Eigensystem calculates the eigen-energies and eigen-vectors of an Operator or matrix. The function comes in several flavors.
val, fun = Eigensystem(A)
A small example:
A = {{ 1 , I , 0 },
{-I , 2 , 1 },
{ 0 , 1 , 5 }}
val, fun = Eigensystem(A)
print(val)
print(fun)
{ 0.31866935639502 , 2.3579263675185 , 5.3234042760865 }
{ { (0 - 0.82050111444738 I) , 0.55903255238504 , -0.11941744665028 } ,
{ (0 - 0.56721932561261 I) , -0.77024207841542 , 0.29152937637548 } ,
{ (0 + 0.070994069063423 I) , 0.30693606176558 , 0.94907855109346 } }
psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi)
calculates the lowest Npsi eigenstates for an operator Hamiltonian. The program uses iterative methods and needs to have a starting point to be defined. Note that the operator Hamiltonian does not know about the number of electrons there are in the system, only the number of orbitals is defined. If these orbitals are occupied or not is not given in the operator. A set of start restrictions is given by a table listing the number of Fermionic modes, Bosonic modes followed by lists including a string that tells the program which orbitals are included in the counting and then a minimum and maximum number of electrons in those orbitals. For example:
StartRestrictions = {Nf, Nb, {"111111",2,2}}
Which would define two electrons in a p-shell. A start restriction of:
StartRestrictions = {Nf, Nb, {"101010",2,2}}
would tell the program to start from two electrons with spin down and either 0,1,2, or 3 electron with spin up. A well defined calculation,but normally not a calculation one would like to do. If you want to start with two electrons with spin down and no electrons with spin up one needs:
StartRestrictions = {Nf, Nb, {"101010",2,2}, {"010101",0,0}}
The final result must not fulfill the start restrictions, The program finds the lowest states in the basis defined by $(H+1)^{\infty} \psi_0$. Whereby $\psi_0$ is some random state that satisfies the start restrictions. If $S_z$ is a conserved quantum number of $H$ one would keep the number of spin up and down electrons fixed. If not other states with different values of $S_z$ will mix.
A small example:
dofile("../definitions.Quanty")
-- we define an Hamiltonian
Hamiltonian = OppU + 0.1 * Oppldots + 0.000001 * (2*OppSz+OppLz)
-- number of states calculated
Npsi=15
-- In order to make sure we have a filling of 2
-- electrons we need to define some restrictions
StartRestrictions = {Nf, Nb, {"111111",2,2}}
-- We now can create the lowest Npsi eigenstates:
psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi)
-- We define a list of some operators to look at their expectation values
oppList={Hamiltonian, OppSsqr, OppLsqr, OppJsqr, OppSz, OppLz}
-- after we've created the eigen states we can look
-- at a list of their expectation values
print(" <E> <S^2> <L^2> <J^2> <S_z> <L_z>");
for i=1,#psiList do
for j=1,#oppList do
io.write(string.format("%7.4f ",psiList[i]*oppList[j]*psiList[i]))
end
io.write("\n")
end
<E> <S^2> <L^2> <J^2> <S_z> <L_z> -2.1033 1.9989 1.9989 0.0000 -0.0000 0.0000 -2.0500 2.0000 2.0000 2.0000 -0.5000 -0.5000 -2.0500 2.0000 2.0000 2.0000 0.0000 -0.0000 -2.0500 2.0000 2.0000 2.0000 0.5000 0.5000 -1.9521 1.9982 2.0036 6.0000 -0.9991 -1.0009 -1.9521 1.9982 2.0036 6.0000 -0.4995 -0.5005 -1.9521 1.9982 2.0036 6.0000 0.0000 -0.0000 -1.9521 1.9982 2.0036 6.0000 0.4996 0.5004 -1.9521 1.9982 2.0036 6.0000 0.9991 1.0009 0.4021 0.0018 5.9964 6.0000 -0.0009 -1.9991 0.4021 0.0018 5.9964 6.0000 -0.0005 -0.9995 0.4021 0.0018 5.9964 6.0000 0.0000 -0.0000 0.4021 0.0018 5.9964 6.0000 0.0005 0.9995 0.4021 0.0018 5.9964 6.0000 0.0009 1.9991 4.0033 0.0011 0.0011 -0.0000 0.0000 -0.0000
Eigensystem(Hamiltonian, psiList)
Instead of starting from a random state that fulfills restrictions one can start from a pre-defined state. This can be useful if you have a large system and only want to do perturbation theory, configuration interaction for example, or if you want to calculate the eigen states of the Hamiltonian after making a small change to the Hamiltonian. In the example before we first calculate the eigen-state of the Hamiltonian without spin-orbit coupling and in a next step start from these eigen-states to calculate the Hamiltonian with spin-orbit coupling.
A small example:
dofile("../definitions.Quanty")
-- we define an Hamiltonian
Hamiltonian = OppU + 0.000001 * (2*OppSz+OppLz)
-- number of states calculated
Npsi=15
StartRestrictions = {Nf, Nb, {"111111",2,2}}
psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi)
oppList={Hamiltonian, OppSsqr, OppLsqr, OppJsqr, OppSz, OppLz}
print(" <E> <S^2> <L^2> <J^2> <S_z> <L_z>");
for i=1,#psiList do
for j=1,#oppList do
io.write(string.format("%7.4f ",psiList[i]*oppList[j]*psiList[i]))
end
io.write("\n")
end
Hamiltonian = Hamiltonian + 0.1 * Oppldots
oppList={Hamiltonian, OppSsqr, OppLsqr, OppJsqr, OppSz, OppLz}
-- Recalculate eigen states with spin orbit
Eigensystem(Hamiltonian,psiList)
print(" <E> <S^2> <L^2> <J^2> <S_z> <L_z>");
for i=1,#psiList do
for j=1,#oppList do
io.write(string.format("%7.4f ",psiList[i]*oppList[j]*psiList[i]))
end
io.write("\n")
end
<E> <S^2> <L^2> <J^2> <S_z> <L_z> -2.0000 2.0000 2.0000 6.0000 -1.0000 -1.0000 -2.0000 2.0000 2.0000 4.0000 -1.0000 0.0000 -2.0000 2.0000 2.0000 4.0000 0.0000 -1.0000 -2.0000 2.0000 2.0000 2.0000 -1.0000 1.0000 -2.0000 2.0000 2.0000 4.0000 0.0000 0.0000 -2.0000 2.0000 2.0000 2.0000 1.0000 -1.0000 -2.0000 2.0000 2.0000 4.0000 0.0000 1.0000 -2.0000 2.0000 2.0000 4.0000 1.0000 0.0000 -2.0000 2.0000 2.0000 6.0000 1.0000 1.0000 0.4000 0.0000 6.0000 6.0000 0.0000 -2.0000 0.4000 0.0000 6.0000 6.0000 0.0000 -1.0000 0.4000 0.0000 6.0000 6.0000 0.0000 0.0000 0.4000 0.0000 6.0000 6.0000 0.0000 1.0000 0.4000 0.0000 6.0000 6.0000 0.0000 2.0000 4.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 <E> <S^2> <L^2> <J^2> <S_z> <L_z> -2.1033 1.9989 1.9989 0.0000 -0.0000 0.0000 -2.0500 2.0000 2.0000 2.0000 -0.5000 -0.5000 -2.0500 2.0000 2.0000 2.0000 0.0000 -0.0000 -2.0500 2.0000 2.0000 2.0000 0.5000 0.5000 -1.9521 1.9982 2.0036 6.0000 -0.9991 -1.0009 -1.9521 1.9982 2.0036 6.0000 -0.4995 -0.5005 -1.9521 1.9982 2.0036 6.0000 0.0000 -0.0000 -1.9521 1.9982 2.0036 6.0000 0.4996 0.5004 -1.9521 1.9982 2.0036 6.0000 0.9991 1.0009 0.4021 0.0018 5.9964 6.0000 -0.0009 -1.9991 0.4021 0.0018 5.9964 6.0000 -0.0005 -0.9995 0.4021 0.0018 5.9964 6.0000 0.0000 -0.0000 0.4021 0.0018 5.9964 6.0000 0.0005 0.9995 0.4021 0.0018 5.9964 6.0000 0.0009 1.9991 4.0033 0.0011 0.0011 -0.0000 0.0000 -0.0000
The last element of Eigensystem can be a table of options. Possible options are: