A systematic CI procedure with modified virtual orbitals

The use of modified virtual orbitals is studied in a systematic conventional CI procedure which offers considerable potential in regard to convergence and extension to larger systems. The method is applied to the HCN molecule by using 37 basis functions, and analysis of energy expectation values, together with the one-electron density, yields some insight into the physical content of CI wavefunctions.     

Dynamically screened local correlation method using enveloping localized orbitals

In this paper we present a local coupled cluster approach based on a dynamical screening scheme, in which amplitudes are either calculated at the coupled cluster level (in this case CCSD) or at the level of perturbation theory, employing a threshold driven procedure based on MP2 energy increments. This way, controllable accuracy and smooth convergence towards the exact result are obtained in the framework of an a posteriori approximation scheme. For the representation of the occupied space a new set of local orbitals is presented with the size of a minimal basis set. This set is atom centered, is nonorthogonal, and has shapes which are fairly independent of the details of the molecular system of interest. Two slightly different versions of combined local coupled cluster and perturbation theory equations are considered. In the limit both converge to the untruncated CCSD result. Benchmark calculations for four systems (heptane, serine, water hexamer, and oxadiazole-2-oxide) are carried out, and decay of the amplitudes, truncation error, and convergence towards the exact CCSD result are analyzed.     

Modification of virtual orbitals

The utility of modifying the virtual orbitals of the Fock operator by introducition of an additional potential is discussed. A particularly convenient form for computational implementations is obtained, and improved methods for the practical solution of the secular problem are recommended.     

Localization of virtual orbitals

The application of the MBPT in the localized representation requires that both the occupied and the virtual orbitals obtained by the canonical HF equation should be localized. The localization of the occupied orbitals is straightforward in general by any localization method. It is shown that by using Boys' method the localized virtual orbitals are spatially well separated and transferable not only in minimal basis sets.     

Usefulness of modified virtual orbitals in multireference CI procedure illustrated by calculations on lithium clusters

The maximization of the exchange interaction between the canonical Hartree–Fock virtual and occupied orbitals leads to a transformed set of virtual orbitals which are well suited as one-electron functions for CI calculations. The procedure, generally known for a long time is seldom applied, despite its simplicity and very low computational demand. However, it is found to be particularly useful in the case of multireference CI, since an improved energy is obtained with a considerable shortening of the CI expansion. Moreover, in the final CI wave function, several configurations appear with considerable weight, thus allowing an easy choice of additional configurations to be inserted in the definition of a new zero-order wave function. The efficiency of the computational procedure is discussed for the case of a Li₆ cluster of D_3h symmetry and for the NaCO and PdCO complexes. Results are reported for the relative stability of four different geometrical arrangements of the Li₆ cluster.     

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.     

Compact formula for Coulomb integrals of Slater orbitals

Institute for Chemical Physics, Russian Academy of Science. Translated from Zhurnal Strukturnoi Khimii, Vol. 33, No. 1, pp. 153–156, January–February, 1992.     

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.     

Transferability of modified atomic orbitals

The modified atomic orbital basis set determined for molecular orbital calculations on a given molecule is shown to be transferable to other molecules. This transferability is examined using the modified atomic orbital basis set determined for the carbon atom in methane for calculations on ethane, propane, butane, cyclohexane, acetylene, ethylene and benzene.     

Fermi and Coulomb holes of molecules in excited states

Correlation holes of electrons with the same (Fermi hole) and different (Coulomb hole) spins in the ground (X¹Σ⁺), first (A¹Σ⁺) and second (B¹II) excited states of LiH were constructed from full configuration interaction (CI ) wave functions. It was found that the shapes of both the Fermi and Coulomb holes in these states are dependent on the location of the reference electron. When the reference electron is chosen to be close to the Li nucleus, the Fermi correlation results in a large negative hole for all three states. However, the A¹Σ⁺ excited state is further characterized by displaying a second hole around the H nucleus, and in the B¹II state, the hole is elongated along the molecular axis. Coulomb correlation shows up strongly in the A¹Σ⁺ state and, in addition, there is clearly correlation of electrons at the two nuclei. These features of the correlation holes were compared with those from a two-Slater-determinant model wave function. The Hartree, Fermi, and Coulomb screening potentials in these states were also studied in the light of possible modeling of the correlation functionals for the excited states. © 1995 John Wiley & Sons, Inc.     

Matrix elements of the Coulomb Green function between Slater orbitals

Analytic expressions are given for integrals of the Coulomb Green function with Slater type atomic orbitals. The results involve hypergeometric functions.Supported by the National Institutes of Health, Grant No. GM23223.     

Calculation of second-order properties with improved virtual orbitals

¹³C---H nuclear spin coupling and ¹³C shielding constants of CH₄, C₂H₆, C₂H₄, C₂H₂ are calculated by means of the second-order perturbation theory and MO LCAO wavefunctions. The aim of this paper is to test the ability of the recently proposed virtual orbitals to accelerte the convergence of the sum over states, compared to the calculation with canonical MO's. The ¹³C---H nuclear spin coupling constants show a significant improvement, the changes of the ¹³C shielding are insignificant.     

Efficient and accurate local single reference correlation methods for high-spin open-shell molecules using pair natural orbitals

A production level implementation of the high-spin open-shell (spin unrestricted) single reference coupled pair, quadratic configuration interaction and coupled cluster methods with up to doubly excited determinants in the framework of the local pair natural orbital (LPNO) concept is reported. This work is an extension of the closed-shell LPNO methods developed earlier [F. Neese, F. Wennmohs, and A. Hansen, J. Chem. Phys. 130, 114108 (2009); F. Neese, A. Hansen, and D. G. Liakos, J. Chem. Phys. 131, 064103 (2009)]. The internal space is spanned by localized orbitals, while the external space for each electron pair is represented by a truncated PNO expansion. The laborious integral transformation associated with the large number of PNOs becomes feasible through the extensive use of density fitting (resolution of the identity (RI)) techniques. Technical complications arising for the open-shell case and the use of quasi-restricted orbitals for the construction of the reference determinant are discussed in detail. As in the closed-shell case, only three cutoff parameters control the average number of PNOs per electron pair, the size of the significant pair list, and the number of contributing auxiliary basis functions per PNO. The chosen threshold default values ensure robustness and the results of the parent canonical methods are reproduced to high accuracy. Comprehensive numerical tests on absolute and relative energies as well as timings consistently show that the outstanding performance of the LPNO methods carries over to the open-shell case with minor modifications. Finally, hyperfine couplings calculated with the variational LPNO-CEPA∕1 method, for which a well-defined expectation value type density exists, indicate the great potential of the LPNO approach for the efficient calculation of molecular properties.     

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.     

Localization of the filled and virtual orbitals in the nucleotide bases

For the four nucleotide bases cytosine, uracil, adenine and guanine both Boys (B) and Edminston-Ruedenberg (ER) localization procedures of the ab initio canonical orbitals have been performed. The results obtained for both σ-π separation and by treating all electrons together show a very good localization for all electrons (one-center lone-pairs and two-center localized orbitals even for π-electrons) and a rather good localization for the virtuals applying both B and ER criteria. The results of the two methods are essentially identical. These results suggest that the application of localized orbitals will open new possibilities for the calculation of correlation in extended systems.     

Pipek–Mezey localization of occupied and virtual orbitals

Recent advances in orbital localization algorithms are used to minimize the Pipek–Mezey localization function for both occupied and virtual Hartree–Fock orbitals. Virtual Pipek–Mezey orbitals for large molecular systems have previously not been considered in the literature. For this work, the Pipek–Mezey (PM) localization function is implemented for both the Mulliken and a Löwdin population analysis. The results show that the standard PM localization function (using either Mulliken or Löwdin population analyses) may yield local occupied orbitals, although for some systems the occupied orbitals are only semilocal as compared to state‐of‐the‐art localized occupied orbitals. For the virtual orbitals, a Löwdin population analysis shows improvement in locality compared to a Mulliken population analysis, but for both Mulliken and Löwdin population analyses, the virtual orbitals are seen to be considerably less local compared to state‐of‐the‐art localized orbitals. © 2013 Wiley Periodicals, Inc.     

Modifications of virtual orbitals in the limited CI calculations for electron-rich molecules

The efficiency of modified virtual orbitals (MVO) of ionic type and of approximate orthogonalized natural orbitals (ONO) in the CI-SD calculations was studied for O₃ and SO₂ molecules and compared with the commonly used canonical virtual orbitals (CVOs). The systems studied represent a class of electron-rich molecules, in which the number of valence electron pairs exceeds substantially the number of formal chemical bonds. We found that the modified orbitals of the types studied appear to be less effective for these systems than in the similar calculations for the AH_n type molecules. Physical reasons for this difference were discussed. The evolution of spatial properties of virtual orbitals within the modification process was analyzed. © 1996 John Wiley & Sons, Inc.     

Analytical calculation of two-center Coulomb and hybrid integrals with orbitals of the Slater type

Journal of Structural Chemistry -     

Unified treatment of two-center Coulomb,hybrid, and exchange integrals over Slater-type orbitals

Using Neumann expansion for 1/r₁₂ in elliptical coordinates a combined formula has been obtained for two-center Coulomb, hybrid, and exchange integrals with Slater-type orbitals. © 1995 John Wiley & Sons, Inc.