Self-consistent Field Equations Research Articles

Implementation of ab initio multiple spawning in the M olpro quantum chemistry package

The ab initio multiple spawning (AIMS) method has been developed to solve the electronic and nuclear Schrodinger equations simultaneously for application to photochemical reaction dynamics. We discuss some details of the implementation of AIMS in the M olpro program package. A few aspects of the implementation are highlighted, including a new multiple timescale integrator and a scheme for solving the coupled-perturbed multiconfiguration self-consistent field (CP-MCSCF) equations in the context of ab initio molecular dynamics. The implementation is very efficient and we demonstrate calculations on the photoisomerization of ethylene using more than 5000 trajectory basis functions. We have included the capability for hybrid quantum mechanics/molecular mechanics (QM/MM) simulations within AIMS, and we investigate the role of an argon solvent in the photoisomerization of ethylene. Somewhat surprisingly, the surrounding argon has little effect on the timescale of non-adiabatic quenching in ethylene.

Nonvariational time-dependent multiconfiguration self-consistent field equations for electronic dynamics in laser-driven molecules

A time-dependent multiconfiguration self-consistent field (TDMCSCF) scheme is developed to describe the time-resolved electron dynamics of a laser-driven many-electron atomic or molecular system, starting directly from the time-dependent Schrodinger equation for the system. This nonvariational formulation aims at the full exploitations of concepts, tools, and facilities of existing, well-developed quantum chemical MCSCF codes. The theory uses, in particular, a unitary representation of time-dependent configuration mixings and orbital transformations. Within a short-time, or adiabatic approximation, the TDMCSCF scheme amounts to a second-order split-operator algorithm involving generically the two noncommuting one-electron and two-electron parts of the time-dependent electronic Hamiltonian. We implement the scheme to calculate the laser-induced dynamics of the two-electron H2 molecule described within a minimal basis, and show how electron correlation is affected by the interaction of the molecule with a strong laser field.

Calculation of the zero-field splitting tensor on the basis of hybrid density functional and Hartree-Fock theory

The zero-field splitting (ZFS) (expressed in terms of the D tensor) is the leading spin-Hamiltonian parameter for systems with a ground state spin S>12. To first order in perturbation theory, the ZFS arises from the direct spin-spin dipole-dipole interaction. To second order, contributions arise from spin-orbit coupling (SOC). The latter contributions are difficult to treat since the SOC mixes states of different multiplicities. This is an aspect of dominant importance for the correct prediction of the D tensor. In this work, the theory of the D tensor is discussed from the point of view of analytic derivative theory. Starting from a general earlier perturbation treatment [F. Neese and E. I. Soloman, Inorg. Chem. 37, 6568 (1998)], straightforward response equations are derived that are readily transferred to the self-consistent field (SCF) Hartree-Fock (HF) or density functional theory (DFT) framework. The main additional effort in such calculations arises from the solution of nine sets of nonstandard coupled-perturbed SCF equations. These equations have been implemented together with the spin-orbit mean-field representation of the SOC operator and a mean-field treatment of the direct spin-spin interaction into the ORCA electronic structure program. A series of test calculations on diatomic molecules with accurately known zero-field splittings shows that the new approach corrects most of the shortcomings of previous DFT based methods and, on average, leads to predictions within 10% of the experimental values. The slope of the correlation line is essentially unity for the B3LYP and BLYP functionals compared to approximately 0.5 in previous treatments.

Self-Consistent Field Study of Polyelectrolyte Brushes

We formulate a self-consistent field theory for polyelectrolyte brushes in the presence of counterions. We numerically solve the self-consistent field equations and study the monomer density profile, the distribution of counterions, and the total charge distribution. We study the scaling relations for the brush height and compare them to the prediction of other theories. We find a weak dependence of the brush height on the grafting density.We fit the counterion distribution outside the brush by the Gouy-Chapman solution for a virtual charged wall. We calculate the amount of counterions outside the brush and find that it saturates as the charge of the polyelectrolytes increases.

The analytical gradient of dual-basis resolution-of-the-identity second-order Møller–Plesset perturbation theory

In this work, we present the analytical gradient of dual-basis second-order Møller–Plesset perturbation theory within the resolution-of-the-identity approximation (DB-RI-MP2). Interestingly, analytical DB-RI-MP2 gradient theory involves significant changes to both the theory and computation of the coupled-perturbed self-consistent field equations (CPSCF). From a theoretical point of view, the number of orbital responses required in DB-RI-MP2 analytical gradient theory has been reduced to the product of the number of occupied and virtual orbitals determined by the rank of the small atomic orbital (AO) basis. From a computational point of view, the DB-CPSCF equations can be solved within this smaller space at a fraction of the computational cost. Additional computational savings can be obtained during the digestion of the four-centered AO integral derivatives and the efficient underlying DB-SCF procedure, which lead to a significant overall reduction in the computational cost necessary for treating molecular systems containing less than 100 atoms. Based on stringent chemical tests and a detailed computational timings analysis, it was found that DB-RI-MP2 reproduces molecular equilibrium structures with an accuracy that approximates RI-MP2 using the target AO basis but at a dramatically reduced cost, thereby enabling more routine use of large AO basis sets during geometry optimizations at the MP2 level of theory.

Variational theory of average-atom and superconfigurations in quantum plasmas

Models of screened ions in equilibrium plasmas with all quantum electrons are important in opacity and equation of state calculations. Although such models have to be derived from variational principles, up to now existing models have not been fully variational. In this paper a fully variational theory respecting virial theorem is proposed-all variables are variational except the parameters defining the equilibrium, i.e., the temperature T, the ion density ni and the atomic number Z. The theory is applied to the quasiclassical Thomas-Fermi (TF) atom, the quantum average atom (QAA), and the superconfigurations (SC) in plasmas. Both the self-consistent-field (SCF) equations for the electronic structure and the condition for the mean ionization Z* are found from minimization of a thermodynamic potential. This potential is constructed using the cluster expansion of the plasma free energy from which the zero and the first-order terms are retained. In the zero order the free energy per ion is that of the quantum homogeneous plasma of an unknown free-electron density n0 = Z* ni occupying the volume 1/ni. In the first order, ions submerged in this plasma are considered and local neutrality is assumed. These ions are considered in the infinite space without imposing the neutrality of the Wigner-Seitz (WS) cell. As in the Inferno model, a central cavity of a radius R is introduced, however, the value of R is unknown a priori. The charge density due to noncentral ions is zero inside the cavity and equals en0 outside. The first-order contribution to free energy per ion is the difference between the free energy of the system "central ion+infinite plasma" and the free energy of the system "infinite plasma." An important part of the approach is an "ionization model" (IM), which is a relation between the mean ionization charge Z* and the first-order structure variables. Both the IM and the local neutrality are respected in the minimization procedure. The correct IM in the TF case is found to be Z-Z*= integral d3 r[n(r)-n0], where n(r) is the first-order electron density. It is shown that in the QAA case the same IM has to be used and that other IMs lead to unphysical solutions. With this IM R becomes in both cases (TF and QAA) equal to the WS radius and the variational calculation leads to SCF equations in an infinite plasma while n0 (or equivalently Z*) is to be found from the condition integral d3r theta(r-R)Vel(r)=0, where theta denotes Heaviside function and Vel(r) is the SCF electrostatic potential. In the SC case results are similar except that averages over all superconfigurations appear. In the TF case the condition for n0 gives the neutrality of the WS sphere and one gets the classical TF ion-in-cell average atom. The situation is different in the QAA and in the SC cases in which the cavity is not neutral and the SCF potential Vel(r) is not zero outside the cavity. Due to the fully variational character of our approach the expression for the thermodynamic pressure in all cases does not require any numerical differentiation and is consistent with the virial theorem.

Converging self-consistent field equations in quantum chemistry – recent achievements and remaining challenges

This paper reviews popular acceleration techniques to converge the non-linear self-consistent field equations appearing in quantum chemistry calculations with localized basis sets. The different methodologies, as well as their advantages and limitations are discussed within the same framework. Several illustrative examples of calculations are presented. This paper attempts to describe recent achievements and remaining challenges in this field.

Strong and weak adsorptions of polyelectrolyte chains onto oppositely charged spheres

We investigate the complexation of long thin polyelectrolyte (PE) chains with oppositely charged spheres. In the limit of strong adsorption, when strongly charged PE chains adapt a definite wrapped conformation on the sphere surface, we analytically solve the linear Poisson-Boltzmann equation and calculate the electrostatic potential and the energy of the complex. We discuss some biological applications of the obtained results. For weak adsorption, when a flexible weakly charged PE chain is localized next to the sphere in solution, we solve the Edwards equation for PE conformations in the Hulthen potential, which is used as an approximation for the screened Debye-Huckel potential of the sphere. We predict the critical conditions for PE adsorption. We find that the critical sphere charge density exhibits a distinctively different dependence on the Debye screening length than for PE adsorption onto a flat surface. We compare our findings with experimental measurements on complexation of various PEs with oppositely charged colloidal particles. We also present some numerical results of the coupled Poisson-Boltzmann and self-consistent field equation for PE adsorption in an assembly of oppositely charged spheres.

Self-consistent field study of the alignment by an electric field of a cylindrical phase of block copolymer

Self-consistent field theory is applied to a film of cylindrical-forming block copolymer subject to a surface field which tends to align the cylinders parallel to electrical plates, and to an external electric field tending to align them perpendicular to the plates. The Maxwell equations and self-consistent field equations are solved exactly, numerically, in real space. By comparing the free energies of different configurations, we show that for weak surface fields, the phase of cylinders parallel to the plates makes a direct transition to a phase in which the cylinders are aligned with the field throughout the sample. For stronger surface fields, there is an intermediate phase in which cylinders in the interior of the film, aligned with the field, terminate near the plates. For surface fields which favor the minority block, there is a boundary layer of hexagonal symmetry at the plates in which the monomers favored by the surface field occupy a larger area than they would if the cylinders extended to the surface.

Localized asymptotic solutions of the self-consistent field equation

Localized asymptotic solutions of the self-consistent field equation

A self-consistent “superatom” of bosonic Rb atoms

It is shown that an ensemble of an arbitrary number of ground state bosonic Rb atoms has stable bound stationary quantum states. The nonlinear Hartree self-consistent field equation is solved for the bound eigenstates of such systems of 10, 100, and 1000 atoms. They have interesting similarities to and differences from electron wave functions in atoms.

Genuine converging solution of self-consistent field equations for extended many-electron systems

Calculations of the ground state of inhomogeneous many-electron systems involve a solving of the Poisson equation for Coulomb potential and the Schrödinger equation for singleparticle orbitals. Due to nonlinearity and complexity this set of equations, one believes in the iterative method for the solution that should consist in consecutive improvement of the potential and the electron density until the self-consistency is attained. Though this approach exists for a long time there are two grave problems accompanying its implementation to infinitely extended systems.The first of them is related with the Poisson equation and lies in possible incompatibility of the boundary conditions for the potential with the electron density distribution renewed by means of the Schrödinger equation. Rigorously speaking, it means the iteration process cannot be continued. The resolution of this difficulty is proposed for both infinite conducting systems in jellium approximation and periodic structures. It provides the existence of self-consistent solution for the potential at every iteration step due to realization of a screening effect.The second problem results from the existence of continuous spectrum of Hamiltonian eigenvalues for unbounded systems. It needs to have a definition of Hilbert space basis with eigenfunctions of continuous spectrum as elements, which would be convenient in numerical applications. It is proposed to insert a limiting transition into the definition of scalar product specifying the Hilbert space. It provides self-adjointness of Hamiltonian and, respectively, the orthogonality of eigenfunctions corresponding to the different eigenvalues. In addition, it allows to normalize them effectively to delta-function and to prove the orthogonality of the 'right' and 'left' eigenfunctions belonging to twofold degenerate eigenvalues.

Self-consistent field theory of protein adsorption in a non-Gaussian polyelectrolyte brush

To describe adsorption of globular protein molecules in a polyelectrolyte brush we use the strong-stretching approximation of the Edwards self-consistent field equation, combined with corrections for a non-Gaussian brush. To describe chemical potentials in this mixture of (globular) species of widely varying sizes (ions, brush polyelectrolyte segments, globular protein molecules), we use the Boublik-Mansoori-Carnahan-Starling-Leland equation of state derived for polydisperse mixtures of spherical particles. The polyelectrolyte chain is described in this approach as a string of beads with the beads of a size related to the chain diameter. We use the one-dimensional Poisson equation to describe the electrostatic field and include the ionizable character of both the brush polyions and the protein molecules. This model explains the experimental observation of high amounts of protein adsorption in a polyacid brush for values above the isoelectric point of the protein as being due to charge reversal of the protein molecules upon entry in the brush. We find a distinct minimum in protein concentration near the edge of the brush. With increasing this barrier to protein transfer becomes larger, but much less so when we increase the ionic strength, a difference that might relate to an experimentally observed difference in the protein release rate in these two cases. A free energy analysis shows that the release of small ions from the brush and the increase of brush ionization are the two driving forces for protein adsorption in a like-charged brush.

An efficient approach for self-consistent-field energy and energy second derivatives in the atomic-orbital basis

Based on self-consistent-field (SCF) perturbation theory, we recast the SCF and the coupled-perturbed SCF (CPSCF) equations for time-independent molecular properties into the atomic-orbital basis. The density matrix and the perturbed density matrix are obtained iteratively by solving linear equations. Only matrix multiplications and additions are required, and this approach can exploit sparse matrix multiplications and thereby offer the possibility of evaluating second-order properties in computational effort that scales linearly with system size. Convergence properties are similar to conventional molecular-orbital-based CPSCF procedures, in terms of the number of derivative Fock matrices that must be constructed. We also carefully address the issue of the numerical accuracy of the calculated second derivatives of the energy, in order to specify the minimum precision necessary in the CPSCF procedure. It is found that much looser tolerances for the perturbed density matrices are adequate when using an expression for the second derivatives that is correct through second order in the CPSCF error.

Volume exclusion effects in the ground-state dominance approximation for polyelectrolyte adsorption on charged interfaces

We consider the problem of polyelectrolyte molecules adsorbing on oppositely charged interfaces. For sufficiently long chains, the ground-state dominance approximation can be used which results in a (semi-)analytical solution of the self-consistent field equations (aSCF). Whereas existing aSCF theory assumes a low polyelectrolyte density, here the required electrostatic corrections for a high polymer density are implemented. Adsorbed polymer excludes volume for the solvent and small ions, a volume effect that also leads to a reduced dielectric permittivity and a resulting polarization term in the exchange potential. Calculations show the influence of volume exclusion on the polymer density profile.

Simple model for grafted polymer brushes.

The first theories of grafted polymer brushes assumed a step profile for the monomer density. Later, the real density profile was obtained from Monte Carlo or molecular dynamics simulations and calculated numerically using a self-consistent field theory. The analytical approximations of the solutions of the self-consistent field equations provided a parabolic dependence of the self-consistent field, which in turn led to a parabolic distribution for the monomer density in neutral brushes. As shown by numerical simulations, this model is not accurate for dense polymer brushes, with highly stretched polymers. In addition, the scaling laws obtained from the analytical approximations of the self-consistent field theory are identical to those derived from the earlier step-profile-approximation and predict a vanishing thickness of the brush at low graft densities, and a thickness exceeding the length of the polymer chains at high graft densities. Here a simple model is suggested to calculate the monomer density and the interaction between surfaces with grafted polymer brushes, based on an approximate calculation of the partition function of the polymer chains. The present model can be employed for both good and poor solvents, is compatible with a parabolic-like profile at moderate graft densities, and leads to an almost steplike density for highly stretched brushes. While the thickness of the brush depends strongly on solvent quality, it is a continuous function in the vicinity of the temperature. In good and moderately poor solvents, the interactions between surfaces with grafted polymer brushes are always repulsive, whereas in poor solvents the interactions are repulsive at small separations and become attractive at intermediate separation distances, in agreement with experiment. At large separations, a very weak repulsion is predicted.

Self-assembly of rod-coil block copolymers.

We present a self-consistent field theory model for the self-assembly behavior of rod-coil block copolymers. The orientational interactions between the rods were modeled through a Maier-Saupe interaction, while the enthalpic interactions between rods and coils were modeled through a standard Flory-Huggins approach. We outline a "real-space" numerical approach to solve the self-consistent field equations for such rod-coil block copolymers. A major focus of our work is upon the nonlamellar phases observed in the experiments on such polymers. To develop a physical understanding of these phases and their regimes of occurrence, we compute the two-dimensional phase diagram for our model. The latter shows significant departures from the one-dimensional phase diagram, but matches qualitatively with the existing experimental results. We also present scaling arguments that rationalize the numerical results for the self-assembly behavior.

Alternative techniques in open‐shell SCF theory

AbstractAn alternative to the Roothaan's open‐shell method that does not involve off‐diagonal Lagrangian multipliers has been proposed. The orthogonality of the closed‐ and open‐shell orbitals is discussed. Two possible schemes of designing the restricted open‐shell Hartree–Fock (HF) wave functions are discussed: In the first scheme the self‐consistent field (SCF) functions are linear combination of Slater determinants with coefficients fixed by symmetry whereas the second one uses the unrestricted HF formalism to build a single open‐shell Slater determinant from which a well‐defined open‐shell Møller–Plesset‐like perturbation theory can be performed. It has been shown that the Euler SCF equations based on the proposed technique lead to an optimum set of orbitals satisfying the correct variational conditions. A structure and correspondence of the proposed equations to those of the conventional approach are discussed. Practical calculations carried out at SCF and perturbation theory levels demonstrate some features of the proposed technique. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Semiflexible polymer solutions. I. Phase behavior and single-chain statistics

We study the thermodynamics and single-chain statistics of wormlike polymer solutions with Maier–Saupe-type interactions using self-consistent-field (SCF) theory. The SCF equations are derived using a systematic field-theoretical approach which yields the SCF equations as the lowest order approximation, but permits fluctuation corrections to be incorporated. We solve the SCF equations using the spheroidal functions, which provides a nonperturbative description of the thermodynamics and single-chain statistics in the nematic state for arbitrary degrees of nematic order. Several types of phase diagrams are predicted, with an emphasis on the limit of metastability (spinodal) associated with each phase. The shape and location of these spinodals suggest interesting scenarios for the phase transition kinetics. A large but finite persistence length is shown to significantly decrease the isotropic–nematic transition temperature relative to that for rigid rods. In the nematic state, the mean-square end-to-end distance in the parallel and perpendicular directions are governed by two separate correlation lengths. An exact relationship between these correlation lengths and the eigenvalues of the spheroidal functions is provided, which reproduces the analytical expressions predicted from earlier studies in the limit of large nematic strength. The dominant contribution to the single-chain thermodynamics is shown to arise from small amplitude undulations in the directions perpendicular to the nematic direction; the presence of hairpins, though crucial for determining the dimensions of the polymer, has insignificant consequences on the single-chain thermodynamics.

Transcorrelated method for electronic systems coupled with variational Monte Carlo calculation

A Jastrow–Slater-type wave function is often used as a trial function for precise calculations of the total energy of electronic systems, where the correlation effect is taken into account by the Jastrow factor that directly depends on the distance between electrons. Since many-body integrals are inevitable there, the calculation totally depends on Monte Carlo sampling, and so, except for very simple cases, it is very difficult to optimize one-body wave functions in the Slater determinant which determine the nodal surfaces of the total wave function. Here we propose and demonstrate that the total wave function is efficiently optimized by coupling an ordinary variational Monte Carlo (VMC) technique with the transcorrelated method, in which the one-body wave functions are definitely obtained by solving Hartree–Fock-type self-consistent-field (SCF) equations derived from the similarity-transformed Hamiltonian. It is shown that the present method reproduces about 90% of the correlation energy for helium-like two-electron systems (H−, He, Li+, and Be2+) and gives much better results than the conventional VMC method using the Hartree–Fock orbitals for a Li atom, a Be atom, and a H2 molecule. It is also shown that the orbital energy appearing in the SCF equations gives a good approximation to the ionization potential.