Convergence optimization of restricted open-shell self-consistent field calculations

Different self-consistent field (SCF) iteration schemes for open-shell systems are discussed. After a brief summary of the well-known level shifting and damping procedure, we describe the quadratically convergent SCF (QCSCF) approach based on the gradient and the Hessian matrix in a space of orbital rotation parameters. An analytical expression for the latter is derived for the general many-shell case. Starting from the expression for the energy change obtained by the QCSCF method, we then present a simplified direct procedure avoiding matrix diagonalization but also the difficulties of the QCSCF method in handling the Hessian matrix. Numerical calculations on some open-shell systems involving transition-metal complexes show that this method leads to rapid and reliable convergence of the iteration process in cases where the usual SCF procedure of iterative diagonalization tends to diverge. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 617–637, 1997     

球胶态颗粒在中等电位条件下双电层相互作用能和力

根据有效表面电位的定义,使用改进的Derjaguin法,推导出在中等电位条件下等同和不等同球形胶态颗粒相互作用能和力的近似表达式.与精确的数值解相比,对等同球颗粒,表面电位≤100mV;对不等同球颗粒,表面电位≤75mV;最大相对误差均小于±10%,表明该近似式是目前较好的.     

Coupled multiconfiguration self-consistent field (MC SCF) perturbation theory

The analytical form of the perturbation theory for the MC SCF method of Veillard and Clementi is presented. The appropriate second-order energy functional which takes into account the self-consistency requirements, leads to a set of coupled first-order perturbed equations determining the perturbed configuration coefficients and orbitals. The second-order energy formula derived from this functional can be given a clear physical interpretation. The present analytical approach is compared with the finite perturbation MC SCF scheme. The possibility of the approximate solution of the coupled MC SCF perturbation equations is also discussed and the so-called uncoupled procedures are devised. In the limit of the single determinant wave function the present formulae are shown to be equivalent to the appropriate Hartree-Fock perturbation results. The differences between the one-configuration SCF and the MC SCF approach are illustrated by the calculation of the electric dipole polarizability of. HZ in the CNDO/2 approximation. It is shown that the one-configuration SCF approaches cannot account for the correct asymptotic properties of the second-order energy for large internuclear distances. This feature of the SCF perturbation theories does not depend on the specific approximations of the CNDO/2 scheme and is corrected by using the MC SCF perturbation theory.     

Approximate second order method for orbital optimization of SCF and MCSCF wavefunctions

A quasi-Newton method involving a diagonal guess orbital hessian with iterative updates has been proposed recently by Almlof for the optimization of closed shell self-consistent field (SCF) wavefunctions. The technique is extended in the present work to more general wavefunctions, ranging from open shell SCF through multiconfigurational SCF. A number of examples are presented to show that convergence for closed and open shell SCF rivals conventional direct inversion in the iterative subspace (DIIS). For multiconfigurational SCF wavefunctions, the method presented here requires more iterations than an exact second order program, but since each iteration is substantially faster, leads to a more efficient overall program. Received: 15 August 1996 / Accepted: 22 January 1997     

Perturbative virtual SCF CI treatment for energy levels of coupled oscillator systems

A perturbative SCF CI treatment to obtain energy levels of coupled oscillator systems is proposed. The method uses the virtual SCF basis set, and the SCF equations are solved by means of a perturbative treatment that provides the diagonal matrix elements involved in the CI calculation. The off-diagonal matrix elements are calculated using a commutation relationship derived from exact quantum theorems. Numerical results for several systems are obtained and compared with those from others SCF, SCF CI , and variational treatments.     

Effect of the Solute Cavity on the Solvation Energy and its Derivatives within the Framework of the Gaussian Charge Scheme

The treatment of the solvation charges using Gaussian functions in the polarizable continuum model results in a smooth potential energy surface. These charges are placed on top of the surface of the solute cavity. In this article, we study the effect of the solute cavity (van der Waals-type or solvent-excluded surface-type) using the Gaussian charge scheme within the framework of the conductor-like polarizable continuum model on (a) the accuracy and computational cost of the self-consistent field (SCF) energy and its gradient and on (b) the calculation of free energies of solvation. For that purpose, we have considered a large set of systems ranging from few atoms to more than 200 atoms in different solvents. Our results at the DFT level using the B3LYP functional and the def2-TZVP basis set show that the choice of the solute cavity does neither affect the accuracy nor the cost of calculations for small systems (< 100 atoms). For larger systems, the use of a vdW-type cavity is recommended, as it prevents small oscillations in the gradient (present when using a SES-type cavity), which affect the convergence of the SCF energy gradient. Regarding the free energies of solvation, we consider a solvent-dependent probe sphere to construct the solvent-accessible surface area required to calculate the nonelectrostatic contribution to the free energy of solvation. For this part, our results for a large set of organic molecules in different solvents agree with available experimental data with an accuracy lower than 1 kcal/mol for both polar and nonpolar solvents.     

Preparative solution crystallization fractionation: a simple and rapid fractionation method for the chemical composition separation of complex ethylene-propylene copolymers

The molecular structure elucidation of complex ethylene-propylene copolymers (EPCs) has benefited tremendously from the ability to combine preparative temperature rising elution fractionation (prep TREF) with various conventional analytical techniques. Recently reported, prep TREF-high-temperature solvent gradient interaction chromatography (HT-SGIC) (Cheruthazhekatt et. al, Macromolecules 45:2025–2034, 2012) is one of the most effective and highly useful coupled methods that allow for the exact measurement of the chemical composition distribution (CCD) present in various components of EPCs. The major drawback of prep TREF involving slow crystallization and elution steps is the long time per experiment. Here, we present a new and by far the simplest and fastest preparative fractionation method for complex polyolefins—preparative solution crystallization fractionation (prep SCF). The scope of the present study was to achieve a fast fractionation of complex bulk samples into an amorphous, semicrystalline and highly crystalline fraction, in sufficient amounts for the subsequent detailed compositional analysis. The effects of two different solvents, xylene and trichlorobenzene (TCB), and their influence on the solution crystallization of chemically different components of EPC were systematically investigated by combining prep SCF with crystallization analysis fractionation (CRYSTAF), FTIR, differential scanning calorimetry (DSC) and HT-SGIC analyses. Significant differences in the chemical composition of similar SCF fractions obtained from xylene and TCB were observed indicating the strong influence of the solvent on solution crystallization. Prep SCF-HT-SGIC results showed that, under similar experimental conditions, TCB as the fractionation solvent provides superior separation of complex semicrystalline ethylene-propylene (EP) components. Very interestingly, for the first time, separation of soluble fractions (30 °C) of iPP, EPC and PE homopolymer components in complex EPC was achieved by prep SCF in TCB. On the other hand, SCF fractionation in xylene provides a soluble fraction that is perfectly amorphous as has been shown by DSC and CRYSTAF. Based on these results, the present SCF approach and an updated method of the combination of prep SCF-HT-SGIC hold significant promise for the fractionation and characterization of similar complex EPCs in a simple way within a short analysis time, by using significantly smaller amounts of solvent compared to the previously reported, rather time-consuming, prep TREF-HT-SGIC combination. No similarly selective solution crystallization fractionations in preparative scale have been reported before.

Improved intermolecular SCF theory and the BSSE problem

Analytical and numerical studies are performed concerning the exclusion of the basis set superposition error (BSSE ) from the SCF calculations of intermolecular interactions. Based on these studies a new procedure is proposed, which consists of the following steps: (1) determine the orbitals by the SCF scheme based on the recent “chemical Hamiltonian approach” (CHA-SCF method), i.e., excluding the delocalization effects caused by BSSE , and then (2) calculate the usual energy expectation value. (This gives results superior to those obtained by the previous nonsymmetric CHA energy formula.) The actual numerical calculations performed for different simple systems (He₂, water dimer) by using various basis sets indicate that the CHA/CE (CHA with “conventional energy” formula) potential curves are well-balanced and are close to those obtained by the Boys–Bernardi (BB ) method and usually (but not necessarily) go slightly beyond the latter. So our method gives results better than (or close to) those given by the BB method by performing only a single ～N⁴ calculation at each geometrical arrangement of the system.     

The trust-region self-consistent field method: towards a black-box optimization in Hartree-Fock and Kohn-Sham theories

The trust-region self-consistent field (TRSCF) method is presented for optimizing the total energy E(SCF) of Hartree-Fock theory and Kohn-Sham density-functional theory. In the TRSCF method, both the Fock/Kohn-Sham matrix diagonalization step to obtain a new density matrix and the step to determine the optimal density matrix in the subspace of the density matrices of the preceding diagonalization steps have been improved. The improvements follow from the recognition that local models to E(SCF) may be introduced by carrying out a Taylor expansion of the energy about the current density matrix. At the point of expansion, the local models have the same gradient as E(SCF) but only an approximate Hessian. The local models are therefore valid only in a restricted region-the trust region-and steps can only be taken with confidence within this region. By restricting the steps of the TRSCF model to be inside the trust region, a monotonic and significant reduction of the total energy is ensured in each iteration of the TRSCF method. Examples are given where the TRSCF method converges monotonically and smoothly, but where the standard DIIS method diverges.     

Accelerated, energy-conserving Born-Oppenheimer molecular dynamics via Fock matrix extrapolation

Born-Oppenheimer molecular dynamics calculations, especially those that exploit information retained from previous time steps in order to accelerate convergence of the electronic structure calculations, can suffer from systematic error in the energy gradient that manifests as a drift in the microcanonical energy. Here, we demonstrate that this is only the case when the self-consistent field (SCF) convergence criterion is set too low; using only a marginally tighter threshold (still two orders of magnitude lower than what is standard for geometry optimizations), the drift disappears completely, for a time scale of several picoseconds. Using a Fock matrix extrapolation technique, SCF convergence is achieved in as few as three iterations per time step, without sacrificing energy conservation. In test calculations for C2F4, (H2O4-, (H2O6, and [Fe(H2O6]2+, we demonstrate energy-conserving Fock matrix extrapolation that reduces the number of SCF cycles by up to 70% and reduces the computer time per molecular dynamics step by 45-55%, relative to simulations performed without extrapolation.     

Multiple stationary point representations in MC SCF calculations

By using a complete second-order Newton-Raphson multiconfigurational self-consistent field (MC SCF) procedure combined with the Fletcher restricted step constraint algorithm and a modification of the surface walking procedure of Simons et al., an MC SCF energy hypersurface at fixed geometry has been examined in considerably more detail than had been done previously. By calculational example, it is shown that there may exist several MC SCF stationary points which fulfill all four structural criteria we require of a state for being a “good” representation of an exact state. The problem with the existence of several stationary point solutions may be reduced if care is taken in the selection of the MC SCF configuration space. Calculational examples also demonstrate that near-lying stationary points exist which fulfill some, but not all, of these four structural criteria. Hence, stationary points should be obtained with a global MC SCF method which automatically eliminates convergence to as many as possible of these unwanted stationary points. Upon convergence, structural criteria which are not automatically fulfilled should be examined in detail.     

Quasi-degenerate perturbation theory with general multiconfiguration self-consistent field reference functions

The quasi-degenerate perturbation theory (QDPT) with complete active space (CAS) self-consistent field (SCF) reference functions is extended to the general multiconfiguration (MC) SCF references functions case. A computational scheme that utilizes both diagrammatic and sum-over-states approaches is presented. The second-order effective Hamiltonian is computed for the external intermediate configurations (including virtual or/and core orbitals) by the diagrammatic approach and for internal intermediate configurations (including only active orbitals) by the configuration interaction matrix-based sum-over-states approach. The method is tested on the calculations of excitation energies of H(2)O, potential energy curves of LiF, and valence excitation energies of H(2)CO. The results show that the present method yields very close results to the corresponding CAS-SCF reference QDPT results and the available experimental values. The deviations from CAS-SCF reference QDPT values are less than 0.1 eV on the average for the excitation energies of H(2)O and less than 1 kcal/mol for the potential energy curves of LiF. In the calculation of the valence excited energies of H(2)CO, the maximum deviation from available experimental values is 0.28 eV.     

Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals

A method is presented for expressing the occupied self-consistent-field (SCF) orbitals of a molecule exactly in terms of chemically deformed atomic minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The molecular orbitals referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact atomic SCF orbitals, and the formulation of the deformed "quasiatomic minimal-basis-sets" is independent of the calculational atomic orbital basis used. The resulting resolution of molecular orbitals in terms of quasiatomic minimal basis set orbitals is therefore intrinsic to the exact molecular wave functions. The deformations are analyzed in terms of interatomic contributions. The Mulliken population analysis is formulated in terms of the quasiatomic minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quantitative ab initio formulation of the qualitative model of virtual valence orbitals, which are useful for calculating electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham density functional theory orbitals and is easily generalized to valence MCSCF orbitals.     

Self-consistent field calculations of excited states using the maximum overlap method (MOM)

We present a simple algorithm, which we call the maximum overlap method (MOM), for finding excited-state solutions to self-consistent field (SCF) equations. Instead of using the aufbau principle, the algorithm maximizes the overlap between the occupied orbitals on successive SCF iterations. This prevents variational collapse to the ground state and guides the SCF process toward the nearest, rather than the lowest energy, solution. The resulting excited-state solutions can be treated in the same way as the ground-state solution and, in particular, derivatives of excited-state energies can be computed using ground-state code. We assess the performance of our method by applying it to a variety of excited-state problems including the calculation of excitation energies, charge-transfer states, and excited-state properties.     

On the Calculation of the Anharmonic Effect of Unimolecular Reactions of HOCO

HOCO is the intermediate of the reaction H + CO₂ → HO + CO. In this study, all the geometries in the collision reaction H + CO₂ were optimized at MP2/6‐311++G** level with Gaussian 03 program and a potential energy surface which shows that three unimolecular reactions were in the process of HOCO → HO + CO. For the three reactions, YL method proposed by L. Yao and S. H. Lin is applied to calculate the anharmonic and harmonic total number of states, the density of states and rate constants. The anharmonic values for rate constants calculated in this study are much lower than harmonic values, which indicate that anharmonic effects are significant and can not be neglected. After convert the experimental lifetime of HOCO into rate constants, the values are close to the calculations in our research, which shows that YL method used in our study is suitable for studying the rate constants of unimolecular reaction.     

MORMIN: A quasi-Newtonian energy minimizer fitting the nuclear overhauser data

In this article, we describe the program MORMIN, which can simultaneously minimize the mechanical energy of a given macromolecular structure, together with a weighted quadratic penalty function of the difference between the observed and computed nuclear Overhauser effect (nOe) peaks. The gradient of the nOe penalty function relatively to the proton coordinates is computed from an exact closed formula of a matrix exponential derivative. To cut CPU time, the molecular system is partitioned into nonoverlapping subsets containing the protons involved in the observed peaks. The algorithm is no longer exact, but if a 1% relative error is accepted it can be run, on a scalar computer, in about the same CPU time as needed for the calculation of the mechanical energy. We have successfully run the program in more than 1000 situations, including cases where the hybrid method failed because of the occurrence of negative eigenvalues. In some cases, the optimization of the Cartesian coordinates could be successfully extended to individual atomic diffusion times. © 1993 John Wiley & Sons, Inc.     

Conformational analysis of primary ethylene ozonide by gradient and multiconfigurational scf calculations

Four conformers of ethylene primary ozonide have been studied by ab initio gradient and MC SCF calculations, using gaussian-type basis functions. The MC SCF results indicate that the conformers are not as close in energy as suggested from single-determinant SCF calculations. The oxygen-oxygen and carbon-oxygen half-chair structures are much lower in energy than the carbon-carbon half-chair.     

An SCF-CI method for determining the potential energy surface of a triatomic molecule

A self-consistent-field (SCF)-configuration interaction (CI) (SCF-CI) method for determining the potential energy surface of a triatomic molecule from the observed vibrational band origins has been suggested. By this method, the SCF-CI procedure in the internal coordinates is used to calculate the vibrational bond origins and their first derivatives with respect to parameters in the potential energy function using the exact vibrational Hamiltonian, and the optimizer LMF in the nonlinear-squares problem is employed to optimize parameters in the potential energy function. This approach is used to optimize the potential energy function of the water molecule. The standard deviation of this fitting to the 70 observed band origins is 1.154cm-1.     

Gradient and mc scf calculations of the conformers and the formation of primary ethylene ozonide

The confonners of primary ethylene ozonide have been studied by ab initio gradient and MC SCF calculations. At the MC SCF level they are more spread in energy than in SCF calculations. The planar conformer, carbon-carbon half chair and the oxygen envelope are much higher m energy than the other conformers. The MC SCF activation energy for cyclo-addition of ozone and ethylene is 91–99 $\text{kJ}\text{mole}$.     

Diminished gradient dependence of density functionals: constraint satisfaction and self-interaction correction

The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation for the exchange-correlation energy functional has two nonempirical constructions, based on satisfaction of universal exact constraints on the hole density or on the energy. We show here that, by identifying one possible free parameter in exchange and a second in correlation, we can continue to satisfy these constraints while diminishing the gradient dependence almost to zero (i.e., almost recovering the local spin density approximation or LSDA). This points out the important role played by the Perdew-Wang 1991 nonempirical hole construction in shaping PBE and later constructions. Only the undiminished PBE is good for atoms and molecules, for reasons we present, but a somewhat diminished PBE could be useful for solids; in particular, the surface energies of solids could be improved. Even for atoms and molecules, a strongly diminished PBE works well when combined with a scaled-down self-interaction correction (although perhaps not significantly better than LSDA). This shows that the undiminished gradient dependence of PBE and related functionals works somewhat like a scaled-down self-interaction correction to LSDA.