Self-consistent Field Approach Research Articles

Electron repulsion integrals for self-energy calculations

A fast algorithm for the calculation of the electron repulsion integrals in an atomic basis is a prerequisite of any ab initio quantum chemistry method. Unlike the case of a self-consistent field (SCF) approach, correlated methods often require a full or partial integral transformation to the molecular basis. The run-time of such an algorithm scales unfavorably as O(Nχ5), where Nχ is the number of the basis function, and additionally poses high requirements on the computer memory. The problem is aggravated in the case of large highly symmetric molecules which can only be modeled by fully taking the symmetry into account (as was recently demonstrated by us in J. Chem. Phys. 135 (2011) 201103). We describe here the algorithm for the calculation of the electron repulsion integrals, the transformation and their use in the correlated Green’s function approach for systems with icosahedral symmetry.

Ab Initio and Density Functional Theory Study of Keto–Enol Equilibria of Deltic Acid in Gas and Aqueous Solution Phase: A Bimolecular Proton Transfer Mechanism

Keto-enol tautomerism in deltic acid (2,3-dihydroxycycloprop-2-en-1-one) has been studied using ab initio methods and the B3LYP functional of density functional theory, as well as complete basis set (CBS-QB3 and CBS-APNO) and G4 methods. Relative and absolute energies were calculated with each of the methods, whereas computations of geometries and harmonic frequencies for dihydroxycyclopropenone and hydroxycyclopropanedione were computed in the gas phase but were limited to HF, MP2, and the B3LYP functional, in combination with the 6-31++G(3df,3pd) basis set. Using the MP2/6-31++G(3df,3pd) gas phase optimized structure, each species was then optimized fully in aqueous solution by using the polarizable continuum model (PCM) self-consistent reaction field approach, in which HF, MP2, and B3LYP levels of theory were utilized, with the same 6-31++G(3df,3pd) basis set. In both gas and aqueous solution phases, the keto form is higher in energy for all of the model chemistries considered. From the B3LYP/6-31++G(3df,3pd) Gibbs free energy, the keto-enol tautomeric equilibrium constant for 2,3-dihydroxycycloprop-2-en-1-one/3-hydroxy-1,2-cyclopropanedione is computed to be K(T)(gas) = 2.768 × 10(-12) and K(T)(aq) = 5.469 × 10(-14). It is concluded that the enol form is overwhelmingly predominant in both environments.

Adsorption of cobalt on graphene: Electron correlation effects from a quantum chemical perspective

In this work, we investigate the adsorption of a single cobalt atom (Co) on graphene by means of the complete active space self-consistent field approach, additionally corrected by the second-order perturbation theory. The local structure of graphene is modeled by a planar hydrocarbon cluster (C$_{24}$H$_{12}$). Systematic treatment of the electron correlations and the possibility to study excited states allow us to reproduce the potential energy curves for different electronic configurations of Co. We find that upon approaching the surface, the ground-state configuration of Co undergoes several transitions, giving rise to two stable states. The first corresponds to the physisorption of the adatom in the high-spin $3d^74s^2$ ($S=3/2$) configuration, while the second results from the chemical bonding formed by strong orbital hybridization, leading to the low-spin $3d^9$ ($S=1/2$) state. Due to the instability of the $3d^9$ configuration, the adsorption energy of Co is small in both cases and does not exceed 0.35 eV. We analyze the obtained results in terms of a simple model Hamiltonian that involves Coulomb repulsion ($U$) and exchange coupling ($J$) parameters for the 3$d$ shell of Co, which we estimate from first-principles calculations. We show that while the exchange interaction remains constant upon adsorption ($\simeq1.1$ eV), the Coulomb repulsion significantly reduces for decreasing distances (from 5.3 to 2.6$\pm$0.2 eV). The screening of $U$ favors higher occupations of the 3$d$ shell and thus is largely responsible for the interconfigurational transitions of Co. Finally, we discuss the limitations of the approaches that are based on density functional theory with respect to transition metal atoms on graphene, and we conclude that a proper account of the electron correlations is crucial for the description of adsorption in such systems.

An electronic structure theory investigation of the physical chemistry of the intermolecular complexes of cyclopropenylidene with hydrogen halides

The proton accepting and donating abilities of cyclopropenylidene (c-C(3)H(2)) on its complexation with hydrogen halides H-X (X = F, Cl, Br) are analyzed using density-functional theory with three functionals (PBE0, B3LYP, and B3LYP-D) and benchmarked against second-order Møller-Plesset (MP2) theory. Standard signatures including, inter alia, dipole moment enhancement, charge transfer from the carbenic lone pair to the antibonding σ*(H-X) orbital, and H-X bond elongation are examined to ascertain the presence of hydrogen bonding in these complexes. The latter property is found to be accompanied with a pronounced red shift in the bond stretching frequency and with a substantial increase in the infrared intensity of the band on complex formation. The MP2/aug-cc-pVTZ c-C(3)H(2)···H-F complex potential energy surface turns out to be an asymmetric deep single well, while asymmetric double wells are found for the c-C(3)H(2)···H-Cl and c-C(3)H(2)···H-Br complexes, with an energy barrier of 4.1 kcal mol(-1) for proton transfer along the hydrogen bond in the latter complex. Hydrogen-bond energy decomposition, with the reduced variational space self-consistent field approach, indicates that there are large polarization and charge-transfer interactions between the interacting partners in c-C(3)H(2)···H-Br compared to the other two complexes. The C···H bonds are found to be predominantly ionic with partial covalent character, unveiled by the quantum theory of atoms in molecules. The present results reveal that the c-C(3)H(2) carbene divalent carbon can act as a proton acceptor and is responsible for the formation of hydrogen bonds in the complexes investigated.

Hybrid Monte Carlo Self-Consistent Field Approach to Model a Thin Layer of a Polyelectrolyte Gel near an Adsorbing Surface

The use of thin layers of a surface bound (polyelectrolyte) hydrogels for measuring the concentration of metal ions from electrolyte solutions is our motivation for modeling such hydrogels. The gels are composed of polymeric species with conformational degrees of freedom on the nanometer scale. The polymer conformations are affected by the presence of cross-links in the gel on a five to ten times larger length scale, and the repulsive interactions generated by the charges along the chains. Here we present a hybrid computational Monte Carlo Self-consistent field (MC-SCF) approach to model such hydrogels. The SCF formalism is used to evaluate the conformational properties of the chains, implementing a freely jointed chain model, in between featureless cross-links. The Monte Carlo simulation method is used to sample the (restricted) translational degrees of freedom of the cross-links in the gel. We consider the case that the polymers in the gel have an affinity for surface positioned at the edge of the simulation volume. The polymer density decays as a power-law from the surface to the gel-density with an exponent close to -4/3. The gel features relatively large density fluctuations which is natural for a gel with a low density (φ ≈ 0.035), a low degree of cross-linking (average of three chainparts per cross-link), and relatively large chains (N = 50) in between the cross-links. Some parts of the gel can break loose from the gel and sample the adjoining volume. Representative snapshots exemplify large density fluctuations, which explain the large pore size distribution observed in experimental counterparts.

Collapse of Polyelectrolyte Star. Theory and Modeling

The collapse of hydrophobic polyelectrolyte stars in aqueous solutions is studied using the Scheutjens–Fleer self-consistent field (SF-SCF) approach. The hydrophobic property of the segments tends to compact the stars, whereas the presence of charges has the opposite effect. As a result, star conformations can be switched from an extended, strongly hydrated, and swollen state to a collapsed state via semicollapsed, quasi-micellar state using control parameters such as the solvent quality, specified by the Flory–Huggins parameter, the pH value or the ionic strength. More specifically, there exists a range of parameters wherein the stars have an inhomogeneous radial structure with a collapsed region, referred to as the core, and a swollen region forming the corona. In such microphase segregated state the fraction of arms of the star that form the core, or alternatively escape into the swollen corona, can be controlled. The SF-SCF analysis is complemented with analytical models to rationalize the complex phase behavior.

Low-lying dipole response: Isospin character and collectivity in68Ni,132Sn, and208Pb

The isospin character, the collective or single-particle nature, and the sensitivity to the slope of the nuclear symmetry energy of the low-energy isovector dipole response (known as pygmy dipole resonance) are nowadays under debate. In the present work we study, within the fully self-consistent non-relativistic mean field (MF) approach based on Skyrme Hartree-Fock plus Random Phase Approximation (RPA), the measured even-even nuclei ${}^{68}$Ni, ${}^{132}$Sn and ${}^{208}$Pb. To analyze the model dependence in the predictions of the pygmy dipole strength, we employ three different Skyrme parameter sets. We find that both the isoscalar and the isovector dipole responses of all three nuclei show a low-energy peak that increases in magnitude, and is shifted to larger excitation energies, with increasing values of the slope of the symmetry energy at saturation. We highlight the fact that the collectivity associated with the RPA state(s) contributing to this peak is different in the isoscalar and isovector case, or in other words it depends on the external probe. While the response of these RPA states to an isovector operator does not show a clear collective nature, the response to an isoscalar operator is recognizably collective, for {\it all} analyzed nuclei and {\it all} studied interactions.

A self-consistent field study of a hydrocarbon droplet at the air–water interface

A molecularly detailed self-consistent field (SCF) approach is applied to describe a sessile hydrocarbon droplet placed at the air-water interface. Predictions of the contact angle for macroscopic droplets follow from using Neumann's equation, wherein the macroscopic interfacial tensions are computed from one-gradient calculations for flat interfaces. A two-gradient cylindrical coordinate system with mirror-like boundary conditions is used to analyse the three dimensional shape of the nano-scale oil droplet at the air-water interface. These small droplets have a finite value of the Laplace pressure and concomitant line tension. It has been calculated that the oil-water and oil-vapour interfacial tensions are curvature dependent and increase slightly with increasing interfacial curvature. In contrast, the line tension tends to decrease with curvature. In all cases there is only a weak influence of the line tension on the droplet shape. We therefore argue that the nano-scale droplets, which are described in the SCF approach, are representative for macroscopic droplets and that the method can be used to efficiently generate accurate information on the spreading of oil droplets at the air-water interface in molecularly more complex situations. As an example, non-ionic surfactants have been included in the system to illustrate how a molecularly more complex situation will change the wetting properties of the sessile drop. This short forecast is aimed to outline and to stress the potential of the method.

Mixed protein–polysaccharide interfacial layers: effect of polysaccharide charge distribution

The influence of the polysaccharide charge distribution on the structure, thickness, and charge reversal of the interfacial layers, formed by adsorbed positively charged protein and oppositely charged polysaccharide, has been investigated using a lattice-based self-consistent field (SCF) approach. We compare the adsorption behaviour of a uniformly charged polysaccharide model with that consisting of a short and a long block carrying different charge densities. For homogeneously charged polysaccharide we observe a resulting interfacial layer that is closer to a mixed protein + polysaccharide film, rather than a multi-layer. We also find that the maximum adsorption of polysaccharide occurs at an optimal value of its charge, above and below which the adsorbed amount decreases. In contrast, for heterogeneously charged chains, as their charge is increasingly located on the shorter block, a much thicker interfacial layer results. In such cases the weakly charged longer blocks extend well away from the surface into the solution. The interfacial film begins to resemble a multi-layer with a primary protein and a distinct secondary polysaccharide layer. When the weakly charged long blocks still have a sufficient amount of negative charge, we also observe a reversal of the sign of surface potential from a positive to a negative value. Our SCF calculated values for the reversed surface potential are of the order of −25 mV, in good agreement with several experimental results involving ζ-potential measurements on particles covered with such protein + polysaccharide films.

The problem of hole localization in inner-shell states of N2 and CO2 revisited with complete active space self-consistent field approach

Potential energy curves for inner-shell states of nitrogen and carbon dioxide molecules are calculated by inner-shell complete active space self-consistent field (CASSCF) method, which is a protocol, recently proposed, to obtain specifically converged inner-shell states at multiconfigurational level. This is possible since the collapse of the wave function to a low-lying state is avoided by a sequence of constrained optimization in the orbital mixing step. The problem of localization of K-shell states is revisited by calculating their energies at CASSCF level based on both localized and delocalized orbitals. The localized basis presents the best results at this level of calculation. Transition energies are also calculated by perturbation theory, by taking the above mentioned MCSCF function as zeroth order wave function. Values for transition energy are in fairly good agreement with experimental ones. Bond dissociation energies for N(2) are considerably high, which means that these states are strongly bound. Potential curves along ground state normal modes of CO(2) indicate the occurrence of Renner-Teller effect in inner-shell states.

Temperature effects on the magnetoplasmon spectrum of a weakly modulated graphene monolayer

In this work, we determine the effects of temperature on the magnetoplasmon spectrum of an electrically modulated graphene monolayer as well as a two-dimensional electron gas (2DEG). The intra-Landau band magnetoplasmon spectrum within the self-consistent field approach is investigated for both the aforementioned systems. Results obtained not only exhibit Shubnikov–de Haas (SdH) oscillations but also commensurability oscillations (Weiss oscillations). These oscillations are periodic as a function of inverse magnetic field. We find that both the magnetic oscillations, SdH and Weiss, have a greater amplitude and are more robust against temperature in graphene compared to a conventional 2DEG. Furthermore, there is a π phase shift between the magnetoplasmon oscillations in the two systems which can be attributed to Dirac electrons in graphene acquiring a Berry’s phase as they traverse a closed path in a magnetic field.

Towards a scalable and accurate quantum approach for describing vibrations of molecule–metal interfaces

We present a theoretical framework for the computation of anharmonic vibrational frequencies for large systems, with a particular focus on determining adsorbate frequencies from first principles. We give a detailed account of our local implementation of the vibrational self-consistent field approach and its correlation corrections. We show that our approach is both robust, accurate and can be easily deployed on computational grids in order to provide an efficient computational tool. We also present results on the vibrational spectrum of hydrogen fluoride on pyrene, on the thiophene molecule in the gas phase, and on small neutral gold clusters.

The generalized active space concept in multiconfigurational self-consistent field methods

A multiconfigurational self-consistent field method based on the concept of generalized active space (GAS) is presented. GAS wave functions are obtained by defining an arbitrary number of active spaces with arbitrary occupation constraints. By a suitable choice of the GAS spaces, numerous ineffective configurations present in a large complete active space (CAS) can be removed, while keeping the important ones in the CI space. As a consequence, the GAS self-consistent field approach retains the accuracy of the CAS self-consistent field (CASSCF) ansatz and, at the same time, can deal with larger active spaces, which would be unaffordable at the CASSCF level. Test calculations on the Gd atom, Gd(2) molecule, and oxoMn(salen) complex are presented. They show that GAS wave functions achieve the same accuracy as CAS wave functions on systems that would be prohibitive at the CAS level.

Ab Initio Study of the Anomalous Solvatochromic Behavior of Large Betaines

The structure and spectroscopic properties of a diluted compound can be deeply affected by its interaction with the neighboring molecules of the solvent, and the associated solvatochromism is an effect that becomes more noticeable with the increase in both the dipole moment of the solute and the polarity of the medium. The correct description of the complex set of interactions that prevail in the solvation process remains a challenge for theoreticians not only when interpreting an observed behavior but also when considering the possible existence of novel properties in untested solute-solvent systems. On the basis of an ab initio study, we examine here how the presence of solvents of different polarities should affect the electronic properties of a family of molecules, formally related to Betaine-30 (aka Reichardt's dye), whose donor (D) and acceptor (A) groups are terminally connected to conjugated chains of different sizes. Because these molecules exhibit elevated ground-state dipole moment that should strongly interact with molecules of a polar solvent, a large hypsochromic shift is predicted for them. However, in a recent gas-phase study of these molecules, we have established the existence of an "inversion" in the spatial localization of their frontier orbitals when the size of the conjugated bridge connecting the D and A groups is progressively increased. This fact has led us to suggest that the increase in size of dissolved betaines should be accompanied by a large variation in their solvatochromic properties. In this work, we first use the self-consistent reaction field approach at the configuration interaction level to estimate the expected bathochromic shift in the absorption spectra (positive solvatochromism) in the largest members of the investigated betaine family when dissolved in different low polarity solvents and then discuss the conformational changes as a consequence of the solute--solvent interactions. We then use these results to interpret the observed solvatochromic properties of push--pull molecules of varying size and discuss the corresponding implications on their photochemical properties.

Anharmonicity of Weakly Bound M+−H2 Complexes

The anharmonicity of weakly bound complexes is studied using the vibrational self-consistent field (VSCF) approach for a series of metal cation dihydrogen (M(+)-H(2)) complexes. The H-H stretching frequency shifts of M(+)-H(2) (M(+) = Li(+), Na(+), B(+), and Al(+)) complexes are calculated with the coupled-cluster method including all single and double excitations with perturbative triples (CCSD(T)) level of theory with the cc-pVTZ basis set. The calculated H-H stretching frequency of Li(+)-H(2), B(+)-H(2), Na(+)-H(2), and Al(+)-H(2) is red-shifted by 121, 202, 74, and 62 cm(-1), respectively, relative to that of unbound H(2). The calculated red shifts and their trends are in good agreement with the available experimental and previously calculated data. Insight into the observed trends is provided by symmetry adapted perturbation theory (SAPT).

High‐Resolution Electronic Spectroscopy Studies of meta‐Aminobenzoic Acid in the Gas Phase Reveal the Origins of its Solvatochromic Behavior

A theoretical and experimental investigation of meta-aminobenzoic acid (MABA) in the gas phase is presented, with the goal of understanding counterintuitive observations on the solvatochromism of this "push-pull" molecule. The adiabatic excitation energies, transition moments, and excited-state structures are examined using the complete active space self-consistent field approach (CASSCF and CASPT2), which shows the first excited electronic state of MABA to be of greater charge transfer character than was found in the para-isomer (PABA). The rotationally resolved electronic spectrum of MABA reveals the existence of two rotamers, owing to asymmetry in the carboxylic acid functional group. Stark measurements in a molecular beam show the change in permanent dipole moment upon excitation to be Δμ≈3.5 D for both rotamers, more than three times larger than that found in PABA. The excited state measurements reported here, along with supporting data from theory, clearly demonstrate how the meta-directing effects of asymmetric substitution in aniline derivatives can drive charge transfer pathways in the isolated molecule.

Reconsidering an analytical gradient expression within a divide-and-conquer self-consistent field approach: Exact formula and its approximate treatment

An analytical energy gradient formula for the density-matrix-based linear-scaling divide-and-conquer (DC) self-consistent field (SCF) method was proposed in a previous paper by Yang and Lee (YL) [J. Chem. Phys. 103, 5674 (1995)]. Since the formula by YL does not correspond to the exact gradient of the DC-SCF energy, we derive the exact formula by direct differentiation, which requires solving the coupled-perturbed equations while including the inter-subsystem coupling terms. Next, we present an alternative formula for approximately evaluating the DC-SCF energy gradient, assuming the variational condition for the subsystem density matrices. Numerical assessments confirmed that the DC-SCF energy gradient values obtained by the present formula are in reasonable agreement with the conventional SCF values when adopting a reliable buffer region. Furthermore, the performance of the present method was found to be better than that of the YL method.

Depletion profiles for dilute solutions of linear chains, stars and H-branched molecules by self-consistent field calculations and Monte Carlo simulations

We have performed self-consistent field (SCF) calculations and Monte Carlo (MC) simulations to analyse the depletion profiles for isolated linear, star-like and H-shaped polymers in good solvent using lattice approximations in both methods. In the SCF approach the intra-molecular excluded-volume effects are accounted for using an approach that resembles Flory's method that leads to the Flory size of the chains. This gives a major improvement over the classical tanh profile, and becomes much closer to the MC results, provided that a Kuhn length of 1.5 is implemented.

Theoretical study on the solvent effect of the nitrosyl cation (NO+ ) generating reaction

Nitrosyl cation (NO+) generating reaction HONO + H+ NO+ + H2O has been theoretically investigated by B3LYP and high-electron-correlation QCISD methods with 6-31G (d,p) basis set. The solvent effects on the geometries, reaction path properties, energies, thermodynamic, and kinetic characters in four solvents (benzene, tetrahydrofuran, acetonitrile, and water) have been calculated using self-consistent reaction field (SCRF) approach with the polarizable continuum model (PCM). The results show that the activation energy barriers and the relative energies of the products are decreased with increase of the polarities of the solvents, and the reaction is favored in polar solvents thermodynamically and kinetically. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

High resolution X-ray emission spectroscopy of water and aqueous ions using the micro-jet technique

Soft X-ray absorption (XA) and emission (XE) spectroscopy is a powerful method for probing the local electronic structure of light elements (e.g. C, O, N, S) and transition metals, which are all of importance for biochemical systems. Here, we report for the first time on the XE spectra of a liquid micro-jet sample in a vacuum environment. We developed a high resolution X-ray emission spectrometer and recorded the spectra of pure water in full agreement with those of the literature, as well as of an aqueous solution of NiCl 2. For the latter system, ground state Hartree–Fock calculations using a self-consistent reaction field (SCRF) approach were carried out to specify the nature of the d-occupied orbital. Our results confirm the dark-channel-fluorescence-yield mechanism that we recently proposed for the case of metal ions in aqueous solutions. The ability to record absorption and emission spectra of an aqueous liquid-jet opens the way for the study of biochemical systems in physiological media.