An open-source program named VeloxChem has been developed for the calculation of electronic real and complex linear response functions at the levels of Hartree–Fock and Kohn–Sham density functional theories. With an object-oriented program structure written in a Python/C++ layered fashion, VeloxChem enables time-efficient prototyping of novel scientific approaches without sacrificing computational efficiency, so that molecular systems involving up to and beyond 500 second-row atoms (or some 10,000 contracted and in part diffuse Gaussian basis functions) can be routinely addressed. In addition, VeloxChem is equipped with a polarizable embedding scheme for the treatment of the classical electrostatic interactions with an environment that in turn is modeled by atomic site charges and polarizabilities. The underlying hybrid message passing interface (MPI)/open multiprocessing (OpenMP) parallelization scheme makes VeloxChem suitable for execution in high-performance computing cluster environments, showing even slightly beyond linear scaling for the Fock matrix construction with use of up to 16,384 central processing unit (CPU) cores. An efficient—with respect to convergence rate and overall computational cost—multifrequency/gradient complex linear response equation solver enables calculations not only of conventional spectra, such as visible/ultraviolet/X-ray electronic absorption and circular dichroism spectra, but also time-resolved linear response signals as due to ultra-short weak laser pulses. VeloxChem distributed under the GNU Lesser General Public License version 2.1 (LGPLv2.1) license and made available for download from the homepage https://veloxchem.org. This article is categorized under: Software > Quantum Chemistry Electronic Structure Theory > Density Functional Theory Theoretical and Physical Chemistry > Spectroscopy.

Within the self-consistent field approximation, computationally tractable expressions for the isotropic second-order hyperpolarizability have been derived and implemented for the calculation of two-photon absorption cross sections. The novel tensor average formulation presented in this work allows for the evaluation of isotropic damped cubic response functions using only similar to 3.3% (one-photon off-resonance regions) and similar to 10% (one-photon resonance regions) of the number of auxiliary Fock matrices required when explicitly calculating all the needed individual tensor components. Numerical examples of the two-photon absorption cross section in the one-photon off-resonance and resonance regions are provided for alanine-tryptophan and 2,5-dibromo-1,4-bis(2-(4-diphenylaminophenyl)vinyl)-benzene. Furthermore, a benchmark set of 22 additional small- and medium-sized organic molecules is considered. In all these calculations, a quantitative assessment is made of the reduced and approximate forms of the cubic response function in the one-photon off-resonance regions and results demonstrate a relative error of less than similar to 5% when using the reduced expression as compared to the full form of the isotropic cubic response function.

For general exchange-correlation functionals with a dependence on the local spin densities and spin-density gradients, we provide computationally tractable expressions for the tensor-averaged quadratic response functions pertinent to the experimental observables in second-harmonic generation (SHG). We demonstrate how the tensor-averaged quantities can be implemented with reference to a derived minimal number of first- and second-order perturbed Fock matrices. Our consideration has the capability of treating a situation of resonance enhancement as it is based on damped response theory and allows for the evaluation of tensor-averaged resonant-convergent quadratic response functions using only similar to 25% (one-photon off-resonance regions) and similar to 50% (one-photon resonance regions) of the number of auxiliary Fock matrices required when explicitly calculating all the needed individual tensor components. Numerical examples of SHG intensities in the one-photon off-resonance region are provided for a sample of makaluvamine derivatives recognized for their large nonlinear optical responses as well as a benchmark set of small- and medium-sized organic molecules.

Within the density Kohn–Sham density functional theory approximation, computa-tionally tractable expressions for the isotropic second-order hyperpolarizability havebeen derived and implemented for the calculation of two-photon absorption crosssections with meta-GGA functionals for the very first time. The tensor average for-mulation presented in this work allows for the evaluation of isotropic damped cubicresponse functions using only ∼3.3 % (one-photon off-resonance regions) and ∼10%(one-photon resonance regions) of the number of kernel integrations required whenexplicitly calculating all the needed individual tensor components of the cubic re-sponse function.1 

We present a a benchmark study on density functional approximation (DFA) per-formances for predicting two-photon absorption (TPA) strengths in π-conjugatedmolecules containing electron-donating/-accepting moieties. We have incorporateda variety of meta-generalized gradient approximation (meta-GGA) functionals, in-cluding SCAN, MN15, and M06-2X, to assess their accuracy in describing the TPAproperties of the chosen set of 48 organic molecules. Quadratic response theory is em-ployed for these functionals, and their performance is compared with the previouslystudied DFAs and the coupled-cluster CC2 model (RI-CC2) is used as a reference. Adetailed analysis of the meta-GGA functionals’ performance is provided, focusing ontheir ability to capture the key electronic and structural features of the π-conjugatedsystems. Furthermore, we discuss the implications of our findings for the design ofnew materials with tailored two-photon activity and the development of more accu-rate and efficient computational methods for TPA predictions