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.     

Coordinate scaling and adiabatic connection formula for ensembles of fractionally occupied excited states

The coordinate scaling for the density matrix of ensembles of fractional occupied states of the density functional theory is explored. The ground-state adiabatic connection formula is extended to the ensemble exchange-correlation energy. © 1995 John Wiley & Sons, Inc.     

Hartree-Fock and Kohn-Sham time-dependent response theory in a second-quantization atomic-orbital formalism suitable for linear scaling

We present a second-quantization based atomic-orbital method for the computation of time-dependent response functions within Hartree-Fock and Kohn-Sham density-functional theories. The method is suited for linear scaling. Illustrative results are presented for excitation energies, one- and two-photon transition moments, polarizabilities, and hyperpolarizabilities for hexagonal BN sheets with up to 180 atoms.     

Computer program for calculating electronic excited states in the valence approximation of the unrestricted Hartree-Fock method

State University, Khar'kov. Translated from Zhurnal Strukturnoi Khimii, Vol. 31, No. 1, pp. 160–161, January–February, 1990.     

Time-dependent approach to electronically excited states of molecules with the multiconfiguration time-dependent Hartree-Fock method

In this paper the authors show how the multiconfiguration time-dependent Hartree-Fock (MCTDHF) method can be used for the calculation of electronic properties of molecules associated with the population of excited states. In contrast to other methods for correlated electron dynamics, such as configuration interaction, MCTDHF does not rely on a solution of the electronic Schrodinger equation prior to the propagation. The authors apply this approach to the calculation of vertical excitation energies, transition dipole moments, and oscillator strengths for two test molecules, lithium hydride and methane.     

Remarks on self-consistent π-electron wavefunctions for excited states of trans-butadiene

The behaviour of π-electron SC wavefunctions and energy levels reproducing directly obtained bond orders is studied for different excited states of trans-butadiene-1,3.     

The implementation of a self-consistent constricted variational density functional theory for the description of excited states

We present here the implementation of a self-consistent approach to the calculation of excitation energies within regular Kohn-Sham density functional theory. The method is based on the n-order constricted variational density functional theory (CV(n)-DFT) [T. Ziegler, M. Seth, M. Krykunov, J. Autschbach, and F. Wang, J. Chem. Phys. 130, 154102 (2009)] and its self-consistent formulation (SCF-CV(∞)-DFT) [J. Cullen, M. Krykunov, and T. Ziegler, Chem. Phys. 391, 11 (2011)]. A full account is given of the way in which SCF-CV(∞)-DFT is implemented. The SCF-CV(∞)-DFT scheme is further applied to transitions from occupied π orbitals to virtual π(?) orbitals. The same series of transitions has been studied previously by high-level ab initio methods. We compare here the performance of SCF-CV(∞)-DFT to that of time dependent density functional theory (TD-DFT), CV(n)-DFT and ΔSCF-DFT, with the ab initio results as a benchmark standard. It is finally demonstrated how adiabatic TD-DFT and ΔSCF-DFT are related through different approximations to SCF-CV(∞)-DFT.     

The formulation of a self-consistent constricted variational density functional theory for the description of excited states

We outline here a self-consistent approach to the calculation of transition energies within density functional theory. The method is based on constricted variational theory (CV-DFT). It constitutes in the first place an improvement over a previous scheme [T. Ziegler, M. Seth, M. Krykunov, J. Autschbach, F. Wang, Chem. Phys. 130 (2009) 154102] in that it includes terms in the variational parameters to any desired order n including n = ∞. For n = 2, CV(n)-DFT is similar to TD-DFT. Adiabatic TD-DFT becomes identical to CV(2)-DFT after the Tamm-Dancoff approximation is applied to both theories. We have termed the new scheme CV(n)-DFT. In the second place, the scheme can be implemented self-consistently, SCF-CV(n)-DFT. The procedure outlined here could also be used to formulate a SCF-CV(n) Hartree-Fock theory. The approach is further kindred to the ΔSCF-DFT procedures predating TD-DFT and we describe how adiabatic TD-DFT and ΔSCF-DFT are related through different approximations to SCF-CV(n)-DFT.     

Time-dependent Hartree-Fock calculations for the excited ‘S’ states of lithium isoelectronic sequence

Time-dependent HF (RPA) calculations have been performed to estimate the transition energies and excited state wavefunctions of Li, Be⁺, B²⁺ and C³⁺ for the transitions 1s ²2s ² S 1s ² ns ² S (n = 3,4, ... 8). The excitation energies and excited state wavefunctions are extracted from the position of the poles of a linearised variational functional. The excitation energies are in excellent agreement with those obtained spectroscopically. The excited state wavefunctions are utilised to find the matrix elements of different operators and the cusp values.     

A self-consistent field procedure for stationary states using an algebraic approach and the maximum entropy formalism

A stationary state of maximal entropy is derived as a solution of a variational procedure. Generators of a continuous group are used as the constraints. The self-consistent hamiltonian is linear in these generators so that the solution of the self-consistency problem is replaced by a solution of an algebraic equation. The familiar Hartree-Fock procedure is a special case.     

Test applications of a new SCF method for excited states

In order to test a recently proposed technique for deriving orthogonality-constrained HF wave functions for excited states, several applications to molecular systems, have been made and the results compared with those provided by other SCF techniques.     

A multiconfigurational time-dependent Hartree-Fock method for excited electronic states. II. Coulomb interaction effects in single conjugated polymer chains

Conjugated polymers have attracted considerable attention in the last few decades due to their potential for optoelectronic applications. A key step that needs optimisation is charge carrier separation following photoexcitation. To understand better the dynamics of the exciton prior to charge separation, we have performed simulations of the formation and dynamics of localised excitations in single conjugated polymer strands. We use a nonadiabatic molecular dynamics method which allows for the coupled evolution of the nuclear degrees of freedom and of multiconfigurational electronic wavefunctions. We show the relaxation of electron-hole pairs to form excitons and oppositely charged polaron pairs and discuss the modifications to the relaxation process predicted by the inclusion of the Coulomb interaction between the carriers. The issue of charge photogeneration in conjugated polymers in dilute solution is also addressed.     

The treatment of solvation by a generalized Born model and a self-consistent charge-density functional theory-based tight-binding method

We present a model to calculate the free energies of solvation of small organic compounds as well as large biomolecules. This model is based on a generalized Born (GB) model and a self-consistent charge-density functional theory-based tight-binding (SCC-DFTB) method with the nonelectrostatic contributions to the free energy of solvation modeled in terms of solvent-accessible surface areas (SA). The parametrization of the SCC-DFTB/GBSA model has been based on 60 neutral and six ionic molecules composed of H, C, N, O, and S, and spanning a wide range of chemical groups. Effective atomic radii as parameters have been obtained through Monte Carlo Simulated Annealing optimization in the parameter space to minimize the differences between the calculated and experimental free energies of solvation. The standard error in the free energies of solvation calculated by the final model is 1.11 kcal mol(-1). We also calculated the free energies of solvation for these molecules using a conductor-like screening model (COSMO) in combination with different levels of theory (AM1, SCC-DFTB, and B3LYP/6-31G*) and compared the results with SCC-DFTB/GBSA. To assess the efficiency of our model for large biomolecules, we calculated the free energy of solvation for a HIV protease-inhibitor complex containing 3,204 atoms using the SCC-DFTB/GBSA and the SCC-DFTB/COSMO models, separately. The computed relative free energies of solvation are comparable, while the SCC-DFTB/GBSA model is three to four times more efficient, in terms of computational cost.     

High-order non-born-oppenheimer electric polarizabilities for excited states of H+2 via the perturbational-variational rayleigh-ritz formalism

Large-order perturbation theory in the form of the perturbational-variational Rayleigh-Ritz (PV RR) formalism has been applied to excited energy levels in the Stark effect for H⁺₂,yielding non-Born-Oppenheimer electric polarizabilities through 16th, 8th, 4th, and 2nd order, respectively, for the first four excited non-rotational states; these non-Born-Oppenheimer excited states correspond to what are normally known as vibrationally excited levels of the electronic ground state. The excited-state polarizability series are found to be more divergent in each corresponding order than the ground-state one, becoming increasingly so for the higher excited states. Methods for obtaining the excited-state polarizabilities to still higher order within the framework of PV RR are discussed.     

Macroscopic solvent polarization-induced reorganization of the electron density in different excited states: A study on formaldehyde molecule by a multiconfiguration self-consistent reaction-field method

A detailed study of the type and extent of electronic reorganization created by macroscopic solvation of a prototypical carbonyl solute is carried out within the framework of the semiempirical multiconfiguration self-consistent reaction-field model. The solvation causes additional polarization of the electronic charge density within the carbonyl group. Orbitalwise breakup of the electron densities on the key atoms throws light on such features as the polarity dependence of inversion barriers and out-of-plane bending. Macroscopic solvation is shown to cause a reverse π-polarization within the carbonyl group, affecting barrier heights on the inversion path. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 415–428, 1998     

Comparison of CIS- and EOM-CCSD-calculated adiabatic excited-state structures. Changes in charge density on going to adiabatic excited states

The CIS and EOM-CCSD adiabatic geometries for the first excited states of a set of small molecules (C2H4, C2H2, H2C=O, H2C=S, CS2, CO2, SO2, NO2) have been calculated using the 6-311++G** basis set to see if the former geometries can be good starting points for optimizations at the latter theoretical level. With most of the molecules, there is fairly good agreement between the results from the two methods, and EOM-CCSD gives good agreement with the available experimental data. A detailed discussion of the lowest-lying singlet excited states in CO2 and CS2 is presented, highlighting the pronounced differences in electronic character and equilibrium structure displayed by these isovalent species. The origins of the structural distortions that are frequently found for the adiabatic excited states are examined with the aid of deformation density plots and the electron localization function (ELF).     

Pseudopotential method for the direct construction of localized orbitals by multiconfiguration self-consistent field theory

The orbital equations for the direct construction of localized fixed orbitals by multiconfiguration self-consistent field theory (MCSCF-FXO) are transformed without approximation into pseudopotential form by a two-step process. First the utilization of a particular family of localization is shown to separate the set of orbital equations into two sets of coupled equations, one describing “valence” orbitals and one describing “core” orbitals. In addition we obtain by appropriate choice of localization potential three different sets of MCSCF-FXO orbitals, namely: maximally screened, “one-center” and “intermediate” orbitals. In the second step the orbital equations are transformed into pseudopotential form and explicit non-local pseudopotentials yielding and core orbitals are obtained. Finally, several different physically motivated approximations to the exact pseudopotentials, and the frozen-core approximation are discussed.