# 7.11.1 The Algebraic Diagrammatic Construction (ADC) Scheme

(June 30, 2021)

The Algebraic Diagrammatic Construction (ADC) of the polarization propagator is an excited state method originating from Green’s function theory. It has first been derived employing the diagrammatic perturbation expansion of the polarization propagator using the Møller-Plesset partition of the Hamiltonian. An alternative derivation is available in terms of the intermediate state representation (ISR), which will be presented in the following.

As starting point for the derivation of ADC equations via ISR serves the exact N electron ground state $\left|\Psi_{0}^{N}\right>$. From $\left|\Psi_{0}^{N}\right>$ a complete set of correlated excited states is obtained by applying physical excitation operators $\hat{C}_{J}$.

 $\left|\bar{\Psi}_{J}^{N}\right>=\hat{C}_{J}\left|\Psi_{0}^{N}\right>$ (7.82)

with

 $\left\{\hat{C}_{J}\right\}=\left\{c^{\dagger}_{a}c_{i};c^{\dagger}_{a}c^{% \dagger}_{b}c_{i}c_{j},i (7.83)

Yet, the resulting excited states do not form an orthonormal basis. To construct an orthonormal basis out of the $|\bar{\Psi}_{J}^{N}\rangle$ the Gram-Schmidt orthogonalization scheme is employed successively on the excited states in the various excitation classes starting from the exact ground state, the singly excited states, the doubly excited states etc.. This procedure eventually yields the basis of intermediate states $\{|\tilde{\Psi}_{J}^{N}\rangle\}$ in which the Hamiltonian of the system can be represented forming the Hermitian ADC matrix

 $M_{IJ}=\left<\tilde{\Psi}_{I}^{N}\right|\hat{H}-E_{0}^{N}\left|\tilde{\Psi}_{J% }^{N}\right>$ (7.84)

Here, the Hamiltonian of the system is shifted by the exact ground state energy $E_{0}^{N}$. The solution of the secular ISR equation

 $\mathbf{M}\mathbf{X}=\mathbf{X}\mathbf{\Omega},\;\;\text{ with }\;\;\mathbf{X}% ^{\dagger}\mathbf{X}=\mathbf{1}$ (7.85)

yields the exact excitation energies $\Omega_{n}$ as eigenvalues. From the eigenvectors the exact excited states in terms of the intermediate states can be constructed as

 $\left|\Psi_{n}^{N}\right>=\sum_{J}X_{nJ}\left|\tilde{\Psi}_{J}^{N}\right>$ (7.86)

This also allows for the calculation of dipole transition moments via

 $T_{n}=\left<\Psi_{n}^{N}\right|\hat{\mu}\left|\Psi_{0}^{N}\right>=\sum_{J}X_{% nJ}^{\dagger}\left<\tilde{\Psi}_{J}^{N}\right|\hat{\mu}\left|\Psi_{0}^{N}% \right>,$ (7.87)

as well as excited state properties via

 $O_{n}=\left<\Psi_{n}^{N}\right|\hat{o}\left|\Psi_{n}^{N}\right>=\sum_{I,J}X_{% nI}^{\dagger}X_{nJ}\left<\tilde{\Psi}_{I}^{N}\right|\hat{o}\left|\Psi_{J}^{N}% \right>,$ (7.88)

where $O_{n}$ is the property associated with operator $\hat{o}$.

Up to now, the exact $N$-electron ground state has been employed in the derivation of the ADC scheme, thereby resulting in exact excitation energies and exact excited state wave functions. Since the exact ground state is usually not known, a suitable approximation must be used in the derivation of the ISR equations. An obvious choice is the $n$th order Møller-Plesset ground state yielding the $n$th order approximation of the ADC scheme. The appropriate ADC equations have been derived in detail up to third order in Refs. 1101, 1103, 1104. Due to the dependency on the Møller-Plesset ground state the $n$th order ADC scheme should only be applied to molecular systems whose ground state is well described by the respective MP($n$) method.