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documentation:tutorials:small_programs_a_quick_start:operators_as_matrices [2016/10/07 18:56] – created Maurits W. Haverkortdocumentation:tutorials:small_programs_a_quick_start:operators_as_matrices [2017/05/03 11:47] (current) – Spelling correction Sven Spachmann
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 +{{indexmenu_n>1}}
 +====== Operators as matrices ======
  
 +###
 +In the first example we show a small script written in Quanty that works and does not require further programming. The script creates the spin, angular momentum and Coulomb operator as matrices on a many body basis. The file conf.in defines the angular momentum of the shell and the number of electrons in this shell. We recommend to understand the quantum mechanics presented in this script, i.e. why are there 15 states for two electrons in the $p$ shell, but not to look into the script too much. You can come back to this after example 5.
 +###
 +
 +###
 +The configuration file is:
 +<code Quanty config.in>
 +-- Angular momentum of the shell (0=="s" 1=="p" 2=="d" 3=="f")
 +l = 1
 +-- Number of electrons
 +Nelec = 1
 +</code>
 +###
 +
 +###
 +The output is:
 +<file Quanty_output Operators_as_Matrices.out>
 +=============================================================
 +--> This small program will print out the possible many-body states for a given electronic configuration
 +--> The obtained states of single Slater determinants will be used as a Hilbert space for several operators.
 +
 +=============================================================
 +
 +Calculating all possible Slater determinants for a shell with angular momentum 1 and a filling of 1 electrons
 +
 +======================== O U T P U T ========================
 +
 +A shell with l= 1 and a filling of 1 electrons has Binomial(6 1) = 6 possible single Slater determinants
 +--> List of possible single Slater determinants which span the basis for all many body states:
 +    1 |100000>
 +    2 |010000>
 +    3 |001000>
 +    4 |000100>
 +    5 |000010>
 +    6 |000001>
 +
 +=============================================================
 +The Sz operator on a basis Single slater determinants for a shell with l = 1 and a filling of 1 electrons is:
 +    1 |100000>  -0.50  0.00  0.00  0.00  0.00  0.00 
 +    2 |010000>   0.00  0.50  0.00  0.00  0.00  0.00 
 +    3 |001000>   0.00  0.00 -0.50  0.00  0.00  0.00 
 +    4 |000100>   0.00  0.00  0.00  0.50  0.00  0.00 
 +    5 |000010>   0.00  0.00  0.00  0.00 -0.50  0.00 
 +    6 |000001>   0.00  0.00  0.00  0.00  0.00  0.50 
 +
 +=============================================================
 +The Lz operator on a basis Single slater determinants for a shell with l = 1 and a filling of 1 electrons is:
 +    1 |100000>  -1.00  0.00  0.00  0.00  0.00  0.00 
 +    2 |010000>   0.00 -1.00  0.00  0.00  0.00  0.00 
 +    3 |001000>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    4 |000100>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    5 |000010>   0.00  0.00  0.00  0.00  1.00  0.00 
 +    6 |000001>   0.00  0.00  0.00  0.00  0.00  1.00 
 +
 +=============================================================
 +The S^2 operator on a basis Single slater determinants for a shell with l = 1 and a filling of 1 electrons is:
 +    1 |100000>   0.75  0.00  0.00  0.00  0.00  0.00 
 +    2 |010000>   0.00  0.75  0.00  0.00  0.00  0.00 
 +    3 |001000>   0.00  0.00  0.75  0.00  0.00  0.00 
 +    4 |000100>   0.00  0.00  0.00  0.75  0.00  0.00 
 +    5 |000010>   0.00  0.00  0.00  0.00  0.75  0.00 
 +    6 |000001>   0.00  0.00  0.00  0.00  0.00  0.75 
 +
 +=============================================================
 +The L^2 operator on a basis Single slater determinants for a shell with l = 1 and a filling of 1 electrons is:
 +    1 |100000>   2.00  0.00  0.00  0.00  0.00  0.00 
 +    2 |010000>   0.00  2.00  0.00  0.00  0.00  0.00 
 +    3 |001000>   0.00  0.00  2.00  0.00  0.00  0.00 
 +    4 |000100>   0.00  0.00  0.00  2.00  0.00  0.00 
 +    5 |000010>   0.00  0.00  0.00  0.00  2.00  0.00 
 +    6 |000001>   0.00  0.00  0.00  0.00  0.00  2.00 
 +
 +=============================================================
 +The Coulomb operator for a shell with l = 1 is given by 2 different operators depending on the angular momentum of the expansion of 1/|r1-r2| in spherical Harmonics
 +=============================================================
 +F[0]
 +    1 |100000>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    2 |010000>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    3 |001000>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    4 |000100>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    5 |000010>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    6 |000001>   0.00  0.00  0.00  0.00  0.00  0.00 
 +=============================================================
 +F[2]
 +    1 |100000>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    2 |010000>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    3 |001000>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    4 |000100>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    5 |000010>   0.00  0.00  0.00  0.00  0.00  0.00 
 +    6 |000001>   0.00  0.00  0.00  0.00  0.00  0.00 
 +
 +=============================================================
 +Timing results
 +   Total_time | NumberOfRuns | Running | Name
 +      0:00:00 |            1 |       0 | Create Determinants
 +      0:00:00 |            1 |       0 | Create operators in second quantization
 +      0:00:00 |            1 |       0 | Create operators in matrix form
 +</file>
 +###
 +
 +###
 +For completeness, the actual code is:
 +<code Quanty Operators_as_Matrices.Quanty>
 +-- script contributed by Yaroslav Kvashnin.
 +
 +-- In order to simplify the program later on we first define some 
 +-- functions.
 +
 +-- Function which swaps two character in a string. It will be used
 +-- to check all possible particles permutations.
 +------------------------------------------------
 +function swap_char(pos1, pos2, str)
 + r=tostring((tonumber(str:sub(pos1,pos1))+3) % 2)
 + z=tostring((tonumber(str:sub(pos2,pos2))+3) % 2)
 + if (r ~= z) then
 +  return str:sub(1, pos1-1) .. r .. str:sub(pos1+1,pos2-1) .. z .. str:sub(pos2+1)
 + else 
 +  return str
 + end
 +end
 +------------------------------------------------
 +
 +-- begin of program
 +print("")
 +print("=============================================================")
 +print("--> This small program will print out the possible many-body states for a given electronic configuration");
 +print("--> The obtained states of single Slater determinants will be used as a Hilbert space for several operators.")
 +print("")
 +print("=============================================================")
 +print("")
 +
 +-- read in the input
 +dofile("conf.in")
 +
 +print("Calculating all possible Slater determinants for a shell with angular momentum "..l.." and a filling of "..Nelec.." electrons")
 +
 +-- Create the basis for quanty, define spin-orbitals and relate them
 +-- to atomic shells. There are 4*l+2 spin-orbitals in a shell with
 +-- angular momentum l. We label the spin-orbitals in the order from
 +-- ml=-l to ml=l alternating down and up.
 +
 +Nf=4*l+2
 +IndexDn={}
 +IndexUp={}
 +j=1
 +for i=0,Nf-1,2 do
 +  IndexDn[j]=i
 +  IndexUp[j]=i+1
 +  j=j+1
 +end
 +
 +-- number of bosons
 +Nb=0
 +
 +print("")
 +print("======================== O U T P U T ========================")
 +print("")
 +
 +-- Number of many particle states.
 +Nstates=math.Binomial(Nf,Nelec)
 +
 +print("A shell with l= "..l.." and a filling of "..Nelec.." electrons has Binomial("..Nf.." "..Nelec..") = "..Nstates.." possible single Slater determinants")
 +
 +if(Nstates>3500) then
 +  print("If you want to calculate more than 3500 states you have to remove this if then else from the script")
 +  os.exit()
 +end
 +if(Nstates>64000) then
 +  print("If you want to calculate more than 64000 states you have to remove this if then else from the script")
 +  print("Furthermore it is recommended that you set the number of bits in the hash key table to a larger number.")
 +  print("Change the line psi[i] = NewWavefunction(Nf, Nb, {{basis[i-1],1}})")
 +  print("to psi[i] = NewWavefunction(Nf, Nb, {{basis[i-1],1}}, {{\"NBitsKey\",20}}).")
 +  os.exit()
 +end
 +
 +TimeStart("Create Determinants")
 +
 +-- Create the first determinant (spin-orbital 1 
 +-- to Nelec filled, Nelec+1 to Nf empty).
 +-- Determinants are stored as strings with 1 
 +-- meaning occupied and 0 meaning empty
 +str=""
 +for i=1,Nelec do
 +  str=str..1
 +end
 +for i=1,Nf-Nelec do
 +  str=str..0
 +end
 +
 +-- Search possible many-body states
 +newstr={}; newstr[0]=str
 +basis={};
 +for i=1,Nstates do
 +  basis[i]=str
 +end
 +kprev={} ; kprev[1]=0 ; kprev[2]=0 ; k=0 ; e=1
 +for i=1,Nf do
 +  for l=kprev[i],kprev[i+1] do
 +    for j=i+1,Nf do
 +      k=k+1
 +      newstr[k]=swap_char(i,j,newstr[l])
 +      info=0
 +      for g=1,Nstates do
 +        if (newstr[k] == basis[g]) then
 +          info=1
 +        end
 +      end
 +      if (info == 0) then
 +        e=e+1
 +        basis[e]=newstr[k]
 +      end
 +    end
 +    if (e == Nstates) then
 +      break
 +    end
 +  end 
 +  kprev[i+2]=k
 +end
 +-- END of search
 +
 +TimeEnd("Create Determinants")
 +
 +-- print the possible many-body single Slater 
 +-- determinant states
 +print("--> List of possible single Slater determinants which span the basis for all many body states:")
 +for i=1,(Nstates) do
 + print(string.format("%5i |%s>",i,basis[i]))
 +end
 +
 +TimeStart("Create operators in second quantization")
 +
 +OppSz   = NewOperator("Sz"   ,Nf,IndexUp,IndexDn)
 +OppLz   = NewOperator("Lz"   ,Nf,IndexUp,IndexDn)
 +OppSsqr = NewOperator("Ssqr" ,Nf,IndexUp,IndexDn)
 +OppLsqr = NewOperator("Lsqr" ,Nf,IndexUp,IndexDn)
 +OppJz   = NewOperator("Jz"   ,Nf,IndexUp,IndexDn)
 +OppJsqr = NewOperator("Jsqr" ,Nf,IndexUp,IndexDn)
 +Oppldots= NewOperator("ldots",Nf,IndexUp,IndexDn)
 +
 +SlaterInts={}
 +OppFk={}
 +Nk=l+1
 +for j=1,Nk do
 +  for i = 1,Nk do
 +   SlaterInts[i]=0
 +  end
 +  SlaterInts[j]=1
 +  OppFk[j] = NewOperator("U", Nf, IndexUp, IndexDn, SlaterInts)
 +end
 +
 +TimeEnd("Create operators in second quantization")
 +
 +TimeStart("Create operators in matrix form")
 +
 +--- create the set of wavefunctions, using the obtained states
 +psi={}
 +for i=1,Nstates do
 + psi[i] = NewWavefunction(Nf,Nb,{{basis[i],1}})
 +end 
 +
 +print("\n=============================================================")
 +print("The Sz operator on a basis Single slater determinants for a shell with l = "..l.." and a filling of "..Nelec.." electrons is:")
 +for i = 1, Nstates do
 +  io.write(string.format("%5i |%s>  ",i,basis[i]))
 +  for j = 1, Nstates do
 +    io.write(string.format("%5.2f ",psi[i] * OppSz * psi[j]))
 +  end
 +  io.write("\n")
 +end
 +
 +print("\n=============================================================")
 +print("The Lz operator on a basis Single slater determinants for a shell with l = "..l.." and a filling of "..Nelec.." electrons is:");
 +for i = 1, Nstates do
 +  io.write(string.format("%5i |%s>  ",i,basis[i]))
 +  for j = 1, Nstates do
 +    io.write(string.format("%5.2f ",psi[i] * OppLz * psi[j]));
 +  end
 +  io.write("\n");
 +end
 +
 +print("\n=============================================================")
 +print("The S^2 operator on a basis Single slater determinants for a shell with l = "..l.." and a filling of "..Nelec.." electrons is:")
 +for i = 1, Nstates do
 +  io.write(string.format("%5i |%s>  ",i,basis[i]))
 +  for j = 1, Nstates do
 +    io.write(string.format("%5.2f ",psi[i] * OppSsqr * psi[j]))
 +  end
 +  io.write("\n")
 +end
 +
 +print("\n=============================================================")
 +print("The L^2 operator on a basis Single slater determinants for a shell with l = "..l.." and a filling of "..Nelec.." electrons is:")
 +for i = 1, Nstates do
 +  io.write(string.format("%5i |%s>  ",i,basis[i]))
 +  for j = 1, Nstates do
 +    io.write(string.format("%5.2f ",psi[i] * OppLsqr * psi[j]))
 +  end
 +  io.write("\n")
 +end
 +
 +print("\n=============================================================")
 +print("The Coulomb operator for a shell with l = "..l.." is given by "..Nk.." different operators depending on the angular momentum of the expansion of 1/|r1-r2| in spherical Harmonics")
 +for k=1,Nk do
 +  print("=============================================================")
 +  print("F["..(2*(k-1)).."]")
 +  for i = 1, Nstates do
 +    io.write(string.format("%5i |%s>  ",i,basis[i]))
 +    for j = 1, Nstates do
 +      io.write(string.format("%5.2f ",psi[i] * OppFk[k] * psi[j]))
 +    end
 +    io.write("\n")
 +  end
 +end
 +
 +TimeEnd("Create operators in matrix form")
 +
 +print("\n=============================================================")
 +TimePrint()
 +</code>
 +###
 +
 +===== Table of contents =====
 +{{indexmenu>.#1|msort}}

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