Heteroatom-vacancy centres in molecular nanodiamonds: a computational study of organic molecules possessing triplet ground states through σ-overlap.

Small molecules possessing a triplet ground state are fundamentally intriguing but also in high demand for applications such as quantum sensing and quantum computing. Such molecules are rare, and most examples involve extended π-systems. Topology and shape of the spin density will be very different for molecules where the triplet state arises from σ-overlap. Drawing inspiration from NV- (anionic nitrogen-vacancy) centres in a diamond crystal, which possess triplet ground states that are robust due to the distortion-preventing crystal lattice, we investigate hetero-atom substituted diamondoids (molecular nanodiamonds) as molecular mimics for NV- centres. It is found that even in these small systems, distortions that stabilize singlet states are energetically costly, and the triplet states are more stable than the singlets. The stabilization of the triplet over the singlet is 13, 16, and 18 kcal mol-1, in anionic C3v-C33H36N- and in the charge-neutral molecules C3v-C33H36O and C3v-C33H36S, respectively, using CAM-B3LYP-D3(BJ)/Def2-QZVPP. Comparable numbers are obtained with other density functional theory (DFT) methods, including double-hybrids. Wavefunction-based approaches on the other hand disagree in their predictions: While the MP2 method applied with the DLPNO approximation predicts a preference for the singlet, density matrix renormalization group (DMRG) calculations qualitatively agree with DFT in their prediction of a triplet ground state, although by a small margin, for C3v-C33H36N- and C3v-C33H36O, but not for C3v-C33H36S. Weighing the evidence, we conclude, with reasonable confidence for C3v-C33H36N- and C3v-C33H36O and lesser confidence for C3v-C33H36S, that the ground state for the molecular nanodiamonds studied is a triplet state.

DFT Modeling of Polyethylene Chains Cross-linked by Selected ns1 and ns2 Metal Atoms.

We analyze the structures, stabilities, and thermochemical properties of polyethylene (PE) oligomer chains cross-linked by metal (M) atoms through C-M-C bonds. Representative PEn-Mm-PEn complexes contain between 7 and 15 carbon atoms in each oligomer and one to three Li, Be, Mg, Zn, Ag, or Au cross-linking metal elements. PEn-Mm-PEn complexes are quasiplanar with nearly parallel PE chains. Their stability is determined by covalent C-M-C bonds accompanied by noncovalent dispersion interactions between PEn chains. Using the CAM-B3LYP+D3BJ+ABC functional, the binding energies of PE15-M-PE15 with respect to two PE15 radicals and metal fragments are -225, -230, -322, -551, -289, and -303 kJ/mol for Li, Ag, Au, Be, Mg, and Zn atoms, respectively. Entropy contributions (109 to 121 kJ/mol at 298.15 K) destabilize all complexes significantly. With two cross-linking metal elements in PE15-M2-PE15 complexes, binding energies are about double. Complexes with several open-shell Li, Ag, or Au doublet atoms have spins located on separated C-M-C bonds. High-spin PE15-Mm1-PE15-Mm2-PE15 complexes of three PE oligomers cross-linked by up to five doublet metal atoms create parallel PE tubes, which are suggested as elementary cells for modeling magnetic polymer tubes.

A simple approach to rotationally invariant machine learning of a vector quantity.

Unlike with the energy, which is a scalar property, machine learning (ML) prediction of vector or tensor properties poses the additional challenge of achieving proper invariance (covariance) with respect to molecular rotation. For the energy gradients needed in molecular dynamics (MD), this symmetry is automatically fulfilled when taking analytic derivative of the energy, which is a scalar invariant (using properly invariant molecular descriptors). However, if the properties cannot be obtained by differentiation, other appropriate methods should be applied to retain the covariance. Several approaches have been suggested to properly treat this issue. For nonadiabatic couplings and polarizabilities, for example, it was possible to construct virtual quantities from which the above tensorial properties are obtained by differentiation and thus guarantee the covariance. Another possible solution is to build the rotational equivariance into the design of a neural network employed in the model. Here, we propose a simpler alternative technique, which does not require construction of auxiliary properties or application of special equivariant ML techniques. We suggest a three-step approach, using the molecular tensor of inertia. In the first step, the molecule is rotated using the eigenvectors of this tensor to its principal axes. In the second step, the ML procedure predicts the vector property relative to this orientation, based on a training set where all vector properties were in this same coordinate system. As the third step, it remains to transform the ML estimate of the vector property back to the original orientation. This rotate-predict-rotate (RPR) procedure should thus guarantee proper covariance of a vector property and is trivially extensible also to tensors such as polarizability. The RPR procedure has an advantage that the accurate models can be trained very fast for thousands of molecular configurations, which might be beneficial where many training sets are required (e.g., in active learning). We have implemented the RPR technique, using the MLatom and Newton-X programs for ML and MD, and performed its assessment on the dipole moment along MD trajectories of 1,2-dichloroethane.

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.

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.

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.

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.

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.

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.

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.

Iterative universal state selective correction for the Brillouin-Wigner multireference coupled-cluster theory.

As a further development of the previously introduced a posteriori Universal State-Selective (USS) corrections [K. Kowalski, J. Chem. Phys. 134, 194107 (2011); J. Brabec et al., ibid. 136, 124102 (2012)], we suggest an iterative form of the USS correction by means of correcting effective Hamiltonian matrix elements. We also formulate USS corrections via the left Bloch equations. The convergence of the USS corrections with excitation level towards the full configuration interaction (FCI) limit is also investigated. Various forms of the USS and simplified diagonal USS corrections at the singles and doubles and perturbative triple levels are numerically assessed on several model systems and on the ozone and tetramethyleneethane molecules. It is shown that the iterative USS correction can successfully replace the previously developed a posteriori Brillouin-Wigner coupled cluster size-extensivity correction, while it is not sensitive to intruder states and performs well also in other cases when the a posteriori one fails, like, e.g., for the asymmetric vibration mode of ozone.

Adiabatic state preparation study of methylene.

Quantum computers attract much attention as they promise to outperform their classical counterparts in solving certain type of problems. One of them with practical applications in quantum chemistry is simulation of complex quantum systems. An essential ingredient of efficient quantum simulation algorithms are initial guesses of the exact wave functions with high enough fidelity. As was proposed in Aspuru-Guzik et al. [Science 309, 1704 (2005)], the exact ground states can in principle be prepared by the adiabatic state preparation method. Here, we apply this approach to preparation of the lowest lying multireference singlet electronic state of methylene and numerically investigate preparation of this state at different molecular geometries. We then propose modifications that lead to speeding up the preparation process. Finally, we decompose the minimal adiabatic state preparation employing the direct mapping in terms of two-qubit interactions.

DMRG-Tailored Coupled Cluster Method in the 4c-Relativistic Domain: General Implementation and Application to the NUHFI and NUF3 Molecules.

Heavy atom compounds represent a challenge for computational chemistry due to the need for simultaneous treatment of relativistic and correlation effects. Often such systems also exhibit strong correlation, which hampers the application of perturbation theory or single-reference coupled cluster (CC) methods. As a viable alternative, we have proposed externally correcting the CC method using the density matrix renormalization group (DMRG) wave functions, yielding the DMRG-tailored CC method. In a previous paper [J. Chem. Phys. 2020, 152, 174107], we reported a first implementation of this method in the relativistic context, which was restricted to molecules with real double group symmetry. In this work, we present a fully general implementation of the method, covering complex and quaternion double groups as well. The 4c-TCC method thus becomes applicable to polyatomic molecules, including heavy atoms. For the assessment of the method, we performed calculations of the chiral uranium compound NUHFI, which was previously studied in the context of the enhancement of parity violation effects. In particular, we performed calculations of a cut of the potential energy surface of this molecule along the stretching of the N-U bond, where the system exhibits strong multireference character. Since there are no experimental data for NUHFI, we have performed also an analogous study of the (more symmetric) NUF3 molecule, where the vibrational frequency of the N-U bond can be compared with spectroscopic data.

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.

An explicitly correlated Mukherjee's state specific coupled cluster method: development and pilot applications.

This paper reports development of the explicitly correlated variant of Mukherjee's state specific multireference coupled cluster method (MkCC-F12). The current implementation is restricted to conventional single and double excitations and to pseudo-double excitations related to the Slater Type Geminal (STG) correlation factor using the SP ansatz. The performance of the MkCCSD-F12 was tested on calculations of singlet methylene, dissociation curve of the fluorine molecule, and the BeH(2) insertion pathway. As expected, the results of the newly developed method reconfirm the significantly faster convergence with respect to the basis set limit compared to the traditional expansion in Slater determinants. Results prove that treating the correlation factor separately for each reference is appropriate.

Communication: Application of state-specific multireference coupled cluster methods to core-level excitations.

The concept of the model space underlying multireference coupled-cluster (MRCC) formulations is a powerful tool to deal with complex correlation effects for various electronic states. Here, we demonstrate that iterative state-specific MRCC methods (SS-MRCC) based on properly defined model spaces can be used to describe core-level excited states even when Hartree-Fock orbitals are utilized. We show that the SS-MRCC models with single and double excitations are comparable in accuracy to high-level single reference equation-of-motion coupled cluster (EOMCC) formalism.