Simple constructive proof of Karadakov's extended pairing theorem

A simple proof is given for Karadakov's recent extension of the pairing theorem to the virtual subspaces. The proof consists of an explicit construction of the virtual orbitals with the required pairing properties, utilizing the pairing of the occupied orbitals.     

Virtual orbital transformation prior to configuration interaction with localized orbitals

A virtual orbital transformation is proposed involving pairing of localized occupied orbitals with virtual orbitals. The virtual orbitals are transformed so that each virtual orbital is as “close” as possible to its occupied counterpart, where closeness is the inverse of the particular definition of localization. The appropriate transformation is derived for the special case of Foster–Boys localization, and an illustrative CNDO /2 calculation on HNO is presented. INDO CI results on the series N₂, CO, BF indicate that use of this transformation may reduce the number of energetically significant configurations.     

A simple extension of the external magnasco-perico localization procedure to the virtual MO-Space

The external localization procedure of Magnasco and Perico is extended to the unoccupied molecular orbitals of the Fock-operator. The formal correspondence between bonding orbitals and localized antibonding MOs is demonstrated. Localized occupied and virtual one-electron functions are calculated within a semiempirical INDO-Hamiltonian and are analyzed; the externally localized occupied MOs are compared with energy localized orbitals computed by the Edmiston and Ruedenberg procedure. Various applications of the fully localized (occupied and virtual) MO set are discussed.     

Using full-CI algorithms in Bethe-Goldstone-type expansions of the correlation energy

In the early 1960s, Nesbet proposed to develop correlation energy in terms of two-, three-, four-, etc., electron contributions. This expansion was, in principle, applicable to a large number of electrons without a size-extensivity error. The now available full-CI algorithms may be used to obtain those expansions in terms of either occupied spin—orbitals or, more efficiently, in terms of sets of occupied or virtual molecular orbitals. Tests on the NH₃ molecule with a DZP basis-set problem show the slow convergence of this approach. © 1997 John Wiley & Sons, Inc.     

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.     

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.     

The Fock Matrix Analysis for Atomic Orbitals in Molecular Orbitals I. A New Look on the Covalent Bond in H2?

The character of the molecular orbitals can be better accounted for in terms of molecular adapted atomic orbitals and the Fock matrix expanded in these atomic orbital sets. A clean‐cut and unique criterion for the diradicals and the covalent bonds can be given for the molecular orbitals in both restricted and unrestricted Hartree‐Fock wavefunctions. Instead of the picture that overlap charge migrates into the bonding region, the new analysis displays another picture that the charge densities for the electrons with α and β spins give rise to two opposite spin density shifts. If the α one shifts from atom A toward atom B then it is vice versa for the β one. The spin density shifts proceed until the bonding molecular orbitals form.     

Perturbative corrections to basis incompleteness in molecular SCF calculations

A unified summary is presented of the mathematical approach developed by McDowell for employing perturbation theory to correct for basis-set incompleteness in ab initio SCF calculations. Revised expressions for the corrections to the wavefunction both in terms of orbitals and spin-orbitals are presented with explicit incorporation of the spin variables. Employing H₂O as an example, we show that this approach is considerably more powerful for computing molecular energies with standard basis sets than was indicated by previous work. In particular at the higher levels of approximation it accurately reproduces the effect of polarization functions in sets such as 6-31G^* and 6-31G^**. The equilibrium molecular structure of H₂O was also computed by this approach and found to give good accuracy. In each case perturbing functions coupled to both occupied and virtual orbitals are required for acceptable results.     

Localization of molecular orbitals on fragments

A non‐iterative algorithm for the localization of molecular orbitals (MOs) from complete active space self consistent field (CASSCF) and for single‐determinantal wave functions on predefined moieties is given. The localized fragment orbitals can be used to analyze chemical reactions between fragments and also the binding of fragments in the product molecule with a fragments‐in‐molecules approach by using a valence bond expansion of the CASSCF wave function. The algorithm is an example of the orthogonal Procrustes problem, which is a matrix optimization problem using the singular value decomposition. It is based on the similarity of the set of MOs for the moieties to the localized MOs of the molecule; the similarity is expressed by overlap matrices between the original fragment MOs and the localized MOs. For CASSCF wave functions, localization is done independently in the space of occupied orbitals and active orbitals, whereas, the space of virtual orbitals is mostly uninteresting. Localization of Hartree–Fock or Kohn–Sham density functional theory orbitals is not straightforward; rather, it needs careful consideration, because in this case some virtual orbitals are needed but the space of virtual orbitals depends on the basis sets used and causes considerable problems due to the diffuse character of most virtual orbitals. © 2012 Wiley Periodicals, Inc.     

A perspective on the localizability of Hartree–Fock orbitals

A common perception about molecular systems with a nonlocal electronic structure (as manifested by a nonlocal Hartree–Fock (HF) density matrix), such as conjugated π-systems, is that they can only be described in terms of nonlocal molecular orbitals. This view is mostly founded on chemical intuition, and further, this view is strengthened by traditional approaches for obtaining local occupied and virtual orbital spaces, such as the occupied Pipek–Mezey orbitals, and projected atomic orbitals. In this article, we discuss the limitations for localizability of HF orbitals in terms of restrictions posed by the delocalized character of the underlying density matrix for the molecular system and by the orthogonality constraint on the molecular orbitals. We show that the locality of the orbitals, in terms of nonvanishing charge distributions of orbitals centered far apart, is much more strongly affected by the orthogonality constraint than by the physical requirement that the occupied orbitals must represent the electron density. Thus, the freedom of carrying out unitary transformations among the orbitals provides the flexibility to obtain highly local occupied and virtual molecular orbitals, even for molecular systems with a nonlocal density matrix, provided that a proper localization function is used. As an additional consideration, we clear up the common misconception that projected atomic orbitals in general are more local than localized orthogonal virtual orbitals.     

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.     

The question of the completeness of the natural orbitals with nonzero occupation numbers for atoms and molecules

Summary The completeness of the natural orbitals with nonzero occupation numbers is examined for several model Hamiltonians and for the helium atom. It is demonstrated that whether the occupied natural orbitals form complete sets depends on the nature of the electron-electron interaction in the model Hamiltonian. Discrepancies in the extant proofs of the exactness and inexactness of the extended Koopmans' theorem are resolved.     

Computational developments in generalized valence bond calculations

In this article a procedure for generating starting orbitals for generalized valence bond (GVB) calculations is presented. This is achieved by selecting orbitals which correspond to specific bonds or electron pairs. These orbitals can be identified from the localized molecular orbitals, for both occupied and virtual orbitals, which are obtained through a unitary transformation of the Hartree-Fock canonical molecular orbitals using the Boys's localization method. A scheme has also been implemented which achieves optimum convergence of the pairwise orbital optimization. An object-oriented GVB program is developed which automatically generates reliable initial GVB orbitals, leading to proper and fast convergence. © 1996 by John Wiley & Sons, Inc.     

Emploi d'une méthode de perturbation-variation pour l'étude de la polarisation de spin dans les radicaux libres aromatiques

A perturbation method has been used to deal with the problem of the interaction of configuration in the free aromatic radicals. We have considered only the mono-excitated configurations which are responsible for the specific effects due to the spin polarization; the corresponding wave functions are built up with the set of molecular orbitals LCAO SCF (occupied and virtual) of the ground-state configuration. We thus obtain a good distribution of spin densities on the rings of the studied radicals: the benzyl and the methylene-naphthyls radicals. The spin density on the extracyclic carbon remains too large as in the case of the SCF representation. This may be explained by the shape of the molecular orbital occupied by the unpaired electron in the SCF configuration, and the structure of the method used which disregards the excitated configurations involving this orbital.     

Optimization of orbital-specific virtuals in local Møller-Plesset perturbation theory

We present an orbital-optimized version of our orbital-specific-virtuals second-order M?ller-Plesset perturbation theory (OSV-MP2). The OSV model is a local correlation ansatz with a small basis of virtual functions for each occupied orbital. It is related to the Pulay-Saeb? approach, in which domains of virtual orbitals are drawn from a single set of projected atomic orbitals; but here the virtual functions associated with a particular occupied orbital are specifically tailored to the correlation effects in which that orbital participates. In this study, the shapes of the OSVs are optimized simultaneously with the OSV-MP2 amplitudes by minimizing the Hylleraas functional or approximations to it. It is found that optimized OSVs are considerably more accurate than the OSVs obtained through singular value decomposition of diagonal blocks of MP2 amplitudes, as used in our earlier work. Orbital-optimized OSV-MP2 recovers smooth potential energy surfaces regardless of the number of virtuals. Full optimization is still computationally demanding, but orbital optimization in a diagonal or Kapuy-type MP2 approximation provides an attractive scheme for determining accurate OSVs.     

A method for population and bonding analyses in calculations with extended basis sets

Summary A method for population and bonding analyses in the calculations with extended basis sets is proposed. The definition and evaluation method of the atomic orbitals in molecular environments (AOIMs) are described. It is shown that the AOIMs can be divided into two subsets, the strongly occupied minimal compact subset {AOIM}_B and the very weakly occupied “Rydberg” subset {AOIM}_R, according to the orbital population obtained from Mulliken analysis with AOIMs as basis sets. The viewpoint of “molecular orbitals consisting of minimal atomic orbital sets” can be optimally realized in terms of {AOIM}_B. The Mulliken population based on AOIMs is reasonable and fairly stable to changes of basis sets. The Mayer bond orders calculated based on {AOIM}_B are quite stable to the changes of basis sets; therefore they can be used to measure objectively the contribution of individual atomic orbitals to bonding.     

Basis-set quality and basis-set bias in molecular property calculations

Quality measures for Gaussian basis sets are proposed that are based on principal angles between the basis set and reference molecular orbitals. The principal angles are obtained from the cosine-sine (CS) decomposition of orthogonal matrices and yield detailed information about basis-set convergence with respect to different regions of space. Principal angles for occupied orbitals show excellent correlation with basis-set errors in ground-state energies. Furthermore, ground-state bias in finite basis sets can be estimated from the relation between principal angles for occupied and Rydberg orbitals. Ground-state bias is observed in basis sets including extensive diffuse augmentation and affects the quality of computed molecular response properties. Principal angles and ground-state bias are investigated for the H-Ne atoms and a series of diatomics using numerical Hartree-Fock calculations as a reference. Convergence of ground-state energies and static polarizabilities is studied for the hierarchies of correlation-consistent and Karlsruhe segmented def2 basis sets including different levels of diffuse augmentation.     

Self-consistent, constrained linear-combination-of-atomic-potentials approach to quantum mechanics

Variational fitting gives a stationary linear-combination of atomic potentials (LCAP) approximation to the Kohn-Sham (KS) potential, V. That potential is central to density-functional theory because it generates all orbitals, occupied as well as virtual. Perturbation theory links two self-consistent field (SCF) calculations that differ by the perturbation. Using the same variational LCAP methods and basis sets in the two SCF calculations gives precise KS potentials for each order. Variational V perturbation theory, developed herein through second order, gives stationary potentials at each order and stationary even-order perturbed energies that precisely link the two SCF calculations. Iterative methods are unnecessary because the dimension of the matrix that must be inverted is the KS basis size, not the number of occupied times virtual orbitals of coupled-perturbed methods. With variational perturbation theory, the precision of derivatives and the fidelity of the LCAP KS potential are not related. Finite differences of SCF calculations allow the precision of analytic derivatives from double-precision code to be verified to roughly seven significant digits. For a simple functional, the fourth derivatives of the energy and the first and second derivative of the KS potentials with respect to orbital occupation are computed for a standard set of molecules and basis sets, with and without constraints on the fit to the KS potential. There is no significant difference between the constrained and unconstrained calculations.     

On the pairing of the natural orbitals for projected broken symmetry states

A pairing scheme for the natural orbitals of projected broken symmetry states, which can be expressed as a superposition of two generally non-orthogonal Slater determinants, is presented. The results obtained here are generalizations of a pairing scheme representation of natural orbitals derived recently by Hendekovi? within the complex molecular orbital method. As a simple example the Kekulé structure of benzene is discussed.     

The spin-flip extended single excitation configuration interaction method

An extension of the spin-flip single excitation configuration interaction (SF-CIS) method is introduced. The extension, abbreviated as SF-XCIS, includes all configurations in which no more than one virtual level of the high spin triplet reference becomes occupied and no more than one doubly occupied level becomes vacant. The number of such configurations is quadratic with molecule size, and the method is implemented in a direct algorithm whose cost scales in the same way with molecule size as CIS itself, thus permitting applications to large systems. Starting from a spin restricted triplet determinant, SF-XCIS yields spin-pure singlet, triplet, and quintet states, and treats both half-occupied reference orbitals in a fully balanced way to allow application to strongly correlated problems. Tests on bond dissociation in the HF molecule, the torsional potential of ethylene, and excited states of polyenes show encouraging improvements using SF-XCIS compared to SF-CIS and a previously suggested extension, the spin-complete CIS model.