The polarisation propagator Π(ω) is a quantity from many-body perturbation theory Fetter1971
. Its relationship to electronically excited states spectra can be understood from the fact that its poles are located exactly at the vertical excitation energies ωn = En − E0 Fetter1971,Schirmer1982
. Here, E0 is the energy of the ground state of the exact N-electron Hamiltonian Schirmer1982,Schirmer2018
may be extracted from Π(ω) as well.
Taking this as a starting point, the algebraic-diagrammatic construction scheme for the polarisation propagator (ADC) examines an alternative representation of the polarisation propagator Schirmer1982
, the so-called intermediate-state representation (ISR). In this formalism, a set of creation and annihilation operators is applied to the exact ground state and the resulting precursor states are orthogonalised block-wise according to excitation class Schirmer1991
. This procedure yields the so-called intermediate states Schirmer1991
by a unitary transformation
The expansion coefficients Schirmer1991
where Ωnm = δnmωn is the diagonal matrix of excitation energies and
From the ADC eigenvectors Schirmer2004,Wormit2014
. Contracting these densities with the MO representation
In this way, e.g., the MO representation of the dipole operator may be contracted with Trofimov2006,Knippenberg2012,Fransson2017
.
As described so far, the above formalism builds the IS basis on top of the exact N-electron ground state and is thus exact as well. For practical calculations, however, the ADC scheme is not applied to the exact ground state, but to a Møller-Plesset ground state at order n of perturbation theory. The resulting ADC method is named ADC(n) and is by construction consistent with an MP(n) ground state. Detailed derivations and the resulting expressions for the ADC matrix Schirmer1982,Schirmer1991,Wormit2014,Dreuw2014,Schirmer2018
and will not be discussed here.
As a result of the construction of ADC(n) as excitations on top of an MP(n) ground state, the matrix Dreuw2014
.
On top of this block structure the individual blocks are sparse as well, see Figure 1b and c. This sparsity is a direct consequence of the selection rules obtained from spin and permutational symmetry in the tensor contractions required for computing eqn:adc_diagonalisation
, adcc follows the conventional approach Dreuw2014,Wormit2014
to use contraction-based, iterative eigensolvers, such as the Jacobi-Davidson Davidson1975
. Furthermore, all tensor operations in the required ADC matrix-vector products are performed on block-sparse tensors. For an optimal performance the spin and permutational symmetry of the ADC equations need to be taken into account when setting up the block tiling along the tensor axes. In this setting the computational scaling of ADC(2) is given as O(N5) where N is the number of orbitals, whereas ADC(2)-x and ADC(3) scale as O(N6). This procedure additionally ensures the numerical stability of the eigensolver with respect to the excitation manifold. That is to say, that (for restricted references) spin-pure guess vectors always lead to eigenvectors Dreuw2014
One important modifications of the ADC scheme as discussed above is the core-valence separation (CVS) Cederbaum1980,Trofimov2000,Wenzel2014b,Wenzel2014a,Wenzel2015
. In this approximate ADC treatment targeting core-excited states, the strong localisation of the core electrons and the weak coupling between core-excited and valence-excited states is exploited to decouple and discard the valence excitations from the ADC matrix. This lowers the number of the actively treated orbitals and thus the computational demand for solving the ADC eigenproblem eqn:adc_diagonalisation
. The validity of this approximation has been analysed in the literature and is backed up by computational studies comparing with experiment Norman2018,Fransson2019
. With this, ADC can be used for considering core-excited states, and subsequent studies have also established the ability of calculating non-resonant X-ray emission spectra Fransson2019
and resonant inelastic X-ray scattering Rehn2017a
. Other variants of ADC include spin-flip Lefrancois2015
, where a modified Davidson guess allows to treat processes of simultaneous excitation and spin-flip, tackling few-reference problems in an elegant and consistent way Lefrancois2016,Lefrancois2017
. Similar to other CI-like methods the range of orbitals which are considered for building the intermediate states may also be artificially truncated. For example, when considering valence-excitations, excitations from the core orbitals may be dropped leading to a frozen-core (FC) approximation. Similarly, high-energy virtual orbitals may be left unpopulated, leading to a frozen-virtual (FV) or restricted-virtual approximation Yang2017
.