Crystal Orbital Formalism Research Articles

Application of second-order Møller–Plesset perturbation theory with resolution-of-identity approximation to periodic systems

Efficient periodic boundary condition (PBC) calculations by the second-order Møller-Plesset perturbation (MP2) method based on crystal orbital formalism are developed by introducing the resolution-of-identity (RI) approximation of four-center two-electron repulsion integrals (ERIs). The formulation and implementation of the PBC RI-MP2 method are presented. In this method, the mixed auxiliary basis functions of the combination of Poisson and Gaussian type functions are used to circumvent the slow convergence of the lattice sum of the long-range ERIs. Test calculations of one-dimensional periodic trans-polyacetylene show that the PBC RI-MP2 method greatly reduces the computational times as well as memory and disk sizes, without the loss of accuracy, compared to the conventional PBC MP2 method.

Density functional crystal orbital study on the normal vibrations and phonon dispersion curves of all-trans polyethylene

Optimized structural parameters and frequencies of the infrared- and Raman-active vibrations are obtained for all-trans polyethylene by using the analytical energy gradient scheme in the density functional crystal orbital formalism. The Slater–Vosko–Wilk–Nusair (SVWN), the Becke–Lee–Yang–Parr (BLYP), and the Becke3–Lee–Yang–Parr (B3LYP) functionals are used with the 3-21G and 6-31G* basis sets. The frequencies calculated with the 6-31G* basis set are found to be in better agreement with the observed frequencies than those calculated with the 3-21G basis set regardless of the exchange-correlation functionals used. The root mean square errors between the calculated and observed frequencies are 21, 20, and 15 cm−1 for the SVWN/6-31G*, the BLYP/6-31G*, and the B3LYP/6-31G* calculations, respectively. Optical branches of the phonon dispersion curves are calculated at the SVWN/6-31G* level by adopting a C7H14 unit as a reference unit cell. The calculated phonon dispersion curves are in reasonable agreement with the curves experimentally determined and with the curves obtained with an empirical force field except for the skeletal stretching branches. Inelastic neutron scattering (INS) spectrum is also calculated by using the force field derived at the SVWN/6-31G* level. The overall intensity profile of the observed INS spectrum is well reproduced by the present calculations in which the effects of the Debye–Waller factors and the phonon wings are taken into account.

Charge distribution in K 3C 60 revisited: Incomplete alkali-to-C 60 electron transfer

The charge distribution in K 3C 60 has been calculated in the theoretical framework of a crystal orbital formalism based on an INDO (intermediate neglect of differential overlap) Hamiltonian. We predict a non-integral potassium-to-C 60 charge transfer (CT) with a clear discrimination between tetrahedral (t) and octahedral (o) alkali atoms. The CT from the o sites exceeds the CT from the t positions. The influence of the incomplete alkali ionization on the binding energy of core electrons is discussed. It is demonstrated that the present theoretical results provide a straightforward explanation of photoemission experiments on K 3C 60.

Theoretical study of the multimode Peierls distortion in the polydecker sandwich compound [Ni(H 5C 3B 2)] ∞

A semiempirical crystal orbital formalism based on an INDO Hamiltonian is employed to calculate the energy and the electronic structure of the one-dimensional polydecker sandwich compound [Ni(H 5C 3B 2) ∞. For the nondistorted system a number of energetically close lying electronic configurations was found. Their behaviour as a function of dimerization coordinates for various kinds of spatial dimerization, including distortions of lattice coordinates, intramonomer coordinates, and a superposition of both types of distortions, was studied. A discussion of the computational results on the basis of simple tight-binding approaches as well as exact analytical results for one-dimensional systems is presented.

Oxygen reactivity in vanadium pentoxide: electronic structure and infrared spectroscopy studies

The activity of V{sub 2}O{sub 5}-lattice oxygen in oxidation reactions has been studied by a semiempirical self-consistent-field Hartree-Fock crystal orbital formalism and by infrared spectroscopy. The electronic structure of oxygen vacancies in V{sub 2}O{sub 5} supports the conclusion that the most stable vacancy corresponds to the oxygen site which has the highest coordination to V atoms. Charge reorganization in the V{sub 2}O{sub 5} lattice upon oxygen vacancy formation shows the tendency of the V atoms around the vacancy to accumulate electronic charge, i.e., to be reduced. The stability associated with different oxygen vacancies is related to the degree of reduction of the transition-metal atoms. The participation of lattice oxygen in V{sub 2}O{sub 5} in the oxidation of dimethyl sulfoxide has been studied by infrared spectroscopy. The results of the infrared study agree with the electronic structure calculations in the sense that the oxygen atoms which participate more readily in the oxidation process correspond to those O centers with the highest V coordination.

A theoretical investigation of electron correlation and relaxation in organometallic polymers

AbstractThe importance of many‐body interactions beyond the mean‐field approximation of the Hartree–Fock (HF) self‐consistent‐field crystal orbital formalism is analyzed in one‐dimensional (lD) transition‐metal (3d) polymers with extended organic π ligands. The correlation energies are expressed in a quasiparticle picture. They are divided into long‐range contributions that are coordinated with the basis of spatially uncorrelated Bloch orbitals and into short‐range correlations derived for local rearrangement processes that are described in terms of a one‐electron basis which breaks the translational symmetry of the lD system. Both contributions (long‐range and short‐range correlations) are fragmented into elements of physical significance (hole and electron self‐energies for the former interactions; relaxation, pair‐relaxation and pairremoval terms for the local virtual excitations). The magnitude of these elements is analyzed as a function of the characters of the one‐electron states in the HF bands, the occupation patterns at the 3d centers, the available particle and hole channels in the elementary fluctuations and the energies and shapes of the various bands. The broad spectrum of possible amplifications and compensations leading to the quasiparticle shifts in metallomacrocycles is discussed. The different mechanisms to change the dispersions and to modify the width of the ϵ(k) curves are studied. It is shown that electron correlation and relaxation in transition‐metal polymers can lead even to a broadening of the energy bands. This behavior is in contrast to the influence of many‐body effects in simpler homogeneous materials where electron correlation is in any case accompanied by a narrowing of the dispersions (i.e., detraction of the group velocities of particles and holes). Possible modifications in the shapes of the one‐particle curves and the quasiparticle bands are also considered in the text [transition from a “normal ϵ(k) dispersion” to an energy band with a negative slope as a result of electron correlation]. Simplified formulas are derived that allow for a rough assessment of the various correction terms even in structurally complicated transition‐metal stacks with extended organic ligands. The approximate relations are used to correct the HF band structures of complex onedimensional metallomacrocycles as well as simpler crystalline materials by means of the quasiparticle approximation.

The band structure of porphyrinatonickel(II). A semiempirical crystal orbital study based on the tight‐binding formalism

AbstractThe band structure of porphyrinatonickel(II) (2) has been studied by means of crystal orbital calculations that are based on the tight‐binding approximation; the computational framework is a recently developed INDO model for transition metal compounds of the 3d series. The porphyrinato polymer has been studied in an eclipsed arrangement (2a) and in a staggered conformation (2b) where neighboring layers are rotated by 41°. The total energy of the metallomacrocycle has been decomposed into one‐ and two‐center contributions; the latter interaction parameters have been fragmented into physically feasible resonance, exchange, and classical electrostatic (electron–electron, electron–core, core–core) interactions. It is shown that individual two‐center potentials between atoms in neighboring layers are prevailingly determined by the electrostatic interaction energy. The NiNi coupling in the chain is highly repulsive; important stabilizing interactions are predicted between the 3d center of one cell and the electronegative N atoms in the neighboring layers. Stabilizing and destabilizing electrostatic interaction potentials largely compensate each other; the net stabilization in the polymer comes from the accumulation of resonance and exchange increments. The unoxidized Ni(II) porphyrinato polymer is an insulator. Several ligand bands (π, σ, and lone‐pair) are predicted on top of bands with significant Ni 3d admixtures; the conduction band of the unoxidized strand is of ligand π* character. The dense manifold of ligand states in the vicinity of the Ni 3d states (3d, 3d, 3dxz/3dyz) prevents the formation of bands in the polymer that are strongly localized at the 3d center. Ni 3d and 3d interact strongly with ligand lone‐pair and σ states. Avoided crossings between ϵ(k) curves in k space lead to compositions in the various bands that differ significantly at the bottom and the top. The INDO crystal orbital formalism predicts a partial oxidation of ligand bands in derivatives of 2 that contain oxidants (e.g., halides). The theoretical findings derived for 2 are compared with available experimental data on highly conducting porphyrinatonicke(II) polymers (tetrabenzo and octamethyltetrabenzo derivatives of 2).

A semiempirical crystal orbital investigation on the one-dimensional polyferrocenylene system

The band structure of polyferrocenylene has been studied in the crystal orbital formalism based on the tight-binding approximation; the computational framework is a semiempirical INDO model designed to reproduce the results of double-zeta ab initio calculations in organic molecules and transition metal compounds. It is shown that 4–6 neighbors are necessary to stabilize the results of the band structure calculations. The total energy of the polymer has been partitioned into elements of physical significance. Metal–metal and metal–ligand potentials have been separated into resonance, exchange, and classical electrostatic contributions. The highest filled band of polyferrocenylene is a ligand π band; Fe 3d bands are predicted at lower energies. The calculated bandwidths span a range between 0.4 to 2.6 eV. Avoided crossing regions in the reciprocal k space lead to a strong metal–ligand intermixing in some of the outer valence bands. It is shown that hole states in partially oxidized polyferrocenylene are probably unstable against the formation of a state with trapped valences. The close relation between this behavior and the soliton formalism is discussed. The various computational results are compared with experimental data.

The band structure of tetrathiosquaratonickel(II)

The band structure of tetrathiosquaratonickel(II), TTSqNi, is studied in the framework of the crystal orbital formalism based on the tight-binding approximation. Computational background is an improved semiempirical INDO model for transition metal compounds. The metal-ligand-metal arrangement in the direction of the stacking axis prevents the formation of broader energy bands. The calculated values for the bandwidth of the filled and empty valence bands are found between ca. 0.1 eV and 1.0 eV. Valence and conduction bands of the polymer are ligand π π ∗ functions with negligible Ni 3d admixtures. Various ligand π and sulfur lone-pair bands are predicted on top of the four occupied Ni 3d bands of the square-planar d 8 center. Strong electron-lattice interactions are expected for the conduction processes in the rather narrow (0.13 eV) conduction band.

Energy gradient in the ab initio Hartree—Fock crystal-orbital formalism of one-dimensional infinite polymers

Energy gradient formulae for one-dimensional polymers are derived using the ab initio Hartree—Fock crystal-orbital formalism. Geometry optimization and vibrational analysis of all-trans polyethylene are performed. The results are in reasonable agreement with experimental values.

A method for the calculation of improved band gaps in the crystal orbital formalism

A computational method for the calculation of improved band gaps in the crystal orbital (CO) formalism based on the Hartee-Fock (HF) procedure is developed. The approach makes use of fractional k-averaged band occupational numbers in the band structure technique. An electron in the recalculated conduction band(s) feel the averageed potential of ( N — 1) electrons. The method is applied to the LiH polymer. Semiempirical INDO CO calculations lead to an ϵ( k) lowering of nearly 3 eV for the improved conduction band in comparison to the dispersion curve of the standard variational approach. The two ϵ( k) curves are characterized due to slightly different dispersions.

Band structures of one-dimensional mixed donor-acceptor systems: A semiempirical INDO crystal orbital study

The band structures of one-dimensional donor-acceptor (DA) stacks have been studied by means of a semiempirical INDO crystal orbital formalism based on a model hamiltonian with variable electronegativities in the active (D and A) groups. An insulator-to-metal transition is predicted for short DA distances and strong DA pairs.

Band structure properties of one-dimensional donor-acceptor model polymers

The band structures of one-dimensional donor-acceptor (DA) stacks have been studied by means of the crystal orbital formalism based on a semi-empirical INDO approximation. The model polymers are composed of quinhydrone moieties ( 1 and 2) with different mutual orientations of D and A. The DA capability was modified by varying the DA distance within the unit cell as well as the effective electronegativity of the active groups of D and A. An insulator-to-metal transition is predicted for 1 at short DA distance and larger differences in the electronegativities of D and A. An avoided crossing between the highest filled and the lowest unfilled ϵ( k) curves is encountered in 2 leading to a finite band gap for all studied DA interaction conditions.

A CNDO/INDO crystal orbital model for transition metal polymers of the 3d series?basis equations

The crystal orbital formalism in the tight-binding approximation is combined with a recently developed CNDO/INDO model for transition metal species of the 3d series in order to allow band structure calculations on the Hartree-Fock (HF) SCF level for one-dimensional (1D) chains with organometallic unit cells. The band structure approach based on the CNDO and INDO approximation can be used for any atom combination up to bromine under the inclusion of the 3d series. The matrix elements for the tight-binding Hamiltonian are derived for an improved CNDO and INDO framework. The total energy of the 1D chain is partitioned into one-center contributions and into two-center increments of the intracell and intercell type. Semiempirical band structure calculations on simple model systems are compared with available ab initio data of high quality.

A CNDO/INDO crystal orbital model for transition metal polymers of the 3d series?the band structure of nickel(II)glyoximate

The band structure of nickel(II)glyoximate has been investigated by means of an INDO crystal orbital formalism based on the tight-binding approximation. Calculations have been performed for unit cell dimensions of 3.223 A which corresponds to the geometry of the partially oxidized chain and of 3.547 A for the unoxidized polymer. The convergence of the lattice sums has been studied. It is shown that resonance and classical electrostatic integrals lead to a fast convergence while the exchange contributions are only slowly reduced with increasing intercell separation. The rotational profile of the title compound shows a pronounced minimum for the staggered 90 ° conformation; this is in line with experimental X-ray data. The conformation of the low-dimensional system is determined by the maximization of stabilizing intercell energies. The intracell stabilization is largest for an orientation where the intercell coupling is most inefficient (α = 50 °−60 °). Ligand π and lone-pair bands are predicted on top of the various Ni bands in the INDO band structure calculations. This sequence is conserved even if the creation of localized 3d states with trapped valences upon partial oxidation is considered. The ground state of the partially oxidized polymer corresponds to a system where charge density from the ligand framework has been removed leading to a typical organic metal or semimetal. The computational results are compared with experimental observations.

An INDO crystal orbital approach to the one-dimensional porphyrinatonickel(II) system

The band structure of the one-dimensional metallomacrocycle porphyrinatonickel(II), Ni(P), has been studied by means of the crystal orbital formalism based on a semi-empirical INDO Hamiltonian. The unoxidized Ni(P) polymer is an insulator, the highest occupied bands as well as the conduction band are localized in the ligand framework. Bands with significant Ni 3 d admixtures are predicted at lower energies. The dense manifold of ligand states in the vicinity of the Ni 3 d basis energies leads to a strong metal-ligand intermixing in Ni(P) bands in the outer valence region.

Structure of infinite polyenes: Ab initio quantum chemical study

Using a minimal basis Hartree–Fock crystal orbital formalism for five structurally different infinite polyene (polyacetylene) models it is concluded that, besides the all-trans configuration, other structures may be energetically equal or even more probable.

On the AB initio crystal orbital method

A new computer realization of the LCAO Hartree—Fock crystal orbital method using contracted Gaussian orbitals is reported. All integrals over the atomic orbitals are calculated explicitly within a finite interaction range. No approximation with respect to exchange is used. The spin-unrestricted Hartree—Fock-type crystal orbital formalism has been programmed, too. Both programs are limited to quasi one-dimensional systems at present, provide nevertheless a useful tool for the theoretical investigation of the electronic structure of simple polymers and one-dimensional models of solids. Basic information on both programs is documented.