measure d-d excitations tenDq = 0 -- the Ni 3d spin-orbit is small but finite zeta_3d = 0*0.081 -- we add a small magnetic field Bz = 0.000001 -- note that all energies are in units of electron volt. (tesla to eV is EnergyUnits.Tesla.value = 5.788.... E-5) -- the total Hamiltonian is the sum of the different operators multiplied with their prefactor Hamiltonian = F0dd*OppF0 + F2dd*OppF2 + F4dd*OppF4 + tenDq*OpptenDq + zeta_3d*Oppldots + Bz*(2*OppSz + OppLz) -- we now can create the lowest Npsi eigenstates: Npsi=45; -- in order to make sure we have a filling of 8 -- electrons we need to define some restrictions StartRestrictions = {NF, NB, {"1111111111",8,8}}; -- we calculate the eigen system psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi); -- In order to get some information on these eigenstates it is good to plot expectation values -- We first define a list of all the operators we would like to calculate the expectation value of oppList={Hamiltonian, OppSsqr, OppLsqr, OppJsqr, OppSz, OppLz, Oppldots, OppF2, OppF4, OppNeg, OppNt2g}; -- next we loop over all operators and all states and print the expectation value print(" <E> <S^2> <L^2> <J^2> <S_z> <L_z> <l.s> <F[2]> <F[4]> <Neg> <Nt2g>"); for i = 1,#psiList do for j = 1,#oppList do expectationvalue = Chop(psiList[i]*oppList[j]*psiList[i]) io.write(string.format("%6.3f ",expectationvalue)) end io.write("\n") end
The output is:
- Groundstate.out
Start of BlockGroundState. Converge 45 states to an energy with relative variance smaller than 1.490116119384766E-06 Start of BlockOperatorPsiSerialRestricted Outer loop 1, Number of Determinants: 45 45 last variance 1.406559456443911E+00 Start of BlockOperatorPsiSerialRestricted Start of BlockGroundState. Converge 45 states to an energy with relative variance smaller than 1.490116119384766E-06 Start of BlockOperatorPsiSerial <E> <S^2> <L^2> <J^2> <S_z> <L_z> <l.s> <F[2]> <F[4]> <Neg> <Nt2g> -1.387 2.000 12.000 20.000 -1.000 -3.000 -1.500 -1.020 -0.878 3.500 4.500 -1.387 2.000 12.000 18.000 -1.000 -2.000 -1.000 -1.020 -0.878 2.500 5.500 -1.387 2.000 12.000 14.000 -0.000 -3.000 -0.000 -1.020 -0.878 3.500 4.500 -1.387 2.000 12.000 16.000 -1.000 -1.000 -0.500 -1.020 -0.878 3.300 4.700 -1.387 2.000 12.000 14.000 0.000 -2.000 0.000 -1.020 -0.878 2.500 5.500 -1.387 2.000 12.000 14.000 -1.000 0.000 0.000 -1.020 -0.878 3.800 4.200 -1.387 2.000 12.000 8.000 1.000 -3.000 1.500 -1.020 -0.878 3.500 4.500 -1.387 2.000 12.000 12.046 -0.977 0.954 0.488 -1.020 -0.878 3.300 4.700 -1.387 2.000 12.000 13.954 -0.023 -0.954 0.012 -1.020 -0.878 3.300 4.700 -1.387 2.000 12.000 10.001 0.980 -1.959 1.000 -1.020 -0.878 2.500 5.500 -1.387 2.000 12.000 13.999 0.000 -0.000 0.000 -1.020 -0.878 3.800 4.200 -1.387 2.000 12.000 10.000 -0.980 1.959 1.000 -1.020 -0.878 2.500 5.500 -1.387 2.000 12.000 8.102 -0.950 2.901 1.474 -1.020 -0.878 3.495 4.505 -1.387 2.000 12.000 13.995 -0.001 1.002 0.001 -1.020 -0.878 3.300 4.700 -1.387 2.000 12.000 11.902 0.951 -0.902 0.524 -1.020 -0.878 3.305 4.695 -1.387 2.000 12.000 14.000 0.000 2.000 0.000 -1.020 -0.878 2.500 5.500 -1.387 2.000 12.000 14.000 1.000 0.000 0.000 -1.020 -0.878 3.800 4.200 -1.387 2.000 12.000 14.000 0.000 3.000 0.000 -1.020 -0.878 3.500 4.500 -1.387 2.000 12.000 16.000 1.000 1.000 -0.500 -1.020 -0.878 3.300 4.700 -1.387 2.000 12.000 18.000 1.000 2.000 -1.000 -1.020 -0.878 2.500 5.500 -1.387 2.000 12.000 20.000 1.000 3.000 -1.500 -1.020 -0.878 3.500 4.500 0.451 0.000 6.000 6.000 0.000 -2.000 0.000 -0.918 -0.776 3.143 4.857 0.451 0.000 6.000 6.000 0.000 -1.000 0.000 -0.918 -0.776 3.429 4.571 0.451 0.000 6.000 6.000 0.000 0.000 -0.000 -0.918 -0.776 2.857 5.143 0.451 0.000 6.000 6.000 0.000 1.000 -0.000 -0.918 -0.776 3.429 4.571 0.451 -0.000 6.000 6.000 0.000 2.000 0.000 -0.918 -0.776 3.143 4.857 0.854 2.000 2.000 6.000 -1.000 -1.000 -0.500 -0.714 -1.048 3.200 4.800 0.854 2.000 2.000 4.000 -1.000 0.000 0.000 -0.714 -1.048 3.200 4.800 0.854 2.000 2.000 2.000 -1.000 1.000 0.500 -0.714 -1.048 3.200 4.800 0.854 2.000 2.000 4.000 -0.000 -1.000 0.000 -0.714 -1.048 3.200 4.800 0.854 2.000 2.000 4.000 0.000 0.000 -0.000 -0.714 -1.048 3.200 4.800 0.854 2.000 2.000 4.000 0.000 1.000 0.000 -0.714 -1.048 3.200 4.800 0.854 2.000 2.000 2.000 1.000 -1.000 0.500 -0.714 -1.048 3.200 4.800 0.854 2.000 2.000 4.000 1.000 0.000 0.000 -0.714 -1.048 3.200 4.800 0.854 2.000 2.000 6.000 1.000 1.000 -0.500 -0.714 -1.048 3.200 4.800 1.497 0.000 20.000 20.000 0.000 -4.000 0.000 -0.776 -0.855 3.000 5.000 1.497 -0.000 20.000 20.000 0.000 -3.000 0.000 -0.776 -0.855 3.500 4.500 1.497 -0.000 20.000 20.000 0.000 -2.000 0.000 -0.776 -0.855 3.357 4.643 1.497 0.000 20.000 20.000 0.000 -1.000 0.000 -0.776 -0.855 3.071 4.929 1.497 0.000 20.000 20.000 0.000 0.000 0.000 -0.776 -0.855 2.943 5.057 1.497 -0.000 20.000 20.000 0.000 1.000 0.000 -0.776 -0.855 3.071 4.929 1.497 0.000 20.000 20.000 0.000 2.000 0.000 -0.776 -0.855 3.357 4.643 1.497 0.000 20.000 20.000 0.000 3.000 0.000 -0.776 -0.855 3.500 4.500 1.497 0.000 20.000 20.000 0.000 4.000 0.000 -0.776 -0.855 3.000 5.000 5.719 0.000 -0.000 -0.000 0.000 0.000 0.000 -0.571 -0.571 3.200 4.800
