Generalising localisation schemes of orthogonal orbitals to the localisation of non-orthogonal orbitals

ABSTRACT

When a set of orthogonal orbitals is localised using orthogonal transformations, the orbital metric is conserved as the unit matrix during the localisation. In this paper we describe how non-orthogonal orbitals may be localised by requiring that the orbital metric is conserved. In particular, we demonstrate how exponential mappings may be used to parametrise orbital transformations such that the orbital metric is conserved. Using this parametrisation, the localisation of non-orthogonal orbitals becomes a generalisation of the localisation for orthogonal orbitals, where the conservation of a unit metric is replaced by the conservation of a non-unit metric. As a result, standard orbital localisation functions and optimisation algorithms that have been used for localising orthogonal orbitals may also be used to localise non-orthogonal orbitals. Numerical illustrations show that the thickness of the orbital tails may be reduced when a set of non-orthogonal orbitals is localised compared to the orthogonal counterpart, and further that the physical interpretation of occupied orbitals for representing chemical bonds is preserved.     

Comparative electronic structure of a lanthanide and actinide diatomic oxide: Nd versus U

Using a modified version of the Alchemy electronic structure code and relativistic pseudopotentials, the electronic structure of the ground and low lying excited states of UO, NdO, and NdO⁺ have been calculated at the Hartree—Fock (HF) and multiconfiguration self-consistent field (MCSCF) levels of theory. Including results from an earlier study of UO⁺ this provides the information for a comparative analysis of a lanthanide and an actinide diatomic oxide. UO and NdO are both described formally as M⁺²O^?2 and the cations as M⁺³O^?2, but the HF and MCSCF calculations show that these systems are considerably less ionic due to large charge back-transfer in the π orbitals. The electronic states putatively arise from the ligand field (oxygen anion) perturbed f⁴, sf³, df³, sdf², or s²f² states of M⁺² and f³, sf² or df² states of M⁺³. Molecular orbital results show a substantial stabilization of the sf³ or s²f² configurations relative to the f⁴ or df³ configurations that are the even or odd parity ground states in the M⁺² free ion. The compact f and d orbitals are more destabilized by the anion field than the diffuse s orbital. The ground states of the neutral species are dominated by orbitals arising from the M⁺²sf³ term, and all the potential energy curves arising from this configuration are similar, which allows an estimate of the vibrational frequencies for UO and NdO of 862 cm^?1 and 836 cm^?1, respectively. For NdO⁺ and UO⁺ the excitation energies for the Ω states were calculated with a valence configuration interaction method using ab initio effective spin—orbit operators to couple the molecular orbital configurations. The results for NdO⁺ are very comparable with the results for UO⁺, and show the vibrational and electronic states to be interleaved.     

One-dimensional configuration sums in vertex models and affine Lie algebra characters

We study the local state probabilities of the vertex models in the face formulation associated with the simple Lie algebras X _n =A _n, B _n, C _n, D _n. The corner transfer matrix method expresses them in terms of one-dimensional configuration sums. We show that the latter are the string functions of X _n ⁽¹⁾ modules. We also present similar results for the restricted face models of types B _n ⁽¹⁾, C _n ⁽¹⁾, D _n ⁽¹⁾.     

Ab initio SCF CI study of the electronic spectra of nitroethylene

Configuration interaction studies of ground, n→π* and π→π* electronically excited states are reported for nitroethylene in its ground-state equilibrium geometry. In case of the n→π* transitions, the two symmetry combinations of oxygen lone pair orbitals are found to give almost the same transition energies of 4·42 eV and 4·48 eV. These are in good agreement with the experimental transition energy of 4·20 eV. The calculated transition energy to the lowest π→π* excited ¹ A′ state of 5·48 eV is also in good agreement with the reported experimental value of 5·12 eV. Numerous other singlet states as well as the triplet states have also been calculated.     

Configuration interaction of Er3+ with a charge transfer configuration in the elpasolite compound Cs2NaErCl6

The discrepancies of the crystal-field analysis in Cs₂NaErCl₆ are lowered by letting the 4f¹¹ configuration interact with a charge transfer configuration enclosing a p electron. 75 levels with a total degeneracy equal to 130 were fitted with a mean deviation of 10.5 cm^?1, compared with 21.4 cm^?1 in the standard model. It is found that: (i) the p electrons involved in the process originate most probably from the six Cl^? ligands; (ii) 14% of the 3p⁶ orbitals are projected onto the central ion coordinate system as an unique orbital with p angular character; (iii) this projected orbital can be coupled with 4fⁿ configurations and handled by tensorial methods; and (iv) the most probable mechanism responsible for the improvement of the crystal-field analysis is a 4f¹¹ 3p⁶-4f¹²3p⁵ interaction. The first configuration represents Er³⁺ surrounded by six Cl^? ligands, whereas in the second one, the rare-earth ion and the ligands have gained and lost one electron respectively. The crystal-field analysis enables the estimation of the location of the charge transfer band.     

Simple one-centre calculation of breathing force constants and equilibrium internuclear distances for NH3, H2O and HF

Total molecular energies, breathing force constants and equilibrium internuclear distances are determined for the NH₃, H₂O and HF molecules using a single determinant wave function of the simplified one-centre form s ² s′² p _x ² p _y ² p _z ², where each of the five Slater orbitals, s, s′, p _x , p _y , p _z , is characterized by an effective orbital exponent ζ and an effective principal quantum number n. Five different calculations are performed for each molecule: in (a) the orbitals are centred on the heavy atom, the parameters n are taken to be integral, and the orbitals p _x , p _y and p _z are given the same ζ values (the spherical approximation); in (b) the orbitals p _x , p _y and p _z are allowed to have different ζ values (the ellipsoidal approximation); (c) and (d) are the same as (a) and (b) except that non-integral n values are allowed; (e) is the same as (d) except that the orbital centre also is taken to be a variational parameter. The values obtained are compared with experimental values (the agreement is surprisingly good) and with values from previous one-centre wave functions. The electronic densities for the various spherical approximations are tabulated.     

The LEDO expansion for diatomic overlap densities: application to the density functional theory of molecules

The limited expansion of differential overlap (LEDO) approach for the expansion of diatomic overlap densities in terms of mono-centre densities is discussed in the context of density functional theory (DFT). It is shown that it leads to a particularly simple construction scheme for major parts of the secular matrix, i.e. the electron-electron interaction and the exchange-correlation potential: using the LEDO expansion coefficients, matrix elements between atomic orbitals located on different centres can be expressed in terms of the corresponding mono-centre elements, thus allowing the reduction of three-centre and four-centre integrals to two-centre integrals. This results in the first DFT method with formal N ² scaling for the construction of the secular matrix, with N being the dimension of the atomic orbital (AO) basis set. Test calculations show that numerical agreement with the results of conventional DFT calculations is excellent.     

Orbital optimisation in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenes

ABSTRACT

We describe the implementation of orbital optimisation for the models in the perfect pairing hierarchy. Orbital optimisation, which is generally necessary to obtain reliable results, is pursued at perfect pairing (PP) and perfect quadruples (PQ) levels of theory for applications on linear polyacenes, which are believed to exhibit strong correlation in the π space. While local minima and σ-π symmetry breaking solutions were found for PP orbitals, no such problems were encountered for PQ orbitals. The PQ orbitals are used for single-point calculations at PP, PQ and perfect hextuples (PH) levels of theory, both only in the π subspace, as well as in the full σπ valence space. It is numerically demonstrated that the inclusion of single excitations is necessary also when optimised orbitals are used. PH is found to yield good agreement with previously published density matrix renormalisation group data in the π space, capturing over 95% of the correlation energy. Full-valence calculations made possible by our novel, efficient code reveal that strong correlations are weaker when larger basis sets or active spaces are employed than in previous calculations. The largest full-valence PH calculations presented correspond to a (192e,192o) problem.     

Lappo-Danilevsky hyperlogarithm and period of Calabi-Yau manifold

We discuss the period of complex structure modulus space for a degree-(n+1) Calabi-Yau hypersurface embedded in complexn-dimensional projective spaceCP ⁿ , in view of a higher logarithm expansion. The main result is that the periods have Lappo-Danilevsky-type hyperlogarithmic structure.     

Classical trajectory calculations for state-resolved Penning ionisation reactions of polycyclic aromatic hydrocarbon C10H8 in collision with He*(23S)

ABSTRACT

The reaction dynamics of Penning ionisation of a polycyclic aromatic hydrocarbon (PAH), naphthalene C₁₀H₈, in collision with the metastable He*(2³S) atom is studied by classical trajectory calculations using an approximate interaction potential energy surface between He* and the molecule, which is constructed based on ab initio calculations for the isovalent Li?+?C₁₀H₈ system. The ionisation width (rate) around the molecular surface are obtained from overlap integrals of the He 1s orbital and the molecular orbital. The calculated collision energy dependences of partial Penning ionisation cross sections (CEDPICS) in the range 50–500?meV at 300?K have reproduced the experimental results semi-quantitatively. The opacity functions, which represent the reaction probability with respect to the impact parameter b, are discussed in connection with collision energy, interaction with He* and the exterior electron density of molecular orbitals. They indicate that the collisional ionisations of C₁₀H₈ can be classified into three types: π electron ionisations with negative collision energy dependences which are predominantly determined by attractive interaction with He*; σ orbitals ionisations of the hardcore type; σ orbital ionisations which reflect interaction potentials around CH bonds. The critical impact parameters b_c become larger with increasing collision energy due to the centrifugal barrier.     

Unified treatment of one-range addition theorems for integer and non-integer n-STO, -GTO and -generalized exponential type orbitals with hyperbolic cosine in position, momentum and four-dimensional spaces

Simpler formulas are derived for one-range addition theorems for the integer and noninteger n generalized exponential type orbitals, momentum space orbitals, and hyperspherical harmonics with hyperbolic cosine (GETO HC, GMSO HC, and GHSH HC) in position, momentum and four-dimensional spaces, respectively. The final results are expressed in terms of one-range addition theorems of complete orthonormal sets of ψα -exponential type orbitals, α - momentum space orbitals and z α -hyperspherical harmonics. We notice that the one-range addition theorems for integer and noninteger n-Slater type orbitals and Gaussian type orbitals in position, momentum and four dimensional spaces are special cases of GETO HC, GMSO HC, and GHSH HC. The theorems presented can be useful in the accurate study of the electronic structure of atomic and molecular systems.     

Effect of Cr3+ ions on the magnetic moment of ferrites of the system CuFe2−x CrxO4

The dependence of the magnetic moment n _0exp of samples of the system CuFe_2−x Cr_xO₄ (x=0.0, 0.2, 0.3, 1.0, 1.4, 1.6, and 2.0) on their Cr³⁺ content is examined here for the first time. It is found that the experimental values of the magnetic moment n _0exp are much smaller than the values calculated from the cation distribution obtained previously (n _{0 theor}). It is suggested that this relationship (n _{0 theor}>n _{0 exp}) is due to a decrease in the magnetic moments of the Cr³⁺ ions resulting both from pairing of the t _2g orbitals of these cations in the octahedral sublattice and from a transfer of spin density from the ligands to the e _g orbitals of these ions. For compositions with x>1.0, the noncollinear magnetic structure also leads to an increase in the difference between n _{0 theor} and n _0exp. Fiz. Tverd. Tela (St. Petersburg) 40, 99–100 (January 1998)     

Orbital ordering in CuGeO3

The combined effect of the ligand crystal field and the exchange interaction on the Cu²⁺ cation in CuGeO₃ is examined. It is shown that, if the magnitude of the exchange interaction exceeds the splitting of the energy levels of the and d _xy orbitals, then an alternate filling of the d orbitals along a chain (orbital ordering) is possible. This effect creates an antiferromagnetic interaction between Cu²⁺ pairs in 90° exchange and a doubling of the lattice period. A Jahn-Teller pseudoeffect causes singletization of the ground state of the antiferromagnetic chain of Cu²⁺ spins. Fiz. Tverd. Tela (St. Petersburg) 40, 1686–1692 (September 1998)     

A theoretical investigation of the interaction between fluorinated dimethyl ethers and molecular chlorine

Halogen bonds have received a great deal of attention in recent years. In this work, the interaction between fluorinated dimethyl ethers (n_F = 0–4) and molecular chlorine has been investigated by the theoretical methods. The two molecules are bonded together by an O···Cl?Cl halogen bond and the interaction energies calculated at the MP2/aug-cc-pVDZ level range between ?15.5 (n_F = 0) and ?6.1 (n_F = 4) kJ mol^?1. The correlations between interaction energies and proton affinity or ionisation potential of the ethers are discussed. The interaction between the molecules results in a small contraction of the CH bond of ethers and an elongation of the Cl?Cl bond. The data are analysed by a natural bond orbital analysis carried out at the wB97XD/6-311++G(d,p) level. The charge transfer from the ethers to Cl₂ is weak, ranges between 0.044 and 0.008 e and occurs mainly to the external Cl atom. The elongation of the Cl?Cl bond is related to the occupation of the σ^*(Cl?Cl) orbital and to the intermolecular hyperconjugation interaction between LP(O) and σ^*(Cl?Cl) orbitals. The interaction between the ethers and chlorine induces an enhancement of the infrared intensity and Raman scattering activity of the ν(Cl?Cl) vibration.     

Ab initio SCF CI study of the electronic spectrum of s-tetrazine

Configuration interaction (CI) studies of ground, n→ π* and π→ π* electronically excited states are reported for s-tetrazine. The first n→ π* singlet excited state (¹ B _3u ), which is responsible for the purple-red colour of the molecule, is calculated at 2·80 eV, compared to the experimental transition energy of 2·22–2·70 eV. The singlet-triplet split of the first n→ π* states (¹ B _3u and ³ B _3u states) is calculated to be 0·76 eV.

The interaction of nitrogen lone pair orbitals (n-orbitals) is studied in terms of the ordering of the n π* excited states and found that the SCF orbital ordering is qualitatively in accord with the ordering of the n π* excited states in the CI level.

The first π→ π* excited state (¹ B _2u ) is calculated at 5·99 eV, slightly above the observed range of absorption. Numerous other high-lying singlet states as well as the triplet states have been calculated and they are used to verify several proposals relating to the excited state dynamics in the photo-physical studies of s-tetrazine.     

Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system ?^{( n )} of observables “up to n loops”, where ?⁽⁰⁾ is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. Received: 9 February 2000 / Accepted: 21 March 2000     

Approximate SCF calculations on the [Mn(H2O)6]2+ complex with a minimal basis set

In the restricted Hartree-Fock scheme approximate SCF-LCAO calculations have been performed for the [Mn(H₂O)]₆ ²⁺ complex using a minimal basis set consisting of nine Slater-type orbitals of the manganese ion and four Slater-type orbitals of the water molecule. The 1s, 2s and 2p orbitals of the manganese ion and the 1s orbital of the oxygen atom are treated as frozen core orbitals. In evaluating the different parts of the Hartree-Fock operators we used a two-centre approximation for the multicentre integrals. A new aspect of the calculations is the use of the Hartree-Fock orbital energies of the free water molecule as a first approximation for the corresponding orbital energies of the complex. The calculations have been done for the ground states and six excited states of the complex with symmetry T _h and also for the ground states of two distorted complexes. From the resulting eigenvectors we calculated the hyperfine interaction of the valence electrons of the central ion with the protons of the water molecules for three different geometries. The excited states give two different ways of finding values of 10 Dq and the Racah parameters B and C. The results are encouraging.     

Atomic orbitals with angularly dependent Z eff to be used in molecular orbitals calculations

A new type of atomic orbital is proposed for use in molecular problems of quantum mechanics. This orbital is formally similar to Slater's orbitals but is characterized by an effective nuclear charge Z _eff which is an explicit function of the angles ? and φ. The fundamental state of the molecule H₂ ⁺ is studied using such an orbital.     

On the strength of electron correlations in high-T c superconducting copper oxides

We analyze the strength of electronic correlations in the half-filled antibonding Cu–O orbitals of high-T _c superconducting copper oxides by considering a Cu₁₂O ₁₇ ^n– cluster withn=8 or 10, respectively. The correlated ground state is calculated by the method of the local approach (LA) in a version which allows the treatment of stronger correlated electrons. As mean-field basis a semiempirical Hamiltonian of the ZDO (zero differential overlap) type has been adopted. It is found that the correlations are particularly strong in the Cu 3d _x ²–y² orbitals. The nonintegral orbital occupation allows for valence fluctuations between Cu⁺ and Cu²⁺ in spite of the remarkable correlations. According to the present model excess holes are located at the oxygen sites. The theoretical findings are compared with the results of spectroscopic investigations. The present electronic-structure analysis allows for a straighforward rationalization of previous experimental measurements. The strong connection between the importance of electronic correlations and the symmetry properties of the electronic wave function is emphasized.