Linear-scaling calculation of static and dynamic polarizabilities in Hartree-Fock and density functional theory for periodic systems

We present a linear-scaling method for analytically calculating static and dynamic polarizabilities with Hartree-Fock and density functional theory, using Gaussian orbitals and periodic boundary conditions. Our approach uses the direct space fast multipole method to evaluate the long-range Coulomb contributions. For exact exchange, we use efficient screening techniques developed for energy calculations. We then demonstrate the capabilities of our approach with benchmark calculations on one-, two-, and three-dimensional systems.     

Coupled-cluster calculation of dispersion contributions to interaction energies and polarizabilities

 The induced dipole dispersion-type contributions to the two-body and nonadditive three-body energies and electric dipole polarizabilities are studied for long-range interactions involving the He, Ne, Ar and Kr atoms and the H₂ and N₂ molecules. The coupled-cluster singles and doubles model and large basis sets are used. Comparison of the energy contributions with data derived from experiment shows in most cases the deviations to be less than 1%; therefore, it may be expected that the calculated polarizability increments are accurately determined and can be used to estimate the accuracy of approximate methods. Received: 20 March 2001 / Accepted: 5 April 2001 / Published online: 27 June 2001     

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.     

Accurate singlet and triplet excitation energies using the Localized Hartree-Fock Kohn-Sham potential

We assess the accuracy of the LHFX Time-Dependent Density-Functional Theory (TD-DFT) approach, which uses Kohn-Sham orbitals and eigenvalues from the Localized Hartree-Fock (LHF) method and the exchange-only adiabatic local density approximation kernel. We compute 172 singlet and triplet excitation energies of π → π^∗, n → π^∗, σ → π^∗ and Rydberg character, for organic molecules of different size. We find that the LHFX method, which is free from the Self-Interaction-Error (SIE) and from empirical parameters, outperforms the state-of-the-art hybrid TD-DFT approaches, and provides the same accuracy for all different classes of excitations. The SIE-free Kohn-Sham orbitals can be thus considered as starting point for TD-DFT developments.     

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.     

Calculation of molecular polarizabilities for large systems within the random phase approximation

We examine the static-field molecular polarizability from a sum over uncoupled Hartree—Fock states (SOS), the Tamm—Dancoff approximation (TDA), and the random phase approximation (RPA). An efficient algorithm for the inversion of the TDA or RPA matrix is outlined, which avoids matrix diagonalization and explicit construction of matrix elements over states, allowing for rapid calculation of molecular polarizabilities. The extension of the method is straightforward; third-order hyperpolarizability is developed as an example. Test cases are reported for molecules represented by an intermediate neglect of differential overlap (INDO) wavefunction.     

Screened potential in π-electron systems and its application to the estimation of excitation energies

A diagrammatic technique was developed for the estimation of the screened potential of -electron systems. The screened potential was expanded in terms of the polarization propagators which were constructed from either the singlet, , or triplet vertex part, . These vertex parts correspond to the singlet or triplet excitations, respectively, in the Random Phase Approximation (RPA) containing exchange diagrams. The excitation energies were calculated by using the screened potential in the framework of RPA with exchange. The excitation energies of several conjugated molecules with or without a hetero atom are in agreement 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.     

Reduction of the virtual space for coupled-cluster excitation energies of large molecules and embedded systems

We investigate how the reduction of the virtual space affects coupled-cluster excitation energies at the approximate singles and doubles coupled-cluster level (CC2). In this reduced-virtual-space (RVS) approach, all virtual orbitals above a certain energy threshold are omitted in the correlation calculation. The effects of the RVS approach are assessed by calculations on the two lowest excitation energies of 11 biochromophores using different sizes of the virtual space. Our set of biochromophores consists of common model systems for the chromophores of the photoactive yellow protein, the green fluorescent protein, and rhodopsin. The RVS calculations show that most of the high-lying virtual orbitals can be neglected without significantly affecting the accuracy of the obtained excitation energies. Omitting all virtual orbitals above 50 eV in the correlation calculation introduces errors in the excitation energies that are smaller than 0.1 eV. By using a RVS energy threshold of 50 eV, the CC2 calculations using triple-ζ basis sets (TZVP) on protonated Schiff base retinal are accelerated by a factor of 6. We demonstrate the applicability of the RVS approach by performing CC2/TZVP calculations on the lowest singlet excitation energy of a rhodopsin model consisting of 165 atoms using RVS thresholds between 20 eV and 120 eV. The calculations on the rhodopsin model show that the RVS errors determined in the gas-phase are a very good approximation to the RVS errors in the protein environment. The RVS approach thus renders purely quantum mechanical treatments of chromophores in protein environments feasible and offers an ab initio alternative to quantum mechanics/molecular mechanics separation schemes.     

On a new partitioning of the Hamiltonian in many-body calculation of pair-correlation energies in closed-shell systems

We introduce here a new partitioning of the Hamiltonian in calculating pair-correlation energies using many-body perturbation theory, by which we are able to eliminate the off-diagonal particle–hole (p–h) ladders exactly to all orders in the perturbation expansion. In this formulation, the particle states turn out to be different for each distinct pair of hole states in the correlation energy calculation. We have also included the contributions of the diagonal particle–particle (p–p) and hole–hole ladders exactly to all orders. The effect of the off-diagonal p–p ladders has been estimated for each pair by computing the third-, foruth- and fifth-order energies. For highly symmetric systems the present partitioning yields in general symmetry-broken orbitals. Here one may use an average kind of partitioning for all the partners of the degenerate sets, which restores the symmetry and at the same time ensures cancellation of the p–h ladders exactly at the lowest order and approximately at the higher orders. Results are presented for a selection of 6π-electron conjugated systems. The correlation energy for each pair is in excellent agreement with that obtained from a partial CI calculation involving all double excitations from this pair. The advantages of implementing the present scheme in larger systems has been discussed.     

On an application of the intermediate Hamiltonian method for molecular systems

An application of the intermediate Hamiltonian method is reported in estimation of the lower bounds to the potential energy curve of the hydrogen molecule ion. An improvement of the method and its limitation are also discussed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 101–107, 1999     

Linear-scaling localized-density-matrix method for the ground and excited states of one-dimensional molecular systems

A linear-scaling localized-density-matrix (LDM) method is developed to evaluate the ground-state reduced single-electron density matrices of one-dimensional molecular systems. The new method may be combined with the existing linear-scaling LDM method for the excited states (Yokojima and Chen, Chem. Phys. Lett. 292 (1998) 379), and thus leads to a linear-scaling calculation method for the properties of both the ground and excited states. The combined method is applied to the polyacetylene oligomers and the linear-scaling of the total computational time is clearly demonstrated.     

New approach to the semiempirical calculation of atomic charges for polypeptides and large molecular systems

Starting from the bond polarization theory (BPT), a new semiempirical method for the calculation of net atomic charges is developed. The bond polarization theory establishes a linear dependence of atomic charges from the bond polarization energy. This energy is calculated from the hybrid orbitals forming a bond and the point charges within the neighborhood. Empirical parameters are introduced for the polarity of an unpolarized bond and for the change of the atomic charge with σ- and π-bond polarization. Because these parameters are linear, they can be calibrated directly using net atomic charges from ab initio calculations. This procedure was performed using the charges from STO3G calculations on a set of 18 amino acids. Using the two parameters for CH, OH, σ-CO, and NH bonds and the three parameters for CC, CO, and CN bonds, the 350 ab initio charges can be reproduced with high accuracy by solving sets of linear equations for the charges. The calculation of charges for large molecular systems including all inter- and intramolecular mutual polarizations requires only a few seconds (up to 100 atoms) or minutes (700 atoms) on a PC. This procedure is well suited for the application in molecular mechanics or molecular dynamics programs to overcome the limitations of most force fields used up to now. One of the weakest points in these programs is the use of fixed or topological charges to define the electrostatic potential. As an application of the new method, we calculated the interaction energy of an ion with valinomycin. This ring molecule forms octahedral oxygen cages around ions like potassium and acts thereby as selective ion carrier. To accomplish this function, valinomycin has to strip off the hydratization spheres of the ions, and therefore its preference for certain types of ions could be deduced from the interaction energies. © 1994 by John Wiley & Sons, Inc.     

Calculations of molecular polarizabilities using the Valence Effective Hamiltonian (VEH) method

In this paper, we investigate the applicability of the Valence Effective Hamiltonian (VEH) method to calculations of theoretical molecular linear polarizabilities by the Sum-Over-States (SOS) methodology. Test calculations are presented on the polyene series. They indicate that the method gives qualitatively the same trends asab initio small basis sets calculations.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Relativistic effective valence shell Hamiltonian method: excitation and ionization energies of heavy metal atoms

The relativistic effective valence shell Hamiltonian H(v) method (through second order) is applied to the computation of the low lying excited and ion states of closed shell heavy metal atoms/ions. The resulting excitation and ionization energies are in favorable agreement with experimental data and with other theoretical calculations. The nuclear magnetic hyperfine constants A and lifetimes tau of excited states are evaluated and they are also in accord with experiment. Some of the calculated quantities have not previously been computed.     

The calculation of ground and excited state molecular polarizabilities: A simple perturbation treatment

Second order perturbation theory has been coupled with the CNDO/S CI method of Del Bene and Jaffé to calculate the ground and excited state polarizabilities of various molecules. It is found that this treatment produces reasonably good polarizabilities with great computational ease.     

Perturbation theories and wave functions for calculation of electronic polarizabilities application to DNA bases

Three different forms of perturbation theories, variational perturbation, finite perturbation and second-order, are evaluated regarding their value for calculation of electronic polarizabilities of small and intermediate size molecules. It is concluded that with the practical constraint of a small basis set the variational perturbation method is the most promising alternative for calculation of polarizabilities. For several small molecules, our calculated polarizabilities indicate that both IEHT and ab initio wave functions give values in close agreement with each other. Variational perturbation calculations of polarizabilities with IEHT wave functions also include the DNA bases.     

Theoretical calculation of vibrational polarizabilities. An application to the study of conformational and solvent effects

The vibrational polarizability tensor is linked to the force constant and dipole moment derivative matrices. The analytical computation of these quantities is possible, in particular by using semi-empirical quantum-chemical methods. The electrostatic influence of liquid surroundings can be analyzed with the help of a cavity model. Applied to a series of molecules, this method of calculation emphasizes the role of non-diagonal elements of the force constant matrix and the variation of the mean vibrational polarizability under the influence of a solvent, especially in the case of non-rigid molecules and hydrogen-bonded complexes.     

An augmented effective core potential basis set for the calculation of molecular polarizabilities

Calculations of molecular polarizabilities require basis sets capable of accurately describing the responses of the electrons to an external perturbation. Unfortunately, basis sets that yield suitable quantitative results have traditionally been all-electron sets with large numbers of primitives, making their use computationally intractable even for moderately sized systems. We present a systematic augmentation of the effective core potential basis set of Stevens et al. [J Chem Phys 81, 12 (1984), Can J Chem 70, 612 (1992)] for 39 main group elements based on the procedure used to construct diffuse and polarization functions in the well-known Sadlej basis sets [Collec Czech Chem Comm 53, 1995 (1988)]. Representative calculations have been performed and we have shown that results to within 1% of all-electron calculations using the Sadlej basis set can be obtained for <1-35% of the computational cost using this new basis set.     

Ab initio calculation of the vertical excitation energies of small helium cluster ions

Quantum chemical ab initio calculations have been performed for the vertical excitation energies and oscillator strengths of all low-lying electronically excited states of small helium cluster ions, He _n ⁺ ,n=2, ..., 7. The geometrical structures of the ions were fixed at the equilibrium geometries of the respective ground states, for He ₄ ⁺ and He ₅ ⁺ also one alternative structure was considered. The low-lying excited states can be classified into two categories: the electronic transition can occur either within the central He ₂ ⁺ or He ₃ ⁺ unit or from the peripheral weakly bound He atoms to this unit. The latter transitions are very weak (f≈0.001), closely spaced, with vertical excitation energies of about 5.7 eV. The He ₂ ⁺ and He ₃ ⁺ units have strong transitions at 9.93 and 5.55 eV, respectively; these transitions are only slightly blue-shifted if He ₂ ⁺ or He ₃ ⁺ are placed as “chromophores” into the centre of a larger He _n ⁺ cluster. The large difference in the vertical excitation energy of the strong transition should enable an experimental decision of the question whether the cluster ions have He ₂ ⁺ or He ₃ ⁺ cores.