## Overview

My research is located right at the interface between **developing electronic structure theories** and their application to the **study of spectroscopic properties** and **the chemistry of challenging molecular complexes** in their ground and electronically excited states. In ongoing collaborative efforts, I am currently targeting the dynamics of molecules in their excited states with **novel multiconfigurational methods** such as **DMRG**, (range-separated) **ensemble DFT** as well as multiconfigurational DFT which allow us to gain insight on the fate of electronically excited states beyond a static picture. In combination with (two-component) **relativistic quantum chemical methods**, such an approach will be applicable not only to (light-element) organic molecules such as carotenoids (in light-harvesting complexes) but also to (late) transition-metal and lanthanide/actinide complexes. Moreover, I am particularly interested in the development of computational tools that allow the prediction of electronic and magnetic properties such as X-Ray, UV-Vis, EPR and NMR spectra of closed and open-shell molecules based on robust, accurate and systematic *ab initio* wave function approaches, preferably taking into account solvent and/or (protein) matrix embedding effects.

## Novel Methods for Multiconfigurational Electronic Structure Problems

### DMRG

A central theme to our research is the development of multiconfigurational methods building on the matrix-product state (MPS) wave function parametrization of the density matrix renormalization group (DMRG) approach. The DMRG approach provides a polynomially-scaling, efficient means to obtain near-exact full CI solutions for (very) large active orbital spaces in a systematic and variational manner. Although the DMRG ansatz in an MPS formulation is capable of efficiently treating the static electron correlation problem, ultimately striving for chemical accuracy in the description of chemical reactions and/or spectroscopic properties requires to take into account dynamical electron correlation as illustrated in a recent publication for a selected set of dissociation reactions of transition metal complexes.

Often, orbital relaxation plays a crucial role for a correct description of chemical properties in ground- and electronically excited states. To this end, we recently devised an efficient second-order DMRG-SCF algorithm based on the Werner-Mayer-Knowles (WMK) algorithm originally proposed for CI reference wave functions which outperforms (in terms of required macro iterations as well as CPU/wall time) existing quasi-second-order algorithms as shown on the left for the Cr_{2} dimer with a CAS(12,12) active orbital space.

Moreover, spin-state energetics are known to be sensitive to subtle electron correlation effects. In a recent work we studied the spin-state energetics of known critical cases of transition metal complexes comprising non-innocent ligands by virtue of our newly developed large-scale second-order perturbation theory DMRG-NEVPT2 model.

The NEVPT2 data for the low-lying triplet states based on zeroth-order DMRG-SCF wave functions employing a CAS(22,22) active orbital space are illustrated on the right in comparison with single-reference DFT data taken from the literature. Future developments need to tackle the computational cost of calculating higher-order reduced density matrices (RDMs) that are required in the perturbation summation. Possible routes could involve the re-design of the RDM algorithm to scale on graphical processing units and/or the controlled introduction of cumulant approximations for the highest required order in the RDM series.

By contrast, in recent works we considered an alternative solution to the dynamical correlation problem for a (zeroth-order) MPS reference wave function. Rather than aiming for a conventional two-step (“diagonalize-and-then-perturb”) approach, we take advantage of a simultaneous treatment of dynamic and static correlations using DFT for the dynamical-correlation part. To avoid the double-counting problem of electron-correlation effects, we make use of a range separation of the two-electron repulsion operator into a short-range (sr) and a long-range (lr) part, an idea originally proposed by Savin. Such an approach has the appealing feature that it (i) does not require the evaluation of higher-order RDMs, (ii) is capable of simultaneously handling dynamic and static correlation, and (iii) combines wave function theory with DFT in a rigorous manner. As discussed below, lrDMRG-srDFT even opens up for a time-independent formulation of a multiconfigurational DFT model that allows for a balanced, simultaneous description of ground- and excited states.

### Ensemble DFT

Time-dependent density-functional theory (TDDFT) is – without doubt – today the standard approach for computing electronic excitation energies in atoms and molecules. Nevertheless, in its usual (approximate) formulation, the reliability of TDDFT suffers from a number of severe shortcomings, for example, it completely misses electronic excitations that cannot be described as single excitations. To this end, we are developing a time-independent alternative to TDDFT, which combines the efficiency of DFT in describing dynamical electron correlation with the advantage of working within a genuine multiconfigurational framework which allows us to overcome the shortcomings of the former time-dependent response formulation. As we have demonstrated in a recent publication, the formulation of a (multi-state) multiconfigurational DFT approach in combination with a DMRG wave function model can be obtained rigorously by using the range separation of the two-electron repulsion in the framework of ensemble DFT. The latter is an extension of DFT to excited states in which the central ingredient of DFT, namely the electron density, is replaced by the so-called “ensemble density” which can be simply obtained from a weighted sum of ground and excited state densities. In a series of publications we have shown how the associated ensemble energy can then be used to calculate individual excitation energies by means of linear interpolation and/or extrapolation methods. In a recent paper we derived a formalism that allows us to obtain ghost-interaction free, weight-independent excitation energies in a rigorous manner. Unphysical ghost interaction between two different states of the ensemble are encountered in the Hartree term through the product of the respective state densities but should, in principle, be counter-balanced by means of the * exact* ensemble exchange-correlation energy which is, however, not the case for approximate functionals. Consequently, as shown exemplarily on the left for excited states of the Be atom, ghost interaction corrected excitation energies (solid lines) show a much faster convergence towards the exact full CI limit (see left; solid straight line) with respect to the range-separation parameter μ regardless of the excitation character (red curves: single excitation; blue curves: double excitation) than their uncorrected counterparts (dashed lines).

**Relevant Publications:**

**Review **“*New approaches for ab initio calculations of molecules with strong electron correlation*“

S. Knecht, E. D. Hedegård, S. Keller, A. Kovyrshin, Y. Ma, A. Muolo, C. J. Stein, M. Reiher, *Chimia*, **70**, 244-251 (2016).

“*Generalized Pauli constraints in small atoms*“

C. Schilling, M. Altunbulak, S. Knecht, A. Lopes, J. D. Whitfield, M. Christandl, D. Gross, and M. Reiher, Phys. Rev. Lett., submitted for publication.

“*An efficient relativistic density-matrix renormalization group implementation in a matrix-product formulation*“

S. Battaglia, S. Keller, and S. Knecht, J. Chem. Theory Comput., 17.12.2017, under revision.

“*Combining extrapolation with ghost interaction correction in range-separated ensemble density functional theory for excited states*“

Md. M. Alam and K. Deur and S. Knecht and E. Fromager, *J. Chem. Phys., ***147**, 204105 (2017).

*Laplace-transformed multi-reference second-order perturbation theories in the atomic and active molecular orbital basis.*

B. Helmich-Paris and S. Knecht, *J. Chem. Phys.*, **146**, 224101 (2017).

“*Second-order self-consistent-field density-matrix renormalization group.*“

Y. Ma, S. Knecht, S. Keller, M. Reiher, *J. Chem. Theory Comput., ***13***, *2533–2549 (2017).

“*Multi-reference perturbation theory with Cholesky decomposition for the density matrix renormalization group*“

L. Freitag, S. Knecht, C. Angeli, M. Reiher, *J. Chem. Theory Comput., ***13,** 451-459 (2017).

“*A nonorthogonal state-interaction approach for matrix product state wave functions*“

S. Knecht, S. Keller, J. Autschbach, M. Reiher, *J. Chem. Theory Comput.*, **12**, 5881–5894 (2016).

“*Ghost interaction correction in ensemble density-functional theory for excited states with and without range separation*“

M. Md. Alam, S. Knecht, E. Fromager, *Phys. Rev. A*, **94**, 012511 (2016).

“*Linear interpolation method in ensemble Kohn-Sham and range-separated density-functional approximations for excited states*“

B. Senjean, S. Knecht, H. J. Aa. Jensen, and E. Fromager,* Phys. Rev. A*, **92**, 012518 (2015).

“*Density matrix renormalization group with efficient dynamical electron correlation through range separation*“

E. D. Hedegård, S. Knecht, J. S. Kielberg, H. J. Aa. Jensen, and M. Reiher, *J. Chem. Phys.*, **142**, 224108 (2015).

“*Four-component density matrix renormalization group*“

S. Knecht, O. Legeza, and M. Reiher, *J. Chem. Phys.*, **140**, 041101 (2014).

“*Multiconfigurational time-dependent density-functional theory based on range separation*“

E. Fromager, S. Knecht, and H. J. Aa. Jensen, *J. Chem. Phys.*, **138**, 084101 (2013).

## Relativistic Quantum Chemical Methods

Relativistic effects are commonly split into kinematic relativistic effects (sometimes also referred to as scalar-relativistic effects) and magnetic effects both of which grow approximately quadratic with the atomic number *Z*. A scalar-relativistic theory therefore considers only kinematic relativistic effects whereas a fully relativistic theory includes both kinematic relativistic and magnetic effects. Kinematic relativistic effects reflect changes of the non-relativistic kinetic energy operator while spin-orbit interaction, which is the dominant magnetic effect (for heavy elements), originates from a coupling of the electron spin to the induced magnetic field resulting from its orbital motion in the field created by the other charged particles, namely the nuclei and remaining electrons.

Encounters of relativistic effects are ubiquitous in the chemistry and physics of heavy elements compounds such that it might be tempting to assume that relativistic effects could safely be ignored for molecules composed only of light elements, e.g., those of the upper half of the periodic table. This is, however, generally not the case, as even a quantitative description of the photochemistry and excited-state dynamics of organic molecules requires to take relativistic effects — in particular spin-orbit coupling — into consideration. Similarly, it has been recognized that achieving or even surpassing chemical accuracy in the thermochemistry of light elements, is only feasible with a proper account of electron correlation and relativistic effects.

A central aspect of our research is therefore the development of relativistic quantum chemical approaches in the context of novel electron correlation methods (see above) with a particular focus on multiconfigurational approaches.

In a recent work, we presented a fully relativistic DMRG approach that not only fully exploits a matrix-product *ansatz* for both, wave function and Hamiltonian but also treats scalar-relativistic effects, spin-orbit coupling and electron correlation effects on an equal footing. This allowed us to study the current density of a Dy(III) complex in a real-space approach in a fully relativistic four-component framework as a function of the active orbital space ranging from a minimal CAS(9,14), panel (a), up to a CAS(27,58), panel (c), in the figure on the left.

Moreover, based on the assumption that spin-orbit coupling could also be treated subsequently in a state-interaction approach as a (minor) perturbation to a correlated (scalar-relativistic) electronic-structure theory, we recently introduced an MPS state-interaction approach whose workflow is outlined below. The results of a two-step procedure depend critically on the convergence of the resulting spin-orbit coupled wave functions with respect to the number of spin-free states considered for the state-interaction approach. Hence, the latter provides a useful check on the quality of the finite state selection by a direct comparison to reference (four-component) data. Future work will require to tackle the orbital relaxation problem for a relativistic multiconfigurational (DMRG) reference wave function as well as to address the dynamical electron correlation issue along the lines of the non-relativistic developments.

**Relevant Publications:**

“*An efficient relativistic density-matrix renormalization group implementation in a matrix-product formulation*“

S. Battaglia, S. Keller, and S. Knecht, J. Chem. Theory Comput., 17.12.2017, under revision.

“*A nonorthogonal state-interaction approach for matrix product state wave functions*“

S. Knecht, S. Keller, J. Autschbach, M. Reiher, *J. Chem. Theory Comput.*, **12**, 5881–5894 (2016).

“*Electron correlation within the relativistic no-pair approximation*“

A. Almoukhalalati, S. Knecht, H. J. Aa. Jensen, K. Dyall, T. Saue, *J. Chem. Phys., ***145**, 074104 (2016).

“*A theoretical benchmark study of the spectroscopic constants of the very heavy rare gas dimers*“

A. Shee, S. Knecht, and T. Saue, *Phys. Chem. Chem. Phys.*, **17**, 10978 (2015).

“*Theoretical Study on ThF ^{+}, a prospective system in search of time-reversal violation*“

M. Denis, M. N. Pedersen, H. J. Aa. Jensen, A. Gomes, M. Nayak, S. Knecht, and T. Fleig,

*New J. Phys.*,

**17**, 043005 (2015).

“*Four-component density matrix renormalization group*“

S. Knecht, O. Legeza, and M. Reiher, *J. Chem. Phys.*, **140**, 041101 (2014).

“*Spin-orbit coupling in actinide cations*“

P. S. Bagus, E. S. Ilton, R. L. Martin, H. J. Aa. Jensen, and S. Knecht, *Chem. Phys. Lett.*, **546**, 58-62 (2012).

## Light-Matter Interaction/Light-Harvesting Materials

### Excited-state spectra of light-harvesting materials

Carotenoids (Cars) are naturally occurring pigments that absorb light in the spectral region in which the sun irradiates maximally (450-550 nm); owing to this characteristic they are often present in natural light- harvesting (LH) pigment-protein complexes where they act as donors of electronic energy to (bacterio)chlorophylls [(B)Chls]. As both Cars and (B)Chls can contribute with various close-lying excited states, many different pathways of energy transfer are possible among the two sets of pigments. In addition Cars play a photoprotective role for the same complexes by quenching dangerous products such as chlorophyll triplet and oxygen singlet states generated by an excess of light. Over 1000 naturally occurring Cars are known. In all cases, the common structural feature is a linear chain of alternating C-C single and C=C double bonds while they differ in the π-electron conjugation length (number of conjugated double bonds, N) and in the type and number of functional groups attached to the carbon backbone.

Due to this structural feature the photophysics of the Cars has generally been inter- preted in terms of two low-lying excited singlet states, called S_{2} (1^{1}B^{+}_{u}) and S_{1} (2^{1}A^{–}_{g}). According to optical selection rules, the one-photon-allowed transition from the ground state S_{0} (1^{1}A^{–}_{g}) goes to S_{2}, which then internally converts to S_{1} in a few hundred fem-toseconds. Singlet-singlet excitation energy transfer from Cars to chlorophylls has been described, depending on the antenna complex involved, as going either from S_{2} to the chlorophyll Q_{x} excited state or from S_{1} to the Q_{y} state; in some complexes, both pathways are active. The energy transfer efficiency of each of the possible pathways depends on the conjugation length N of the Car and on the functional groups attached to the carbon backbone.

In recent publications we tackled the challenging excited-state issue of Cars by making use of a hybrid DFT/MRCI approach. In addition, we assessed the performance of the single-reference TDDFT approach within the Tamm-Dancoff approximation with respect to both transition energies and transition densities by comparison to the DFT/MRCI results. As illustrated on the left, for the best-performing (judging from a comparison of a wide range of functionals of different type) meta-GGA functional TPSS, the overall agreement is qualitatively and quantitatively good for the S_{1} and S_{3} states with errors oscillating around 0.1 eV with the only evident exception being the S_{3} state of the carbonyl-containing Car peridinin for which a much larger error is found. The description of the bright S_{2} state is slightly worse with maximum errors of around 0.2 eV for the longer Cars. The better behavior found for the S_{1} and S_{3} states with respect to S_{2} state could be ascribed to a fortuitous cancellation of errors that occurs in a TDA description for states exhibiting a non-negligible double excitation character.

### Resoncance Raman spectroscopy

A spectroscopic characterization of molecules, relevant in a biological context, e.g., pro- teins or nucleic acids, is a challenging task. Considering vibrational spectroscopic techniques, e.g., infrared or Raman spectroscopy, the number of peaks may become pro- hibitively large for an assignment due to close-lying or overlapping peaks. By contrast resonance Raman (RR) spectroscopy is a selective vibrational spectroscopic technique because the resonance conditions act like a filter such that only certain peaks are selectively enhanced in the vibrational spectrum. In RR spectroscopy the energy of the incident light of a given wavelength matches the energy of an electronic transition in a molecule. As a result, only those vibrational frequencies associated with the targeted electronic transition are visible in the RR spectrum. For a given molecule or part of a molecular complex, RR spectroscopy therefore provides selective access to information about excited state structure and dynamics.

In a recent work, we reported the calculation of RR spectra within the framework of MPS wave functions by making extensive use of state-specific gradients obtained for DMRG-SCF wave functions. As shown on the left, our new computational model allowed us to study valence correlation effects on the RR spectrum of uracil (in gas phase) by considering active orbital spaces of different sizes and compositions which would not have been possible with a “traditional” wave-function based electron correlation approach.

### Mössbauer spectroscopy

Molecular properties of heavy element compounds can be affected by subtle relativistic effects such as, for example, spin-orbit coupling. Numerous examples for such phenomena are known, especially if the observables are probed at a heavy atomic nucleus. One such property is the contact density, i.e., the electron (number) density at the center of an atomic nucleus. The contact density can be related to the chemical isomer shift that is observed in Mössbauer spectra if the frequency of the γ-radiation absorbed by a nucleus (absorber nucleus) in a solid is not equal to the one emitted by a source nucleus of the same element. Rephrased in terms of relative energy, the energy difference between ground and excited states of the absorbing compound is different from that of the emitting compound. This relative energy shift is usually expressed in terms of the speed of the source relative to the absorber which creates the Doppler shift necessary to bring the emitter and absorber into resonance. In recent works, we considered the application of (scalar-)relativistic electron correlation approaches towards the prediction of Mössbauer spectra and calibration of computational models for the calculation of the latter for prominent atomic nuclei such as Fe and Hg.

**Relevant Publications:**

“*Excited state characterization of carbonyl containing carotenoids: a comparison between single and multi reference descriptions*“

R. Spezia, S. Knecht, B. Mennucci, *Phys. Chem. Chem. Phys., 16, 17156-17166 (2017).*

“*Multiconfigurational effects in theoretical resonance Raman spectra*“

Y. Ma, S. Knecht, M. Reiher, *ChemPhysChem, ***18**, 384–393 (2017).

“*Carotenoids and light-harvesting: from DFT/MRCI to the Tamm-Dancoff approximation*“

O. Andreussi, S. Knecht, C. M. Marian, J. Kongsted, and B. Mennucci, *J. Chem. Theory Comput.*, **11**, 655-666 (2015).

“*Assessment of charge-transfer excitations with time-dependent, range-separated density functional theory based on long-range MP2 and multiconfigurational self-consistent field wave functions*“

E. D. Hedegård, F. Heiden, S. Knecht, E. Fromager, and H. J. Aa. Jensen, *J. Chem. Phys.*, **139**, 184308 (2013).

“*On the photophysics of carotenoids: A multireference DFT study of peridinin*“

S. Knecht, C. M. Marian, J. Kongsted, and B. Mennucci, *J. Phys. Chem. B.*, **117**, 13808-13815 (2013).

“*Toward reliable prediction of the energy ladder in multichromophoric systems: A benchmark study on the FMO light-harvesting complex*“

N. Holmgaard List, C. Curutchet, S. Knecht, B. Mennucci, and J. Kongsted, *J. Chem. Theory Comput.*, **9**, 4928-4938 (2013).

“*An interpretation of the absorption and emission spectra of the gold dimer using modern theoretical tools*“

K. R. Geethalakshmi, F. Ruiperez, S. Knecht, J. M. Ugalde, M. Morse, and I. Infante, *Phys. Chem. Chem. Phys.*, **14**, 8732-8741 (2012).

“*Nuclear size effects in rotational spectra: a tale with a twist*“

S. Knecht and T. Saue, *Chem. Phys.*, **401**,103-112 (2012).

“*Mössbauer spectroscopy for heavy elements: a relativistic benchmark study of mercury*“

S. Knecht, S. Fux, R. van Meer, L. Visscher, M. Reiher, and T. Saue, *Theor. Chem. Acc.*, **129**, 631-650 (2011).

## Static and Response Properties for (Open-Shell) Molecules

The calculation of (first-order) molecular properties is intimately connected with the evaluation of expectation values for a given property operator and optimized wave function(s). The evaluation becomes cumbersome if the bra and ket wave functions are nonorthogonal and do not share the same molecular orbital basis, which is, for example, the case for state-specific orbital-optimized MPSs. This requires the development of a general state-interaction framework for MPS wave functions introduced above. The latter can be employed to, e.g., determine spin-orbit coupling matrix elements between spin-free (scalar-relativistic) wave functions. This is particularly useful in the context of predicting magnetic properties such as electronic g-factors thus bridging the gap from theory to experimental EPR spectroscopy. A deviation from the free-electron g value is solely a relativistic effect and its calculation necessitates spin-orbit coupled wave functions.

The caveat concerning the convergence of spin-orbit properties in a state-interaction procedure becomes apparent in the study of g-factors of the NpO_{2}^{+} molecule. Whereas a reasonable convergence of the g-factors can be achieved by considering nearly 40 interacting spin-free MPSs, a similarly accurate value could be obtained from a simple expectation-value calculation involving only a single relativistic four-component CI wave function and its time-reversal complement (see left).

An obvious alternative to an MPS state-interaction approach is the development of a genuine relativistic framework, as indicated above, combining an MPS wave function optimization with a variational account of spin-orbit coupling. This will open up the way to fully relativistic studies on optical (x-ray, UV-Vis, etc.) and magnetic properties (EPR, paramagnetic NMR, …) of open-shell molecular complexes, as was first indicated in a real-space calculation of the current density of a dysprosium-containing lanthanide chelating tag for paramagnetic labeling of biomolecules.

**Relevant Publications:**

“*An efficient relativistic density-matrix renormalization group implementation in a matrix-product formulation*“

S. Battaglia, S. Keller, and S. Knecht, J. Chem. Theory Comput., 17.12.2017, under revision.

“*A nonorthogonal state-interaction approach for matrix product state wave functions*“

S. Knecht, S. Keller, J. Autschbach, M. Reiher, *J. Chem. Theory Comput.*, **12**, 5881–5894 (2016).

“*Theoretical Study on ThF ^{+}, a prospective system in search of time-reversal violation*“

M. Denis, M. N. Pedersen, H. J. Aa. Jensen, A. Gomes, M. Nayak, S. Knecht, and T. Fleig,

*New J. Phys.*,

**17**, 043005 (2015).

“*Spin-orbit coupling in actinide cations*“

P. S. Bagus, E. S. Ilton, R. L. Martin, H. J. Aa. Jensen, and S. Knecht, *Chem. Phys. Lett.*, **546**, 58-62 (2012).

## Embedding Approaches

Most molecular calculations are – for simplicity – carried out for isolated molecular systems. Clearly, this poses a severe limitation towards an unambiguous understanding of the chemistry of molecular complexes at a molecular level since a comparison to experimental data, which is often recorded either in solution or other complex environments, then hinges on a favorable error compensation and/or negligible solvation effects for the spectroscopic property in question. Considering our recent efforts towards novel novel multiconfigurational methods**, **first important steps towards a combination with promising and flexible embedding schemes were recently taken within our DMRG framework. Its combination with the dynamic electron correlation theories and, equally important, with both the one- and two-step relativistic DMRG theories that we are currently working on will pave the way for a unique framework to carry out in a focused nature tailor-made spectroscopic studies of embedded molecular systems.

**Relevant Publications:**

“*Self-consistent embedding of density-matrix renormalization group wavefunctions in a density functional environment*“

T. Dresselhaus, J. Neugebauer, S. Knecht, S. Keller, Y. Ma, and M. Reiher, *J. Chem. Phys.*, **142**, 044111 (2015).

“*Polarizable Embedding with a Multiconfiguration Short-Range Density Functional Theory Linear Response Method*“

E. Hedegård, J. M. Olsen, S. Knecht, J. Kongsted, and H. J. Aa. Jensen, *J. Chem. Phys.*, **142**, 114113 (2015).

“*Benchmarking TD-DFT for excited state geometries of organic molecules in gas-phase and in solution*“

C. A. Guido, S. Knecht, J. Kongsted, and B. Mennucci, *J. Chem. Theory Comput.*, **9**, 2209-2220 (2013)