A generalized formula for the energies of alternant molecular orbitals. I. Homonuclear molecules

Using the method of alternant molecular orbitals (AMO ) it is shown that the energies of AMO 's (E_k), for any alternant homonuclear molecule having a singlet ground state, are connected with the energies of the MO 's (e_k) obtained by the conventional Hartree–Fock (HF ) method by the formula \documentclass{article}\pagestyle{empty}\begin{document}$ E_{k\alpha (\beta )} = \pm \sqrt {\Delta ^2 + e_k ^2 } $\end{document}, where Δ is the correlation correction. The formula is applicable in the semiempirical LCAO form used in the Pariser–Parr–Pople theory, by Hubbard's approximation of γ integrals.     

A generalized formula for the energies of alternant molecular orbitals. II. Heteronuclear molecules

Using the method of alternant molecular orbitals (AMO ), it is shown that the energies of AMOS (E_kσ) for an arbitrary heteronuclear alternant system, having a singlet ground state, are connected with the energies of MOS (e_{k(k )}) obtained by means of the conventional Hartree–Fock (HF ) method (SCF -LCAO -MO -PPP ) via the formula: In the general case, the determination of the correlation corrections δ_i,kσ is connected with the solving of a complicated system of integral equations, which is considerably simplified if the Hubbard approximation is accepted for the electron interaction. The energy spectrum of a chain with two atoms in the elementary cell (AB)_n is considered as an example. It is shown that if nontrivial solutions exist (δ_i,kσ ≠ 0), the correlation correction for AMOS of different spin are different (δ_i,kσ ≠ δ_i,kβ), from which it follows, that the width of the energy gap ΔE_∞ for AMOS with different spin is different: ΔE_∞,α ≠ ΔE_∞,β.     

Localized molecular orbitals for polyatomic molecules

Molecular wavefunctions have been generated by the PRDDO (Partial Retention of Diatomic Differential Overlap) method for the monocyclic aromatic rings containing six π-electrons (C₄H ₄ ^?2 , C₅H ₅ ^? , C₆H₆, C₇H ₇ ⁺ , and C₈H ₈ ⁺² ) and ten π-electron species (C₈H ₈ ^?2 , C₉H ₉ ^? , C₁₀H₁₀). The eigenvalue spectra of the canonical molecular orbitals are presented. Localized molecular orbitals (LMO's) generated using the Boys criterion are reported for localizations involving all occupied molecular orbitals (complete localizations) and localizations of the π orbitals only. We find evidence for σ-π separation in the complete localizations for some of these molecules even though the Boys criterion is often biased against such results. We demonstrate for C₆H₆ and find for the other molecules that the π-orbital localizations are indeterminate (i.e. there are an infinite number of equally satisfactory LMO structures between two limiting extremes). This result may be viewed as a corollary of Hückel's (4n+2) rule for aromaticity.     

Localized molecular orbitals for aromatic molecules

Localized molecular orbitals are calculated using the method of Boys for the aromatic molecules C₆H₆, C₆H₅X and the p-, m-, and o-forms of C₆H₄XY, where X,Y = CN,OH,F. The calculations are performed both with and without the constraint of σ, π-separation in the localization. The localized π-orbitals are multicenter bonds. If the σ- and π-orbitals are localized together, two different structures are found, Kekulé-type structures and structures with a set of six two center and a set of three three center bonds. The C-X bond turns out to be a single bond if X = CN and a double bond, if X = OH or F.     

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.     

On the use of localized orbitals and restricted alternant orbitals in chemical systems

An analysis of the transformation of localized orbitals into restricted alternant orbitals is proposed. This approach has the advantage of expressing the wave-function in an orbital product while some electron correlation is introduced permitting the study of dissociation reactions. All applications of the orbital technique may be made as easily as with RHF, but with the additional possibility of studying chemical radicals. Some illustrations of this fact are shown for the molecules HF, H₂O, NH₃, CH₄, C₂H₆ and for the dissociation reactions of CH₄ and C₂H₆ generating CH₃ radicals.     

Spin-Free configuration-interaction study of molecules using single-parameter alternat molecular orbitals

A procedure is outlined for a programmable spin-free configuration-interaction (CI ) study in molecules using single-parameter alternant molecular orbitals for generating various configurations. The configurations were chosen to form bases for the irreducible representation {2^N/2–2, 1^2S} of the general linear group GL(n). Using a transformation to biorthogonal space the CI matrix elements of a spin-free Hamiltonian were generated. The procedure has been used to obtain the π-electron energies for the ^3,1A_g and ^3,1B_u states of cis- and trans-butadiene.     

Dispersion energy evaluated by using locally projected occupied and excited molecular orbitals for molecular interaction

The dispersion terms are evaluated with the perturbation theory based on the locally projected molecular orbitals. A series of model systems, including some of the S22 set, is examined, and the calculated binding energies are compared with the published results. The basis set dependence is also examined. The dispersion energy correction is evaluated by taking into account the double excitations only of the dispersion type electron configurations and is added to the 3rd order single excitation perturbation energy, which is a good approximation to the counterpoise (CP) corrected Hartree-Fock (HF) binding energy. The procedure is the approximate "CP corrected HF + D" method. It ensures that the evaluated binding energy is approximately free of the basis set superposition error without the CP procedure. If the augmented basis functions are used, the evaluated binding energies for the predominantly dispersion-bound systems, such as rare gas dimers and halogen bonded clusters, agree with those of the reference calculations within 1 kcal mol(-1) (4 kJ mol(-1)). The limitation of the present method is also discussed.     

Reference program for molecular calculations with Slater-type orbitals

A program for computing all the integrals appearing in molecular calculation with Slater-type orbitals is reported. The program is mainly intended as a reference for testing and comparing other algorithms and techniques. An analysis of the performance of the program is presented, paying special attention to the computational cost and the accuracy of the results. Results are also compared with others obtained with Gaussian basis sets of similar quality. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1284–1293, 1998     

Variationally optimized basis orbitals for biological molecules

Numerical atomic basis orbitals are variationally optimized for biological molecules such as proteins, polysaccharides, and deoxyribonucleic acid within a density functional theory. Based on a statistical treatment of results of a fully variational optimization of basis orbitals (full optimized basis orbitals) for 43 biological model molecules, simple sets of preoptimized basis orbitals classified under the local chemical environment (simple preoptimized basis orbitals) are constructed for hydrogen, carbon, nitrogen, oxygen, phosphorous, and sulfur atoms, each of which contains double valence plus polarization basis function. For a wide variety of molecules we show that the simple preoptimized orbitals provide well convergent energy and physical quantities comparable to those calculated by the full optimized orbitals, which demonstrates that the simple preoptimized orbitals possess substantial transferability for biological molecules.     

Localized orbitals for the oxygen and nitric oxide molecules

Localized orbitals are derived for the ground states of the oxygen and nitric oxide molecules by applying localization methods separately to the orbitals containing electrons of alpha- and betaspin. Both the intrinsic energy localization and the uniform localization methods are used. The resulting localized orbitals are in good agreement with the formulae suggested by Linnett.

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Zusammenfassung Für die Grundzustände der Moleküle O₂ und NO werden lokalisierte Orbitale abgeleitet, indem die Lokalisierungsmethoden getrennt für die Orbitale mit- und-Spin angewendet werden. Dabei werden sowohl die eigentlichen Energielokalisierungsmethoden als auch die gleichförmigen Lokalisierungsmethoden benutzt. Die erhaltenen lokalisierten Orbitale stimmen gut mit den von Linnett vorgeschlagenen Formeln überein.

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Résumé Obtention d'orbitales localisées pour l'état fondamental des molécules d'oxygène et d'oxyde nitreux par localisation indépendante des orbitales de spin et des orbitales de spin. Emploi simultané de la localisation selon l'énergie intrinsèque et de la localisation uniforme. Les orbitales localisées résultantes sont en bon accord avec les formules suggérées par Linnett.