Hilbert space multireference coupled cluster tailored by matrix product states.

In the past decade, the quantum chemical version of the density matrix renormalization group method has established itself as the method of choice for strongly correlated molecular systems. However, despite its favorable scaling, in practice, it is not suitable for computations of dynamic correlation. Several approaches to include that in post-DMRG methods exist; in our group, we focused on the tailored coupled cluster (TCC) approach. This method works well in many situations; however, in exactly degenerate cases (with two or more determinants of equal weight), it exhibits a bias toward the reference determinant representing the Fermi vacuum. Although sometimes it is possible to use a compensation scheme to avoid this bias for energy differences, it is certainly a drawback. In order to overcome this bias of the TCC method, we have developed a Hilbert-space multireference version of tailored CC, which can treat several determinants on an equal footing. We have implemented and compared the performance of three Hilbert-space multireference coupled cluster (MRCC) variants-the state universal one and the Brillouin-Wigner and Mukherjee's state specific ones. We have assessed these approaches on the cyclobutadiene and tetramethyleneethane molecules, which are both diradicals with exactly degenerate determinants at a certain geometry. We have also investigated the sensitivity of the results on the orbital rotation of the highest occupied and lowest unoccupied molecular orbital (HOMO-LUMO) pair, as it is well known that Hilbert-space MRCC methods are not invariant to such transformations.

Vertical Excitation Energies and Lifetimes of the Two Lowest Singlet Excited States of Cytosine, 5-Aza-cytosine, and the Triazine Family: Quantum Mechanics-Molecular Mechanics Studies.

A swarm of semi-classical quantum mechanics/molecular mechanics molecular-dynamics simulations where OM2/MNDO is combined with the Gromacs program for consideration of explicit water is performed, solving the time-dependent Schrödinger equation in each step of the trajectories together with the Tully's fewest switches algorithm. Within this stochastic treatment, time dependent probabilities of the three lowest electronic states are determined. The fact that nucleobases are quickly deactivated is confirmed in the cytosine case where our best lifetime estimation is τ1=0.82 ps for the model with 100 water molecules with the SPCE force field and a time step of 0.1 fs. Lifetimes of the remaining molecules are visibly longer: 5-azacytosine, 2,4-diamino-1,3,5-triazine (DT), and 2,4,6-triamino-1,3,5-triazine (TT) molecules have an S1 → S0 de-excitation time of slightly above 10 ps. The lifetimes of the triazine family increases with the increasing number of exocyclic amino groups, that is, s-triazine < 2-amino-1,3,5-triazine < DT < TT. This can be explained by a higher mobility of the carbon-bonded hydrogen atoms in comparison with heavier amino groups since their movement is slowed down due to a substantially higher mass than hydrogen atoms, which can easier reach the out-of-plane positions required in the conical intersection structures. Moreover, bulkier NH2 ligands suffer due to greater friction caused by the surrounding water environment. These mechanical aspects caused a change in the explored lifetime dependences in comparison with our previous gas-phase study.

Newton-X Platform: New Software Developments for Surface Hopping and Nuclear Ensembles.

Newton-X is an open-source computational platform to perform nonadiabatic molecular dynamics based on surface hopping and spectrum simulations using the nuclear ensemble approach. Both are among the most common methodologies in computational chemistry for photophysical and photochemical investigations. This paper describes the main features of these methods and how they are implemented in Newton-X. It emphasizes the newest developments, including zero-point-energy leakage correction, dynamics on complex-valued potential energy surfaces, dynamics induced by incoherent light, dynamics based on machine-learning potentials, exciton dynamics of multiple chromophores, and supervised and unsupervised machine learning techniques. Newton-X is interfaced with several third-party quantum-chemistry programs, spanning a broad spectrum of electronic structure methods.

Theoretical Investigation of the Effect of Alkylation and Bromination on Intersystem Crossing in BODIPY-Based Photosensitizers.

Halogenated and alkylated BODIPY derivatives are reported as suitable candidates for their use as photosensitizers in photodynamic therapy due to their efficient intersystem crossing (ISC) between states of different spin multiplicities. Spin-orbit couplings (SOCs) are evaluated using an effective one-electron spin-orbit Hamiltonian for brominated and alkylated BODIPY derivatives to investigate the quantitative effect of alkyl and bromine substituents on ISC. BODIPY derivatives containing bromine atoms have been found to have significantly stronger SOCs than alkylated BODIPY derivatives outside the Frank-Condon region while they are nearly the same at local minima. Based on calculated time-dependent density functional theory (TD-DFT) vertical excitation energies and SOCs, excited-state dynamics of three BODIPY derivatives were further explored with TD-DFT surface hopping molecular dynamics employing a simple accelerated approach. Derivatives containing bromine atoms have been found to have very similar lifetimes, which are much shorter than those of the derivatives possessing just the alkyl moieties. However, both bromine atoms and alkyl moieties reduce the HOMO/LUMO gap, thus assisting the derivatives to behave as efficient photosensitizers.

Ground state of the Fe(ii)-porphyrin model system corresponds to quintet: a DFT and DMRG-based tailored CC study.

Fe(ii)-porphyrins play an important role in many reactions relevant to material science and biological processes, due to their closely lying spin states. Although the prevalent opinion is that these systems posses the triplet ground state, the recent experiment on Fe(ii)-phthalocyanine under conditions matching those of an isolated molecule points toward the quintet ground state. We present a thorough DFT and DMRG-based tailored CC study of Fe(ii)-porphyrin model, in which we address all previously discussed correlation effects. We examine the importance of geometrical parameters, the Fe-N distances in particular, and conclude that the system possesses the quintet ground state.

Toward DMRG-tailored coupled cluster method in the 4c-relativistic domain.

There are three essential problems in computational relativistic chemistry: Electrons moving at relativistic speeds, close lying states, and dynamical correlation. Currently available quantum-chemical methods are capable of solving systems with one or two of these issues. However, there is a significant class of molecules in which all the three effects are present. These are the heavier transition metal compounds, lanthanides, and actinides with open d or f shells. For such systems, sufficiently accurate numerical methods are not available, which hinders the application of theoretical chemistry in this field. In this paper, we combine two numerical methods in order to address this challenging class of molecules. These are the relativistic versions of coupled cluster methods and the density matrix renormalization group (DMRG) method. To the best of our knowledge, this is the first relativistic implementation of the coupled cluster method externally corrected by DMRG. The method brings a significant reduction of computational costs as we demonstrate on the system of TlH, AsH, and SbH.

Near-Linear Scaling in DMRG-Based Tailored Coupled Clusters: An Implementation of DLPNO-TCCSD and DLPNO-TCCSD(T).

We present a new implementation of density matrix renormalization group based tailored coupled clusters method (TCCSD), which employs the domain-based local pair natural orbital approach (DLPNO). Compared to the previous local pair natural orbital (LPNO) version of the method, the new implementation is more accurate, offers more favorable scaling, and provides more consistent behavior across the variety of systems. On top of the singles and doubles, we include the perturbative triples correction (T), which is able to retrieve even more dynamic correlation. The methods were tested on three systems: tetramethyleneethane, oxo-Mn(Salen), and iron(II)-porphyrin model. The first two were revisited to assess the performance with respect to LPNO-TCCSD. For oxo-Mn(Salen), we retrieved between 99.8 and 99.9% of the total canonical correlation energy which is an improvement of 0.2% over the LPNO version in less than 63% of the total LPNO runtime. Similar results were obtained for iron(II)-porphyrin. When the perturbative triples correction was employed, irrespective of the active space size or system, the obtained energy differences between two spin states were within the chemical accuracy of 1 kcal/mol using the default DLPNO settings.

Toward the efficient local tailored coupled cluster approximation and the peculiar case of oxo-Mn(Salen).

We introduce a new implementation of the coupled cluster method with single and double excitations tailored by the matrix product state wave functions (DMRG-TCCSD), which employs the local pair natural orbital (LPNO) approach. By exploiting locality in the coupled cluster stage of the calculation, we were able to remove some of the limitations that hindered the application of the canonical version of the method to larger systems and/or with larger basis sets. We assessed the accuracy of the approximation using two systems: tetramethyleneethane (TME) and oxo-Mn(Salen). Using the default cut-off parameters, we were able to recover over 99.7% and 99.8% of the canonical correlation energy for the triplet and singlet state of TME, respectively. In the case of oxo-Mn(Salen), we found that the amount of retrieved canonical correlation energy depends on the size of the complete active space (CAS)-we retrieved over 99.6% for the larger 27 orbital CAS and over 99.8% for the smaller 22 orbital CAS. The use of LPNO-TCCSD allowed us to perform these calculations up to quadruple-ζ basis set, amounting to 1178 basis functions. Moreover, we examined dependence of the ground state of oxo-Mn(Salen) on the CAS composition. We found that the inclusion of 4dxy orbital plays an important role in stabilizing the singlet state at the DMRG-CASSCF level via double-shell effect. However, by including dynamic correlation, the ground state was found to be triplet regardless of the size of the basis set or the composition of CAS, which is in agreement with previous findings by canonical DMRG-TCCSD in smaller basis.

Photophysics of BODIPY-Based Photosensitizer for Photodynamic Therapy: Surface Hopping and Classical Molecular Dynamics.

Halogenated BODIPY derivatives are emerging as important candidates for photodynamic therapy of cancer cells due to their high triplet quantum yield. We probed fundamental photophysical properties and interactions with biological environments of such photosensitizers. To this end, we employed static TD-DFT quantum chemical calculations as well as TD-DFT surface hopping molecular dynamics on potential energy surfaces resulting from the eigenstates of the total electronic Hamiltonian including the spin-orbit (SO) coupling. Matrix elements of an effective one-electron spin-orbit Hamiltonian between singlet and triplet configuration interaction singles (CIS) auxiliary wave functions are calculated using a new code capable of dealing with singlets and both restricted and unrestricted triplets built up from up to three different and independent sets of (singlet, alpha, and beta) molecular orbitals. The interaction with a biological environment was addressed by using classical molecular dynamics (MD) in a scheme that implicitly accounts for electronically excited states. For the surface hopping trajectories, an accelerated MD approach was used, in which the SO couplings are scaled up, to make the calculations computationally feasible, and the lifetimes are extrapolated back to unscaled SO couplings. The lifetime of the first excited singlet state estimated by semiclassical surface hopping simulations is 139 ± 75 ps. Classical MD demonstrates that halogenated BODIPY in the ground state, in contrast to the unsubstituted one, is stable in the headgroup region of minimalistic cell membrane models, and while in the triplet state, the molecule relocates to the membrane interior ready for further steps of photodynamic therapy.

Quantum information-based analysis of electron-deficient bonds.

Recently, the correlation theory of the chemical bond was developed, which applies concepts of quantum information theory for the characterization of chemical bonds, based on the multiorbital correlations within the molecule. Here, for the first time, we extend the use of this mathematical toolbox for the description of electron-deficient bonds. We start by verifying the theory on the textbook example of a molecule with three-center two-electron bonds, namely, diborane(6). We then show that the correlation theory of the chemical bond is able to properly describe the bonding situation in more exotic molecules which have been synthesized and characterized only recently, in particular, the diborane molecule with four hydrogen atoms [diborane(4)] and a neutral zerovalent s-block beryllium complex, whose surprising stability was attributed to a strong three-center two-electron π bond stretching across the C-Be-C core. Our approach is of high importance especially in the light of a constant chase after novel compounds with extraordinary properties where the bonding is expected to be unusual.

Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer.

In this article, we investigate the numerical and theoretical aspects of the coupled-cluster method tailored by matrix-product states. We investigate formal properties of the used method, such as energy size consistency and the equivalence of linked and unlinked formulation. The existing mathematical analysis is here elaborated in a quantum chemical framework. In particular, we highlight the use of what we have defined as a complete active space-external space gap describing the basis splitting between the complete active space and the external part generalizing the concept of a HOMO-LUMO gap. Furthermore, the behavior of the energy error for an optimal basis splitting, i.e., an active space choice minimizing the density matrix renormalization group-tailored coupled-cluster singles doubles error, is discussed. We show numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose a complete active space. Moreover, the dependence of the ground-state energy error on the complete active space has been analyzed numerically in order to find an optimal split between the complete active space and external space by minimizing the density matrix renormalization group-tailored coupled-cluster error.

Perturbative triples correction to domain-based local pair natural orbital variants of Mukherjee's state specific coupled cluster method.

In this article we report an implementation of the perturbative triples correction to Mukherjee's state-specific multireference coupled cluster method based on the domain-based pair natural orbital approach (DLPNO-MkCC). We tested the performance of DLPNO-MkCCSD(T) in calculations involving tetramethyleneethane and isomers of naphthynes. These tests show that more than 97% of triples energy was recovered with respect to the canonical MkCCSD(T) method, which together with the DLPNO-MkCCSD part accounts for about 99.70-99.85% of the total correlation energy. The applicability of the method was demonstrated on calculations of singlet-triplet gaps for several large systems: triangulene, dynemicin A, and a beryllium complex.

An Isolated Molecule of Iron(II) Phthalocyanin Exhibits Quintet Ground-State: A Nexus between Theory and Experiment.

Iron(II) phthalocyanine (FePc) is an important member of the phthalocyanines family with potential applications in the fields of electrocatalysis, magnetic switching, electrochemical sensing, and phototheranostics. Despite the importance of electronic properties of FePc in these applications, a reliable determination of its ground-state is still challenging. Here we present combined state of the art computational methods and experimental approaches, that is, Mössbauer spectroscopy and Superconducting Quantum Interference Device (SQUID) magnetic measurements to identify the ground state of FePc. While the nature of the ground state obtained with density functional theory (DFT) depends on the functional, giving mostly the triplet state, multi-reference complete active space second-order perturbation theory (CASPT2) and density matrix renormalization group (DMRG) methods assign quintet as the FePc ground-state in gas-phase. This has been confirmed by the hyperfine parameters obtained from 57 Fe Mössbauer spectroscopy performed in frozen monochlorobenzene. The use of monochlorobenzene guarantees an isolated nature of the FePc as indicated by a zero Weiss temperature. The results open doors for exploring the ground state of other metal porphyrin molecules and their controlled spin transitions via external stimuli.

Orientation of Laurdan in Phospholipid Bilayers Influences Its Fluorescence: Quantum Mechanics and Classical Molecular Dynamics Study.

Fluidity of lipid membranes is known to play an important role in the functioning of living organisms. The fluorescent probe Laurdan embedded in a lipid membrane is typically used to assess the fluidity state of lipid bilayers by utilizing the sensitivity of Laurdan emission to the properties of its lipid environment. In particular, Laurdan fluorescence is sensitive to gel vs liquid⁻crystalline phases of lipids, which is demonstrated in different emission of the dye in these two phases. Still, the exact mechanism of the environment effects on Laurdan emission is not understood. Herein, we utilize dipalmitoylphosphatidylcholine (DPPC) and dioleoylphosphatidylcholine (DOPC) lipid bilayers, which at room temperature represent gel and liquid⁻crystalline phases, respectively. We simulate absorption and emission spectra of Laurdan in both DOPC and DPPC bilayers with quantum chemical and classical molecular dynamics methods. We demonstrate that Laurdan is incorporated in heterogeneous fashion in both DOPC and DPPC bilayers, and that its fluorescence depends on the details of this embedding.

The Intricate Case of Tetramethyleneethane: A Full Configuration Interaction Quantum Monte Carlo Benchmark and Multireference Coupled Cluster Studies.

We have performed a full configuration interaction (FCI) quality benchmark calculation for the tetramethyleneethane molecule in the cc-pVTZ basis set employing a subset of complete active space second order perturbation theory, CASPT2(6,6), natural orbitals for the FCI quantum Monte Carlo calculation. The results are in an excellent agreement with the previous large scale diffusion Monte Carlo calculations by Pozun et al. and available experimental results. Our computations verified that there is a maximum on the potential energy surface (PES) of the ground singlet state (1A) 45° torsional angle, and the corresponding vertical singlet-triplet energy gap is 0.01 eV. We have employed this benchmark for the assessment of the accuracy of Mukherjee's coupled clusters with up to triple excitations (MkCCSDT) and CCSD tailored by the density matrix renormalization group method (DMRG). Multireference MkCCSDT with CAS(2,2) model space, though giving good values for the singlet-triplet energy gap, is not able to properly describe the shape of the multireference singlet PES. Similarly, DMRG(24,25) is not able to correctly capture the shape of the singlet surface, due to the missing dynamic correlation. On the other hand, the DMRG-tailored CCSD method describes the shape of the ground singlet state with excellent accuracy but for the correct ordering requires computation of the zero-spin-projection component of the triplet state (3B1).

Domain-Based Local Pair Natural Orbital Version of Mukherjee's State-Specific Coupled Cluster Method.

This article reports development of a local variant of Mukherjee's state-specific multireference coupled cluster method based on the domain-based pair natural orbital approach (DLPNO-MkCC). The current implementation is restricted to connected single and double excitations and model space with up to biexcited references. The performance of the DLPNO-MkCCSD was tested on calculations of tetramethyleneethane. The results show that above 99.9% of the correlation energy was recovered, with respect to the conventional MkCC method. To demonstrate the applicability of the method to large systems, singlet-triplet gaps of triangulene and bis(1-(2,6-diisopropylphenyl)-3,3,5,5-tetramethylpyrrolidine-2-ylidene)beryllium complex were studied. For the last system (105 atoms), we were able to perform a calculation in cc-pVTZ with 2158 basis functions on a single CPU in less than 9 days.

A new approach to molecular dynamics with non-adiabatic and spin-orbit effects with applications to QM/MM simulations of thiophene and selenophene.

We present surface hopping dynamics on potential energy surfaces resulting from the spin-orbit splitting, i.e., surfaces corresponding to the eigenstates of the total electronic Hamiltonian including the spin-orbit coupling. In this approach, difficulties arise because of random phases of degenerate eigenvectors and possibility of crossings of the resulting mixed states. Our implementation solves these problems and allows propagation of the coefficients both in the representation of the spin free Hamiltonian and directly in the "diagonal representation" of the mixed states. We also provide a detailed discussion of the state crossing and point out several peculiarities that were not mentioned in the previous literature. We also incorporate the effect of the environment via the quantum mechanics/molecular mechanics approach. As a test case, we apply our methodology to deactivation of thiophene and selenophene in the gas phase, ethanol solution, and bulk liquid phase. First, 100 trajectories without spin-orbit coupling have been calculated for thiophene starting both in S1 and S2 states. A subset of 32 initial conditions starting in the S2 state was then used for gas phase simulations with spin-orbit coupling utilizing the 3-step integrator of SHARC, our implementation of the 3-step propagator in Newton-X and two new "one-step" approaches. Subsequently, we carried out simulations in ethanol solution and bulk liquid phase for both thiophene and selenophene. For both molecules, the deactivation of the S2 state proceeds via the ring opening pathway. The total population of triplet states reaches around 15% and 40% after 80 fs for thiophene and selenophene, respectively. However, it only begins growing after the ring opening is initiated; hence, the triplet states do not directly contribute to the deactivation mechanism. For thiophene, the resulting deactivation lifetime of the S2 state was 68 fs in the gas phase, 76 fs in ethanol solution, and 78 fs in the liquid phase, in a good agreement with the experimental value of 80 fs (liquid phase). For selenophene, the obtained S2 lifetime was 60 fs in the gas phase and 62 fs for both ethanol solution and liquid phase. The higher rate of intersystem crossing to the triplet states in selenophene is likely the reason for the lower fluorescence observed in selenium containing polymer compounds.

Perturbative universal state-selective correction for state-specific multi-reference coupled cluster methods.

In this work, we report an extension of our previous development of the universal state-selective (USS) multireference coupled-cluster (MRCC) formalism. It was shown [Brabec et al., J. Chem. Phys. 136, 124102 (2012)] and [Banik et al., J. Chem. Phys. 142, 114106 (2015)] that the USS(2) approach significantly improves the accuracy of Brillouin-Wigner and Mukherjee MRCC formulations, however, the numerical and storage costs associated with calculating highly excited intermediates pose a significant challenge, which can restrict the applicability of the USS(2) method. Therefore, we introduce a perturbative variant of the USS(2) approach (USS(pt)), which substantially reduces numerical overhead of the full USS(2) correction while preserving its accuracy. Since the new USS(pt) implementation calculates the triple and quadruple projections in on-the-fly manner, the memory bottleneck associated with the need of storing expensive recursive intermediates is entirely eliminated. On the example of several benchmark systems, we demonstrate accuracies of USS(pt) and USS(2) approaches and their efficiency in describing quasidegenerate electronic states. It is also shown that the USS(pt) method significantly alleviates problems associated with the lack of invariance of MRCC theories upon the rotation of active orbitals.