On the use of corresponding orbitals for the construction of mutually orthogonal orbital sets

Consider two orbital sets χ_k, k = 1…m and η₁, 1 = 1…n, which are mutually nonorthogonal. Provided that n > m, at least n ? m orbitals of the set {η} can be orthogonalized to the set {χ} by a transformation within the set {η}. The orthogonalization of the remaining orbitals of {η} to the set {χ} requires a transformation in which the χk appear explicitly. The orthogonalization of one orbital set to another is relevant for SCF optimizations in a truncated basis set, in the presence of frozen occupied orbitals. Examples are frozen core calculations, ECP calculations, and embedded cluster calculations, where the cluster is embedded in a frozen environment. A simple orthogonalization scheme, which makes use of a corresponding orbital transformation, is presented. It is suggested that with a small, well-defined extension of the set {η} the complete orthogonalization can be done with a transformation in which the {χ} do not appear explicitly. © 1993 John Wiley & Sons, Inc.     

原子价壳层电子量子拓扑指数与元素电负性的关系

在基态原子价壳层电子隐核图的基础上, 基于拓扑化学原理以及原子价壳层电子结构特征, 构建了原子价壳层电子量子拓扑指数(AEI), 它对基态原子实现唯一性表征, 结合原子价壳层电子平均化能(∑n_iE_i/∑n_i)等参数, 建立了一套新的元素电负性标度: X_N＝－0.588710AEI₁＋0.761214AEI₂＋0.154982(∑n_iE_i/∑n_i)－0.080929. 该式给出了周期表中氢至镅共95种元素的电负性, 结果表明新电负性标度X_N与Pauling电负性标度颇为一致. 进一步从原子价轨道量子拓扑指数确定了sp, sp², sp³杂化轨道的电负性. 新标度在元素和物质的结构-性质研究中具有一定的适用性.     

A new algorithm for inactive orbital optimization in valence bond theory

This paper presents an efficient algorithm for energy gradients in valence bond self-consistent field (VBSCF) method with non-orthogonal orbitals. The frozen core approximation method is extended to the case of non-orthogonal orbitals. The expressions for the total energy and its gradients are presented by introducing auxiliary orbitals, where inactive orbitals are orthogonal, while active orbitals are non-orthogonal themselves but orthogonal to inactive orbitals. It is shown that our new algorithm has a low scaling of (N _a + 1)m ⁴, where N _a and m are the numbers of the active orbitals and basis functions, respectively, and is more efficient than the existing VBSCF algorithms.     

Electronegativity Scaled as a Average Attracting Energy of Valence‐Shell Electrons in a Ground‐State Free Atom

The total capability of an atom attracting valence electrons can be measured by the sum of ionization energies of valence electron in a ground‐state free atom plus its electron affinity called Total Attracting Energy, TAE = Σn_iE_i + EA, where, E_i is the ionization energy of the ith valence‐shell electron in a ground‐state free atom, n_i is the number of valence‐shell electron bearing energy E_i, and EA is the electron affinity. And the electronegativity χ_CL is proportional to the average of TAE, AAE = TAE_av, divided by Σn_i, the number of atomic valence‐shell electrons. χ_CL = 0.1813 TAE_av = 0.1813 AAE = 0.1813 TAE/Σn_i, = 0.1813 (Σn_iE_I + EA)/Σn_i. Further, the atomic valence orbital electronegativity can be also obtained from the TAE value of an atom. Some discussions were made on several special aspects such as scale of rare gases, comparisons with Pauling's and Allen's scales, etc.     

Orbital electronegativity and analytical representation of atom valence state energy

An analytical representation of atom valence state energy (E(n_j), j = 1,…?4; E(n_j) is a nonlinear function of orbital occupancy numbers n_j) is proposed and explicitly derived for H? Ar; the values of electronegativity calculated based on E(n_j) agree within truncation error with those of Hinze and Jaffe. However, in our representation, orbital electronegativity χ and hardness parameters η of a given orbital always include nonlinear contributions from other orbitals, hence accounting for their influence on χ and η. An atomic charge calculation procedure based on E(n_j) is also described and shown to perform well.     

Selection rules in alternant systems

The CI space X_n generated by n electrons moving over 2n spin orbitals is considered. It is shown that the transitions between different eigenstates ψ_i ? X_n of alternant and weakly alternant Hamiltonians are governed by some special selection rules. These selection rules are characteristic to alternant systems, and they do not apply to nonalternant systems. The set of all such selection rules can be easily derived from the splitting theorem. In particular, the selection rules associated with spin independent alternant systems are considered. As an example, the PPP Hamiltonian ?_p describing netural alternant hydrocarbons is treated. In the case of electron dipole transitions between eigenstates ψ_i ? X_n of the Hamiltonian ?_p, the selection rules obtained are in agreement with the selection rules derived previously by Pariser and McLachlan.     

Bestimmung der Stabilit?tskonstanten von mehrkernigen Komplexen mit Hilfe einer spektrophotometrischen Methode

Zusammenfassung Es wurde eine allgemeine Gleichung für die Berechnung der Stabilitätskonstante von Komplexen L _mH_iM_nZ_j (Z=OH^–, usw.) abgeleitet und für Maximumbedingungen der Jobschen Kurve die Schwarzenbachsche graphische Methode so modifiziert, daß sie die allgemeine Bestimmung der Zusammensetzung und Stabilität von Komplexen L _mH_iM_nZ_j ermöglicht.

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Summary A general equation has been derived for the calculation of stability constants of complexes L _mH_iM_nZ_j (Z=OH^–, etc.). The graphic method of Schwarzenbach has been modified for maximum conditions of Job's curve in order to make possible the general determination of the composition and stability of complexes L _mH_iM_nZ_j.

     

Distribution of Valence Electron Density in C n H m and BnHm Framework and Polyhedral Molecules and Orbital-Reduntant Bonds

In framework molecular cations and radical cations of adamantane C₁₀H _m ^q+ and also in polyhedral molecules and molecular ions C₅H₅ ⁺, C₆H₆ ^{2 +}, B₅H₉, and B₁₀H₁₀ ^{2 -}, the charge density of valence electrons in the central areas of C _n and B _n cavities and faces is significant. In the molecule of adamantane C₁₀H₁₆, the valence electron density in central areas of the cavity and faces of the C₁₀ framework is small as compared to the electron density along its edges C-C. These distinctions are due to the fact that, in the electronic structure of C _n H ^q _m cations and radical cations and also of B _n H _m molecules and molecular ions, there is an additional orbital interaction involving vacant valence orbitals of C⁺ or B (orbital-reduntant bonds); the absence of vacant valence orbitals of C atoms in neutral adamantane molecule excludes additional orbital interactions in excess of C-H and C-C.     

Isolation of a Perfectly Linear Uranium(II) Metallocene

Reduction of the uranium(III) metallocene [(η⁵‐C₅ⁱPr₅)₂UI] ( 1 ) with potassium graphite produces the “second‐generation” uranocene [(η⁵‐C₅ⁱPr₅)₂U] ( 2 ), which contains uranium in the formal divalent oxidation state. The geometry of 2 is that of a perfectly linear bis(cyclopentadienyl) sandwich complex, with the ground‐state valence electron configuration of uranium(II) revealed by electronic spectroscopy and density functional theory to be 5f³ 6d¹. Appreciable covalent contributions to the metal‐ligand bonds were determined from a computational study of 2 , including participation from the uranium 5f and 6d orbitals. Whereas three unpaired electrons in 2 occupy orbitals with essentially pure 5f character, the fourth electron resides in an orbital defined by strong 7s‐6d mixing.     

A series of 1-D zinc(II) coordination polymers with 3-D supramolecular networks: synthesis,structural investigation,and NBO analysis

Abstract

N,N′-Bis(pyridin-4-ylmethylene)naphthalene-1,5-diamine (L) acts as a bipyridine analogue linker ligand towards {Zn₇(μ₄-O)₂(OAc)₁₀}, {Zn₂(NCS)₂(OAc)₂}, and {Zn(N₃)₂} nodes and allows construction of three new 1-D coordination polymers, the linear chain [Zn₇(μ₄-O)₂(OAc)₁₀(L)]_n (1), [Zn(NCS)(OAc)(L)]_n (2) in ladder-type geometry and the zigzag chain [Zn(N₃)₂(L)]_n (3). Structural characterization reveals that in 1 acetate anionic ligands connect seven Zn(II) ions through the bridging coordination modes μ₃-η¹,η² and μ₂-η¹,η¹. The resulting heptanuclear node is located on an inversion center and therefore consists of four crystallographically distinct cations; their coordination spheres correspond to distorted octahedra or tetrahedra. The Zn(II) ions in polymer 2 exhibit distorted trigonal bipyramidal {ZnN₃O₂} coordination; μ₂-η¹,η¹ coordinated acetate and terminal thiocyanate ligands lead to inversion-symmetric [Zn₂(NCS)₂(OAc)₂] secondary building units (SBU), which are further linked by the N,N′-bipyridine analogue L. Terminal coordination of two anionic azide ligands and the bridging bipyridine L result in coordination polymer 3, in which the cations adopt distorted tetrahedral {ZnN₄} coordination. In all crystalline solids 1–3, adjacent 1-D chains interact through π–π stacking and non-classical (C???H···O, C???H···π) hydrogen bonds, leading to 3-D supramolecular architectures. Differences in their 3-D arrangement are due to variations in the anionic co-ligands, subtle conformational differences in the semi-rigid linker and the variable coordination sphere about the zinc cations. Thermogravimetric investigations indicate differences in both thermal stability and decomposition mode. Natural bond orbital (NBO) analysis provides a convenient basis for investigating the intramolecular bonding interactions and delocalization effects in these molecular systems. Finally, solids 1–3 exhibit intense luminescence at room temperature.     

Efficient generation of Heisenberg Hamiltonian matrices for VB calculations of potential energy surfaces

The spin‐Hamiltonian valence bond theory relies upon covalent configurations formed by singly occupied orbitals differing by their spin counterparts. This theory has been proven to be successful in studying potential energy surfaces of the ground and lowest excited states in organic molecules when used as a part of the hybrid molecular mechanics—valence bond method. The method allows one to consider systems with large active spaces formed by n electrons in n orbitals and relies upon a specially proposed graphical unitary group approach. At the same time, the restriction of the equality of the numbers of electrons and orbitals in the active space is too severe: it excludes from the consideration a lot of interesting applications. We can mention here carbocations and systems with heteroatoms. Moreover, the structure of the method makes it difficult to study charge‐transfer excited states because they are formed by ionic configurations. In the present work we tackle these problems by significant extension of the spin‐Hamiltonian approach. We consider (i) more general active space formed by n ± m electrons in n orbitals and (ii) states with the charge transfer. The main problem addressed is the generation of Hamiltonian matrices for these general cases. We propose a scheme combining operators of electron exchange and hopping, generating all nonzero matrix elements step‐by‐step. This scheme provides a very efficient way to generate the Hamiltonians, thus extending the applicability of spin‐Hamiltonian valence bond theory. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

An optimized orbital transformation able to reproduce partially prefixed overlaps

Given a function space spanned by a basis {?_i}, we are interested in finding another basis {g_i} for which the overlaps (g_i | g_i) assume arbitrarily prefixed values in a subset ?? of the full set of the pairs of indices (i, j). The other overlaps are let free. We show how it is possible to perform this linear transformation ?_i → g_i minimizing the “distortion” J = Σ_i(g_i – ?_i | g_i – ?_i).     

Theorie des Orbitals Moleculaires: Recherche d'une Base STO Homogene pour les Atomes Alcalins et Leur Serie Isoelectronique

A method of direct treatment of valence electrons is proposed for the various ²S, ²P,²D, ²F states of the lithium, sodium, and potassium atoms and the corresponding isoelectronic ions. The function describing the outer electron, which is orthogonal to the wave functions of the electrons in the core, is given as a linear combination of generalized Laguerre functions, with effective charges Q = 1, 2 …? equal to the charges of the core. A simpler analytical STO function, with non-integer principal quantum number n is then proposed.     

Electronic charge density and mean kinetic energy of lithium orbitals in LiH,LiH+, Li2, Li2+, LiH2+, and Li2H+

The electron density near the lithium nucleus in the species LiH, LiH⁺, Li₂, Li₂⁺, LiH₂⁺, and Li₂H⁺ was analyzed by transforming the SCF molecular orbitals into a sum of atomic contribnutions, for both core and valence orbitals. These “hybrid-atomic” orbitals were used to compare: electron densities, orbital polarizations, and orbital mean kinetic energies with the corresponding lithium atom quantities. Core-orbital electron densities at the lithium nucleus were observed to increase by up to 0.5% relative to the lithium atom 1s orbital. Lithium cores also exhibited polarization but, surprisingly, in the direction away from the internuclear region. Similar dramatic changes were seen in the electron densities of the valence orbitals of lithium: The electron density at the nucleus for these orbitals increased two-fold for homonuclear species and twenty-fold for heteronuclear triatomic species relative to the electron density at the nucleus in lithium atom. The polarization of the valence orbital electronic charge, in the vicinity of the lithium nucleus, was also away from the internuclear region. The mean “hybrid-atomic” orbital kinetic energies associated with the lithium atom in the molecules also showed changes relative to the free lithium atom. Such changes, accompanying bond formation, were relatively small for the lithium core orbitals (within 0.2% of the value for lithium atom). The orbital kinetic energies for the lithium valence electrons, however, increased considerably relative to the lithium atom: By a factor of about 2 in homonuclear diatomics, by a factor of 7 in heteronuclear diatomics, and by a factor of 11 in the triatomic species. In summary, the total electronic density (core plus valence) at the lithium nucleus remained remarkably constant for all of the species studied, regardless of the effective charge on lithium. Thus, the drastic changes noted in the individual lithium orbitals occurred in a cooperative fashion so as to preserve a constant total electron density in the vicinity of the lithium nucleus. In all cases, bond formation was accompanied by an increase in the orbital kinetic energy of the lithium valence orbital. We suggest that these two observations represent important and significant features of chemical bonding which have not previously been emphasized.     

An approximate frozen core approach to valence only molecular calculations

A simple valence electron-only theory based on an approximate frozen core approach and an exact core-valence strong orthogonality condition is developed for atomic and molecular systems. A unique reduced basis is introduced in which both core and valence orbitals are expanded. The core representation is roughly approximated, and the valence orbital overlap with the corresponding all-electron reference functions is nearly exact. The size of the reduced basis in terms of primitive functions is practically the same as that adopted by effective core potential methods in which the valence orbitals have the correct nodal properties. Results obtained with the present approach are presented for LiO, BeO and CaO molecules, and compared with the corresponding all-electron frozen core calculations. In addition, a detailed investigation on Li _n Be clusters (n=1,..., 6) is carried out.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Model potentials for molecular calculations. II. The spd-MP set for transition metal atoms Sc through Hg

Model potential parameters and valence orbitals were generated for the transition metal atoms Sc through Hg. They are named the spd-MPs and are supplementary to the sd-MPs presented in the preceding article. The outermost core np electrons were treated explicitly together with valence nd and (n + 1)s electrons, and the remaining electrons were replaced by a model potential. The model potential parameters and valence orbitals were determined in the same way as the sd-MPs. Major relativistic effects (via the mass velocity and Darwin terms) were also incorporated in the spd-MPs for the second-and third-row transition metal atoms. The results of numerical nonrelativistic Hartree-Fock (HF) calculations for the first-row transition metal atoms and of the quasirelativistic HF calculations with Cowan and Griffin's method for the second-row and third-row transition metal atoms were used as reference data in determination of the spd-MPs.     

The Valence Orbitals of the Alkaline-Earth Atoms

Quantum chemical calculations of the alkaline-earth oxides, imides and dihydrides of the alkaline-earth atoms (Ae=Be, Mg, Ca, Sr, Ba) and the calcium cluster Ca₆H₉[N(SiMe₃)₂]₃(pmdta)₃ (pmdta=N,N,N′,N′′,N′′-pentamethyldiethylenetriamine) have been carried out by using density functional theory. Analysis of the electronic structures by charge and energy partitioning methods suggests that the valence orbitals of the lighter atoms Be and Mg are the (n)s and (n)p orbitals. In contrast, the valence orbitals of the heavier atoms Ca, Sr and Ba comprise the (n)s and (n−1)d orbitals. The alkaline-earth metals Be and Mg build covalent bonds like typical main-group elements, whereas Ca, Sr and Ba covalently bind like transition metals. The results not only shed new light on the covalent bonds of the heavier alkaline-earth metals, but are also very important for understanding and designing experimental studies.     

Enhanced Zintl anions by carbon doping in Al-Na clusters and new magic structure Al6Na4C

This article investigated the low-energy structures of Al₆Na _mC (m = 2, 4, 6, 8) clusters and their electronic structures by using genetic algorithm combined with density functional theory and configuration interaction methods. The computations show that the C atoms prefer sitting at the center, whereas the Na atoms tend to locate at the outside of the clusters. The valence molecular orbitals (MOs) agree well with the prediction of the jellium model. The stronger attraction of the central carbon to the valence electrons will depress the potential energies locally, which makes the 2S level go obviously lower and the 2P and 1D orbitals form a sub-band. The 26 valence electrons in Al₆Na₄C form closed 1S²1P⁶2S²1D¹⁰2P⁶ shells and correspond to a new magic structure. The MOs and electron localization function show that the sodium cores are exposed at the outside of the valence electrons and form naked cations. The contraction of the valence electrons because of the carbon doping enhances the charges on the Al₆C moieties, and the Na⁺ cores on the peripheries are ionically bonded to the Zintl anions (Al₆C)^q−. The Al₆Na₄C has a tetrahedral structure with symmetry T_d, and it may be used as building blocks to synthesize Zintl solid.     

Harmonic oscillations in a polymerization

A simple radical polymerization is proposed in this paper, with step‐by‐step chain growth (R_i + M → R_i+1), and termination by transfer to a third body (R_i + S → polymer) such as the solvent. It is assumed that, for a certain critical degree of polymerization n, the propagator R_n reacts with substrate H to produce a deactivator (V) of the first propagator (H + R_n → R_n + V; V + R₁ → P₁) R₁. Assuming that monomer, M, and precursor concentrations are constant, and assuming that the deactivator reaches fast a steady state, the resulting kinetic equations are formally linear, but they admit, perturbations r_j(t) of the steady‐state concentrations of the propagators R₁, R₂, …, R_n, which are periodic functions of time. Even more, they can be purely sinusoidal functions (which have been called “harmonic,” in analogy to the solutions of the well‐known classical harmonic oscillator) with phase shift between perturbations r_j(t) = R_j? (R_j)₀ and r_j+1(t) = R_j+1 ? (R_j+1)₀. Based on these periodic solutions and aiming to a model as simple as possible, a theoretical analysis is made, resulting in that the minimum value for n would be n = 3. Of course, these harmonic oscillations “driven by trimer” are equally found in the group of all the remaining propagators with polymerization degree higher than 3 (variable Y = ∑ R_j). © 2009 Wiley Periodicals, Inc. Int J Chem Kinet 41: 507–511, 2009