Nesting multiple equations

Question

I want to be able to follow a derivation cleanly from eqn 1 to eqn 2(a or b) to eqn 3a(a or b) to eqn 4aa(a or b), etc. if it makes sense. I am currently using the built-in sub-equation environment to try to do this, but it only has a default counter of 3 and that is not enough for me, as eqn 3 keeps getting Repeated in my long derivation.

GPT suggested that I make a custom env with an unlimited counter that mirrors subeqn in function, but it hasn’t been working for me. I’m wondering if the sub-equation environment is really what I’m looking for, or if there is a better way that I don’t know of that I should be considering.

\usepackage{physics}
\usepackage{breqn}
\title{FCI Questions}
\author{Patryk Kozlowski}
\date{\today} %% Change "\today" by another date manually
\begin{document}
\maketitle
\section{0 differences between two determines}
\begin{equation}
    \mel{\Psi }{V}{\Psi }
    =v^{\alpha\beta\gamma\delta}\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
        a^{\dag}_{\alpha  }a^{\dag}_{\beta }a_{\gamma }a_{\delta }
    \left(\prod_{\kappa^{\prime}=\left(\kappa_{1}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
\end{equation}
\begin{subequations}
    \begin{align}
        =\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
            a^{\dag}_{\alpha }a^{\dag}_{\beta }a_{\gamma }\delta _{\delta \kappa _{1}}
        \left(\prod_{\kappa^{\prime}=\left(\kappa_{2}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
    \end{align}
    \begin{align}
        -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
            a^{\dag}_{\alpha }a^{\dag}_{\beta }a_{\gamma }a^{\dag}_{\kappa _{1}}a_{\delta }
        \left(\prod_{\kappa^{\prime}=\left(\kappa_{2}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
    \end{align}
\end{subequations}
\begin{subequations}
    \begin{subequations}
        \begin{align}
            =\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
                a^{\dag}_{\alpha }a^{\dag}_{\beta }\delta _{\delta \kappa _{1}}\delta _{\gamma \kappa _{2}}
            \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
        \end{align}
        \begin{align}
            -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
                a^{\dag}_{\alpha }a^{\dag}_{\beta }\delta _{\delta_{\kappa _{1}}}a^{\dag}_{\kappa _{2}}a_{\gamma }
            \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
        \end{align}
    \end{subequations}
    \begin{subequations}
        \begin{align}
            -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
                a^{\dag}_{'}a^{\dag}_{\beta }\delta _{\gamma \kappa _{1}}a_{\delta }
            \left(\prod_{\kappa^{\prime}=\left(\kappa_{2}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
        \end{align}
        \begin{align}
            +\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
                a^{\dag}_{\alpha }a^{\dag}_{\beta }a^{\dag}_{\kappa _{1}}a_{\gamma }a_{\delta }
            \left(\prod_{\kappa^{\prime}=\left(\kappa_{2}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
        \end{align}
    \end{subequations}
\end{subequations}
\begin{subequations}
    \begin{subequations}
        \begin{align}
            =\delta _{\alpha \kappa _{1}}\delta _{\beta \kappa _{2}}\delta _{\gamma \kappa _{2}}\delta _{\delta \kappa _{1}}
        \end{align}
        \begin{align}
            -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                a_{\kappa _{2}}a_{\kappa _{1}}a^{\dag}_{\kappa _{2}}a^{\dag}_{\alpha }a^{\dag}_{\beta }\delta _{\delta \kappa _{1}}a_{\gamma }
            \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
        \end{align}
    \end{subequations}
    \begin{subequations}
        \begin{align}
            -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
                \delta _{\gamma \kappa _{1}}a^{\dag}_{\alpha }a^{\dag}_{\beta }\delta _{\delta \kappa _{2}}
            \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
        \end{align}
        \begin{align}
            +\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
                \delta _{\gamma \kappa _{1}}a^{\dag}_{\alpha }a^{\dag}_{\beta }a^{\dag}_{\kappa _{2}}a_{\delta }
            \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
        \end{align}
    \end{subequations}
\end{subequations}
\begin{subequations}
    \begin{subequations}
        \begin{align}
            =\delta _{\alpha \kappa _{1}}\delta _{\beta \kappa _{2}}\delta _{\gamma \kappa _{2}}\delta _{\delta \kappa _{1}}
        \end{align}
        \begin{subequations}
            \begin{align}
                +\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                    \delta _{\delta \kappa _{1}}\delta _{\alpha \kappa _{1}}a^{\dag}_{\beta }a_{\gamma }
                \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
            \end{align}
            \begin{align}
                -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                    \delta _{\delta \kappa _{1}}a^{\dag}_{\alpha }a_{\kappa _{1}}a^{\dag}_{\beta }a_{\gamma }
                \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
            \end{align}
        \end{subequations}
    \end{subequations}
    \begin{subequations}
        \begin{align}
            -\delta _{\alpha \kappa _{1}}\delta _{\beta \kappa _{2}}\delta _{\gamma\kappa _{1}}\delta _{\delta \kappa _{2}} 
        \end{align}
        \begin{align}
            -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                \delta _{\gamma \kappa _{1}}a_{\kappa _{1}}a^{\dag}_{\alpha }a^{\dag}_{\beta }a_{\delta }
            \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
        \end{align}
    \end{subequations}
\end{subequations}
\begin{subequations}
    \begin{subequations}
        \begin{align}
            =\delta _{\alpha \kappa _{1}}\delta _{\beta \kappa _{2}}\delta _{\gamma \kappa _{2}}\delta _{\delta \kappa _{1}}
        \end{align}
        \begin{subequations}
            \begin{align}
                +\delta _{\delta \kappa _{1}}\delta _{\alpha \kappa _{1}}\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                    a^{\dag}_{\beta }a_{\gamma }
                \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
            \end{align}
            \begin{subequations}
                \begin{align} 
                    -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                        \delta _{\delta \kappa _{1}}a^{\dag}_{\alpha }
                        \delta _{\beta \kappa _{1}}
                        a_{\gamma }
                    \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
                \end{align}
                \begin{align}
                    +\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                        \delta _{\delta  \kappa _{1}}a^{\dag}_{\alpha }a_{\kappa _{1}}a^{\dag}_{\beta }a_{\gamma }
                    \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
                \end{align}
                
            \end{subequations}
        \end{subequations}
    \end{subequations}
    \begin{subequations}
        \begin{align}
            -\delta _{\alpha \kappa _{1}}\delta _{\beta \kappa _{2}}\delta _{\gamma\kappa _{1}}\delta _{\delta \kappa _{2}} 
        \end{align}
        \begin{subequations}
            \begin{align}
                -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                    \delta _{\gamma  \kappa _{1}}\delta _{\alpha  \kappa _{1}}a^{\dag}_{\beta }a_{\gamma }
                \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
            \end{align}
            \begin{align}
                +\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                    \delta _{\gamma  \kappa _{1}}a^{\dag}_{\alpha }a_{\kappa _{1}}a^{\dag}_{\beta }a_{\gamma }
                \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
            \end{align}
        \end{subequations}
    \end{subequations}
\end{subequations}
\begin{subequations}
    \begin{subequations}
        \begin{align}
            =\delta _{\alpha  \kappa _{1}}\delta _{\beta  \kappa _{2}}\delta _{\gamma  \kappa _{2}}\delta _{\delta  \kappa _{1}}
        \end{align}
        \begin{subequations}
            \begin{align}
                +\delta _{\alpha  \kappa _{1}}\delta _{\delta  \kappa _{1}}\mel{\Psi }{a^{\dag}_{\kappa _{1}}a^{\dag}_{\kappa _{2}}H_{0}a_{\kappa _{2}}a_{\kappa _{1}}}{\Psi }
            \end{align}
            \begin{subequations}
                \begin{align}
                    -\delta _{\beta  \kappa _{1}}\delta _{\delta  \kappa _{1}}\mel{\Psi }{a^{\dag}_{\kappa _{1}}a^{\dag}_{
                        _{2}
                    }H_{0}a_{
                        _{2}
                    }a_{1}
                    a_{
                        \kappa _{2}
                    }
                    }{\Psi }
                \end{align}
                \begin{subequations}
                    \begin{align}
                        +\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                            \delta _{\delta  \kappa _{1}}a^{\dag}_{\alpha }\delta _{\beta  \kappa _{1}}a_{\gamma }
                        \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
                    \end{align}
                    \begin{align}
                        -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                            \delta _{\delta  \kappa _{1}}a^{\dag}_{\alpha }a^{\dag}_{\beta }a_{\kappa _{1}}a_{\gamma }
                        \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
                    \end{align}
                \end{subequations}
            \end{subequations}
        \end{subequations}
    \end{subequations}
    \begin{align}
        -\delta _{\alpha  \kappa _{1}}\delta _{\beta  \kappa _{2}}\delta _{\gamma  \kappa _{1}}\delta _{\delta  \kappa _{2}}
    \end{align}
    \begin{subequations}
        \begin{align}
            -\delta _{\gamma  \kappa _{1}}\delta _{\alpha  \kappa _{1}}\mel{\Psi}{a^{\dag}_{\kappa _{1}}a^{\dag}_{\kappa _{2}}H_{0}a_{\kappa _{2}}a_{\kappa _{1}}}{\Psi }
        \end{align}
        \begin{subequations}
            \begin{align}
                +\delta _{\gamma  \kappa _{1}}\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{3}\right)}a_{\kappa}\right)
                    a^{\dag}_{\alpha }\delta _{\beta  \kappa _{1}}a_{\gamma }
                \left(\prod_{\kappa^{\prime}=\left(\kappa_{}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
            \end{align}
            \begin{align}
                -0
            \end{align}
        \end{subequations}
    \end{subequations}
\end{subequations}
\begin{subequations}
    \begin{subequations}
        \begin{align}
            =\delta _{\alpha  \kappa _{1}}\delta _{\beta  \kappa _{2}}\delta _{\gamma  \kappa _{2}}\delta _{\delta  \kappa _{1}}
        \end{align}
        \begin{subequations}
            \begin{align}
                +\delta _{\alpha  \kappa _{1}}\delta _{\delta  \kappa _{1}}\mel{\Psi }{a^{\dag}_{\kappa _{1}}a^{\dag}_{\kappa _{2}}H_{0}a_{\kappa _{2}}a_{\kappa _{1}}}{\Psi }
            \end{align}
            \begin{subequations}
                \begin{align}
                    -\delta _{\beta  \kappa _{1}}\delta _{\delta  \kappa _{1}}\mel{\Psi }{a^{\dag}_{\kappa _{1}}a^{\dag}_{
                        _{2}
                    }H_{0}a_{
                        _{2}
                    }a_{1}
                    a_{
                        \kappa _{2}
                    }
                    }{\Psi }
                \end{align}
                \begin{subequations}
                    \begin{align}
                        +\delta _{\beta  \kappa _{1}}\delta _{\delta  \kappa _{1}}\mel{\Psi }{a^{\dag}_{\kappa _{1}}a^{\dag}_{\kappa _{2}}H_{0}a_{\kappa _{2}}a_{\kappa _{1}}}{\Psi }
                    \end{align}
                    \begin{align}
                        -0
                    \end{align}
                \end{subequations}
            \end{subequations}
        \end{subequations}
    \end{subequations}
    \begin{align}
        -\delta _{\alpha  \kappa _{1}}\delta _{\beta  \kappa _{2}}\delta _{\gamma  \kappa _{1}}\delta _{\delta  \kappa _{2}}
    \end{align}
    \begin{subequations}
        \begin{align}
            -\delta _{\gamma  \kappa _{1}}\delta _{\alpha  \kappa _{1}}\mel{\Psi}{a^{\dag}_{\kappa _{1}}a^{\dag}_{\kappa _{2}}H_{0}a_{\kappa _{2}}a_{\kappa _{1}}}{\Psi }
        \end{align}
        \begin{subequations}
            \begin{align}
                +\delta _{\gamma  \kappa _{1}}\delta _{\beta  \kappa _{1}}\mel{\Psi}{a^{\dag}_{\kappa _{1}a^{\dag}_{\kappa _{2}}}H_{0}  a_{\kappa _{2}}a_{\kappa _{1}}}{\Psi }
            \end{align}
            \begin{align}
                -0
            \end{align}
        \end{subequations}
    \end{subequations}
\end{subequations}
\begin{subequations}
    \begin{subequations}
        \begin{align}
            =\delta _{\alpha  \kappa _{1}}\delta _{\beta  \kappa _{2}}\delta _{\gamma  \kappa _{2}}\delta _{\delta  \kappa _{1}}
        \end{align}
        \begin{align}
        -\delta _{\alpha  \kappa _{1}}\delta _{\beta  \kappa _{2}}\delta _{\gamma  \kappa _{1}}\delta _{\delta  \kappa _{2}}
    \end{align}
    \end{subequations}
    \begin{subequations}
        \begin{align}
                +\delta _{\alpha  \kappa _{1}}\delta _{\delta  \kappa _{1}}\mel{\Psi }{a^{\dag}_{\kappa _{1}}a^{\dag}_{\kappa _{2}}H_{0}a_{\kappa _{2}}a_{\kappa _{1}}}{\Psi }
            \end{align}
        \begin{align}
                +\delta _{\gamma  \kappa _{1}}\delta _{\beta  \kappa _{1}}\mel{\Psi}{a^{\dag}_{\kappa _{1}a^{\dag}_{\kappa _{2}}}H_{0}  a_{\kappa _{2}}a_{\kappa _{1}}}{\Psi }
            \end{align}
        \begin{align}
            -\delta _{\gamma  \kappa _{1}}\delta _{\alpha  \kappa _{1}}\mel{\Psi}{a^{\dag}_{\kappa _{1}}a^{\dag}_{\kappa _{2}}H_{0}a_{\kappa _{2}}a_{\kappa _{1}}}{\Psi }
        \end{align}
    \end{subequations}
\end{subequations}
\text{adding the integrals back in.}
\begin{subequations}
    \begin{subequations}
        \begin{align}
            =v^{\kappa _{1}\kappa _{2}\kappa _{2}\kappa _{1}}
        \end{align}
        \begin{align}
            -v^{\kappa _{1}\kappa _{2}\kappa _{1}\kappa _{2}}
        \end{align}
    \end{subequations}
    \begin{subequations}
        \begin{align}
            +v^{\kappa _{1}\beta \gamma \kappa _{1}}
        \end{align}
        \begin{align}
            +v^{\alpha \kappa _{1}\kappa _{1}\delta }
        \end{align}
        \begin{align}
            -v^{\kappa _{1}\beta \kappa _{1}\delta }
        \end{align}
    \end{subequations}
\end{subequations}
\text{equation 3 part b similar to the Condon roles. not sure were to go with this, or with part a}
\section{math_drafts.pdf}
I'm confused about the steps you took to get from
\begin{equation}
    \sum_{\kappa }h^{\kappa \kappa }  
\end{equation}
\begin{equation}
    \sum_{\kappa } h^{(\kappa )(\kappa )}\delta _{[\kappa ],[\kappa ]}
\end{equation}
\end{document}

Answer 1

One possible way is to use align and keep it simple. Taking the first few lines of formulas presented in the lengthy example then:

\documentclass[12pt]{article}

\usepackage{amsmath}
\usepackage{physics}

\begin{document}

\section{differences between two determines}
\begin{align}
\mel{\Psi }{V}{\Psi } &= v^{\alpha\beta\gamma\delta}\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
        a^{\dag}_{\alpha  }a^{\dag}_{\beta }a_{\gamma }a_{\delta }
    \left(\prod_{\kappa^{\prime}=\left(\kappa_{1}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0} \\
&= \bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right) a^{\dag}_{\alpha }a^{\dag}_{\beta }a_{\gamma }\delta _{\delta \kappa _{1}} \left(\prod_{\kappa^{\prime}= \left(\kappa_{2}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0} \\
& \hspace{5mm}  -\bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
            a^{\dag}_{\alpha }a^{\dag}_{\beta }a_{\gamma }a^{\dag}_{\kappa _{1}}a_{\delta }
        \left(\prod_{\kappa^{\prime}=\left(\kappa_{2}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0} \\
&= \bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
                a^{\dag}_{\alpha }a^{\dag}_{\beta }\delta _{\delta \kappa _{1}}\delta _{\gamma \kappa _{2}}
            \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0} \\
& \hspace{5mm} - \bra{0}\left(\prod_{\kappa=\left(\kappa_{n}\dots\kappa_{1}\right)}a_{\kappa}\right)
                a^{\dag}_{\alpha }a^{\dag}_{\beta }\delta _{\delta_{\kappa _{1}}}a^{\dag}_{\kappa _{2}}a_{\gamma }            \left(\prod_{\kappa^{\prime}=\left(\kappa_{3}\dots\kappa_{n}\right)}a^{\dag}_{\kappa^{\prime}}\right)\ket{0}
\end{align}


\end{document}

It should be noted that the equal signs are aligned and the lines without equal signs are slightly offset. The equation numbering, in this example, just counts the number of equations. The output is:

As a side note: If this is just for study then keeping the format in a simple format may often do more good than bad.

There are various ways to make, or use, equation counters that are of the form 1.xx for section one and 2.xx for section two and so on.

Answer 2

As @Leucippus does in their answer, I suggest you drop the subequations approach and employ a single, multi-page align environment. I would also replace all tall parentheses (sized via \left( and \right)) with \bigg-sized parentheses, for a more balanced appearance of the material.

You find it easier to track the derivations by inserting colored dots or other visual markers at the start of each equation group.

\documentclass{article}
\usepackage{physics,breqn}

\begin{document}

\section{0 differences between two determines}

\allowdisplaybreaks % allow page breaks in long 'align' env.

\begin{align} % use a single 'align' env.
\mel{\Psi}{V}{\Psi}
  &=v^{\alpha\beta\gamma\delta}
  \bra{0}
  \biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_1)}\mkern-10mu a_{\kappa}\biggr)
  a^{\dag}_{\alpha} a^{\dag}_{\beta} a_{\gamma} a_{\delta}
  \biggl(\prod_{\,\kappa'=(\kappa_1\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr) 
  \ket{0}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &=\bra{0}
  \biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_1)}\mkern-10mu a_{\kappa}\biggr)
  a^{\dag}_{\alpha}a^{\dag}_{\beta}a_{\gamma}\delta_{\delta\kappa_1}
  \biggl(\prod_{\,\kappa'=(\kappa_2\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
  %\begin{align}
  &\quad-\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_1)}\mkern-10mu a_{\kappa}\biggr)
  a^{\dag}_{\alpha}a^{\dag}_{\beta}a_{\gamma}a^{\dag}_{\kappa_1}a_{\delta}
  \biggl(\prod_{\,\kappa'=(\kappa_2\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\end{subequations}
%\begin{subequations}
%\begin{subequations}
  %\begin{align}
  &=\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_1)}\mkern-10mu a_{\kappa}\biggr)
  a^{\dag}_{\alpha}a^{\dag}_{\beta}\delta_{\delta\kappa_1}\delta_{\gamma \kappa_2}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
  %\begin{align}
  &\quad-\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_1)}\mkern-10mu a_{\kappa}\biggr)
  a^{\dag}_{\alpha}a^{\dag}_{\beta}\delta_{\delta_{\kappa_1}}a^{\dag}_{\kappa_2}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\end{subequations}
%\begin{subequations}
  %\begin{align}
  &\quad-\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_1)}\mkern-10mu a_{\kappa}\biggr)
  a^{\dag}_{'}a^{\dag}_{\beta}\delta_{\gamma \kappa_1}a_{\delta}
  \biggl(\prod_{\,\kappa'=(\kappa_2\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
  %\begin{align}
  &\quad+\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_1)}\mkern-10mu a_{\kappa}\biggr)
  a^{\dag}_{\alpha}a^{\dag}_{\beta}a^{\dag}_{\kappa_1}a_{\gamma}a_{\delta}
  \biggl(\prod_{\,\kappa'=(\kappa_2\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\end{subequations}
%\end{subequations}
%\begin{subequations}
%\begin{subequations}
  %\begin{align}
  &=\delta_{\alpha \kappa_1}\delta_{\beta \kappa_2}\delta_{\gamma \kappa_2}\delta_{\delta\kappa_1}
  \\
  %\end{align}
  %\begin{align}
  &\quad-\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  a_{\kappa_2}a_{\kappa_1}a^{\dag}_{\kappa_2}a^{\dag}_{\alpha}a^{\dag}_{\beta}
  \delta_{\delta\kappa_1}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\end{subequations}
%\begin{subequations}
  %\begin{align}
  &\quad-\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_1)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\gamma \kappa_1}a^{\dag}_{\alpha}a^{\dag}_{\beta}\delta_{\delta\kappa_2}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
  %\begin{align}
  &\quad+\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_1)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\gamma \kappa_1}a^{\dag}_{\alpha}a^{\dag}_{\beta}a^{\dag}_{\kappa_2}a_{\delta}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\end{subequations}
%\end{subequations}
%\begin{subequations}
%\begin{subequations}
  %\begin{align}
  &=\delta_{\alpha \kappa_1}\delta_{\beta \kappa_2}\delta_{\gamma \kappa_2}\delta_{\delta\kappa_1}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad+\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\delta\kappa_1}\delta_{\alpha \kappa_1}a^{\dag}_{\beta}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
  %\begin{align}
  &\quad-\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\delta\kappa_1}a^{\dag}_{\alpha}a_{\kappa_1}a^{\dag}_{\beta}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\end{subequations}
%\end{subequations}
%\begin{subequations}
  %\begin{align}
  &\quad-\delta_{\alpha \kappa_1}\delta_{\beta \kappa_2}\delta_{\gamma\kappa_1}\delta_{\delta\kappa_2}
  \\ 
  %\end{align}
  %\begin{align}
  &\quad-\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\gamma \kappa_1}a_{\kappa_1}a^{\dag}_{\alpha}a^{\dag}_{\beta}a_{\delta}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\end{subequations}
%\end{subequations}
%\begin{subequations}
%\begin{subequations}
  %\begin{align}
  &=\delta_{\alpha \kappa_1}\delta_{\beta \kappa_2}\delta_{\gamma \kappa_2}\delta_{\delta\kappa_1}
  \\*
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad+\delta_{\delta\kappa_1}\delta_{\alpha \kappa_1}\bra{0}
   \biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  a^{\dag}_{\beta}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align} 
  &\quad-\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\delta\kappa_1}a^{\dag}_{\alpha}\delta_{\beta \kappa_1}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
  %\begin{align}
  &\quad+\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\delta\kappa_1}a^{\dag}_{\alpha}a_{\kappa_1}a^{\dag}_{\beta}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\end{subequations}
%\end{subequations}
%\end{subequations}
%\begin{subequations}
  %\begin{align}
  &\quad-\delta_{\alpha \kappa_1}\delta_{\beta \kappa_2}\delta_{\gamma\kappa_1}\delta_{\delta\kappa_2} 
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad-\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\gamma\kappa_1}\delta_{\alpha \kappa_1}a^{\dag}_{\beta}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
  %\begin{align}
  &\quad+\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\gamma\kappa_1}a^{\dag}_{\alpha}a_{\kappa_1}a^{\dag}_{\beta}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\end{subequations}
%\end{subequations}
%\end{subequations}
%\begin{subequations}
%\begin{subequations}
  %\begin{align}
  &=\delta_{\alpha \kappa_1}\delta_{\beta\kappa_2}\delta_{\gamma\kappa_2}\delta_{\delta\kappa_1} 
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad+\delta_{\alpha \kappa_1}\delta_{\delta\kappa_1}\mel{\Psi}{a^{\dag}_{\kappa_1}
   a^{\dag}_{\kappa_2}H_0a_{\kappa_2}a_{\kappa_1}}{\Psi}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad-\delta_{\beta\kappa_1}\delta_{\delta\kappa_1}\mel{\Psi}{a^{\dag}_{\kappa_1}
   a^{\dag}_{
  _2
  }H_0a_{
  _2
  }a_1
  a_{
  \kappa_2
  }
  }{\Psi}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad+\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\delta\kappa_1}a^{\dag}_{\alpha}\delta_{\beta\kappa_1}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
  %\begin{align}
  &\quad-\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  \delta_{\delta\kappa_1}a^{\dag}_{\alpha} a^{\dag}_{\beta}a_{\kappa_1}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_3\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
%\end{subequations}
%\end{subequations}
%\end{subequations}
%\end{subequations}
  %\begin{align}
  &\quad-\delta_{\alpha \kappa_1}\delta_{\beta\kappa_2}\delta_{\gamma\kappa_1}\delta_{\delta\kappa_2}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad-\delta_{\gamma\kappa_1}\delta_{\alpha \kappa_1}\mel{\Psi}{a^{\dag}_{\kappa_1}a^{\dag}_{\kappa_2}H_0a_{\kappa_2}a_{\kappa_1}}{\Psi} 
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad+\delta_{\gamma\kappa_1}\bra{0}\biggl(\prod_{\,\kappa=(\kappa_n\dots\kappa_3)}\mkern-10mu a_{\kappa}\biggr)
  a^{\dag}_{\alpha}\delta_{\beta\kappa_1}a_{\gamma}
  \biggl(\prod_{\,\kappa'=(\kappa_{}\dots\kappa_n)}\mkern-10mu a^{\dag}_{\kappa'}\biggr)\ket{0}
  \\
  %\end{align}
  %\begin{align}
  -0
  %\\
  %\end{align}
%\end{subequations}
%\end{subequations}
%\end{subequations}
%\begin{subequations}
%\begin{subequations}
  %\begin{align}
  &=\delta_{\alpha \kappa_1}\delta_{\beta\kappa_2}\delta_{\gamma\kappa_2}\delta_{\delta\kappa_1}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad+\delta_{\alpha \kappa_1}\delta_{\delta\kappa_1}\mel{\Psi}{a^{\dag}_{\kappa_1}
   a^{\dag}_{\kappa_2}H_0a_{\kappa_2}a_{\kappa_1}}{\Psi}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad-\delta_{\beta\kappa_1}\delta_{\delta\kappa_1}\mel{\Psi}{a^{\dag}_{\kappa_1}
   a^{\dag}_{
  _2
  }H_0a_{
  _2
  }a_1
  a_{
  \kappa_2
  }
  }{\Psi}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad+\delta_{\beta\kappa_1}\delta_{\delta\kappa_1}\mel{\Psi}{a^{\dag}_{\kappa_1}
   a^{\dag}_{\kappa_2}H_0a_{\kappa_2}a_{\kappa_1}}{\Psi}
  \\
  %\end{align}
  %\begin{align}
  -0
  %\\
  %\end{align}
%\end{subequations}
%\end{subequations}
%\end{subequations}
%\end{subequations}
  %\begin{align}
  &=\delta_{\alpha \kappa_1}\delta_{\beta\kappa_2}\delta_{\gamma\kappa_1}\delta_{\delta\kappa_2}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad-\delta_{\gamma\kappa_1}\delta_{\alpha \kappa_1}\mel{\Psi}{a^{\dag}_{\kappa_1}
   a^{\dag}_{\kappa_2}H_0a_{\kappa_2}a_{\kappa_1}}{\Psi}
  \\
  %\end{align}
%\begin{subequations}
  %\begin{align}
  &\quad+\delta_{\gamma\kappa_1}\delta_{\beta\kappa_1}
   \mel{\Psi}{a^{\dag}_{\kappa_1a^{\dag}_{\kappa_2}}H_0  a_{\kappa_2}a_{\kappa_1}}{\Psi}
  \\
  %\end{align}
  %\begin{align}
  -0
  %\\
  %\end{align}
%\end{subequations}
%\end{subequations}
%\end{subequations}
%\begin{subequations}
%\begin{subequations}
  %\begin{align}
  &=\delta_{\alpha \kappa_1}\delta_{\beta\kappa_2}\delta_{\gamma\kappa_2}\delta_{\delta\kappa_1}
  \\
  %\end{align}
  %\begin{align}
  &\quad-\delta_{\alpha \kappa_1}\delta_{\beta\kappa_2}\delta_{\gamma\kappa_1}\delta_{\delta\kappa_2}
  \\
  %\end{align}
%\end{subequations}
%\begin{subequations}
  %\begin{align}
  &\quad+\delta_{\alpha \kappa_1}\delta_{\delta\kappa_1}
   \mel{\Psi}{a^{\dag}_{\kappa_1}a^{\dag}_{\kappa_2}H_0a_{\kappa_2}a_{\kappa_1}}{\Psi}
  \\
  %\end{align}
  %\begin{align}
  &\quad+\delta_{\gamma\kappa_1}\delta_{\beta\kappa_1}
   \mel{\Psi}{a^{\dag}_{\kappa_1a^{\dag}_{\kappa_2}}H_0  a_{\kappa_2}a_{\kappa_1}}{\Psi}
  \\
  %\end{align}
  %\begin{align}
  &\quad-\delta_{\gamma\kappa_1}\delta_{\alpha \kappa_1}
   \mel{\Psi}{a^{\dag}_{\kappa_1}a^{\dag}_{\kappa_2}H_0a_{\kappa_2}a_{\kappa_1}}{\Psi}
  \\
  %\end{align}
%\end{subequations}
%\end{subequations}
\intertext{adding the integrals back in.}
%\begin{subequations}
%\begin{subequations}
  %\begin{align}
  &=v^{\kappa_1\kappa_2\kappa_2\kappa_1}
  \\
  %\end{align}
  %\begin{align}
  &\quad-v^{\kappa_1\kappa_2\kappa_1\kappa_2}
  \\
  %\end{align}
%\end{subequations}
%\begin{subequations}
  %\begin{align}
  &\quad+v^{\kappa_1\beta \gamma \kappa_1}
  \\
  %\end{align}
  %\begin{align}
  &\quad+v^{\alpha \kappa_1\kappa_1\delta }
  \\
  %\end{align}
  %\begin{align}
  &\quad-v^{\kappa_1\beta \kappa_1\delta }
  %\end{align}
%\end{subequations}
%\end{subequations}
\end{align}

Equation 3 part b similar to the Condon roles. not sure were to go with this, or with part a.


\section{math\_drafts.pdf}

I'm confused about the steps you took to get from
  \begin{gather}
  \sum_{\kappa }h^{\kappa \kappa }  \\
  \sum_{\kappa } h^{(\kappa )(\kappa )}\delta_{[\kappa ],[\kappa ]}
  \end{gather}

\end{document}

Answer 3

Not a solution, but showing how to reproduce subequation numbering.

Note that amsmath uses the counter parentequation to store the old equaton number then changes \theequation to produce subequation numbers. That way they can use normal equations inside the subequations environment. For \tag it is easier just to create a subequation counter. Then again, maybe not.

\documentclass{article}
\usepackage{amsmath}

\newcounter{subequation}[equation]
\renewcommand{\thesubequation}{\theequation\alph{subequation}}

\begin{document}

\begin{align}
  x &= a \\
    &= b+c \tag{\stepcounter{subequation}\thesubequation}
\end{align}
%
\begin{subequations}
\begin{align}
  x &= a \tag{\theparentequation} \\
    &= b+c
\end{align}
\end{subequations}

\end{document}