The generalized separated electron pair model. 1. An application to NH3

The utility of the separated electron pair (SEP ) model (strongly orthogonal geminals) is examined quantitatively, for pyramidal and planar nuclear configurations of the NH₃ molecule. The best SEP wave function computed for each species is capable of recovering about half of the correlation energy obtained by a fairly accurate configuration interaction (CI ) calculation, (corresponding to roughly 25% of the total molecular correlation energy). It is illustrated that the model can be systematically extended with only a modest effort to yield more accurate results (about 40% of the total correlation energy). The fact that the corrections to the SEP model have a simple physical interpretation suggests that this model may be a useful starting point for “brute force” CI calculations on larger chemical systems.     

The generalized separated electron pair model. III. An application to three localization schemes for CO

The separated electron pair (SEP ) model (strongly orthogonal geminals) and methods for its systematic extension have been applied to three different localization schemes for C?O. The optimum SEP wave function is obtained for the particular localization scheme that involves three equivalent bent bonds. The major corrections to the SEP model arise from one-electron transfer terms. Two-electron transfer terms were important only for those pairs that were not well localized. It was found that the separate definitions of the total intrapair and interpair correlation energies did not depend strongly on the choice of localization scheme.     

A stationary property of the APSG wave function

The method of antisymmetrized product of strongly orthogonal geminals (APSG) emerges by optimizing the one-electron functions used to construct two-electron functions (geminals), the latter being expanded (with coefficients selected variationally) in mutually exclusive subspaces of the former ones. Accordingly, E _APSG is stationary with respect to the expansion coefficients and to unitary transformations of the one-electron orbitals. We show that the APSG energy is also stationary to the unitary transformation of identical geminals. For non-identical geminals, this statement holds only approximately.     

Electron group functions for the analysis of the electronic structures of molecules

The electronic structure of a vast majority of molecular systems can be understood in terms of electron groups and their wave functions. They serve as a natural basis for bringing intuitive chemical and physical concepts into quantum chemical calculations. This article considers the general electron group functions formalism as well as its simple geminal version. We try to characterize the wave function with the group structure and its capabilities in actual calculations. For this purpose we implement a variational method based on the wave function in the form of an antisymmetrized product of strongly orthogonal group functions and perform a series of electronic structure calculations for small molecules and model systems. The most important point studied is the relation between the choice of electron groups and the results obtained. We consider energetic characteristics as well as optimal geometry parameters. In view of practical importance, the structure of variationally optimized local one-electron states is considered in detail as well as intuitive characteristics of chemical bonds.     

Die horizontale Korrelation in π-Elektronensystemen und ihre Beschreibung durch Elektronenpaarfunktionen

Different types of pair functions (geminal products and their linear combinations) are tested with respect to their ability to describe the “horizontal correlation” of the π-electrons of butadiene. The validity of the π-electron approximation is not discussed and “full configuration interaction” within the limited LCAO basis is used as the standard to which the model calculations are referred. An APSG-function (APSG = antisymmetrized product of strongly orthogonal geminals) built up from equivalent (localized) geminals, which contains only one variational parameter is able to account for about 90% of the “horizontal correlation energy”. Both APSG and APIG functions constructed from delocalized geminals, are much less favorable. Criteria of the goodness of an approximate wave function are a) the energy b) comparison of its one- and two-particle density matrices with those obtained from “full CI”. The good results with the localized APSG function are related to the fact that electron correlation between electrons of opposite spin is (in this molecule) essential only within either of the “double bonds” of the “canonical structure”. The pertinent results are quite insensitive to different parametrization of the integrals.     

Strongly orthogonal geminals: size-extensive and variational reference states

Properties and some applications of strongly orthogonal geminals (APSG) are reviewed emphasizing the motivations for their use and their shortcomings. An overview presents some techniques capable of improving the APSG function.     

GF Calculations for some tricyclic molecules: C3H6, C3H4, C2H4NH,C2H4O and C2H4S

We have functions expressed as antisymmetrized products of strongly orthogonal geminals have been evaluated for some three membered ring molecules. GF results are compared with previously computed SCF-MO results, obtained employing the same atomic basis. Transferability features of bonds and inner shells are shown.     

Sensitivity of an exothermic chemical wave front to a departure from local equilibrium

We study the propagation of an exothermic chemical wave front in a reactive dilute gas and show that the particle velocity distribution departs from the Maxwellian form in the front zone. The analytical corrections to the balance equations for concentrations, temperature, and stream velocity induced by the departure from local equilibrium are derived from a perturbative solution of the Boltzmann equation. Our analytical predictions of the front properties, including its propagation speed, compare well with microscopic simulations of the particle dynamics.     

Is spin-component scaled second-order Møller-Plesset perturbation theory an appropriate method for the study of noncovalent interactions in molecules?

Testing of the spin-component scaled second-order M?ller-Plesset (SCS-MP2) method for the computation of noncovalent interaction energies is done with a database of 165 biologically relevant complexes. The effects of the spin-scaling procedure (i.e., MP2 vs SCS-MP2), the basis set size, and the corrections for basis set superposition error (BSSE) are systematically examined. When using two-point basis set extrapolations for the correlation energy, augmentation of the atomic orbital basis with computationally costly diffuse functions is found to be obsolete. In general, SCS-MP2 also improves results for noncovalent interactions statistically on MP2, and significant outliers are removed. Moreover, it is shown that effects of BSSE and one-particle basis set incompleteness almost cancel each other in the case of triple-zeta sets (SCS-MP2/TZVPP or SCS-MP2/cc-pVTZ without counterpoise correction), which opens a practical route to efficient computations for large systems. We recommend SCS-MP2 as the preferred quantum chemical wave function based method for the noncovalent interactions in large biologically relevant systems when reasonable coupled-cluster with single and double and perturbative triple excitations (CCSD(T)) calculations cannot be performed anymore. A comparison to MP2 and CCSD(T) interaction energies for n-alkane dimers, however, indicates (and this also holds to a lesser extent for hydrogen-bonded systems) limitations of SCS-MP2 when treating chemically "saturated" interactions. The different behavior of second-order perturbation theory for saturated and for stacked pi-systems is discussed.     

Second-order correction to perfect pairing: an inexpensive electronic structure method for the treatment of strong electron-electron correlations

We have formulated a second-order perturbative correction for perfect-pairing wave functions [PP2] based on similarity-transformed perturbation techniques in coupled cluster theory. The perfect-pairing approximation is used to obtain a simple reference wave function which can qualitatively describe bond breaking, diradicals, and other highly correlated systems, and the perturbative correction accounts for the dynamical correlation. An efficient implementation of this correction using the resolution of the identity approximation enables PP2 to be computed at a cost only a few times larger than that of canonical MP2 for systems with hundreds of active electrons and tens of heavy atoms. PP2 significantly improves on MP2 predictions in various systems with a challenging electronic structure.     

The interaction of chemical bonds. III. Perturbed strictly localized geminals in LMO basis

The formalism of strictly localized geminals (SLGs ) is summarized. It is shown that the SLG wave function serves as an appropriate multiconfigurational reference state that can easily be improved by perturbational, CI - or coupled cluster-type procedures. The possibility of expanding the geminals in the basis set of localized Hartree-Fock molecular orbitals (LMOs ) is discussed. Sample calculations on H₄, CH₄, H₂O, and He…?He systems are reported. © 1994 John Wiley & Sons, Inc.     

A truncated version of reduced multireference coupled-cluster method with singles and doubles and noniterative triples: application to F2 and Ni(CO)n (n=1, 2, and 4)

A perturbatively truncated version of the reduced multireference coupled-cluster method with singles and doubles and noniterative triples RMR CCSD(T) is described. In the standard RMR CCSD method, the effect of all triples and quadruples that are singles or doubles relative to references spanning a chosen multireference (MR) model space is accounted for via the external corrections based on the MR CISD wave function. In the full version of RMR CCSD(T), the remaining triples are then handled via perturbative corrections as in the standard, single-reference (SR) CCSD(T) method. By using a perturbative threshold in the selection of MR CISD configuration space, we arrive at the truncated version of RMR CCSD(T), in which the dimension of the MR CISD problem is significantly reduced, thus leaving more triples to be treated perturbatively. This significantly reduces the computational cost. We illustrate this approach on the F2 molecule, in which case the computational cost of the truncated version of RMR CCSD(T) is only about 10%-20% higher than that of the standard CCSD(T), while still eliminating the failure of CCSD(T) in the bond breaking region of geometries. To demonstrate the capabilities of the method, we have also used it to examine the structure and binding energy of transition metal complexes Ni(CO)n with n=1, 2, and 4. In particular, Ni(CO)2 is shown to be bent rather than linear, as implied by some earlier studies. The RMR CCSD(T) binding energy differs from the SR CCSD(T) one by 1-2 kcal/mol, while the energy barrier separating the linear and bent structures of Ni(CO)2 is smaller than 1 kcal/mol.     

Communication: efficient counterpoise corrections by a perturbative approach

We investigate the use of Hartree-Fock and density functional perturbative corrections for estimating the counterpoise correction (CPC) for interaction energies at the self-consistent field level. We test our approach using several popular basis sets on the S22 set of weakly bound systems, which can exhibit large basis set superposition errors. Our results show that the perturbative approaches typically recover over 95% of the CPC and can be up to twelve times faster to compute than the conventional methods and therefore provide an attractive alternative to calculating CPCs in the conventional way.     

A Chemist's View of BCS Theory for Low Tc Superconductivity and Its Relationship to Charge Transfer and High Tc Superconductivity

Based on a chemist's understanding of electron transfer, attempts are made to interpret the physicists' BCS theory for low Tc superconductivity in terms of quantum chemical views. To show how to improve and apply some of the principles of the BCS theory for high Tc superconductors, especially those having to do with many-body aspects of correlation, coherence and long-range order, we (1) extend the two-partner, donor-acceptor electron transfer to large periodic systems with cyclic boundary conditions and with double-well potentials beyond Peierls distortion. (2) introduce second-order Jahn-Teller stabilization and introduce vibronic mixing as configuration interaction. To take advantage of existing bonds in high Tc superconductors we propose a linear combination of bonding (two-electron) geminals to form molecular bonding geminals which we call “Vibronic Geminals” after mixing with different running waves of bond structure vibrations. To take advantage of the valence bonds and double-well potentials due to anti-symmetric and other vibrations, we propose a ‘Covalon’ type model for the propagation and tunneling of such bonds which transform as Bosons.     

All-pair wave function and reduced variational equation for electronic systems

A new type of wave function is proposed for atomic and molecular systems. This all-pair function is constructed of N(N – 1)/2 identical geminals for N electrons. For systems with the highest multiplicity this is the full space part of the wave function. For closed shell systems it has to be multiplied by a Slater determinant according to the antisymmetry condition. In the case of maximal multiplicity a reduced variational equation is derived for the geminal. This equation is independent of the dimensionality of the system and contains the particle number as a multiplicative factor only. The method is extended to the closed shell case where a restriction has to be fulfilled. The reduction of the variational equation can be done only approximately. The use of identical geminals can be treated as a first approximation. An extension of the method, called the pair interdependent configuration interaction (PICI), is proposed. The special features of the method are discussed briefly.     

Nonadiabatic corrections to the wave function and energy

Nonadiabatic corrections in molecules composed of a few atoms are considered. It is demonstrated that a systematic perturbative expansion around the adiabatic solution is possible, with the expansion parameter being the electron-nucleus mass ratio to the 3/4 power. Closed form formulas for the leading corrections to the wave function and to the energy are derived. Their applicability is demonstrated by a comparison of numerical results for the hydrogen molecule with the former nonadiabatic calculations and the experimental values. Good agreement with the recent experiment is achieved for the ground state dissociation energy of both H(2) and D(2).     

Extended group function calculations for H2O,NH3 and CH4

Wave functions expressed as antisymmetrized products of strongly orthogonal geminals have been evaluated for H₂O, NH₃ and CH₄. The geminals have been expressed as linear combinations of 2 × 2 detors constructed with localized SCF -MO 's. Several ground state observables have been computed together with the electric polarizabilities and magnetic susceptibilities. In addition, a configuration interaction calculation limited to all possible double group excitations has been carried out.     

Quadratically convergent iteration procedure for geminal determinations

A general formalism is presented for the determination of the optimum natural orbitals within the various strongly orthogonal geminals of a wavefunction describing a 2N-electron closed-shell molecule or atom. The relationship to Hartree–Fock–Roothaan theory is established; the algorithm that is developed is quadratically convergent to the desired result, and does not ignore off diagonal Lagrangian multipliers, or require an infinite series of 2 × 2 orthogonal transformations of the original basis.     

Simple coupled-cluster singles and doubles method with perturbative inclusion of triples and explicitly correlated geminals: The CCSD(T)R12 model

To approach the complete basis set limit of the "gold-standard" coupled-cluster singles and doubles plus perturbative triples [CCSD(T)] method, we extend the recently proposed perturbative explicitly correlated coupled-cluster singles and doubles method, CCSD(2)(R12) [E. F. Valeev, Phys. Chem. Chem. Phys. 8, 106 (2008)], to account for the effect of connected three-electron correlations. The natural choice of the zeroth-order Hamiltonian produces a perturbation expansion with rigorously separable second-order energy corrections due to the explicitly correlated geminals and conventional triple and higher excitations. The resulting CCSD(T)(R12) energy is defined as a sum of the standard CCSD(T) energy and an amplitude-dependent geminal correction. The method is technically very simple: Its implementation requires no modification of the standard CCSD(T) program and the formal cost of the geminal correction is small. We investigate the performance of the open-shell version of the CCSD(T)(R12) method as a possible replacement of the standard complete-basis-set CCSD(T) energies in the high accuracy extrapolated ab initio thermochemistry model of Stanton et al. [J. Chem. Phys. 121, 11599 (2004)]. Correlation contributions to the heat of formation computed with the new method in an aug-cc-pCVXZ basis set have mean absolute basis set errors of 2.8 and 1.0 kJmol when X is T and Q, respectively. The corresponding errors of the standard CCSD(T) method are 9.1, 4.0, and 2.1 kJmol when X=T, Q, and 5. Simple two-point basis set extrapolations of standard CCSD(T) energies perform better than the explicitly correlated method for absolute correlation energies and atomization energies, but no such advantage found when computing heats of formation. A simple Schwenke-type two-point extrapolation of the CCSD(T)(R12)aug-cc-pCVXZ energies with X=T,Q yields the most accurate heats of formation found in this work, in error on average by 0.5 kJmol and at most by 1.7 kJmol.