Variational Pair-Density Functional Theory: Dealing with Strong Correlation at the Protein Scale.

Multiconfigurational pair-density functional theory (MC-PDFT) offers a promising solution to the challenges faced by traditional density functional theory (DFT) in addressing molecular systems containing transition metals, open-shells, or strong correlations in general. By utilizing both the density and on-top pair-density, MC-PDFT can make use of a more flexible multiconfigurational wave function to capture the necessary static correlation, while the pair-density functional also includes the effect of dynamic correlation. So far, MC-PDFT has been used after a multiconfigurational self-consistent field (MCSCF) step, using the orbitals and configuration interaction coefficients from the converged MCSCF wave function to compute PDFT energies and properties. Here, instead, we propose to perform a direct optimization of the wave function using the pair-density functionals, resulting in a variational formulation of MC-PDFT. We derive the expressions for the wave function gradient and illustrate their similarity to standard MCSCF equations. Furthermore, we illustrate the accuracy on a set of singlet-triplet gaps as well as dissociation curves. Our findings highlight one of MC-PDFT's standout features: a reduced dependency on the active space size compared to conventional multiconfigurational wave function methodologies. Additionally, we show that the computational cost of MC-PDFT is potentially lower than MCSCF and often on-par with standard Kohn-Sham DFT, which is demonstrated by performing a MC-PDFT calculation of the entire ferredoxin protein with 1447 atoms and nearly 12 000 basis functions.

Multiconfigurational Pair-Density Functional Theory Is More Complex than You May Think.

Multiconfigurational pair-density functional theory (MC-PDFT) is a promising way to describe both strong and dynamic correlations in an inexpensive way. The functionals in MC-PDFT are often "translated" from standard spin density functionals. However, these translated functionals can in principle lead to "translated spin densities" with a nonzero imaginary component. Current developments so far neglect this imaginary part by simply setting it to zero. In this work, we show how this imaginary component is actually needed to reproduce the correct physical behavior in a range of cases, especially low-spin open shells. We showcase the resulting formalism on both local density approximation and generalized gradient approximation functionals and illustrate the numerical behavior by benchmarking a number of singlet-triplet splittings (ST gaps) of organic diradicals and low-lying excited states of some common organic molecules. The results demonstrate that this scheme improves existing translated functionals and gives more accurate results, even with minimal active spaces.

Complex Linear Response Functions for a Multiconfigurational Self-Consistent Field Wave Function in a High Performance Computing Environment.

We present novel developments for the highly efficient evaluation of complex linear response functions of a multiconfigurational self-consistent field (MCSCF) wave function as implemented in MultiPsi. Specifically, expressions for the direct evaluation of linear response properties at given frequencies using the complex polarization propagator (CPP) approach have been implemented, within both the Tamm-Dancoff approximation (TDA) and the random phase approximation (RPA). Purely real algebra with symmetric and antisymmetric trial vectors in a shared subspace is used wherein the linear response equations are solved. Two bottlenecks of large scale MC-CPP calculations, namely, the memory footprint and computational time, are addressed. The former is addressed by limiting the size of the subspace of trial vectors by using singular value decomposition (SVD) on either orbital or CI subspaces. The latter is addressed using an efficient parallel implementation as well as the strategy of dynamically adding linear response equations at near-convergence to neighboring roots. Furthermore, a novel methodology for decomposing MC-CPP spectra in terms of intuitive orbital excitations in an approximate fashion is presented. The performance of the code is illustrated with several numerical examples, including the X-ray spectrum of a molecule with nearly one hundred atoms. Additionally, for X-ray spectroscopy, the effect of including or excluding the core orbital in the active space on small covalent metal complexes is discussed.

Magnetic circular dichroism within the algebraic diagrammatic construction scheme of the polarization propagator up to third order.

We present an implementation of the B term of Magnetic Circular Dichroism (MCD) within the Algebraic Diagrammatic Construction (ADC) scheme of the polarization propagator and its Intermediate State Representation. As illustrative results, the MCD spectra of the ADC variants ADC(2), ADC(2)-x, and ADC(3) of the molecular systems uracil, 2-thiouracil, 4-thiouracil, purine, hypoxanthine 1,4-naphthoquinone, 9,10-anthraquinone, and 1-naphthylamine are computed and compared with results obtained by using the Resolution-of-Identity Coupled-Cluster Singles and Approximate Doubles method, with literature Time-Dependent Density Functional Theory results, and with available experimental data.

Ab Initio Excited-State Electronic Circular Dichroism Spectra Exploiting the Third-Order Algebraic-Diagrammatic Construction Scheme for the Polarization Propagator.

Excited-state rotatory strengths are reported for the first time at a correlated ab initio level, here with the algebraic diagrammatic construction scheme of the polarization propagator up to the third order. To demonstrate the capabilities of this computational approach, the gas phase S1 electronic circular dichroism spectra of the bicyclic ketones (1R)-camphor, (1R)-norcamphor, and (1R)-fenchone have been calculated at the ADC(3) level of theory. Furthermore, the solution excited-state spectra of the energetically lowest conformer of R-(+)-1,1'-bi(2-naphthol) have been computed with inclusion of a polarizable continuum model at the ADC(2) level of theory.

Electronic circular dichroism spectra using the algebraic diagrammatic construction schemes of the polarization propagator up to third order.

Expressions for the calculation of rotatory strengths using the algebraic diagrammatic construction (ADC) scheme of the polarization propagator in both length and velocity gauges have been implemented. This enables the simulation of electronic circular dichroism (ECD) spectra at the ADC level up to third order of perturbation theory. The ADC(n) methods produce rotatory strengths of comparable accuracy to those obtained with coupled cluster methods of corresponding approximation levels, as evaluated for methyloxirane, methylthiirane, dimethyloxirane, dimethylthiirane, hydrogen peroxide, and dihydrogen disulfide. ECD spectra of (1R)-camphor, (1R)-norcamphor, and (1R)-fenchone computed at the third order ADC(3) level of theory are shown to agree very favorably with experimental gas phase spectra, demonstrating the usefulness of ADC for the calculation of chiro-optical properties of organic molecules. ADC(2) in combination with the polarizable continuum model is shown to successfully reproduce the ECD spectrum of the L-epinephrine enantiomer in water, further demonstrating the applicability of this approach.

Efficient implementation of isotropic cubic response functions for two-photon absorption cross sections within the self-consistent field approximation.

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 ∼3.3% (one-photon off-resonance regions) and ∼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 ∼5% when using the reduced expression as compared to the full form of the isotropic cubic response function.