DOCI
Doubly-occupied configurations
DOCI is the abbreviation of doubly-occupied CI, in which the number of Slater determinants in the full CI expansion is limited to those in which every spatial orbital is doubly occupied. This means that DOCI is actually synonymous with seniority zero. Generally, we can write the DOCI wave function as a weighted sum over all doubly-occupied configurations \(\ket{\kappa}\): \[\begin{equation} \require{physics} \ket{\text{DOCI}} % = \sum_{\kappa} C_{\kappa} \ket{\kappa} % \thinspace , \end{equation}\] where every configuration \(\ket{\kappa}\) is characterized by spatial orbital doubly-occupation numbers \(m_{\kappa, p}=0, 1\) for every spatial orbital \(p\). If \(m_{\kappa, p}=1\), the spatial orbital \(p\) is said to be doubly occupied. The number of DOCI configurations is \[\begin{equation} \dim{\mathcal{F}_{\text{DOCI}}} % = \binom{K}{N_P} \end{equation}\] since we have to choose \(N_P\) doubly-occupied spatial orbitals from a set of \(K\) of them. We can express every configuration \(\ket{\kappa}\) as \[\begin{equation} \ket{\kappa} % = \qty( % \prod_i^K \qty( \hat{P}^+_i )^{m_{\kappa, i}} % ) \ket{\text{vac}} % \thinspace , \end{equation}\] where the sum of spatial orbital occupation numbers must be equal to the number of electron pairs: \[\begin{equation} \sum_p^K m_{\kappa,p} = N_P % \thinspace , \end{equation}\] such that the DOCI wave function for \(N_P\) pairs of electrons in \(K\) spatial orbitals becomes \[\begin{equation} \label{eq:DOCI} \ket{\text{DOCI}} % = \sum_{\kappa} % C_{\kappa} \qty( % \prod_p^K \qty( \hat{P}^+_p )^{m_{\kappa, p}} % ) \ket{\text{vac}} % \thinspace . \end{equation}\] Furthermore, by using \[\begin{equation} \bra{\kappa} % = \bra{\text{vac}} % \prod_p^K \qty( \hat{P}^-_p )^{m_{\kappa, p}} % \thinspace , \end{equation}\] we can show that every DOCI configuration \(\ket{\kappa}\) is normalized: \[\begin{equation} \label{eq:doci_configuration_normalized} \braket{\kappa} = 1 % \thinspace , \end{equation}\] from which follows that the overlap between two DOCI wave functions \(\ket{\Phi_1}\) and \(\ket{\Phi_2}\) \[\begin{align} & \ket{\Phi_1} = \sum_\kappa C_\kappa \ket{\kappa} \\ & \ket{\Phi_2} = \sum_\mu D_\mu \ket{\mu} \end{align}\] is \[\begin{equation} \label{eq:overlap_DOCIs} \braket{\Phi_1}{\Phi_2} % = \sum_\kappa C_\kappa^* D_\kappa % \thinspace , \end{equation}\] where \(\kappa\) labels all configurations that are both present in \(\ket{\Phi_1}\) and \(\ket{\Phi_2}\).
The doubly-occupied Hamiltonian
The DOCI Hamiltonian is a special case of the seniority-preserving electronic Hamiltonian (cfr. equation \(\eqref{eq:seniority-preserving_hamiltonian}\)). If we would only use DOCI wave functions, the terms involving the spin-raising and spin-lowering operators will vanish. Furthermore, for seniority zero (DOCI) wave functions \[\begin{equation} \hat{N}_{p \alpha} \ket{\Omega = 0} % = \hat{N}_{p \beta} \ket{\Omega = 0} % = \tfrac{1}{2} \hat{N}_p \ket{\Omega = 0} % \thinspace , \end{equation}\] which leads to (De Baerdemacker 2017) \[\begin{equation} \label{eq:h_doci} \hat{\mathcal{H}}_\text{DOCI} % = \sum_p^K h_{pp} \hat{N}_p % + \sum_{pq}^K g_{pq} \hat{P}^+_p \hat{P}^-_q % + \frac{1}{4} \sum_{p \neq q}^K % \gamma_{pq} \hat{N}_p \hat{N}_q % \thinspace , \end{equation}\] in which we have defined the following shorthand notations for two-electron integrals: \[\begin{equation} g_{pq} = g_{pqpq} % \thinspace , \end{equation}\] and \[\begin{align} \gamma_{pq} % &= 2 g_{ppqq} - g_{pqqp} \\ &= 2 J_{pq} - K_{pq} % \thinspace . \end{align}\] Furthermore, we have for these integrals: \[\begin{equation} \gamma_{pp} = g_{pp} % \thinspace , \end{equation}\] and for complex conjugates, we have \[\begin{equation} g_{pq}^* = g_{qp} % \thinspace , \end{equation}\] and \[\begin{equation} \gamma_{pq}^* = \gamma_{qp} % \thinspace . \end{equation}\]
The effect of the DOCI Hamiltonian on an RHF determinant \(\ket{\text{RHF}} = \ket{\Phi^0}\) is \[\begin{equation} \label{eq:H_DOCI_RHF} \hat{\mathcal{H}}_\text{DOCI} \ket*{\Phi^0} % = \qty( % 2 \sum_i^{N_P} h_{ii} % + \sum_{ij}^{N_P} \gamma_{ij}) \ket*{\Phi^0} % + \qty( % \sum_i^{N_P} \sum_{a= N_P+1}^K g_{ai} % ) % \ket{\Phi_i^a} % \thinspace , \end{equation}\] in which \(\ket{\Phi_i^a}\) corresponds to the pair-excited Slater determinant in which the electrons in the doubly-occupied orbital \(i\) are both excited to the initially virtual orbital \(a\). We also use the notation \(\ket{\Phi_i^a} = \ket*{\Phi_{i\bar{i}}^{a\bar{a}}}\). Its effect on a singly pair-excited determinant \(\ket{\Phi_i^a}\) is \[\begin{equation} \begin{split} \hat{\mathcal{H}}_\text{DOCI} \ket{\Phi_i^a} % = &\thinspace g_{ia} \ket{\Phi_0} % + \qty[ % \sum_{j \neq i}^{N_P} \qty( % 2 h_{jj} + \gamma_{aj} + \gamma_{ja} + % \sum_{k \neq i}^{N_P} \gamma_{jk} % ) % + 2 h_{aa} + g_{aa} % ] \ket{\Phi_i^a} \\ &+ \sum_{j \neq i}^{N_P} g_{ij} \ket{\Phi_j^a} % + \sum_{ b = N_P + 1, b \neq a}^K g_{ba} \ket{\Phi_i^b} % + \sum_{ b = N_P + 1, b \neq a}^K \sum_{j \neq i}^{N_P} % g_{bj} \ket{\Phi_{ij}^{ab}} % \thinspace , \end{split} \end{equation}\] or, after some rearrangements: \[\begin{equation} \begin{split} \hat{\mathcal{H}}_\text{DOCI} \ket{\Phi_i^a} = & \thinspace g_{ia} \ket{\Phi_0} % + \sum_{j \neq i}^{N_P} % g_{ij} \ket{\Phi_j^a} % + \sum_{ b = N_P + 1, b \neq a}^K g_{ba} \ket{\Phi_i^b} % + \sum_{b = N_P + 1, b \neq a}^K \sum_{j \neq i}^{N_P} % g_{bj} \ket{\Phi_{ij}^{ab}} \\ & + \Bigg[ % 2(h_{aa} - h_{ii}) % + (g_{aa} + g_{ii}) % - (\gamma_{ai} + \gamma_{ia}) \\ & + \sum_j^{N_P} \qty( % 2 h_{jj} + \gamma_{ja} + \gamma_{aj} % - \gamma_{ij} - \gamma_{ji} % + \sum_k^{N_P} \gamma_{jk}) % \Bigg] \ket{\Phi_i^a} % \thinspace , \end{split} \end{equation}\] which are both expressions that are equal to the ones listed in (De Baerdemacker 2017).
DOCI Hamiltonian matrix elements
In the same way we were interested in the FCI Hamiltonian elements, we are interested in the DOCI Hamiltonian elements \[\begin{equation} \matrixel{ I_\alpha I_\beta }{ % \hat{\mathcal{H}}_\text{DOCI} % }{ J_\alpha J_\beta } % \thinspace , \end{equation}\] in which \(I_\alpha\) and \(I_\beta\) are the addresses of the \(\alpha\)- and \(\beta\)-string of the occupation number vector \(I\) and likewise for \(J\) (which should both seniority-zero wave functions). Since in DOCI, every spatial orbital is supposed to be doubly occupied, this means that the \(\alpha\)- and \(\beta\)-strings must be equal. A general DOCI Hamiltonian element is thus given by \[\begin{equation} \mathcal{H}_{\text{DOCI}, IJ} % = \matrixel{I}{ \hat{\mathcal{H}}_\text{DOCI} }{J} % \thinspace , \end{equation}\] in which it is understood that \(I\) and \(J\) are the addresses of doubly-occupied occupation number vectors, each consisting themselves of equal \(\alpha\)- and \(\beta\)-parts. In the calculation of the diagonal contributions, we encounter the term \[\begin{equation} \sum_{pq}^K g_{pq} % \matrixel{I}{ \hat{P}^+_p \hat{P}^-_q }{I} % \thinspace , \end{equation}\] and I feel it is instructive to explain how to work out this term further. By the commutator in equation \(\eqref{eq:commutator_P-_prod_P+}\), the matrix element \(\matrixel{I}{\hat{P}^+_p \hat{P}^-_q}{I}\) is non-zero only when in \(\ket{I}\), orbitals \(p\) and \(q\) are occupied, otherwise we would end up annihilating on the vacuum. This means that we get \[\begin{equation} \sum_{pq}^K % g_{pq} \matrixel{I}{ \hat{P}^+_p \hat{P}^-_q }{I} % = \sum_{p,q \in I}^K g_{pq} % \matrixel{I}{ \hat{P}^+_p \hat{P}^-_q }{I} % \thinspace , \end{equation}\] in which we have indicated explicitly in the summation that \(p\) and \(q\) should be occupied in \(\ket{I}\). By now using the elementary commutator in equation \(\eqref{eq:commutator_P+p_P-q}\), we encounter an expression of the form \[\begin{equation} \matrixel{I}{ \hat{P}^-_q \hat{P}^+_p }{I} % \thinspace , \end{equation}\] which vanishes because of the Pauli principle (cfr. equation \(\eqref{eq:Pauli_principle_pair_operators}\)). We have left out some of the easier steps, but in the end we find that the diagonal contribution (i.e. \(I = J\)) to the DOCI Hamiltonian matrix is equal to \[\begin{equation} \mathcal{H}_{\text{DOCI}, II} % = 2 \sum_{p \in I}^K h_{pp} % + \sum_{p \in I \neq q \in I}^K \gamma_{pq} % + \sum_{p \in I}^K g_{pp} % \thinspace . \end{equation}\]
The off-diagonal elements (i.e. \(I \neq J\)) for two ONVs that differ in the pair-occupation for one orbital, i.e. \[\begin{align} & \ket{I} % = \ket{ % k_1, \ldots, 0_{w \alpha}, \ldots, % 1_{x \alpha}, \dots, % 0_{w \beta}, \ldots, % 1_{x_\beta}, \ldots, k_{2K} % } \\ & \ket{J} % = \ket{ % k_1, \ldots, 1_{w \alpha}, \ldots, % 0_{x \alpha}, \dots, % 1_{w \beta}, \ldots, % 0_{x_\beta}, \ldots, k_{2K} % } % \thinspace , \end{align}\] are equal to \[\begin{equation} \label{eq:H_DOCI_IJ} \mathcal{H}_{\text{DOCI}, IJ} = g_{wx} % \thinspace . \end{equation}\] Let’s shed some light on its derivation. The first and the third term are zero, since \(\ket{I} \neq \ket{J}\). For the second term, we have apply a procedure similar to the derivation of the Slater-Condon rules, which results in us having a non-vanishing contribution only if \(\hat{a}_{q \beta} \hat{a}_{q \alpha} \ket{J} = \pm \hat{a}_{p \beta} \hat{a}_{p \alpha} \ket{I}\). This means that in the summation \[\begin{equation} \sum_{pq}^K % g_{pqpq} % \matrixel{I}{ % \hat{a}^\dagger_{p \alpha} \hat{a}^\dagger_{p \beta} % \hat{a}_{q \beta} \hat{a}_{q \alpha} % }{J} \end{equation}\] we must have that \(p = x\) and \(q = w\), leading to \[\begin{equation} \mathcal{H}_{\text{DOCI}, IJ} % = g_{xwxw} % \matrixel{I}{ % \hat{a}^\dagger_{x \alpha} \hat{a}^\dagger_{x \beta} % \hat{a}_{w \beta} \hat{a}_{w \alpha} % }{J} % \thinspace . \end{equation}\] Since we want any resulting phase factors to be corresponding to either \(J\) or \(I\) (and not \(J\) or \(I\) in which an annihilation has already taken place), we apply two elementary anticommutators, leading to \[\begin{align} \mathcal{H}_{\text{DOCI}, IJ} % &= g_{xwxw} % \matrixel{I}{ % \hat{a}^\dagger_{x \beta} \hat{a}^\dagger_{x \alpha} % \hat{a}_{w \alpha} \hat{a}_{w \beta} % }{J} \\ &= g_{xwxw} % \Gamma^J_{w \beta} \Gamma^J_{w \alpha} % \Gamma^I_{x \beta} \Gamma^I_{x \alpha} % \thinspace . \end{align}\] Let’s take a look at the product of the phase factors. Written out explicitly, we have \[\begin{align} \Gamma^J_{w \beta} \Gamma^J_{w \alpha} % \Gamma^I_{x \beta} \Gamma^I_{x \alpha} % &= \qty( % \prod_p^K (-1)^{k_{p \alpha}} % \prod_p^{w-1} (-1)^{k_{p \beta}} % ) % \qty( % \prod_p^{w-1} (-1)^{k_{p \alpha}} % ) \times \notag \\ & \hspace{12pt} % \qty( % \prod_p^K (-1)^{k_{p \alpha}} \prod_p^{x-1} (-1)^{k_{p \beta}} % ) % \qty( % \prod_p^{x-1} (-1)^{k_{p \alpha}} % ) \\ &= 1 % \thinspace , \end{align}\] in which we get the reduction to \(1\) because we find every product twice, and \((-1)^2 = 1\). We thus finally have \[\begin{equation} \mathcal{H}_{\text{DOCI}, IJ} % = g_{xwxw} % \thinspace , \end{equation}\] in which the strings \(\ket{I}\) are coupled with the strings \(\ket{J}\) of the following form: \[\begin{equation} \label{eq:coupling_equation_DOCI} \ket{J} = \hat{P}^+_w \hat{P}^-_x \ket{I} % \thinspace . \end{equation}\]
DOCI matrix-vector products
Similarly to the matrix-vector products discussed for FCI, we can calculate \[\begin{equation} \sigma_I % = \sum_J % \matrixel{I}{ \hat{\mathcal{H}}_\text{DOCI} }{J} % C_J % \thinspace . \end{equation}\] Splitting up in diagonal and off-diagonal contributions, we get \[\begin{align} \sigma_I % &= \matrixel{I}{ \hat{\mathcal{H}}_\text{DOCI} }{I} C_I % + \sum_{J \neq I} % \matrixel{I}{ \hat{\mathcal{H}}_\text{DOCI} }{J} C_J \\ &= % \qty( % 2 \sum_{p \in I}^K h_{pp} % + \sum_{p \in I \neq q \in I}^K \gamma_{pq} % + \sum_{p \in I}^K g_{pp} % ) C_I % + \sum_{J \neq I} g_{xwxw} C_J % \thinspace , \end{align}\] in which \[\begin{equation} \ket{J} % = \hat{P}^+_w \hat{P}^-_x \ket{I} % \tag{\ref{eq:coupling_equation_DOCI}} % \thinspace . \end{equation}\] A more detailed discussion about this ONV coupling equation can be found in section \(\ref{sec:DOCI_Hamiltonian_elements}\).
Density matrices for DOCI
Using the definitions for the 1- and 2-DMS in equations \(\eqref{eq:1-DM}\) and \(\eqref{eq:2-DM}\) in conjunction with the seniority preserving formulas \(\eqref{eq:seniority_preserving_E_pq}\) and \(\eqref{eq:seniority_preserving_e_pqrs}\), we can derive that the DOCI, and by extension every seniority zero wave function, 1-DM is diagonal: \[\begin{align} D_{pq} % &= \delta_{pq} \ev{ \hat{N}_p }{\Psi} \label{eq:1-DM_DOCI} \\ &= \delta_{pq} D_{pp} % \thinspace , \end{align}\] and for the 2-DM we find \[\begin{align} d_{pqrs} % &= 2 \delta_{pr} \delta_{qs} % \ev{ \hat{P}^+_p \hat{P}^-_q }{\Psi} % + \delta_{pq} \delta_{rs} (1 - \delta_{qr}) % \ev{ \hat{N}_p \hat{N}_r }{\Psi} \notag \\ & \hspace{12pt} - \frac{1}{2} \delta_{ps} \delta_{qr} (1 - \delta_{pq}) % \ev{\hat{N}_p \hat{N}_q}{\Psi} \label{eq:2-DM_DOCI} \\ &= \delta_{pq} \delta_{pr} \delta_{ps} d_{pppp} % + \delta_{pr} \delta_{qs} d_{pqpq} % + \delta_{pq} \delta_{rs} d_{pprr} % + \delta_{ps} \delta_{qr} d_{pqqp} % \thinspace . \end{align}\] The validity of these expressions can be tested by checking (and it is the case) that the expectation value of the DOCI Hamiltonian is equal to the expectation value of the electronic Hamiltonian (i.e. the energy expression) in which we fill in these density matrices, i.e.: \[\begin{align} E_{\text{DOCI}} % &= \ev{ \hat{\mathcal{H}}_{\text{DOCI}} }{\Psi} % = \sum_{pq}^K h_{pq} D_{pq} % + \frac{1}{2} \sum_{pqrs}^K g_{pqrs} d_{pqrs} \\ &= \sum_p^K h_{pp} D_{pp} % + \frac{1}{2} \sum_p^K g_{pppp} d_{pppp} % + \frac{1}{2} \sum_{pq}^K ( % g_{pqpq} d_{pqpq} % + g_{ppqq} d_{ppqq} % + g_{pqqp} d_{pqqp} % ) \label{eq:E_DOCI_DMs} \thinspace . \end{align}\] We can verify that the 1- and 2-DM obey the usual relation, i.e. equation \(\eqref{eq:1-2-DM_spatial_orbitals}\) is fulfilled.
The matrix elements appearing in \(\eqref{eq:2-DM_DOCI}\) are sometimes represented differently (De Baerdemacker 2017) as: \[\begin{align} & \Pi_{pq} = \ev{ \hat{P}^+_p \hat{P}^-_q }{\Psi} % \label{eq:Pi_pq} \\ & \Delta_{pq} = \ev{ \hat{N}_p \hat{N}_q }{\Psi} % \label{eq:Delta_pq} \thinspace , \end{align}\] leading to \[\begin{equation} \label{eq:2-DM_DOCI_Pi_Delta} d_{pqrs} = % 2 \delta_{pr} \delta_{qs} % \Pi_{pq} % + \delta_{pq} \delta_{rs} (1 - \delta_{qr}) % \Delta_{pr} % - \frac{1}{2} \delta_{ps} \delta_{qr} (1 - \delta_{pq}) % \Delta_{pq} \end{equation}\] and the expression for the energy \[\begin{equation} E_{\text{DOCI}} % = \sum_p^K h_{pp} D_{pp} % + \sum_{pq}^K g_{pqpq} \Pi_{pq} % + \frac{1}{4} \sum_{pq}^K % (2 g_{ppqq} - g_{pqqp}) % \Delta_{pq} % - \frac{1}{4} \sum_p^K g_{pppp} \Delta_{pp} % \thinspace . \end{equation}\]
We can work out the expressions for the \(1\)- and \(2\)-DMs a little more, leading to formulas that are actually implementable in computer code. For the 1-DM, we find that the only non-zero elements are: \[\begin{equation} \label{eq:DOCI_1-DM} D_{pp} = 2 \sum_{I}^\text{dim} \abs{c_I}^2 \thinspace , \end{equation}\] in which it is understood that the only contributions in this matrix element are those where \(p \in I\), i.e. the orbital \(p\) must be occupied in the basis vector \(\ket{I}\). Likewise, for the non-zero elements of the 2-DM, we have \[\begin{align} &d_{pppp} = 2 \sum_{I}^\text{dim} \abs{c_I}^2 % \label{eq:DOCI_2-DM_pppp} \\ &d_{pqpq} = 2 \sum_{I}^\text{dim} c_I^* c_J % \label{eq:DOCI_2-DM_pqpq} \\ &d_{ppqq} = 4 \sum_{I}^\text{dim} \abs{c_I}^2 % \label{eq:DOCI_2-DM_ppqq} \\ &d_{pqqp} = -2 \sum_{I}^\text{dim} \abs{c_I}^2 % \label{eq:DOCI_2-DM_pqqp} % \thinspace , \end{align}\] in which the following is understood:Since RHF is inside the DOCI subspace, we should find that RHF is a special case of DOCI. Indeed, looking at the formulas for the 1- and 2-DM for RHF in equations \(\eqref{eq:1-DM_RHF}\) and \(\eqref{eq:2-DM_RHF}\), we find these similarities, as the sum in the DOCI matrix elements only has one contribution (which is equal to 1) because the only basis vector in the ‘linear combination’ of the RHF wave function is the RHF Slater determinant itself.
We will now list the separate spin contributions of the DOCI DMs. By equation \(\eqref{eq:seniority_preserving_E_pq_generalized}\), we have for the 1-DM: \[\begin{equation} D^{\alpha \alpha}_{pq} % = \delta_{pq} \ev{ \hat{N}_{p \alpha} }{\Psi} % \thinspace , \end{equation}\] such that the non-zero elements are given by \[\begin{equation} D^{\alpha \alpha}_{pp} % = \sum_{I}^\text{dim} \abs{c_I}^2 % \thinspace , \end{equation}\] in which it is understood that \(p \in I\). Furthermore, since every orbital \(p\) is either virtual or doubly occupied, we also have \[\begin{equation} D^{\beta \beta}_{pq} = D^{\alpha \alpha}_{pq} % \thinspace , \end{equation}\] which ultimately leads to equation \(\eqref{eq:DOCI_1-DM}\) by summing over all DOCI 1-DM spin contributions.
For the spin-separated 2-DMs, we find the following: \[\begin{align} & d^{\alpha \alpha \alpha \alpha}_{pqrs} % = \delta_{pq} \delta_{rs} % \ev{ % \hat{N}_{p \alpha} \hat{N}_{r \alpha} % }{\Psi} % - \delta_{ps} \delta_{qr} % \ev{ % \hat{N}_{p \alpha} \hat{N}_{q \alpha} % }{\Psi} \\ % & d^{\alpha \alpha \beta \beta}_{pqrs} = % \delta_{pq} \delta_{rs} % \ev{ % \hat{N}_{p \alpha} \hat{N}_{r \beta} % }{\Psi} % + \delta_{pr} \delta_{qs} (1 - \delta_{qr}) % \ev{ % \hat{P}^+_p \hat{P}^-_q % }{\Psi} \\ % & d^{\beta \beta \alpha \alpha}_{pqrs} % = \delta_{pq} \delta_{rs} % \ev{ % \hat{N}_{p \beta} \hat{N}_{r \alpha} % }{\Psi} % + \delta_{pr} \delta_{qs} (1 - \delta_{qr}) % \ev{ % \hat{P}^+_p \hat{P}^-_q % }{\Psi} \\ % & d^{\beta \beta \beta \beta}_{pqrs} % = \delta_{pq} \delta_{rs} % \ev{ % \hat{N}_{p \beta} \hat{N}_{r \beta} % }{\Psi} % - \delta_{ps} \delta_{qr} % \ev{ % \hat{N}_{p \beta} \hat{N}_{q \beta} % }{\Psi} % \thinspace . \end{align}\] We will now list the non-zero elements for the four spin-separated 2-DMs, where we keep in mind that \(p \neq q\). For \(\vb{d}^{\alpha \alpha \alpha \alpha}\) we find: \[\begin{align} & d^{\alpha \alpha \alpha \alpha}_{ppqq} % = \sum_{I}^\text{dim} \abs{c_I}^2 \\ % & d^{\alpha \alpha \alpha \alpha}_{pqqp} % = -\sum_{I}^\text{dim} \abs{c_I}^2 % \thinspace . \end{align}\] For \(\vb{d}^{\alpha \alpha \beta \beta}\), we have \[\begin{align} & d^{\alpha \alpha \beta \beta}_{pppp} % = \sum_{I}^\text{dim} \abs{c_I}^2 \\ & d^{\alpha \alpha \beta \beta}_{ppqq} % = \sum_{I}^\text{dim} \abs{c_I}^2 \\ & d^{\alpha \alpha \beta \beta}_{pqpq} % = \sum_{I}^\text{dim} c_I^* c_J % \thinspace . \end{align}\] In all previous formulas, we apply the same `understandings’ as in section \(\ref{sec:1-2-DMs_DOCI}\). Furthermore, specifically for DOCI, we also have \[\begin{align} & d^{\beta \beta \alpha \alpha}_{pqrs} % = d^{\alpha \alpha \beta \beta}_{pqrs} \\ & d^{\beta \beta \beta \beta}_{pqrs} % = d^{\alpha \alpha \alpha \alpha}_{pqrs} % \thinspace , \end{align}\] from which we can find equations \(\eqref{eq:DOCI_2-DM_pppp}\) through \(\eqref{eq:DOCI_2-DM_pqqp}\) by summing over all DOCI 2-DM spin contributions.