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.     

SCF and localized orbitals in ethylene: MBPT/CC results and comparison with one-million configuration Cl

Many-body perturbation theory (MBPT) and coupled-cluster (CC) calculations are performed on the ethylene molecule employing canonical SCF and simple bond-orbital localized orbitals (LO). Full fourth-order MBPT [i.e. SDTQ MBPT(4)], CC doubles (CCD) and CC singles and doubles (CCSD) energies are compared with the over one-million configuration ‘bench-mark” Cl calculation of Saxe et al. Though the SCF and LO reference determinant energies differ by 0.29706 hartree, the CCSD energy difference is only 1.7 mhartrees (mh). Our most extensive SCF orbital calculation, CCSD plus fourth-order triples, is found to be lower in energy than the CI result by 5.3 mh.     

Model study of the impact of orbital choice on the accuracy of coupled-cluster energies. I. Single-reference-state formulation

The impact of the choice of molecular orbital sets on the results of single-reference-state coupled-cluster (CC) methods was studied for the H4 model. This model offers a straightforward way of taking into account all possible symmetry-adapted orbitals. Moreover, the degree of quasi-degeneracy of its ground state can be varied over a wide range by changing its geometry. The CCD, CCSD, and CCSDT approaches are considered. Surfaces representing the dependence of the energy on the parameters defining the orbitals are obtained. It is documented that for every method there exist alternative orbital sets which allow one to obtain more accurate energies than the standard (HF, BO, and NO) ones. However, for many of the former orbital sets, one obtains relatively large one-body amplitudes or one may encounter problems with solving the CC equations by conventional methods. An interesting variety of orbitals which might be useful for studies of quasi-degenerate states by the CCD method was found. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 205–219, 1998     

Cost reduction of high-order coupled-cluster methods via active-space and orbital transformation techniques

We discuss several techniques which have the potential to decrease the computational expenses of high-order coupled-cluster (CC) methods with a reasonable loss in accuracy. In particular, the CC singles, doubles, and triples (CCSDT) as well as the CC singles, doubles, triples, and perturbative quadruples [CCSDT(Q)] methods are considered, which are frequently used in high-precision model chemistries for the calculation of iterative triples and quadruples corrections. First, we study the possibilities for using active spaces to decrease the computational costs. In this case, an active space is defined and some indices of cluster amplitudes are restricted to be in the space. Second, the application of transformed virtual orbitals is investigated. In this framework, to reduce the computation time the dimension of the properly transformed virtual one-particle space is truncated. We have found that the orbital transformation techniques outperform the active-space approaches. Using the transformation techniques, the computational time can be reduced in average by an order of magnitude without significant loss in accuracy. It is demonstrated that high-order CC calculations are possible for considerably larger systems than before using the implemented techniques.     

Analytical evaluation of first-order electrical properties based on the spin-free Dirac-Coulomb Hamiltonian

We report an analytical scheme for the calculation of first-order electrical properties using the spin-free Dirac-Coulomb (SFDC) Hamiltonian, thereby exploiting the well-developed density-matrix formulations in nonrelativistic coupled-cluster (CC) derivative theory. Orbital relaxation effects are fully accounted for by including the relaxation of the correlated orbitals with respect to orbitals of all types, viz., frozen-core, occupied, virtual, and negative energy state orbitals. To demonstrate the applicability of the presented scheme, we report benchmark calculations for first-order electrical properties of the hydrogen halides, HX with X = F, Cl, Br, I, At, and a first application to the iodo(fluoro)methanes, CH(n)F(3 - n)I, n = 0-3. The results obtained from the SFDC calculations are compared to those from nonrelativistic calculations, those obtained via leading-order direct perturbation theory as well as those from full Dirac-Coulomb calculations. It is shown that the full inclusion of spin-free (SF) relativistic effects is necessary to obtain accurate first-order electrical properties in the presence of fifth-row elements. The SFDC scheme is also recommended for applications to systems containing lighter elements because it introduces no extra cost in the rate-determining steps of a CC calculation in comparison to the nonrelativistic case. On the other hand, spin-orbit contributions are generally small for first-order electrical properties of closed-shell molecules and may be handled efficiently by means of perturbation theory.     

pCCSD: parameterized coupled-cluster theory with single and double excitations

The primary characteristics of single reference coupled-cluster (CC) theory are size-extensivity and size-consistency, invariance under orbital rotations of the occupied or virtual space, the exactness of CC theory for N electron systems when the cluster operator is truncated to N-tuple excitations, and the relative insensitivity of CC theory to the choice of the reference determinant. In this work, we propose a continuous class of methods which display the desirable features of the coupled-cluster approach with single and double excitations (CCSD). These methods are closely related to the CCSD method itself and are inspired by the coupled electron pair approximation (CEPA). It is demonstrated that one can systematically improve upon CCSD and obtain geometries, harmonic vibrational frequencies, and total energies from a parameterized version of CCSD or pCCSD(α,β) by selecting a specific member from this continuous family of approaches. In particular, one finds that one such approach, the pCCSD(-1,1) method, is a significant improvement over CCSD for the calculation of equilibrium structures and harmonic frequencies. Moreover, this method behaves surprisingly well in the calculation of potential energy surfaces for single bond dissociation. It appears that this methodology has significant promise for chemical applications and may be particularly useful in applications to larger molecules within the framework of a high accuracy local correlation approach.     

Toward more efficient CCSD(T) calculations of intermolecular interactions in model hydrogen-bonded and stacked dimers

Interaction energies of the model H-bonded complexes, the formamide and formamidine dimers, as well as the stacked formaldehyde and ethylene dimers are calculated by the coupled cluster CCSD(T) method. These systems serve as a model for H-bonded and stacking interactions, typical in molecules participating in biological systems. We use the optimized virtual orbital space (OVOS) technique, by which the dimension of the space of virtual orbitals in coupled cluster CCSD(T) calculations can be significantly reduced. We demonstrate that when the space of virtual orbitals is reduced to 50% of the full space, which means reducing computational demands by 1 order of magnitude, the interaction energies for both H-bonded and stacked dimers are affected by no more than 0.1 kcal/mol. This error is much smaller than the error when interaction energies are calculated using limited basis sets.     

Comparison of explicitly correlated local coupled-cluster methods with various choices of virtual orbitals

Explicitly correlated local coupled-cluster (LCCSD-F12) methods with pair natural orbitals (PNOs), orbital specific virtual orbitals (OSVs), and projected atomic orbitals (PAOs) are compared. In all cases pair-specific virtual subspaces (domains) are used, and the convergence of the correlation energy as a function of the domain sizes is studied. Furthermore, the performance of the methods for reaction energies of 52 reactions involving 58 small and medium sized molecules is investigated. It is demonstrated that for all choices of virtual orbitals much smaller domains are needed in the explicitly correlated methods than without the explicitly correlated terms, since the latter correct a large part of the domain error, as found previously. For PNO-LCCSD-F12 with VTZ-F12 basis sets on the average only 20 PNOs per pair are needed to obtain reaction energies with a root mean square deviation of less than 1 kJ mol(-1) from complete basis set estimates. With OSVs or PAOs at least 4 times larger domains are needed for the same accuracy. A new hybrid method that combines the advantages of the OSV and PNO methods is proposed and tested. While in the current work the different local methods are only simulated using a conventional CCSD program, the implications for low-order scaling local implementations of the various methods are discussed.     

Quasiparticle virtual orbitals in electron propagator calculations

The computational limits of accurate electron propagator methods for the calculation of electron binding energies of large molecules are usually determined by the rank of the virtual orbital space. Electron density difference matrices that correspond to these transition energies in the second-order quasiparticle approximation may be used to obtain a virtual orbital space of reduced rank that introduces only minor deviations with respect to the results produced with the full, original set of virtual orbitals. Numerical tests show the superior accuracy and efficiency of this approach compared to the usual practice of omission of virtual orbitals with the highest energies.     

The optimization of molecular orbitals for coupled cluster wavefunctions

The purpose of this work is to make the coupled cluster (CC) energy stationary with respect to molecular-orbital (MO) variations in the reference configuration. To achieve this, we have used the Z vector, the solution of a set of perturbation-independent CPHF-like equations, to rotate the MOs. A new energy and gradient calculation is carried out with these non-SCF orbitals to obtain a new Z vector. The process is repeated until the orbitals are optimized (Z = 0), i.e. the contribution to the analytic CC gradient coming from orbital relaxation (CPHF) is zero. At the CCD level the orbitals thus obtained are approximate Brueckner orbitals. At the CCSD level, convergence problems were found in the iterative procedure to optimize the orbitals. Results obtained for several molecules show that CCD wavefunctions constructed from these optimized orbitals are of CCSD quality. We conclude that the presence of exp(t₁) in the CCSD model accounts for most relaxation effects and there is not much to gain by orbital optimization in CCSD wavefunctions.     

Explaining the “large numbers” via a unified (classical) theory of strong and gravitational interactions

An advantage of modified virtual orbitals of Hartree–Fock method is discussed in the calculation of the second-order perturbation energy. All the modified virtual orbitals can be fitted for the intermediate virtual states in the perturbation expansion, only if the molecular orbitals are expanded in terms of infinite basis functions and the set of molecular orbitals is infinite and complete. If the molecular orbitals are expanded in terms of finite basis functions, only the modified virtual orbitals with lower energies are appropriate to describe the intermediate virtual states, but the modified virtual orbitals with higher energies become inadequate. To explain the usefulness of the modified virtual orbitals, the calculation by the modified Hartree–Fock method without CI are compared with the calculation by the traditional Hartree–Fock method with complete CI .     

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.     

Simple test of the utility of modified Hartree-Fock orbitals: The He atom

A method proposed recently for the generation of modified Hartree-Fock virtual orbitals has been tested by performing a simple configuration-interaction calculation for the ground state of the He atom. A very compact truncated basis set was constructed by using an a priori criterion based on the probability density within the region of the occupied orbital. The best results were obtained by making the virtual orbital energies more positive.     

Extended floating spherical Gaussian basis sets for molecules. Alternative correlating orbitals for molecular energy calculations

A procedure previously described for representing large basis SCF results in terms of a smaller floating spherical Gaussian orbital (FSGO) basis set is generalized to apply to the virtual orbitals from the SCF calculation. This provides a method for systematically reducing the dimensions of the virtual space or replacing the virtual orbitals with a simpler, compact basis set. The method is illustrated by application to Lill.     

Effect of hyperconjugation on ionization energies of hydroxyalkyl radicals

On the basis of electronic structure calculations and molecular orbital analysis, we offer a physical explanation of the observed large decrease (0.9 eV) in ionization energies (IE) in going from hydroxymethyl to hydroxyethyl radical. The effect is attributed to hyperconjugative interactions between the sigma CH orbitals of the methyl group in hydroxyethyl, the singly occupied p orbital of carbon, and the lone pair p orbital of oxygen. Analyses of vertical and adiabatic IEs and hyperconjugation energies computed by the natural bond orbital (NBO) procedure reveal that the decrease is due to the destabilization of the singly occupied molecular orbital in hydroxyethyl radical as well as structural relaxation of the cation maximizing the hyperconjugative interactions. The stabilization is achieved due to the contraction of the CO and CC bonds, whereas large changes in torsional angles bear little effect on the total hyperconjugation energies and, consequently, IEs.     

An orbital-invariant and strictly size extensive post-Hartree-Fock correlation functional

A strictly size extensive post-Hartree-Fock correlation functional being invariant with respect to orbital transformations within the occupied and virtual subspaces is presented. While avoiding the necessity to solve additional Z vector equations for the calculation of properties and energy gradients, this functional reproduces almost exactly the results of coupled-cluster singles doubles (CCSD) calculations. In particular, it is demonstrated that the method is rigorous in the sense that it can be systematically improved by the perturbative inclusion of triple excitations in the same way as CCSD. As to the computational cost, the presented approach is somewhat more expensive than the CCSD if the energy is variationally optimized with respect to both the orbitals and the excitation amplitudes. Replacement of orbital optimization by the Brueckner condition reduces the computational cost by a factor of two, thus making the method less expensive than CCSD.     

Coupled-cluster with active space selected higher amplitudes: performance of seminatural orbitals for ground and excited state calculations

The active space approach for coupled-cluster models is generalized using the general active space concept and implemented in a string-based general coupled-cluster code. Particular attention is devoted to the choice of orbitals on which the subspace division is based. Seminatural orbitals are proposed for that purpose. These orbitals are obtained by diagonalizing only the hole-hole and particle-particle block of the one-electron density of a lower-order method. The seminatural orbitals are shown to be a good replacement for complete active space self-consistent field orbitals and avoid the ambiguities with respect to the reference determinant introduced by the latter orbitals. The seminatural orbitals also perform well in excited state calculations, including excited states with strong double excitation contributions, which usually are difficult to describe with standard coupled-cluster methods. A set of vertical excitation energies is obtained and benchmarked against full configuration interaction calculations, and alternative hierarchies of active space coupled-cluster models are proposed. As a simple application the spectroscopic constants of the C(2) B (1)Delta(g) and B(') (1)Sigma(g) (+) states are calculated using active space coupled-cluster methods and basis sets up to quadruple-zeta quality in connection with extrapolation and additivity schemes.     

Natural linear-scaled coupled-cluster theory with local transferable triple excitations: applications to peptides

The natural linear-scaled coupled-cluster (NLSCC) method ( Flocke, N.; Bartlett, R. J. J. Chem. Phys. 2004, 121, 10935 ) is extended to include approximate triple excitations via a coupled-cluster with single, double, and triple excitation method (CCSDT-3). The triples contribution can potentially be embedded in a larger singles and doubles region. NLSCC exploits the extensivity of the CC wave function to represent it in terms of transferable natural localized molecular orbitals (NLMOs) or functional groups thereof that are obtained from small quantum mechanical (QM) regions. Both occupied and virtual NLMOs are local because they derive from the single-particle density matrix. Noncanonical triples amplitudes are avoided by applying the unitary localization matrix to the canonical CC wave function for a QM region. A generalized NLMO code interfaced to the ACES II quantum chemistry software package provides NLMOs for the relevant number of atoms in a given functional group. Applications include linear polyglycine and the pentapeptide met-enkephalin, which was chosen as a more realistic three-dimensional system with nontrivial side chains. The results show that the triples contributions are quite large for aromatic bonds suggesting an interesting active space method for triples in which different bonds require different excitation levels. The NLSCC approach recovers a very large percentage (>99%) of the CCSD or CCSDT-3 correlation energy.