Response function theory of electron correlation

The full perturbation expansion for the response (or density—density correlation) function is examined in order to provide a useful general theory of excitation energies, oscillator strengths, dynamic polarizabilities, etc., that is more accurate than the random phase approximation. It is first shown how the formal partition of the diagrammatic version of the perturbation expansion into reducible and irreducible diagrams is generally useless as the latter category contains all the difficult terms which have heretofore resisted analysis in all but a haphazard form. It is then shown how the diagram for the response function can be partitioned into “correlated” and “uncorrelated” subsets. Restricting attention to the particle—hole blocks of the full response function, the “uncorrelated” diagrams desecribe the propagation of a particle—hole pair in an N-electron system where the particle and hole are each interacting with the remaining electrons but they are not interacting with each other. The “correlated” diagrams are those containing the hole—particle interactions, and, by defining a new class of reducible and irreducible diagrams, these are all summed to provide a perturbation expansion of the effective two-body hole—particle interaction that appears in the inverse of the response function. The “uncorrelated” diagrams are further partitioned into two sets, one of which is summed to all orders, while the other set is inverted in an order by order fashion. The final result presents a perturbation expansion for the inverse of the response function that is analogous to the Dyson equation for one-electron Green functions. Maintaining the perturbation expansion through first order for the inverse of the response function yields the eigenvalue equation of the familiar random phase approximation, while truncation at second order provides the most advanced theories that have been generated by the equations-of-motion method.     

Parametrization of the equations-of-motion method for conjugated hydrocarbons

The equations-of-motion (EOM ) method has been parametrized for the evaluation of the “π → π*” transition energies and moments of conjugated hydrocarbons. In this new semiempirical scheme, the effect of the dynamical screening by σ electrons is explicitly included.     

Application of many-body green's functions to the calculation of molecular ionization potentials

We present a method for calculating the one-particle Green's function for molecules. The scheme is essentially that proposed by Schneider and Taylor [1]. The Green's function is obtained through the Dyson equation. Closed expressions result by using the functional derivation technique to truncate an infinite set of coupled equations. A physical interpretation of the approximation is given and a connection with the equations-of-motion method is pointed out. In a numerical application the ionization potentials are obtained for the molecules N₂, H₂O, NH₃ and CH₄.     

A variational principle for transition and density matrices and approximations to the equations-of-motion

A general variational principle for transition and density matrices is proposed. The principle is closely related to Rowe's variational treatment of the equations-of-motion method. It permits the simultaneous construction of coupled approximations for two eigenstates, and it is a straightforward extension of the usual variational method.     

The equations-of-motion method for F2: Transition energies,oscillator strengths and born cross sections

The equations-of-motion method has been used to study various electronic states of F₂. The transition energies have been found in both the random phase approximation (RPA) and higher random approximation (HRPA) using single particle—hole components in the excitation operators. We have also computed generalized oscillator strengths (Born cross sections) for the scattering of high energy electrons by F₂.     

Dissipative particle dynamics at isothermal, isobaric, isoenergetic, and isoenthalpic conditions using Shardlow-like splitting algorithms

Numerical integration schemes based upon the Shardlow-splitting algorithm (SSA) are presented for dissipative particle dynamics (DPD) approaches at various fixed conditions, including a constant-enthalpy (DPD-H) method that is developed by combining the equations-of-motion for a barostat with the equations-of-motion for the constant-energy (DPD-E) method. The DPD-H variant is developed for both a deterministic (Hoover) and stochastic (Langevin) barostat, where a barostat temperature is defined to satisfy the fluctuation-dissipation theorem for the Langevin barostat. For each variant, the Shardlow-splitting algorithm is formulated for both a velocity-Verlet scheme and an implicit scheme, where the velocity-Verlet scheme consistently performed better. The application of the Shardlow-splitting algorithm is particularly critical for the DPD-E and DPD-H variants, since it allows more temporally practical simulations to be carried out. The equivalence of the DPD variants is verified using both a standard DPD fluid model and a coarse-grain solid model. For both models, the DPD-E and DPD-H variants are further verified by instantaneously heating a slab of particles in the simulation cell, and subsequent monitoring of the evolution of the corresponding thermodynamic variables as the system approaches an equilibrated state while maintaining their respective constant-energy and constant-enthalpy conditions. The original SSA formulated for systems of equal-mass particles has been extended to systems of unequal-mass particles. The Fokker-Planck equation and derivations of the fluctuation-dissipation theorem for each DPD variant are also included for completeness.     

AB initio study of the low-lying excited states of cyclopropene by the equations-of-motion method

The lowest singlet and triplet excited states of cyclopropene have been investigated with the equations-of-motion method. The first spectral band is due to a singlet transition arising from π → 3s,3p excitations. The other two band systems are associated with clusters of transitions predominantly of Rydberg nature or strong admixtures of valence-valence and valence-Rydberg excitations. The first triplet transition is mainly of π → π character.     

Linear dependence and energy conservation in Gaussian wavepacket basis sets

We propose a method for dealing with the problem of linear dependence in quantum dynamics simulations employing over-complete Gaussian wavepacket (GWP) basis sets. In particular, by periodically projecting out redundant basis functions using the matching pursuit algorithm whilst simultaneously introducing GWPs which avoid linear dependence with the current basis set, we find that numerical conditioning of the equations-of-motion can be readily controlled. In applications to particle tunnelling in one- and two-dimensional potentials, this method allows us to reproduce the exact quantum-mechanical results with fewer GWP basis functions than similar calculations with non-adaptive basis sets, a result which we trace back to the improved energy conservation of our adaptive approach.     

Dynamical screening by Σ-electrons in π-electron transitions: Treatment in the equations-of-motion method

Dynamical screening in π-electron systems is studied by the equations of motion method. By using a partitioning technique on the equations of motion we can obtain simple expressions for the effect of dynamical screening directly on the transition energies and transition moments in π-electron systems. These results are used to study the effect of such screening in the N → V transition in ethylene. This procedure can be used to extend the equations-of-motion method to larger π-electron systems.     

Shake-up peak positions and intensities by many-body theory methods

The valence shell X-ray photoelectron spectrum of N₂ is calculated using the equations-of-motion—Green's function method. The inclusion of shake-up basis operator configurations in the primary operator space along with the simple ionization operator configurations allows for the calculation of shake-up peak positions on an equal footing with the simple ionization energies. The important shake-up basis operator configurations are identified using configuration selection techniques similar to those which have been successfully employed in large scale configuration interaction problems, thus minimizing the size of the matrices to be diagonalized. The relative peak intensities are calculated within the plane wave approximation. The intensity equations are analyzed indicating that the relative peak intensities are more sensitive to ground state correlation effects than the peak positions. Modifications of the theory to improve the calculations of shake-up energies are discussed.     

Padé spectrum decompositions of quantum distribution functions and optimal hierarchical equations of motion construction for quantum open systems

Pade? spectrum decomposition is an optimal sum-over-poles expansion scheme of Fermi function and Bose function [J. Hu, R. X. Xu, and Y. J. Yan, J. Chem. Phys. 133, 101106 (2010)]. In this work, we report two additional members to this family, from which the best among all sum-over-poles methods could be chosen for different cases of application. Methods are developed for determining these three Pade? spectrum decomposition expansions at machine precision via simple algorithms. We exemplify the applications of present development with optimal construction of hierarchical equations-of-motion formulations for nonperturbative quantum dissipation and quantum transport dynamics. Numerical demonstrations are given for two systems. One is the transient transport current to an interacting quantum-dots system, together with the involved high-order co-tunneling dynamics. Another is the non-Markovian dynamics of a spin-boson system.     

Ab initio calculation of the low-lying vibrational states of C2H2(A) in full dimensionality

We report full-dimensional calculations of vibrational energies of trans-C2H2(A) using the code MULTIMODE and with a full-dimensional potential energy surface obtained by fitting singles and doubles coupled-cluster equations-of-motion (EOM-CCSD) energies using a [3s 2p 1d] atomic natural orbital basis. The EOM-CCSD calculations were done with the code "ACES II". We compare the properties of the potential surface to previous calculations at the trans minimum and also compare the vibrational energies to experimental ones.     

Time correlation function and finite field approaches to the calculation of the fifth order Raman response in liquid xenon

The fifth order, two-dimensional Raman response in liquid xenon is calculated via a time correlation function (TCF) theory and the numerically exact finite field method. Both employ classical molecular dynamics simulations. The results are shown to be in excellent agreement, suggesting the efficacy of the TCF approach, in which the response function is written approximately in terms of a single classical multitime TCF.     

Ab initio study of the J(CC) indirect nuclear spin-spin coupling constants in cyclobutane and related systems

Non-empirical calculations using the equations-of-motion approach, which incorporates the main portion of the electron correlation effects, are reported for the carbon-carbon nuclear spin-spin coupling constants in cyclobutane, bicyclobutane, tricyclobutane, cyclobutene, cyclobutyne, cyclobutadiene, bicyclobutene, methylenecyclopropane, and methylenecyclopropene. The results provide an overall picture of the influences exerted on sign and magnitude of the J(CC) by progressive condensation, unsaturation, and branching rearrangement of the cyclobutane frame.     

Theoretical studies of molecular ions. Vertical detachment energy of OH−

The ¹Σ⁺→²Π_i vertical electron detachment energy of OH^? is studied using a basis of twenty Slater-type orbitals in our equations-of-motion (EOM) theory of molecular electron affinities and ionization potentials. The delicate balance between the contributions of orbital reorganization effects and correlation energy change to the calculated negative-ion detachment energy is demonstrated clearly. Comparisons are made with the results of very precise experimental photodetachment measurements and with other theoretical predictions.     

Reduced eigensystem representation of the linear density-density response function

The linear density-density response function represents a formulation of the generalized density response of a molecular (or extended) system to arbitrary perturbing potentials. We have recently established an approach for reducing the dimension of the (in principle infinite) eigenspace representation (the moment expansion) and generalized it to arbitrary self-adjoint, positive-definite, and compact linear operators. Here, we present a modified representation—the reduced eigensystem representation—which allows to define a trivial criterion for the convergence of the approximation to the density response. By means of this novel eigensystem-like structure, the remarkable reduction of the dimensionality becomes apparent for the calculation of the density-density response function.     

Moment expansion of the linear density‐density response function

We present a low rank moment expansion of the linear density‐density response function. The general interacting (fully nonlocal) density‐density response function is calculated by means of its spectral decomposition via an iterative Lanczos diagonalization technique within linear density functional perturbation theory. We derive a unitary transformation in the space of the eigenfunctions yielding subspaces with well‐defined moments. This transformation generates the irreducible representations of the density‐density response function with respect to rotations within SO(3). This allows to separate the contributions to the electronic response density from different multipole moments of the perturbation. Our representation maximally condenses the physically relevant information of the density‐density response function required for intermolecular interactions, yielding a considerable reduction in dimensionality. We illustrate the performance and accuracy of our scheme by computing the electronic response density of a water molecule to a complex interaction potential. © 2015 Wiley Periodicals, Inc.     

Applications of a time correlation function theory for the fifth-order Raman response function I: atomic liquids

Multidimensional spectroscopy has the ability to provide great insight into the complex dynamics and time-resolved structure of liquids. Theoretically describing these experiments requires calculating the nonlinear-response function, which is a combination of quantum-mechanical time correlation functions R5(t1,t2) was expressed with a two-time, computationally tractable, classical TCF. Writing the response function in terms of classical TCFs brings the full power of atomistically detailed molecular dynamics to the problem. In this paper, the new TCF theory is employed to calculate the fifth-order Raman response function for liquid xenon and investigate several of the polarization conditions for which experiments can be performed on an isotropic system. The theory is shown to reproduce line-shape characteristics predicted by earlier theoretical work.     

Transient-time correlation function applied to mixed shear and elongational flows

The transient-time correlation function (TTCF) method is used to calculate the nonlinear response of a homogeneous atomic fluid close to equilibrium. The TTCF response of the pressure tensor subjected to a time-independent planar mixed flow of shear and elongation is compared to directly averaged non-equilibrium molecular dynamics (NEMD) simulations. We discuss the consequence of noise in simulations with a small rate of deformation. The generalized viscosity for planar mixed flow is also calculated with TTCF. We find that for small rates of deformation, TTCF is far more efficient than direct averages of NEMD simulations. Therefore, TTCF can be applied to fluids with deformation rates which are much smaller than those commonly used in NEMD simulations. Ultimately, TTCF applied to molecular systems is amenable to direct comparison between NEMD simulations and experiments and so in principle can be used to study the rheology of polymer melts in industrial processes.