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.     

Combined Open Shell Hartree–Fock Theory of Atomic–Molecular and Nuclear Systems

In this study, the combined Hartree–Fock (HF) and Hartree–Fock–Roothaan equations are derived for multideterminantal single configuration states with any number of open shells of atoms, molecules and nuclei. It is shown that the postulated orbital-dependent energy and Fock operators are invariant to the unitary transformation of orbitals. This new methodology is based entirely on the spin-restricted HF theory. As an application of combined open shell theory of atomic–molecular and nuclear systems presented in this paper, we have solved Hartree–Fock–Roothaan equations for the ground state of electronic configuration C(1s ²2s ²2p ²) using Slater type orbitals as a basis.     

Research in the method of large molecular calculations utilizing transferability of LMO

The localized molecular orbitals of some saturated hydrocarbons and their derivatives have been formed using ab initio method and M. P. [1–2] localization procedure. Two models, SLMO and ELMO , a set of parameters of LMO Fock matrix elements, and a technique called “Group Effect” are proposed. Based on these, we developed a procedure to simulate the ab initio calculations on large molecules. Some test calculations have been done. The results are compared with those of the ab initio method. In general, absolute errors of orbital energies are about 10^?3 a.u., and the relative errors of total energies are about 10^?4. For the original applications, we applied this procedure to some large systems of alkane and their derivatives as well as three Crown-ether compounds. Satisfactory results are obtained.     

Tamm states in zigzag carbon nanotubes: Analytic estimates in the Hückel approximation

The frontier molecular orbitals of [C _n ] _q hydrocarbons, fragments of (n, 0) nanotubes, were considered. Asymptotic estimates for two different sets of frontier molecular orbitals were obtained in the Hückel approximation. The problem was shown to be equivalent to the search for model polyene levels with alternating resonance parameters (β and β′ = γβ). The γ parameter was determined by the symmetry type of molecular orbitals and took on values from 0 to 2. The problem was reduced to an analysis of a regular linear polyene with a modified terminal atom by taking alternant symmetry into account. The states of the system were classified according to quasi-momentum Θ values, which could be complex. Solutions with a complex Θ value were localized at tube ends and could be interpreted as Tamm states. The criterion of the appearance of Tamm solutions for the problem under consideration was formulated, γ < 1 − 1/(q + 2). The conclusion was drawn that these states for an arbitrary [C _n ] _q fragment lied close to the Fermi level, and their energies were described by the asymptotic equation α ± βρ^{(q + 1)}. Delocalized levels were always situated farther from the Fermi level than Tamm states.     

Use of addition theorems in evaluation of multicenter nuclear-attraction and electron-repulsion integrals with integer and noninteger n Slater-type orbitals

Multicenter integrals appearing in the Hartree–Fock–Roothaan equations for molecules are calculated using different kinds of series expansion formulas obtained from the expansions of integer and noninteger n Slater-type orbitals, in terms of Ψ ^α -exponential-type orbitals (where α=1, 0, –1, –2,...) at a displaced center, that form complete orthonormal sets and are represented by linear combinations of integer n Slater-type orbitals. The convergence of these series is tested by calculating concrete cases. The accuracy of the results is quite high for quantum numbers, screening constants, and location of orbitals. Received: 13 February 2002 / Accepted: 11 March 2002 / Published online: 4 July 2002     

Quadratically convergent calculation of localized molecular orbitals

Two iterative procedures for the transformation of canonical self-consistent field molecular orbitals to intrinsic localized molecular orbitals are proposed. A first-order method based on a series of (n × n) unitary transformations may be applied to orbitals which are far from convergence. The second method, based on Newton's method, yields quadratic convergence. Numerical results based on Boys' criterion are presented for water, carbon monoxide, boron fluoride, nitric oxide, and methylacetylene. A composite method may be used to obtain rapid convergence for large molecules for which it is not practical to calculate the entire hessian matrix. The performance of the composite method is demonstrated by application to the dinitrogen tetroxide molecule. Highly converged localized molecular orbitals may be obtained for most molecules with five to eight first-order iterations followed by three or four iterations based on either the second-order or composite method.     

A new method for analysis of disulfide-containing proteins by matrix-assisted laser desorption ionization (MALDI) mass spectrometry

A simple and high-throughput method for the identification of disulfide-containing peptides utilizing peptide-matrix adducts is described. Some commonly used matrices in MALDI mass spectrometry were found to specifically react with sulfhydryl groups within peptide, thus allowing the observation of the peptide-matrix adduct ion [M+n+n′ matrix+H]⁺ or [M+n+n′ matrix+Na]⁺ (n = the number of cysteine residues, n′=1, 2,…, n) in MALDI mass spectra after chemical reduction of disulfide-linked peptides. Among several matrices tested, α-cyano-4-hydroxycinnamic acid (CHCA, molecular mass 189 Da) and α-cyano-3-hydroxycinnamic acid (3-HCCA) were found to be more effective for MALDI analysis of disulfide-containing peptides/proteins. Two reduced cysteines involved in a disulfide bridge resulted in a mass shift of 189 Da per cysteine, so the number of disulfide bonds could then be determined, while for the other matrices (sinapinic acid, ferulic acid, and caffeic acid), a similar addition reaction could not occur unless the reaction was carried out under alkaline conditions. The underlying mechanism of the reaction of the matrix addition at sulfhydryl groups is proposed, and several factors that might affect the formation of the peptide-matrix adducts were investigated. In general, this method is fast, effective, and robust to identify disulfide bonds in proteins/peptides.     

Optimization of virtual orbitals in the framework of a multiconfiguration spin-coupled wave function

A new method is introduced for the optimization of nonorthogonal virtual orbitals for use in general multiconfiguration spin-coupled wave functions. The use of a number of highly effective approximations greatly reduces the computational effort involved, the most important being the use of a second-order perturbation expression for the energy and an approximate expression for the elements of the Hessian. As a result, the overall scheme scales very favourably with respect to the numbers of active electrons and of basis functions, making it suitable for the accurate study of large systems. Benchmark calculations are presented for the dissociation of LiH(X¹Σ⁺) and Li₂(X¹Σ⁺ _g ) using a highly compact four-configuration wave function. Standard spin-coupled valence bond expansions in the same virtual space are required to be significantly larger before equivalent results are obtained. The results are shown to compare very favourably with full valence complete active space self-consistent field calculations using an identical basis, and binding energies are within 4% of the values obtained from full configuration interaction calculations in the same basis set. Received: 10 June 1997 / Accepted: 7 October 1997     

On the evaluation of the characteristic polynomial of a chemical graph

The evaluation of the characteristic polynomial of a chemical graph is considered. It is shown that the operation count of the Le Verrier–Faddeev–Frame method, which is presently considered to be the most efficient method for the calculation of the characteristic polynomial, is of the order n⁴. Here n is the order of the adjacency matrix A or equivalently, the number of vertices in the graph G. Two new algorithms are described which both have the operation count of the order n³. These algorithms are stable, fast, and efficient. A related problem of finding a characteristic polynomial from the known eigenvalues λ_i of the adjacency matrix is also considered. An algorithm is described which requires only n(n ? 1)/2 operations for the solution of this problem.     

Intrinsic local constituents of molecular electronic wave functions. I. Exact representation of the density matrix in terms of chemically deformed and oriented atomic minimal basis set orbitals

A coherent, intrinsic, basis-set-independent analysis is developed for the invariants of the first-order density matrix of an accurate molecular electronic wavefunction. From the hierarchical ordering of the natural orbitals, the zeroth-order orbital space is deduced, which generates the zeroth-order wavefunction, typically an MCSCF function in the full valence space. It is shown that intrinsically embedded in such wavefunctions are elements that are local in bond regions and elements that are local in atomic regions. Basis-set-independent methods are given that extract and exhibit the intrinsic bond orbitals and the intrinsic minimal-basis quasi-atomic orbitals in terms of which the wavefunction can be exactly constructed. The quasi-atomic orbitals are furthermore oriented by a basis-set independent method (viz. maximization of the sum of the fourth powers of all off-diagonal density matrix elements) so as to exhibit clearly the chemical interactions. The unbiased nature of the method allows for the adaptation of the localized and directed orbitals to changing geometries. Contribution of the Mark S. Gordon 65th Birthday Fegtschrift Issue.     

Partial Hessian vibrational analysis: the localization of the molecular vibrational energy and entropy

The partial Hessian vibrational analysis (PHVA), in which only a subblock of the Hesssian matrix is diagonalized to yield vibrational frequencies for partially optimized systems, is extended to the calculation of vibrational enthalpy and entropy changes for chemical reactions. The utility of this method is demonstrated for various deprotonation reactions by reproducing full HVA values to within 0.1–0.4 kcal/mol, depending on the number atoms included in the PHVA. When combined with the hybrid effective fragment potential method [Gordon MS, et al. (2001) J Phys Chem A 105:293–307], the PHVA method can provide (harmonic) free-energy changes for localized chemical reactions in very large systems. Received: 21 September 2001 / Accepted: 30 October 2001 / Published online: 22 March 2002     

Adsorption behavior of 181W and 93mMo as lighter homologues of seaborgium (Sg) in HF/HNO3 on anion-exchange resin

Adsorption of carrier-free radiotracers ¹⁸¹W and ^93mMo produced in the ¹⁸¹Ta(p, n) and ^natNb(p, n) reactions, respectively, on anion-exchange resin was studied in mixed solution of HF and HNO₃ in a concentration range of 10⁻⁴–10⁻¹ M HF/0.1 M HNO₃. Distribution coefficients (K _d) of ¹⁸¹W and ^93mMo at 70 °C showed the V-shaped variation with the minimum at around 10⁻¹ M HF/0.1 M HNO₃, although variation of the K _d values for ^93mMo was quite small compared with that for ¹⁸¹W. Formation of oxofluoro complexes for W and Mo is briefly discussed.     

Symmetry simplifications of space types in configuration interaction induced by orbital degeneracy

Symmetry simplifications are introduced in configuration interaction (CI ) by reducing the number of symmetry-allowed space types if there is degeneracy in some of the molecular orbitals by constructing the unique space types. A new symmetry group which we call the configuration symmetry group is defined and is shown to be expressible as a generalized wreath product group. Generating functions are derived for enumerating the equivalence classes of space types. A double coset method is expounded which constructs the representatives of all equivalence classes of space types using the cycle index of generalized wreath product and the double cosets of label subgroup with generalized wreath product in the symmetric group S_n, if n is twice the number of occupied and virtual orbitals. Method is illustrated with CI using the localized orbitals of polyenes, CI in benzene, and atomic CI for several reference states.     

Synthesis,characterization, and quantum chemical calculational studies on 3-<Emphasis Type="Italic">p</Emphasis>-methylphenyl-4-amino- 1, 2, 4-triazole-5-thione

The title compound of 3-p-methylphenyl-4-amino-1, 2, 4-triazole-5-thione was synthesized and characterized by elemental analysis, IR, electronic spectra, and X-ray single crystal diffraction. Quantum chemical calculations of the structure, natural bond orbital, and thermodynamic functions of the title compound were performed by using B3LYP/6-311G** and HF-6-311G** methods. Both the methods can well simulate the molecular structure. Vibrational frequencies were predicted, assigned and compared with the experimental values, and B3LYP/6-311G** method is superior to HF/6-311G** method to predict the vibrational frequencies. Electronic absorption spectra calculated by B3LYP/6-311G** method have some red shifts compared with the experimental ones and natural bond orbitals analyses indicate that the two absorption bands are mainly derived from the contribution of n → π^* and π → π^* transitions. On the basis of vibrational analyses, the thermodynamic properties of the title compound at different temperatures have been calculated, revealing the correlations between C ⁰ _p,m , S ⁰ _m , H ⁰ _m , and temperatures.     

Localized Orbitals vs. Pseudopotential-Plane Waves Basis Sets: Performances and Accuracy for Molecular Magnetic Systems

 Density functional theory, in combination with a) a careful choice of the exchange-correlation part of the total energy and b) localized basis sets for the electronic orbitals, has become the method of choice for calculating the exchange-couplings in magnetic molecular complexes. Orbital expansion on plane waves can be seen as an alternative basis set especially suited to allow optimization of newly synthesized materials of unknown geometries. However, little is known on the predictive power of this scheme to yield quantitative values for exchange coupling constants J as small as a few hundredths of eV (50–300 cm⁻¹). We have used density functional theory and a plane waves basis set to calculate the exchange couplings J of three homodinuclear Cu-based molecular complexes with experimental values ranging from +40 cm⁻¹ to −300 cm⁻¹. The plane waves basis set proves as accurate as the localized basis set, thereby suggesting that this approach can be reliably employed to predict and rationalize the magnetic properties of molecular-based materials. Corresponding author. E-mail: Carlo.Massobrio@ipcms.u-strasbg.fr Received August 5, 2002; accepted August 9, 2002     

Theoretical study on cooperative and extra-additive behavior of hydrogen-bonded clusters

The stabilization energy △ E(n) and four typical properties of hydrogen bond F-H…F in chain-like and cyclic (HF)n clusters (n = 1-5) have been calculated using MP2 and three DF levels of theory with the Gaussian 98 program, and 6-31++G** bases set. The results demonstrate that the extra-additive or cooperative behavior in (HF)n clusters is very obvious. In addition, we studied much larger chain-like (HF)n (n = 6, 9, 12,18, 24) clusters using one of these DF methods.     

From determinants to spin eigenfunctions—a simple algorithm

Calculation of blocks of matrix elements between determinants associated with two fixed orbital configurations is very easy. A simple method to obtain the whole blocks and then transform them to the spinadapted basis is described. The method is suitable for many-particle operators, the number of operations to obtain matrix elements being independent of the number of orbitals or electrons. Some applications of the proposed algorithm, and possible extensions to eigenfunctions of ?² and other operators, are discussed.     

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     

Ab initio calculation of dynamic polarizability and dielectric constant of carbon and silicon cubic crystals

Valence and conduction bands of carbon silicon cubic systems are first obtained by a process called linear combination of atomic orbitals self-consistent field (LCAO-SCF), both at the Hartree-Fock (HF) and local density approximation (LDA) levels. Then, the crystalline orbitals are used in a sum-over-states (SOS) method to calculate the corresponding dielectric constants related to electronic polarizabilities. This method allows parallel computations with large granularity of the optical properties and leads to uncoupled HF and LDA results. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1253–1263, 1997