Higher Random Phase Approximation Research Articles

Response theory in the multipole reaction field model for equilibrium and nonequilibrium solvation: Exact theory and the second order polarization propagator approximation

Exact response functions are derived for a multipole solvent reaction field model of equilibrium and nonequilibrium solvation using the generalized Ehrenfest theorem and assuming a spherical cavity surrounding the solute. The starting point is the Schrödinger equation and we shortly review how the reaction field is introduced into the Schrödinger equation in order to clearly identify the limitations of describing solute–solvent interactions with a reaction field model. The solvent is described as an isotropic homogeneous linear dielectric medium characterized by a static and an optical dielectric constant. From the exact response functions we derive linear response functions within the higher random phase and the second order polarization propagator approximation. Excitation energies, oscillator strengths, and polarizabilities are then calculated for solvated H2S and furan using the augmented correlation consistent triple-ζ (aug-cc-pVTZ) and double-ζ (aug-cc-pVDZ) basis sets for H2S and furan, respectively. We have also calculated excitation energies and oscillator strengths for H2S with standard (vacuum) ab initio methods using a variety of basis set, as there has been no previously reported values of these quantities calculated with the second order polarization propagator approximation. The second order polarization propagator approximation gives excitation energies and oscillator strengths close to values obtained by coupled cluster methods for a solvated H2S molecule, whereas the higher random phase approximation tends to overestimate the value of these quantities. The solvent effect of the excitation energies follow the same trends for all of the reaction field ab initio methods used in the present study, but some oscillator strengths show different solvent effects whether they are calculated with correlated or with noncorrelated ab initio methods. The calculated polarizabilities show the same solvent effect independent of any inclusion of dynamical electron correlation. It is also shown that the equilibrium solvation model is not appropriate for high-frequency perturbations.

Higher RPA and second-order polarization propagator calculations of coupling constants in acetylene

The higher random-phase approximation which includes electronic correlation at a level between the first-order (RPA, coupled Hartree—Fock) and the second-order polarization propagator approximation is found to remove the well-known deficiencies of RPA. In a numerical example (acetylene) the nuclear spin—spin coupling constants computed at the HRPA level even seem to be close to the second-order results. This is surprising since the excitation energies determined in HRPA often are too large. By examining the individual contributions, we conclude, that the numerical similarity found here between HRPA and the full second-order approximation may not always hold.

A b i n i t i o calculations of oscillator and rotatory strengths in the random-phase approximation: Twisted mono-olefins

A b initio (STO-nG) computations of ordinary and rotatory intensities of low-lying electronic transitions are presented for twisted ethylene and twisted trans-2-butene in the random-phase approximation (RPA). The intensities are computed in both dipole length and dipole velocity forms, as well as the mixed form for the oscillator strength, and the convergence of these formally equivalent results is examined in the RPA and several other methods for constructing the electronic excitation: the virtual orbital, or single-transition, approximation (STA), the monoexcited configuration-interaction, or Tamm–Dancoff, approximation (TDA), and one version of the higher RPA (HRPA). We show that the RPA has consistent advantages over the TDA for calculation of CD as well as ordinary intensities. Our computations confirm that a localized, ethylenic chromophore is indeed adequate to account for the low-lying CD spectrum in mono-olefins. Further, even with minimal valence-shell basis sets, our RPA rotatory strengths agree essentially completely in both sign and magnitude with the experimental results of Mason and Schnepp on trans-cyclooctene.

An order analysis of the particle–hole propagator

The particle–hole propagator is examined in terms of orders in electron repulsion. The order analysis is accomplished by expressing the equation of motion for the particle–hole Green’s function in the superoperator formalism and by using the inner projection technique to represent the superoperator resolvent. A particular choice of the projection manifold leads to a propagator which is consistent through third order in electron repulsion and in addition contains terms which are important for the description of the collective motions in medium size systems. Our decoupling approach is compared with a diagrammatic perturbation expansion, and with the higher random-phase approximation, and the self-consistent polarization propagator approximation. The latter two approximations, with two-particle, two-hole corrections, are shown to be consistent through second order in electron repulsion. Finally, we demonstrate that approximate propagator methods give a more balanced description of an excitation process than approximate configuration interaction calculations.

Direct Calculation of First- and Second-Order Density Matrices. The Higher RPA Method

We present a procedure for approximately determining the first- and second-order density matrices of pure-state N-fermion systems. A higher random-phase approximation is used to derive a method in which the resulting density matrices are approximated self-consistently in an iterative scheme. As test calculations, the method is applied to the ground states of the helium, lithium, and beryllium atoms. The possibility and importance of obtaining error bounds involving the density matrices are discussed.

Application of the RPA and Higher RPA to the V and T States of Ethylene

We have applied our proposed higher random-phase approximation (HRPA) to the T and V states of ethylene. In the HRPA, unlike the RPA, one solves for the excitation frequencies and the ground-state correlations self-consistently. We also develop a simplified scheme (SHRPA) for solving the equations of the HRPA, using only molecular integrals sufficient for the usual RPA calculations. The HRPA removes the triplet instability which often occurs in the RPA. The excitation energy for the N → T transition is now in good agreement with experiment. The N → V transition energy increases by 15% over its RPA Value. The N → V oscillator strength changes only very slightly. These results are also useful in explaining the appearance and ordering of states obtained in recent direct open-shell SCF calculations.

Higher Random-Phase Approximation as an Approximation to the Equations of Motion

Starting from the equations of motion expressed as ground-state expectation values, we have derived a higher-order random-phase approximation (RPA) for excitation frequencies of low-lying states. The matrix elements in the expectation value are obtained up to terms linear in the ground-state correlation coefficients. We represent the ground state as eU|HF〉, where U is a linear combination of two particle-hole operators, and |HF〉 is the Hartree-Fock ground state. We then retain terms only up to those linear in the correlation coefficients in the equation determining the ground state. This equation and that for the excitation energy are then solved self-consistently. We do not make the quasiboson approximation in this procedure, and explicitly discuss the overcounting characteristics of this approximation. The resulting equations have the same form as those of the RPA, but this higher RPA removes many deficiencies of the RPA.

Equations-of-Motion Method and the Extended Shell Model

This paper presents the equations-of-motion method as a useful and flexible tool in the study of nuclear spectroscopy. It is partly a review, but also it introduces a new and much more powerful equations-of-motion technique which supercedes the older linearization methods. The older methods worked with operator equations. To obtain closed expressions they had to be linearized in a rather arbitrary manner. The present approach works with the ground-state expectation of operator equations and thereby avoids all problems of linearization. Thus, like the Green's function method, the equations-of-motion method becomes potentially exact. It has many advantages over Green's function methods, however, among which are its greater compactness, simplicity, and the physical insight it yields.The method is first applied to rederive the random phase approximation (RPA) and the quasi-particle RPA (QRPA) and to show precisely what terms they neglect. It is demonstrated that some of these terms have coherent phases. A higher RPA and QRPA are then derived to include these terms. The corrections have some interesting effects: notably, there is a reduction of the effective interaction strength and a stabilization of the nucleus against sudden phase transitions. The equations-of-motion method is also used to generalize, in a very simple and natural way, the Hartree-Fock (HF) and Hartree-Bogolyubov (HB) concepts of independent particles and quasi-particles to nonsimple ground states.The equations-of-motion method is presented as a simple extension of the shell model to the treatment of excitationg of a correlated ground state. By concentrating on the quantities of direct physical interest, the complexity of workins with correlated wavefunctions is avoided.

Two- and Four-Quasiparticle States in Spherical Vibrational Nuclei

Expressions are derived in the modified Tamm-Dancoff (TD) approximation which take into account the mixing of zero-, two-, and four-quasiparticle states for the description of the collective vibrational states of spherical nuclei. The proposed method is free from two very important defects inherent in the higher random-phase approximation (HRPA), namely, nonorthonormality and redunancy of the four-quasiparticle basic states. The relative merits of the present method over the existing one (HRPA) are discussed.

The higher random phase approximation and the stability of the energy spectrum of the nuclear shell model

Assuming only two-body nuclear forces, we study the dynamical two and many-nucleon correlations in spherical nuclei with the method of the Heisenberg equations of motion of the one-nucleon operators. After self-consistently solving the Hartree-Fock problem for the shell model we study the effect of the left-over non linear coupling terms using the method of the Higher Random Phase Approximation (HRPA). In particular, we formulate the secular problem and solve it in the Second Random Phase Approximation (SRPA) for spherical nuclei. The numerical work has been carried out for the example of the16O nucleus. The original Hartree-Fock shell model spectrum has been found rather stable against the inclusion of the « triple » correlations. The density fluctuation terms have been shown to be rather small, but, generally, capable of introducing a certain damping of some of the single-particle states as a consequence of the non-hermiticity of the extended SRPA secular matrix.

Higher Random Phase Approximation and Energy Spectra of Spherical Nuclei

A method which is an extension of the usual random phase approximation (RPA) is proposed in order to study the spectra of spherical nuclei. In contrast to the usual RPA method based on the linearized equations of motion of the one-nucleon excitation operators, in the higher random phase approximation (HRPA) the many-nucleon excitations are also included. Nuclear states are superpositions of the one- and many-nucleon excitations (i.e., particle-hole pairs). We have especially studied the modes consisting of one- and two-nucleon excitations. In this second RPA, one solves the secular problem obtained from the closed system of linearized equations of motion for the one particle-hole pair operators ("doubles") and the two-pair operators ("quadruples"). States of definite nuclear spin, parity, and isotopic spin are considered. The method of the second RPA is illustrated in the example of the 6.06-Mev, ${J}^{\ensuremath{\pi}}={0}^{+}$, $T=0$ state of ${\mathrm{O}}^{16}$. Satisfactory semiquantitative agreement with the experiment is obtained for this case. The second RPA is also discussed in connection with the "aligned coupling scheme." The importance of the method for the study of the vibrational ${4}^{+}$ states is indicated.