# 7.9.1 The Algebraic Diagrammatic Construction (ADC) Scheme

The Algebraic Diagrammatic Construction (ADC) scheme 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.810 An alternative derivation is available in terms of the intermediate state representation (ISR),809 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 $\hat{C}_{J}$.

 $\left|\bar{\Psi}_{J}^{N}\right>=\hat{C}_{J}\left|\Psi_{0}^{N}\right>$ (7.70)

with

 $\left\{\hat{C}_{J}\right\}=\left\{c^{\dagger}_{a}c_{i};c^{\dagger}_{a}c^{% \dagger}_{b}c_{i}c_{j},i (7.71)

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

 $M_{IJ}=\left<\tilde{\Psi}_{I}^{N}\right|\hat{H}-E_{0}^{N}\left|\tilde{\Psi}_{J% }^{N}\right>$ (7.72)

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

 $\mathbf{M}\mathbf{X}=\mathbf{X}\mathbf{\Omega},\;\;\text{ with }\;\;\mathbf{X}% ^{\dagger}\mathbf{X}=\mathbf{1}$ (7.73)

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

 $\left|\Psi_{n}^{N}\right>=\sum_{J}X_{nJ}\left|\tilde{\Psi}_{J}^{N}\right>$ (7.74)

This also allows for the calculation of dipole transition moments via

 $T_{n}=\left<\Psi_{n}^{N}\right|\hat{\mu}\left|\Psi_{0}^{N}\right>=\sum_{J}X_{% nJ}^{\dagger}\left<\tilde{\Psi}_{J}^{N}\right|\hat{\mu}\left|\Psi_{0}^{N}% \right>,$ (7.75)

as well as excited state properties via

 $O_{n}=\left<\Psi_{n}^{N}\right|\hat{o}\left|\Psi_{n}^{N}\right>=\sum_{I,J}X_{% nI}^{\dagger}X_{nJ}\left<\tilde{\Psi}_{I}^{N}\right|\hat{o}\left|\Psi_{J}^{N}% \right>,$ (7.76)

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. 921, 922, 923. 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.

As in Møller-Plesset perturbation theory, the first ADC scheme which goes beyond the non-correlated wave function methods in Section 7.2 is ADC(2). ADC(2) is available in a strict and an extended variant which are usually referred to as ADC(2)-s and ADC(2)-x, respectively. The strict variant ADC(2)-s scales with the 5th power of the basis set. The quality of ADC(2)-s excitation energies and corresponding excited states is comparable to the quality of those obtained with CIS(D) (Section 7.7) or CC2. More precisely, excited states with mostly single excitation character are well-described by ADC(2)-s, while excited states with double excitation character are usually found to be too high in energy. The ADC(2)-x variant which scales as the sixth power of the basis set improves the treatment of doubly excited states, but at the cost of introducing an imbalance between singly and doubly excited states. As result, the excitation energies of doubly excited states are substantially decreased in ADC(2)-x relative to the states possessing mostly single excitation character with the excitation energies of both types of states exhibiting relatively large errors. Still, ADC(2)-x calculations can be used as a diagnostic tool for the importance doubly excited states in the low-energy region of the spectrum by comparing to ADC(2)-s results. A significantly better description of both singly and doubly excited states is provided by the third order ADC scheme ADC(3). The accuracy of excitation energies obtained with ADC(3) is almost comparable to CC3, but at computational costs that scale with the sixth power of the basis set only.345