Spin-projected EHF method. IV. Comparison of potential curves given by different one-electron methods

Some aspects of the computer realization of the spin-projected extended Hartree–Fock (EHF ) method at the ab initio level are briefly discussed for the algorithm of solution developed in the previous papers of this series. Calculations have been performed for the BH molecule by using a small basis of contracted Gaussian lobes with the purpose of comparing the potential curves given by the different one-electron methods RHF , UHF , UHF with subsequent spin projection, and EHF . It is concluded that the UHF and, in particular, the EHF methods give a qualitatively correct shape of the potential curve; the RHF method shows the known incorrect dissociation behavior while the potential curve obtained by subsequent spin projection of the UHF wave function exhibits spurious extrema at intermediate internuclear separations.     

Constrained active space unrestricted mean-field methods for controlling spin-contamination

We have recently proposed a novel approach for obtaining high-spin restricted open-shell Hartree-Fock wave functions by imposing constraints on the unrestricted Hartree-Fock (UHF) method [T. Tsuchimochi and G. E. Scuseria, J. Chem. Phys. 133, 141102 (2010)]. We here extend these ideas to the case where the constraints are released in an active space but imposed elsewhere. If the active space is properly chosen, our constrained UHF (CUHF) method greatly benefits from a controlled broken-symmetry effect while avoiding the massive spin contamination of traditional UHF. We also revisit and apply Lo?wdin's projection operator to CUHF and obtain multireference wave functions with moderate computational cost. We report singlet-triplet energy splittings showing that our constrained scheme outperforms fully unrestricted methods. This constrained approach can be readily used in spin density functional theory with similar favorable effects.     

An analysis of the topology of the electron charge density and the reactant–product electronic structure variation along the intrinsic reaction coordinate

In this work the topology of the electron charge density and the variations in the reactant and product electronic structures are analyzed along the Fukui intrinsic reaction coordinate (IRC). The systems studied are the ionic and the Menschutkin S_N2 reactions. This study is performed at ab initio RHF and MP2 levels, and density functional level, employing the B3LYP functional. The basis set in all cases is of split valence type and includes diffuse and polarization functions in nonhydrogen atoms 6‐31+G*. As a measure of the variations of reactant and product electronic structures, we calculate at the RHF level, the overlap integral between the total wavefunction and the wavefunction based on the reactant (or product) localized fragment orbitals. This integral can be interpreted, in Hilbert space, as the cosine of the angle between the vector representing the electronic structure of the molecule in each point of the IRC and that of reactant (or product) electronic structure. The calculated molecular properties were analyzed in light of the valence bond approach, and qualitative differences were noted depending on the property studied. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

The influence of basis sets on wave function tails

Wave function tails are analyzed quantitatively by investigating the dependence of exterior electron density (EED ) on basis sets; the EED is defined as the integrated electron density outside the repulsive molecular surface. Ab initio MO calculations with large scale basis sets were performed to establish the benchmark order of EED values for valence orbitals of some simple molecules. It is found that very popular basis sets, such as 4-31G, which are determined by energy optimization, are inferior in describing the wave function tails to some similar size basis sets, such as MIDI -4, which are obtained by least-squares fit to near Hartree-Fock atomic functions. Further the EED values for atomic 2s functions are shown to be unfavorably smaller than those for atomic 2p functions when the same value is used for the exponent α in the GTO basis sets. This indicates that the frequently used constraint α_s = α_p is not appropriate for describing wave function tails with medium-size basis sets. Deficiencies in the energy-optimized basis sets are found to become more serious for molecules including heavier atoms.     

The ab initio calculation of molecular electric, magnetic and geometric properties

We give an account of some recent advances in the development of ab initio methods for the calculation of molecular response properties, involving electric, magnetic, and geometric perturbations. Particular attention is given to properties in which the basis functions depend explicitly both on time and on the applied perturbations such as perturbations involving nuclear displacements or external magnetic fields when London atomic orbitals are used. We summarize a general framework based on the quasienergy for the calculation of arbitrary-order molecular properties using the elements of the density matrix in the atomic-orbital basis as the basic variables. We demonstrate that the necessary perturbed density matrices of arbitrary order can be determined from a set of linear equations that have the same formal structure as the set of linear equations encountered when determining the linear response equations (or time-dependent self-consistent-field equations). Additional components needed to calculate properties involving perturbation-dependent basis sets are flexible one- and two-electron integral techniques for geometric or magnetic-field differentiated integrals; in Kohn-Sham density-functional theory (KS-DFT), we also need to calculate derivatives of the exchange-correlation functional. We describe a recent proposal for evaluating these contributions based on automatic differentiation. Within this framework, it is now possible to calculate any molecular property for an arbitrary self-consistent-field reference state, including two- and four-component relativistic self-consistent-field wave functions. Examples of calculations that can be performed with this formulation are presented.     

Constrained optimization in delocalized internal coordinates

Using the recently introduced delocalized internal coordinates, in conjunction with the classical method of Lagrange multipliers, an algorithm for constrained optimization is presented in which the desired constraints do not have to be satisfied in the starting geometry. The method used is related to a previous algorithm by the same author for constrained optimization in Cartesian coordinates [J. Comput. Chem., 13 , 240 (1992)], but is simpler and far more efficient. Any internal (distance or angle/torsion) constraint can be imposed between any atoms in the system whether or not the atoms involved are formally bonded. Imposed constraints can be satisfied exactly. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :1079–1095, 1997     

Calculations concerning Kato's equations

Calculations of Kato's equations have been carried out as a function of the coordinates of the system using a number of approximate wave functions in order to determine the effects of the energy, electron density, constraints, and the functional form of the wave functions on how well Kato's equations are obeyed. It is shown that by putting constraints on the functional form of some wave functions, the wave functions will give the exact value of Kato's equations.     

Wave functions derived from experiment. V. Investigation of electron densities,electrostatic potentials,and electron localization functions for noncentrosymmetric crystals

The constrained Hartree-Fock method using experimental X-ray diffraction data is extended and applied to the case of noncentrosymmetric molecular crystals. A new way to estimate the errors in derived properties as a derivative with respect to added Gaussian noise is also described. Three molecular crystals are examined: ammonia [NH(3)], urea [CO(NH(2))(2)], and alloxan [(CO)(4)(NH)(2)]. The energetic and electrical properties of these molecules in the crystalline state are presented. In all cases, an enhancement of the dipole moment is observed upon application of the experimental constraint. It is found that the phases of the structure factors are robustly determined by the constrained Hartree-Fock model, even in the presence of simulated noise. Plots of the electron density, electrostatic potential, and the electron localization function for the molecules in the crystal are displayed. In general, relative to the Hartree-Fock model, there is a depletion of charge around hydrogen atoms and lone pair regions, and a build-up of charge within the molecular framework near nuclei, directed along the bonds. The electron localization function plots reveal an increase in the pair density between vicinal hydrogen atoms.     

RHF and two-configuration SCF calculations are inappropriate for conjugated diradicals

Restricted Hartee Fock (RHF) and two-configuration self-consistent field (TCSCF) calculations provide qualitatively correct molecular orbitals for the two open-shell electrons in diradicals. Nevertheless, these calculations fail to give correct relative energies and in some cases they even lead to incorrect geometries. Examples of these failures are given for both singlet and triplet states of some conjugated diradicals. In several cases these failures are related to the “doublet instability problem” in RHF calculations on radicals. It is argued that unrestricted Hartee-Fock (UHF) calculations on triplet states are more likely that RHF to provide accurate geometries.     

PIDAZTA: Structurally Constrained Chelators for the Efficient Formation of Stable Gallium-68 Complexes at Physiological pH

Two structurally constrained chelators based on a fused bicyclic scaffold, 4-amino-4-methylperhydro-pyrido[1,2-a][1,4]diazepin-N,N′,N′-triacetic acids [(4R*,10aS*)-PIDAZTA ( L1 ) and (4R*,10aR*)-PIDAZTA ( L2 )], were designed for the preparation of Ga^III-based radiopharmaceuticals. The stereochemistry of the ligand scaffold has a deep impact on the properties of the complexes, with unexpected [Ga( L2 )OH] species being superior in terms of both thermodynamic stability and inertness. This peculiar behavior was rationalized on the basis of molecular modeling and appears to be related to a better fit in size of Ga^III into the cavity of L2 . Fast and efficient formation of the Ga^III chelates at room temperature was observed at pH values between 7 and 8, which enables ⁶⁸Ga radiolabeling under truly physiological conditions (pH 7.4).     

Approximate transferability in alkanenitriles

The atomic and bond properties of a series of alkanenitriles were calculated in order to analyze the transferability of the CN, methyl, and methylene groups. The calculations were carried out using the atoms in molecules (AIM) theory on RHF/6‐31++G**// RHF/6‐31G** wave functions obtained for compounds CN–R (R ranging from H to C₁₁H₂₃). Linear correlations between L(Ω) and N(Ω) were used to establish N(CH₂) and N(CH₃) nearly transferable values. Average values and maximum differences to the mean value of several properties were used for classifying the CN group. It shows a transferable behavior along the CN–R series for R>Et. The methyl group presents specific properties when R<Pr. The methylene groups can be classified considering both their position with respect to the end of the chain and the position with respect to the CN group. The atomic energy displays a dependence on the molecular size. Although this behavior does not allow to consider this property as transferable, both the ab initio total electronic molecular energies and the experimental heats of formation can be fitted, by linear regression analysis, as a function of the number of methylene groups. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Matrix elements of multi-electron operator in restricted Hartree-Fock theory

A theorem has been established for the matrix elements of a general t-electron operator between N-dimensional determinantal wave functions arising in the solution of the atomic and molecular multi-electron problem by the restricted Hartree-Fock (RHF) method (1 ≤ t ≤ N). The required matrix elements of this operator are sums of matrix elements over t-dimensional basic determinantal wave functions. The final results are especially useful in the determination of multi-electron properties for atoms and molecules when the Hylleraas approach in RHF theory is employed.     

Theoretical approach to fluorine substitution in X2NO and X2CN free radicals. Comparison between ab initio UHF and RHF + perturbation treatments

Non-empirical calculations have been performed to analyze the effects of fluorine substitution on the geometry and electronic properties of two series of π and σ radicals. Both UHF and RHF + perturbation methods have been used and the results are compared as a function of the basis set quality. As concerns geometry and spin-free electronic properties the results are independent of the UHF or RHF formalism, but highly sensitive to the basis set. The STO-3G basis is unable to reproduce even general trends. Polarization functions always play a relevant role and correlation effects seem not negligible at least for fluorine-containing radicals. The molecular shape of π radicals changes from a planar to a pyramidal geometry when increasing the electronegativity of the substituents. On the contrary, σ radicals always remain planar. Unprojected UHF spin densities are closer to the RHF + perturbation results for small spin contamination (X₂NO). On the contrary, it is the projected UHF spin density which is in better agreement with the RHF + perturbation value for large spin contamination (X₂CN). No simple correlation can be found between spin densities and gross atomic spin populations. For H₂NO the spin density at nitrogen is smaller than at the oxygen nucleus, but substitution may enhance or reverse this trend.     

The Hartree-Fock dissociation of F2

We examine the stability of the restricted Hartree-Fock (RHF) wave function for F₂ in the vicinity of the equilibrium internuclear distance (R=R _e ) and the shape of the unrestricted Hartree-Fock (UHF) potential energy curve for the same system. The results depend on the basis set: With a split valence plus polarization basis, 6-31G(d), the RHF wave function is unstable at R _e , and the UHF potential curve is purely dissociative. When the basis is extended to 6-311+G(3d) or 6-311+G(3df), the RHF wave function becomes stable, and the UHF potential curve acquires a local maximum for R slightly (0.02 å) greater than R _e . The local maximum, however, is only 0.1 kcal/mol higher than the local minimum at R=R _e .     

Atomic dipole moments calculated using analytical molecular second-moment gradients

We have implemented analytical second-moment gradients for Hartree-Fock and multiconfigurational self-consistent-field wave functions. The code is used to calculate atomic dipole moments based on the generalized atomic polar tensor (GAPT) formalism [Phys. Rev. Lett. 62, 1469 (1989)], and the proposal of Dinur and Hagler (DH) for the calculation of atomic multipoles [J. Chem. Phys. 91, 2949 (1989)]. Both approaches display smooth basis-set convergence toward a well-defined basis-set limit and give reasonable electron correlation effects on the calculated atomic properties. However, the atomic charges and atomic dipole moments obtained from the GAPT partitioning scheme are unable to provide even qualitatively meaningful molecular quadrupole moments for some molecules, and thus the atomic multipole moments calculated in this scheme cannot be considered well suited for analyzing the electron density in molecules and for calculating intermolecular interaction energies. In contrast, the DH approach gives atomic charges and dipole moments that by definition exactly reproduce the molecular quadrupole moments. The approach of DH is, however, restricted to planar molecules and thus suffers from not being applicable to molecules of arbitrary shape. Both the GAPT and DH approaches give rather poor results for octupole and hexadecapole moments, indicating that at least atomic quadrupole moments are required for an accurate representation of the molecular charge distribution in terms of atomic electric moments.     

磷的d轨道在H~3PO分子中的作用

用分子片轨道在分子环境中发生极化的概念研究d轨道在H~3PO分子中的作用。H~3PO分子被分为两个分子片---H~3P和O.在RHF/6-31G^*水平上计算出分子环境中的极化了的分子片轨道(FOM)。再剔除d函数为主的FOM,用剩余的FOM为基进行构型优化,得到与RHF/6-31G^*相近的结果。这一结果说明磷原子的d函数在H~3PO分子中仅仅起一个极化函数的作用,而不是起价轨道作用。     

Role of coupling operators on open shell RHF equations

We have derived the expressions for the extremum condition of E, corresponding to any wave function. These expressions are given as a function of the spin orbitals. We have carried out the derivation considering the spin orbitals as vectors belonging to an orthonormal basis. The corresponding variational equations have been derived introducing the condition that the norm of the wave function is constant, as the only additional constraint.From the expression obtained for the first variation of the matrix elements of H, as a function of the spin orbitals, we have derived the RHF equations for a simple case.In the present procedure, the couplings between orbitals of different shells appear directly, being defined explicitly, and they may be taken as corresponding with the elements of a Hermitian matrix.The calculations that we have carried out show that the coupling operators defined in the paper give results which are variationally correct.     

Variational second order density matrix study of F3-: importance of subspace constraints for size-consistency

Variational second order density matrix theory under "two-positivity" constraints tends to dissociate molecules into unphysical fractionally charged products with too low energies. We aim to construct a qualitatively correct potential energy surface for F(3)(-) by applying subspace energy constraints on mono- and diatomic subspaces of the molecular basis space. Monoatomic subspace constraints do not guarantee correct dissociation: the constraints are thus geometry dependent. Furthermore, the number of subspace constraints needed for correct dissociation does not grow linearly with the number of atoms. The subspace constraints do impose correct chemical properties in the dissociation limit and size-consistency, but the structure of the resulting second order density matrix method does not exactly correspond to a system of noninteracting units.     

A general,recursive, and open‐ended response code

We present a new implementation of a recent open‐ended response theory formulation for time‐ and perturbation‐dependent basis sets (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) at the Hartree–Fock and density functional levels of theory. A novel feature of the new implementation is the use of recursive programming techniques, making it possible to write highly compact code for the analytic calculation of any response property at any valid choice of rule for the order of perturbation at which to include perturbed density matrices. The formalism is expressed in terms of the density matrix in the atomic orbital basis, allowing the recursive scheme presented here to be used in linear‐scaling formulations of response theory as well as with two‐ and four‐component relativistic wave functions. To demonstrate the new code, we present calculations of the third geometrical derivatives of the frequency‐dependent second hyperpolarizability for HSOH at the Hartree–Fock level of theory, a seventh‐order energy derivative involving basis sets that are both time and perturbation dependent. © 2014 Wiley Periodicals, Inc.