# ADC: Ab Initio Polarization Propagator¶

Section author: Michael F. Herbst

Algebraic-diagrammatic construction methods for the polarization propagator (ADC) determine correlated excitation energies by investigating the pole structure of said propagator. For this the propagator is expressed in a representation constructed from so-called intermediate states, which in turn are based upon a correlated Møller–Plesset (MP) ground state. The original derivation of the ADC scheme was purely diagrammatic [Schirmer:1982] and the connect to the intermediate states was developed only later [Trofimov:2006]. In general $$n$$-th order ADC theory, ADC($$n$$), is constructed upon an $$n$$-th order MP ground state. In this sense one can consider an ADC($$n$$) treatment of excited states consistent to an MP($$n$$) perturbation expansion of the ground state.

In ADC methods the residue calculus of the propagator is translated into an eigenvalue problem with respect to the so-called shifted Hamiltonian or ADC matrix. Denoting this matrix as $$\mathbf{A}$$, the eigenproblem can be written in terms of several blocks

$\begin{split}\begin{pmatrix} \mathbf{A_{SS}} & \mathbf{A_{SD}}\\ \mathbf{A_{DS}} & \mathbf{A_{DD}} \end{pmatrix} \begin{pmatrix} \mathbf{X_S}\\ \mathbf{X_D} \end{pmatrix} =\omega \begin{pmatrix} \mathbf{X_S}\\ \mathbf{X_D} \end{pmatrix}\end{split}$

where S refers to the single and D to the double excitation manifolds. This matrix is typically sparse and thus may be diagonalised iteratively, for example using Davidson’s method [Dreuw:2014:82]. An alternative viewpoint has been addressed for example in [Haettig:2002], where ADC(2) is related to other response theories such as CC2-LR, CIS(D) and CIS(D$$_n$$). In this sense one may consider the ADC matrix the correlated response matrix to a response problem based on CIS and apply the simultaneous expansion method (SEM), in which the σ-vectors (ADC matrix-vector products) are constructed several times.

The structure and order of the blocks in the equation above depend on the ADC level employed. With this also the computational cost changes. The key computational step, namely the formation of the matrix-vector products scales as $${\cal O}(N^5)$$ for ADC(2) and $${\cal O}(N^6)$$ for ADC(2)-x and ADC(3). Several additional approximations, such as frozen-core, frozen-virtual may be applied to reduce the cost of the problem. Using the core-valence separation (CVS) approximation one may specifically target core-valence-excitations at a substantial reduction in cost. With the spin-flip modification few-reference ground states can be tackled starting from a triplet reference by simultaneously exciting an electron and flipping its spin. A more detailed overview of such modifications gives [Dreuw:2014:82] and the adcc theory documentation.

Section author: Michael F. Herbst

Several ADC methods are available in PSI4 for the computation of excited states, see ADC capabilities of Psi4. The methods are implemented in two distinct codes, one is part of PSI4 itself and will be referred to as the built-in implementation, the other is available via an interface to the adcc python module. The two backends follow different approaches to compute ADC excited states and as a result details and supported keywords differ. After a more general introduction, specific aspects of the two implementations will be highlighted in section Interface to adcc and Built-in ADC(2) code.

Method

Backend

References

Exc. Energies

Props

Supported values for kind keyword

RHF, UHF

yes

yes

any, singlet, triplet, spin_flip

RHF, UHF

yes

yes

any, singlet, triplet, spin_flip

built-in

RHF

yes

singlet

RHF, UHF

yes

yes

any, singlet, triplet, spin_flip

RHF, UHF

yes

yes | any, singlet, triplet, spin_flip

RHF, UHF

yes

yes | any, singlet, triplet

RHF, UHF

yes

yes

any, singlet, triplet

RHF, UHF

yes

yes

any, singlet, triplet

RHF, UHF

yes

yes

any, singlet, triplet

The leftmost column of table ADC capabilities of Psi4 provides the supported ADC methods. If only excitation energies are desired, one can simply pass one of the listed method strings to the function energy(). For example, energy('adc(2)-x') will compute excitation energies at ADC(2)-x level. Properties such as oscillator strengths, transition or state dipole moments are available by calling the function properties() with appropriate arguments. Most commonly users will want to compute at least oscillator strengths along with the excitation energies, resulting in a call like properties('adc(2)', properties=["oscillator_strength"]).

Section author: Michael F. Herbst

Running an ADC calculation with PSI4 requires the call to properties() as discussed above as well as one or more mandatory keyword arguments.

The most important keyword argument is ROOTS_PER_IRREP, which is an array with the number of excited states desired for each irreducible representation. Most ADC methods are only supported at C1 symmetry at the moment, such that this option should in most cases be set to an array with a single element only. For example one can run an ADC(2) calculation for 10 (singlet) excited states using:

set roots_per_irrep 


where the molecule section was dropped for brevity.

Selecting the excitation manifold. To select between the possible excitation manifolds, use the KIND keyword. For restricted references by default only singlet excited states are computed, corresponding to the keyword value 'singlet'. To compute triplet states, select 'triplet'. To compute both without making a spin distinction, select 'any'. The latter is default for unrestricted references.

The special KIND value 'spin_flip' selects a spin-flip computation where a simultaneous flip of spin and excitation is performed. This is only available for unrestricted references and not for CVS-ADC(n) methods, see table ADC capabilities of Psi4.

Using the core-valence separation. For tackling core-valence excitations using the CVS-ADC(n) methods, the keyword argument NUM_CORE_ORBITALS is additionally required. It is used to specify the number of (spatial) orbitals to put into the core space and thus select as target orbitals for a core-valence excitation process. A value of 2 indicates, for example, that the two lowest-energy $$\alpha$$ and the two lowest-energy $$\beta$$ orbitals are placed in the core space. Since the implemented ADC procedures tackle the lowest-energy excitations, the value should be specified such that the targeted core orbital is just inside the core space.

Example: Consider furane, $$C_4H_4O$$. In order to tackle the oxygen 1s edge, i.e simulate a O 1s XAS spectrum, one may just set NUM_CORE_ORBITALS to 1. This will select the oxygen 1s orbital for the core space as it is energetically the lowest. For C 1s core excitations the NUM_CORE_ORBITALS value needs to be set to 5, such that both the O 1s and all four C 1s orbitals are part of the core space.

Other keywords and examples. Apart from the mentioned keywords, the following are common:

### REFERENCE¶

Reference wavefunction type

• Type: string

• Possible Values: RHF, UHF

• Default: RHF

### R_CONVERGENCE¶

Convergence threshold for ADC matrix diagonalisation. Negative values keep the * adcc default (1e-6)

### NUM_GUESSES¶

Number of guess vectors to generate and use. Negative values keep * the adcc default (currently 2 * ROOTS_PER_IRREP). This option is only available for the adcc backend.

• Type: integer

• Default: -1

### CUTOFF_AMPS_PRINT¶

Tolerance for extracted or printed amplitudes. This option is only available for the adcc backend.

• Type: double

• Default: 0.01

The full list is provided in appendix ADC and many more sample input files can be found in the adc and adcc subfolders of psi4/samples. Note, that not all keywords are supported by all backends.

Switching between ADC backends. Psi4 currently defaults to the built-in implementation for all ADC(2) energy calculations. You can explicitly set the QC_MODULE option to 'adcc' enforce using adcc also for this case.

Code author: Michael F. Herbst

Section author: Michael F. Herbst

For most implemented ADC methods PSI4 relies on an interface to the adcc python package. The approach of adcc is to directly diagonalise the ADC matrix $$\mathbf{A}$$ in an iterative diagonalisation procedure, usually a Jacobi-preconditioned Davidson. Expensive parts of the ADC matrix-vector product are precomputed and stored in memory. This approach is general in the sense that it can be applied to a large range of ADC methods and variants. So far levels up to ADC(3) and CVS-ADC(3) are available and additional approximations such as FREEZE_CORE and NUM_FROZEN_UOCC are supported with all ADC methods using the adcc backend.

Currently adcc is only capable of performing in-core calculations, for which, however, permutational symmetry and spin symmetry is taken into account for both tensor computations and tensor storage. Inside adcc some heuristic checks for overly excessive memory requirements are implemented, resulting in a warning in case a successful execution is unlikely. There are no guarantees for the memory to be sufficient in case such a warning is not displayed.

The ADC wavefunction object. After running the ADC calculation in adcc, the interface code sets a number of variables in the returned Wavefunction in case they are computed. In the following the <method> prefix refers to the ADC method (such as adc(1), adc(3), cvs-adc(2)-x). For example excitation energies for ADC(2) are thus available via the variable ADC(2) excitation energies.

• Ground state energy terms like MP2 correlation energy, MP3 correlation energy, MP2 total energy, MP3 total energy, current correlation energy and current energy.

• number of iterations: The number of iterations the iterative solver required to converge.

• number of excited states: The number of excited states, which were computed.

• <method> excitation energies: The obtained excitation energies as a Matrix.

• MP2 dipole X and the other components: Ground state dipole moments at MP(2) level.

• <method> transition dipoles, <method> oscillator strengths, <method> rotational strengths and <method> dipoles: The respective properties as Matrix

The following attribute is set on returned wavefunctions:

• adcc_state: The adcc.ExcitedStates object used by adcc to store the ADC(n) excitation energies and all precomputed data in the format used by adcc. Provides direct access to analysis and plotting capabilities from adcc. For example adcc_state.plot_spectrum() plots a broadened excited states spectrum in matplotlib. See the adcc calculations documentation for details.

Tips for convergence issues. If you encounter convergence issues inside adcc, the following parameters are worth tweaking:

• MAX_NUM_VECS: Specifies the maximal number of subspace vectors in the Jacobi-Davidson scheme before a restart occurs. The defaults are usually good, but do not be shy to increase this value if you encounter convergence problems.

• NUM_GUESSES: By default adcc uses twice as many guess vectors as states to be computed. Sometimes increasing this value by a few vectors can be helpful. If you encounter a convergence to zero eigenvalues, than decreasing this parameter might solve the problems.

Code author: Masaaki Saitow

Section author: Masaaki Saitow

The ADC code built into PSI4 is capable of ADC(2) computations of singlet excited states only. It makes use of the libtrans library for efficient and flexible integral-transformation and also the libdpd library to utilize molecular symmetry in the tensorial manipulations in framework of the direct-product decomposition algorithm. By this feature, the Ritz space and intermediate tensors are blocked according to the irreducible representations of the point group, and the excited states that belong to different symmetry are sought separately.

In the output of ADC, the ADC(2) results may look as follows:

->  1 B1 state   :  0.2565095 (a.u.),  6.9799824 (eV)
Non-iterative:  0.2565636 (a.u.),  6.9814532 (eV)
Occ Vir        Coefficient
---------------------------------------------
3   0        -0.9017047264
3   2         0.3038332241
3   1         0.2907567119
3   5        -0.0790167706
3   4        -0.0425829926

Converged in   4 iteration.
Squared norm of the S component:  0.9315336
The S vector is rotated up to  8.102 (deg.)


in which the ADC(2) excitation energy is indicated with arrow symbol and the pseudo-perturbative value, which is calculated in very similar fashion to the CIS(D) energy, is also presented on the following line. In this implementation, the ADC(2) secular matrix is treated effectively by renormalization of the double excitation manifold into the single excitation manifold. So, the effective secular equation is solved for several times for the specific state due to the eigenvalue dependence of the effective response matrix. Only the S component of the transition amplitude is obtained explicitly and the squared norm of the S block and the rotation angle from the corresponding CIS vector are given below the element of the amplitude. The difference between the ADC(2) value and its non-iterative counterpart is mostly negligible if the mixture among the CIS excited states is small and the quasi-degeneracy in the excited state is tolerably weak. But if there is a significant discrepancy in these energies, or the rotation angle is visibly large, special care may have to be taken for the strong effects caused by the higher excited states.

Partial Renormalization Scheme

The built-in ADC code is capable of performing the partially-renormalized ADC(2) computation, termed PR-ADC(2). In the perturbative treatment of the singly-excited state, the doubly and triply excited configurations are accounted for as in the case of CIS(D). In the language of CIS(D), the former is regarded to introduce the orbital relaxation (OR) effect while the latter is argued to give rise to the differential correlation (DC) correction to the excited state. In the PR-ADC(2) scheme, the DC term is corrected according to the ground state PR-MP2 correlation, in which the correlation between the electron pairs is accounted for in size-consistent and unitary-invariant fashion by modulating the MP1 amplitude. By utilizing the PR scheme, substantial resistance against quasi-degeneracy is readily granted as discussed in Ref. [Saitow:2012].

Theory of the built-in ADC(2) implementation

For the built-in ADC(2) implementation some very essential points shall be emphasized. In ADC(2) specifically one may write the response equation as

$\begin{split}\begin{pmatrix} \mathbf{A_{SS}^{(2)}} & \mathbf{A_{SD}^{(1)}}\\ \mathbf{A_{DS}^{(1)}} & \mathbf{A_{DD}^{(0)}} \end{pmatrix} \begin{pmatrix} \mathbf{X_S}\\ \mathbf{X_D} \end{pmatrix} =\omega \begin{pmatrix} \mathbf{X_S}\\ \mathbf{X_D} \end{pmatrix}\end{split}$

where the superscript on each matrix block indicates the order of the fluctuation. Instead of solving the above equation explicitly, the large D manifold is treated effectively as

$[\mathbf{A_{SS}^{(2)}}+ \mathbf{A_{SD}^{(1)}}^{\dagger}(\omega- \mathbf{A_{DD}^{(0)}})^{-1}\mathbf{A_{DS}^{(1)}}]\mathbf{X_{S}}= \omega\mathbf{X_{S}}.$

This form of the ADC(2) equation requires 7–10 iterations for convergence on only one root. But thanks to Newton-Raphson acceleration,

$\omega^{n+1}=\omega^{n}- \frac{\omega^n-\mathbf{X_{S}}(\omega^n)^{\dagger} [\mathbf{A_{SS}^{(2)}}+ \mathbf{A_{SD}^{(1)}}^{\dagger}(\omega^n-\mathbf{A_{DD}^{(0)}})^{-1} \mathbf{A_{DS}^{(1)}}]\mathbf{X_{S}}(\omega^n)}{1+\mathbf{X_{S}} (\omega^n)^{\dagger}[\mathbf{A_{SD}^{(1)}}^{\dagger} (\omega^n-\mathbf{A_{DD}^{(0)}})^{-2}\mathbf{A_{DS}^{(1)}}]\mathbf{X_{S}} (\omega^n)}$

the computational time reduces to shorter than half of the simple iterative procedure. Construction of the denominator of the second term in the above equation is less computationally expensive than construction of one $$\sigma$$-vector with respect to the effective response matrix. The non-iterative excitation energy stated above is calculated as a diagonal element of the Davidson mini-Hamiltonian matrix in the SEM as,

$\omega^{Non-Iterative}= \mathbf{X_{CIS}}^{\dagger}[\mathbf{A_{SS}^{(2)}}+ \mathbf{A_{SD}^{(1)}}^{\dagger}(\omega^{CIS}-\mathbf{A_{DD}^{(0)}})^{-1} \mathbf{A_{DS}^{(1)}}]\mathbf{X_{CIS}}$

where $$\omega^{CIS}$$ and $$\mathbf{X_{CIS}}$$ denote the CIS excitation energy and wave function, respectively. The explicit form of the σ-vector is provided in a note accompanying the source code, in the file psi4/psi4/src/psi4/adc/sigma.pdf.