Prediction of MOF Performance in Vacuum Swing Adsorption Systems for Postcombustion CO2 Capture Based on Integrated Molecular Simulations, Process Optimizations, and Machine Learning Models.

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.

Constricted Variational Density Functional Theory Approach to the Description of Excited States.

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

Excited State Studies of Polyacenes Using the All-Order Constricted Variational Density Functional Theory with Orbital Relaxation.

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.

Applications of time-dependent and time-independent density functional theory to Rydberg transitions.

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.

Introducing constricted variational density functional theory in its relaxed self-consistent formulation (RSCF-CV-DFT) as an alternative to adiabatic time dependent density functional theory for studies of charge transfer transitions.

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.

The implementation of a self-consistent constricted variational density functional theory for the description of excited states.

We present here the implementation of a self-consistent approach to the calculation of excitation energies within regular Kohn-Sham density functional theory. The method is based on the n-order constricted variational density functional theory (CV(n)-DFT) [T. Ziegler, M. Seth, M. Krykunov, J. Autschbach, and F. Wang, J. Chem. Phys. 130, 154102 (2009)] and its self-consistent formulation (SCF-CV(∞)-DFT) [J. Cullen, M. Krykunov, and T. Ziegler, Chem. Phys. 391, 11 (2011)]. A full account is given of the way in which SCF-CV(∞)-DFT is implemented. The SCF-CV(∞)-DFT scheme is further applied to transitions from occupied π orbitals to virtual π(∗) orbitals. The same series of transitions has been studied previously by high-level ab initio methods. We compare here the performance of SCF-CV(∞)-DFT to that of time dependent density functional theory (TD-DFT), CV(n)-DFT and ΔSCF-DFT, with the ab initio results as a benchmark standard. It is finally demonstrated how adiabatic TD-DFT and ΔSCF-DFT are related through different approximations to SCF-CV(∞)-DFT.

On the calculation of charge transfer transitions with standard density functionals using constrained variational density functional theory.

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.

Implementation of a hybrid DFT method for calculating NMR shieldings using Slater-type orbitals with spin-orbital coupling included. Applications to 187Os, 195Pt, and 13C in heavy-metal complexes.

We report on the implementation of an algorithm for the calculation of the NMR shielding tensor. Our scheme is based on the Hartree-Fock method and the zeroth-order regular approximation (ZORA) Hamiltonian with spin-orbital coupling included. Gauge-including atomic orbitals (GIAOs) are employed to ensure the origin invariance of the results. Unlike the previous implementation by Fukui and Baba [J. Chem. Phys. 2002, 117, 7836], our computational scheme makes use of Slater-type orbitals. We have employed this method in B3LYP calculations of the (13)C, (195)Pt, and (187)Os NMR chemical shifts in 5d metal carbonyls, Pt(II) square-planar complexes, and osmium phosphines, respectively. The calculated NMR chemical shifts are compared to the results obtained with the BP86 and BLYP functionals, as well as the Hartree-Fock method. Comparisons are also given to experimental values. For the (195)Pt chemical shifts, we have found a small improvement with respect to experiment for the B3LYP results over the BP86 and BLYP values. For the other systems, use of the B3LYP method does not improve the agreement with experiment compared to results from pure functionals such as BP86 and BLYP.

On the relation between time-dependent and variational density functional theory approaches for the determination of excitation energies and transition moments.

It is shown that it is possible to derive the basic eigenvalue equation of adiabatic time-dependent density functional theory within the Tamm-Dancoff approximation (TD-DFT/TD) from a variational principle. The variational principle is applied to the regular Kohn-Sham formulation of DFT energy expression for a single Slater determinant and leads to the same energy spectrum as TD-DFT/TD. It is further shown that this variational approach affords the same electric and magnetic transition moments as TD-DFT/TD. The variational scheme can also be applied without the Tamm-Dancoff approximation. Practical implementations of TD-DFT are limited to second order response theory which introduces errors in transition energies for charge transfer and Rydberg excitations. It is indicated that higher order terms can be incorporated into the variational approach. It is also discussed how the current variational method is related to traditional DFT schemes based on variational principles such as DeltaSCF-DFT, and how they can be combined.

Application of magnetically perturbed time-dependent density functional theory to magnetic circular dichroism. II. Calculation of A terms.

The magnetically perturbed time-dependent density functional theory is used to derive equations for the magnetic circular dichroism (MCD) of degenerate transitions of closed shell molecules. The MCD of this type of transition can be divided into two contributions. The dominant contribution is usually that from A terms that arise because of the breaking of the degeneracy of the excited state in the presence of the magnetic field. The second contribution comes from B terms that arise because of the perturbation of the transition dipole by the magnetic field. The formalism is applied to ten tetrahedral d(0) transition metal oxy- and thioanions. The MCD parameters of these systems are reproduced quite well by the calculations. Simulated spectra derived from the calculated parameters are in good agreement with the observed spectra.

A revised electronic Hessian for approximate time-dependent density functional theory.

Time-dependent density functional theory (TD-DFT) at the generalized gradient level of approximation (GGA) has shown systematic errors in the calculated excitation energies. This is especially the case for energies representing electron transitions between two separated regions of space or between orbitals of different spatial extents. It will be shown that these limitations can be attributed to the electronic ground state Hessian G(GGA). Specifically, we shall demonstrate that the Hessian G(GGA) can be used to describe changes in energy due to small perturbations of the electron density (Deltarho), but it should not be applied to one-electron excitations involving the density rearrangement (Deltarho) of a full electron charge. This is in contrast to Hartree-Fock theory where G(HF) has a trust region that is accurate for both small perturbations and one-electron excitations. The large trust radius of G(HF) can be traced back to the complete cancellation of Coulomb and exchange terms in Hartree-Fock (HF) theory representing self-interaction (complete self-interaction cancellation, CSIC). On the other hand, it is shown that the small trust radius for G(GGA) can be attributed to the fact that CSIC is assumed for GGA in the derivation of G(GGA) although GGA (and many other approximate DFT schemes) exhibits incomplete self-interaction cancellation (ISIC). It is further shown that one can derive a new matrix G(R-DFT) with the same trust region as G(HF) by taking terms due to ISIC properly into account. Further, with TD-DFT based on G(R-DFT), energies for state-to-state transitions represented by a one-electron excitation (psi(i)-->psi(a)) are approximately calculated as DeltaE(ai). Here DeltaE(ai) is the energy difference between the ground state Kohn-Sham Slater determinant and the energy of a Kohn-Sham Slater determinant where psi(i) has been replaced by psi(a). We make use of the new Hessian in two numerical applications involving charge-transfer excitations. It is concluded that higher than second order response theory (involving ISIC terms) must be used in approximate TD-DFT, in order to describe charge-transfer excitations.

Bond Type Restricted Property Weighted Radial Distribution Functions for Accurate Machine Learning Prediction of Atomization Energies.

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.

A quantum chemical approach to the design of chiral negative index materials.

This paper presents methodology developed for the computational modeling and design of negative refractive index materials (NIMs) based on molecular chirality. An application of the methodology is illustrated by ab initio computations on two organometallic molecules which constitute the monomer units of a chiral polymer. Comparisons with experimental data for the polymer are made. Even though the resulting chirality parameter for the pristine material is small, it is shown that negative index can be achieved by introducing sharp plasmonic resonances with metal nanoparticle inclusions.

A New Split Charge Equilibration Model and REPEAT Electrostatic Potential Fitted Charges for Periodic Frameworks with a Net Charge.

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 SQEAB (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 SQEAB 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.

Self-Consistent Constricted Variational Theory RSCF-CV(∞)-DFT and Its Restrictions To Obtain a Numerically Stable ΔSCF-DFT-like Method: Theory and Calculations for Triplet States.

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.

Applications of Time-Dependent and Time-Independent Density Functional Theory to Electronic Transitions in Tetrahedral d(0) Metal Oxides.

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⁻, CrO4²â», and VO4³â», as well as the 4d congeners RuO4, TcO4⁻, and MoO4²â», and 5d homologues OsO4, ReO4⁻, and WO4²â». 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.

Derivation of the RPA (Random Phase Approximation) Equation of ATDDFT (Adiabatic Time Dependent Density Functional Ground State Response Theory) from an Excited State Variational Approach Based on the Ground State Functional.

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.

Applications of Time Dependent and Time Independent Density Functional Theory to the First π to π* Transition in Cyanine Dyes.

The first π → π* transition in a number of cyanine dyes was studied using both time dependent and time independent density functional methods using a coupled cluster (CC2) method as the benchmark scheme. On the basis of 10 different functionals, it was concluded that adiabatic time dependent density functional theory (ATDDFT) almost independently of the functional gives rise to a singlet-triplet separation that is too large by up to 1 eV, leading to too high singlet energies and too low triplet energies. This trend is even clearer when the Tamm-Dancoff (TD) approximation is introduced and can in ATDDFT/TD be traced back to the representation of the singlet-triplet separation by a HF-type exchange integral between π and π*. The time independent DFT methods (ΔSCF and RSCF-CV-DFT) afford triplet energies that are functional independent and close to those obtained by ATDDFT. However, both the singlet energies and the singlet-triplet separations increases with the fraction α of HF exchange. This trend can readily be explained in terms of the modest magnitude of a KS-exchange integral between π and π* in comparison to the much larger HF-exchange integral. It was shown that a fraction α of 0.5 affords good estimates of both the singlet energies and the singlet-triplet separations in comparison to several ab initio benchmarks.

Self-consistent Formulation of Constricted Variational Density Functional Theory with Orbital Relaxation. Implementation and Applications.

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*).

Accurate Theoretical Description of the (1)La and (1)Lb Excited States in Acenes Using the All Order Constricted Variational Density Functional Theory Method and the Local Density Approximation.

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.