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.     

Extended Hartree-Fock (EHF) theory in chemical reactions

The roles of orbital, spin and permutation symmetries in the extended Hartree-Fock (EHF) wavefunction are investigated in relation to the applications of group theory to chemical reactions. The utility of the magnetically ordered set for an extended HF calculation is pointed out. The relative stabilities among linear Hückel and Möbius three-center three-electron (3,3) systems are investigated by the generalized Hartree-Fock (GHF) and EHF methods in order to confirm the reliability of the valence-bond (VB) selection rule for free radical reactions.     

Natural Bond Orbital Study on the Nature of Binding in the H2 Molecule

We use the natural bond orbital (NBO) method to decompose a MO wavefunction into the intuitive valence bond (VB) structures. At least two natural orbital type MO are required to describe the essential binding of the H₂ molecule at all inter nuclear distances. At first the MO wavefunction is transformed into an unrestricted Hartree-Fock wave-function consisted of non-orthogonal localized orbitals u' and v', and then the NBO method is used to decompose u' and v' into the physical meaningful orthogonal localized orbitals. Our results show that the orbitals u' and v' are decomposed into an atomic and an overlap parts. The latter part gives rise to the conventional ionic structure in the VB picture.     

Spin–orbit coupling of spin‐frustrated systems

We have investigated the effects of spin–orbit (SO) interactions on noncollinear molecular magnetism by combining the classical Dzyaloshinsky–Moriya (DM) model and ab initio generalized spin orbital (GSO) method. We have derived an estimation scheme of the magnetic anisotropy energy (MAE) and the Dzyaloshinsky vector based on the SO first‐order perturbation theory (SOPT1) for GSO Hartree–Fock (GHF) solutions. We found that the fundamental results of GHF‐SOPT1 method can be reproduced by diagonalizing the core Hamiltonian plus SO terms, and that the spin topologies of odd‐ring systems can be determined by the topological indices of the singly occupied molecular orbitals. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Electron correlation effects on the shape of the Kohn–Sham molecular orbital

Even systems in which strong electron correlation effects are present, such as the large near-degeneracy correlation in a dissociating electron pair bond exemplified by stretched H₂, are represented in the Kohn–Sham (KS) model of non-interacting electrons by a determinantal wavefunction built from the KS molecular orbitals. As a contribution to the discussion on the status and meaning of the KS orbitals we investigate, for the prototype system of H₂ at large bond distance, and also for a one-dimensional molecular model, how the electron correlation effects show up in the shape of the KS σ _g orbital. KS orbitals φ^HL and φ^FCI obtained from the correlated Heitler-London and full configuration interaction wavefunctions are compared to the orbital φ^LCAO, the traditional linear combination of atomic orbitals (LCAO) form of the (approximate) Hartree-Fock orbital. Electron correlation manifests itself in an essentially non-LCAO structure of the KS orbitals φ^HL and φ^FCI around the bond midpoint, which shows up particularly clearly in the Laplacian of the KS orbital. There are corresponding features in the kinetic energy density t _s of the KS system (a well around the bond midpoint) and in the one-electron KS potential v _s (a peak). The KS features are lacking in the Hartree-Fock orbital, in a minimal LCAO approximation as well as in the exact one. Received: 11 December 1996 / Accepted: 10 January 1997     

Molecular orbital studies of dinitrogen tetroxide and related molecules

Hartree-Fock LCAO MO calculations for N₂O₄ have been performed in a basis of symmetry orbitals formed from a minimal Slater basis set. Effects of rounding and truncation errors were minimized by the use of the symmetry basis, which also allowed the order or tilling of molecular orbitals to be specified independently of orbital energies. Convergence difficulties were overcome by combined use of the conjugate gradients method and Roothaan's iterative procedure; the method of steepest descents was less effective than either of these. Multicentre ‘non-NDDO’ two-electron integrals were evaluated by the gaussian expansion technique. The wavefunction obtained for the lowest state is NN antibonding, largely as a result of the filling of the 6b_1u antibonding sigma orbital in preference to the 6a_g bonding sigma orbital. There is only a small amount of NN pi-bonding. A bond energy analysis shows that the lowest state is markedly stabilized by NNO three-centre interactions.     

Picture change error correction in the radial distributions of canonical orbital densities and total electron density of radon atom: the effect of the size of nucleus and the basis set limit

The 2nd order Douglas-Kroll-Hess (DKH2) and the Infinite Order Two Component (IOTC) radial distributions of electron density of canonical Hartree-Fock (HF) orbitals of radon atom are presented. Furthermore, the total electron density is revisited. The picture change error (PCE) correction is investigated by analytical means. The point charge model of nucleus and the Gaussian nucleus model are employed. The basis set is extrapolated by means of including tight s and also p Gaussians within the original triple zeta basis set. It is found that the DKH1 PCE corrected DKH2 total electron and s orbital contact densities are negative for the point charge model of nucleus if tight enough s Gaussians are included in the basis set. It is shown that this failure is caused due to the missing terms of the second order Douglas-Kroll transformation for the DKH2 electron density. PCE is found the most striking in the DKH2/IOTC electron density of s orbitals close to the nucleus. The radial distributions of the 2-component p _1/2 orbital densities are considerably affected by PCE at the nucleus as well. Furthermore, the PCE corrected DKH2/IOTC scalar p orbital densities have a non-zero value of electron density at nucleus and can be considered as an spin-orbit (SO) average of the p _1/2 and p _3/2 orbitals. The d and f orbitals are affected by PCE in the vicinity of the nucleus only little. The PCE corrected DKH2 and IOTC radial distributions of orbital densities are nodeless, which is completely in agreement with the radial distribution of the analytic or numeric DCH orbital densities.     

Open-shell localized Hartree-Fock method based on the generalized adiabatic connection Kohn-Sham formalism for a self-consistent treatment of excited states

An effective exact-exchange Kohn-Sham approach for the treatment of excited electronic states, the generalized adiabatic connection open-shell localized Hartree-Fock (GAC-OSLHF) method is presented. The GAC-OSLHF method is based on the generalized adiabatic connection Kohn-Sham formalism and therefore capable of treating excited electronic states, which are not the energetically lowest of their symmetry. The method is self-interaction free and allows for a fully self-consistent computation of excited valence as well as Rydberg states. Results for atoms and small- and medium-size molecules are presented and compared to restricted open-shell Hartree-Fock (ROHF) and time-dependent density-functional results as well as to experimental data. While GAC-OSLHF and ROHF results are quite close to each other, the GAC-OSLHF method shows a much better convergence behavior. Moreover, the GAC-OSLHF method as a Kohn-Sham method, in contrast to the ROHF approach, represents a framework which allows also for a treatment of correlation besides an exchange by appropriate functionals. In contrast to the common time-dependent density-functional methods, the GAC-OSLHF approach is capable of treating doubly or multiply excited states and can be easily applied to molecules with an open-shell ground state. On the nodal planes of the energetically highest occupied orbital, the local multiplicative GAC-OSLHF exchange potential asymptotically approaches a different, i.e., nonzero, value than in other regions, an asymptotic behavior which is known from exact Kohn-Sham exchange potentials of ground states of molecules.     

A time-dependent molecular orbital approach to electron transfer in ion–metal surface collisions

A time-dependent molecular orbital method has been developed to study charge transfer in collisions of ions with metal surfaces at energies between 1 and 100 au. A set of localized basis functions consisting of generalized Wannier functions for the surface and s- and p-atomic functions for the ion, is used to separate the system into primary and secondary regions. An effective Hamiltonian and time-dependent equations for the electron density matrix are obtained in the primary region, where most charge transfer occurs. The equations for the electron density matrix are solved with a linearization scheme. The method is suitable to study atomic orbital orientation for collisions of ions and surfaces. A model calculation for Na⁺ + W(110) collisions with a prescribed trajectory is presented. The interaction potentials between the W(110) surface and Na⁺ 3s and 3p orbitals are calculated from Na⁺ pseudopotentials. Results show that the yield of neutralized atoms in 3p states changes as the collision energy is lowered.     

Studies of unoccupied orbitals of BF3 and BCL3 by electron transmission spectroscopy and multiple scattering Xα calculations

Electron transmission spectroscopy (ETS ) and bound-state and continuum multiple-scattering Xα (MS -Xα) calculations are employed to characterize the unoccupied a′₂, a′₁ and e′ orbitals of BF₃ and BCl₃. The a′₂ orbital of BF₃, which produces a peak about 7 eV below threshold in the x-ray absorption spectrum (XAS ), generates a scattering resonance at 3.5 eV in ETS. Similarly, the e′ orbital that lies about 2 eV above threshold in XAS occurs about 13?16 eV above threshold in ETS . Dissociation of F^? from BF₃ due to electron attachment is attributed to a core-excited shape resonance involving an e″ → a″₂ excitation and electron capture into the a′₂ orbital. In BCl₃ all the unoccupied orbitals lie at lower energy than in BF₃ and are closely spaced, making definitive spectral assignments difficult. Both Hartree-Fock (HF ) and MS -Xα methods apparently underestimate the stability of the unoccupied e′ orbital of BCl₃. Vibronic coupling due to out-of-plane bending may significantly affect the spectral intensities. Feshbach resonances are observed for BCl₃ at energies close to those observed in the vacum-UV absorption spectrum. The lower energies of the unoccupied orbitals of BCl₃ are consistent with their stronger bonding to nucleophiles.     

Natural spin orbital analysis of diatomic molecular wave functions in terms of generalized diatomic orbitals I. Outline of the method. Results for the ground state of H2

The method of linear combinations of generalized diatomic orbitals (LCGDO) is combined with the method of configuration interaction (CI). CI wave functions obtained in this way are finally submitted to a natural spin orbital analysis; the resulting natural spin orbitals are expansions in terms of generalized diatomic orbitals.For the ground state of H₂, a one-determinantal-approach with a single completely optimized one-electron basis function nearly reproduces the Hartree-Fock-result. The two-determinantal approach with two optimized basis functions of type _g and _u nearly gives the optimized double configuration SCF result.     

On the choice of active space orbitals in MCSCF calculations

We describe a procedure which may be used to aid selection of the active space in multiconfigurational self-consistent field (MCSCF) calculations for general chemical systems. Starting from a restricted Hartree-Fock calculation, we define a hierarchy of interacting virtual orbitals for every occupied orbital. The most strongly interacting orbitals are then taken to constitute the active space in a configuration interaction (CI) calculation. The natural orbital occupation numbers obtained from the CI calculation are then used to choose the active space to be used in a subsequent MCSCF calculation. We illustrate our method on a number of systems (Li₂, B₂, C₂, carbonyl oxide and the transition state for oxidation of H₂S by dioxirane). In all these cases, ‘intuitive’ active spaces are inadequate, as are active spaces derived from the natural orbitals of unrestricted Hartree-Fock calculations.     

Quantum-chemical investigation of nature of neutral intermolecular hydrogen bond in complex H3PO...HF

Results are presented from modeling the complex H₃PO...HF by the ab initio Hartree-Fock-Roothaan method. A generalized method is proposed for the quantitative fragmentary analysis of molecular orbitals (MOs). It is shown that when the complex is formed, the chemical bond between the phosphine oxide and the HF molecule is formed by means of a shift of the electron pair of the -bond of the HF to a 2p orbital of the O atom and the formation of a bonding three-center MO localized on the O, H, and F atoms, and also through a shift of an electron pair from the O atom to a 2p orbital of the F atom and the formation of a nonbonding MO localized on the O and F atoms.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 28, No. 1, pp. 38–41, January–February, 1992.     

Parallel, linear-scaling building-block and embedding method based on localized orbitals and orbital-specific basis sets

We present a linear scaling method for the energy minimization step of semiempirical and first-principles Hartree-Fock and Kohn-Sham calculations. It is based on the self-consistent calculation of the optimum localized orbitals of any localization method of choice and on the use of orbital-specific basis sets. The full set of localized orbitals of a large molecule is seen as an orbital mosaic where each tessera is made of only a few of them. The orbital tesserae are computed out of a set of embedded cluster pseudoeigenvalue coupled equations which are solved in a building-block self-consistent fashion. In each iteration, the embedded cluster equations are solved independently of each other and, as a result, the method is parallel at a high level of the calculation. In addition to full system calculations, the method enables to perform simpler, much less demanding embedded cluster calculations, where only a fraction of the localized molecular orbitals are variational while the rest is frozen, taking advantage of the transferability of the localized orbitals of a given localization method between similar molecules. Monitoring single point energy calculations of large poly(ethylene oxide) molecules and three dimensional carbon monoxide clusters using an extended Huckel Hamiltonian are presented.     

A numerically stable procedure for calculating Møller-Plesset energy derivatives,derived using the theory of Lagrangians

Summary When Møller-Plesset energy derivatives are determined in the canonical Hartree-Fock basis, singularities or instabilities may arise due to degeneracies among the occupied or unoccupied orbitals. If a non-canonical basis is used these singularities disappear. Numerically stable expressions are presented for the molecular gradient and Hessian of the second-order Møller-Plesset energy, obtained by differentiating a fully variational Lagrangian of the energy constructed in a non-canonical representation. By using a non-canonical representation, singularities and instabilities are avoided, and the variational property of the Lagrangian ensures that Wigner's 2n + 1 rule is satisfied for the orbital derivatives and that the multipliers satisfy the stronger 2n + 2 rule. It is shown that the most expensive step in the calculation of the Hessian scales as Mn ₄o, where M is the number of independent Cartesian distortions, n the total number of orbitals, and o the number of occupied orbitals.     

The calculation of ionization energies by perturbation,configuration interaction and approximate coupled pair techniques and comparisons with green's function methods for Ne,H2O and N2

The vertical valence ionization potentials of Ne, H₂O and N₂ have been calculated by Rayleigh-Schrödinger perturbation and configuration interaction methods. The calculations were carried out in the space of a single determinant reference state and its single and double excitations, using both the N and N - 1 electron Hartree-Fock orbitals as hole/particle bases. The perturbation series for the ion state were generally found to converge fairly slowly in the N electron Hartree-Fock (frozen) orbital basis, but considerably faster in the appropriate N - 1 electron RHF (relaxed) orbital basis. In certain cases, however, due to near-degeneracy effects, partial, and even complete, breakdown of the (non-degenerate) perturbation treatment was observed. The effects of higher excitations on the ionization potentials were estimated by the approximate coupled pair techniques CPA′ and CPA″ as well as by a Davidson type correction formula. The final, fully converged CPA″ results are generally in good agreement with those from PNO-CEPA and Green's function calculations as well as experiment.     

Away from generalized gradient approximation: orbital-dependent exchange-correlation functionals

The local-density approximation of density functional theory (DFT) is remarkably accurate, for instance, for geometries and frequencies, and the generalized gradient approximations have also made bond energies quite reliable. Sometimes, however, one meets with failure in individual cases. One of the possible routes towards better functionals would be the incorporation of orbital dependence (which is an implicit density dependency) in the functionals. We discuss this approach both for energies and for response properties. One possibility is the use of the Hartree-Fock-type exchange energy expression as orbital-dependent functional. We will argue that in spite of the increasing popularity of this approach, it does not offer any advantage over Hartree-Fock for energies. We will advocate not to apply the separation of exchange and correlation, which is so ingrained in quantum chemistry, but to model both simultaneously. For response properties the energies and shapes of the virtual orbitals are crucial. We will discuss the benefits that Kohn-Sham potentials can offer which are derived from either an orbital-dependent energy functional, including the exact-exchange functional, or which can be obtained directly as orbital-dependent functional. We highlight the similarity of the Hartree-Fock and Kohn-Sham occupied orbitals and orbital energies, and the essentially different meanings the virtual orbitals and orbital energies have in these two models. We will show that these differences are beneficial for DFT in the case of localized excitations (in a small molecule or in a fragment), but are detrimental for charge-transfer excitations. Again, orbital dependency, in this case in the exchange-correlation kernel, offers a solution.