Prediction of photodynamics of 200 nm excited cyclobutanone with linear response electronic structure and ab initio multiple spawning.

Simulations of photochemical reaction dynamics have been a challenge to the theoretical chemistry community for some time. In an effort to determine the predictive character of current approaches, we predict the results of an upcoming ultrafast diffraction experiment on the photodynamics of cyclobutanone after excitation to the lowest lying Rydberg state (S2). A picosecond of nonadiabatic dynamics is described with ab initio multiple spawning. We use both time dependent density functional theory (TDDFT) and equation-of-motion coupled cluster singles and doubles (EOM-CCSD) theory for the underlying electronic structure theory. We find that the lifetime of the S2 state is more than a picosecond (with both TDDFT and EOM-CCSD). The predicted ultrafast electron diffraction spectrum exhibits numerous structural features, but weak time dependence over the course of the simulations.

Communication: Non-adiabatic derivative coupling elements for the coupled cluster singles and doubles model.

We present an efficient implementation of analytical non-adiabatic derivative coupling elements for the coupled cluster singles and doubles model. The derivative coupling elements are evaluated in a biorthonormal formulation in which the nuclear derivative acts on the right electronic state, where this state is biorthonormal with respect to the set of left states. This stands in contrast to earlier implementations based on normalized states and a gradient formula for the derivative coupling. As an illustration of the implementation, we determine a minimum energy conical intersection between the nπ* and ππ* states in the nucleobase thymine.

Geometric phase in coupled cluster theory.

It has been well-established that the topography around conical intersections between excited electronic states is incorrectly described by coupled cluster and many other single reference theories (the intersections are "defective"). Despite this, we show both analytically and numerically that the geometric phase effect (GPE) is correctly reproduced upon traversing a path around a defective excited-state conical intersection (CI) in coupled cluster theory. The theoretical analysis is carried out by using a non-Hermitian generalization of the linear vibronic coupling approach. Interestingly, the approach qualitatively explains the characteristic (incorrect) shape of the defective CIs and CI seams. Moreover, the validity of the approach and the presence of the GPE indicate that defective CIs are local (and not global) artifacts. This implies that a sufficiently accurate coupled cluster method could predict nuclear dynamics, including geometric phase effects, as long as the nuclear wavepacket never gets too close to the conical intersections.

Efficient implementation of molecular CCSD gradients with Cholesky-decomposed electron repulsion integrals.

We present an efficient implementation of ground and excited state coupled cluster singles and doubles (CCSD) gradients based on Cholesky-decomposed electron repulsion integrals. Cholesky decomposition and density fitting are both inner projection methods, and, thus, similar implementation schemes can be applied for both methods. One well-known advantage of inner projection methods, which we exploit in our implementation, is that one can avoid storing large V3O and V4 arrays by instead considering three-index intermediates. Furthermore, our implementation does not require the formation and storage of Cholesky vector derivatives. The new implementation is shown to perform well, with less than 10% of the time spent calculating the gradients in geometry optimizations. Furthermore, the computational time per optimization cycle is significantly lower compared to other implementations based on an inner projection method. We showcase the capabilities of the implementation by optimizing the geometry of the retinal molecule (C20H28O) at the CCSD/aug-cc-pVDZ level of theory.

Equation-of-motion coupled-cluster method with double electron-attaching operators: Theory, implementation, and benchmarks.

We report a production-level implementation of the equation-of-motion (EOM) coupled-cluster (CC) method with double electron-attaching (DEA) EOM operators of 2p and 3p1h types, EOM-DEA-CCSD. This ansatz, suitable for treating electronic structure patterns that can be described as two-electrons-in-many orbitals, represents a useful addition to the EOM-CC family of methods. We analyze the performance of EOM-DEA-CCSD for energy differences and molecular properties. By considering reduced quantities, such as state and transition one-particle density matrices, we compare EOM-DEA-CCSD wave functions with wave functions computed by other EOM-CCSD methods. The benchmarks illustrate that EOM-DEA-CCSD is capable of treating diradicals, bond-breaking, and some types of conical intersections.

Accelerated multimodel Newton-type algorithms for faster convergence of ground and excited state coupled cluster equations.

We introduce a multimodel approach to solve coupled cluster equations, employing a quasi-Newton algorithm for the ground state and an Olsen algorithm for the excited states. In these algorithms, both of which can be viewed as Newton algorithms, the Jacobian matrix of a lower level coupled cluster model is used in Newton equations associated with the target model. Improvements in convergence then imply savings for sufficiently large molecular systems, since the computational cost of macroiterations scales more steeply with system size than the cost of microiterations. The multimodel approach is suitable when there is a lower level Jacobian matrix that is much more accurate than the zeroth order approximation. Applying the approach to the CC3 equations, using the CCSD approximation of the Jacobian, we show that the time spent to determine the ground and valence excited states can be significantly reduced. We also find improved convergence for core excited states, indicating that similar savings will be obtained with an explicit implementation of the core-valence separated CCSD Jacobian transformation.

eT 1.0: An open source electronic structure program with emphasis on coupled cluster and multilevel methods.

The eT program is an open source electronic structure package with emphasis on coupled cluster and multilevel methods. It includes efficient spin adapted implementations of ground and excited singlet states, as well as equation of motion oscillator strengths, for CCS, CC2, CCSD, and CC3. Furthermore, eT provides unique capabilities such as multilevel Hartree-Fock and multilevel CC2, real-time propagation for CCS and CCSD, and efficient CC3 oscillator strengths. With a coupled cluster code based on an efficient Cholesky decomposition algorithm for the electronic repulsion integrals, eT has similar advantages as codes using density fitting, but with strict error control. Here, we present the main features of the program and demonstrate its performance through example calculations. Because of its availability, performance, and unique capabilities, we expect eT to become a valuable resource to the electronic structure community.

An efficient algorithm for Cholesky decomposition of electron repulsion integrals.

Approximating the electron repulsion integrals using inner projections is a well-established approach to reduce the computational demands of electronic structure calculations. Here, we present a two-step Cholesky decomposition algorithm where only the elements of the Cholesky basis (the pivots) are determined in the pivoting procedure. This allows for improved screening, significantly reducing memory usage and computational cost. After the pivots have been determined, the Cholesky vectors are constructed using the inner projection formulation. We also propose a partitioned decomposition approach where the Cholesky basis is chosen from a reduced set generated by decomposing diagonal blocks of the matrix. The algorithm extends the application range of the methodology and is well suited for multilevel methods. We apply the algorithm to systems with up to 80 000 atomic orbitals. The accuracy of the integral approximations is demonstrated for a formaldehyde-water system using a new Cholesky-based CCSD implementation.

Spin adapted implementation of EOM-CCSD for triplet excited states: Probing intersystem crossings of acetylacetone at the carbon and oxygen K-edges.

We present an equation of motion coupled cluster singles and doubles approach for computing transient absorption spectra from a triplet excited state. The implementation determines the left and right excitation vectors by explicitly spin-adapting the triplet excitation space. As an illustrative application, we compute transient state X-ray absorption spectra at the carbon and oxygen K-edges for the acetylacetone molecule.

Crossing conditions in coupled cluster theory.

We derive the crossing conditions at conical intersections between electronic states in coupled cluster theory and show that if the coupled cluster Jacobian matrix is nondefective, two (three) independent conditions are correctly placed on the nuclear degrees of freedom for an inherently real (complex) Hamiltonian. Calculations using coupled cluster theory on a 21A'/31A' conical intersection in hypofluorous acid illustrate the nonphysical artifacts associated with defects at accidental same-symmetry intersections. In particular, the observed intersection seam is folded about a space of the correct dimensionality, indicating that minor modifications to the theory are required for it to provide a correct description of conical intersections in general. We find that an accidental symmetry allowed 11A″/21A″ intersection in hydrogen sulfide is properly described, showing no artifacts as well as linearity of the energy gap to first order in the branching plane.

Coupled Cluster Theory for Nonadiabatic Dynamics: Nuclear Gradients and Nonadiabatic Couplings in Similarity Constrained Coupled Cluster Theory.

Coupled cluster theory is one of the most accurate electronic structure methods for predicting ground and excited state chemistry. However, the presence of numerical artifacts at electronic degeneracies, such as complex energies, has made it difficult to apply the method in nonadiabatic dynamics simulations. While it has already been shown that such numerical artifacts can be fully removed by using similarity constrained coupled cluster (SCC) theory [J. Phys. Chem. Lett. 2017, 8(19), 4801-4807], simulating dynamics requires efficient implementations of gradients and nonadiabatic couplings. Here, we present an implementation of nuclear gradients and nonadiabatic derivative couplings at the similarity constrained coupled cluster singles and doubles (SCCSD) level of theory, thereby making possible nonadiabatic dynamics simulations using a coupled cluster theory that provides a correct description of conical intersections between excited states. We present a few numerical examples that show good agreement with literature values and discuss some limitations of the method.

Biorthonormal Formalism for Nonadiabatic Coupled Cluster Dynamics.

In coupled cluster theory, the electronic states are biorthonormal in the sense that the left states are orthonormal to the right states. Here, we present an extension of this formalism to a left and right total molecular wave function. Starting from left and right Born-Huang expansions, we derive projected Schrödinger equations for the left and right nuclear wave functions. Observables may be extracted from the resulting wave function pair using standard expressions. The formalism is shown to be invariant under electronic basis transformations, such as normalization of the electronic states. Consequently, the nonadiabatic coupling elements can be expressed with biorthonormal electronic wave functions. Calculating normalization factors that scale as full configuration interaction is not necessary, contrary to claims in the literature. For nonadiabatic nuclear dynamics, we need expressions for the derivative couplings in the biorthonormal formalism. These are derived in a Lagrangian framework.

Multilevel CC2 and CCSD in Reduced Orbital Spaces: Electronic Excitations in Large Molecular Systems.

We present efficient implementations of the multilevel CC2 (MLCC2) and multilevel CCSD (MLCCSD) models. As the system size increases, MLCC2 and MLCCSD exhibit the scaling of the lower-level coupled cluster model. To treat large systems, we combine MLCC2 and MLCCSD with a reduced-space approach in which the multilevel coupled cluster calculation is performed in a significantly truncated molecular orbital basis. The truncation scheme is based on the selection of an active region of the molecular system and the subsequent construction of localized Hartree-Fock orbitals. These orbitals are used in the multilevel coupled cluster calculation. The electron repulsion integrals are Cholesky decomposed using a screening protocol that guarantees accuracy in the truncated molecular orbital basis and reduces computational cost. The Cholesky factors are constructed directly in the truncated basis, ensuring low storage requirements. Systems for which Hartree-Fock is too expensive can be treated by using a multilevel Hartree-Fock reference. With the reduced-space approach, we can handle systems with more than a thousand atoms. This is demonstrated for paranitroaniline in aqueous solution.

Linear-Scaling Implementation of Multilevel Hartree-Fock Theory.

We introduce a new algorithm for the construction of the two-electron contributions to the Fock matrix in multilevel Hartree-Fock (MLHF) theory. In MLHF, the density of an active molecular region is optimized, while the density of an inactive region is fixed. The MLHF equations are solved in a reduced molecular orbital (MO) basis localized to the active region. The locality of the MOs can be exploited to reduce the computational cost of the Fock matrix: the cost related to the inactive density becomes linear scaling, while the iterative cost related to the active density is independent of the system size. We demonstrate the performance of this new algorithm on a variety of systems, including amino acid chains, water clusters, and solvated systems.

An Orbital Invariant Similarity Constrained Coupled Cluster Model.

We present a similarity constrained coupled cluster method able to describe conical intersections between two excited electronic states of the same symmetry. For a given pair of states, this singles and doubles method (SCCSD) is unique and orbital invariant. The computational cost scales as the sixth power with respect to the number of orbitals, and preliminary calculations indicate that the excitation energy difference relative to CCSD is within the error range of CCSD (approximately 0.10 eV). We also analyze the size-scaling properties of the orthogonality condition. For a projected orthogonality condition we show, and demonstrate numerically, that the method is rigorously size-intensive.

Resolving the Notorious Case of Conical Intersections for Coupled Cluster Dynamics.

The motion of electrons and nuclei in photochemical events often involves conical intersections, or degeneracies between electronic states. They serve as funnels in nuclear relaxation processes where the electrons and nuclei couple nonadiabatically. Accurate ab initio quantum chemical models are essential for interpreting experimental measurements of such phenomena. In this Letter, we resolve a long-standing problem in coupled cluster theory, presenting the first formulation of the theory that correctly describes conical intersections between excited electronic states of the same symmetry. This new development demonstrates that the highly accurate coupled cluster theory can be applied to describe dynamics on excited electronic states involving conical intersections.