cc_ade-theory.ipynb

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   "source": [
    "### Algebraic Evaluation of Coupled-Cluster Diagrams"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7ef6c81",
   "metadata": {},
   "source": [
    "In _cogus_ coupled-cluster equations are derived from an approach utilising second quantization in a purely algebraic way. An alternative, although equivalent, approach is though diagrammatic methods. Here we will explore this approach, again with some help from Sympy.\n",
    "\n",
    "__THE EXPANSION__\n",
    "\n",
    "The coupled-cluster exponential ansatz requires an expansion to a specified order of the series $e^x$. The exponential series is well known and is $e^x = 1 + x +\\frac{1}{2!} x^2 + \\frac{1}{3!}x^3 + ... + \\frac{1}{n!} x^n$. Coupled-cluster requires an evaluation of the exponential in terms of the cluster amplitude operators $T_n$, for example CCSD requires $e^{(T_1+T_2)}$. The first few often used expansions are"
   ]
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   "source": [
    "__SD-singles__ $\\frac{1}{6}T_1^{3} + \\frac{1}{2}T_1^{2} + T_1T_2 + T_1 + T_2 + 1~~~~~~$ \n",
    "*we will see later that the biggest interaction number for a Hamiltonian is +1 so in order to make a singles ie excitation of 1 the smallest vertex allowed is 0. The smallest interaction number of a Hamiltonian is -2 so the largest term cannot have a combined level of greater than 3 ie $T_1^3$ or $T_1 T_2$*\n",
    "\n",
    "__SD-doubles__ $\\frac{1}{24}T_1^{4} + \\frac{1}{6}T_1^{3} + \\frac{1}{2}T_1^{2}T_2 + \\frac{1}{2}T_1^{2} + T_1T_2 + T_1 + \\frac{1}{2}T_2^{2} + T_2 + 1 ~~~~~~$ \n",
    "*to make a doubles ie 2 the smallest combined level for a term is 1, the largest term that will contract with a -2 Hamiltonian to make a doubles ie 2 is a combined level no greater than 4 ie $T_1^4$ or $T_2^2$*\n",
    "\n",
    "__SDT-triples__ $\\frac{1}{6}T_1^{3}T_2 + \\frac{1}{2}T_1^{2}T_2 + \\frac{1}{2}T_1^{2}T_3 + \\frac{1}{2}T_1T_2^{2} + T_1T_2 + T_1T_3 + \\frac{1}{2}T_2^{2} + T_2T_3 + T_2 + T_3~~~~~~ $  \n",
    "*to make a triples ie 3 the smallest combined level for a term is 2, additionally $T_1$ and any power of $T_1$ cannot contract to produce a triples diagram having only one interaction vertex, the largest term that will contract with a -2 Hamiltonian to make a triples ie 3 is a combined level no greater than 5 ie $T_3 T_2$, $T_3 T_1^2$ or $T_2 T_1^3$*\n",
    "\n",
    "__SDTQ-quadruples__ $\\frac{1}{6}T_1^{3}T_2 + \\frac{1}{6}T_1^{3}T_3 + \\frac{1}{4}T_1^{2}T_2^{2} + \\frac{1}{2}T_1^{2}T_2 + \\frac{1}{2}T_1^{2}T_3 + \\frac{1}{2}T_1^{2}T_4 + \\frac{1}{2}T_1T_2^{2} + T_1T_2T_3 + T_1T_2 + T_1T_3 + T_1T_4 +$\n",
    "$\\frac{1}{6}T_2^{3} + \\frac{1}{2}T_2^{2} + T_2T_3 + T_2T_4 + T_2 + \\frac{1}{2}T_3^{2} + T_3 + T_4~~~~~~$\n",
    "*to make a quadruple ie 4 the smallest combined level for a term is 3, additionally $T_1$, $T_2$ and any power of $T_1$ cannot contract to produce a quadruples diagram, the largest term that will contract with a -2 Hamiltonian to make a quadruple ie 4 is a combined level no greater than 6 ie $T_4 T_2$, $T_3 T_1^3$ or $T_2 T_1^4$ etc*"
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"
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    "__THE DIAGRAM__\n",
    "\n",
    "We now introduce some diagrammatic notation. This is a typical diagram\n",
    "\n",
    "![Screenshot%20from%202022-06-18%2008-58-04.png](attachment:Screenshot%20from%202022-06-18%2008-58-04.png)\n",
    "\n",
    "The solid lines at the bottom represent an amplitude operator, in this case the base lines have 4 attached vertical lines each and so represent $T_2$ amplitudes, a $T_1$ operator will have 2 lines and $T_3$ will have 6 etc. Each pair of lines has one up arrow (+ particle) and one down arrow (- hole). We will denote $T_1$ by '__+-__', $T_2$ by '__++--__' and $T_3$ by '__+++---__'. An expansion term will be written with __'|'__ seperating the conponent operators so $T_1 T_2$ will be '__+-|++--__'. We will develop an algebra of the sign strings or for short s-algebra.\n",
    "\n",
    "The number of excitation lines (+) for $T_1 T_2$ is 3, if we are considering singles (S) then we expect an excitation of +1 for each term hence we define the _interaction number_ (expected excitation - actual excitation) as 1 - 3 = __-2__  for this term.\n",
    "\n",
    "Connected to the interaction vertices in the diagram are the Hamiltonian components (in the diagram example above this is __++--__), these can be"
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"
    }
   },
   "cell_type": "markdown",
   "id": "d765eb23",
   "metadata": {},
   "source": [
    "![Screenshot%20from%202022-06-18%2009-21-40.png](attachment:Screenshot%20from%202022-06-18%2009-21-40.png)\n",
    "\n",
    "Since these diagrams interact with the amplitude below them it is the lines below the vertex that are of interest and determine the sign-strings. The _interaction number_ for each diagram is determined as 'number of +-lines above vertex - number of +-lines below vertex'.These diagrams are\n",
    "+ $f_a^c$ which we will call __pp__ has a single '+'-line below vertex so it's string is __+__, and 1 '+'-line above vertex and one below so it's _interaction number_ is 0\n",
    "+ $f_i^k$ which we will call __hh__ has a single '-'-line below vertex so it's string is __-__, and 0 +-line above vertex and 0 below so it's _interaction number_ is 0\n",
    "+ $f_a^i$ which we will call __ph__ has a no lines below vertex so it's string is __0__, and one +-line above vertex and 0 below so it's _interaction number_ is +1\n",
    "+ $f_i^a$ which we will call __hp__ has a one '+'-line and one '-'-line below vertex so it's string is __+-__, and 0 +-line above vertex and one below so it's _interaction number_ is -1\n",
    "+ $\\langle ij\\Vert kl \\rangle$ which we will call __hhhh__ has two '+'-lines below vertex so is __++__ , there are two '+'-lines above the vertex and two below so it's _interaction number_ is 0\n",
    "+ $\\langle ab\\Vert cd \\rangle$ which we will call __pppp__ has two '-'-lines below vertex so is __--__ , there are no '+'-lines above the vertex and none below so it's _interaction number_ is 0\n",
    "+ $\\langle ai\\Vert bj \\rangle$ which we will call __phph__ has one '+'-line below vertex and one '-'-line so is __+-__ , there is one '+'-line above the vertex and one below so it's _interaction number_ is 0\n",
    "+ $\\langle ab\\Vert ck \\rangle$ which we will call __ppph__ has one '-'-line below vertex so is __-__ , there are two '+'-lines above the vertex and one below so it's _interaction number_ is +1\n",
    "+ $\\langle ak\\Vert bc \\rangle$ which we will call __phpp__ has two '+'-lines below vertex and one '-'-line so is __++-__ , there are one '+'-line above the vertex and two below so it's _interaction number_ is -1\n",
    "+ $\\langle ia\\Vert kl \\rangle$ which we will call __hphh__ has one '-'-line below vertex so is __-__ , there are one '+'-lines above the vertex and none below so it's _interaction number_ is +1\n",
    "+ $\\langle ij\\Vert ka \\rangle$ which we will call __hhhp__ has one '+'-lines below vertex and two '-'-lines so is __+--__ , there are no '+'-lines above the vertex and one below so it's _interaction number_ is -1\n",
    "+ $\\langle ab\\Vert ij \\rangle$ which we will call __pphh__ has no '+'-lines below vertex and no '-'-lines so is __0__ , there are two '+'-lines above the vertex and none below so it's _interaction number_ is +2\n",
    "+ $\\langle ij\\Vert ab \\rangle$ which we will call __hhpp__ has two '+'-lines below vertex and two '-'-lines so is __++--__ , there are no '+'-lines above the vertex and two below so it's _interaction number_ is -2\n",
    "\n",
    "If we again consider $T_1 T_2$ which has an _interaction number_ of __-2__ then the only Hamiltonian elements it can connect to are those which also have an _interaction number_ of __-2__ ie __hhpp__. So out task is to compute the _interaction number_ of the term, find the possible compatible Hamiltonian elements, then determine all possible valid combinations that the amplitude operators and the Hamiltonion operator can connect by. For example $T_1 T_2$ can connect to __hhpp__ as follows __+|+--__, __-|++-__ or __+-|+-__."
   ]
  },
  {
   "attachments": {
    "f1.png": {
     "image/png": 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"
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   "cell_type": "markdown",
   "id": "19c0b45b",
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   "source": [
    "__sign of diagram__\n",
    "\n",
    "To determine the sign of a diagram we need to know the number of hole lines (down arrow) or __-__. This is number of __-__ signs in the operator sign-string plus the level of expansion. For $D_{5a}$ __+|-__ this is 1 + 2 (doubles) 3.\n",
    "We also need to know the number of loops, (paired external lines or quasi-loop). For each amplitude in the operator sign-string add the number of __+-__ pairs and add to the level of expansion eg $D_{5a}$ is __+|-__ there are 0 + 2 __+-__ pairs, so for this diagram the loops plus down arrow lines is 3 + 2=5 and $(-1)^5 = -1$ so diagram has a -ve sign. As an example let's look at doubles $T_1 T_2$ $D_{5a}$ which is __+-|-__: there are 2 __-__ signs in the operator and we are looking at the doubles expansion which implies 2+2=4, there is 1 __+-__ in the operator and again the expansion is doubles so 1+2=3. This gives a total of 4+3=7 and $(-1)^7=-1$. There is one special case when the operator string is __++--__, as this can be written __+-|+-__ it counts as 2 not 1 loops.\n",
    "\n",
    "__coefficient of the diagram__\n",
    "\n",
    "To determine the multiplicative factor we need to determine the number of _equivalent internal lines_. Two internal lines are considered equivalent if they connect the same two vertices, going in the same direction. It is clear that only lines attached to the same amplitude can be equivalent and they must connect to the same operator. It is not possible for the 1-body operators to contract in such a way to create equivalent lines. However we can see that eg $S_{2b}$ does as it is $\\langle ak\\Vert cd \\rangle t^{cd}_{ik}$ - clearly __c__ and __d__ are equivalent as both connect the same vertices and so this term has a factor of $\\frac{1}{2}$. Additionally if two _vertices_ are equivalent then they two contribute a factor of $\\frac{1}{2}$. Two vertices are equivalent if they have the same number of line pairs and are connected in equivalent ways to the Hamiltonian operator vertex. The example given in Shavitt and Bartlett is\n",
    "\n",
    "![f1.png](attachment:f1.png)\n",
    "\n",
    "Here diagrams 1 & 2 are equivalent vertices but 3 is not.\n",
    "\n",
    "To determine the number of equivalent lines add the number of __++__ and __--__ in the same partition of the operator sign-string. So, for example, in diagram $S_{4a}$ which is a $T_1 T_2$ vertex, we have __++--__ combinations of __+|+--__, __-|++-__ and __+-|+-__. The first two of these strings have __--__ and __++__ in the same amplitude respectively and so have a factor of $\\frac{1}{2}$, but __+-|+-__ does not.\n",
    "\n",
    "To determine the number of equivalent vertices add the number of pairs of __|+|__ or __|-|__ in the operator sign string. For example in $D_{6a}$ which is __+|+__ (1 pair=factor $\\frac{1}{2}$) into __+-|+-__, $D_{7a}$ __+|+|--__ into __+-|+-|++--__ (has both equivalent lines and vertices), $D_{8b}$ __+|-|-__ in __+-|+-|+-__, $D_9$ __+|-|+|-__  into __+-|+-|+-|+-__(two equivalent vertices) and $T_{10b}$ which is a $\\frac{1}{6} T_1^3 T_2$ vertex with  __+|+|-|-__ into __+-|+-|+-|++--__ similar to $D_9$. Note only pairs on similar amplitude types count, hence $T_{4a}$ which is $T_1 T_2$ and __+|+__ into __+-|++--__ does not count. The pairs of __|+-|__ also count as in $D_{3b}$ which is __+-|+-__ onto __++--|++--__.\n",
    "\n",
    "__distinct permutations__\n",
    "\n",
    "Shavitt & Bartlett- 'Sum over all distinct permutations $\\hat{P}$ of labels of inequivalent external particle lines and of inequivalent external hole lines, including a parity factor $(−1)^{\\sigma(\\hat{P})}$'. Consider diagram $D_{4a}$ which is $\\displaystyle \\sum_{c} \\langle ab \\Vert  cj \\rangle t^c_i$. We have the external lines _i_ and _j_ split between two elements leading to the permutation $\\hat{P}(ij)$. In the more complex case of triples consider $T_{1b}$ which is $\\displaystyle \\sum_{l} \\langle lc\\Vert jk \\rangle t^{ab}_{il}$, here we have external lines (ab)c and (jk)i split between elements. We write the label that appears by itself _i_ and _c_ and the two that appear together ie _jk_ and _ab_ as $\\hat{P}(i/jk)\\hat{P}(c/ab)$. $\\hat{P}(i/jk)$ is defined as 1 - $\\hat{P}(ij)$ - $\\hat{P}(ik)$. As another example take $T_{2d}$ which is $\\displaystyle \\sum_{lm} \\langle lm\\Vert ij \\rangle t^{abc}_{lmk}$ - appearing by itself is just _k_ and the pair of external line appearing together are _ij_ giving a permutation of $\\hat{P}(k/ij)$. The idea extends to quadruples and beyond, for example in the quadruples $Q_{5a}$ is $\\displaystyle \\sum_{ef} \\langle bc\\Vert ef \\rangle t^{ae}_{ij} t^{fd}_{kl}$ the external labels are split over three terms giving $\\hat{P}(bc/a/d)$ and $\\hat{P}(ij/kl)$.\n",
    "- - -\n",
    "permutations: $\\hat{P}(ab)$ is defined as = 1 -$\\hat{P}_{ab}~~~~~~~$. Where $\\hat{P}_{ab}$ means to swap the indices a and b.\\\n",
    "$\\hat{P}$(ab)$\\hat{P}$(ij) = 1 - $\\hat{P}_{ab}$ - $\\hat{P}_{ij}$ + $\\hat{P}_{ab}\\hat{P}_{ij}$. Sometimes we write $\\hat{P}$(ab)$\\hat{P}$(ij) as $\\hat{P}$(ab|ij)\\\n",
    "$\\hat{P}$(a/bc) is defined as $1 - \\hat{P}_{ab} - \\hat{P}_{ac}$\\\n",
    "$\\hat{P}$(a/bc/d) = $1 - \\hat{P}_{ab} - \\hat{P}_{ac} - \\hat{P}_{ad} - \\hat{P}_{bd} - \\hat{P}_{cd} + \\hat{P}_{ab} \\hat{P}_{cd} + \\hat{P}_{ac} \\hat{P}_{bd} + \\hat{P}_{bd} \\hat{P}_{ad} + \\hat{P}_{cd} \\hat{P}_{ad} + \\hat{P}_{ab} \\hat{P}_{ad} + \\hat{P}_{ac} \\hat{P}_{ad}$\n"
   ]
  },
  {
   "attachments": {
    "f2-3.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "8c66a5d6",
   "metadata": {},
   "source": [
    "We can see now why $T_1$ alone cannot contribute to the triples and above. For example, the triples need a __+++__ interaction and a $T_1$ contibutes __+__ meaning we need a __++__ Hamiltonian interaction which can only be __pphh__ which has no contraction lines. If we try $T_1^2$ __+-|+-__ this contributes __++__ so we need a __+__ which can be __ph__, __ppph__ or __hphh__, but __ph__ has no contraction lines and __ppph__ and __hphh__ have only one so cannot contract with the two $T_1$ amplitudes to form a connected diagram. $T_1 T_1 T_1$ contributes __+++__ so we need a 0 interaction number Hamiltonian ie __pp__, __hh__, __hhhh__, __pppp__ or __phph__. These have contraction lines either one or two so again cannot contract with the three amplitude term. Similar analysis applies to the quadruples to show $T_2$ cannot contract.\n",
    "\n",
    "__Labelling the diagrams__\n",
    "\n",
    "These are the 'rules' for labelling the cluster diagram, they are not the only ways of assigning labels but are designed to produce results similar to Shavitt & Bartlett. If we can produce a set of rules then we can translate those into coded algorithms.\n",
    "\n",
    "The 1-body operators are __pp__, __hh__, __ph__ and __hp__ which are __+__, __-__, __0__ and __+-__ with interaction numbers 0, 0, +1 and -1.\n",
    "\n",
    "Notation: we will write $t^{(+)}$ to mean a continuous arbitary sequence of '+' external labels, similarly $t_{(-)}$ will denote a continuous arbitary sequence of '-' external labels. 'e' and 'f' will denote an internal '+' label and 'm' and 'n' will denote an internal '-' label. $t^{[+)}$ will denote a continuous sequence of '+' external labels starting from the last referenced label + 1 eg $t^{(+)} t^{e[+)}$ if applied to $T_2$ amplitudes would mean $t^{ab} t^{ed}$. We extend the last notation so that $t^{[[+)}$ implies that the the first two available labels are skipped and $[$ will be used to denote a skipped label. If a sequence of labels is not of arbitary length then a string of length 'n' is denoted $t^{(+)^n}$. These algoritms for labelling the diagrams are labelled __a.n__ and as a recipe can be programmed.\n",
    "- - -\n",
    "__1-body Hamiltonian elements__\n",
    "- - -\n",
    "__pp(+)__ with an interaction number of zero and a single interaction line can contract with only one amplitude as  in the following diagrams\n",
    "\n",
    "| __S__   |   __D__   | __T__ | __Q__ |\n",
    "|:---------|:----|:----|:----|\n",
    "| $T_1~~~~~S_{3a}~~~~~t^c_i f_{ac}$| $T_2~~~~~D_{2a}~~~~~t^{ac}_{ij} f_{bc}$ | $T_3~~~~~T_{2a}~~~~~t^{abd}_{ijk} f_{cd}$ | $T_4~~~~~Q_{2a}~~~~~t^{abce}_{ijkl} f_{de}$ |\n",
    "\n",
    "The above are the label assignments from Shavitt & Bartlett, the order of diagrams is arbitrary so the labelling is not unique.\n",
    "\n",
    "The __pp__ diagrams can be represented as $$\\boldsymbol{t^{(+)e}_{(-)} f_{(+)e}}~~~~~a.1$$\n",
    "\n",
    "Example __Diagram $T_{2a}$__: assign __+$^d$__ to the operator -> the __(+)__ will be a sequence of two external '+' labels giving __+$^a$+$^b$+$^d$---__ -> the __(-)__ will be a sequence of three '-' external labels ____+$^a$+$^b$+$^d$-$^i$-$^j$-$^k$__ -> the operator has a sequence of one '+' external operators contingous with the previous set and the internal label ie __+$^c$+$^d$__\n",
    "- - -\n",
    "__hh(-)__ is similar to pp it contracts in with the same ampliudes\n",
    "\n",
    "| __S__   |   __D__   | __T__ | __Q__ |\n",
    "|:---------|:----|:----|:----|\n",
    "| $T_1~~~~~S_{3b}~~~~~t^a_k f_{ki}$| $T_2~~~~~D_{2b}~~~~~t^{ab}_{ik} f_{kj}$ | $T_3~~~~~T_{2b}~~~~~t^{abc}_{ijl} f_{lk}$ | $T_4~~~~~Q_{2b}~~~~~t^{abcd}_{ijkm} f_{ml}$ |\n",
    "\n",
    "The __hh__ diagrams can be represented as $$\\boldsymbol{t^{(+)}_{(-)m} f_{m(-)}}~~~~~a.2$$\n",
    "\n",
    "Example __Diagram $T_{2b}$__: assign __-$^l$__ to the operator -> the __(-)__ will be a sequence of two '-' external labels giving __+++-$^i$-$^j$-$^l$__ > the __(+)__ will be a sequence of '+' external labels __+$^a$+$^b$+$^c$-$^i$-$^j$-$^l$__ -> the operator has the  internal label and a sequence of one '-' external label ie __+$^l$+$^k$__.\n",
    "- - -\n",
    "__ph(0)__ can only contract with $T_0 = 1$ in the singles where we write $f^a_i$\n",
    "\n",
    "| __S__ |\n",
    "|:------|\n",
    "| $f_{ai}$ |\n",
    "\n",
    "- - -\n",
    "__hp(+-)__ with interaction number -1 can contract with\n",
    "\n",
    "| __S__  | __D__ | __T__| __Q__ |\n",
    "|:-------|:------|:-----|:------|\n",
    "| $T_2~~~~~S_{2a}~~~~~t^{ac}_{ik} f_{kc}$ | $T_1 T_2~~~~~D_{5a}~~~~~t^c_i t^{ab}_{kj} f_{kc}$ | $T_2 T_2~~~~~T_{3a}~~~~~t^{ad}_{ij} t^{bc}_{lk} f_{ld}$ | $T_1 T_4~~~~~Q_{4a}~~~~~t^e_i t^{abcd}_{mjkl} f_{me}$ |\n",
    "|   | $T_1 T_2~~~~~D_{5b}~~~~~t^a_k t^{cb}_{ij} f_{kc}$ | $T_1 T_3~~~~~T_{6g}~~~~~t^d_i t^{abc}_{ljk} f_{ld}$| $T_1 T_4~~~~~Q_{4b}~~~~~t^a_m t^{ebcd}_{ijkl} f_{me}$|\n",
    "|  |  | $T_1 T_3~~~~~T_{6h}~~~~~t^a_l t^{dbc}_{ijk} f_{ld}$ | $T_2 T_3~~~~~Q_{6a}~~~~~t^{ae}_{ij} t^{bcd}_{mkl} f_{me}$|\n",
    "| | | | $T_2 T_3~~~~~Q_{6b}~~~~~t^{ab}_{im} t^{ecd}_{jkl} f_{me}$|\n",
    "\n",
    "The only time __hp__ can  contract with a single amplitude is in the singles where it can can contract with $T_2$. In all higher expansions with an interaction number of -1 it needs an amplitude one greater than the expansion level eg in triples it needs __++++----__ which cannot appear in the triples. For contractions with two amplitudes we have diagrams like $D_{5a}$, $T_{3a}$ and $Q_{4a}$\n",
    "\n",
    "![f2-3.png](attachment:f2-3.png)\n",
    "\n",
    "Note that the internal labels will always appear on adjacent amplitudes.These diagrams can be represented as\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{(+)}_{m(-)}f_{me}}~~~~~a.3$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{e(+)}_{(-)} f_{me}}~~~~~a.4$$\n",
    "\n",
    "The other case is when the Hamiltonian contracts with just one amplitude\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)m} f_{me}}~~~~~a.5$$\n",
    "\n",
    "Example __Diagram $S_{2a}$__: assign __+$^c$-$^k$__ to the operator -> __(+)__ will require one '+' external label and __(-)__ will required one '-' external label __+$^a$+$^c$-$^i$-$^k$__ -> label Hamiltonian operator as __-$^k$+$^c$__\n",
    "\n",
    "Example __Diagram $T_{3a}$__: this is __+|-__ onto __++--|++--__ so assign __+$^d$|-$^l$__ to the operator ->  for the first amplitude the (+) requires one label and the (-) will require two labels __+$^a$+$^d$-$^i$-$^j$|++--__ -> in the second amplitude (+) will require two labels and the (-) one  __+$^a$+$^d$-$^i$-$^j$|+$^b$+$^c$-$^l$-$^k$__-> the Hamiltonian is then __-$^l$+$^d$__. __Diagram $Q_{6b}$__: this is __-|+__ onto __++--|+++---__ so assign __-$^m$|+$^e$__ to the operator -> for the first amplitude the (+) requires two labels and the (-) one __+$^a$+$^b$-$^i$-$^m$|+++---__ -> in the second amplitude the (+) will require two labels and the (-) three __+$^a$+$^b$-$^i$-$^m$|+$^e$+$^c$+$^d$-$^j$-$^k$-$^l$__ -> the Hamiltonian is the __+$^m$-$^e$__."
   ]
  },
  {
   "attachments": {
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"
    }
   },
   "cell_type": "markdown",
   "id": "a4cbfd47",
   "metadata": {},
   "source": [
    "- - -\n",
    "__2-body Hamiltonian elements__\n",
    "- - -\n",
    "__+ (ppph)__ This element has an interaction number of +1 so cannot contract in the singles. It can only contract with a one amplitude. The diagram for this element is\n",
    "\n",
    "![f3.png](attachment:f3.png)\n",
    "\n",
    "The 2-body Hamiltonian is labelled $\\langle$ node$_1^{out}$ node$_2^{out}$\\Vert node$_1^{in}$ node$_2^{in} \\rangle$ which is $\\langle ab\\Vert  \\boldsymbol{c}i \\rangle$.\n",
    "\n",
    "| __S__ | __D__ | __T__ | __Q__ |\n",
    "|:---------------|:--------------------------|:------------------|:-----------------|\n",
    "|$$~~~~~~~~~$$ | $$T_1~~~~~D_{4a}~~~~~t^c_i \\langle ab\\Vert cj \\rangle~~~~~~~~~~$$ | $$T_2~~~~~T_{1a}~~~~~t^{ad}_{ij} \\langle bc\\Vert dk \\rangle~~~~~~~~~~~~$$ | $$T_3~~~~~Q_{1a}~~~~~t^{abe}_{ijk} \\langle cd\\Vert el \\rangle$$ |\n",
    "\n",
    "Our representation is $$\\boldsymbol{t^{(+)e}_{(-)} \\scriptsize{\\langle (+)\\Vert e(-) \\rangle}}~~~~~a.6$$\n",
    "\n",
    "Example, __Diagram $Q_{1a}$__: this is __+__ on __+++---__,the __+__ of the operator is assigned __+$^e$__ -> the (+) in the amplitude requires two labels and the (-) requires three __+$^a$+$^b$+$^e$-$^i$-$^j$-$^k$__ -> the bra requires two more contiguous '+' labels as $\\langle cd|$ -> the ket requires the assigned internal label 'e' and a contiguous '-' label 'l' giving $|el \\rangle$. \n",
    "- - -\n",
    "__- (hphh)__ This element has an interaction number of +1 and there are no single amplitude terms in the singles that require a +1 interaction. It can only contract with one amplitude. Its diagram is"
   ]
  },
  {
   "attachments": {
    "f4.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "01489c9b",
   "metadata": {},
   "source": [
    "![f4.png](attachment:f4.png)\n",
    "\n",
    "| __S__ | __D__ | __T__ | __Q__ |\n",
    "|:---------------|:--------------------------|:------------------|:-----------------|\n",
    "    |$$~~~~~~~~~$$ | $$T_1~~~~~D_{4b}~~~~~t^a_k \\langle kb\\Vert ij \\rangle~~~~~~~~~~$$ | $$T_2~~~~~T_{1b}~~~~~t^{ab}_{il} \\langle lc\\Vert jk \\rangle~~~~~~~~~~~~$$ | $$T_3~~~~~Q_{1b}~~~~~t^{abc}_{ijm} \\langle md\\Vert kl \\rangle$$ |\n",
    "\n",
    "Our representation is $$\\boldsymbol{t^{(+)}_{(-)m} \\scriptsize{\\langle m(+)\\Vert (-) \\rangle}}~~~~~a.7$$\n",
    "\n",
    "Example, __Diagram $Q_{1b}$__: this is __-__ contracted with __+++---__, the __-__ of the operator is assigned __-$^m$__ -> the (-) in the amplitude requires two labels and the (+) three __+$^a$+$^b$+$^c$-$^i$-$^j$-$^m$__ -> the bra requires the assigned label and a contiguous '+' label $\\langle md|$ -> the ket requires two contiguous '-' labels $|kl \\rangle$.\n"
   ]
  },
  {
   "attachments": {
    "f12.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "5a338e85",
   "metadata": {},
   "source": [
    "- - -\n",
    "__++ (hhhh)__ for this to contract in a singles it would have to be $T_1$ as __hhhh__  interaction number is zero but the operator and the amplitude $T_1$ are incompatible (there is no __-__ in the Hamiltonian). The diagram is\n",
    "\n",
    "![f12.png](attachment:f12.png)\n",
    "\n",
    "| __S__ | __D__ | __T__ | __Q__ |\n",
    "|:---------------|:--------------------------|:------------------|:-----------------|               \n",
    "|$$~~~~~~~~~$$ | $$T_2~~~~~~~D_{2c}~~~~~t^{cd}_{ij}  \\langle ab\\Vert cd \\rangle~~~~~~~~~~$$ | $$T_3~~~~~~~T_{2c}~~~~~t^{dec}_{ijk} \\langle ab\\Vert de \\rangle$$ |  $$T_4~~~~~~Q_{2c}~~~~t^{efcd}_{ijkl} \\langle ab\\Vert ef \\rangle$$|\n",
    "|$$~~~~~~~~~$$ | $$T_1 T_1~~~~~D_{6a}~~~~~t^c_i t^d_j \\langle ab\\Vert cd \\rangle~~~~~~~~~~$$ | $$T_1 T_2~~~~~T_{4a}~~~~~t^d_i t^{ec}_{jk} \\langle ab\\Vert de \\rangle$$  |  $$T_2 T_2~~~~~Q_{5a}~~~~~t^{ae}_{ij} t^{fd}_{kl} \\langle bc\\Vert ef \\rangle$$|"
   ]
  },
  {
   "attachments": {
    "f5.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "e4f2dbef",
   "metadata": {},
   "source": [
    "The diagrams $D_{2c}$ and $Q_{5a}$ are \n",
    "\n",
    "![f5.png](attachment:f5.png)\n",
    "\n",
    "Noting that the internal lines are adjacent we may propose $$\\boldsymbol{t^{ef[[+)}_{(-)} \\scriptsize{\\langle [[\\Vert ef\\rangle}~~~~~a.8}$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{f[[+)}_{(-)} \\scriptsize{\\langle [[\\Vert  ef\\rangle}}~~~~~a.9$$\n",
    "\n",
    "Examples\n",
    " __Diagram $Q_{2c}$__ this is __++__ contracted with __+++---__, assign __+$^e$+$^f$__ to the operator -> these labels are assigned to the '+' amplitude with contiguous '+' external labels excluding the first two __+$^c$+$^d$--__ > contiguous '-' labels are assigned __$^c$+$^d$-$^i$-$^j$__ -> the two skipped '+' external labels are assigned to the bra $\\langle ab|$ -> the two internal labels are assigned to the ket $|cd \\rangle$.\n",
    "\n",
    "For __diagram $Q_{5a}$__ this is __+|+__ contracted with __++--|++--__, assign __+$^e$+$^f$__ to the operator -> external labels are assigned contiguously to the first amplitude in this case __$^a$+--|++--__ -> the first internal is assigned and contiguous '-' externals as __+$^a$+$^e$-$^i$-$^j$|++--__ -> the second internal is added to the second amplitude followed by contiguous '+' external labels skipping the first that are two available __+$^a$+$^e$-$^i$-$^j$|+$^f$+$^d$--__ -> contiguous '-' external labels are assigned to the second amplitude +$^a$+$^e$-$^i$-$^j$|+$^f$+$^d$-$^k$-$^l$ -> the two skipped external labels are assigned to the bra $\\langle bc|$ -> the two internal labels are assigned to the ket $|ef \\rangle$.\n",
    "- - - - - - \n",
    "__-- (pppp)__ this has the same interaction number as __hhhh__ and so contracts with the same expansion terms. It's labelling is similar to __hhhh__ but with bra and ket definitions reversed for the Hamiltonian term. \n",
    "\n",
    "| __S__ | __D__ | __T__ | __Q__ |\n",
    "|:---------------|:--------------------------|:------------------|:-----------------|               \n",
    "|$$~~~~~~~~~$$ | $$T_2~~~~~~~D_{2d}~~~~~t^{ab}_{kl}  \\langle kl\\Vert ij \\rangle~~~~~~~~~~$$|$$T_3~~~~~~~T_{2d}~~~~~t^{abc}_{lmk} \\langle lm\\Vert ij \\rangle$$ |  $$T_4~~~~~~Q_{2d}~~~~t^{abcd}_{mnkl} \\langle mn\\Vert ij \\rangle$$|\n",
    "|$$~~~~~~~~~$$ | $$T_1 T_1~~~~~D_{6b}~~~~~t^a_k t^b_l \\langle kl\\Vert ij \\rangle~~~~~~~~~~$$ | $$T_1 T_2~~~~~T_{4b}~~~~~t^a_l t^{bc}_{mk} \\langle lm\\Vert ij \\rangle$$  |  $$T_2 T_2~~~~~Q_{5b}~~~~~t^{ab}_{im} t^{cd}_{nl} \\langle mn\\Vert jk \\rangle$$|"
   ]
  },
  {
   "attachments": {
    "f6.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "f347f4df",
   "metadata": {},
   "source": [
    "The diagrams $D_{2d}$ and $Q_{5b}$ are ordered\n",
    "\n",
    "![f6.png](attachment:f6.png)\n",
    "\n",
    "we can propose\n",
    "$$\\boldsymbol{t^{(+)}_{mn[[-)} \\scriptsize{\\langle mn\\Vert [[ \\rangle}}~~~~~a.10$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{(+)}_{n[[-)} \\scriptsize{\\langle mn\\Vert [[ \\rangle}}~~~~~a.11$$\n",
    "\n",
    "Examples\n",
    " __Diagram $Q_{2d}$__ this is __--__ contracted with __++++----__, assign __-$^m$-$^n$__ to the operator -> assign contiguous '+' labels to the amplitude __+$^a$+$^b$+$^c$+$^d$----__ -> assign contiguous '-' internal labels __+$^a$+$^b$+$^c$+$^d$-$^m$-$^n$--__ -> assign contiguous '-' external labels skipping the first two available __+$^a$+$^b$+$^c$+$^d$-$^m$-$^n$-$^k$-$^l$__ -> the bra is the internal labels $\\langle mn|$ -> the ket are the two skipped external '-' labels $|ij \\rangle$,\n",
    "\n",
    " __Diagram $Q_{5b}$__ this is __-|-__ contracted with __++--|++--__, assign __-$^m$-$^n$__ to the operator -> assign contiguous '+' external labels to the  amplitudes __+$^a$+$^b$--|+$^c$+$^d$--__ -> to the first amplitude assign contiguous '-' external labels except for last position __+$^a$+$^b$-$^i$-|+$^c$+$^d$--__ -> assign first '-' internal label to remaining first amplitude sign __+$^a$+$^b$-$^i$-$^m$|+$^c$+$^d$--__ -> assign second internal '-' label to second amplitude __+$^a$+$^b$-$^i$-$^m$|+$^c$+$^d$-$^n$-__ -> assign to remaining '-' external labels on second amplitude contiguous labels skipping the first two available __+$^a$+$^b$-$^i$-$^m$|+$^c$+$^d$-$^n$-$^l$__ -> the bra is the internal labels $\\langle mn|$ -> the ket is the two skipped labels $|jk \\rangle$.\n",
    "\n",
    "- - -\n",
    "__+- (phph)__ has an interaction number of zero and so the same as __hhhh__ and __pppp__. It connects with\n",
    "\n",
    "| __S__ | __D__ | __T__ | __Q__ |\n",
    "|:---------------|:--------------------------|:------------------|:-----------------|     \n",
    "|$$T_1~~~~~~~S_{3c}~~~~~t^c_k \\langle ak\\Vert ic \\rangle$$|$$T_2~~~~~~D_{2e}~~~~~t^{ac}_{ik} \\langle kb\\Vert cj \\rangle$$| $$T_3~~~~~~~T_{2e}~~~~~t^{dbc}_{ljk} \\langle al\\Vert id \\rangle$$  | $$T_4~~~~~~Q_{2e}~~~~~t^{ebcd}_{imkl} \\langle am\\Vert ej \\rangle$$ |\n",
    "|   |  $$T_1 T_1~~~~~D_{6c}~~~~~t^c_i t^a_k \\langle kb\\Vert cj \\rangle$$|$$T_1 T_2~~~~~T_{4c}~~~~~t^d_j t^{bc}_{lk} \\langle al\\Vert id \\rangle$$  | $$T_1 T_3~~~~~Q_{3c}~~~~~t^e_j t^{bcd}_{mkl} \\langle am\\Vert ie \\rangle $$  |\n",
    "|  |  |  $$T_1 T_2~~~~~T_{4d}~~~~~t^b_l t^{dc}_{jk} \\langle al\\Vert id \\rangle$$| $$T_1 T_3~~~~~Q_{3d}~~~~~t^b_m t^{ecd}_{jkl} \\langle am\\Vert ie \\rangle $$|\n",
    "| |  |   | $$T_2 T_2~~~~~Q_{5c}~~~~~t^{ae}_{ij} t^{bd}_{ml} \\langle mc\\Vert ek \\rangle $$|\n"
   ]
  },
  {
   "attachments": {
    "f7.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "22ab887c",
   "metadata": {},
   "source": [
    "Diagrams $S_{3c}$, $D_{2e}$, and $T_{2e}$ as \n",
    "\n",
    "![f7.png](attachment:f7.png)\n",
    "\n",
    "For the case where __phph__ contracts with only one amplitude we can propose $$\\boldsymbol{t^{e[+)}_{m[-)} \\scriptsize{\\langle [m\\Vert [e \\rangle}  }~~~~~a.12$$\n",
    "Examples, __Diagram $T_{2e}$__ this is __+-__ contracted to __+++---__, assign __+$^d$-$^l$__ to the operator -> assign the '+' internal label and contiguous '+' external labels skipping the first one __+$^d$+$^b$+$^c$---__-> assign the '-' internal label and contiguous '-' external labels again skipping the first available one __+$^d$+$^b$+$^c$-$^l$-$^j$-$^k$__-> the bra is the skipped external '+' label with the '-' internal label $\\langle al|$ -> the ket is the skipped '-' external label and the '+' internal label $|id \\rangle$.\n",
    "\n",
    "__Diagram $D_{2e}$__ this is __+-__ contracted with __++--__, assign __+$^c$-$^k$__ to the operator -> assign the '+' internal label and contiguous '+' external labels skipping the first one __+$^c$+$^b$--__ -> assign the '-' internal label and contiguous '-' external labels skipping the first one  __+$^c$+$^b$-$^k$-$^j$__ -> the bra is $\\langle ak|$ -> the ket is $|ic \\rangle$ _We note that the full expression from Shavitt & Bartlett is $\\hat{P}(ij|ab) t^{ac}_{ik} \\langle kb\\Vert cj \\rangle$ which we can write $\\hat{P}(ij|ab) t^{bc}_{jk} \\langle ka\\Vert ci \\rangle$ or  $\\hat{P}(ij|ab) t^{cb}_{kj} \\langle ka\\Vert ci \\rangle$ or $\\hat{P}(ij|ab) t^{cb}_{kj} \\langle ak\\Vert ic \\rangle$ which is the expression we get._"
   ]
  },
  {
   "attachments": {
    "f8.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "ecf8ae68",
   "metadata": {},
   "source": [
    "Diagrams $T_{4c}$ and $Q_{5c}$\n",
    "\n",
    "![f8.png](attachment:f8.png)\n",
    "\n",
    "We may propose $$\\boldsymbol{t^{(+)e}_{(-)} t^{[+)}_{m[-)}\\scriptsize{\\langle [^+m\\Vert [^-e \\rangle}}~~~~~a.13$$\n",
    "\n",
    "and $$\\boldsymbol{t^{(+)}_{(-)m} t^{e[+)}_{[-)}\\scriptsize{\\langle m[^+\\Vert [^-e \\rangle}}~~~~~a.14$$\n",
    "\n",
    "__Diagram $Q_{5c}$__: this is __+|-__ onto __++--|++--__ so assign to the operator __+$^e$-$^m$__ -> assign '+' external labels to the first amplitude except the last position which is assigned the internal '+' amplitude __+$^a$+$^e$--|++--__ -> assign contiguous '-' external labels to the first amplitude __+$^a$+$^e$-$^i$-$^j$|++--__ -> assign contiguous '+' external labels to the second amplitude skipping the first available label __+$^a$+$^e$-$^i$-$^j$|+$^c$+$^d$--__ -> assign internal '-' labels and contiguous '-' external labels skipping the first available __+$^a$+$^e$-$^i$-$^j$|+$^c$+$^d$-$^m$-$^l$__ -> the bra is $\\langle bm|$ -> the ket is $|ke \\rangle$.\n",
    "\n",
    "Examples: __Diagram $T_{4d}$__: this is __-|+__ onto __+-|++--__ so assign to the operator __+$^d$-$^l$__ -> assign contiguous '+' external labels to all positions in the first amplitude  __+$^a$-|++--__ -> assign contiguous '-' external labels to the first amplitude except for the last which is assigned the '-' internal label __+$^a$-$^l$|++--__ -> assign the internal '+' label to the first position and assign contiguous '+' external labels to the second amplitude skipping the first available label __+$^a$-$^l$|+$^d$+$^c$--__ -> assign contiguous '-' external labels skipping the first available __+$^a$-$^l$|+$^d$+$^c$-$^j$-$^k$__ -> the bra is constructed from the skipped '+' external label and the '-' internal label $\\langle bl|$ -> the ket is constructed from the skipped '-' external label and the '+' internal label $| id \\rangle$.\n",
    "the '+' internal label "
   ]
  },
  {
   "attachments": {
    "f9.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "4d3c565d",
   "metadata": {},
   "source": [
    "- - -\n",
    "__++-(phpp)__ this has an interaction number of -1 and can contract with up to three vertices. It's diagram is\n",
    "\n",
    "![f9.png](attachment:f9.png)\n",
    "\n",
    "| __S__ | __D__ | __T__ | __Q__ |\n",
    "|:---------------|:--------------------------|:------------------|:-----------------|     \n",
    "|$$T_2~~~~~S_{2b}~~~~~t^{cd}_{ik} \\langle ak\\Vert cd \\rangle$$ |$$T_1 T_2~~~~~D_{5c}~~~~~t^c_i t^{db}_{kj} \\langle ak\\Vert cd \\rangle$$ |$$T_1 T_3~~~~~T_{6c}~~~~~t^d_i t^{ebc}_{ljk} \\langle al\\Vert de \\rangle$$ |$$T_1 T_1 T_3~~~~~Q_{10a}~~~~~t^e_i t^f_j t^{bcd}_{mkl} \\langle am\\Vert ef \\rangle$$ |\n",
    "|$$T_1 T_1~~~~~S_{5b}~~~~~t^c_i t^d_k \\langle ak\\Vert cd \\rangle$$ |$$T_1^3~~~~~D_{8a}~~~~~t^c_i t^a_k t^d_j \\langle kb\\Vert cd \\rangle$$ |$$T_2 T_2~~~~~T_{3b}~~~~~t^{ad}_{il}t^{ec}_{jk}\\langle lb\\Vert de \\rangle$$ | |\n",
    "|  |   | $$T_2 T_2~~~~~T_{3d}~~~~~t^{ab}_{il} t^{de}_{jk} \\langle lc\\Vert de \\rangle$$| |\n",
    "\n",
    "The situation with 3 interaction lines is more complex. There is only 1 diagram in which __phpp__ can act on a single amplitude and that is $S_{2b}$. We propose $$\\boldsymbol{t^{ef}_{m(-)} \\scriptsize{\\langle ma\\Vert ef \\rangle}}~~~~~a.15$$\n",
    "\n",
    "Example __Diagram $S_{2b}$__: this is __++-__ contracted with __++--__, assign to to the operator labels __+$^c$+$^d$-$^k$__ -> assign '+' internal labels and '-' internal labels contiguously to the amplitude  __+$^c$+$^d$-$^k$-__ -> assign '-' external labels to the amplitude  __+$^c$+$^d$-$^k$-$^i$__-> the bra is $\\langle ka|$ -> the ket is $|cd \\rangle$,\n",
    "\n",
    "When acting on 2 amplitudes there are four possible combination __+-|+__, __+|+-__, __++|-__ and __-|++__ leading to\n",
    "\n",
    "$$\\boldsymbol{t^{(+)[+)^1 e}_{(-)m} t^{f(+)}_{(-)} \\scriptsize{\\langle m[\\Vert ef \\rangle}}~~~~~a.16$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{f[+)}_{m(-)} \\scriptsize{\\langle m[\\Vert ef \\rangle}}~~~~~a.17$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)ef}_{(-)} t^{[+)}_{m(-)} \\scriptsize{\\langle [m\\Vert ef \\rangle}}~~~~~a.18$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{ef[+)}_{(-)} \\scriptsize{\\langle m[\\Vert ef \\rangle}}~~~~~a.19$$\n",
    "\n",
    "Example __Diagram $T_{3b}$__ is __+-|+__ on a __++--|++--__, assign __+$^d$+$^e$-$^l$__ labels to the operator -> assign required contiguous '+' external labels to first amplitude __+$^a$+--|++--__ -> the $^{[+)^1}$ means we skip a label and add '+' string of length one, but it is not required in this case however the skipped label is noted -> assign the first '+' internal label to the last position of the first amplitude __+$^a$+$^d$--|++--__ -> assign '-' external labels in sequence to all but the last position which is reserved for the '-' internal label __+$^a$+$^d$-$^i$-$^l$|++--__ -> assign second '+' external label to second amplitude followed by contiguous '+' external labels (remember we have skipped one) __+$^a$+$^d$-$^i$-$^l$|+$^e$+$^c$--__ -> assign '-' external labels contiguously __+$^a$+$^d$-$^i$-$^l$|+$^e$+$^c$-$^j$-$^k$__ -> the bra is the '-' internal label and the skipped external '+' label $\\langle lb|$ -> the ket is the internal '+' labels $|de \\rangle$.\n",
    "\n",
    "Example __Diagram $T_{6c}$__ is __+|+-__ contracted with __+-|+++---__, assign labels to the operator __+$^d$+$^e$-$^l$__ -> in the first amplitude as the first internal'+' goes in the last place there are no external labels to assign __+$^d$-|+++---__ -> assign contiguous '-' external labels to the first amplitude __+$^d$-$^i$|+++---__ ->assign second internal '+' labels to the frst position of the second amplitude __+$^d$-$^i$|+$^e$++---__ followed by contiguous '+' external labels skipping the first available __+$^d$-$^i$|+$^e$+$^b$+$^c$---__ -> assign '-' internal label to second amplitude followed by contiguous '-' external labels __+$^d$-$^i$|+$^e$+$^b$+$^c$-$^l$-$^j$-$^k$__ -> the bra is $\\langle la$ -> the ket is $|de \\rangle$\n",
    "\n",
    "Example __Diagram $T_{3d}$__ this is __-|++__ onto ____++--|++--__, assign to the operator __-$^l$+$^d$+$^e$__ -> assign contiguous '+' external labels and contiguous '-' external labels except for the last '-' label in the first amplitude __+$^a$+$^b$-$^i$-$^l$|++--__ -> assign the '-' internal labels to the second amplitude followed by contiguous '+' external labels skipping one (there are none to assign in this case but we still skip one) __+$^a$+$^b$-$^i$-$^l$|+$^d$+$^e$-$^j$-$^k$__ -> the bra is $\\langle lc$ -> the ket is $|de \\rangle$.\n",
    "\n",
    "When __phpp__ contracts with 3 amplitudes it can only do so in three ways corresponding to __-|+|+__, __+|-|+__ or __+|+|-__. These situations correspond to \n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{e(+)}_{(-)} t^{f[+)}_{(-)} \\scriptsize{\\langle m[\\Vert ef \\rangle}}~~~~~a.20$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{(+)}_{m(-)} t^{f[+)}_{(-)} \\scriptsize{\\langle m[\\Vert ef \\rangle}}~~~~~a.21$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{[+)f}_{(-)} t^{(+)}_{m(-)} \\scriptsize{\\langle [m\\Vert ef \\rangle}}~~~~~a.22$$\n",
    "\n",
    "Example __$T_{3d}$__ this is \n",
    "For $D_{8a}$ all the amplitudes are of the same type and we can assign as rule 9 __+$^c$-|+$^d$-|+$^k$__ and again the __-__ externals as __+$^c$-$^i$|+$^d$-$^j$|+-$^k$__. With the additional rule that we ignore amplitude one, we assign __a__ to the bra as there was no assignment to the first counting amplitude, then __b__ goes to the amplitude __+$^c$-$^i$|+$^d$-$^j$|+$^b$-$^k$__ the assigned internal is on the second counting amplitude so we write $\\langle ak\\Vert cd \\rangle$ There is a permutation factor $\\hat{P}(ab)$ on this diagram so we may rewrite our expression as $t^c_i t^d_j t^a_k \\langle  kb\\Vert cd \\rangle$ in agreement with Shavitt & Bartlett.\n",
    "\n",
    "Example $Q_{10a}$ is __+|+|-__ on __+-|+-|+++---__ so we assign internal labels __+$^e$-|+$^f$-|+++-$^m$--__, assign __-__ external labels __+$^e$-$^i$|+$^f$-$^j$|+++-$^m$-$^k$-$^l$__ the first counting amplitude needs no external __+__ so __a__ goes to the bra and we label external __+__'s as __+$^e$-$^i$|+$^f$-$^j$|+$^b$+$^c$+$^d$-$^m$-$^k$-$^l$__, the ket is __ef__ and as the first internal __-__ label was assigned to the second counting amplitude we write $\\langle am\\Vert ef \\rangle$"
   ]
  },
  {
   "attachments": {
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     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "968701ae",
   "metadata": {},
   "source": [
    "- - -\n",
    "__+--(hhhp)__ has an interaction number of -1 and can contract with at most three vertices. It's diagram is\n",
    "\n",
    "![f10.png](attachment:f10.png)\n",
    "\n",
    "| __S__ | __D__ | __T__ | __Q__ |\n",
    "|:---------------|:--------------------------|:------------------|:-----------------|     \n",
    "| $$T_2~~~~~S_{2c}~~~~~t^{ac}_{kl} \\langle kl\\Vert ic \\rangle $$   | $$T_1 T_2~~~~~D_{5d}~~~~~t^a_k t^{cb}_{lj} \\langle kl\\Vert ic \\rangle$$ |$$ T_2^2~~~~~T_{3c}~~~~~t^{ad}_{il} t^{bc}_{mk} \\langle lm\\Vert  dj \\rangle$$ | $$T_2 T_3~~~~~Q_{6j}~~~~~t^{ab}_{mn} t^{ecd}_{ijk} \\langle mn\\Vert ie \\rangle$$|\n",
    "|  $$T_1^2~~~~~S_{5c}~~~~~t^a_k t^c_l \\langle kl\\Vert ic \\rangle$$  | $$T_1 T_2~~~~~D_{5f}~~~~~ t^c_i t^{ab}_{kl} \\langle kl\\Vert cj \\rangle$$ | $$T_2^2~~~~~T_{3e}~~~~~t^{ad}_{ij} t^{bc}_{lm} \\langle lm\\Vert dk \\rangle$$| |\n",
    "|    | $$ T_1 T_2~~~~~D_{5h}~~~~~t^c_k t^{ab}_{lj} \\langle kl\\Vert ci \\rangle$$ | $$T_1 T_3~~~~~T_{6f}~~~~~t^d_l t^{abc}_{mjk} \\langle lm\\Vert di \\rangle$$ |  |\n",
    "|   |  $$ T_1^3~~~~~~~~~D_{8b}~~~~~t^c_i t^a_k t^b_l \\langle kl\\Vert cj \\rangle$$ | $$T_1^2 T_2~~~~~T_{7b}~~~~~t^d_i t^a_l t^{bc}_{mk} \\langle lm\\Vert dj \\rangle$$  |  |\n",
    "\n",
    "As with __phpp__ this operator can only contract with a one amplitude in the singles ($S_{2c}$) as \n",
    "\n",
    "$$\\boldsymbol{t^{e(+)}_{mn} \\scriptsize{}\\langle mn\\Vert ei \\rangle}~~~~~a.23$$\n",
    "\n",
    "Example __diagram $S_{2c}$__ this is __--+__ contracted with __++--__, assigning internal labels to the operator __-$^k$-$^l$+$^c$__ -> assign the '+' internal labels to the amplitude with contiguous '+' external labels as required __+$^c$+$^a$--___ -> assign the '-' internal labels to the amplitude __+$^c$+$^a$-$^k$-$^l$___ -> the bra is the '-' internal amplitudes $\\langle kl|$ -> the ket is the internal '+' label and the continguous '-' external label $|ci \\rangle$.\n",
    "\n",
    "When acting on two amplitudes there are four possible combinations __+-|-__, __-|+-__, __+|--__ and __--|+__ summarised as\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)m} t^{(+)}_{n[-)} \\scriptsize{\\langle mn\\Vert [e \\rangle}}~~~~~a.24$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{e(+)}_{n[-)} \\scriptsize{\\langle mn\\Vert e[ \\rangle}}~~~~~a.25$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{(+)}_{mn[-)} \\scriptsize{\\langle mn\\Vert e[ \\rangle}}~~~~~a.26$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)mn} t^{e(+)}_{[-)} \\scriptsize{\\langle mn\\Vert [e \\rangle}}~~~~~a.27$$\n",
    "\n",
    "Example __diagram $T_{3c}$__ is a __+-|-__ into __++--|++--__. We assign internal labels __+$^d$-$^l$-$^m$__ to the operator -> assign '+' external labels in order to the first amplitude as required and end with the '+' internal label __+$^a$+$^d$--|++--__ -> assign contiguous '-' external labels as required to the first amplitude assigning the first '-' internal label to the last position __$^a$+$^d$-$^i$-$^l$|++--__ -> assign '+' external labels to the second amplitude -> __$^a$+$^d$-$^i$-$^l$|+$^b$+$^c$--__ -> assign the second '-' internal label to the first position and the '-' external labels skipping the first available one __$^a$+$^d$-$^i$-$^l$|+$^b$+$^c$-$^m$-$^k$__ -> the bra is the '-' internal labels $\\langle lm|$ -> the ket is the skipped '-' external label and the '+' internal label $|jd \\rangle$, On the face of it there is a sign difference between this expression an Shavitt & Bartlett, however this is because of slight difference in the diagrams resulting in a '-' sign for S&B and a '+' sign for us. In detail the diagrams are shown below with S&B on the left having one more loop than the equivalent diagram on the right which we use\n"
   ]
  },
  {
   "attachments": {
    "f11.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "cbfd77f5",
   "metadata": {},
   "source": [
    "![f11.png](attachment:f11.png)\n",
    "\n",
    "Example __diagram $T_{6d}$__, this is __-|+-__ contracted with __+-|+++---__, assign to the operator labels __-$^l$-$^m$+$^d$__ -> assign contiguous '+' external labels to the first amplitude and '-' external labels except for the last position which is the first '-' internal label __+$^a$-$^l$|+++---__ -> assign the '+' internal label to the second amplitude followed by contigous '+' external labels __+$^a$-$^l$|+$^d$+$^b$+$^c$---__ -> assign the second '-' followed by contiguous '-' external labels skipping one __+$^a$-$^l$|+$^d$+$^b$+$^c$-$^m$-$^j$-$^k$__  -> the bra is the internal '-' labels $\\langle lm|$ -> the ket is the internal '+' label and the skipped external '-' label $|di \\rangle$.\n",
    "\n",
    "Example __diagram $Q_{6c}$__,this is __+|--__ contracted with __++--|+++---__, assign to the operator labels __-$^m$-$^n$+$^e$__ -> assign contiguous '+' external labels to the first amplitude except the last label which is assigned the '+' internal label __+$^a$+$^e$--|+++---__ -> assign '-' external labels to the first amplitude __+$^a$+$^e$-$^i$-$^j$|+++---__ -> assign contiguous '+' external labels to the second amplitude __+$^a$+$^e$-$^i$-$^j$|+$^b$+$^c$+$^d$---__ -> assign the two '-' internal labels and then contiguous '-' external labels skipping the first available label __+$^a$+$^e$-$^i$-$^j$|+$^b$+$^c$+$^d$-$^m$-$^n$-$^l$__ -> the bra is the internal '-' labels $\\langle mn|$ -> the ket is the internal '+' label and the skipped external -'-' label $|ek \\rangle$.\n",
    "\n",
    "When __hhhp__ contracts with three amplitudes it can do so as __+|-|-__, __-|+|-__ and __-|-|+__. These correspond to\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{(+)}_{m(-)} t^{(+)}_{n[-)} \\scriptsize{ \\langle mn\\Vert e[ \\rangle}}~~~~~a.28$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{(+)e}_{[-)} t^{(+)}_{n(-)} \\scriptsize{ \\langle mn\\Vert [e \\rangle}}~~~~~a.29$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{(+)}_{[-)n} t^{e(+)}_{(-)} \\scriptsize{ \\langle mn\\Vert [e \\rangle}}~~~~~a.30$$\n",
    "\n",
    "Example $T_{7b}$ is a __+|-|-__ into __+-|+-|++--__. We assign internal labels as __+$^d$-$^l$-$^m$-__ -> assign '+' external labels in order to the first amplitude except for the last '+' sign which is assiged the '+' internal label __+$^d$-|+-|++--__ -> assign external '-' labels to first amplitude __+$^d$-$^i$|+-|++--__ -> to the second amplitude assign contiguous '+' external labels -> __+$^d$-$^i$|+$^a$-|++--__ -> to the second amplitude assign the first '-' internal label followed by contiguous '-' external labels __+$^d$-$^i$|+$^a$-$^l$|++--__ -> to the third amplitude assign contiguous '+' external labels  __+$^d$-$^i$|+$^a$-$^l$|+$^b$+$^c$--__ -> assign the second '-' internal label followed by contiguous '-' external labels skipping the first available label __+$^d$-$^i$|+$^a$-$^l$|+$^b$+$^c$-$^m$-$^k$__ -> the bra is the internal '-' labels $\\langle lm|$ -> the ket is the internal '+' label followed by the skipped external '-' label $|dj \\rangle$"
   ]
  },
  {
   "attachments": {
    "f13.png": {
     "image/png": "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"
    }
   },
   "cell_type": "markdown",
   "id": "6f96757d",
   "metadata": {},
   "source": [
    "- - -\n",
    "(0)__pphh__ has an interaction number of 0 and does not contract with the amplitudes,\n",
    "\n",
    "- - -\n",
    "__++--(hhpp)__ has an interaction number of -2 and can contract with up to four amplitudes. It's diagram is\n",
    "\n",
    "![f13.png](attachment:f13.png)\n",
    "\n",
    "With an interaction number of -2 it cannot contract with a single amplitude until the triples.\n",
    "\n",
    "When __hhpp__ contracts with two amplitudes eg\n",
    "\n",
    "| __S__ | __D__ | __T__ | __Q__ |\n",
    "|:---------------|:--------------------------|:------------------|:-----------------|     \n",
    "|$$T_1 T_2~~~~~S_{4a}~~~~~t^c_i t^{ad}_{kl} \\langle kl\\Vert cd \\rangle$$|$$T_2^2~~~~~D_{3a}~~~~~t^{cd}_{ij} t^{ab}_{kl} \\langle kl\\Vert cd \\rangle$$ |$$T_2 T_3~~~~~~T_{5f}~~~~~t^{de}_{ij} t^{abc}_{lmk} \\langle lm\\Vert de \\rangle$$ |$$T_3^2~~~~~Q_{8b}~~~~~t^{aej}_{ijk} t^{bcd}_{mnl} \\langle mn\\Vert ef \\rangle$$ |\n",
    "|$$T_1 T_2~~~~~S_{4b}~~~~~t^a_k t^{cd}_{il} \\langle kl\\Vert cd \\rangle$$|$$T_2^2~~~~~D_{3b}~~~~~t^{ac}_{ik} t^{bd}_{jl} \\langle kl\\Vert cd \\rangle$$  |$$T_2 T_3~~~~~~T_{5g}~~~~~t^{ab}_{lm} t^{dec}_{ijk} \\langle lm\\Vert de \\rangle$$ |$$T_3^2~~~~~Q_{8c}~~~~~t^{aef}_{imj} t^{bcd}_{nkl} \\langle mn\\Vert ef \\rangle$$ |\n",
    "|$$T_1 T_2~~~~~S_{4c}~~~~~t^c_k t^{da}_{li} \\langle kl\\Vert cd \\rangle$$|$$T_2^2~~~~~D_{3c}~~~~~t^{dc}_{ik} t^{ab}_{lj} \\langle kl\\Vert cd \\rangle$$  | |$$T_3^2~~~~~Q_{8d}~~~~~t^{abe}_{imn} t^{fcd}_{jkl} \\langle mn\\Vert ef \\rangle$$ |\n",
    "\n",
    "There are seven possible combinations __+|+--__, __+--|+__, __++|--__, __--|++__, __-|++-__, __++-|-__ and __+-|+-__ corresponding to\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{f(+)}_{mn(-)} \\scriptsize{\\langle nm\\Vert ef \\rangle}}~~~~~a.31$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)mn} t^{f(+)}_{(-)} \\scriptsize{\\langle nm\\Vert ef \\rangle}}~~~~~a.32$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)ef}_{(-)} t^{(+)}_{mn(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.33$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)mn} t^{ef(+)}_{(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.34$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{ef(+)}_{n(-)} \\scriptsize{\\langle nm\\Vert ef \\rangle}}~~~~~a.35$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)ef}_{(-)n} t^{(+)}_{m(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.36$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)m} t^{f(+)}_{n(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.37$$\n",
    "\n",
    "Example __diagram $D_{3c}$__ this is __++-|-__ contracted with __++--|++--__, we assign internal labels to the Hamiltonian s-string __+$^c$+$^d$-$^k$|-$^l$__ -> from a.35 to the first amplitude assign __+$^c$+$^d$-$^i$-$^k$__ -> to the second amplitude assign __+$^a$+$^b$-$^l$-$^j$__ -> the bra is $\\langle lk|$ -> the ket is $|cd \\rangle$ \n",
    "\n",
    "__hhpp__ can contract with three amplitudes as for example\n",
    "\n",
    "| __S__ | __D__ | __T__ | __Q__ |\n",
    "|:---------------|:--------------------------|:------------------|:-----------------|     \n",
    "|$$T_1^3~~~~~S_{6}~~~~~t^c_i t^a_k t^d_l \\langle kl\\Vert cd \\rangle$$| $$T_1^2 T_2~~~~~D_{7a}~~~~~t^c_i t^d_j t^{ab}_{kl} \\langle kl\\Vert cd \\rangle$$|$$T_1 T_2 T_3~~~~~T_{9c}~~~~~t^d_i t^e_k t^{abc}_{ljm} \\langle lm\\Vert de \\rangle$$|$$T_1 T_2 T_3~~~~~Q_{10c}~~~~~t^e_i t^{ab}_{mn} t^{fcd}_{jkl} \\langle mn\\Vert ef \\rangle$$  |\n",
    "| | $$T_1^2 T_2~~~~~D_{7c}~~~~~t^c_i t^a_k t^{db}_{lk} \\langle kl\\Vert cd \\rangle$$|$$T_1 T_2^2~~~~~T_{8a}~~~~~t^d_l t^{ab}_{im} t^{ec}_{jk} \\langle lm\\Vert de \\rangle$$ | $$T_1^2 T_4~~~~~Q_{11a}~~~~~t^e_m t^f_i t^{abcd}_{njkl} \\langle mn\\Vert ef \\rangle$$ |\n",
    "| | | |$$T_1 T_2 T_3~~~~~Q_{13f}~~~~~t^b_n t^{ae}_{im} t^{fcd}_{jkl}\\langle mn\\Vert ef \\rangle$$|\n",
    "\n",
    "There are 12 possible combinations __+|+|--__, __+|--|+__, __--|+|+__, __-|-|++__, __-|++|-__, __++|-|-__, __+|-|+-__, __-|+|+-__, __+|+-|-__, __-|+-|+__, __+-|+|-__ and __+-|-|+__. Note that __++|...__ and __--|...__ , since our amplitudes are ordered with the lowest order to the left, can start with at least $T_2$ so the minimum interaction number of the term is 3x$T_2$=6. As the Hamiltonian has an interaction number of -2 this means these contractions can only appear at the Q-level and above. \n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{f(+)}_{(-)} t^{(+)}_{mn(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.38$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{(+)}_{mn(-)} t^{f(+)}_{(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.39$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)mn} t^{(+)e}_{(-)} t^{f(+)}_{(-)} \\scriptsize{\\langle mn \\Vert ef \\rangle}}~~~~~~a.40$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{(+)}_{n(-)} t^{ef(+)}_{(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.41$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{ef(+)}_{(-)} t^{(+)}_{n(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.42$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)ef}_{(-)} t^{(+)}_{(-)m} t^{(+)}_{n(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.43$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{(+)}_{m(-)} t^{f(+)}_{n(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.44$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{e(+)}_{(-)} t^{f(+)}_{n(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.45$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{f(+)}_{m(-)} t^{(+)}_{n(-)} \\scriptsize{\\langle nm\\Vert ef \\rangle}}~~~~~a.46$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{e(+)}_{(-)n} t^{f(+)}_{(-)} \\scriptsize{\\langle nm\\Vert ef \\rangle}}~~~~~a.47$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)m} t^{(+)f}_{(-)} t^{(+)}_{n(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.48$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)m} t^{(+)}_{(-)n} t^{f(+)}_{(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.49$$\n",
    "\n",
    "Example __diagram Q_{13f}__ this is __-|+-|+__ contracted with __+-|++--|+++---__, assign internal labels to the operator __-$^m$|+$^e$-$^n$|+$^f$__ -> this is a.46 so the first amplitude is __+$^a$-$^m$__ -> the second amplitude is __+$^e$+$^b$-$^i$-$^n$__ -> and the third amplitude is __+$^f$+$^c$+$^d$-$^j$-$^k$-$^l$__ -> the bra is $\\langle nm|$ -> the ket is $|ef \\rangle$. The final result is $t^a_m t^{eb}_{in} t^{fcd}_{jkl} \\langle nm\\Vert ef \\rangle$. The permutations are i/jkl and a/b/cd. We can show the result obtained is equivalent to Shavitt & Bartlett labelling. Interchange a<>b gives $-t^b_m t^{ea}_{in} t^{fcd}_{jkl} \\langle nm\\Vert ef \\rangle$, interchange m<>n $-t^b_n t^{ea}_{im} t^{fcd}_{jkl} \\langle mn\\Vert ef \\rangle$, interchanging superscript on $T_2$ amplitude gives $t^b_n t^{ae}_{im} t^{fcd}_{jkl} \\langle mn\\Vert ef \\rangle$ which is the Shavitt and Bartlett labelling. We introduce a minus sign in a.45 and a.46 by swapping the bra labels, this is because we draw the diagrams as a 'w'-type of __+-__ where as Shavitt & Bartlett use a 'o'-type of representation - this leads to one less loop and so a change of sign.\n",
    "\n",
    "\n",
    "With four amplitudes __+|+|-|-__, __+|-|-|+__, __+|-|+|-__, __-|-|+|+__, __-|+|-|+__ and __-|+|+|-__.\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{(+)f}_{(-)} t^{(+)}_{(-)m} t^{(+)}_{n(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.50$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{(+)}_{m(-)} t^{(+)}_{(-)n} t^{f(+)}_{(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.51$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)e}_{(-)} t^{(+)}_{m(-)} t^{(+)f}_{(-)} t^{(+)}_{n(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.52$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{(+)}_{(-)n} t^{(+)e}_{(-)} t^{f(+)}_{(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.53$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{e(+)}_{(-)} t^{(+)}_{(-)n} t^{f(+)}_{(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.54$$\n",
    "\n",
    "$$\\boldsymbol{t^{(+)}_{(-)m} t^{e(+)}_{(-)} t^{(+)f}_{(-)} t^{(+)}_{n(-)} \\scriptsize{\\langle mn\\Vert ef \\rangle}}~~~~~a.55$$\n"
   ]
  },
  {
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"
    }
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   "id": "ecf8122e",
   "metadata": {},
   "source": [
    "__SUMMARY__\\\n",
    "The steps in determining coupled-cluster amplitudes are then\n",
    "+ Determine the exponential expansion for the required level of coupled-cluster theory in terms of $T_n$ operators.\n",
    "+ For each term in the expansion write down it's s-string, determine the interaction number and so get the possible Hamiltonians that can contract with the term.\n",
    "+ Determine all possible permutations of the Hamiltonian s-string that are contractable with the terms s-string.\n",
    "+ For each contractable s-string and amplitude s-string pair infer the sign and factor of the diagram that the pair represent.\n",
    "+ Using the appropriate label algorithm assign labels to the contracted s-string.\n",
    "+ Determine the permutation of external labels\n",
    "\n",
    "As an example consider $T_2 T_4$ contracted with __phpp__ as __++|-__ at the P level. \n",
    "+ the s-strings are __++--|++++----__ and __++|-__, the interaction number of the term is 2+4=6 and __phpp__ has an interaction number of -1. Giving a total interaction number of 5 which as we're at P level is correct.\n",
    "+ The number of __-__ signs in the operator s-string is 1 and the level of expansion is 5 giving a sum of 6, there are no __+-__ pairs in the operator s-string and again the level of expansion is 5. The overall sum is 6+5 so the sign is $(-1)^{11} = -1$\n",
    "+ there is a __++__ in the first amplitude of the operator string contributing a factor of $\\frac{1}{2}$. There are no __|+|__ or __|-|__ pairs in the operator amplitude so no equivalent vertices factor.\n",
    "+ The label algorithm is a.17 or $t^{(+)ef}_{(-)} t^{[+)}_{m(-)} \\langle [m\\Vert ef \\rangle$ giving $t^{fg}_{ij} t^{bcde}_{nklm} \\langle an\\Vert fg \\rangle$\n",
    "+ the permutations are $\\hat{P}(ij/k) \\hat{P}(a/bc)$\n",
    "+ we can then write $\\boldsymbol{-\\frac{1}{2} \\hat{P}(ij/k) \\hat{P}(a/bc) t^{fg}_{ij} t^{bcde}_{nklm} \\langle an\\Vert fg \\rangle}$\n",
    "\n",
    "The visual representation of the diagram is \n",
    "\n",
    "![f14.png](attachment:f14.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e02c6366",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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