Postcombustion CO2 capture and storage (CCS) is a key technological approach to reducing greenhouse gas emission while we transition to carbon-free energy production. However, current solvent-based CO2 capture processes are considered too energetically expensive for widespread deployment. Vacuum swing adsorption (VSA) is a low-energy CCS that has the potential for industrial implementation if the right sorbents can be found. Metal-organic framework (MOF) materials are often promoted as sorbents for low-energy CCS by highlighting select adsorption properties without a clear understanding of how they perform in real-world VSA processes. In this work, atomistic simulations have been fully integrated with a detailed VSA simulator, validated at the pilot scale, to screen 1632 experimentally characterized MOFs. A total of 482 materials were found to meet the 95% CO2 purity and 90% CO2 recovery targets (95/90-PRTs)-365 of which have parasitic energies below that of solvent-based capture (∼290 kWhe/MT CO2) with a low value of 217 kWhe/MT CO2. Machine learning models were developed using common adsorption metrics to predict a material's ability to meet the 95/90-PRT with an overall prediction accuracy of 91%. It was found that accurate parasitic energy and productivity estimates of a VSA process require full process simulations.

Metal-organic frameworks (MOFs) have garnered interest as potential solid sorbent materials for postcombustion CO2 capture. With a seemingly infinite design space, high-throughput computational screening of MOFs has developed into an effective tool for the development of new materials. In this work, machine learning (ML) has been used to develop accurate quantitative structure-property relationship (QSPR) models to rapidly predict the CO2 working capacity and CO2/N-2 selectivity at the low-pressure conditions relevant to postcombustion carbon capture (0.15 bar CO2, 0.85 bar N-2). A database of over 340 000 MOFs constructed from hundreds of types of building units arranged in over 1000 net topologies was used to train and test the models. Neural network ML models were optimized using six geometric descriptors along with three so-called chemical descriptors, namely, the atomic property-weighted radial distribution function (AP-RDF) and some variants thereof, the bag-of-atoms, and the chemical motif density descriptors. The ML models built using geometric descriptors alone resulted in test set correlation R-2 values of only 0.71 and 0.75 for CO2 working capacity and CO2/N-2 selectivity, respectively. ML models built with a single type of chemical descriptor all outperformed the geometry-only models giving R-2 values ranging from 0.83 to 0.94 with the AP-RDF model being the most accurate. Overall, the best model was built using a combination of AP-RDF, chemical motif, and geometric descriptors (R-2 = 0.96 when predicting the CO2 working capacity and R-2 = 0.95 for the selectivity). To date, these are the most accurate ML models for predicting low-pressure gas uptake of MOFs. The combined model was able to capture 994 of the true top 1000 MOFs (from a test set of similar to 70 000) within the top 5000 MOFs as predicted by the model with CO, working capacity as the target. Thus, if the ML model were used to prescreen materials for more compute intensive GCMC simulations, then it would result in a greater than 10 times speed up while still capturing >99% of high-performing materials. These results highlight the importance of chemical descriptors in predicting low-pressure gas adsorption properties in nanoporous materials.

This work is devoted to the development of quantitative structure property relationship (QSPR) models using machine learning to predict CO2 working capacity and CO2/H-2 selectivity for precombustion carbon capture using a topologically diverse database of hypothetical metal organic framework (MOF) structures (358 400 MOFs, 1166 network topologies). Such a diversity of the networks topology is much higher than previously used (<20 network topologies) for rapid and accurate recognition of high-performing MOFs for other gas-separation applications. The gradient boosted trees regression method allowed us to use 80% of the database as a training set, while the rest was used for the validation and test set. The QSPR models are first built using purely geometric descriptors of MOFs such as gravimetric surface area and void fraction. Additional models which account for chemical features of MOFs are constructed using atomic property weighted radial distribution functions (AP-RDFs) with a novel normalization to accommodate the size diversity of the MOF database. It is shown that the best models for CO2 working capacities (R-2 = 0.944) and CO2/H-2 selectivities (R-2 = 0.872) are built from a combination of six geometric descriptors and three AP-RDF descriptors. However, more important is that our QSPR models can identify top 1000 high-performing MOFs in just top 3000 or 5000 MOFs. This work shows that QSPR modeling can account for the topological diversity of MOFs and accelerate the screening for top-performing MOFs for precombustion carbon capture.

Understanding the performance of machine learning algorithms is essential for designing more accurate and efficient statistical models. It is not always possible to unravel the reasoning of neural networks. Here, we propose a method for calculating machine learning kernels in closed and analytic form by combining atomic property weighted radial distribution function (AP-RDF) descriptor with a Gaussian kernel. This allowed us to analyze and improve the performance of the Bag-of-Bonds descriptor when the bond type restriction is included in AP-RDF. The improvement is achieved for the prediction of molecular atomization energies (MAE = 1.7 kcal/mol for QM7 data set) and is due to the incorporation of a tensor product into the kernel, which captures the multidimensional representation of the AP-RDF. On the other hand, the numerical version of the AP-RDF is a constant size descriptor, making it more computationally efficient than Bag-of-Bonds. We have also discussed a connection between molecular quantum similarity and machine learning kernels with first-principles kinds of descriptors.

Periodic frameworks that possess a net charge, such as zeolites, are an important class of materials in wide use. For guest-host interactions to be simulated in these materials, partial atomic charges are often used. In this work, we investigate two methods for the generation of partial atomic charges in periodic systems having a net framework charge. We first examine the validity of generating REPEAT electrostatic potential fitted charges derived from periodic electronic structure calculations, where a constant background charge is added to neutralize the net charge on the framework. The constant background charge obviates the need to add neutralizing counterions, which may induce artifacts such as polarization in the infinite periodic system. The second method we explore is the split charge equilibration (SQE) method for the rapid generation of partial atomic charges. The original formulation of the SQE method cannot be applied to systems with a net charge. In this work, we reformulate the SQE method by transforming the split charges into an atomic charge basis that allows for non-neutral systems to be treated. The new SQE model, which we call SQE(AB) (for atomic basis), was validated with a series of tests using both charged and neutral metal organic frameworks and zeolites. It was shown that SQEAB gives equivalent results to those of the original SQE model for neutral systems. We then demonstrated that the SQE(AB) method is able to "capture" the chemical structure of a charged framework better than that of the charge equilibration model by comparing to REPEAT electrostatic potential fitted charges.

In this paper, the relaxed self-consistent field infinite order constricted variational density functional theory (RSCF-CV(∞)-DFT) for triplet calculations is presented. Here, we focus on two main features of our implementation. First, as an extension of our previous work by Krykunov and Ziegler ( J. Chem. Theory Comput. 2013 , 9 , 2761 ), the optimization of the transition matrix representing the orbital transition is implemented and applied for vertical triplet excitations. Second, restricting the transition matrix, we introduce RSCF-CV(∞)-DFT-based numerically stable ΔSCF-DFT-like methods, the most general of them being SVD-RSCF-CV(∞)-DFT. The reliability of the different methods, RSCF-CV(∞)-DFT and its restricted versions, is examined using the benchmark test set of Silva-Junior et al. ( J. Chem. Phys. 2008 , 129 , 104103 ). The obtained excitation energies validate our approach and implementation for RSCF-CV(∞)-DFT and also show that SVD-RSCF-CV(∞)-DFT mimics very well ΔSCF-DFT, as the root-mean-square deviations between these methods are less than 0.1 eV for all functionals examined.

In Delta SCF density functional theory studies of a i -> a transition one performs separate fully self-consistent field calculations on the ground state configuration (i)(n) (n = 1,2) and the excited state configuration (i)(n - 1)a. The excitation energy for the transition i -> a is subsequently determined as the Kohn-Sham energy difference Delta E-i -> a = E[i(n - 1)a] - E[i(n)] between the ground state (i)(n) and the excited state configuration (i)(n - 1)a. The Delta SCF scheme has been applied extensively and works well for lower energy excitations provided that they can be represented by a single orbital replacement or transition i -> a. However, for excitations of higher energy Delta SCF tends to become numerically unstable with a variational collapse to transitions of lower energy. We demonstrate here a numerically stable Delta SCF scheme for local functionals that is guaranteed not to collapse on excited configurations of lower energy as well as the ground state. The new scheme is based on constricted variational density functional theory in which the canonical ground state orbitals are allowed to relax (R-CV(infinity)-DFT). Since it is restricted to a single orbital replacement i -> a it is termed SOR-R-CV(infinity)-DFT.

For the polyacenes series from naphthalene to hexacene, we present the vertical singlet excitation energies 1 (1)La and 1 (1)Lb, as well as the first triplet excitation energies obtained by the all-order constricted variational density functional theory with orbital relaxation (R-CV(∞)-DFT). R-CV(∞)-DFT is a further development of variational density functional theory (CV(∞)-DFT), which has already been successfully applied for the calculation of the vertical singlet excitation energies (1)La and (1)Lb for polyacenes,15 and we show that one obtains consistent excitation energies using the local density approximation as a functional for singlet as well as for triplet excitations when going beyond the linear response theory. Furthermore, we apply self-consistent field density functional theory (ΔSCF-DFT) and compare the obtained excitation energies for the first triplet excitations T1, where, due to the character of the transition, ΔSCF-DFT and R-CV(∞)-DFT become numerically equivalent, and for the singlet excitations 1 (1)La and 1 (1)Lb, where the two methods differ.

We present benchmark calculations on excitation energies based on time-dependent density functional theory (TDDFT) as well as orbital relaxed self-consistent and constricted variational DFT (RSCF-CV-DFT) with and without use of the Tamm-Dancoff approximation. The compilation contains results for the 3d complexes MnO₄⁻, CrO₄²⁻, and VO₄³⁻, as well as the 4d congeners RuO₄, TcO₄⁻, and MoO₄²⁻, and 5d homologues OsO₄, ReO₄⁻, and WO₄²⁻. Considerations have been given to the local density approximation (LDA) and the functionals BP86 and PBE based on the generalized gradient approximation (GGA), as well as the hybrids B3LYP, BHLYP, and PBE0 and the length corrected functional LCBP86. We find for the 3d complexes that RSCF-CV-DFT fares better than TDDFT. Thus, in the case of RSCF-CV-DFT, the average root-mean-square deviations (RMSDs) are 0.25-0.3 eV for GGAs, 0.1 eV for B3LYP, and 0.45 eV for BHLYP. TDDFT affords RMSDs that on average range from 0.3 eV for local functionals to 0.7 eV for BHLYP with the largest fraction of Hartree-Fock (HF) exchange. TDDFT is seen to fare better among the heavier tetraoxo systems. For the 4d and 5d systems, the three functionals B3LYP, PBE0 with an intermediate fraction of HF exchange, and LCBP86 have the lowest RMSD of 0.2 eV, whereas the local functionals (LDA, BP86, BPE) and BHLYP with the highest HF fraction and LCBP86* have a somewhat larger RMSD of 0.3 eV. Nearly the same performance is observed for RSCF-CV-DFT with respect to the different functionals in the case of the 4d and 5d systems. Thus, for the heavier tetraoxo systems, the two DFT schemes are comparable in accuracy.

The random phase approximation (RPA) equation of adiabatic time dependent density functional ground state response theory (ATDDFT) has been used extensively in studies of excited states. It extracts information about excited states from frequency dependent ground state response properties and avoids, thus, in an elegant way, direct Kohn-Sham calculations on excited states in accordance with the status of DFT as a ground state theory. Thus, excitation energies can be found as resonance poles of frequency dependent ground state polarizability from the eigenvalues of the RPA equation. ATDDFT is approximate in that it makes use of a frequency independent energy kernel derived from the ground state functional. It is shown in this study that one can derive the RPA equation of ATDDFT from a purely variational approach in which stationary states above the ground state are located using our constricted variational DFT (CV-DFT) method and the ground state functional. Thus, locating stationary states above the ground state due to one-electron excitations with a ground state functional is completely equivalent to solving the RPA equation of TDDFT employing the same functional. The present study is an extension of a previous work in which we demonstrated the equivalence between ATDDFT and CV-DFT within the Tamm-Dancoff approximation.

We have recently introduced a new version of the constricted nth-order variational density functional method (CV(n)-DFT) in which the occupied excited-state orbitals are allowed to relax in response to the change of both the Coulomb and exchange-correlation potential in going from the ground state to the excited state. The new scheme is termed the relaxed self-consistent field nth-order constricted variational density functional (RSCF-CV(n)-DFT) method. We demonstrate in the current study that it is possible to extend RSCF-CV()-DFT to double excitations. It is further shown that it is possible within certain limitations to derive a second-order CV(2)-DFT theory that can be considered as an extension of adiabatic time-dependent density functional theory (TDDFT) to double excitations within the Tamm-Dancoff approximation.

We have applied the relaxed and self-consistent extension of constricted variational density functional theory (RSCF-CV-DFT) for the calculation of the lowest charge transfer transitions in the molecular complex X-TCNE between X = benzene and TCNE = tetracyanoethylene. Use was made of functionals with a fixed fraction (α) of Hartree-Fock exchange ranging from α = 0 to α = 0.5 as well as functionals with a long range correction (LC) that introduces Hartree-Fock exchange for longer inter-electronic distances. A detailed comparison and analysis is given for each functional between the performance of RSCF-CV-DFT and adiabatic time-dependent density functional theory (TDDFT) within the Tamm-Dancoff approximation. It is shown that in this particular case, all functionals afford the same reasonable agreement with experiment for RSCF-CV-DFT whereas only the LC-functionals afford a fair agreement with experiment using TDDFT. We have in addition calculated the CT transition energy for X-TCNE with X = toluene, o-xylene, and naphthalene employing the same functionals as for X = benzene. It is shown that the calculated charge transfer excitation energies are in as good agreement with experiment as those obtained from highly optimized LC-functionals using adiabatic TDDFT. We finally discuss the relation between the optimization of length separation parameters and orbital relaxation in the RSCF-CV-DFT scheme.

We have benchmarked the performance of time-independent density functional theory (ΔSCF and RSCF-CV-DFT) in studies on Rydberg transitions employing five different standard functionals and a diffuse basis. Our survey is based on 71 triplet or singlet Rydberg transitions distributed over nine different species: CO (7), CH2O (8), C2H2 (8), H2O (10), C2H4 (13), Be (6), Mg (6), and Zn (8). The best performance comes from the long-range corrected functional LCBP86 (ω = 0.4.) with an average root-mean-square deviation (RMSD) of 0.23 eV. Of similar accuracy are LDA and B3LYP, both with a RMSD of 0.24 eV. The largest RMSD of 0.32 eV comes from BP86 and LCBP86* (ω = 0.75). The performance of ΔSCF is considerably better than that of adiabatic time-dependent density functional theory (ATDDFT) and matches that of highly optimized long-range corrected functionals. However, it is not as accurate as ATDDFT based on highly tuned functionals. The reasonable success of ΔSCF is based on its well-documented ability to afford good estimates of ionization potentials (IP) and electron affinities (EA) even for simple local functionals after orbital relaxation has been taken into account. In ATDDFT based on semilocal functionals, both IP and -EA are poorly described, with errors of up to 5 eV. In the transition energy (ΔE = IP - EA), these errors are canceled to some degree. However, ΔE still carries an error exceeding 1 eV.

We introduce here a new version of the constricted nth order variational density functional method (CV(n)-DFT) in which the occupied excited state orbitals are allowed to relax in response to the change of both the Coulomb and exchange-correlation potential in going from the ground state to the excited state. The new scheme is termed the relaxed self-consistent field nth order constricted variational density functional (RSCF-CV(n)-DFT) method. We have applied the RSCF-CV(n)-DFT scheme to the nσ→π* transitions in which an electron is moved from an occupied lone-pair orbital nσ to a virtual π* orbital. A total of 34 transitions involving 16 different compounds were considered using the LDA, B3LYP, and BHLYP functionals. The DFT-based results were compared to the "best estimates" (BE) from high level ab initio calculations. With energy terms included to second order in the variational parameters (CV(2)-DFT), our theory is equivalent to the adiabatic version of time dependent DFT . We find that calculated excitation energies for CV(2)-DFT using LDA and BHLYP differ substantially from BE with root-mean-square-deviations (RMSD) of 0.87 and 0.65 eV, respectively, whereas B3LYP affords an excellent fit with BE at RMSD = 0.33 eV. Resorting next to CV(∞)-DFT where energy terms to all orders in the variational parameters are included results for all three functionals in too high excitation energies with RMSD = 1.62, 1.14, and 1.48 eV for LDA, B3LYP, and BHLYP, respectively. Allowing next for a relaxation of the orbitals (nσ,π*) that participate directly in the transition (SCF-CV(n)-DFT) leads to an improvement with RMSD = 0.49 eV (LDA), 0.50 eV (B3LYP), and 1.12 eV (BHLYP). The best results are obtained with full relaxation of all orbitals (RSCF-CV(n)) where now RMSD = 0.61 eV (LDA), 0.32 eV (B3LYP), and 0.52 eV (BHLYP). We discuss finally the relation between RSCF-CV(n) and Slater's ΔSCF method and demonstrate that the two schemes affords quite similar results in those cases where the excitation can be described by a single orbital displacement (nσ→π*).

We present the results of calculations on the vertical singlet (1)La and (1)Lb excitation energies in acenes within time dependent density functional theory (TDDFT), second order constricted variational DFT (CV(2)-DFT), and all order constricted variational DFT (CV(∞)-DFT) using the local density approximation LDA(VWN). For the linear acenes it is shown that the application of the Tamm-Dancoff (TD) approximation to TDDFT (TDDFT-TD) substantially improves the agreement with experiment compared to pure TDDFT. This improvement leads to the correct ordering of the (1)La and (1)Lb excitation energies in naphthalene. As TDDFT-TD is equivalent to the second order CV(2)-TD method one might hope for further improvements by going to all orders in CV(∞)-TD. Indeed, for linear acenes the application of the CV(∞)-TD method brings the agreement with experiment to within 0.1 eV for both types of excitations using the simple LDA functional. The CV(∞)-TD method based on LDA is also shown to be accurate for 15 nonlinear acenes with root-mean-square deviations of 0.24 eV for (1)La and 0.17 eV for (1)Lb.

We review the theoretical foundation of constricted variational density functional theory and illustrate its scope through applications.

We have applied the constricted variational density functional method (CV(n)-DFT) to n → π* transitions in which an electron is moved from an occupied lone-pair orbital n to a virtual π* orbital. A total of 34 transitions involving 16 different compounds were considered using the local density approximation (LDA), Becke, three-parameter, Lee-Yang-Parr (B3LYP), and BHLYP functionals. The DFT-based results were compared to the "best estimates" (BE) from high-level ab initio calculations. With energy terms included to second order in the variational parameters (CV(2)-DFT), our theory is equivalent to the adiabatic version of time-dependent density functional theory (DFT). We find that calculated excitation energies for CV(2)-DFT using LDA and BHLYP differ substantially from BE with root-mean-square-deviations (rmsd) of 0.86 and 0.69 eV, respectively, whereas B3LYP affords an excellent fit with BE at rmsd = 0.18 eV. Resorting next to CV(∞)-DFT, where energy terms to all orders in the variational parameters are included, results in all three functionals in too high excitation energies with rmsd = 1.69, 1.14, and 0.93 eV for LDA, B3LYP, and BHLYP, respectively. Adding in orbital relaxation considerably improves the results with rmsd = 0.54, 0.30, and 0.48 eV for LDA, B3LYP, and BHLYP, respectively. It is concluded that CV(∞)-DFT with orbital relaxation is a robust method for which the accuracy is less functionally dependent than that of CV(2)-DFT or adiabatic TDDFT.

We present a theoretical method that combines density functional theory (DFT) and full configuration interaction (CI). Such a combination is based on the impurity model to (i) incorporate the advantages of DFT in description of the dynamic correlation and (ii) reduce the number of the two-electron integrals. The impurity model has been introduced with the help of a special type of the hybrid functional, namely, the exact exchange for correlated electrons (EECE), and the orthogonal fragment orbitals formalism. The EECE functional, originally based on the formulation of the local density approximation + U (LDA + U) method without the empirical parameter U, has been rewritten in this work in a form that incorporates the projected electronelectron repulsion energy operator. This representation has been compared with other types of nonglobal hybrids: position dependent and range separated hybrids. Further, the obtained Hamiltonian of the generalized Anderson model has been transformed to the form that affords the implementation of the linear scaling density-matrix renormalization group algorithm. (c) 2011 Wiley Periodicals, Inc.

It is well known that time-dependent density functional theory (TD-DFT) based on standard gradient corrected functionals affords both a quantitative and qualitative incorrect picture of charge transfer transitions between two spatially separated regions. It is shown here that the well known failure can be traced back to the use of linear response theory. Further, it is demonstrated that the inclusion of higher order terms readily affords a qualitatively correct picture even for simple functionals based on the local density approximation. The inclusion of these terms is done within the framework of a newly developed variational approach to excitation energies called constrained variational density functional theory (CV-DFT). To second order [CV(2)-DFT] this theory is identical to adiabatic TD-DFT within the Tamm-Dancoff approximation. With inclusion of fourth order corrections [CV(4)-DFT] it affords a qualitative correct description of charge transfer transitions. It is finally demonstrated that the relaxation of the ground state Kohn-Sham orbitals to first order in response to the change in density on excitation together with CV(4)-DFT affords charge transfer excitations in good agreement with experiment. The new relaxed theory is termed R-CV(4)-DFT. The relaxed scheme represents an effective way in which to introduce double replacements into the description of single electron excitations, something that would otherwise require a frequency dependent kernel.