Nonorthogonal Atomic Orbitals Research Articles

Chemical Bonding in the C2 Molecule

Bonding in the C2 molecule is investigated with CAS(8,8) wave functions using canonical MOs. In a subsequent step, orthogonal atomic orbitals are constructed by localizing the CASSCF MOs on the two carbon atoms with an orthogonal transformation. This orbital transformation causes an orthogonal transformation of the configuration state functions (CSF) spanning the function space of the singlet ground state of C2. Instead of CSFs built from canonical MOs, one obtains CSFs of orthogonal deformed atomic orbitals (AO). This approach resembles the orthogonal valence bond (OVB) methods’ CSFs, which are very different from the conventional VB, based on non-orthogonal AOs. To become used to the different argumentation, the bonding situations in ethane (single bond), ethene (double bond), and the nitrogen molecule (triple bond) are also studied. The complex bonding situation in C2 is caused by the possibility to excite an electron with a spin flip from the doubly occupied 2s AO into the 2p subshell, and the resulting high-spin 5Su state of the carbon atom allows for a better reduction of the Pauli repulsion. However, the electron structure around the equilibrium distance does not allow one to say that C2 in its ground state has a double, or triple, or even a quadruple bond.

Tight-binding calculations of optical matrix elements for conductivity using nonorthogonal atomic orbitals: Anomalous Hall conductivity in bcc Fe

We present a general formula for the tight-binding representation of momentum matrix elements needed for calculating the conductivity based on the Kubo-Greenwood formula using atomic orbitals, which are in general not orthogonal to other orbitals at different sites. In particular, the position matrix element is demonstrated to be important for delivering the exact momentum matrix element. This general formula, applicable to both orthonormal and non-orthonormal bases, solely needs the information of the position matrix elements and the ingredients that have already contained in the tight-binding representation. We then study the anomalous Hall conductivity in the standard example, ferromagnetic bcc Fe, by a first-principles tight-binding Hamiltonian. By assuming the commutation relation $\hat{\vec{p}} = (i m_e/\hbar) [\hat{H},\hat{\vec{r}}]$, the obtained frequency-dependent Hall conductivity is found to be in good agreement with existing theoretical and experimental results. Better agreement with experiments can be reached by introducing a reasonable bandwidth renormalization, evidencing the strong correlation among 3$d$ orbitals in bcc Fe. Since a tight-binding Hamiltonian can be straightforwardly obtained after finishing a first-principles calculation using atomic basis functions that are generated before the self-consistent calculation, the derived formula is particularly useful for those first-principles calculations.

Chemical bonding: the orthogonal valence-bond view.

Chemical bonding is the stabilization of a molecular system by charge- and spin-reorganization processes in chemical reactions. These processes are said to be local, because the number of atoms involved is very small. With multi-configurational self-consistent field (MCSCF) wave functions, these processes can be calculated, but the local information is hidden by the delocalized molecular orbitals (MO) used to construct the wave functions. The transformation of such wave functions into valence bond (VB) wave functions, which are based on localized orbitals, reveals the hidden information; this transformation is called a VB reading of MCSCF wave functions. The two-electron VB wave functions describing the Lewis electron pair that connects two atoms are frequently called covalent or neutral, suggesting that these wave functions describe an electronic situation where two electrons are never located at the same atom; such electronic situations and the wave functions describing them are called ionic. When the distance between two atoms decreases, however, every covalent VB wave function composed of non-orthogonal atomic orbitals changes its character from neutral to ionic. However, this change in the character of conventional VB wave functions is hidden by its mathematical form. Orthogonal VB wave functions composed of orthonormalized orbitals never change their character. When localized fragment orbitals are used instead of atomic orbitals, one can decide which local information is revealed and which remains hidden. In this paper, we analyze four chemical reactions by transforming the MCSCF wave functions into orthogonal VB wave functions; we show how the reactions are influenced by changing the atoms involved or by changing their local symmetry. Using orthogonal instead of non-orthogonal orbitals is not just a technical issue; it also changes the interpretation, revealing the properties of wave functions that remain otherwise undetected.

The orbital minimization method for electronic structure calculations with finite-range atomic basis sets

The implementation of the orbital minimization method (OMM) for solving the self-consistent Kohn–Sham (KS) problem for electronic structure calculations in a basis of non-orthogonal numerical atomic orbitals of finite-range is reported. We explore the possibilities for using the OMM as an exact cubic-scaling solver for the KS problem, and compare its performance with that of explicit diagonalization in realistic systems. We analyze the efficiency of the method depending on the choice of line search algorithm and on two free parameters, the scale of the kinetic energy preconditioning and the eigenspectrum shift. The results of several timing tests are then discussed, showing that the OMM can achieve a noticeable speedup with respect to diagonalization even for minimal basis sets for which the number of occupied eigenstates represents a significant fraction of the total basis size (>15%). We investigate the hard and soft parallel scaling of the method on multiple cores, finding a performance equal to or better than diagonalization depending on the details of the OMM implementation. Finally, we discuss the possibility of making use of the natural sparsity of the operator matrices for this type of basis, leading to a method that scales linearly with basis size.

Efficient and accurate linear algebraic methods for large-scale electronic structure calculations with nonorthogonal atomic orbitals

The need for large-scale electronic structure calculations arises recently in the field of material physics, and efficient and accurate algebraic methods for large simultaneous linear equations become greatly important. We investigate the generalized shifted conjugate orthogonal conjugate gradient method, the generalized Lanczos method, and the generalized Arnoldi method. They are the solver methods of large simultaneous linear equations of the one-electron Schr\"odinger equation and map the whole Hilbert space to a small subspace called the Krylov subspace. These methods are applied to systems of fcc Au with the NRL tight-binding Hamiltonian [F. Kirchhoff et al., Phys. Rev. B 63, 195101 (2001)]. We compare results by these methods and the exact calculation and show them to be equally accurate. The system size dependence of the CPU time is also discussed. The generalized Lanczos method and the generalized Arnoldi method are the most suitable for the large-scale molecular dynamics simulations from the viewpoint of CPU time and memory size.

Orthogonal natural atomic orbitals form an appropriate one‐electron basis for expanding CASSCF wave functions into localized bonding schemes and their weights

Localized bonding schemes and their weights have been obtained for the pi-electron system of nitrone by expanding complete active space self-consistent field wave functions into a set of Slater determinants composed of orthogonal natural atomic orbitals (NAOs) of Weinhold and Landis (Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective, 2005). Thus, the derived bonding schemes are close to orthogonal valence bond structures. The calculated sequence of bonding scheme weights accords with the sequence of genuine resonance structure weights derived previously by Ohanessian and Hiberty (Chem Phys Lett 1987, 137, 437), who employed nonorthogonal atomic orbitals. This accord supports the notion that NAOs form an appropriate orthogonal one-electron basis for expanding complete active space self-consistent field wave functions into meaningful bonding schemes and their weights.

First-principles calculations of conductance within a plane wave basis set vianon-orthogonal Wannier-type atomic orbitals

We present a plane-wave/pseudopotential implementation of a method to calculate theelectron transport properties of nanostructures. The conductance is calculated via theLandauer formula within the formalism of Green’s functions. Non-orthogonalWannier-type atomic orbitals are obtained by sequential unitary rotations of virtual andoccupied Kohn–Sham orbitals, followed by two-step variational localization. We usethese non-orthogonal Wannier-type atomic orbitals to partition the Kohn–ShamHamiltonian into electrode–contact–electrode submatrices. The electrode partsof the system are modelled by two metal clusters with additional Lorentzianbroadening of discrete energy levels. We examined our implementation by modellingthe transport properties of Na atomic wires. Our results indicate that with theappropriate level broadening the small cluster model for contacts reproducesodd–even oscillations of the conductance as a function of the nanowire length.

Three‐electron two‐center bonding in cycloimmonium ylides

AbstractWe explore the possibility that a 3‐electron‐2‐center bonding exists in cycloimmonium ylides. To detect this bonding in a polyatomic system, 3‐electron‐1‐hole density operators, characterizing a Pauling 3‐electron bond, are used in the framework of second quantization formalism. The weights of 3‐electron resonance structures are calculated and compared with the weights of 2‐electron structures for the ylide bond of pyridinium dicyanomethylide; the correlations of (↑↓) and (↑) electronic events, involved in the 3‐electron resonance structures, are also investigated. The calculations are performed in various approximation levels, and both orthogonal and nonorthogonal natural atomic orbitals are adopted. All calculations show that a 3‐electron bond exists between N and C atoms of ylide bond, but this bonding is not extended in C atoms of the pyridinium group. The interactions of α,β electrons (at the configuration interaction [CI] level) increase the localization of electrons, the weights of 3‐electron resonance structures, and thus the probability for 3‐electron bonding. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Re-examination of the NDDO approximation and introduction of a new model beyond it

The NDDO (neglect of diatomic differential overlap) approximation, a widely used basis for many semi-empirical molecular orbital (MO) approaches, is re-examined based on non-empirical frozen-core calculations on small molecules. An improvement going beyond the NDDO approximation is proposed. Our study shows that under the NDDO approximation, when the remaining non-DDO-type two-electron repulsion integrals (TERIs) are calculated using the basis set from the Löwdin orthogonalization of the valence atomic orbitals, the resulting total energies are much higher than those from the corresponding frozen-core ab initio calculations. On the other hand, when the remaining non-DDO TERIs are calculated using non-orthogonal valence atomic orbitals (similar to the Roby model), for most of the molecules calculated, the total energies are significantly lower than those from the corresponding ab initio calculations. Furthermore, we also find that for some molecules, the total energies thus calculated are higher than the corresponding ab initio results. The nonsystematic variation of the absolute errors in the total energy calculations is due to the fact that the core-electron and the electron-electron interactions are not treated in a balanced way in the NDDO approximation. A new model, which overcomes the deficiencies in the NDDO model, is proposed. In this model, a first-order correction term is added to the electron-electron Coulomb interactions, thereby improving the balance between the core-electron and the electron-electron interactions. Non-empirical test calculations show that the total energies from the new model are consistently higher than those from the ab initio calculation but closer to the ab initio results. We expect that the proposed new model would be useful in developing new high-quality semi-empirical MO approaches.

Orbital non-orthogonality in surface-state theory

The introduction of non-orthogonal atomic orbitals, into the tight-binding approximation, gives rise to the existence of overlap terms in electronic band-structure calculations. In contrast to the zero-overlap situation, it is helpful to employ a tensorial Green-function approach to access the surface density of states, and investigate the effects of overlap on its behaviour.

Significance of nonorthogonality in tight-binding models. II. The possibility of high-Tc superconductivity in intercalation compounds.

Using the L\"owdin transformation from nonorthogonal atomic orbitals to orthogonal Wannier orbitals for the tight-binding model with many-body interactions, we derive the t-J model for graphite. We demonstrate the significance of nonorthogonality on the parameters t and J. We obtain a mean-field phase diagram for the critical temperature ${\mathit{T}}_{\mathit{c}}$ versus doping fraction using an auxiliary slave-boson method. We compare the stabilization of the superconducting phase transition via a weak interlayer coupling with a Kosterlitz-Thouless two-dimensional dilute Bose gas condensation. It is concluded that graphite intercalation compounds form a testing ground for the t-J model of high-${\mathit{T}}_{\mathit{c}}$ superconductivity independent of the cuprates.

“Slide-rule” basis sets: application to the 3S state of helium-like species

The energy of a single determinant wavefunction with maximum total spin is usually determined by imposing the computationally convenient requirement of orthogonality on the atomic orbital functions, which are themselves expanded over a suitable basis set in the self-consistent-field (SCF) model. An alternative to orthogonalisation, yielding the same total energies, is presented here in which, for helium-like species, the chosen basis (a scaled even-tempered basis of N 1s Slater-type orbitals) is partitioned into three disjoint subsets XXX, XXX and XXX. For subsets XXX and XXX containing one element, it is demonstrated that the total energy is invariant to the choice of partitioning, and that the two (non-orthogonal) atomic orbitals may be expanded over N-1 basis functions contained in XXX ∪ XXX and XXX ∪ XXX, respectively. The total energies for the 3S state, as a function of overall scale factor, are compared with those obtained from full configuration interaction calculations, and the radial distribution functions for the 2 1S and 3S states are presented for helium.

Theory of the multicenter bond.

Tight-binding theory, as applied to tetrahedral semiconductors by Harrison, has been extended to crystal structures of more general coordination. Following Harrison's application to ${\mathrm{SiO}}_{2}$, the total energy is formulated in terms of multicenter bonding units, the analog of the two-center bond in tetrahedral structures. The universal matrix elements, which are obtained from fits to free-electron bands and depend only on the internuclear distance, correspond to matrix elements between orthogonal Wannier functions in the zinc-blende structure, and do not apply directly to other structures. Matrix elements of nonorthogonal atomic orbitals, which are applicable to other structures, are formulated in terms of the universal matrix elements in the tetrahedral structure. These matrix elements are then applicable to new structures, and permit an extension of the theory without introducing new parameters, or modifying the theory in any fundamental way. Several properties, including the total energy, minimum bond length, and force constant, are calculated for units of ${\mathrm{Zn}}_{3}$${\mathrm{P}}_{2}$ and elemental boron, and are found to agree with experiment much as did the theory's original application. The tight-binding bands of ${\mathrm{Zn}}_{3}$${\mathrm{P}}_{2}$, using a fluorite structure as an approximation to the true crystal structure, are compared to a pseudopotential calculation of Lin-Chung. It is found that second neighbors are necessary to obtain the correct band structure, but the total energy is not sensitive to the choice of neighbors.

Electronic structure of boron

Using a recently developed tight-binding theory with universal matrix elements, the electronic structure of α-rhombohedral boron and B 4C are calculated. The universal matrix elements, which correspond to matrix elements between Wannier functions in the zincblende structure, were found to be inappropriate for the unusual icosahedral unit found in α-rhombohedral boron and B 4C. However, a simple, approximate prescription relating universal matrix elements to matrix elements between nonorthogonal atomic orbitals is derived. The nonorthogonality does not separate into an additive, two-body repulsion as it does in the zincblende structure, and so must be included explicitly in the diagonalization of the Hamiltonian. This prescription is found to well characterize the internal energy levels, the equilibrium spacing and force constant of an icosahedron of boron. The cohesion and equilibrium spacings calculated for an icosahedron and the remaining bonding units in α-rhombohedral boron and B 4C agree well with experimental values: the cohesion is overestimated by the same ratio as for carbonrow semiconductors. Normal mode frequencies calculated for B 12H = 12, B 12D = 12 and B 12Cl = 12 are in good agreement with measured frequencies.

A model calculation of superexchange interaction and comparison among different methods

A perturbational formalism developed previously is applied to a 3-center 4-electron model of superexchange interaction. Both Löwdin's orthogonalized basis and biorthogonal basis set constructed from nonorthogonal atomic orbitals are used to obtain the exchange parameter and results are compared with complete configuration interaction calculations. The model shows ferromagnetic coupling for bond angles around 90° and an antiferromagnetic one for large and small angles. This qualitative feature is independent of the type of calculation and coincides with the results obtained by Van Kalkeren et al. based on several other methods. Simplified revisions of the perturbational formula are examined.

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.

Electronic states in amorphous metals

The electronic states in a model amorphous metal (dense random packing hard-sphere model) are studied in the cases of orthogonal and nonorthogonal atomic orbitals. Compared to the infinite FCC crystal, the density of states shows relatively small change in the bandwidth and the L point singularity seems to be preserved. Peak positions are, however, seen to shift towards the lower-energy side. A 'roton' like behaviour in the wave vector-dependent spectra is also found. These features are discussed in detail by comparison with the FCC case.

Second quantization for systems with a constant number of particles in the Dirac notation

AbstractThe relation between the completeness condition for an appropriate one‐particle basis set and the occupation number representation (second quantization) is shown for the time‐independent case. The explicit expressions for the basic symmetric operators are derived in the Dirac bra–ket notation. The physical meaning of these operators, the algebra as well as the connections with the one‐electron density matrix and with the projector on the Fermi sea in the one‐electron approximation, follow directly from these expressions. The generalization for a nonorthogonal basis and the algebra for corresponding basic operators are formulated. The connection with the notion of the molecular diagrams of different kinds for the nonorthogonal atomic orbitals is shown. The Mulliken populations and the Chirgwin–Coulson bond orders are equal to the diagonal and offdiagonal elements of the molecular diagram 1, respectively. The matrix elements of the projector on the Fermi sea in the one‐electron approximation in the representation of nonorthogonal atomic orbitals are elements of the molecular diagram 2.

Non-orthogonal atomic orbitals: Kinetic energy

Minimum kinetic energy deorthogonalization leads to valence atomc orbitals whose kinetic energy is a simple function of the orbital radius, R f ; 1/ R i = ( i|1/ r| i), and the angular momentum quantum number, l.

Non-orthogonal atomic orbitals

The procedure of Raffenetti and Ruedenberg for defining non-orthogonal atomic orbitals is generalized to provide minimum-kinetic-energy atomic orbitals for use in applications of pseudopotential theory in atoms and molecules.