Two by two rotations of orbitals for minimizing the energy of LCAO-MO wave functions

A new method is studied for finding the molecular orbitals which minimize the energy of an LCAO-MO wavefunction. The method makes use of successive rotations of pairs of orbitals. It can be applied to multi-determinant as well as to single-determinant wavefunctions. Criteria are found to minimize excited states.

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Zusammenfassung Es wird eine neue Methode zur Bestimmung der M.O.s nach dem Kriterium minimaler Gesamtenergie beschrieben. Dabei werden jeweils zwei M.O.s sukzessive transformiert. Das Verfahren bleibt auch dann anwendbar, wenn für die Zustandsfunktion eine Linearkombination von HS-Determinanten angesetzt wird. Zum Schluß werden noch einige Kriterien für die Bestimmung von angeregten Zuständen angegeben.

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Résumé Une nouvelle méthode est étudiée ici pour trouver les orbitales moléculaires qui minimisent l'énergie d'une fonction d'onde LCAO-MO. La méthode utilise des rotations successives de couples d'orbitales. La méthode peut minimiser aussi des fonctions d'onde multi-déter-minantales et les états excités.

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The author is indebted with Professor B. Pullman and Dr. A. Pullman for their suggestions and helpful advises. He aknowledges also Dr. Berthier for useful discussions.

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This work prepared in part at the Institut de Biologie Physico-Chimique, Université de Paris, was supported by a grant CY-3073 of the US Public Health Service (National Cancer Institute) to that Institute.     

Molecular orbital calculations based on linear combinations of fragment orbitals

A semiempirical MO method based on localized fragment orbitals has been developed, which is particularly suited for the construction of orbital correlation diagrams for the discussion of the electronic structure of complex molecules in terms of fragments and their interactions. The method allows for the inclusion of experimental ionization potentials and electron affinities of the fragments within the calculation of the Fock matrix elements and may thus form the basis of an interpretation of photoelectron spectra, comparable to the interpretation of UV spectra by means of the MIM method of Longuet-Higgins and Murrell. Several levels of approximation are discussed using the acrolein molecule as an example.     

Local pair natural orbitals for excited states

We explore how in response calculations for excitation energies with wavefunction based (e.g., coupled cluster) methods the number of double excitation amplitudes can be reduced by means of truncated pair natural orbital (PNO) expansions and localized occupied orbitals. Using the CIS(D) approximation as a test model, we find that the number of double excitation amplitudes can be reduced dramatically with minor impact on the accuracy if the excited state wavefunction is expanded in state-specific PNOs generated from an approximate first-order guess wavefunction. As for ground states, the PNO truncation error can also for excitation energies be controlled by a single threshold related to generalized natural occupation numbers. The best performance is found with occupied orbitals which are localized by the Pipek-Mezey localization. For a large test set of excited states we find with this localization that already a PNO threshold of 10(-8)-10(-7), corresponding to an average of only 40-80 PNOs per pair, is sufficient to keep the PNO truncation error for vertical excitation energies below 0.01 eV. This is a significantly more rapid convergence with the number doubles amplitudes than in domain-based local response approaches. We demonstrate that the number of significant excited state PNOs scales asymptotically linearly with the system size in the worst case of completely delocalized excitations and sub-linearly whenever the chromophore does not increase with the system size. Moreover, we observe that the flexibility of state-specific PNOs to adapt to the character of an excitation allows for an almost unbiased treatment of local, delocalized and charge transfer excited states.     

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.     

On the use of corresponding orbitals for the construction of mutually orthogonal orbital sets

Consider two orbital sets χ_k, k = 1…m and η₁, 1 = 1…n, which are mutually nonorthogonal. Provided that n > m, at least n ? m orbitals of the set {η} can be orthogonalized to the set {χ} by a transformation within the set {η}. The orthogonalization of the remaining orbitals of {η} to the set {χ} requires a transformation in which the χk appear explicitly. The orthogonalization of one orbital set to another is relevant for SCF optimizations in a truncated basis set, in the presence of frozen occupied orbitals. Examples are frozen core calculations, ECP calculations, and embedded cluster calculations, where the cluster is embedded in a frozen environment. A simple orthogonalization scheme, which makes use of a corresponding orbital transformation, is presented. It is suggested that with a small, well-defined extension of the set {η} the complete orthogonalization can be done with a transformation in which the {χ} do not appear explicitly. © 1993 John Wiley & Sons, Inc.     

The use of population localised orbitals to interpret molecular orbital calculations

An efficient and general method is derived to calculate population localised molecular orbitals (LMO's) from a given SCF eigenvector matrix, by reduction to an eigenvalue problem. Applications to both localised molecules (NH₃ and C₂H₂) and delocalised ones (B₂H₆, C₆H₆ and butadiene) are discussed in some detail. It is shown that unequal occupation of atomic energy levels leads to non-orthogonal LMO's. The consequences of non-orthogonal atomic hybrid orbitals are discussed, formulas for their overlap in terms of atomic occupation numbers are derived and it is shown that the occupation numbers are connected to LMO atomic orbital coefficients by various sum rules.     

The use of single exponential orbitals in one-center molecular orbital calculations

Single-exponential Slater type orbitals of the form ψ₁ = (1 + L₁(r, θ) + L₂(r, θ) +…?+ L_n(r, θ)) exp (? αr) are examined for their potential use as one-center molecular orbitals. These are then to be used as molecular fragments in a LCMO study. The system examined is HeH^{+ +} with calculated energies and dipole moments being compared to the exact values. These functions behave best in the region of chemical interest (the bonding region) and thus demonstrate a possible usefulness in LCMO calculations and in the field of one-electron diatomics.     

Generation of orbitals that control molecular reactivity: projected reactive orbital approach

A method that generalizes the notion of frontier orbital (FO) theory is introduced. The method is based on the projected reactive orbitals (PROs). Although PROs have been shown to describe local reactivity better than FOs in high-symmetry systems, the PRO method needs an arbitrary choice of a reference atomic orbital (AO), causing ambiguity of the method and poor applicability to low-symmetry molecules. To overcome these difficulties, we examined three different kinds of methods for uniquely determining the reference AO, one of which (Method 1) was reported by other authors (Kurita, Y.; Takayama, C. J. Phys. Chem. A 1997, 101, 5593-5595). We specifically applied the methods to the prediction of basicities of heteroaromatic amines. The study showed that the newly developed reactivity-index maximization method (Method 3) yields the most reasonable PRO.     

Reduced hamiltonian orbitals. II. Optimal orbital basis sets for the many-electron problem

Optimal orbital exponents are approximated by minimization of the reduced Hamiltonian orbital ground state energy. They appear to be as good as and are obtained at much less expense than the values derived by the usual SCF exponent optimization scheme. Partitioning of energy into 0-energy, 1-energy, and 2-energy (Absar and Coleman, Int. J. Quant. Chem. 10 , 319 (1976); Chem. Phys. Lett. 39 , 60 (1976)) is used to study the variation in the electronic energy surface upon variation of orbital exponents. The 1-energy operator, the natural orbitals of which are the reduced Hamiltonian orbitals, is compared with the SCF operator.     

Cluster molecular orbitals and an orbital symmetry view of solid phase transitions in perovskites

The molecular orbitals, normalization constants and energies of the M₈(O_h), M₄(T_d) and M₆(O_h) clusters are derived and tabulated through the d-atomic orbitals. A vector method, adapted to computer application, is devised to compute s, p and d overlap between variously oriented orbitals at atoms that do not have co-directional local axes. Mixing of σ, π and δ orbitals to give the same irreducible representation is also included. As illustrations, the orbitals of Sr₈, La₈, TiO₆ and AlO₆ clusters are computed by the Mulliken—Wolfsberg and Helmholz approximations. During solid phase transitions in the perovskite structures of SrTiO₃ and LaAlO₃, the TiO₆ octahedron rotates about the C₄ axis whereas the AlO₆ octahedron rotates about the C₃ axis. This difference is explained qualitatively in terms of the relative symmetries of the cluster HOMOs and LUMOs using the second-order Jahn—Teller effect. Allusions are made to the application of this cluster symmetry approach to other systems.     

Natural spin orbital analysis of diatomic molecular wave functions in terms of generalized diatomic orbitals

Starting from the demi-H ₂ ⁺ -model for Rydberg states, ab initio calculations of the energy and the wave function for some excited states of H₂ have been carried out with the help of diatomic orbitals. The potential curves and wave functions for the following states: 2¹ _g< ^/+ , 3¹ _g< ^/+ , 1³ _g< ^/+ , 2³ _g< ^/+ , 1¹ _u< ^/+ , 2¹ _u< ^/+ , 1³ _u< ^/+ , 2³ _u< ^/+ , 3³ _u< ^/+ , 1¹ _g , 1³ _g , 1¹ _u , and 1³ _u , have been calculated by a complete CI (configuration interaction) calculation in the sense that all configurations of the state symmetry have been used which can be formed from a given basis set. From the wave functions thus obtained the natural spin orbitals are calculated subsequently to the variational calculations. The dependence of the occupation numbers of the natural spin orbitals on internuclear distance is interpreted according to the model and is used for the explanation of the special features like double minima and maxima which occur in the potential curves of H₂. For the curves of the occupation numbers a non-crossing rule in analogy to that for potential curves is valid. The potential curves for the states 1³ _g and 1³ _u have been improved by the use of linear combinations of diatomic orbitals with different nuclear charges, which allow a flexible transition to linear combinations of atomic orbitals.Dedicated to Professor Iwan N. Stranski on the occasion of his 80th birthday.     

Natural spin orbital analysis of diatomic molecular wave functions in terms of generalized diatomic orbitals

The original idea of the model applied to HeH⁺ excited states is: One electron occupies a diatomic orbital similar to the HeH⁺⁺ ground state 1s function. The other electron occupies an orbital which can be represented by a linear combination of functions similar to H ₂ ⁺ excited state functions. One or two screening parameters are variationally optimized to compensate for the smallness of the one-electron basis.CI calculations have been performed for five excited HeH⁺ states covering a wide range of internuclear distances. The CI wave functions have been submitted to a natural spin orbital analysis. The strongly occupied NSOs are compared with the original model functions.Dedicated to Professor Hermann Hartmann on occasion of his 65th birthday on May 4th, 1979.     

Local orbital eigenvalues and basis set balance

Consideration of the chemical potential of an electron in a wavefunction suggests that a quantity called the local orbital eigenvalue and its variation in space provides a method of testing the balance of a basis set as a function of spatial position. The Hartree-Fock method as applied to the helium and neon atoms is used as an example.     

Calculation of the brueckner orbitals and generalized natural orbitals

The diagrammatic-perturbation approach for the construction of the one-particle Hermitian pseudoeigenvalue problem determining the Brueckner orbitals and/or generalized natural orbitals is elaborated.     

On the use of frozen orbitals in molecular orbital cluster calculations: Cl on Si(111)

By using the corresponding orbital transformation a quantitative criterion is found for the MOs that can be kept fixed during calculations using a perturbative SCF procedure. Results obtained keeping fixed a subset of molecular orbitals are quantitatively in agreement with those obtained by using the canonical SCF procedure but with a considerable saving of computer time in the SCF step.     

Extremely localized nonorthogonal orbitals by the pairing theorem

Using the concepts of L?wdin pairing theorem, a method is developed to calculate extremely localized, but nonorthogonal, sets of molecular orbitals and their strictly localized counterparts. The method is very suitable to study to what extent a given model of bonding in a given molecule can be considered adequate from the point of view of the actual LCAO-MO (Hartree Fock or DFT) wave function and is expected to be useful for doing local approximations of electron correlation.     

Bond strength and bond angles for hybrid orbitals composed of arbitrary sets of orbital angular momentum quantum number

According to the formulas obtained in the preceding paper, it may be used for all types of the hybridization with any set of azimuthal quantum numbers l, l = 0 through l = 5, and a complete theoretical data of bond angles and bond strengths are shown in this paper.