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documentation:language_reference:objects:responsefunction:start [2024/09/16 11:44] – created Maurits W. Haverkortdocumentation:language_reference:objects:responsefunction:start [2024/10/07 10:00] (current) – Sina Shokri
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 ====== Response function ====== ====== Response function ======
  
-### +One simple way to represent a spectrum, for example one that is calculated using the function //[[documentation:language_reference:functions:createspectra|CreateSpectra()]]//, is a sum over the poles multiplied by the residues: 
-Operators define quantum mechanical operators implemented in second quantization. The most general operator one can implement is: +\begin{equation} 
-$$ \begin{align} +    I(\omega) = \sum_{k} \frac{R_k}{\omega - \omega_k + i \gamma/2} \qquad \qquad \qquad    (1) 
-O&=\alpha^{(0,0)} \\ +\end{equation} 
- &+\sum_{\tau} \alpha^{(0,1)}_{\tau} a^{\phantom{\dagger}}_{\tau} + \alpha^{(1,0)}_{\tau} a^{\dagger}_{\tau} \\ +A more compact way to store this spectrum, as a function of $\omega$ (and $\gamma$), is by just using two arrays for the values of $ \{R_k\} $ (residues) and $ \{\omega_k\} $ (poles). This is precicely the purpose of the object //Response Function//. In other words, response functions in Quanty are functions that, given a complex number as input ($\omega + i \gamma/2$) return a complex number (single valued functions). Additionally, the output could be a matrix (matrix functions) when the response function is defined using array of matrices, instead of array of numbers. Response functions fullfil the Kramers Kronig relations and "causality".
- &+\sum_{\tau_1,\tau_2} \alpha^{(0,2)}_{\tau_1,\tau_2} a^{\phantom{\dagger}}_{\tau_1}a^{\phantom{\dagger}}_{\tau_2} + \alpha^{(1,1)}_{\tau_1,\tau_2} a^{\dagger}_{\tau_1}a^{\phantom{\dagger}}_{\tau_2} + \alpha^{(2,0)}_{\tau_1,\tau_2} a^{\dagger}_{\tau_1}a^{\dagger}_{\tau_2} \\ +
-& + \sum_{\tau_1,\tau_2,\tau_3} \alpha^{(0,3)}_{\tau_1,\tau_2,\tau_3} a^{\phantom{\dagger}}_{\tau_1}a^{\phantom{\dagger}}_{\tau_2}a^{\phantom{\dagger}}_{\tau_3} + \alpha^{(1,2)}_{\tau_1,\tau_2,\tau_3} a^{\dagger}_{\tau_1}a^{\phantom{\dagger}}_{\tau_2}a^{\phantom{\dagger}}_{\tau_3}  + \alpha^{(2,1)}_{\tau_1,\tau_2,\tau_3} a^{\dagger}_{\tau_1}a^{\dagger}_{\tau_2}a^{\phantom{\dagger}}_{\tau_3}  + \alpha^{(3,0)}_{\tau_1,\tau_2,\tau_3} a^{\dagger}_{\tau_1}a^{\dagger}_{\tau_2}a^{\dagger}_{\tau_3} \\ +
-&+ \dots +
-\end{align} $$ +
-Note that as $a^{\dagger}_{\tau_1}a^{\phantom{\dagger}}_{\tau_2} = \delta_{\tau_1,\tau_2}-a^{\phantom{\dagger}}_{\tau_2}a^{\dagger}_{\tau_1}$ for Fermions and $a^{\dagger}_{\tau_1}a^{\phantom{\dagger}}_{\tau_2} = a^{\phantom{\dagger}}_{\tau_2}a^{\dagger}_{\tau_1}-\delta_{\tau_1,\tau_2}$ for Bosons the equation above allows for any operator to be defined. Operator defines an object that can store the most general operator. Several methods that act on this object are implemented. Furthermore one can access several properties of Operators.  +
-###+
  
-### +The response functions can also be defined by matrices such that the function is given by  
-Operators can act on Fermions and Bosons. An operator //O// has //O.NF// Fermions and //O.NB// Bosons. The Fermions are indexed from //0// to //O.NF-1//, the Bosons are indexed from //O.NF// to //O.NF+O.NB-1//. +$$ A_0 + B_0^* \frac{ 1 }{\omega - H + i \gamma / 2} \Bigg|_{[0,0]} B_0^{T} $$  
-###+where $A_0$, $B_0$ and $H$ are matrices. Any unitary transformation that leaves the [0,0] element of $H$ unchanged results in the same spectrum. Hence $H$ can have different forms, whereby only a few elements/blocks are non-zero. In Quanty, there are 4 different forms (types) for the matrix $H$: 
 +  * list of poles (//ListOfPoles//) 
 +  * tri-diagonal (//Tri//) 
 +  * Anderson (//And//) 
 +  * natural impurity orbital (//NaturalImpurityOrbital//)
  
 +These types are related to each other by unitary transformations and the Quanty function //[[documentation/language_reference/objects/responsefunction/functions/changetype|ChangeType()]]// can be used to transform between these types. List of poles is exactly the representation in Eq. (1).
  
-===== Table of contents ===== +In order to define a response function, one needs to set the meta table to "ResponseFunctionMeta" in Lua. Note that there is no type-check, (i.e. setting the meta table is optional) for the functions acting on these tables. 
-{{indexmenu>.#2}}+ 
 +==== Definitions ==== 
 +<code Quanty Example.Quanty> 
 +-- Defining single-valued response functions 
 + 
 +a = {0, -1,-0.5, 0,   0.5,  1,  1.5} 
 +b = {  0.2, 0.1, 0.1, 0.1, 0.2, 0.3} 
 +GA_L = {a,b,mu=0,type="ListOfPoles", name="A"} 
 +setmetatable(GA_L, ResponseFunctionMeta) 
 + 
 +a = {0, 1,   1,   1,   1,   1,  1} 
 +b = {   1, 0.5, 0.5, 0.5, 0.5, 0.5} 
 +GB_T = {a,b,mu=0,type="Tri", name="B"} 
 +setmetatable(GB_T, ResponseFunctionMeta) 
 + 
 +a = {0, 1, 1.5,   2, 2.5,   3, 3.5} 
 +b = {   1, 0.5, 0.5, 0.5, 0.5, 0.5} 
 +GC_A = {a,b,mu=0,type="And", name="C"} 
 +setmetatable(GC_A, ResponseFunctionMeta) 
 + 
 +acon = {0,         1,   1,   1,   1,   1,  1} 
 +bcon = {   sqrt(1/2), 0.5, 0.5, 0.5, 0.5, 0.5} 
 +Gcon = {acon,bcon,mu=0,type="Tri"} 
 +setmetatable(Gcon, ResponseFunctionMeta) 
 +aval = {0,        -1,  -1,  -1,  -1,  -1, -1} 
 +bval = {   sqrt(1/2), 0.5, 0.5, 0.5, 0.5, 0.5} 
 +Gval = {aval,bval,mu=0,type="Tri"} 
 +setmetatable(Gval, ResponseFunctionMeta) 
 +a0=0 
 +b0=1 
 +GD_N = {{a0,b0},val=Gval,con=Gcon,mu=0,type="Nat", name="D"} 
 +setmetatable(GD_N, ResponseFunctionMeta) 
 + 
 +-- For a more efficient storage, one can convert from Table to Userdata 
 +GA_l = ResponseFunction.ToUserdata(GA_L) 
 +GB_t = ResponseFunction.ToUserdata(GB_T) 
 +GC_a = ResponseFunction.ToUserdata(GC_A) 
 +GD_n = ResponseFunction.ToUserdata(GD_N) 
 + 
 +print("GA_L:") 
 +print(GA_L) 
 +print("GA_l:") 
 +print(GA_l) 
 + 
 +print("GB_T:") 
 +print(GB_T) 
 +print("GB_t:") 
 +print(GB_t) 
 + 
 +print("GC_A:") 
 +print(GC_A) 
 +print("GC_a:") 
 +print(GC_a) 
 + 
 +print("GD_N:") 
 +print(GD_N) 
 +print("GD_n:") 
 +print(GD_n) 
 + 
 +-- response functions in Quanty are functions that, given  
 +-- a complex number as input w + I gamma/2 return a complex 
 +-- number (single valued functions) or a matrix (matrix functions) 
 +omega = 0.764 
 +gamma = 0.1 
 +print("omega = ", omega) 
 +print("gamma = ", gamma) 
 + 
 +print("") 
 +print("GA_L(omega, gamma) = ", GA_L(omega, gamma)) 
 +print("GA_l(omega, gamma) = ", GA_l(omega, gamma)) 
 + 
 +print("GB_T(omega, gamma) = ", GB_T(omega, gamma)) 
 +print("GB_t(omega, gamma) = ", GB_t(omega, gamma)) 
 + 
 +print("GC_A(omega, gamma) = ", GC_A(omega, gamma)) 
 +print("GC_a(omega, gamma) = ", GC_a(omega, gamma)) 
 + 
 +print("GD_N(omega, gamma) = ", GD_N(omega, gamma)) 
 +print("GD_n(omega, gamma) = ", GD_n(omega, gamma)) 
 + 
 + 
 +-- Defining matrix response functions 
 + 
 +A0 = {{0,0,0},{0,0,0},{0,0,0}} 
 +setmetatable(A0, MatrixMeta) 
 +A1 = {{1,2,3},{2,5,6},{3,6,9}} 
 +setmetatable(A1, MatrixMeta) 
 +A2 = {{2,2,3},{2,5,6},{3,6,9}} 
 +setmetatable(A2, MatrixMeta) 
 +A3 = {{3,2,3},{2,5,6},{3,6,9}} 
 +setmetatable(A3, MatrixMeta) 
 +a1 = -1 
 +a2 = 0 
 +a3 = 1 
 +av1 = -0.5 
 +av2 = -1.0 
 +av3 = -1.5 
 +ac1 = 0.5 
 +ac2 = 1.0 
 +ac3 = 1.5 
 +B0s = {{1,0,0},{0,1,0},{0,0,1}} 
 +setmetatable(B0s, MatrixMeta) 
 +t=sqrt(0.5) 
 +B0vs = {{t,0,0},{0,t,0},{0,0,t}} 
 +setmetatable(B0vs, MatrixMeta) 
 +B0cs = {{t,0,0},{0,t,0},{0,0,t}} 
 +setmetatable(B0cs, MatrixMeta) 
 +B0 = B0s * B0s 
 +B0v = B0vs * B0vs 
 +B0c = B0cs * B0cs 
 +B1s = {{1,I,3},{-I,5,6},{3,6,9}} 
 +setmetatable(B1s, MatrixMeta) 
 +B1 = B1s * B1s 
 +B2s = {{2,0,3},{0,5,6},{3,6,9}} 
 +setmetatable(B2s, MatrixMeta) 
 +B2 = B2s * B2s 
 +B3s = {{3,0,3},{0,5,6},{3,6,9}} 
 +setmetatable(B3s, MatrixMeta) 
 +B3 = B3s * B3s 
 + 
 +MA_L = { {A0,a1,a2,a3}, {B1,B2,B3}, mu=0, type="ListOfPoles", name="A"} 
 +setmetatable(MA_L, ResponseFunctionMeta) 
 + 
 +MB_T = { {A0,A1,A2,A3}, {B0,B1,B2}, mu=0, type="Tri", name="B"} 
 +setmetatable(MB_T, ResponseFunctionMeta) 
 + 
 +MC_A = { {A0,A1,A2,A3}, {B0,B1,B2}, mu=0, type="And", name="C"} 
 +setmetatable(MC_A, ResponseFunctionMeta) 
 + 
 +MD_Nv = { {A0,av1,av2,av3}, {B1,B2,B3}, mu=0, type="ListOfPoles", name="Dv"} 
 +setmetatable(MD_Nv, ResponseFunctionMeta) 
 + 
 +MD_Nc = { {A0,ac1,ac2,ac3}, {B1,B2,B3}, mu=0, type="ListOfPoles", name="Dc"} 
 +setmetatable(MD_Nc, ResponseFunctionMeta) 
 + 
 +MD_N = {{A0,B0},val=MD_Nv,con=MD_Nc,mu=0,type="Nat", name="D"} 
 +setmetatable(MD_N, ResponseFunctionMeta) 
 + 
 +-- For a more efficient storage, one can convert from Table to Userdata 
 +MA_l = ResponseFunction.ToUserdata(MA_L) 
 +MB_t = ResponseFunction.ToUserdata(MB_T) 
 +MC_a = ResponseFunction.ToUserdata(MC_A) 
 +MD_n = ResponseFunction.ToUserdata(MD_N) 
 + 
 + 
 +print("MA_L:") 
 +print(MA_L) 
 +print("MA_l:") 
 +print(MA_l) 
 + 
 +print("MB_T:") 
 +print(MB_T) 
 +print("MB_t:") 
 +print(MB_t) 
 + 
 +print("MC_A:") 
 +print(MC_A) 
 +print("MC_a:") 
 +print(MC_a) 
 + 
 +print("MD_N:") 
 +print(MD_N) 
 +print("MD_n:") 
 +print(MD_n) 
 + 
 + 
 +-- response functions in Quanty are functions that, given  
 +-- a complex number as input w + I gamma/2 return a complex 
 +-- number (single valued functions) or a matrix (matrix functions) 
 +omega = 0.764 
 +gamma = 0.1 
 +print("omega = ", omega) 
 +print("gamma = ", gamma) 
 + 
 +print("") 
 +print("MA_L(omega, gamma) = ") 
 +print(MA_L(omega, gamma)) 
 +print("MA_l(omega, gamma) = ") 
 +print(MA_l(omega, gamma)) 
 + 
 +print("") 
 +print("MB_T(omega, gamma) = ") 
 +print(MB_T(omega, gamma)) 
 +print("MB_t(omega, gamma) = ") 
 +print(MB_t(omega, gamma)) 
 + 
 +print("") 
 +print("MC_A(omega, gamma) = ") 
 +print(MC_A(omega, gamma)) 
 +print("MC_a(omega, gamma) = ") 
 +print(MC_a(omega, gamma)) 
 + 
 +print("") 
 +print("MD_N(omega, gamma) = ") 
 +print(MD_N(omega, gamma)) 
 +print("MD_n(omega, gamma) = ") 
 +print(MD_n(omega, gamma)) 
 +</code>
  
-==== Result ==== 
 <file Quanty_Output> <file Quanty_Output>
-text produced as output+GA_L: 
 +{ { 0 , -1 , -0.5 , 0 , 0.5 , 1 , 1.5 } ,  
 +  { 0.2 , 0.1 , 0.1 , 0.1 , 0.2 , 0.3 } , 
 +  type = ListOfPoles , 
 +  mu = 0 , 
 +  name = A } 
 +GA_l: 
 +ResponseFunction in userdata format use ToTable() in order to get a table form 
 +GB_T: 
 +{ { 0 , 1 , 1 , 1 , 1 , 1 , 1 } ,  
 +  { 1 , 0.5 , 0.5 , 0.5 , 0.5 , 0.5 } , 
 +  type = Tri , 
 +  mu = 0 , 
 +  name = B } 
 +GB_t: 
 +ResponseFunction in userdata format use ToTable() in order to get a table form 
 +GC_A: 
 +{ { 0 , 1 , 1.5 , 2 , 2.5 , 3 , 3.5 } ,  
 +  { 1 , 0.5 , 0.5 , 0.5 , 0.5 , 0.5 } , 
 +  type = And , 
 +  mu = 0 , 
 +  name = C } 
 +GC_a: 
 +ResponseFunction in userdata format use ToTable() in order to get a table form 
 +GD_N: 
 +{ { 0 , 1 } , 
 +  type = Nat , 
 +  val = { { 0 , -1 , -1 , -1 , -1 , -1 , -1 } ,  
 +  { 0.70710678118655 , 0.5 , 0.5 , 0.5 , 0.5 , 0.5 } , 
 +  mu = 0 , 
 +  type = Tri } , 
 +  con = { { 0 , 1 , 1 , 1 , 1 , 1 , 1 } ,  
 +  { 0.70710678118655 , 0.5 , 0.5 , 0.5 , 0.5 , 0.5 } , 
 +  mu = 0 , 
 +  type = Tri } , 
 +  name = D , 
 +  mu = 0 } 
 +GD_n: 
 +ResponseFunction in userdata format use ToTable() in order to get a table form 
 +omega = 0.764 
 +gamma = 0.1 
 + 
 +GA_L(omega, gamma) = (-0.52850742098896 - 0.28351791776891 I) 
 +GA_l(omega, gamma) = (-0.52850742098896 - 0.28351791776891 I) 
 +GB_T(omega, gamma) = (-1.6762624946074 - 5.2043002596032 I) 
 +GB_t(omega, gamma) = (-1.6762624946074 - 5.2043002596032 I) 
 +GC_A(omega, gamma) = (1.5075603166492 - 0.20713674761056 I) 
 +GC_a(omega, gamma) = (1.5075603166492 - 0.20713674761056 I) 
 +GD_N(omega, gamma) = (-0.52770560643508 - 2.61282805234 I) 
 +GD_n(omega, gamma) = (-0.52770560643508 - 2.61282805234 I) 
 +MA_L: 
 +{ { { { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } } , -1 , 0 , 1 } ,  
 +  { { { 11 , (18 + 6 I) , (30 + 6 I) } ,  
 +  { (18 - 6 I) , 62 , (84 - 3 I) } ,  
 +  { (30 - 6 I) , (84 + 3 I) , 126 } } ,  
 +  { { 13 , 18 , 33 } ,  
 +  { 18 , 61 , 84 } ,  
 +  { 33 , 84 , 126 } } ,  
 +  { { 18 , 18 , 36 } ,  
 +  { 18 , 61 , 84 } ,  
 +  { 36 , 84 , 126 } } } , 
 +  type = ListOfPoles , 
 +  mu = 0 , 
 +  name = A } 
 +MA_l: 
 +ResponseFunction in userdata format use ToTable() in order to get a table form 
 +MB_T: 
 +{ { { { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } } ,  
 +  { { 1 , 2 , 3 } ,  
 +  { 2 , 5 , 6 } ,  
 +  { 3 , 6 , 9 } } ,  
 +  { { 2 , 2 , 3 } ,  
 +  { 2 , 5 , 6 } ,  
 +  { 3 , 6 , 9 } } ,  
 +  { { 3 , 2 , 3 } ,  
 +  { 2 , 5 , 6 } ,  
 +  { 3 , 6 , 9 } } } ,  
 +  { { { 1 , 0 , 0 } ,  
 +  { 0 , 1 , 0 } ,  
 +  { 0 , 0 , 1 } } ,  
 +  { { 11 , (18 + 6 I) , (30 + 6 I) } ,  
 +  { (18 - 6 I) , 62 , (84 - 3 I) } ,  
 +  { (30 - 6 I) , (84 + 3 I) , 126 } } ,  
 +  { { 13 , 18 , 33 } ,  
 +  { 18 , 61 , 84 } ,  
 +  { 33 , 84 , 126 } } } , 
 +  type = Tri , 
 +  mu = 0 , 
 +  name = B } 
 +MB_t: 
 +ResponseFunction in userdata format use ToTable() in order to get a table form 
 +MC_A: 
 +{ { { { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } } ,  
 +  { { 1 , 2 , 3 } ,  
 +  { 2 , 5 , 6 } ,  
 +  { 3 , 6 , 9 } } ,  
 +  { { 2 , 2 , 3 } ,  
 +  { 2 , 5 , 6 } ,  
 +  { 3 , 6 , 9 } } ,  
 +  { { 3 , 2 , 3 } ,  
 +  { 2 , 5 , 6 } ,  
 +  { 3 , 6 , 9 } } } ,  
 +  { { { 1 , 0 , 0 } ,  
 +  { 0 , 1 , 0 } ,  
 +  { 0 , 0 , 1 } } ,  
 +  { { 11 , (18 + 6 I) , (30 + 6 I) } ,  
 +  { (18 - 6 I) , 62 , (84 - 3 I) } ,  
 +  { (30 - 6 I) , (84 + 3 I) , 126 } } ,  
 +  { { 13 , 18 , 33 } ,  
 +  { 18 , 61 , 84 } ,  
 +  { 33 , 84 , 126 } } } , 
 +  type = And , 
 +  mu = 0 , 
 +  name = C } 
 +MC_a: 
 +ResponseFunction in userdata format use ToTable() in order to get a table form 
 +MD_N: 
 +{ { { { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } } ,  
 +  { { 1 , 0 , 0 } ,  
 +  { 0 , 1 , 0 } ,  
 +  { 0 , 0 , 1 } } } , 
 +  type = Nat , 
 +  val = { { { { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } } , -0.5 , -1 , -1.5 } ,  
 +  { { { 11 , (18 + 6 I) , (30 + 6 I) } ,  
 +  { (18 - 6 I) , 62 , (84 - 3 I) } ,  
 +  { (30 - 6 I) , (84 + 3 I) , 126 } } ,  
 +  { { 13 , 18 , 33 } ,  
 +  { 18 , 61 , 84 } ,  
 +  { 33 , 84 , 126 } } ,  
 +  { { 18 , 18 , 36 } ,  
 +  { 18 , 61 , 84 } ,  
 +  { 36 , 84 , 126 } } } , 
 +  type = ListOfPoles , 
 +  mu = 0 , 
 +  name = Dv } , 
 +  con = { { { { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } ,  
 +  { 0 , 0 , 0 } } , 0.5 , 1 , 1.5 } ,  
 +  { { { 11 , (18 + 6 I) , (30 + 6 I) } ,  
 +  { (18 - 6 I) , 62 , (84 - 3 I) } ,  
 +  { (30 - 6 I) , (84 + 3 I) , 126 } } ,  
 +  { { 13 , 18 , 33 } ,  
 +  { 18 , 61 , 84 } ,  
 +  { 33 , 84 , 126 } } ,  
 +  { { 18 , 18 , 36 } ,  
 +  { 18 , 61 , 84 } ,  
 +  { 36 , 84 , 126 } } } , 
 +  type = ListOfPoles , 
 +  mu = 0 , 
 +  name = Dc } , 
 +  name = D , 
 +  mu = 0 } 
 +MD_n: 
 +ResponseFunction in userdata format use ToTable() in order to get a table form 
 +omega = 0.764 
 +gamma = 0.1 
 + 
 +MA_L(omega, gamma) =  
 +{ { (-49.82074737235 - 16.750435144161 I) , (-39.242754250336 - 13.890672203015 I) , (-85.890426598686 - 30.827754261851 I) } ,  
 +  { (-39.435420350993 - 20.687932234176 I) , (-132.74935751632 - 58.607579693628 I) , (-183.63057392827 - 82.382725361235 I) } ,  
 +  { (-86.083092699343 - 37.625014293012 I) , (-183.53424087794 - 78.984095345655 I) , (-275.37361110465 - 121.02511553017 I) } } 
 +MA_l(omega, gamma) =  
 +{ { (-49.82074737235 - 16.750435144161 I) , (-39.242754250336 - 13.890672203015 I) , (-85.890426598686 - 30.827754261851 I) } ,  
 +  { (-39.435420350993 - 20.687932234176 I) , (-132.74935751632 - 58.607579693628 I) , (-183.63057392827 - 82.382725361235 I) } ,  
 +  { (-86.083092699343 - 37.625014293012 I) , (-183.53424087794 - 78.984095345655 I) , (-275.37361110465 - 121.02511553017 I) } } 
 + 
 +MB_T(omega, gamma) =  
 +{ { (0.99841211757801 - 0.4318886700871 I) , (-0.92217346376056 - 0.59364349550167 I) , (0.33618642304746 + 0.56252370645231 I) } ,  
 +  { (-0.51999398150154 + 0.87279523667066 I) , (0.69431808205569 - 0.1748977600005 I) , (-0.34355189835894 - 0.03809026037919 I) } ,  
 +  { (0.13982839704936 - 0.53101630282404 I) , (-0.25718128095213 + 0.17938506335344 I) , (0.080137084707226 - 0.049212538037938 I) } } 
 +MB_t(omega, gamma) =  
 +{ { (0.99841211757801 - 0.4318886700871 I) , (-0.92217346376056 - 0.59364349550167 I) , (0.33618642304746 + 0.56252370645231 I) } ,  
 +  { (-0.51999398150154 + 0.87279523667066 I) , (0.69431808205569 - 0.1748977600005 I) , (-0.34355189835894 - 0.03809026037919 I) } ,  
 +  { (0.13982839704936 - 0.53101630282404 I) , (-0.25718128095213 + 0.17938506335344 I) , (0.080137084707226 - 0.049212538037938 I) } } 
 + 
 +MC_A(omega, gamma) =  
 +{ { (0.0033404740201663 - 0.049014208032374 I) , (-0.0092579857609902 - 0.018133779674219 I) , (0.0041464743284782 + 0.021581712320061 I) } ,  
 +  { (0.001131475130102 - 0.019625764066449 I) , (-0.006620607647268 - 0.0094591143626733 I) , (0.0044705980344151 + 0.010254424901895 I) } ,  
 +  { (0.0018776712535675 + 0.024036826455971 I) , (0.0083397043445284 + 0.01003150710377 I) , (-0.0056913744700838 - 0.011590149052798 I) } } 
 +MC_a(omega, gamma) =  
 +{ { (0.0033404740201663 - 0.049014208032374 I) , (-0.0092579857609902 - 0.018133779674219 I) , (0.0041464743284782 + 0.021581712320061 I) } ,  
 +  { (0.001131475130102 - 0.019625764066449 I) , (-0.006620607647268 - 0.0094591143626733 I) , (0.0044705980344151 + 0.010254424901895 I) } ,  
 +  { (0.0018776712535675 + 0.024036826455971 I) , (0.0083397043445284 + 0.01003150710377 I) , (-0.0056913744700838 - 0.011590149052798 I) } } 
 + 
 +MD_N(omega, gamma) =  
 +{ { (-12.839446772164 - 21.169048830916 I) , (5.1855777994224 - 3.9320983625761 I) , (-10.184899840804 - 27.575394683943 I) } ,  
 +  { (-3.5000860033155 - 57.291484603566 I) , (7.3025877301526 - 104.46376534286 I) , (1.7613982402315 - 156.19487348125 I) } ,  
 +  { (-18.870563643542 - 80.934780924933 I) , (6.1042301416005 - 129.51518036075 I) , (5.8992212863723 - 214.2825403815 I) } } 
 +MD_n(omega, gamma) =  
 +{ { (-12.839446772164 - 21.169048830916 I) , (5.1855777994224 - 3.9320983625761 I) , (-10.184899840804 - 27.575394683943 I) } ,  
 +  { (-3.5000860033155 - 57.291484603566 I) , (7.3025877301526 - 104.46376534286 I) , (1.7613982402315 - 156.19487348125 I) } ,  
 +  { (-18.870563643542 - 80.934780924933 I) , (6.1042301416005 - 129.51518036075 I) , (5.8992212863723 - 214.2825403815 I) } }
 </file> </file>
  
 ===== Table of contents ===== ===== Table of contents =====
-{{indexmenu>.#1|msort}}+{{indexmenu>.#2}}
  

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