# The projected Schrödinger equation framework

Suppose we have put forth a [wave function model](#wave-function-models) $\require{physics} \ket{\Psi(\vb{p})}$, parametrized by $m$ wave function parameters collected in a vector $\vb{p}$. Ultimately, the goal is to minimize the energy function
\begin{equation}
    E(\boldsymbol{\eta}, \vb{p})
    =
    \frac{
        \ev{
            \hat{\mathcal{H}}(\boldsymbol{\eta})
        }{\Psi(\vb{p})}
    }{ \braket{\Psi(\vb{p})} }
    \thinspace ,
\end{equation}
where $\boldsymbol{\eta}$ is an external perturbation, subject to the constraint that at a certain optimal point $\vb{p}^\star$ the wave function is normalized:
\begin{equation}
    \braket{\Psi(\vb{p}^\star)} = 1
    \thinspace .
\end{equation}
Within the Born-Oppenheimer approximation, this amounts to solving the electronic Schrödinger problem:
\begin{equation}
    \hat{\mathcal{H}}(\boldsymbol{\eta})
        \ket{\Psi(\vb{p}^\star)}
    = E(\boldsymbol{\eta}, \vb{p}^\star)
        \ket{\Psi(\vb{p}^\star)}
    \thinspace ,
\end{equation}
which means trying to find the eigenvectors $\ket{\Psi(\vb{p}^\star)}$ with corresponding eigenvalues $E(\boldsymbol{\eta}, \vb{p}^\star)$ of the electronic Hamiltonian $\hat{\mathcal{H}}(\boldsymbol{\eta})$.

Borrowing the concept from numerical analysis (cfr. [Galerkin methods](https://en.wikipedia.org/wiki/Galerkin_method)), we can convert this eigenvalue problem to a weak formulation, where we introduce a projection space or test set of $S+1$ vectors:
\begin{equation}
    \mathcal{P} = \qty{ \ket{0}, \ket{1}, \ldots, \ket{S} }
\end{equation}
and instead of searching for the eigenvectors of the Hamiltonian, we only require the projected Schrödinger equations to hold. These are $S+1$ possibly non-linear equations in the wave function parameters $\vb{p}$:
\begin{equation}
    \begin{cases}
        \matrixel{0}{
            \hat{\mathcal{H}}(\boldsymbol{\eta})
        }{\Psi(\vb{p})}
        = E \braket{0}{\Psi(\vb{p})} \\
        %
        \matrixel{1}{
            \hat{\mathcal{H}}(\boldsymbol{\eta})
        }{\Psi(\vb{p})}
        = E \braket{1}{\Psi(\vb{p})} \\
        %
        \phantom{
            \matrixel{1}{
                \hat{\mathcal{H}}(\boldsymbol{\eta})
            }{\Psi(\vb{p})}
        }
        \thinspace \thinspace \thinspace \vdots \\
        %
        \matrixel{S}{
            \hat{\mathcal{H}}(\boldsymbol{\eta})
        }{\Psi(\vb{p})}
        = E \braket{S}{\Psi(\vb{p})}
        \thinspace ,
    \end{cases}
\end{equation}
which should be fulfilled at the optimal point $\vb{p} = \vb{p}^\star$ and for all values of the perturbation $\forall \boldsymbol{\eta}$. Informally, we could coin this as requiring the Schrödinger equation only to hold inside the projection space $\mathcal{P}$.

At this point, some remarks are due:

- We should note that this projection set $\mathcal{P}$ is very general: it can solely consist of [occupation number vectors (ONVs)](#fock-space), but it can in principle contain any kind of [Fock space](#fock-space) vectors.
- We will call the vector $\ket{0}$ the *reference*, but we should note that this is just an arbitray choice.
- We should realize that by applying a suitable linear combination of [creation and annihilation operators](#one-electron-excitation-operators), the whole projection set $\mathcal{P}$ can be constructed from this reference vector $\ket{0}$.

We can now introduce the PSE energy function as being related to the reference determinant:
\begin{equation}
    E(\boldsymbol{\eta}, \vb{p})
    =
    \frac{
        \matrixel{0}{
            \hat{\mathcal{H}}(\boldsymbol{\eta})
        }{\Psi(\vb{p})}
    }{ \braket{0}{\Psi(\vb{p})} }
\end{equation}
together with the (non-linear) equations ($a=1 \cdots S$)
\begin{equation}
    f_a(\boldsymbol{\eta}, \vb{p}^\star)
    =
    \matrixel{a}{
        \hat{\mathcal{H}}(\boldsymbol{\eta})
        - E(\boldsymbol{\eta}, \vb{p}^\star)
    }{\Psi(\vb{p}^\star)}
    = 0 \thinspace ,
\end{equation}
and we can verify that they constitute the same set of equations as introduced before.

Optimizing a wave function model in this PSE framework amounts to solving the projected Schrödinger equations for the optimal parameters $\vb{p}^\star$, whose corresponding energy can then be calculated from the PSE energy function. We should note that the optimal parameters $\vb{p}^\star(\boldsymbol{\eta})$ are implicitly dependent on the value of the external perturbation $\boldsymbol{\eta}$.^[Where we feel it is instructional, we will include this implicit dependence in the formulas. However, in order to keep the notation clear, we will drop this implicit dependence almost everywhere.] At the given value of the perturbation $\boldsymbol{\eta}$, the energy may be calculated from the energy function:
\begin{equation}
    \mathcal{E}(\boldsymbol{\eta})
    = E(\boldsymbol{\eta}, \vb{p}^\star)
    \thinspace .
\end{equation}
It is important to note that, since the PSEs are non-stationary (i.e. non-variational), the energy $\mathcal{E}(\boldsymbol{\eta})$ is not an upper bound to the exact energy. As a remark, in order to determine the optimal parameters $\vb{p}^\star$, we must have at least as many PSEs as there are wave function parameters:
\begin{equation}
    S \geq m
\end{equation}
or else we have wave function parameters that can be chosen as zero (i.e. which are redundant).

This projected Schrödinger equation approach is of course [related](#the-projected-schrödinger-coupled-cluster-equations) to the amplitude equations in the coupled-cluster method.

We may combine the [variational Lagrangian formalism](#the-variational-lagrangian) with the PSE framework to yield the [projected Schrödinger Lagrangian](#the-projected-schrödinger-Lagrangian).
