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.     

Theoretical study of the electronic spectroscopy of CO adsorbed on Pt(111)

The excited states of CO adsorbed on the Pt(111) surface are studied using a time-dependent density functional theory formalism. To reduce the computational cost, electronic excitations are computed within a reduced single excitation space. Using cluster models of the surface, excitation energies are computed for CO in the on-top, threefold, and bridge binding sites. On adsorption, there is a lowering of the 5sigma orbital energy. This leads to a large blueshift in the 5sigma- -> pi(CO*) excitation energy for all adsorption sites. The 1pi and 4sigma orbital energies are lowered to a lesser extent, and smaller shifts in the corresponding excitation energies are predicted. For the larger clusters, pi* excitations at lower energies are observed. These transitions correspond to excitations to virtual orbitals of pi* character which lie below the pi* orbitals of gas phase CO. These orbitals are associated predominantly with the metal atoms of the cluster. The excitation energies are also found to be sensitive to changes in the adsorption geometry. The electronic spectrum of CO on Pt(111) is simulated and the assignment of the bands observed in experimental electron energy loss spectroscopy discussed.     

Local explicitly correlated second-order Møller-Plesset perturbation theory with pair natural orbitals

We explore using a pair natural orbital analysis of approximate first-order pair functions as means to truncate the space of both virtual and complementary auxiliary orbitals in the context of explicitly correlated F12 methods using localised occupied orbitals. We demonstrate that this offers an attractive procedure and that only 10-40 virtual orbitals per significant pair are required to obtain second-order valence correlation energies to within 1-2% of the basis set limit. Moreover, for this level of virtual truncation, only 10-40 complementary auxiliary orbitals per pair are required for an accurate resolution of the identity in the computation of the three- and four-electron integrals that arise in explicitly correlated methods.     

Frozen natural orbital coupled-cluster theory: forces and application to decomposition of nitroethane

The frozen natural orbital (FNO) coupled-cluster method increases the speed of coupled-cluster (CC) calculations by an order of magnitude with no consequential error along a potential energy surface. This method allows the virtual space of a correlated calculation to be reduced by about half, significantly reducing the time spent performing the coupled-cluster (CC) calculation. This paper reports the derivation and implementation of analytical gradients for FNO-CC, including all orbital relaxation for both noncanonical and semicanonical perturbed orbitals. These derivatives introduce several new orbital relaxation contributions to the CC density matrices. FNO-CCSD(T) and FNO-LambdaCCSD(T) are applied to a test set of equilibrium structures, verifying that these methods are capable of reproducing geometries and vibrational frequencies accurately, as well as energies. Several decomposition pathways of nitroethane are investigated using CCSD(T) and LambdaCCSD(T) with 60% of the FNO virtual orbitals in a cc-pVTZ basis, and find differences on the order of 5 kcalmol with reordering of the transition state energies when compared to B3LYP 6-311 + G(3df, 2p).     

Local orbitals by minimizing powers of the orbital variance

It is demonstrated that a set of local orthonormal Hartree-Fock (HF) molecular orbitals can be obtained for both the occupied and virtual orbital spaces by minimizing powers of the orbital variance using the trust-region algorithm. For a power exponent equal to one, the Boys localization function is obtained. For increasing power exponents, the penalty for delocalized orbitals is increased and smaller maximum orbital spreads are encountered. Calculations on superbenzene, C(60), and a fragment of the titin protein show that for a power exponent equal to one, delocalized outlier orbitals may be encountered. These disappear when the exponent is larger than one. For a small penalty, the occupied orbitals are more local than the virtual ones. When the penalty is increased, the locality of the occupied and virtual orbitals becomes similar. In fact, when increasing the cardinal number for Dunning's correlation consistent basis sets, it is seen that for larger penalties, the virtual orbitals become more local than the occupied ones. We also show that the local virtual HF orbitals are significantly more local than the redundant projected atomic orbitals, which often have been used to span the virtual orbital space in local correlated wave function calculations. Our local molecular orbitals thus appear to be a good candidate for local correlation methods.     

Coexistence of 1,3-butadiene conformers in ionization energies and Dyson orbitals

The minimum-energy structures on the torsional potential-energy surface of 1,3-butadiene have been studied quantum mechanically using a range of models including ab initio Hartree-Fock and second-order M?ller-Plesset theories, outer valence Green's function, and density-functional theory with a hybrid functional and statistical average orbital potential model in order to understand the binding-energy (ionization energy) spectra and orbital cross sections observed by experiments. The unique full geometry optimization process locates the s-trans-1,3-butadiene as the global minimum structure and the s-gauche-1,3-butadiene as the local minimum structure. The latter possesses the dihedral angle of the central carbon bond of 32.81 degrees in agreement with the range of 30 degrees-41 degrees obtained by other theoretical models. Ionization energies in the outer valence space of the conformer pair have been obtained using Hartree-Fock, outer valence Green's function, and density-functional (statistical average orbital potentials) models, respectively. The Hartree-Fock results indicate that electron correlation (and orbital relaxation) effects become more significant towards the inner shell. The spectroscopic pole strengths calculated in the Green's function model are in the range of 0.85-0.91, suggesting that the independent particle picture is a good approximation in the present study. The binding energies from the density-functional (statisticaly averaged orbital potential) model are in good agreement with photoelectron spectroscopy, and the simulated Dyson orbitals in momentum space approximated by the density-functional orbitals using plane-wave impulse approximation agree well with those from experimental electron momentum spectroscopy. The coexistence of the conformer pair under the experimental conditions is supported by the approximated experimental binding-energy spectra due to the split conformer orbital energies, as well as the orbital momentum distributions of the mixed conformer pair observed in the orbital cross sections of electron momentum spectroscopy.     

An extended-valence MC SCF procedure: determination of the dissociation energies of C2, N2, O2, and F2

A series of MC SCF calculations have been carried out on C₂, N₂, O₂, and F₂ with the goal of obtaining compact wavefunctions which recover a significant fraction of the electron correlation effects important for bond dissociation. The active orbital space is varied in size, with the largest spaces including the molecular orbitals derived from 2s, 2p, 3s, 3p and 4p atomic orbitals. Several basis sets ranging in size from 5s3p to 5s4p2d1f are investigated to determine the flexibility in the basis set needed with various choices of the active orbital space. The best extended-valence MC SCF (EVMC) dissociation energies are 0.2–0.5 eV less than the experimental values, indicating that further enlargement of the active orbital space is necessary to achieve 0.1 eV accuracy in the computed dissociation energies. The EVMC calculations reveal that, for the calculation of the dissociation energies, inclusion of non-valence orbitals is much more important for O₂ and F₂ than for C₂ and N₂. The EVMC results are compared with the predictions of full fourth-order perturbation theory, coupled cluster theory, and with the best available CI calculations.     

Core molecular orbital contribution to N2O isomerization as studied using theoretical electron momentum spectroscopy

Core molecular orbital contribution to the electronic structure of N2O isomers has been studied using quantum mechanical density functional theory combined with a plane wave impulse approximation method. Momentum distributions of wave functions for inner shell molecular orbitals of the linear NNO, cyclic and linear NON isomers of N2O are calculated through the (e, 2e) differential cross sections in momentum space. This is possible because this momentum distribution is directly proportional to the modulus squared of the momentum space wave function for the molecular orbital in question. While the momentum distributions of the NNO and cyclic N2O isomers demonstrate strong atomic orbital characteristics in their core space, the outer core molecular orbitals of the linear NON isomer exhibit configuration interactions between them and the valence molecular orbitals. It is suggested that the frozen core approximation breaks down in the prediction of the electronic structure of such an isomer. Core molecular orbital contributions to the electronic structure can alter the order of total energies of the isomers and lead to incorrect conclusions of the stability among the isomers. As a result, full electron calculations should be employed in the study of N2O isomerization.     

Away from generalized gradient approximation: orbital-dependent exchange-correlation functionals

The local-density approximation of density functional theory (DFT) is remarkably accurate, for instance, for geometries and frequencies, and the generalized gradient approximations have also made bond energies quite reliable. Sometimes, however, one meets with failure in individual cases. One of the possible routes towards better functionals would be the incorporation of orbital dependence (which is an implicit density dependency) in the functionals. We discuss this approach both for energies and for response properties. One possibility is the use of the Hartree-Fock-type exchange energy expression as orbital-dependent functional. We will argue that in spite of the increasing popularity of this approach, it does not offer any advantage over Hartree-Fock for energies. We will advocate not to apply the separation of exchange and correlation, which is so ingrained in quantum chemistry, but to model both simultaneously. For response properties the energies and shapes of the virtual orbitals are crucial. We will discuss the benefits that Kohn-Sham potentials can offer which are derived from either an orbital-dependent energy functional, including the exact-exchange functional, or which can be obtained directly as orbital-dependent functional. We highlight the similarity of the Hartree-Fock and Kohn-Sham occupied orbitals and orbital energies, and the essentially different meanings the virtual orbitals and orbital energies have in these two models. We will show that these differences are beneficial for DFT in the case of localized excitations (in a small molecule or in a fragment), but are detrimental for charge-transfer excitations. Again, orbital dependency, in this case in the exchange-correlation kernel, offers a solution.     

Large-scale RPA calculations of chiroptical properties of organic molecules: Program RPAC

The computational considerations involved in calculating ordinary and rotatory intensities and electronic excitation energies in the random phase approximation (RPA ) are examined. We employ a localized orbital formulation in order to analyze the results in terms of local and charge-transfer excitations. Occupied orbitals are localized by the Foster–Boys procedure. The virtual space is transformed into a localized “valence” set that maximizes dipole strengths with the occupied counterparts, and a delocalized remainder. The two-electron integral transformation is performed with an efficient algorithm, based on Diercksen's, that generates only the particle–hole-type integrals required in the RPA . The lowest solutions of the RPA equations are obtained iteratively using a modification of the Davidson-Liu simultaneous vector expansion method. This allows the inclusion of the entire set of particle–hole states supported by a basis set of up to 102 orbitals. Calculations at this level give better excitation energies and intensities than SDCI methods, at substantial savings in computational effort. Comparative timings, computed results and analysis in terms of localized orbitals are given for planar and distorted ethylene using extended atomic orbital bases including diffuse functions. The results for planar ethylene are in excellent agreement with experiment.     

Tensor factorizations of local second-order Møller-Plesset theory

Efficient electronic structure methods can be built around efficient tensor representations of the wavefunction. Here we first describe a general view of tensor factorization for the compact representation of electronic wavefunctions. Next, we use this language to construct a low-complexity representation of the doubles amplitudes in local second-order M?ller-Plesset perturbation theory. We introduce two approximations--the direct orbital-specific virtual approximation and the full orbital-specific virtual approximation. In these approximations, each occupied orbital is associated with a small set of correlating virtual orbitals. Conceptually, the representation lies between the projected atomic orbital representation in Pulay-Saeb? local correlation theories and pair natural orbital correlation theories. We have tested the orbital-specific virtual approximations on a variety of systems and properties including total energies, reaction energies, and potential energy curves. Compared to the Pulay-Saeb? ansatz, we find that these approximations exhibit favorable accuracy and computational times while yielding smooth potential energy curves.     

Fast localized orthonormal virtual orbitals which depend smoothly on nuclear coordinates

We present here an algorithm for computing stable, well-defined localized orthonormal virtual orbitals which depend smoothly on nuclear coordinates. The algorithm is very fast, limited only by diagonalization of two matrices with dimension the size of the number of virtual orbitals. Furthermore, we require no more than quadratic (in the number of electrons) storage. The basic premise behind our algorithm is that one can decompose any given atomic-orbital (AO) vector space as a minimal basis space (which includes the occupied and valence virtual spaces) and a hard-virtual (HV) space (which includes everything else). The valence virtual space localizes easily with standard methods, while the hard-virtual space is constructed to be atom centered and automatically local. The orbitals presented here may be computed almost as quickly as projecting the AO basis onto the virtual space and are almost as local (according to orbital variance), while our orbitals are orthonormal (rather than redundant and nonorthogonal). We expect this algorithm to find use in local-correlation methods.     

Electronic charge density and mean kinetic energy of lithium orbitals in LiH,LiH+, Li2, Li2+, LiH2+, and Li2H+

The electron density near the lithium nucleus in the species LiH, LiH⁺, Li₂, Li₂⁺, LiH₂⁺, and Li₂H⁺ was analyzed by transforming the SCF molecular orbitals into a sum of atomic contribnutions, for both core and valence orbitals. These “hybrid-atomic” orbitals were used to compare: electron densities, orbital polarizations, and orbital mean kinetic energies with the corresponding lithium atom quantities. Core-orbital electron densities at the lithium nucleus were observed to increase by up to 0.5% relative to the lithium atom 1s orbital. Lithium cores also exhibited polarization but, surprisingly, in the direction away from the internuclear region. Similar dramatic changes were seen in the electron densities of the valence orbitals of lithium: The electron density at the nucleus for these orbitals increased two-fold for homonuclear species and twenty-fold for heteronuclear triatomic species relative to the electron density at the nucleus in lithium atom. The polarization of the valence orbital electronic charge, in the vicinity of the lithium nucleus, was also away from the internuclear region. The mean “hybrid-atomic” orbital kinetic energies associated with the lithium atom in the molecules also showed changes relative to the free lithium atom. Such changes, accompanying bond formation, were relatively small for the lithium core orbitals (within 0.2% of the value for lithium atom). The orbital kinetic energies for the lithium valence electrons, however, increased considerably relative to the lithium atom: By a factor of about 2 in homonuclear diatomics, by a factor of 7 in heteronuclear diatomics, and by a factor of 11 in the triatomic species. In summary, the total electronic density (core plus valence) at the lithium nucleus remained remarkably constant for all of the species studied, regardless of the effective charge on lithium. Thus, the drastic changes noted in the individual lithium orbitals occurred in a cooperative fashion so as to preserve a constant total electron density in the vicinity of the lithium nucleus. In all cases, bond formation was accompanied by an increase in the orbital kinetic energy of the lithium valence orbital. We suggest that these two observations represent important and significant features of chemical bonding which have not previously been emphasized.     

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.     

Asymptotic extrapolation scheme for large-scale calculations with hybrid coupled cluster and molecular dynamics simulations

In this paper we discuss a simple extrapolation scheme based on the asymptotic behavior of the electronic energies considered as functions of cutoff factor for orbital energies corresponding to virtual orbitals. The performance of this approach is illustrated in the context of large-scale dynamic simulations for excitation energies of the cytosine molecule in its native DNA environment. We demonstrate that the extrapolation errors are significantly smaller than the excitation-energy fluctuations, due to the fluctuating environment.     

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.     

Independent particle theory with electron correlation

We formulate an effective independent particle model where the effective Hamiltonian is composed of the Fock operator and a correlation potential. Within the model the kinetic energy and the exchange energy can be expressed exactly leaving the correlation energy functional as the remaining unknown. Our efforts concentrate on finding a correlation potential such that exact ionization potentials and electron affinities can be reproduced as orbital energies. The equation-of-motion coupled-cluster approach enables us to define an effective Hamiltonian from which a correlation potential can be extracted. We also make the connection to electron propagator theory. The disadvantage of the latter is the inherit energy dependence of the potential resulting in a different Hamiltonian for each orbital. Alternatively, the Fock space coupled-cluster approach employs an effective Hamiltonian which is energy independent and universal for all orbitals. A correlation potential is extracted which yields the exact ionization potentials and electron affinities and a set of associated molecular orbitals. We also describe the close relationship to Brueckner theory.     

Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals

A method is presented for expressing the occupied self-consistent-field (SCF) orbitals of a molecule exactly in terms of chemically deformed atomic minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The molecular orbitals referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact atomic SCF orbitals, and the formulation of the deformed "quasiatomic minimal-basis-sets" is independent of the calculational atomic orbital basis used. The resulting resolution of molecular orbitals in terms of quasiatomic minimal basis set orbitals is therefore intrinsic to the exact molecular wave functions. The deformations are analyzed in terms of interatomic contributions. The Mulliken population analysis is formulated in terms of the quasiatomic minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quantitative ab initio formulation of the qualitative model of virtual valence orbitals, which are useful for calculating electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham density functional theory orbitals and is easily generalized to valence MCSCF orbitals.     

Full spectral decomposition of ring currents

Within the ipsocentric method for calculation of molecular magnetic response, projection of perturbed orbitals onto the virtual orbital space allows partition of induced current density into contributions from individual virtual excitations between occupied and unoccupied orbitals, enabling detailed assignment of the origin of currents in, e.g., benzene, cyclooctatetraene, borazine, coronene, and corannulene. Whereas delocalized currents in benzene and planar cyclooctatetraene are described by transitions within the valence space, localized currents in the borazine pi system involve excitations outside the valence space.