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SeQuant.m
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(* ::Package:: *)
BeginPackage["SeQuant`"];
(* turn on Assert *)
On[Assert];
(* Setting up internal things *)
Off[General::spell];
Off[General::spell1];
$RecursionLimit = 8192;
$IterationLimit = Infinity;
(*SetOptions[$FrontEnd,CommonDefaultFormatTypes->{"Output"->TraditionalForm}];*)
NCM = NonCommutativeMultiply;
(* ::Section::Closed:: *)
(* Global Parameters *)
(*
Special Operators
*)
Unprotect[defaultHamiltonianOpers];
Unprotect[ignoreDisconnectedOpers];
Unprotect[ignoreConnectedOpers];
defaultHamiltonianOpers = {"F","g","\!\(\*OverscriptBox[\(g\), \(_\)]\)"};
(* maps antisymmetric label to nonsymmetric, and vice versa *)
antisymm2nonsymmSQMlabel[label_String]:=If[label=="\!\(\*OverscriptBox[\(g\), \(_\)]\)","g",label];
nonsymm2antisymmSQMlabel[label_String]:=If[label=="g","\!\(\*OverscriptBox[\(g\), \(_\)]\)",label];
ignoreDisconnectedOpers = {};
ignoreConnectedOpers = {};
Protect[ignoreConnectedOpers];
Protect[ignoreDisconnectedOpers];
Protect[defaultHamiltonianOpers];
(*
Global SeQuant Parameters
*)
If[ !ValueQ[SeQuantDebugLevel],
SeQuantDebugLevel = 0
];
SeQuantVacuumChoices["Physical"] = 0;
SeQuantVacuumChoices["SingleConfiguration"] = 1;
SeQuantVacuumChoices["MultiConfiguration"] = 2;
If[ !ValueQ[SeQuantVacuum],
SeQuantVacuum = SeQuantVacuumChoices["SingleConfiguration"]
];
(* Definition of spaces *)
occ = particleSpace[occupied];
virt = particleSpace[virtual];
othervirt = particleSpace[othervirtual];
any = Union[occ,virt];
allvirt = Union[virt,othervirt];
allany = Union[occ,allvirt];
(* Definition of spaces with spin *)
occA = Append[occ,particleSpin[A]];
virtA=Append[virt,particleSpin[A]];
othervirtA=Append[othervirt,particleSpin[A]];
allvirtA=Append[allvirt,particleSpin[A]];
anyA = Append[any,particleSpin[A]];
allanyA = Append[allany,particleSpin[A]];
occB = Append[occ,particleSpin[B]];
virtB=Append[virt,particleSpin[B]];
othervirtB=Append[othervirt,particleSpin[B]];
allvirtB=Append[allvirt,particleSpin[B]];
anyB = Append[any,particleSpin[B]];
allanyB = Append[allany,particleSpin[B]];
(* this converts spin tag into m_s *)
Ms[particleSpin[A]]=1/2;
Ms[particleSpin[B]]=-1/2;
Ms[particleSpin[none]]=0;
stringAppendSpin[str_,s_particleSpin]:=If[s===particleSpin[none],str, "\!\(\*SubscriptBox[\("<>str<>"\), \("<>If[s===particleSpin[A],"\[Alpha]","\[Beta]"]<>"\)]\)"];
spacesEqualIgnoreParticleType[a_particleSpace,b_particleSpace] :=
(Select[a,Head[#]=!=particleType&]==Select[b,Head[#]=!=particleType&]);
spacesOverlapIgnoreParticleType[a_particleSpace,b_particleSpace] :=
(Select[Intersection[a,b],Head[#]=!=particleType&]=!=particleSpace[]);
spacesOverlapIgnoreSpinAndParticleType[a_particleSpace,b_particleSpace] :=
(Select[Intersection[a,b],Head[#]=!=particleType&&Head[#]=!=particleSpin&]=!=particleSpace[]);
spacesOverlap[a_particleSpace,b_particleSpace] :=
(spacesOverlapIgnoreSpinAndParticleType[a,b] && Select[a,Head[#]===particleType&]===Select[b,Head[#]===particleType&]&& Select[a,Head[#]===particleSpin&]===Select[b,Head[#]===particleSpin&]);
(* return +1 if above, -1 if below, 0 if neither or both *)
spaceWRTFermiLevel[a_particleSpace] :=
If[ spacesOverlapIgnoreSpinAndParticleType[a,allvirt],
If[ spacesOverlapIgnoreSpinAndParticleType[a,occ],
0,
+1
],
If[ spacesOverlapIgnoreSpinAndParticleType[a,occ],
-1,
0
]
];
(* Global index for most recently generated index
Generated indices are named iXX, where i is generated by DefaultSpaceSymbol *)
GlobalIndexCounter = 0;
DefaultSpaceSymbol[occ] = "i";
DefaultSpaceSymbol[virt] = "a";
DefaultSpaceSymbol[allvirt] = "\[Alpha]";
DefaultSpaceSymbol[othervirt] = "\[Alpha]'";
DefaultSpaceSymbol[any] = "p";
DefaultSpaceSymbol[allany] = "\[Kappa]";
DefaultSpaceSymbol[occA]="i";
DefaultSpaceSymbol[virtA]="a";
DefaultSpaceSymbol[allvirtA]="\[Gamma]";
DefaultSpaceSymbol[othervirtA]="\[Gamma]'";
DefaultSpaceSymbol[anyA]="p";
DefaultSpaceSymbol[allanyA]="\[Kappa]";
DefaultSpaceSymbol[occB]="i";
DefaultSpaceSymbol[virtB]="a";
DefaultSpaceSymbol[allvirtB]="\[Gamma]";
DefaultSpaceSymbol[othervirtB]="\[Gamma]'";
DefaultSpaceSymbol[anyB]="p";
DefaultSpaceSymbol[allanyB]="\[Kappa]";
(* ::Section::Closed:: *)
(* particleIndex Class *)
(* create particleIndex with space and symbol i *)
createParticleIndex[i_String,space_particleSpace] :=
Module[ {},
particleIndex[i,space]
];
(* create particleIndex with DefaultSpaceSymbol *)
createParticleIndex[space_particleSpace] :=
createParticleIndex[ToString[Subscript[DefaultSpaceSymbol[space],
GlobalIndexCounter++],TraditionalForm],space];
(* takes A and replaces name and space with i and s, respectively *)
createParticleIndex[A_particleIndex,i_String,s_particleSpace] :=
Module[ {},
If[ Length[A]==3,
particleIndex[i,s,A[[3]]],
particleIndex[i,s]
]
];
(* if a has particleIndex b ignoring indexType *)
indexQ[a_,b_particleIndex] :=
MemberQ[a,c_particleIndex/;(c[[1]]==b[[1]])&&(c[[2]]==b[[2]]),Infinity];
(* if particleIndex a has indexType[cre]*)
indexCreQ[a_particleIndex] :=
MemberQ[a,indexType[cre]];
(* if particleIndex a has indexType[ann]*)
indexAnnQ[a_particleIndex] :=
MemberQ[a,indexType[ann]];
(* if particleIndex a has indexType[bra]*)
indexBraQ[a_particleIndex] :=
MemberQ[a,indexType[bra]];
(* if particleIndex a has indexType[ket]*)
indexKetQ[a_particleIndex] :=
MemberQ[a,indexType[ket]];
(* return indexType of particleIndex with indexType *)
indexType[a_particleIndex] :=
Cases[a,_indexType][[1,1]]
(* return particleSpace of particleIndex *)
indexSpace[a_particleIndex] :=
Cases[a,_particleSpace][[1]];
(* return spin-free particleSpace of particleIndex *)
indexSpaceSpinFree[a_particleIndex] :=
DeleteCases[indexSpace[a],_particleSpin];
(* return the particleType of a particleIndex *)
indexParticle[a_particleIndex] :=
Module[ {typeList},
typeList = Cases[a[[2]],_particleType];
If[ typeList=={},
Return[particleType[default]],
Return[typeList[[1]]]
]
];
(* return the particleSpin of a particleIndex *)
indexSpin[a_particleIndex] :=
Module[ {typeList},
typeList = Cases[a[[2]],_particleSpin];
If[ typeList=={},
Return[particleSpin[none]],
Return[typeList[[1]]]
]
];
(* return the symbol of particleIndex *)
indexSymbol[a_particleIndex] :=
a[[1]];
(* remove indexType *)
indexLight[a_particleIndex] :=
createParticleIndex[indexSymbol[a],indexSpace[a]];
(* if particleIndex equiv ignoring indexType *)
indexEquiv[a_particleIndex,b_particleIndex] :=
(a[[1]]===b[[1]])&&(a[[2]]===b[[2]]);
Unprotect[Equal];
Equal[a_particleIndex,b_particleIndex] :=
(a[[1]]===b[[1]])&&(a[[2]]===b[[2]]);
Protect[Equal];
(* canonical ordering of particleIndex objects is defined as follows: order by spin, then the rest of particleSpace, then alphabetically by the symbol.
particleSpace objects ordered by *)
sqOrder[A_particleIndex,B_particleIndex] :=
If[ Ms[indexSpin[A]]!=Ms[indexSpin[B]],
If[Ms[indexSpin[A]]>Ms[indexSpin[B]],1,-1],
If[ Order[indexSpaceSpinFree[A],indexSpaceSpinFree[B]]==0,
Order[A[[1]],B[[1]]],
Order[indexSpaceSpinFree[A],indexSpaceSpinFree[B]]
]];
sqOrder[a_,b_]:=Order[a,b];
(* this functions throws out all multiple occurences of the same index from inds *)
uniqueIndexList[inds_List] :=
Module[ {tmpinds},
(* Old code *)
(*tmpinds={inds[[1]]};
Do[
ind=inds[[i]];
If[!indexQ[tmpinds,ind],
tmpinds=Append[tmpinds,ind]
],
{i,Length[inds]}
];*)
(* Union does it now *)
tmpinds = Union[inds];
Return[tmpinds];
];
(* "tag " and "untag " all indices in expr which also appear in intInds. uses indexQ *)
tagIndex[index_particleIndex] :=
Append[index,mysecrettag[]];
tagIndex[expr_] :=
expr;
untagIndex[index_particleIndex] :=
DeleteCases[index,_mysecrettag];
untagIndex[expr_] :=
expr;
tagIndices[expr_,inds_List] :=
Module[ {result,ninds,pos,ind},
ninds = Length[inds];
result = expr;
Do[
ind = inds[[i]];
pos = Position[result,c_particleIndex/;indexEquiv[c,ind]];
result = MapAt[tagIndex,result,pos],{i,1,ninds}];
Return[result];
];
untagIndices[expr_,inds_List] :=
Module[ {result,ninds,pos,ind},
ninds = Length[inds];
result = expr;
Do[
ind = inds[[i]];
pos = Position[result,c_particleIndex/;indexEquiv[c,ind]];
result = MapAt[untagIndex,result,pos],{i,1,ninds}];
Return[result];
];
(* ::Section::Closed:: *)
(* SQS Class *)
(*
SQS is a string of creation/annihilation operators
only normal ordered strings can be utilized at the moment
*)
inorder = normalOrder[True];
noorder = normalOrder[False];
createSQS[creInds_List,annInds_List,norm_normalOrder:inorder] :=
Module[ {},
If[ SeQuantVacuum === SeQuantVacuumChoices["MultiConfiguration"],
Return[mSQS[norm, flattenSQS[SQS[creInds,annInds,norm]]]]
];
Return[flattenSQS[SQS[creInds,annInds,norm]]]
];
createSQS[creInds_List, annInds_List ] :=
Return[flattenSQS[SQS[creInds,annInds]]]
SQS::wrongdepth = "Argument has wrong depth";
flattenSQS[a_SQS] :=
Module[ {nc,na,result,creInds,annInds,x},
Off[Append::normal];
If[ Depth[a]<5,
Return[a]
];
Clear[result];
Clear[creInds];
Clear[annInds];
Clear[x];
na = Length[a[[2]]];
nc = Length[a[[1]]];
creInds = Cases[a[[1]],x_particleIndex->Append[x,indexType[cre]]];
annInds = Cases[a[[2]],x_particleIndex->Append[x,indexType[ann]]];
result = FlattenAt[SQS[FlattenAt[{creInds,Reverse[annInds]},{{1},{2}}]],{1}];
Return[result];
];
(* SQS in MultiConfiguration case has non normal ordered form *)
(*
createmultiSQS[creInds_List,annInds_List,norm_normalOrder:inorder] :=
Module[ {},
Return[multiSQS[norm, createSQS[creInds,annInds,norm]]]
];
*)
(* find all index types in a *)
indexTypesSQS[a_SQS] :=
Module[ {ninds,ind,types,type},
ninds = Length[a];
types = {};
Do[
type = indexParticle[a[[ind]]];
types = Append[types,type]
,{ind,1,ninds}
];
types = Union[Flatten[types]];
Return[types];
];
(* find the number of ptype operator in a *)
numIndicesOfType[a_SQS,ptype_particleType] :=
Module[ {ninds,result},
ninds = Length[a];
result = 0;
Do[
result+=If[ indexParticle[a[[i]]]===ptype,
1,
0
]
,{i,1,ninds}
];
Return[result];
];
(*Find the first index of ptype, else return -1*)
firstIndexOfType[a_SQS,ptype_particleType] :=
Module[ {ninds,ind,index},
ninds = Length[a];
ind = -1;
Do[
index = a[[i]];
If[ indexParticle[index]===ptype,
ind = i;
Break[]
]
,{i,1,ninds}
];
Return[ind];
];
(* selects creation/annihilation indices from SQS
while the indexType was droped
*)
creIndices[a_SQS] :=
Cases[a,x_particleIndex/;indexCreQ[x]->indexLight[x] ];
annIndices[a_SQS] :=
Reverse[Cases[a,x_particleIndex/;indexAnnQ[x]->indexLight[x] ]];
(* ::Section::Closed:: *)
(* SQM class *)
(*
SQM is a matrix element of a particle-symmetric operator, i.e. it treats particles of same type are indistinguishable.
Operators are further classified according to the permutational symmetry of equivalent indices within the bra (or, due to particle-symmetric property, within the ket)
as antisymm, symm, and nonsymmetric. These symmetries are utilized to reduce expressions to their simplest form.
*)
antisymm = indexSymm[-1];
symm = indexSymm[1];
nonsymm = indexSymm[0];
createSQM[O_String,braInds_List,ketInds_List,symm_indexSymm:antisymm] :=
Module[ {},
Return[flattenSQM[SQM[O,braInds,ketInds,symm]]]
];
SQM::wrongdepth = "Argument has wrong depth";
flattenSQM[a_SQM] :=
Module[ {nb,nk,result,braInds,ketInds,x},
Off[Append::normal];
If[ Depth[a]<5,
Return[a]
];
Clear[result];
Clear[braInds];
Clear[ketInds];
Clear[x];
nb = Length[a[[2]]];
nk = Length[a[[3]]];
braInds = Cases[a[[2]],x_particleIndex->Append[x,indexType[bra] ] ];
ketInds = Cases[a[[3]],x_particleIndex->Append[x,indexType[ket] ] ];
result = FlattenAt[SQM[FlattenAt[{braInds,ketInds},{{1},{2}}]],{1}];
result = Prepend[result,OHead[a[[1]],a[[4]]]];
Return[result];
];
(* deltaIndex, Kronecker delta, is a special SQM and has its own Head *)
createDeltaIndex[idx1_particleIndex,idx2_particleIndex]:=deltaIndex[particleIndex[idx1[[1]],indexSpace[idx1],indexType[bra]],particleIndex[idx2[[1]],indexSpace[idx2],indexType[ket]]];
(* rhoIndex, complement to Kronecker delta (i.e. rho^p_q = 1 - delta^p_q), is a special SQM and has its own Head *)
createRhoIndex[idx1_particleIndex,idx2_particleIndex]:=rhoIndex[particleIndex[idx1[[1]],indexSpace[idx1],indexType[bra]],particleIndex[idx2[[1]],indexSpace[idx2],indexType[ket]]];
(* selects bra/ket indices from SQM
while the indexType was dropped
*)
braIndices[a_SQM] :=
Cases[a,x_particleIndex/;indexBraQ[x]->indexLight[x] ];
ketIndices[a_SQM] :=
Cases[a,x_particleIndex/;indexKetQ[x]->indexLight[x] ];
(*
the following two functions are used to deal with \Eta SQM in MultiConfiguration
*)
(* expand \eta to \lamda and \delta *)
expandEta[a_SQM] :=
Module[{bra, ket, result},
bra = braIndices[a];
ket = ketIndices[a];
Assert[Length[bra]==1&&Length[ket]==1];
result = Plus[deltaIndex[bra[[1]], ket[[1]]], -createSQM["\[Lambda]", bra, ket, antisymm] ];
Return[result];
];
(* expand every \eta in expr *)
expandExp[expr_] :=
expr /. x_SQM /; x[[1, 1]] == "\[Eta]" -> expandEta[x];
(* convert cumulant to density matricies *)
(* this function is not universal,
it only works for rank 1 and 2 cumulant
*)
convertLambda[a_SQM] :=
Module[{bra, ket, result},
bra = braIndices[a];
ket = ketIndices[a];
Assert[ Length[bra] === Length[ket] ];
If [ Length[bra] === 1,
result = createSQM["\[Gamma]",bra,ket,antisymm];
Return[result]
];
If [Length[bra] === 2,
If [Depth[a] === 4,
result = createSQM["\[Gamma]",bra,ket,antisymm] - createSQM["\[Gamma]",{bra[[1]]},{ket[[1]]},antisymm]*createSQM["\[Gamma]",{bra[[2]]},{ket[[2]]},antisymm]
+ createSQM["\[Gamma]",{bra[[1]]},{ket[[2]]},antisymm]*createSQM["\[Gamma]",{bra[[2]]},{ket[[1]]},antisymm];
Return[result]
];
(*in spin free case, the rank one density with different spin will be zero *)
If [Depth[a] === 5,
result = createSQM["\[Gamma]",bra,ket,antisymm] - spinConvertGamma[{bra[[1]]},{ket[[1]]}]*spinConvertGamma[{bra[[2]]},{ket[[2]]}]
+ spinConvertGamma[{bra[[1]]},{ket[[2]]}]*spinConvertGamma[{bra[[2]]},{ket[[1]]}];
Return[result]
];
];
If[ Length[bra] > 2 ,
Return[a]
];
];
spinConvertGamma[bra_List, ket_List] :=
Module[{tmpbra, tmpket, spinsame, result},
spinsame = True;
tmpbra = bra[[1]];
tmpket = ket[[1]];
If[ tmpbra[[2,-1]] =!= tmpket[[2,-1]],
spinsame = False;
];
If [ spinsame,
result = createSQM["\[Gamma]",bra,ket,antisymm],
result = 0;
];
Return[result];
];
convertExp[expr_] :=
expr /. x_SQM /; x[[1,1]] == "\[Lambda]" -> convertLambda[x]
(* ::Section::Closed:: *)
(* Visualize function *)
(* these functions display particleIndex in string *)
visualizeIndex[a_particleIndex] :=
stringAppendSpin[ToString[a[[1]],StandardForm],indexSpin[a]];
Format[particleIndex[a__],TraditionalForm] :=
visualizeIndex[particleIndex[a]];
(* these functions display SQ expressions in tensor notation *)
visualizeSQE[a_*b_] :=
visualizeSQE[a]*visualizeSQE[b];
visualizeSQE[a_**b_] :=
visualizeSQE[a]**visualizeSQE[b];
visualizeSQE[a_^n_Integer] :=
visualizeSQE[a]^n;
visualizeSQE[a_+b_] :=
visualizeSQE[a]+visualizeSQE[b];
visualizeSQE[a_/;(Head[a]=!=SQS&&Head[a]=!=deltaIndex&&Head[a]=!=rhoIndex&&Head[a]=!=SQM&&Head[a]=!=mSQS)] :=
a;
visualizeSQE[a_deltaIndex] :=
Subsuperscript["\[Delta]",visualizeIndex[a[[1]]],visualizeIndex[a[[2]]]];
(* rho^p_q = 1 - delta^p_q, i.e. a complement to the Kronecker delta (obtained by reflection) *)
visualizeSQE[a_rhoIndex] :=
Subsuperscript["\[Rho]",visualizeIndex[a[[1]]],visualizeIndex[a[[2]]]];
(* converts a list indices into a pair of merged index labels,
all creators are in superscript and all annihilators in subscript *)
makeSupSubIndexStrings[indices_,SuperscriptQ_,padLeft_Symbol,reverseSubscript_Symbol]:=Module[{supInds,subInds,nSupInds,nSubInds,padding},
supInds = ""; nSupInds = 0;
subInds = ""; nSubInds = 0;
Do[
If[ SuperscriptQ[indices[[i]]],
(supInds = StringJoin[supInds,visualizeIndex[indices[[i]]] ]; ++nSupInds),
(subInds = If[reverseSubscript, StringJoin[visualizeIndex[indices[[i]]],subInds],
StringJoin[subInds,visualizeIndex[indices[[i]]]]
]; ++nSubInds)
],{i,1,Length[indices]}
];
padding=If[nSubInds>nSupInds, StringPadLeft["", nSubInds-nSupInds, "\[UnderBracket]"], ""];
supInds=If[padLeft,StringJoin[padding,supInds],StringJoin[supInds,padding]];
padding=If[nSubInds<nSupInds, StringPadLeft["", nSupInds-nSubInds, "\[UnderBracket]"], ""];
subInds=If[padLeft,StringJoin[padding,subInds],StringJoin[subInds,padding]];
Return[{supInds,subInds}]
];
visualizeSQE[a_SQS] :=
Module[ {bodyLabel,i,supInds,subInds},
(* convention labels strings normal-ordered wrt to nonphysical vacuum as tilde{a} *)
bodyLabel = If[SeQuantVacuum==SeQuantVacuumChoices["Physical"],"a","\[ATilde]"];
(* superscript (creator) indices are right padded, i.e. a^+_p a_s a_r should be typeset as Subsuperscript[a, rs, \[SpaceIndicator]p] *)
(* subscript (annihilator) indices are right padded, i.e. a^+_p a^+_q a_s should be typeset as Subsuperscript[a, \[SpaceIndicator]s, pq] *)
{supInds, subInds} = makeSupSubIndexStrings[a, indexCreQ, True, True];
Return[Subsuperscript[bodyLabel,subInds,supInds]]
];
visualizeSQE[a_mSQS] :=
Module[ {sqs,bodyLabel,i,supInds,subInds},
If [ Length[a[[2]]] === 0,
Return[a[[2]]]
];
bodyLabel = If[ a[[1]] === normalOrder[True],
OverTilde["a"],
"a"
];
sqs = a[[2]];
(* superscript (creator) indices are right padded, i.e. a^+_p a_s a_r should be typeset as Subsuperscript[a, rs, \[SpaceIndicator]p] *)
(* subscript (annihilator) indices are right padded, i.e. a^+_p a^+_q a_s should be typeset as Subsuperscript[a, \[SpaceIndicator]s, pq] *)
{supInds, subInds} = makeSupSubIndexStrings[sqs, indexCreQ, True, True];
Return[Subsuperscript[bodyLabel,subInds,supInds]]
];
visualizeSQE[a_SQM] :=
Module[ {bodyLabel,i,supInds,subInds},
bodyLabel = a[[1,1]];
(* bra and left indices are left padded *)
{supInds, subInds} = makeSupSubIndexStrings[a, indexKetQ, False, False];
Return[Subsuperscript[bodyLabel,subInds,supInds]]
];
Format[deltaIndex[a__],TraditionalForm] :=
visualizeSQE[deltaIndex[a]];
Format[rhoIndex[a__],TraditionalForm] :=
visualizeSQE[rhoIndex[a]];
Format[SQM[a__],TraditionalForm] :=
visualizeSQE[SQM[a]];
Format[SQS[a__],TraditionalForm] :=
visualizeSQE[SQS[a]];
Format[mSQS[a__],TraditionalForm] :=
visualizeSQE[mSQS[a]];
(* ::Section::Closed:: *)
(* Contraction functions *)
(*
Low-level contraction routines
contractIndex contracts 2 indices
contractSQS contracts a List of SQ strings of operators
deltaIndex is our representation of Kroneker delta
*)
contractIndex[L_particleIndex,R_particleIndex] :=
Module[ {typeL,typeR, fermiL, fermiR,intIndex,result},
(* check if particles are the same *)
typeL = Cases[L[[2]],_particleType];
typeR = Cases[R[[2]],_particleType];
If[ typeL=!=typeR,
Return[0]
];
(*if both creation or both annihilation - result is 0*)
If[ L[[3]]==R[[3]],
Return[0]
];
(* the case of a multiconfiguration vacuum cannot be handled by simple Wick-type rules *)
If[ SeQuantVacuum==SeQuantVacuumChoices["MultiConfiguration"],
Print["Wick theorem for MultiConfiguration vacuum is not expressed as simple contraction"];
Abort[]
];
(* discard zero contractions for the case of a determinant vacuum *)
If[ SeQuantVacuum==SeQuantVacuumChoices["SingleConfiguration"],
fermiL = spaceWRTFermiLevel[L[[2]]];
If[ L[[3]]==indexType[ann]&&fermiL==-1,
Return[0]
];
If[ L[[3]]==indexType[cre]&&fermiL==+1,
Return[0]
];
fermiR = spaceWRTFermiLevel[R[[2]]];
If[ R[[3]]==indexType[cre]&&fermiR==-1,
Return[0]
];
If[ R[[3]]==indexType[ann]&&fermiR==+1,
Return[0]
];
];
(* discard zero contractions for the case of a physical vacuum *)
If[ SeQuantVacuum==SeQuantVacuumChoices["Physical"],
If[ L[[3]]==indexType[cre],
Return[0]
];
];
If[ !spacesOverlap[L[[2]],R[[2]]],
Return[0]
];
If[ SeQuantDebugLevel>=5,
Print["In contractIndex: ",L//TraditionalForm," and ", R//TraditionalForm]
];
(* single-configuration contractions involving general (non-particle and non-hole) indices involve particle and hole density matrices *)
If[ SeQuantVacuum==SeQuantVacuumChoices["SingleConfiguration"]&&(fermiL==0&&fermiR==0),
(* must be unoccupied *)
If[ R[[3]]==indexType[cre],
intIndex = createParticleIndex[allvirt];
result = deltaIndex[Drop[L,{3}],intIndex ] * deltaIndex[intIndex,Drop[R,{3}] ];
];
(* must be occupied *)
If[ R[[3]]==indexType[ann],
intIndex = createParticleIndex[occ];
result = deltaIndex[Drop[R,{3}],intIndex ] * deltaIndex[intIndex,Drop[L,{3}] ];
];
Return[result];
];
If[ L[[1]]==R[[1]],
Return[1]
];
If[ R[[3]]===indexType[cre],
Return[deltaIndex[Drop[L,{3}],Drop[R,{3}] ] ],
Return[deltaIndex[Drop[R,{3}],Drop[L,{3}] ] ]
];
];
(* contractIndex function used in MultiConfiguration
return True if it is nonzero contration
return False if it is zero contraction
*)
contractIndex[a_List] :=
Module[{bra, ket, spinmatch, tmpresult, result},
If [ Depth[a] === 4,
result = !MemberQ[a, y_particleIndex /; spaceWRTFermiLevel[indexSpace[y]] === +1];
Return[result]
];
(* for spin free cases *)
(* only rank one cumulant will be zero if the spin is different *)
If [ Depth[a] === 5,
tmpresult = !MemberQ[a, y_particleIndex /; spaceWRTFermiLevel[indexSpace[y]] === +1];
If[ !tmpresult,
Return[tmpresult];
];
spinmatch = True;
If[ Length[a] === 2,
ket = a[[1]];
bra = a[[2]];
If [ bra[[2,-1]] =!= ket[[2,-1]],
spinmatch = False;
]
];
result = tmpresult && spinmatch;
Return[result];
];
];
(* if a has member deltaIndex b *)
deltaQ[a_,b_deltaIndex] :=
MemberQ[a,c_deltaIndex/;(indexQ[c,b[[1]] ]&&indexQ[c,b[[2]] ])];
(* ::Subsection:: *)
(* CR Class *)
(*
CR is the head for a general result of a contraction (unless it's a zero)
such a result consists of a prefactor and a noncommutative product of SQ strings:
CR[pfac,body]
*)
factorIntoCR[a_,0] :=
0;
factorIntoCR[a_,b_CR] :=
CR[a*b[[1]],b[[2]]];
factorIntoCR[a_,b:Plus[__]] :=
Module[ {result,nterms},
result = 0;
nterms = Length[b];
Do[result+=factorIntoCR[a,b[[i]]],{i,1,nterms}];
Return[result];
];
AddCR[a_CR,b_CR] :=
CR[1,a+b];
(* returns c if it does not equal to t *)
uniqueCR[t_CR,c_CR] :=
If[ t=!=c,
c,
0
];
(* returns members of c that do not equal to t *)
uniqueCR[t_CR,c_Plus] :=
Module[ {uc},
uc = 0;
Do[uc+=uniqueCR[t,c[[i]]],
{i,1,Length[c]}
];
Return[uc];
];
(* returns c if it does not appear in t *)
uniqueCR[t_Plus,c_CR] :=
If[ FreeQ[t,c],
c,
0
];
(* returns the terms in c that do not appear in t *)
uniqueCR[t_Plus,c_Plus] :=
Module[ {uc},
uc = 0;
Do[uc+=uniqueCR[t,c[[i]]],
{i,1,Length[c]}
];
Return[uc];
];
(* return the number of CR in Plus *)
numberCR[t_Plus] :=
Module[ {len, num},
num = 0;
len = Length[t];
Do[
If[ Head[t[[i]] ] === CR,
num += 1
];
,{i,1,len}
];
Return[num]
];
numberCR[t_CR] :=
1;
(* ::Subsection::Closed:: *)
(* Contract SQS *)
(*
If fullContract
find the first index
contract recursively, i.e. result = contraction[i,j]*contractSQS[str/ij]
else
loop over all pairs of indices i and j
result += normalOrderedForm[str] + contraction[i,j]*contractSQS[str/ij]
*)
contractSQS[str:NCM[__SQS],ptype_particleType,contractOptions_List] :=
Module[ {result,newcontractOptions,sqsL,sqsR,indexL,indexR,indL,indR,iR,indRoff,strLoff,contr1,contr2,newstr,sqsLlast,sqsLfirst,indLfirst,indLlast,newcontrib,f,nstr},
If[ Depth[str]==2,
Return[CR[1,1]]
];
If[ SeQuantDebugLevel>=5,
Print["called contract SQS for str=",str//TraditionalForm," ptype=",ptype//TraditionalForm]
];
(*Recursively drop empty strings*)
result = {};
Do[
sqsL = sL;
If[ Length[str[[sL]]]==0,
result = contractSQS[Drop[str,{sL}],ptype,contractOptions ];
Break[]
],
{sL,1,Length[str]}
];
If[ result=!={},
Return[result]
];
(*Cannot contract single string*)
If[ Length[str]==1,
If [ SeQuantVacuum==SeQuantVacuumChoices["MultiConfiguration"],
result = If[ !(finalfullContract/.contractOptions),
CR[1,str[[1]]],
0
],
result = If[ !(fullContract/.contractOptions),
CR[1,str[[1]]],
0
]
];
Return[result]
];
If[ SeQuantDebugLevel>=7,
Print["in contractSQS: dropped all empty strings str=",str//TraditionalForm]
];
nstr = Length[str];
If [ SeQuantVacuum==SeQuantVacuumChoices["MultiConfiguration"],
If [ nstr === 2,
(* If the last contraction, do fullContraction if finalfullContraction is True *)
If [ finalfullContract/. contractOptions,
newcontractOptions = Cases[contractOptions, a_ /; FreeQ[a, fullContract]];
newcontractOptions = Append[ newcontractOptions, fullContract->True ],
newcontractOptions = Cases[contractOptions, a_ /; FreeQ[a, fullContract]];
newcontractOptions = Append[ newcontractOptions, fullContract->False ]
];
result = contractSQS[str[[1]],str[[2]],newcontractOptions];
Return[result],
(* If not the last contraciton, do not do fullContraction *)
If [ fullContract/. contractOptions,
newcontractOptions = Cases[contractOptions, a_ /; FreeQ[a, fullContract]];
newcontractOptions = Append[ newcontractOptions, fullContract->False ],
newcontractOptions = contractOptions
];
result = contractSQS[str[[1]],str[[2]], newcontractOptions];
newstr = Take[ str, {3,nstr}];
f[a_CR] := (
If[ a[[2]] === 1,
CR[ a[[1]], newstr ],
CR[ a[[1]], a[[2]]**newstr ]
]
);
(* when there is only one CR[] result, map function no longer works *)
If [ numberCR[result] === 1,
result = f[result],
result = Map[f, result]
];
If[ SeQuantDebugLevel>=5,
Print["new contraction ", result//TraditionalForm];
];
result = contractSQS[result, ptype, newcontractOptions]
];
Return[result];
];
(*Find the first string that has an index of ptype*)
indRoff = 0;
sqsL = -1;
Do[
indL = firstIndexOfType[str[[sL]],ptype];
If[ indL!=-1,
sqsL = sL;
Break[]
],
{sL,1,Length[str]-1}
];
If[ SeQuantDebugLevel>=7,
Print["in contractSQS: first substring that contains index of ptype = ",sqsL]
];
(*exit if such string is not found*)
If[ sqsL==-1,
result = If[ (fullContract/.contractOptions),
0,
CR[1,str]
];
Return[result]
];
(* If not doing full contraction, then first add the strings reordered into normal order (taking into account the signs) *)
If[ SeQuantDebugLevel>=7,
Print["in contractSQS: ",(fullContract/.contractOptions)]
];
result = If[ !(fullContract/.contractOptions),
normalOrderedForm[str,ptype],
0
];
(* Only one string needs to be considered if full contraction is sought *)
sqsLfirst = sqsL;
sqsLlast = If[ !(fullContract/.contractOptions),
Length[str]-1,
sqsL
];
Do[
sqsL = sL;
indL = firstIndexOfType[str[[sqsL]],ptype];
If[ indL==-1,
Continue[]
];
(* only 1 index need to be considered if seeking a full contraction *)
indLfirst = indL;
indLlast = If[ !(fullContract/.contractOptions),
Length[str[[sqsL]]],
indL
];
(* strLoff keeps track of the distance between the first ptype index in strL and the end of the string. It's used to initialize indRoff.
indRoff is the distance between left and right indices *)
strLoff = numIndicesOfType[str[[sqsL]],ptype];
Do[
indL = iL;
indexL = str[[sqsL,indL]];
If[ indexParticle[indexL]=!=ptype,
Continue[]
];
strLoff--;
indRoff = strLoff+1;
If[ SeQuantDebugLevel>=7,
Print["Chose left index: indexL = ",indexL//TraditionalForm," indRoff = ",indRoff];
];
(*Contract with every other index of ptype*)
Do[
iR = 0;
Do[
indexR = str[[sqsR,indR]];
If[ SeQuantDebugLevel>=7,
Print["Chose right index: indexL = ",indexL//TraditionalForm," indexR = ",indexR//TraditionalForm," indRoff = ",indRoff];
];
If[ indexParticle[indexR]=!=ptype,
Continue[],
iR+=1
];
If[ SeQuantDebugLevel>=5,
Print["Calling contractIndex for ",indexL//TraditionalForm," and ", indexR//TraditionalForm]
];