The "chain of spheres" (COS) algorithm, as part of the RIJCOSX SCF procedure, approximates the exchange term by performing analytic integration with respect to the coordinates of only one of the two electrons, whereas for the remaining coordinates, integration is carried out numerically. In the present work, we attempt to enhance the efficiency of the method by minimizing numerical errors in the COS procedure. The main idea is based on the work of Friesner and consists of finding a fitting matrix, Q, which leads the numerical and analytically evaluated overlap matrices to coincide. Using Q, the evaluation of exchange integrals can indeed be improved. Improved results and timings are obtained with the present default grid setup for both single point calculations and geometry optimizations. The fitting procedure results in a reduction of grid sizes necessary for achieving chemical accuracy. We demonstrate this by testing a number of grids and comparing results to the fully analytic and the earlier COS approximations. This turns out to be favourable for total and reaction energies, for which chemical accuracy can now be reached with a corresponding ~30% speedup over the original RIJCOSX procedure for single point energies. Results are slightly less favourable for the accuracy of geometry optimizations, but the procedure is still shown to yield geometries with errors well below the method inherent errors of the employed theoretical framework.
In this work, we present and implement the theory for calculating fluorescence rates and absorption and emission spectra from first principles, using the path integral approach. We discuss some approximations and modifications to the full set of equations that improve speed and numerical stability for the case when a large number of modes are considered. New methods to approximate the excited state potential energy surface are also discussed and it is shown that for most purposes, these can be used instead of a full geometry optimization to obtain the rates mentioned above. A few examples are presented and the overall performance of the method is discussed. It is shown that the rates and spectra computed in this way are well within the acceptable range of errors and can be used in future predictions, particularly for screening purposes, with the only limitation on size being that of the electronic structure calculation itself.
The newly developed back transformed pair natural orbital based similarity transformed equation of motion (bt-STEOM) method at the coupled cluster singles and doubles level (CCSD) is combined with an appropriate modification of our earlier active space selection scheme for STEOM. The resulting method is benchmarked for valence, Rydberg, and charge transfer excited states of Thiel's test set and other test systems. The bt-PNO-STEOM-CCSD method gives very similar results to canonical STEOM-CCSD for both singlet and triplet excited states. It performs in a balanced manner for all these types of excited states, while the EOM-CCSD method performs especially well for Rydberg excited states and the CC2 method excels at obtaining accurate results for valence excited states. Both EOM-CCSD and CC2 perform worse than bt-PNO-STEOM-CCSD for charge transfer states for the test cases studied.
In
this work, we present a general method for predicting phosphorescence
rates and spectra for molecules using time-dependent density functional
theory (TD-DFT) and a path integral approach for the dynamics that
relies on the harmonic oscillator approximation for the nuclear movement.
We first discuss the theory involved in including spin–orbit
coupling (SOC) among singlet and triplet excited states and then how
to compute the corrected transition dipole moments and phosphorescence
rates. We investigate the dependence of these rates on some TD-DFT
parameters, such as the nature of the functional, the number of roots,
and the Tamm–Dancoff approximation. After that, we evaluate
the effect of different SOC integral schemes and show that our best
method is applicable to a large number of systems with different excited
state characters.
The chain-of-spheres method (COS) for approximating two-electron integrals is applied to Hartree-Fock and density functional theory calculations of nuclear magnetic resonance chemical shielding tensors, based on gauge-including atomic orbitals. The accuracy of the approximation is compared to that of the resolution of the identity (RI) approach, using a benchmark test set of 15 small molecules. Reasonable auxiliary basis sets and grid sizes are selected on the basis of a careful investigation of how approximating each of the two-electron terms in the self-consistent field (SCF) and coupled perturbed SCF equations affects the calculated shielding constants. It is found that the errors are linearly additive but can have either sign. The mean absolute relative error due to applying the RI/COS approximations with the chosen settings to all two-electron terms is on the order of 0.01% and therefore negligible compared to the errors due to basis set incompleteness (∼1%) and the method used (10-50%). Several larger organic systems are used to assess the efficiency of the RI approximation for both Coulomb- and exchange-type integrals (RIJK) as well as a combination of RI for Coulomb and COS for exchange contributions (RIJCOSX). The RIJK approximation is more efficient for small molecules, while for systems of over 100 electrons and 1000 basis functions, the RIJCOSX approximation is superior.
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