价键方法中的非活性轨道优化新算法

本文通过引进一组正交的辅助非活性轨道和与它正交的辅助活性轨道,将价键理论方法中的冻核近似推广到轨道非正交的情形,得到了体系能量及其对非活性轨道的梯度解析表达式,简化了价键自洽场方法中非正交轨道能量梯度的计算．该方法的标度为（Na＋1）m^4,其中Na和m分别是活性轨道和基函数的个数．分析表明,与现有的其他算法相比较,该方法具有更低的计算标度,因而计算效率更高．     

The application of cholesky decomposition in valence bond calculation

The Cholesky decomposition (CD) technique, used to approximate the two‐electron repulsion integrals (ERIs), is applied to the valence bond self‐consistent field (VBSCF) method. Test calculations on ethylene, C_2nH_2n+2, and C_2nH_4n?2 molecules (n = 1–7) show that the performance of the VBSCF method is much improved using the CD technique, and thus, the integral transformation from basis functions to VB orbitals is no longer the bottleneck in VBSCF calculations. The errors of the CD‐based ERIs and of the total energy are controlled by the CD threshold, for which a value of 10^?6 ensures to control the total energy error within 10^?6 Hartree. © 2016 Wiley Periodicals, Inc.     

Reply to comment on the paper “An efficient Algorithm for Energy Gradients and Orbital Optimization in Valence Bond Theory”

van Lenthe, Broer, and Rashid made comments on our 2009 paper [Song et al., J. Comput. Chem. 2009, 30, 399] by criticizing that we did not properly reference the work by Broer and Nieuwpoort in 1988 [Broer and Nieuwpoort, Theor. Chim. Acta. 1988, 73, 405], and we favorably compared our valence bond self‐consistent field (VBSCF) algorithm with theirs. However, both criticisms are unjustified insignificant. The Broer–Nieuwpoort algorithm, properly cited in our paper, is for the evaluations of matrix elements between determinants of nonorthogonal orbitals. Stating that this algorithm “can be used for an orbital optimization” afterwards [van Lenthe et al., submitted] is not a plausible way to require more credits or even criticize others. While we stand by our statement that our algorithms scales at O(m⁴) and van Lenthe et al.'s approximate Newton Raphson algorithm scales at O(mN⁵) (here m and N are the numbers of basis functions and electrons), as we discussed in our original paper, it becomes obvious that any strict comparison among different algorithms is difficult, unproductive, and counteractive. © 2012 Wiley Periodicals, Inc.     

Graphical representation of a new algorithm for nonorthogonalab initio valence bond calculations

A pictorial representation of the algorithm using successive expansion method for the nonorthogonal VB calculations is given. With the help of this representation and the graph analysis, the efficiency of this algorithm is improved and theN! problem is reduced by a factor of about (N!)^1/2. Anab initio VB program for valence bond self-consistent-field (VBSCF) calculations has been implemented based on this algorithm. Some VBSCF calculations have been performed for systems of up to 14 electrons. The statistics of the CPU time of the calculations indicate that this new group-theoretical approach is quite practical.     

An integral direct, distributed-data, parallel MP2 algorithm

Summary A scalable integral direct, distributed-data parallel algorithm for four-index transformation is presented. The algorithm was implemented in the context of the second-order M?ller-Plesset (MP2) energy evaluation, yet it is easily adopted for other electron correlation methods, where only MO integrals with two indices in the virtual orbitals space are required. The major computational steps of the MP2 energy are the two-electron integral evaluationO(N ⁴) and transformation into the MO basisO(ON ⁴), whereN is the number of basis functions, andO the number of occupied orbitals, respectively. The associated maximal communication costs scale asO(n _Σ O ² V N), whereV andn _Σ denote the number of virtual orbitals, and the number of symmetry-unique shells. The largest local and global memory requirements areO(N ²) for the MO coefficients andO(OV N) for the three-quarter transformed integrals, respectively. Several aspects of the implementation such as symmetry-treatment, integral prescreening, and the distribution of data and computational tasks are discussed. The parallel efficiency of the algorithm is demonstrated by calculations on the phenanthrene molecule, with 762 primitive Gaussians, contracted to 412 basis functions. The calculations were performed on an IBM SP2 with 48 nodes. The measured wall clock time on 48 nodes is less than 15 min for this calculation, and the speedup relative to single-node execution is estimated to 527. This superlinear speedup is a result of exploiting both the compute power and the aggregate memory of the parallel computer. The latter reduces the number of passes through the AO integral list, and hence the operation count of the calculation. The test calculations also show that the evaluation of the two-electron integrals dominates the calculation, despite the higher scaling of the transformation step.     

INDO Calculation of Spin Densities in Radical Cations of a Series of Substituted Nitro- and m-Dinitrobenzenes

The spin densities in radical cations of 22 substituted nitro- and m-dinitrobenzenes were calculated by the INDO method. For radical anions of substituted nitrobenzenes, a good linear correlation was obtained between the spin densities s_Ns_N and experimental hyperfine coupling constants with the nitrogen atoms of the NO₂ groups (a ^N): a ^N = K _N s _N s _N, where K _N = 428.58 (R ² = 0.96). For radical anions of substi- tuted m-dinitrobenzenes, no satisfactory agreement between the calculated and experimental a ^N constants was attained.     

Thermodynamics of Silicon Nitride. Standard molar enthalpy of formation of amorphous Si3N4 at 298.15 K

The standard molar enthalpy of formation Δ_f H _m ⁰=–760±12 kJ for amorphous silicon nitride a-Si₃N₄ has been determined from fluorine combustion calorimetry measurements of the massic energy of the reaction: a-Si₃N₄(s)+6F₂(g)=3SiF₄(g)+2N₂(g). This value combined with Δ_f H _m ⁰= –828.9±3.4 kJ for a-Si₃N₄ indicates that determined for the first time molar enthalpy change for the transition from amorphous to α-crystalline form Δ_trs H _m ⁰=69±13 kJ is very evident, in spite of its large uncertainty range resulting from impurity corrections. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Photoelectron spectra andab initio calculations of the electronic structure of amino-n-sulfenyl chlorides

The electronic structure of some amino-N-sulfenyl chlorides and related compounds is studied by photoelectron spectroscopy and ab initio calculations. Similar values of IP(n _S ) and IP(n _N ) and the total energy minimum indicate that in stable conformations the n _S and n _N orbitals are orthogonal. These conformers are characterized by an effective n_N-σ _S-Cl ^* interaction. The relationship between the intramolecular shift of charge and ionization potential values and the spatial structure of amino-N-sulfenyl chlorides is analyzed. A. E. Arbuzov Institute of Organic and Physical Chemistry, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 35, No. 2, pp. 69–73, March–April, 1994. Translated by L. Smolina     

Unrestricted CNDO-MO calculations. II. MnO 4 2− and CrO 4 3−

The electronic structures of the tetrahedral molecule ions MnO ₄ ^2– and CrO ₄ ^3– have been investigated within an unrestricted CNDO-MO approximation [Theoret. Chim. Acta (Berl.)20, 317 (1971)]. Calculations assuming the unpaired electron occupies the 3a ₁, 2e, and 4t2 molecular orbitals indicate that the 3a ₁ and2e orbitals have similar orbital energies and that the 4t ₂ orbital is at a higher energy. The experimentally indicated2e orbital for the unpaired electron is obtained with expanded O^1– type atomic orbitals for oxygen and valence metal orbitals of the expanded 3d and plus one ion 4p types. The metal 4s orbitals must be held to the neutral atom type. The optimum valence orbitals above with a slightly contracted 4s type metal orbitals yield the minimum total energy and places the unpaired electron in the 3a ₁ orbital. Since the contracted 4s metal orbital produces results that are not in agreement with experimental data, the method used apparently does not adequately take into account the increased electron-electron repulsions that contracted 4s orbitals produce.     

Photoelectron spectrum and quantum chemical analysis of the structure of bis(1,3,6-trimethyluracilyl-5)methane

Photoelectron spectra of bis(1,3,6-trimethyluracilyl-5)methane (I) and 1,3,6-trimethyluracil (II) were studied; AM1 optimization of geometric characteristics was carried out. The total energy minimum and the best agreement between the values of IP_m and -ɛ_m were obtained for conformations with nearly orthogonal location of uracilyl fragments. In such conformations, the highest occupied orbitals are pseudodegenerate. To interpret the photoelectron spectra, we employed ab initio calculations in STO-3G and 4-31G basis sets. For uracil and its derivatives, all methods give the π, π, n⁻, n⁺, π sequence of the highest orbitals. A. E. Arbuzov Institute of Organic and Physical Chemistry, Kazan Scientific Center, Russian Academy of Sciences. Translated fromZhurnal Struktumoi Khimii, Vol. 36, No. 1, pp. 102–107, January–February, 1995. Translated by L. Smolina     

Construction of orthogonal subspaces

An orthogonalization procedure is presented that allows construction of at least (n?m) vectors orthogonal to {X_j}, j equals; 1, m, by linear combinations solely among {η_i}, i equals; 1, n, n>m, and 〈X_j/η_i〉≠0. An important application of the procedure is in effective core potential methods for which valence orbitals can be constructed that are orthogonal to the core orbitals and yet involve no component of the core. Thus, a separate calculation for only the valence electrons can be performed without any explicit reference to the core electrons (orbitals).     

The Fock Matrix Analysis for Atomic Orbitals in Molecular Orbitals II. The Electronic Structure of N2 Molecules

Weinhold's natural hybrid orbitals can be chosen as the molecular adapted atomic orbitals to build the canonical molecular orbitals of N₂ molecules. The molecular Fock matrix expanded in the natural hybrid orbitals can reveal deeper insight of the electronic structure and reaction of the N₂ molecule. For example, the magnitude of F_ab can signify the bonding character of the paired electrons as well as the diradical character of the unpaired electrons for both σ‐ and π‐types. Discarding the concept of the overlap between non‐orthogonal atomic orbitals, the different orbitals for different spins in the unrestricted Hartree‐Fock wavefunction reveal that there are three pairs of opposite spin density flows between two atoms, which proceed until the bonding molecular orbitals form.     

Extremely localized molecular orbitals (ELMO): a non-orthogonal Hartree-Fock method

A new optimization method for extremely localized molecular orbitals (ELMO) is derived in a non-orthogonal formalism. The method is based on a quasi Newton-Raphson algorithm in which an approximate diagonal-blocked Hessian matrix is calculated through the Fock matrix. The Hessian matrix inverse is updated at each iteration by a variable metric updating procedure to account for the intrinsically small coupling between the orbitals. The updated orbitals are obtained with approximately n ² operations. No n ³ processes such as matrix diagonalization, matrix multiplication or orbital orthogonalization are employed. The use of localized orbitals allows for the creation of high-quality initial “guess” orbitals from optimized molecular orbitals of small systems and thus reduces the number of iterations to converge. The delocalization effects are included by a Jacobi correction (JC) which allows the accurate calculation of the total energy with a limited number of operations. This extension, referred to as ELMO(JC), is a variational method that reproduces the Hartree-Fock (HF) energy with an error of less than 2 kcal/mol for a reduced total cost compared to standard HF methods. The small number of variables, even for a very large system, and the limited number of operations potentially makes ELMO a method of choice to study large systems. Received: 30 December 1996 / Accepted: 5 June 1997     

The origin of energy functional in Roothaan open shell SCF theory

Different methods of averaging of energy over the states of electronic configurations γ^N (n_γ = 1, 2, 3 and N = 1, 2, …, 2n_γ ? 1) leading to Roothaan' energy expression are considered. The consequent values of vector coupling coefficients (VCC ) in energy functionals for various states as well as for average values of energy are presented. It is shown also that in molecular systems of cubic and tetragonal symmetry having electronic configurations t^N (N = 2–4) and e² there exist states for which VCC are dependent on the choice of basis set of degenerate open-shell molecular orbitals. The origin of such “non-Roothaan” terms and peculiarities of its calculation by the restricted Hartree–Fock method are discussed.     

Activating Bi p-orbitals in Dispersed Clusters of Amorphous BiOx for Electrocatalytic Nitrogen Reduction

Catalytic strategies based on main group metals are significantly less advanced than those of transition metal catalysis, leaving untapped areas of potentially fruitful research. We here demonstrate an effective approach for the modulation of Bi 6p energy levels during the construction of atomically dispersed clusters of amorphous BiO_x. Bi oxidation state is proposed to strongly affects the nitrogen fixation activity, with the half-occupied p_z orbitals of the Bi²⁺ ions being highly efficient toward electron injection into the inert N₂ molecule. With sufficient catalytic sites to adsorb and activate N₂, the bonding between N₂ and catalyst is able to be in situ identified. The catalyst shows an outstanding Faraday efficiency (≈30 %) and high yield (≈113 μg h⁻¹ mg⁻¹_cat) in NH₃ production, outperforming most of the existing catalysts in aqueous solution. These results lay the basis for developing the potential of p-block elements for catalysis of multi-electron reactions.     

Spin-Projected EHF equations

Extended Hartree-Fock (EHF ) equations are developed for the general open-shell case using a modified pair-orthogonality-constrained variation (POCV ) method. The EHF energy is expressed in terms of corresponding orbitals that are required to remain orthogonal and paired for all arbitrary infinitesimal variations. The Euler equations for each set of orbitals are reduced to unique pseudosecular equations, the LCAO form of which may easily be derived. The Euler equations and the expressions obtained for the off-diagonal elements of the ?^γδ (γ, δ = a or b) matrices for the closed-shell case are identical to those obtained by Mayer, who used the generalized Brillouin theorem method. However, the present method yields equations for both closed- and open-shell cases and for any spin state.     

Convergence of an s‐Wave calculation of the He ground state

The configuration interaction (CI) method using a large Laguerre basis restricted to ? = 0 orbitals is applied to the calculation of the He ground state. The maximum number of orbitals included was 60. The numerical evidence suggests that the energy converges as ΔE^N ≈ A/N^7/2 + B/N^8/2 + …, where N is the number of Laguerre basis functions. The electron–electron δ‐function expectation converges as Δδ^N ≈ A/N^5/2 + B/N^6/2 + …, and the variational limit for the ? = 0 basis is estimated as 0.1557637174(2) a. It was seen that extrapolation of the energy to the variational limit is dependent on the basis dimension at which the exponent in the Laguerre basis was optimized. In effect, it may be best to choose a nonoptimal exponent if one wishes to extrapolate to the variational limit. An investigation of the natural orbital asymptotics revealed the energy converged as ΔE^N ≈ A/N⁶ + B/N⁷ + …, while the electron–electron δ‐function expectation converged as Δδ^N ≈ A/N⁴ + B/N⁵ + …. The asymptotics of expectation values other than the energy showed fluctuations that depended on whether N was even or odd. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Semi-empirically derived petrophysical and thermodynamical coefficients of permselective shales—Implications on ore mineralization

Phenomenological coefficients of shale–electrolyte systems may offer a glimpse into probable matrix-permeability and solute-exclusionary relationships. Shales from unexposed Upper Cretaceous Period Mancos Shale and from Permian Period Abo Formation were cut into thin wafers, placed in custom built osmometers and a chemical potential applied across them giving rise to induced osmotic flow. This in turn spawned matrix unique constants namely mechanical filtration coefficient L_P (m³ N⁻¹ s⁻¹), diffusional mobility per unit osmotic pressure L_PD (m³ N⁻¹ s⁻¹), osmotic flow coefficient; L_D (m³ N⁻¹ s⁻¹), reflection coefficient σ (dimensionless) at zero gradients of temperature and hydrostatic pressure. Considering intrinsic relationships between these constants where and L_PD = −σL_P, we have ascertained that the bentonitic fossiliferous Mancos shale had a lower L_P and a higher σ compared to the kaolinitic and siliceous shale from Abo Formation indicating a higher degree of compaction post-diagenesis (lower porosity) and higher filtration efficiency. Mechanistic processes involved in solute transport and matrix morphology indicate key multi-scale transformations, ionic- and atomic-exchange competitions on high energy sites like cation-exchange sites, isomorphic substitution at argillaceous mineral edges, atomic-clipping within basal spacing, preferential pathway migration, dead-end pores that give rise to localized solute exclusionary processes and solute attenuation giving rise to anomalous osmotic gradients.     

Two-electron repulsion integrals over Gaussian s functions

We present an efficient scheme to evaluate the [ 0 ]^(m) integrals that arise in many ab initio quantum chemical two-electron integral algorithms. The total number of floating-point operations (FLOPS ) required by the scheme has been carefully minimized, both for cases where multipole expansions of the integrals are admissable and for cases where this is not so. The algorithm is based on the use of a modified Chebyshev interpolation formula to compute the function exp(?T) and the integral F_m(T) = ∫₀¹u^2mexp(?Tu²) du very cheaply.