Local CC2 electronic excitation energies for large molecules with density fitting

A new local method for the computation of electronic excitation energies of singlet states in extended molecular systems is presented. It is based on the CC2 model and local approximations to the wave functions. In the proposed method the singles excitations are treated nonlocally and local restrictions are imposed on doubles amplitudes only. The accuracy of the new method was tested by calculating several lowest excited states for 14 molecules and comparing them with canonical CC2 values. Deviations of the local excitation energies from the canonical reference values do not exceed 0.05 eV for all test molecules and all states in the lower energy range investigated in this work. The method uses the density-fitting approximation for all two-electron integrals, which considerably simplifies the computational complexity of the individual diagrams. A combination of the local approximations and the powerful density-fitting technique leads to a low-scaling method, capable to treat molecular systems comprised of 100 atoms and more in a basis of a polarized double zeta quality. A test calculation for a system consisting of 127 atoms and 370 active electrons without symmetry is presented to show the efficiency of the new method.     

Multireference coupled-cluster methods using an incomplete model space: Application to ionization potentials and excitation energies of formaldehyde

A recent fully linked multireference coupled-cluster method using an incomplete model space is applied to the direct calculation of the difference energies of formaldehyde. For the calculation of excitation energies (EE) use is made of a reference space composed of particle-hole excited configurations built from a set of active orbitals. Ionization potentials are obtained from a model space of singly ionized configurations. Results are compared with experiment and previous calculations.     

Molecular applications of the intermediate Hamiltonian Fock-space coupled-cluster method for calculation of excitation energies

The intermediate Hamiltonian Fock-space coupled-cluster (FS-CC) method with singles and doubles is applied to calculate vertical excitation energies (EEs) for some molecular systems. The calculations are performed for several small molecules, such as H2O, N2, and CO, and for larger systems, such as C2H4, C4H6, and C6H6. Due to the intermediate Hamiltonian formulation, which provides a robust computational scheme for solving the FS-CC equations, and the efficient factorization strategy, relatively large basis sets and model spaces are employed permitting a comparison of the calculated vertical EEs with the experimental data.     

A new fragment-based approach for calculating electronic excitation energies of large systems

We present a new fragment-based scheme to calculate the excited states of large systems without necessity of a Hartree-Fock (HF) solution of the whole system. This method is based on the implementation of the renormalized excitonic method [M. A. Hajj et al., Phys. Rev. B 72, 224412 (2005)] at ab initio level, which assumes that the excitation of the whole system can be expressed by a linear combination of various local excitations. We decomposed the whole system into several blocks and then constructed the effective Hamiltonians for the intra- and inter-block interactions with block canonical molecular orbitals instead of widely used localized molecular orbitals. Accordingly, we avoided the prerequisite HF solution and the localization procedure of the molecular orbitals in the popular local correlation methods. Test calculations were implemented for hydrogen molecule chains at the full configuration interaction, symmetry adapted cluster/symmetry adapted cluster configuration interaction, HF/configuration interaction singles (CIS) levels and more realistic polyene systems at the HF/CIS level. The calculated vertical excitation energies for lowest excited states are in reasonable accordance with those determined by the calculations of the whole systems with traditional methods, showing that our new fragment-based method can give good estimates for low-lying energy spectra of both weak and moderate interaction systems with economic computational costs.     

Intermediate Hamiltonian Fock-space multireference coupled-cluster method with full triples for calculation of excitation energies

The intermediate Hamiltonian multireference coupled-cluster (CC) method with singles, doubles, and triples within the excited (1,1) sector of Fock space (FS) is implemented and formulated to calculate excitation energies (EEs). Due to the intermediate Hamiltonian formulation, which provides a robust computational scheme for solving the FS-CC equations, coupled to an efficient factorization strategy, relatively large basis sets and model spaces are employed permitting basis set converged comparisons of the calculated vertical EEs, which can be compared to the experimental data for the N(2) and CO molecules. The issue of charge-transfer separability is also addressed.     

A note on the direct calculation of excitation energies by quasi-degenerate MBPT and coupled-cluster theory

For excitation energy calculations using quasi-degenerate MBPT or coupled-cluster (CC) theory, the hitherto chosen strategy has been to have particle-hole (p-h) determinants forming the model space and to use standard formalisms originally developed for complete model spaces. In view of our recent analysis, the p-h determinants constitute an incomplete model space for which intermediate normalization (IN) of the wave-operator Ω is not appropriate if a linked formulation is desired. The theoretical status of earlier applications which had used formulae appropriate to IN, yet ignored disconnected diagrams, is analyzed. It is shown that if only excitation energies, rather than the full Ω, are desired, then no theoretical error is made. The situation is, however, fortuitous in that for mp-mh model spaces with m > 1, a similar procedure using IN would necessarily generate disconnected diagrams.     

MRD-CI studies of vertical excitation energies of unsaturated hydrocarbon molecules

Systematic MRD-CI calculations using the AM1 Hamiltonian have been carried out for two polyenes and eight aromatic hydrocarbons ranging from benzene to ovalene (C₃₂H₁₄). Twenty singlet–singlet excitation energies in these compounds were calculated and compared with experimental data and ab initio STO-3G results. On an absolute scale, the AM1/MRD-CI approach underestimates the excitation energies to states with dominant covalent character by an average of 1.1 eV, whereas the errors for ionic states are between ?1.0 and 1.0 eV. The STO-3G calculated data are much too high by ≈ 1 eV and ≈ 5 eV, respectively. The inclusion of σπ-correlation effects through second-order Epstein–Nesbet perturbation theory combined with the use of localized orbitals leads to a significant improvement of the ab initio calculated state energies. In an analogous AM1 treatment, negligible corrections for the σπ correlations are found, which is attributed to the implicit account in the parameters and approximation of the semiempirical Hamiltonian. The possible error sources of the calculational methods are discussed. © 1994 by John Wiley & Sons, Inc.     

Bond energies for molecules, clusters, and deposit systems

A new approach is suggested to the assignment of bond energies in molecules and clusters. It uses a shareholder principle for the redistribution of the shifts in atomic energies, which arise in a molecule, on the bonds. The scheme is directly suitable for semiempirical methods, where only one- and two-center terms occur. MSINDO calculations are performed to demonstrate the suitability of the approach for molecules and clusters. As an application the bonding in a deposit system is analyzed for the case of copper on magnesium oxide. It is found that copper atoms do not only bind to the preferred oxygen sites but also substantially to the magnesium sites. The copper-copper bonds are the strongest and will determine the structure of copper clusters on magnesium oxide surfaces.     

Linear-scaling formation of Kohn-Sham Hamiltonian: application to the calculation of excitation energies and polarizabilities of large molecular systems

We present calculations of excitation energies and polarizabilities in large molecular systems at the local-density and generalized-gradient approximation levels of density-functional theory (DFT). Our results are obtained using a linear-scaling DFT implementation in the program system DALTON for the formation of the Kohn-Sham Hamiltonian. For the Coulomb contribution, we introduce a modification of the fast multipole method to calculations over Gaussian charge distributions. It affords a simpler implementation than the original continuous fast multipole method by partitioning the electrostatic Coulomb interactions into "classical" and "nonclassical" terms which are explicitly evaluated by linear-scaling multipole techniques and a modified two-electron integral code, respectively. As an illustration of the code, we have studied the singlet and triplet excitation energies as well as the static and dynamic polarizabilities of polyethylenes, polyenes, polyynes, and graphite sheets with an emphasis on the trends observed with system size.     

An efficient fragment-based approach for predicting the ground-state energies and structures of large molecules

An efficient fragment-based approach for predicting the ground-state energies and structures of large molecules at the Hartree-Fock (HF) and post-HF levels is described. The physical foundation of this approach is attributed to the "quantum locality" of the electron correlation energy and the HF total energy, which is revealed by a new energy decomposition analysis of the HF total energy proposed in this work. This approach is based on the molecular fractionation with conjugated caps (MFCC) scheme (Zhang, D. W.; Zhang, J. Z. H. J. Chem. Phys. 2003, 119, 3599), by which a macromolecule is partitioned into various capped fragments and conjugated caps formed by two adjacent caps. We find that the MFCC scheme, if corrected by the interaction between non-neighboring fragments, can be used to predict the total energy of large molecules only from energy calculations on a series of small subsystems. The approach, named as energy-corrected MFCC (EC-MFCC), computationally achieves linear scaling with the molecular size. Our test calculations on a broad range of medium- and large molecules demonstrate that this approach is able to reproduce the conventional HF and second-order Moller-Plesset perturbation theory (MP2) energies within a few millihartree in most cases. With the EC-MFCC optimization algorithm described in this work, we have obtained the optimized structures of long oligomers of trans-polyacetylene and BN nanotubes with up to about 400 atoms, which are beyond the reach of traditional computational methods. In addition, the EC-MFCC approach is also applied to estimate the heats of formation for a series of organic compounds. This approach provides an appealing approach alternative to the traditional additivity rules based on either bond or group contributions for the estimation of thermochemical properties.     

Generalized energy-based fragmentation approach for computing the ground-state energies and properties of large molecules

We present a generalized energy-based fragmentation (GEBF) approach for approximately predicting the ground-state energies and molecular properties of large molecules, especially those charged and polar molecules. In this approach, the total energy (or properties) of a large molecule can be approximately obtained from energy (or properties) calculations on various small subsystems, each of which is constructed to contain a certain fragment and its local surroundings within a given distance. In the quantum chemistry calculation of a given subsystem, those distant atoms (outside this subsystem) are modeled as background point charges at the corresponding nuclear centers. This treatment allows long-range electrostatic interaction and polarization effects between distant fragments to be taken into account approximately, which are very important for polar and charged molecules. We also propose a new fragmentation scheme for constructing subsystems. Our test calculations at the Hartree-Fock and second-order M?ller-Plesser perturbation theory levels demonstrate that the approach could yield satisfactory ground-state energies, the dipole moments, and static polarizabilities for polar and charged molecules such as water clusters and proteins.     

Calculation of vertical excitation energies of closed-shell molecules in the CNDO and INDO approximations

The Green's function method for the calculation of vertical excitation energies is adapted to the CNDO and INDO approximations by introducing an effective interaction into the irreducible vertex part. The computational scheme is explicitly developed for closed-shell molecules and applied to H₂O, H₂CO, HCOOH, HCONH₂.     

A statistical approach to resonance energies of large molecules

For large conjugated molecules, resonance energies of SCF MO quality are not available. Here we outline a method of determining molecular resonance energies by combining a graph theoretical approach to aromaticity with a statistical analysis of random Kekule valence structures. The approach involves construction of random Kekule valence forms and subsequent enumeration of conjugated circuits within each such structure.     

Application of a spin-adapted coupled-cluster based linear response theory: Direct calculation of π -π* singlet and triplet excitation energies of conjugated systems

A procedure is tested for directly calculating exciation energies for spin-conserving and spin-forbidden transitions using a spin-adapted coupled-cluster based linear response theory. The excited states are generated from the ground state through an excitation operator S, a combination of various nh—np excitations of spin-rank zero and one for singlet and triplet excitations.     

Highly efficient implementation of pseudospectral time‐dependent density‐functional theory for the calculation of excitation energies of large molecules

We have developed and implemented pseudospectral time‐dependent density‐functional theory (TDDFT) in the quantum mechanics package Jaguar to calculate restricted singlet and restricted triplet, as well as unrestricted excitation energies with either full linear response (FLR) or the Tamm–Dancoff approximation (TDA) with the pseudospectral length scales, pseudospectral atomic corrections, and pseudospectral multigrid strategy included in the implementations to improve the chemical accuracy and to speed the pseudospectral calculations. The calculations based on pseudospectral time‐dependent density‐functional theory with full linear response (PS‐FLR‐TDDFT) and within the Tamm–Dancoff approximation (PS‐TDA‐TDDFT) for G2 set molecules using B3LYP/6‐31G*^* show mean and maximum absolute deviations of 0.0015 eV and 0.0081 eV, 0.0007 eV and 0.0064 eV, 0.0004 eV and 0.0022 eV for restricted singlet excitation energies, restricted triplet excitation energies, and unrestricted excitation energies, respectively; compared with the results calculated from the conventional spectral method. The application of PS‐FLR‐TDDFT to OLED molecules and organic dyes, as well as the comparisons for results calculated from PS‐FLR‐TDDFT and best estimations demonstrate that the accuracy of both PS‐FLR‐TDDFT and PS‐TDA‐TDDFT. Calculations for a set of medium‐sized molecules, including C_n fullerenes and nanotubes, using the B3LYP functional and 6‐31G^** basis set show PS‐TDA‐TDDFT provides 19‐ to 34‐fold speedups for C_n fullerenes with 450–1470 basis functions, 11‐ to 32‐fold speedups for nanotubes with 660–3180 basis functions, and 9‐ to 16‐fold speedups for organic molecules with 540–1340 basis functions compared to fully analytic calculations without sacrificing chemical accuracy. The calculations on a set of larger molecules, including the antibiotic drug Ramoplanin, the 46‐residue crambin protein, fullerenes up to C₅₄₀ and nanotubes up to 14×(6,6), using the B3LYP functional and 6‐31G^** basis set with up to 8100 basis functions show that PS‐FLR‐TDDFT CPU time scales as N^2.05 with the number of basis functions. © 2016 Wiley Periodicals, Inc.     

Localized electron pair theory for the calculation of ground state energies of large molecules

The quantum mechanical energy is examined in which groups of one, two, three, and four localized electron pairs found within a molecule are separately computed. From these results, the interaction energies of the electron pairs taken one, two, three, and four at a time form the terms of a convergent molecular mechanics like expansion of the molecular ground state energy. This procedure can be used with any size consistent quantum mechanical method. The computational time for large molecules depends chiefly upon the order needed in the energy expansion to obtain sufficient convergence and not on the particular quantum mechanical method used. Preliminary results within the framework of a semiempirical CNDO/2 model Hamiltonian show at the Hartree–Fock and Møller–Plesset perturbation levels that relative energies converge to within a few tenths of a kcal/mol of the exact values at the four body level for molecules that have little delocalization. In strained ring and aromatic systems, convergence is however not nearly as rapid. Results can be improved somewhat by using larger interacting fragments containing two or more electron pairs over three or more atomic centers. © 1992 by John Wiley & Sons, Inc.     

Fock space multireference coupled cluster theory: noniterative inclusion of triples for excitation energies

In this paper we propose and numerically implement a specific scheme for calculating the excitation energies (EEs) within the Fock space multireference coupled cluster framework, which includes the contributions from noniterative triples cluster amplitudes. These contribute to the EEs at the third order. We present results for CH⁺ and N2, and study the effects of these noniterative triples on EEs. Received: 28 July 1997 / Accepted: 8 December 1997     

Efficient singular-value decomposition of the coupled-cluster triple excitation amplitudes

We demonstrate a novel technique to obtain singular-value decomposition (SVD) of the coupled-cluster triple excitations amplitudes, . The presented method is based on the Golub-Kahan bidiagonalization strategy and does not require to be stored. The computational cost of the method is comparable to several coupled cluster singles and doubles (CCSD) iterations. Moreover, the number of singular vectors to be found can be predetermined by the user and only those singular vectors which correspond to the largest singular values are obtained at convergence. We show how the subspace of the most important singular vectors obtained from an approximate triple amplitudes tensor can be used to solve equations of the CC3 method. The new method is tested for a set of small and medium-sized molecular systems in basis sets ranging in quality from double- to quintuple-zeta. It is found that to reach the chemical accuracy (≈1 kJ/mol) in the total CC3 energies as little as 5 − 15% of SVD vectors are required. This corresponds to the compression of the amplitudes by a factor of about 0.0001 − 0.005 . Significant savings are obtained also in calculation of interaction energies or rotational barriers, as well as in bond-breaking processes. © 2019 Wiley Periodicals, Inc.     

Fock space multireference coupled cluster method with full inclusion of connected triples for excitation energies

We report the initial Fock space multireference coupled cluster method with the full inclusion of single, double, and triple excitations (FS-CCSDT) for the (1,1) sector. We present pilot applications for calculating excitation energies for the N(2) molecule and the Ne atom. The performance of the current model, along with the FS-CCSD one, has been studied in comparison with the equation-of-motion coupled-cluster and the similarity transformed methods.     

Relationship between orbital energy gaps and excitation energies for long‐chain systems

The difference between the excitation energies and corresponding orbital energy gaps, the exciton binding energy, is investigated based on time‐dependent (TD) density functional theory (DFT) for long‐chain systems: all‐trans polyacetylenes and linear oligoacenes. The optimized geometries of these systems indicate that bond length alternations significantly depend on long‐range exchange interactions. In TDDFT formalism, the exciton binding energy comes from the two‐electron interactions between occupied and unoccupied orbitals through the Coulomb‐exchange‐correlation integral kernels. TDDFT calculations show that the exciton binding energy is significant when long‐range exchange interactions are involved. Spin‐flip (SF) TDDFT calculations are then carried out to clarify double‐excitation effects in these excitation energies. The calculated SF‐TDDFT results indicate that double‐excitation effects significantly contribute to the excitations of long‐chain systems. The discrepancies between the vertical ionization potential minus electron affinity (IP–EA) values and the HOMO–LUMO excitation energies are also evaluated for the infinitely long polyacetylene and oligoacene using the least‐square fits to estimate the exciton binding energy of infinitely long systems. It is found that long‐range exchange interactions are required to give the exciton binding energy of the infinitely long systems. Consequently, it is concluded that long‐range exchange interactions neglected in many DFT calculations play a crucial role in the exciton binding energies of long‐chain systems, while double‐excitation correlation effects are also significant to hold the energy balance of the excitations. © 2016 Wiley Periodicals, Inc.