Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package.

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

TD-DFT spin-adiabats with analytic nonadiabatic derivative couplings.

We present an algorithm for efficient calculation of analytic nonadiabatic derivative couplings between spin-adiabatic, time-dependent density functional theory states within the Tamm-Dancoff approximation. Our derivation is based on the direct differentiation of the Kohn-Sham pseudowavefunction using the framework of Ou et al. Our implementation is limited to the case of a system with an even number of electrons in a closed shell ground state, and we validate our algorithm against finite difference at an S1/T2 crossing of benzaldehyde. Through the introduction of a magnetic field spin-coupling operator, we break time-reversal symmetry to generate complex valued nonadiabatic derivative couplings. Although the nonadiabatic derivative couplings are complex valued, we find that a phase rotation can generate an almost entirely real-valued derivative coupling vector for the case of benzaldehyde.

Birefringent Stable Glass with Predominantly Isotropic Molecular Orientation.

Birefringence in stable glasses produced by physical vapor deposition often implies molecular alignment similar to liquid crystals. As such, it remains unclear whether these glasses share the same energy landscape as liquid-quenched glasses that have been aged for millions of years. Here, we produce stable glasses of 9-(3,5-di(naphthalen-1-yl)phenyl)anthracene molecules that retain three-dimensional shapes and do not preferentially align in a specific direction. Using a combination of angle- and polarization-dependent photoluminescence and ellipsometry experiments, we show that these stable glasses possess a predominantly isotropic molecular orientation while being optically birefringent. The intrinsic birefringence strongly correlates with increased density, showing that molecular ordering is not required to produce stable glasses or optical birefringence, and provides important insights into the process of stable glass formation via surface-mediated equilibration. To our knowledge, such novel amorphous packing has never been reported in the past.

Ultrafast Electronic Relaxation through a Conical Intersection: Nonadiabatic Dynamics Disentangled through an Oscillator Strength-Based Diabatization Framework.

We employ surface hopping trajectories to model the short-time dynamics of gas-phase and partially solvated 4-(N,N-dimethylamino)benzonitrile (DMABN), a dual fluorescent molecule that is known to undergo a nonadiabatic transition through a conical intersection. To compare theory vs time-resolved fluorescence measurements, we calculate the mixed quantum-classical density matrix and the ensemble averaged transition dipole moment. We introduce a diabatization scheme based on the oscillator strength to convert the TDDFT adiabatic states into diabatic states of La and Lb character. Somewhat surprisingly, we find that the rate of relaxation reported by emission to the ground state is almost 50% slower than the adiabatic population relaxation. Although our calculated adiabatic rates are largely consistent with previous theoretical calculations and no obvious effects of decoherence are seen, the diabatization procedure introduced here enables an explicit picture of dynamics in the branching plane, raising tantalizing questions about geometric phase effects in systems with dozens of atoms.

An Efficient, Augmented Surface Hopping Algorithm That Includes Decoherence for Use in Large-Scale Simulations.

We propose and implement a highly efficient augmented surface hopping algorithm that (i) can be used for large simulations (with many nuclei and many electronic states) and (ii) includes the effects of decoherence without parametrization. Our protocol is based on three key modifications of the surface hopping methodology: (a) a novel separation of classical and quantum degrees of freedom that treats avoided and trivial crossings efficiently, (b) a multidimensional approximation of the time derivative matrix that avoids explicit construction of the derivative coupling at most time steps, and (c) an efficient approximation for the augmented fewest-switches surface hopping decoherence rate. We will show that this protocol can be several orders of magnitude more efficient than the traditional protocol for large multidimensional problems. Furthermore, the marginal cost for including decoherence effects is now negligible.

The requisite electronic structure theory to describe photoexcited nonadiabatic dynamics: nonadiabatic derivative couplings and diabatic electronic couplings.

Electronically photoexcited dynamics are complicated because there are so many different relaxation pathways: fluorescence, phosphorescence, radiationless decay, electon transfer, etc. In practice, to model photoexcited systems is a very difficult enterprise, requiring accurate and very efficient tools in both electronic structure theory and nonadiabatic chemical dynamics. Moreover, these theoretical tools are not traditional tools. On the one hand, the electronic structure tools involve couplings between electonic states (rather than typical single state energies and gradients). On the other hand, the dynamics tools involve propagating nuclei on multiple potential energy surfaces (rather than the usual ground state dynamics). In this Account, we review recent developments in electronic structure theory as directly applicable for modeling photoexcited systems. In particular, we focus on how one may evaluate the couplings between two different electronic states. These couplings come in two flavors. If we order states energetically, the resulting adiabatic states are coupled via derivative couplings. Derivative couplings capture how electronic wave functions change as a function of nuclear geometry and can usually be calculated with straightforward tools from analytic gradient theory. One nuance arises, however, in the context of time-dependent density functional theory (TD-DFT): how do we evaluate derivative couplings between TD-DFT excited states (which are tricky, because no wave function is available)? This conundrum was recently solved, and we review the solution below. We also discuss the solution to a second, pesky problem of origin dependence, whereby the derivative couplings do not (strictly) satisfy translation variance, which can lead to a lack of momentum conservation. Apart from adiabatic states, if we order states according to their electronic character, the resulting diabatic states are coupled via electronic or diabatic couplings. The couplings between diabatic states |ΞA⟩ and |ΞB⟩ are just the simple matrix elements, ⟨ΞA|H|ΞB⟩. A difficulty arises, however, because constructing exactly diabatic states is formally impossible and constructing quasi-diabatic states is not unique. To that end, we review recent advances in localized diabatization, which is one approach for generating adiabatic-to-diabatic (ATD) transformations. We also highlight outstanding questions in the arena of diabatization, especially how to generate multiple globally stable diabatic surfaces.

Derivative Couplings between Time-Dependent Density Functional Theory Excited States in the Random-Phase Approximation Based on Pseudo-Wavefunctions: Behavior around Conical Intersections.

In this paper, we present a formalism for derivative couplings between time-dependent density functional theory (TD-DFT) excited states within the randomphase approximation (RPA) using analytic gradient theory. Our formalism is based on a pseudo-wavefunction approach in a companion paper (DOI 10.1021/jp505767b), and can be checked against finite-difference overlaps. Our approach recovers the correct properties of derivative couplings around a conical intersection (CI), which is a crucial prerequisite for any derivative coupling expression. As an example, we study the test case of protonated formaldimine (CH2NH2(+)).

Calculating Derivative Couplings between Time-Dependent Hartree-Fock Excited States with Pseudo-Wavefunctions.

A pseudo-wavefunction description of time-dependent Hartree-Fock (TDHF) states is proposed and used to develop an analytic expression for derivative couplings between TDHF excited states based on the Hellmann-Feynman theorem. The resulting expression includes Pulay terms associated with using an atom-centered basis as well as a correction to ensure translational invariance. We demonstrate that our formalism recovers the well-known Chernyak-Mukamel expression near a crossing and in the limit of a complete basis, and thus our approach is consistent with time-dependent response theory. In a companion paper (DOI 10.1021/jp5057682 ), we investigate these derivative couplings near conical intersections and show that they behave correctly.

Exploring non-Condon effects in a covalent tetracene dimer: how important are vibrations in determining the electronic coupling for singlet fission?

Singlet fission (SF) offers opportunities for wavelength-selective processing of solar photons with an end goal of achieving higher efficiency inexpensive photovoltaic or solar-fuels-producing devices. In order to evaluate new molecular design strategies and for theoretical exploration of dynamics, it is important to put in place tools for efficient calculation of the electronic coupling between single-exciton reactant and multiexciton product states. For maximum utility, the couplings should be calculated at multiple nuclear geometries (rather than assumed constant everywhere, i.e., the Condon approximation) and we must be able to evaluate couplings for covalently linked multichromophore systems. With these requirements in mind, here we discuss the simplest methodology possible for rapid calculation of diabatic one-electron coupling matrix elements-based on Boys localization and rediagonalization of molecular orbitals. We focus on a covalent species called BT1 that juxtaposes two tetracene units in a partially cofacial geometry via a norbornyl bridge. In BT1, at the equilibrium C2v structure, the "nonhorizontal" couplings between HOMOs and LUMOs (t(HL) and t(LH)) vanish by symmetry. We then explore the impact of molecular vibrations through the calculation of t(AB) coupling gradients along 183 normal modes of motion. Rules are established for the types of motions (irreducible representations in the C2v point group) that turn on tHL and tLH values as well as for the patterns that emerge in constructive versus destructive interference of pathways to the SF product. For the best modes, calculated electronic coupling magnitudes for SF (at root-mean-squared deviation in position at 298 K), are within a factor of 2 of that seen for noncovalent tetracene dimers relevant to the molecular crystal. An overall "effective" electronic coupling is also given, based on the Stuchebrukhov formalism for non-Condon electron transfer rates.

Derivative couplings between TDDFT excited states obtained by direct differentiation in the Tamm-Dancoff approximation.

Working within the Tamm-Dancoff approximation, we calculate the derivative couplings between time-dependent density-functional theory excited states by assuming that the Kohn-Sham superposition of singly excited determinants represents a true electronic wavefunction. All Pulay terms are included in our derivative coupling expression. The reasonability of our approach can be established by noting that, for closely separated electronic states in the infinite basis limit, our final expression agrees exactly with the Chernyak-Mukamel expression (with transition densities from response theory). Finally, we also validate our approach empirically by analyzing the behavior of the derivative couplings around the T1/T2 conical intersection of benzaldehyde.

Analysis of localized diabatic states beyond the condon approximation for excitation energy transfer processes.

In a previous paper [ Fatehi , S. ; et al. J. Chem. Phys. 2013 , 139 , 124112 ], we demonstrated a practical method by which analytic derivative couplings of Boys-localized CIS states can be obtained. In this paper, we now apply that same method to the analysis of triplet-triplet energy transfer systems studied by Closs and collaborators [ Closs , G. L. ; et al. J. Am. Chem. Soc. 1988 , 110 , 2652 ]. For the systems examined, we are able to conclude that (i) the derivative coupling in the BoysOV basis is negligible, and (ii) the diabatic coupling will likely change little over the configuration space explored at room temperature. Furthermore, we propose and evaluate an approximation that allows for the inexpensive calculation of accurate diabatic energy gradients, called the "strictly diabatic" approximation. This work highlights the effectiveness of diabatic state analytic gradient theory in realistic systems and demonstrates that localized diabatic states can serve as an acceptable approximation to strictly diabatic states.

Derivative couplings and analytic gradients for diabatic states, with an implementation for Boys-localized configuration-interaction singles.

We demonstrate that Boys-localized diabatic states do indeed exhibit small derivative couplings, as is required of quasidiabatic states. In doing so, we present a general formalism for calculating derivative couplings and analytic gradients for diabatic states. We then develop additional equations specific to the case of Boys-localized configuration-interaction singles (CIS)--in particular, the analytic gradient of the CIS dipole matrix--and we validate our implementation against finite-difference results. In a forthcoming paper, we will publish additional algorithmic and computational details and apply our method to the Closs energy-transfer systems as a further test of the validity of Boys-localized diabatic states.

Communication: an inexpensive, variational, almost black-box, almost size-consistent correction to configuration interaction singles for valence excited states.

Configuration interaction singles (CIS) describe excited electronic states only qualitatively and improvements are imperative as a means of recovering chemical accuracy. In particular, variational improvements would be ideal to account for state crossings and electronic relaxation. To accomplish such an objective, in this communication we present a new suite of algorithms, abbreviated VOO-CIS for variationally orbital optimized CIS. We show below that VOO-CIS yields a uniform improvement to CIS, rebalancing the energies of CT states versus non-CT states within the same framework. Furthermore, VOO-CIS finds energetic corrections for CT states that are even larger than those predicted by CIS(D). The computational cost of VOO-CIS depends strongly on the number of excited states requested (n), but otherwise should be proportional to the cost of CIS itself.

Optimal diabatic states based on solvation parameters.

A new method for obtaining diabatic electronic states of a molecular system in a condensed environment is proposed and evaluated. This technique, which we denote as Edmiston-Ruedenberg (ER)-ε diabatization, forms diabatic states as a linear combination of adiabatic states by minimizing an approximation to the total coupling between states in a medium with temperature T and with a characteristic Pekar factor C. ER-ε diabatization represents an improvement upon previous localized diabatization methods for two reasons: first, it is sensitive to the energy separation between adiabatic states, thus accounting for fluctuations in energy and effectively preventing over-mixing. Second, it responds to the strength of system-solvent interactions via parameters for the dielectric constant and temperature of the medium, which is physically reasonable. Here, we apply the ER-ε technique to both intramolecular and intermolecular excitation energy transfer systems. We find that ER-ε diabatic states satisfy three important properties: (1) they have small derivative couplings everywhere; (2) they have small diabatic couplings at avoided crossings, and (3) they have negligible diabatic couplings everywhere else. As such, ER-ε states are good candidates for so-called "optimal diabatic states."

Analytic derivative couplings between configuration-interaction-singles states with built-in electron-translation factors for translational invariance.

We present a method for analytically calculating the derivative couplings between a pair of configuration-interaction-singles (CIS) excited states obtained in an atom-centered basis. Our theory is exact and has been derived using two completely independent approaches: one inspired by the Hellmann-Feynman theorem and the other following from direct differentiation. (The former is new, while the latter is in the spirit of existing approaches in the literature.) Our expression for the derivative couplings incorporates all Pulay effects associated with the use of an atom-centered basis, and the computational cost is minimal, roughly comparable to that of a single CIS energy gradient. We have validated our method against CIS finite-difference results and have applied it to the lowest lying excited states of naphthalene; we find that naphthalene derivative couplings include Pulay contributions sufficient to have a qualitative effect. Going beyond standard problems in analytic gradient theory, we have also constructed a correction, based on perturbative electron-translation factors, for including electronic momentum and eliminating spurious components of the derivative couplings that break translational symmetry. This correction is general and can be applied to any level of electronic structure theory.

Diabatic couplings for charge recombination via Boys localization and spin-flip configuration interaction singles.

We describe a straightforward technique for obtaining diabatic couplings applicable to charge transfer from or charge recombination to the electronic ground state. Our method is nearly black box, requiring minimal chemical intuition from the user, and merges two well-established approaches in electronic structure theory: first, smooth and balanced adiabatic states are generated using spin-flip-configuration interaction singles (SF-CIS) based on a triplet HF state; second, Boys localization is applied to rotate all adiabatic states into charge-localized diabatic states. The method is computationally inexpensive, scaling only with the cost of CIS, and does not require a choice of active space, which is usually required for such intrinsically multiconfigurational problems. Molecular LiF in vacuum and LiF solvated by a single water molecule are examined as model systems. We find nearly smooth diabatic potential energy surfaces and couplings and we find that the Condon approximation is obeyed approximately for this model problem.