# Configuration Interaction (CI)

- Author:
David Turban, University of Cambridge

## Introduction

The aim of the CI functionality is to evaluate the electronic Hamiltonian with respect to a set of reference configurations obtained from constrained DFT (cDFT) and LR-TDDFT. This makes it possible to obtain transition rates between different electronic states using Fermi’s Golden Rule (e.g. rate of charge transfer at donor/acceptor-interface in an organic solar cell). Also, CI can help to construct eigenstates of a system in situations where the basic ground- and excited state methods fail due to deficiencies of the approximate exchange-correlation functional.

A classic example for such a case is the binding curve of \(\text{H}_{2}^{+}\). The LDA functional gives an incorrect dissociation limit with a binding energy that is significantly too small. The reason is that there is a spurious self-interaction of the single electron with itself, such that a delocalisation of the electron over both atoms will always yield a lower energy, even at infinite separation. Even more advanced hybrid functionals (like B3LYP) do not solve this problem since they only include part of the exact Hartree-Fock exchange which would cancel the self-interaction. In this scenario CI offers a way forward. By choosing the two states with the electron fully localised on either atom as references one ensures the correct long-range limit. Finally, CI is used to evaluate the Hamiltonian in the reference basis and obtain approximate eigenstates. This gives a very good match with LDA at short range and also retains the physical dissociation limit.

## Theory

As a first step we assume that one intends to find the Hamiltonian matrix element \(\langle\Psi_{B}|\hat{H}|\Psi_{A}\rangle\) between two cDFT states. In cDFT the constrained solutions are obtained as ground states of the electronic Hamiltonian augmented with a constraining potential that pushes charge (and/or spin) around the system:

Here the potential is written as the product of its magnitude \(V_{c}\) and a weighting operator \(\hat{w}_{c}\) which specifically acts on the donor and acceptor regions of the system with appropriate signs. In ONETEP the weighting operator is built from local orbitals (like PAOs or NGWFs) that define the donor and acceptor regions. The eigenvalue \(F\) is the energy \(E\) of the constrained solution plus a correction due to the (unphysical) constraining potential:

The magnitude of the potential \(V_{c}\) takes the role of a Lagrange multiplier that is chosen such that the amount of displaced charge/spin matches the population target \(N_{c}\). It should be noted that cDFT states are generally not eigenstates of the electronic Hamiltonian \(\hat{H}\).

Using the cDFT potentials we now obtain

which reduces the problem to calculating overlaps of states and matrix elements of the cDFT potentials. At this point the expression still contain the full many-body wave functions \(\Psi\). To obtain a practical computational scheme we replace them with Kohn-Sham (KS) determinants \(\Phi\) and approximate

We will assume that all orbitals are chosen to be real functions, and not worry about complex conjugation in inner products. The standard result for the overlap of two Slater determinants is given by

where the functions \(\psi\) denote KS orbitals. The result is simply the determinant of the matrix \(S_{AB}\) of overlaps between KS orbitals of states A and B. In a similar fashion we can evaluate matrix elements of a potential operator:

In the third line the determinant is expanded along the \(i\)-th column. \(C_{j,i}\) denotes cofactors of \(S_{AB}\) which are sign-adapted determinants of the submatrices formed by deleting the \(j\)-th row and \(i\)-th column of \(S_{AB}\). A well-known theorem in linear algebra states that the matrix of cofactors of an invertible matrix is equal to the transpose of the inverse of the matrix times its determinant.

Next, we discuss how a constrained reference state can be coupled to an excited state from LR-TDDFT. In LR-TDDFT the excited states are represented as superpositions of single-particle excitations from an occupied to an unoccupied orbital. This information is contained in the response density matrix \(R_{jb}\). A particular non-zero entry indicates that a transition from valence orbital \(\psi_{j}\) to conduction orbital \(\psi_{b}\) contributes to the excited state. In the following indices \(i,j,k,\ldots\) will denote valence orbitals and indices \(b,c,d,\ldots\) conduction orbitals. A natural choice for a DFT wave function of such an excitation that retains the response density by construction is

\(|\Phi_{j}^{b}\rangle\) denotes a Slater determinant constructed from the valence orbitals, except for the single valence orbital \(j\) replaced with conduction orbital \(b\). For the following we assume that state B was obtained as a LR-TDDFT excitation, and A is a (constrained) ground state as before. For the overlap we calculate

The matrix \(T_{AB}^{vc}\) represents the overlap of the valence orbitals of state A with the conduction orbitals of state B. The derivation of the overlap with a potential operator is a bit more involved but proceeds along similar lines:

In the first determinant two columns are distinct from the overlap \(S_{AB}\), we therefore expand along both. This leads to an expression including the second cofactors \(C_{kl,ij}\). It follows from Jacobi’s theorem that

Putting everything together we finally obtain

where \(W_{AB}^{vc}\) refers to matrix elements of \(\hat{V}\) between valence orbitals of state A and conduction orbitals of state B.

We note that in general the Hamiltonian matrix obtained in the way shown is not symmetric due to the approximations inherent in the DFT formalism. Hence, the Hamiltonian must be symmetrised before eigenstates can be obtained.

## Implementation

The CI functionality is implemented in `couplings_mod`

. For each
reference state the density kernel and NGWFs are read from the
corresponding files. Additionally, the cDFT-potentials are read from
file for a cDFT reference state. For an excited state from LR-TDDFT,
conduction kernel, conduction NGWFs and the response kernel are read. A
set of orthonormal orbitals representing the valence space is obtained
from the NGWF representation by solving the eigenvalue problem

and restricting to the occupied subspace. Here \(K^{\alpha\beta}\) is the valence density kernel and \(S_{\beta\gamma}\) the overlap matrix of valence NGWFs. Orthonormal conduction orbitals are obtained in an equivalent manner. The actual CI calculations then proceeds in this basis as outlined in the theory section. It should be noted that the orbitals obtained in this way generally do not correspond to the KS orbitals (they do not result from a diagonalisation of the Hamiltonian). However, both are related through an orthogonal transformation. Hence, the determinants are identical, therefore all results are unaffected by this choice of basis.

The transformation to a orthonormal basis comes with an inherent \(N^{3}\) scaling of the method. The computational effort is expected to be comparable with a properties calculation (which involves a diagonalisation of the Hamiltonian).

## Performing a calculation

This section explains how to set up a CI calculation, and points out a couple of important things to look out for.

First perform calculations for desired reference states. For each state the density kernel and NGWFs have to be written to files (

`.dkn`

and`.tightbox_ngwfs`

). For cDFT reference states the potentials and projectors are required (`.cdft`

and`.tightbox_hub_projs`

). For LR-TDDFT states conduction kernel and NGWFs are required (`.dkn_cond`

and`.tightbox_ngwfs_cond`

), as well as the response density matrix.It is currently required that all reference states use the same unit cell, grid, geometry and identical atomic species with the same number of NGWFs and the same pseudopotentials.

**NOTE:**The current implementation is not compatible with PAW!Now set up a new input file for the CI calculation. It is recommended to copy the input file of one of the cDFT reference calculations. This ensures that the setup of the CI run is consistent with the reference calculations (in particular with the correct projectors). If a LR-TDDFT reference state is used, also copy the conduction species block into the file. Set

`TASK`

to`COUPLINGS`

.Add the block

`couplings_states`

. This tells the CI calculation which reference states to use. Here is an example:Each line corresponds to one reference state. The first column is short identifier for the state (currently unused). The second column indicates whether it is a cDFT or LR-TDDFT state, the third column contains the root name (i.e. name of original input file without extension). For a LR-TDDFT state, the fourth column determines the index of the excitation to be used (set to 0 for cDFT states). Finally, the fifth column is the energy in Hartree. It should be made sure that all energies are referenced to the same zero point.The output is written to matrix files (using

`dense_write`

). The names of these files consist of the root name of the CI run with extensions`_ci_ham`

and`_ci_ham_sym`

for the CI Hamiltonian and its symmetrised version, respectively. Eigenvalues and -states of the (symmetrised) CI Hamiltonian are written to files with extensions`_ci_eigvals`

and`_ci_eigstates`

(column-wise). If`output_detail : VERBOSE`

is chosen, the results are also written to standard output.

## References

Extracting electron transfer coupling elements from constrained density functional theory, Q. Wu and T. Van Voorhis, J. Chem. Phys.

**125**, 164105 (2006)Exciton/Charge-Transfer Electronic Couplings in Organic Semiconductors, S. Difley and T. Van Voorhis, JCTC

**7**, 594 (2011)Determinants and matrices, A.C. Aitken, University mathematical texts vol. 20, Oliver and Boyd (1958)