ADC: Ab Initio Polarization Propagator¶
Section author: Michael F. Herbst
Module: Keywords, PSI Variables, ADC
Algebraicdiagrammatic 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 socalled 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 socalled shifted Hamiltonian or ADC matrix. Denoting this matrix as \(\mathbf{A}\), the eigenproblem can be written in terms of several blocks
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 CC2LR, 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 σvectors (ADC matrixvector 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 matrixvector 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 frozencore, frozenvirtual may be applied to reduce the cost of the problem. Using the corevalence separation (CVS) approximation one may specifically target corevalenceexcitations at a substantial reduction in cost. With the spinflip modification fewreference 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.
Available ADC methods¶
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 builtin 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 Builtin ADC(2) code.
Method 
Backend 
References 
Exc. Energies 
Props 
Supported values for kind keyword 

ADC(1) 
adcc 
RHF, UHF 
yes 
yes 
any, singlet, triplet, spin_flip 
ADC(2) 
adcc 
RHF, UHF 
yes 
yes 
any, singlet, triplet, spin_flip 
builtin 
RHF 
yes 
— 
singlet 

ADC(2)x 
adcc 
RHF, UHF 
yes 
yes 
any, singlet, triplet, spin_flip 
ADC(3) 
adcc 
RHF, UHF 
yes 
yes  any, singlet, triplet, spin_flip 

CVSADC(1) 
adcc 
RHF, UHF 
yes 
yes  any, singlet, triplet 

CVSADC(2) 
adcc 
RHF, UHF 
yes 
yes 
any, singlet, triplet 
CVSADC(2)x 
adcc 
RHF, UHF 
yes 
yes 
any, singlet, triplet 
CVSADC(3) 
adcc 
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"])
.
Running ADC calculations¶
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 [10]
properties('adc(2)', properties=["oscillator_strength"])
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 spinflip computation where a simultaneous flip of spin
and excitation is performed. This is only available
for unrestricted references and not for CVSADC(n)
methods,
see table ADC capabilities of Psi4.
Using the corevalence separation.
For tackling corevalence excitations using the CVSADC(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 corevalence excitation process.
A value of 2
indicates, for example,
that the two lowestenergy \(\alpha\) and the two
lowestenergy \(\beta\) orbitals are placed in the core space.
Since the implemented ADC procedures tackle the
lowestenergy 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:
R_CONVERGENCE¶
Convergence threshold for ADC matrix diagonalisation. Negative values keep the * adcc default (1e6)
Type: conv double
Default: 1
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 builtin implementation for all ADC(2) energy calculations.
You can explicitly set the QC_MODULE option to 'adcc'
enforce using adcc also for this case.
Interface to adcc¶
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 Jacobipreconditioned Davidson. Expensive parts of the ADC matrixvector 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 CVSADC(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 incore 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.
More detailed documentation about adcc and its features can be found at https://adcconnect.org, especially the theory section. If you are using adcc from PSI4 for your calculations, please cite both PSI4 as well as adcc [Herbst2020] in your published work.
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)
, cvsadc(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
andcurrent 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 aMatrix
.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 asMatrix
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 exampleadcc_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 JacobiDavidson 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.
Builtin ADC(2) code¶
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 integraltransformation and also the libdpd library to utilize molecular symmetry in the tensorial manipulations in framework of the directproduct 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)
Noniterative: 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 pseudoperturbative 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 noniterative counterpart is mostly negligible if the mixture among the CIS excited states is small and the quasidegeneracy 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 builtin ADC code is capable of performing the partiallyrenormalized ADC(2) computation, termed PRADC(2). In the perturbative treatment of the singlyexcited 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 PRADC(2) scheme, the DC term is corrected according to the ground state PRMP2 correlation, in which the correlation between the electron pairs is accounted for in sizeconsistent and unitaryinvariant fashion by modulating the MP1 amplitude. By utilizing the PR scheme, substantial resistance against quasidegeneracy is readily granted as discussed in Ref. [Saitow:2012].
Theory of the builtin ADC(2) implementation
For the builtin ADC(2) implementation some very essential points shall be emphasized. In ADC(2) specifically one may write the response equation as
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
This form of the ADC(2) equation requires 7–10 iterations for convergence on only one root. But thanks to NewtonRaphson acceleration,
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 noniterative excitation energy stated above is calculated as a diagonal element of the Davidson miniHamiltonian matrix in the SEM as,
where \(\omega^{CIS}\) and \(\mathbf{X_{CIS}}\) denote the CIS excitation energy and wave function, respectively. The explicit form of the σvector is provided in a note accompanying the source code, in the file psi4/psi4/src/psi4/adc/sigma.pdf.