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In electronic structure theory, variational computing offers a valuable paradigm for the approximation of electronic ground states. However, for historical reasons, this principle is mostly restricted to model-chemistries in pre-defined fixed basis sets. Especially in quantum computation, these model-chemistries are far from an accurate description of the initial electronic Hamiltonian. Throwing down the gauntlet, we here demonstrate a fully variational approach to the electronic structure problem by variationally optimizing the orbitals representing the second quantized Hamiltonian alongside a quantum circuit that generates the many-electron wavefunction. To this end, the orbitals are represented within an adaptive multi-wavelet format, guaranteeing numerical precision. We then showcase explicit numerical protocols and highlight the quantum circuit's effects on determining the optimal orbital basis.

We present a measurement-cost efficient implementation of Strongly-Contracted $N$-Electron Valence Perturbation Theory (SC-NEVPT2) for use on near-term quantum devices. At the heart of our algorithm we exploit the properties of adaptive Informationally Complete positive operator valued measures (IC-POVMs) to recycle the measurement outcomes from a ground state energy estimation on a quantum device to reconstruct the matrix elements of the three- and four-body reduced density matrices for use in a subsequent CPU-driven NEVPT2 calculation. The proposed scheme is capable of producing results in good agreement with corresponding conventional NEVPT2 simulations, while significantly reducing the cost of quantum measurements and allowing for embarrassingly parallel estimations of higher-order RDMs in classical post-processing. Our scheme shows favourable scaling of the total number of shots with respect to system size. This paves the way for routine inclusion of dynamic electron correlation effects in hybrid quantum-classical computing pipelines., Comment: 12 pages, 6 figures

Quantum chemistry simulations offer a cost-effective way for computational design of BODIPY photosensitizers with potential use in photodynamic therapy (PDT). However, accurate predictions of photophysical properties, such as excitation energies, pose a challenge for the popular time-dependent density functional theory (TDDFT) and equation-of-motion coupled cluster with singles and doubles (EOM-CCSD) methods. By contrast, reliable descriptions can be achieved by multi-reference quantum chemistry methods, though unfortunately, their computational cost grows exponentially with the number of correlated electrons. Alternatively, quantum computing holds a great potential for exact simulation of photophysical properties in a computationally more efficient way. To this end, we introduce the state-specific $\Delta$UCCSD-VQE (unitary coupled cluster with singles and doubles variational quantum eigensolver) and $\Delta$ADAPT-VQE methods in which the electronically excited state is calculated via a non-Aufbau electronic configuration. The accuracy and capability of the developed methods are assessed against experimentally determined excitation energies for six BODIPY-derivatives. We show that the proposed methods predict accurate vertical excitation energies that are not only in good agreement with experimental reference data but also outperform popular quantum chemistry methods, such as TDDFT and EOM-CCSD. Spurred by its impressive performance and simplicity, we are confident that $\Delta$ADAPT will emerge as the method of choice for guiding the rational design of photosensitizers for PDT and photocatalysis in the era of near-term quantum computing.

We test the performance of the Polarizable Embedding Variational Quantum Eigensolver Self-Consistent-Field (PE-VQE-SCF) model for computing electric field gradients with comparisons to conventional complete active space self-consistent-field (CASSCF) calculations and experimental results. We compute quadrupole coupling constants for ice VIII and ice IX. We find that the inclusion of the environment is crucial for obtaining results that match the experimental data. The calculations for ice VIII are within the experimental uncertainty for both CASSCF and VQE-SCF for oxygen and lie close to the experimental value for ice IX as well. With the VQE-SCF, which is based on an Adaptive Derivative-Assembled Problem-Tailored (ADAPT) ansatz, we find that the inclusion of the environment and the size of the different basis sets do not directly affect the gate counts. However, by including an explicit environment, the wavefunction and, therefore, the optimization problem becomes more complicated, which usually results in the need to include more operators from the operator pool, thereby increasing the depth of the circuit., Comment: 18 pages, 5 figures

The near-term utility of quantum computers is hindered by hardware constraints in the form of noise. One path to achieving noise resilience in hybrid quantum algorithms is to decrease the required circuit depth -- the number of applied gates -- to solve a given problem. This work demonstrates how to reduce circuit depth by combining the transcorrelated (TC) approach with adaptive quantum ans\"atze and their implementations in the context of variational quantum imaginary time evolution (AVQITE). The combined TC-AVQITE method is used to calculate ground state energies across the potential energy surfaces of H$_4$, LiH, and H$_2$O. In particular, H$_4$ is a notoriously difficult case where unitary coupled cluster theory, including singles and doubles excitations, fails to provide accurate results. Adding TC yields energies close to the complete basis set (CBS) limit while reducing the number of necessary operators -- and thus circuit depth -- in the adaptive ans\"atze. The reduced circuit depth furthermore makes our algorithm more noise-resilient and accelerates convergence. Our study demonstrates that combining the TC method with adaptive ans\"atze yields compact, noise-resilient, and easy-to-optimize quantum circuits that yield accurate quantum chemistry results close to the CBS limit.

We formulate and implement the Variational Quantum Eigensolver Self Consistent Field (VQE-SCF) algorithm in combination with polarizable embedding (PE), thereby extending PE to the regime of quantum computing. We test the resulting algorithm, PE-VQE-SCF, on quantum simulators and demonstrate that the computational stress on the quantum device is only slightly increased in terms of gate counts compared to regular VQE-SCF. On the other hand, no increase in shot noise was observed. We illustrate how PE-VQE-SCF may lead to the modeling of real chemical systems using a simulation of the reaction barrier of the Diels-Alder reaction between furan and ethene as an example., Comment: 19 pages, 5 figures, submitted to Journal of Chemical Physics

Nuclear quantum effects such as zero-point energy and hydrogen tunnelling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets whereas realistic simulations beyond the Born--Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H$_2$ and D$_2$ molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by a several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles (NEO-UCCSD) while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method -- an important step toward accurate simulations involving non-classical nuclei and non--Born--Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects and we demonstrate that inclusion of correlation energy systematically improves the prediction of $\Delta$ZPE.

Automated active space selection is arguably one of the most challenging and essential aspects of multiconfigurational methods. In this work we propose an effective quantum information-assisted complete active space optimization (QICAS) scheme. What sets QICAS apart from other correlation-based selection schemes is (i) the use of unique measures from quantum information that assess the correlation in electronic structures in an unambiguous and predictive manner, and (ii) an orbital optimization step that minimizes the correlation discarded by the active space approximation. Equipped with these features QICAS yields for smaller correlated molecules sets of optimized orbitals with respect to which the CASCI energy reaches the corresponding CASSCF energy within chemical accuracy. For more challenging systems such as the Chromium dimer, QICAS offers an excellent starting point for CASSCF by greatly reducing the number of iterations required for numerical convergence. Accordingly, our study validates a profound empirical conjecture: the energetically optimal non-active spaces are predominantly those that contain the least entanglement., Comment: Close to published version

In recent years, there has been tremendous progress in the development of quantum computing hardware, algorithms and services leading to the expectation that in the near future quantum computers will be capable of performing simulations for natural science applications, operations research, and machine learning at scales mostly inaccessible to classical computers. Whereas the impact of quantum computing has already started to be recognized in fields such as cryptanalysis, natural science simulations, and optimization among others, very little is known about the full potential of quantum computing simulations and machine learning in the realm of healthcare and life science (HCLS). Herein, we discuss the transformational changes we expect from the use of quantum computation for HCLS research, more specifically in the field of cell-centric therapeutics. Moreover, we identify and elaborate open problems in cell engineering, tissue modeling, perturbation modeling, and bio-topology while discussing candidate quantum algorithms for research on these topics and their potential advantages over classical computational approaches., Comment: 6 figures

We present a self consistent field approach (SCF) within the Adaptive Derivative-Assembled Problem-Tailored Ansatz Variational Quantum Eigensolver (ADAPT-VQE) framework for efficient quantum simulations of chemical systems on near-term quantum computers. To this end, our ADAPT-VQE-SCF approach combines the idea of generating an ansatz with a small number of parameters, resulting in shallow-depth quantum circuits with a direct minimization of an energy expression which is correct to second order with respect to changes in the molecular orbital basis. Our numerical analysis, including calculations for the transition metal complex ferrocene (Fe$\rm (C_5H_5)_2$), indicates that convergence in the self-consistent orbital optimization loop can be reached without a considerable increase in the number of two-qubit gates in the quantum circuit by comparison to a VQE optimization in the initial molecular orbital basis. Moreover, the orbital optimization can be carried out simultaneously within each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us to implement a routine analogous to CASSCF, a cornerstone of state-of-the-art computational chemistry, in a hardware-efficient manner on near-term quantum computers. Hence, ADAPT-VQE-SCF paves the way towards a paradigm shift for quantitative quantum-chemistry simulations on quantum computers by requiring fewer qubits and opening up for the use of large and flexible atomic orbital basis sets in contrast to earlier methods that are predominantly based on the idea of full active spaces with minimal basis sets., Comment: 21 pages, 5 figures

Fermion-to-qubit mappings are used to represent fermionic modes on quantum computers, an essential first step in many quantum algorithms for electronic structure calculations. In this work, we present a formalism to design flexible fermion-to-qubit mappings from ternary trees. We discuss in an intuitive manner the connection between the generating trees' structure and certain properties of the resulting mapping, such as Pauli weight and the delocalisation of mode occupation. Moreover, we introduce a recipe that guarantees Fock basis states are mapped to computational basis states in qubit space, a desirable property for many applications in quantum computing. Based on this formalism, we introduce the Bonsai algorithm, which takes as input the potentially limited topology of the qubit connectivity of a quantum device and returns a tailored fermion-to-qubit mapping that reduces the SWAP overhead with respect to other paradigmatic mappings. We illustrate the algorithm by producing mappings for the heavy-hexagon topology widely used in IBM quantum computers. The resulting mappings have a favourable Pauli weight scaling $\mathcal{O}(\sqrt{N})$ on this connectivity, while ensuring that no SWAP gates are necessary for single excitation operations.

Scientific and technological advances in medicine and systems biology have unequivocally shown that health and disease must be viewed in the context of the interplay among multiple molecular and environmental factors. Understanding the effects of cellular interconnection on disease progression may lead to the identification of novel disease genes and pathways, and hence influence precision diagnostics and therapeutics. To accomplish this goal, the emerging field of network medicine applies network science approaches to investigate disease pathogenesis, integrating information from relevant Omics databases, including protein-protein interaction, correlation-based, gene regulatory, and Bayesian networks. However, this requires analysing and computing large amounts of data. Moreover, if we are to efficiently search for new drugs and new drug combinations, there is a pressing need for computational methods that could allow us to access the immense chemical compound space until now largely unexplored. Finally, at the microscopic level, drug-target chemistry simulation is ultimately a quantum problem, and hence it requires a quantum solution. As we will discuss, quantum computing may be a key ingredient in enabling the full potential of network medicine. We propose to combine network medicine and quantum algorithms in a novel research field, quantum network medicine, to lay the foundations of a new era of disease prevention and drug design.

The second quantum revolution is all about exploiting the quantum nature of atoms and molecules to execute quantum information processing tasks. To support this growing endeavor and by anticipating the key role of quantum chemistry therein, our work establishes a toolbox for systematically exploring, quantifying and dissecting correlation effects in quantum chemical systems. By utilizing the geometric picture of quantum states we compare -- on a unified basis and in an operationally meaningful way -- total, quantum and classical correlation and entanglement in molecular ground states. To maximize the quantum informational resourcefulness of molecules an orbital optimization scheme is provided, leading to a paradigm-shifting insight: A single covalent bond equates to the entanglement $2\ln(2)$. This novel and more versatile perspective on electronic structure suggests a generalization of valence bond theory, overcoming deficiencies of modern chemical bonding theories., Comment: Close to published version

Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple, yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic \textit{and} spin-orbit two-electron picture-change effects (PCE) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, $viz.$ Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need of a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group-18 (closed-shell) and group-16 (open-shell) diatomic molecules, achieving a consistent $\approx\!10^{-5}$ Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnF$_{n}$, n=2,4,6), as well as equation-of-motion coupled cluster calculations of X-ray core ionization energies of $5d$ and $6d$-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone towards a universal and reliable relativistic two-component quantum chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost., Comment: The following article has been submitted to The Journal of Chemical Physics. After it is published, it will be found at this https://aip.scitation.org/toc/jcp/current

DIRAC is a freely distributed general-purpose program system for 1-, 2- and 4-component relativistic molecular calculations at the level of Hartree--Fock, Kohn--Sham (including range-separated theory), multiconfigurational self-consistent-field, multireference configuration interaction, coupled cluster and electron propagator theory. At the self-consistent-field level a highly original scheme, based on quaternion algebra, is implemented for the treatment of both spatial and time reversal symmetry. DIRAC features a very general module for the calculation of molecular properties that to a large extent may be defined by the user and further analyzed through a powerful visualization module. It allows the inclusion of environmental effects through three different classes of increasingly sophisticated embedding approaches: the implicit solvation polarizable continuum model, the explicit polarizable embedding, and frozen density embedding models. DIRAC was one of the earliest codes for relativistic molecular calculations and remains a reference in its field., Comment: The following article has been submitted to The Journal of Chemical Physics. After it is published, it will be found at https://aip.scitation.org/toc/jcp/current

We present an approximate scheme for analytical gradients and nonadiabatic couplings for calculating state-average density matrix renormalization group self-consistent-field wavefunction. Our formalism follows closely the state-average complete active space self-consistent-field (SA-CASSCF) \emph{ansatz}, which employs a Lagrangian, and the corresponding Lagrange multipliers are obtained from a solution of the coupled-perturbed CASSCF (CP-CASSCF) equations. We introduce a definition of the matrix product state (MPS) Lagrange multipliers based on a single-site tensor in a mixed-canonical form of the MPS, such that a sweep procedure is avoided in the solution of the CP-CASSCF equations. We apply our implementation to the optimization of a conical intersection in 1,2-dioxetanone, where we are able to fully reproduce the SA-CASSCF result up to arbitrary accuracy., Comment: 17 pages, 3 figures, 2 tables

The density matrix renormalization group (DMRG) is a powerful method to treat static correlation. Here we present an inexpensive way to add additional dynamic correlation energy to a DMRG self-consistent field (DMRG) wave function using pair-density functional theory (PDFT). We applied this new approach, called DMRG-PDFT, to study singlet-triplet gaps in polyacenes and polyacetylenes that require active spaces larger than the feasibility limit of the conventional complete active-space self-consistent field (CASSCF) method. The results match reasonably well the most reliable literature values and have only a moderate dependence on the compression of the initial DMRG wave function. Furthermore, DMRG-PDFT is significantly less expensive than other commonly applied ways of adding additional correlation to DMRG, such as DMRG followed by multireference perturbation theory or multireference configuration interaction.

We present an implementation of the relativistic quantum-chemical density matrix renormalization group (DMRG) approach based on a matrix-product formalism. Our approach allows us to optimize matrix product state (MPS) wave functions including a variational description of scalar-relativistic effects and spin-orbit coupling from which we can calculate, for example, first-order electric and magnetic properties in a relativistic framework. While complementing our pilot implementation (S. Knecht et al., J. Chem. Phys., 140, 041101 (2014)) this work exploits all features provided by its underlying non-relativistic DMRG implementation based on an matrix product state and operator formalism. We illustrate the capabilities of our relativistic DMRG approach by studying the ground-state magnetization as well as current density of a paramagnetic $f^9$ dysprosium complex as a function of the active orbital space employed in the MPS wave function optimization., Comment: 41 pages, 15 figures

The natural occupation numbers of fermionic systems are subject to non-trivial constraints, which include and extend the original Pauli principle. A recent mathematical breakthrough has clarified their mathematical structure and has opened up the possibility of a systematic analysis. Early investigations have found evidence that these constraints are exactly saturated in several physically relevant systems, e.g., in a certain electronic state of the Beryllium atom. It has been suggested that in such cases, the constraints, rather than the details of the Hamiltonian, dictate the system's qualitative behaviour. Here, we revisit this question with state-of-the-art numerical methods for small atoms. We find that the constraints are, in fact, not exactly saturated, but that they lie much closer to the surface defined by the constraints than the geometry of the problem would suggest. While the results seem incompatible with the statement that the generalized Pauli constraints drive the behaviour of these systems, they suggest that the qualitatively correct wave-function expansions can in some systems already be obtained on the basis of a limited number of Slater determinants, which is in line with numerical evidence from quantum chemistry., Comment: published version

The extrapolation technique of Savin [J. Chem. Phys. 140, 18A509 (2014)], which was initially applied to range-separated ground-state-density-functional Hamiltonians, is adapted in this work to ghost-interaction-corrected (GIC) range-separated ensemble density-functional theory (eDFT) for excited states. While standard extrapolations rely on energies that decay as $\mu^{-2}$ in the large range-separation-parameter $\mu$ limit, we show analytically that (approximate) range-separated GIC ensemble energies converge more rapidly (as $\mu^{-3}$) towards their pure wavefunction theory values ($\mu\rightarrow+\infty$ limit), thus requiring a different extrapolation correction. The purpose of such a correction is to further improve on the convergence and, consequently, to obtain more accurate excitation energies for a finite (and, in practice, relatively small) $\mu$ value. As a proof of concept, we apply the extrapolation method to He and small molecular systems (viz. H$_{2}$, HeH$^{+}$ and LiH), thus considering different types of excitations like Rydberg, charge transfer and double excitations. Potential energy profiles of the first three and four singlet $\Sigma^+$ excitation energies in HeH$^{+}$ and H$_{2}$, respectively, are studied with a particular focus on avoided crossings for the latter. Finally, the extraction of individual state energies from the ensemble energy is discussed in the context of range-separated eDFT, as a perspective., Comment: 17 pages, 10 figures, regular article

Carotenoids can play multiple roles in biological photoreceptors thanks to their rich photophysics. In the present work, we have investigated six of the most common carbonyl containing carotenoids: Echinenone, Canthaxanthin, Astaxanthin, Fucoxanthin, Capsanthin and Capsorubin. Their excitation properties are investigated by means of a hybrid density functional theory (DFT) and multireference configuration interaction (MRCI) approach to elucidate the role of the carbonyl group: the bright transition is of {\pi}{\pi}* character, as expected, but the presence of a C=O moiety reduces the energy of n{\pi}* transitions which may become closer to the {\pi}{\pi}* transition, in particular as the conjugation chain decreases. This can be related to the presence of a low-lying charge transfer state typical of short carbonyl- containing carotenoids. The DFT/MRCI results are finally used to benchmark single- reference time-dependent DFT-based methods: among the investigated functionals, the meta- GGA (and in particular M11L and MN12L) functionals show to perform the best for all six investigated systems., Comment: 15 pages, 10 figures

In the present article, we show how to formulate the partially contracted n-electron valence second order perturbation theory (NEVPT2) energies in the atomic and active molecular orbital basis by employing the Laplace transformation of orbital-energy denominators (OED). As atomic-orbital (AO) basis functions are inherently localized and the number of active orbitals is comparatively small, our formulation is particularly suited for a linearly-scaling NEVPT2 implementation. Some of the NEVPT2 energy contributions can be formulated completely in the AO basis as single-reference second-order M{\o}ller-Plesset perturbation theory and benefit from sparse active-pseudo density matrices - particularly if the active molecular orbitals are localized only in parts of a molecule. Furthermore, we show that for multi-reference perturbation theories it is particularly challenging to find optimal parameters of the numerical Laplace transformation as the fit range may vary among the 8 different OEDs by many orders of magnitude. Selecting the number of quadrature points for each OED separately according to an accuracy-based criterion allows us to control the errors in the NEVPT2 energies reliably.

We present a matrix-product state (MPS)-based quadratically convergent density-matrix renormalization group self-consistent-field (DMRG-SCF) approach. Following a proposal by Werner and Knowles (JCP 82, 5053, (1985)), our DMRG-SCF algorithm is based on a direct minimization of an energy expression which is correct to second-order with respect to changes in the molecular orbital basis. We exploit a simultaneous optimization of the MPS wave function and molecular orbitals in order to achieve quadratic convergence. In contrast to previously reported (augmented Hessian) Newton-Raphson and super-configuration-interaction algorithms for DMRG-SCF, energy convergence beyond a quadratic scaling is possible in our ansatz. Discarding the set of redundant active-active orbital rotations, the DMRG-SCF energy converges typically within two to four cycles of the self-consistent procedure, Comment: 40 pages, 5 figures, 3 tables

We present a state-interaction approach for matrix product state (MPS) wave functions in a nonorthogonal molecular orbital basis. Our approach allows us to calculate for example transition and spin-orbit coupling matrix elements between arbitrary electronic states provided that they share the same one-electron basis functions and active orbital space, respectively. The key element is the transformation of the MPS wave functions of different states from a nonorthogonal to a biorthonormal molecular orbital basis representation exploiting a sequence of non-unitary transformations following a proposal by Malmqvist (Int. J. Quantum Chem. 30, 479 (1986)). This is well-known for traditional wave-function parametrizations but has not yet been exploited for MPS wave functions., Comment: 34 pages, 1 figure, submitted to J. Chem. Theory Comput

We present a second-order N-electron valence state perturbation theory (NEVPT2) based on a density matrix renormalization group (DMRG) reference wave function that exploits a Cholesky decomposition of the two-electron repulsion integrals (CD-DMRG-NEVPT2). With a parameter-free multireference perturbation theory approach at hand, the latter allows us to efficiently describe static and dynamic correlation in large molecular systems. We demonstrate the applicability of CD-DMRG-NEVPT2 for spin-state energetics of spin-crossover complexes involving calculations with more than 1000 atomic basis functions. We first assess in a study of a heme model the accuracy of the strongly- and partially-contracted variant of CD-DMRG-NEVPT2 before embarking on resolving a controversy about the spin ground state of a cobalt tropocoronand complex., Comment: 9 pages, 4 figures, 2 tables

Ensemble density-functional theory (eDFT) suffers from the so-called "ghost interaction" error when approximate exchange-correlation functionals are used. In this work, we present a rigorous ghost interaction correction (GIC) scheme in the context of range-separated eDFT. The method relies on an exact decomposition of the ensemble short-range exchange-correlation energy into a multideterminantal exact exchange term, which involves the long-range interacting ensemble density matrix instead of the Kohn--Sham (KS) one, and a complementary density-functional correlation energy. A generalized adiabatic connection formula is derived for the latter. In order to perform practical calculations, the complementary correlation functional has been simply modeled by its ground-state local density approximation (LDA) while long-range interacting ground- and excited-state wavefunctions have been obtained self-consistently by combining a long-range configuration interaction calculation with a short-range LDA potential. We show that GIC reduces the curvature of approximate range-separated ensemble energies drastically while providing considerably more accurate excitation energies, even for charge-transfer and double excitations. Interestingly, the method performs well also in the context of standard KS-eDFT, which is recovered when the range-separation parameter is set to zero., Comment: 10 pages, 7 figures

Reliable quantum chemical methods for the description of molecules with dense-lying frontier orbitals are needed in the context of many chemical compounds and reactions. Here, we review developments that led to our newcomputational toolbo x which implements the quantum chemical density matrix renormalization group in a second-generation algorithm. We present an overview of the different components of this toolbox., Comment: 19 pages, 1 table

The combination of a recently proposed linear interpolation method (LIM) [Senjean et al., Phys. Rev. A 92, 012518 (2015)], which enables the calculation of weight-independent excitation energies in range-separated ensemble density-functional approximations, with the extrapolation scheme of Savin [J. Chem. Phys. 140, 18A509 (2014)] is presented in this work. It is shown that LIM excitation energies vary quadratically with the inverse of the range-separation parameter mu when the latter is large. As a result, the extrapolation scheme, which is usually applied to long-range interacting energies, can be adapted straightforwardly to LIM. This extrapolated LIM (ELIM) has been tested on a small test set consisting of He, Be, H2 and HeH+. Relatively accurate results have been obtained for the first singlet excitation energies with the typical mu=0.4 value. The improvement of LIM after extrapolation is remarkable, in particular for the doubly-excited 2^1Sigma+g state in the stretched H2 molecule. Three-state ensemble calculations in H2 also show that ELIM does not necessarily improves relative excitation energies, even though individual excitation energies are more accurate after extrapolation. Finally, an alternative decomposition of the short-range ensemble exchange-correlation energy is proposed in order to correct for ghost-interaction errors in multideterminant range-separated ensemble density-functional theory calculations. The implementation and calibration of such a scheme is currently in progress., Comment: 25 pages, 7 figures

Gross-Oliveira-Kohn density functional theory (GOK-DFT) for ensembles is in principle very attractive, but has been hard to use in practice. A novel, practical model based on GOK-DFT for the calculation of electronic excitation energies is discussed. The new model relies on two modifications of GOK-DFT: use of range separation and use of the slope of the linearly-interpolated ensemble energy, rather than orbital energies. The range-separated approach is appealing as it enables the rigorous formulation of a multi-determinant state-averaged DFT method. In the exact theory, the short-range density functional, that complements the long-range wavefunction-based ensemble energy contribution, should vary with the ensemble weights even when the density is held fixed. This weight dependence ensures that the range-separated ensemble energy varies linearly with the ensemble weights. When the (weight-independent) ground-state short-range exchange-correlation functional is used in this context, curvature appears thus leading to an approximate weight-dependent excitation energy. In order to obtain unambiguous approximate excitation energies, we propose to interpolate linearly the ensemble energy between equiensembles. It is shown that such a linear interpolation method (LIM) can be rationalized and that it effectively introduces weight dependence effects. As proof of principle, LIM has been applied to He, Be, H$_2$ in both equilibrium and stretched geometries as well as the stretched HeH$^+$ molecule. Very promising results have been obtained for both single (including charge transfer) and double excitations with spin-independent short-range local and semi-local functionals. Even at the Kohn--Sham ensemble DFT level, that is recovered when the range-separation parameter is set to zero, LIM performs better than standard time-dependent DFT., Comment: 26 pages, 8 figures

We present a new hybrid multiconfigurational method based on the concept of range-separation that combines the density matrix renormalization group approach with density functional theory. This new method is designed for the simultaneous description of dynamical and static electron-correlation effects in multiconfigurational electronic structure problems., Comment: 13 pages, 4 figures, 2 tables

We present the first implementation of a density matrix renormalization group algorithm embedded in an environment described by density functional theory. The frozen density embedding scheme is used with a freeze-and-thaw strategy for a self-consistent polarization of the orbital-optimized wavefunction and the environmental densities with respect to each other., Comment: 4 pages, 3 figures

We present the first implementation of the relativistic quantum chemical two- and four-component density matrix renormalization group algorithm that includes a variational description of scalar-relativistic effects and spin-orbit coupling. Numerical results based on the four-component Dirac-Coulomb Hamiltonian are presented for the standard reference molecule for correlated relativistic benchmarks: thallium hydride., Comment: 5 pages, 2 figures, 2 tables

Charge transfer excitations can be described within TD-DFT, not only by means of long-range corrected exchange functionals but also with a combination of wave function theory and TD-DFT based on range separation. The latter approach enables a rigorous formulation of multi-determinantal TD-DFT schemes where excitation classes, which are absent in conventional TD-DFT spectra (like for example double excitations), can be addressed. This paper investigates the combination of both the long-range MCSCF and SOPPA ans\"atze with a short-range DFT (srDFT) description. We find that the combinations of SOPPA or MCSCF with TD-DFT yield better results than could be expected from the pure wave function schemes. For the Time-Dependent MCSCF short-range DFT ansatz (TD-MC-srDFT) excitation energies calculated over a larger benchmark set of molecules with predominantly single reference character yield good agreement with their reference values, and are in general comparable to the long-range corrected functional CAM-B3LYP. The SOPPA-srDFT scheme is tested for a subset of molecules used for benchmarking TD-MC-srDFT and performs slightly better against the reference data for this small subset. Beyond the proof-of-principle calculations comprising the first part of this contribution, we additionally studied the low-lying singlet excited states (S1 and S2) of the retinal chromophore. The chromophore displays multireference character in the ground state and both excited states exhibit considerable double excitation character, which in turn cannot be described within standard TD-DFT, due to the adiabatic approximation. However, a TD-MC-srDFT approach can account for the multireference character, and excitation energies are obtained with accuracy comparable to CASPT2, although using a much smaller active space., Comment: 13 pages

Multi-configuration range-separated density-functional theory is extended to the time-dependent regime. An exact variational formulation is derived. The approximation, which consists in combining a long-range Multi-Configuration-Self-Consistent Field (MCSCF) treatment with an adiabatic short-range density-functional (DFT) description, is then considered. The resulting time-dependent multi-configuration short-range DFT (TD-MC-srDFT) model is applied to the calculation of singlet excitation energies in H2, Be and ferrocene, considering both short-range local density (srLDA) and generalized gradient (srGGA) approximations. In contrast to regular TD-DFT, TD-MC-srDFT can describe double excitations. As expected, when modeling long-range interactions with the MCSCF model instead of the adiabatic Buijse-Baerends density-matrix functional as recently proposed by Pernal [K. Pernal, J. Chem. Phys. 136, 184105 (2012)], the description of both the 1^1D doubly-excited state in Be and the 1^1\Sigma^+_u state in the stretched H2 molecule are improved, although the latter is still significantly underestimated. Exploratory TD-MC-srDFT/GGA calculations for ferrocene yield in general excitation energies at least as good as TD-DFT/CAM-B3LYP, and superior to wave-function (TD-MCSCF, symmetry adapted cluster-configuration interaction) and TD-DFT results based on LDA, GGA, and hybrid functionals., Comment: 41 pages, 4 figures, 2 tables

Last years witnessed a remarkable interest in application of quantum computing for solving problems in quantum chemistry more efficiently than classical computers allow. Very recently, even first proof-of-principle experimental realizations have been reported. However, so far only the non-relativistic regime (i.e. Schroedinger equation) has been explored, while it is well known that relativistic effects can be very important in chemistry. In this letter we present the first quantum algorithm for relativistic computations of molecular energies. We show how to efficiently solve the eigenproblem of the Dirac-Coulomb Hamiltonian on a quantum computer and demonstrate the functionality of the proposed procedure by numerical simulations of computations of the spin-orbit splitting in the SbH molecule. Finally, we propose quantum circuits with 3 qubits and 9 or 10 CNOTs, which implement a proof-of-principle relativistic quantum chemical calculation for this molecule and might be suitable for an experimental realization.