n

) Family 7.11.2 The Algebraic Diagrammatic Construction (ADC) Scheme 7.11.4 Resolution of the Identity ADC Methods

7.11.3 IP- and EA-ADC

(February 4, 2022)

Similar to ADC scheme of the polarization propagator (ADC), the procedure can also be applied to the $(N-1)$ - and $(N+1)$ -parts of the electron propagator, which is given in its spectral representation as

G_{pq}(\omega)=\underbrace{\sum_{n}\frac{\bigl{\langle}\Psi_{0}^{N}\bigl{% \lvert}c_{p}\bigr{\rvert}\Psi_{n}^{N+1}\bigr{\rangle}\bigl{\langle}\Psi_{n}^{N% +1}\bigl{\lvert}c_{q}^{\dagger}\bigr{\rvert}\Psi_{0}^{N}\bigr{\rangle}}{\omega% +E_{0}^{N}-E_{n}^{N+1}}}_{(N+1)\text{-electron (EA) part}}+\underbrace{\sum_{n% }\frac{\bigl{\langle}\Psi_{0}^{N}\bigl{\lvert}c_{q}^{\dagger}\bigr{\rvert}\Psi% _{n}^{N-1}\bigr{\rangle}\bigl{\langle}\Psi_{n}^{N-1}\bigl{\lvert}c_{p}\bigr{% \rvert}\Psi_{0}^{N}\bigr{\rangle}}{\omega+E_{n}^{N-1}-E_{0}^{N}}}_{(N-1)\text{% -electron (IP) part}}.

(7.90)

Doing so, the (non-Dyson) IP- and EA-ADC methods up to third order of perturbation theory have been derived.^{999, 1134} As in the case of the ADC scheme of the polarization propagator, the derivation of the same working equations is possible via the ISR formalism, the only difference to the procedure shown in the previous section 7.11.2 being the excitation operators

	$\displaystyle\left\{\hat{C}_{J}^{N-1}\right\}$	$\displaystyle=\left\{c_{i};c^{\dagger}_{a}c_{i}c_{j},i<j;\ldots\right\}$	IP-ADC		(7.91)
	$\displaystyle\left\{\hat{C}_{J}^{N+1}\right\}$	$\displaystyle=\left\{c^{\dagger}_{a};c^{\dagger}_{a}c^{\dagger}_{b}c_{i},a<b;% \ldots\right\}$	EA-ADC		(7.92)

replacing the electron number-conserving one $\hat{C}_{J}$ in Eq. (7.83).

Diagonalization of the IP- and EA-ADC secular matrices $\mathbf{M}$ yields electron-detachment energies (or ionization potentials, IPs) and electron-attachment energies (or negative electron affinities, EAs), respectively. In addition, relative spectral intensities of $(N-1)$ - and $(N+1)$ -transitions are accessible as pole strengths $P_{n}$ , which are computed according to

P_{n}=\sum_{p}\left|x_{pn}\right|^{2},

(7.93)

where the $x_{pn}$ are the spectroscopic factors computed by means of the IP- and EA-ADC eigenvectors $\mathbf{X}$ and the matrix of effective transition amplitudes $\mathbf{f}$ using the relations

	$\displaystyle x_{pn}^{N-1}$	$\displaystyle=\sum_{J}X_{Jn}f^{N-1}_{Jp}=\sum_{J}X_{Jn}\left<\tilde{\Psi}_{n}^% {N-1}\Bigl{\|}c_{p}\Bigr{\|}\Psi_{0}^{N}\right>$	IP-ADC		(7.94)
	$\displaystyle x_{pn}^{N+1}$	$\displaystyle=\sum_{J}X_{Jn}f^{N+1}_{Jp}=\sum_{J}X_{Jn}\left<\tilde{\Psi}_{J}^% {N+1}\Bigl{\|}c_{p}^{\dagger}\Bigr{\|}\Psi_{0}^{N}\right>$	$\displaystyle\text{EA-ADC}.$		(7.95)

When requesting electron-detached or electron-attached states, the pole strengths are automatically computed. For IP- and EA-ADC(2) calculations, second-order pole strengths are used [i.e., the IP- and EA-ADC(2) $\mathbf{f}$ matrix is employed in their computation]. As suggested in Ref. 999, for computational reasons the same second-order pole strengths are computed in case of strict IP- and EA-ADC(3) calculations, i.e., using the second-order ground state density throughout the $\mathbf{M}$ matrix equations, ADC_DENSITY_ORDER = 2. When requesting a higher-order ground state density to be used, e.g., by setting ADC_DENSITY_ORDER = 3 (corresponding to the $\Sigma(4)$ scheme¹¹³⁴) or ADC_DENSITY_ORDER = 4 (corresponding to the $\Sigma(4+)$ scheme,¹¹³⁴ also denoted as standard IP- and EA-ADC(3) schemes), the corresponding pole strengths are used, i.e., third-order pole strenghts in case of $\Sigma(4)$ and improved third-order pole strengths in case of $\Sigma(4+)$ .

The spectroscopic factors also allow for computing the Dyson orbitals $|\phi_{n}\rangle$ connected to electron detachment and attachment processes

\left|\phi_{n}\right>=\sum_{p}x_{pn}\left|\varphi_{p}\right>,

(7.96)

where in the latter relation the $\varphi_{p}$ denote HF orbitals. Dyson orbital output is triggered by ADC_DO_DYSON = TRUE together with STATE_ANALYSIS = TRUE. Also see Section 10.2.9 for further details.

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7.11.3 IP- and EA-ADC