Second-order Perturbation Correction Research Articles

Polaronic correction to the first excited electronic energy level in semiconductor quantum dots with parabolic confinement

We show that the second-order perturbative correction to the first excited state energy of an electron in a multi-dimensional parabolic quantum dot can be written in a simple closed analytical form. We apply our theory to several realistic semiconductor quantum dots and show that the first excited state polaron self-energy corrections in these dots can be considerable if their sizes are of the order of a few nanometers.

An optimized second-order perturbative study of the asymmetrical quantum anharmonic oscillator

We study the energy eigenvalues of a cubic-quartic anharmonic oscillator with the operator method introduced by Feranchuk and Komarov (Ann. Phys. (N.Y.), 238 (1995) 370). Based on the minimization of the second-order perturbative correction and the energy-variance, a simple optimized second-order perturbative calculation is found to yield very accurate energies especially for the excited states: the asymptotic error is shown to be -0.002 %. When the same method is applied to an oscillator with the sextic anharmonic potentialx2 /2 + X%j, i = » similar accuracy is obtained for the excited states with an asymptotic error of about -0.003%.

A CASPT2 study of the valence and lowest Rydberg electronic states of benzene and phenol

The valence excited states and the 3s, 3p, and 3d (united atom) Rydberg states of benzene and phenol have been obtained by the CASPT2 method, which computes a second-order perturbation correction to complete active space self-consistent field (CASSCF) energies. All non-zero dipole oscillator strengths are also computed, at the CASSCF level. For benzene, 16 singlet and 16 triplet states with excitation energies up to ca. 7.86 eV (63 400 cm–1) are obtained. Of these, 12 singlet and three triplet energies are experimentally known well enough to allow meaningful comparison. The average error is around 0.1 eV. The highest of these singlet states (21 E2g) is the highest valence ππ* state predicted by elementary π-electron theory. Its energy is then considerably lower than has been suggested from laser flash experiments, but in perfect agreement with a reinterpretation of that experiment. For phenol, 27 singlet states are obtained, in the range 4.53–7.84 eV (63 300 cm−1). Only the lowest has a well-known experimental energy, which agrees with the computed result within 0.03 eV. The ionization energy is in error by 0.05 eV.

Dynamical effect corrections to hole self-energies in CuO2 planes

Abstract The total density of states for the three-band Hubbard model of CuO 2 planes is calculated including fully self-consistent second-order perturbative corrections to the self-energy at the copper orbitals. We apply a configurational mixing approach in a Bethe-lattice formulation to evaluate these quantum fluctuations in a local approximation. For realistic values of the model parameters we obtain a destruction of the antiferromagnetism at a small dopant hole concentration of 3–5% due to the occurrence of a spin disordered state with local moments. The charge transfer gap is slightly reduced as compared with the Hartree-Fock results due to the incorporated dynamical effects.

Ab initio Hartree—Fock studies of the binding energy of a soliton to a sodium counterion in doped trans-polyacetylene

We have performed split valence ab initio Hartree—Fock calculations of the total binding energy, E B, of charged solitons to donor impurities in doped trans-polyacetylene. The total energy is calculated for a system with the soliton pinned to the impurity and for a system where the soliton is isolated from the impurity. E B is taken to be the difference between the optimized total energies of these two systems. For a sodium counterion we obtain a total binding energy of E B = 3.18 eV. This value reduces to 2.25 eV when Møller—Plesset second-order perturbation corrections are included. The energy of the soliton level relative to the valence band edge is found to decrease by 0.22 eV in the presence of a sodium counterion, a result which correlates well with optical absorption data.

Second-order polarization energies for Rydberg states

The energy level shifts for an electron in a high-nL Rydberg state of an atom such as helium are well described by an asymptotic potential with terms proportional to the multipole moments of the core. We obtain exact analytic expressions as a function of n and L for second-order perturbation corrections to the energy due to the asymptotic potential. The results are of importance in the analysis of high-precision variational calculations and experimental transition energies for Rydberg states.

Effects of electron correlation upon molecular interactions: Correction of the electrostatic interaction between DNA bases

A procedure is advanced for correcting the electrostatic portion of the interaction energy for the effects of intramolecular electron correlation. This expression, which specifically treats the dipole-dipole term, involves the interaction between the Hartree-Fock dipole moment of one subunit and the second-order perturbation correction of its partner (plus the converse). The algorithm is tested on 28 different DNA base pairs, whose interaction energies have been evaluated previously as a sum of the (superposition-corrected) SCF interaction plus the (London) dispersion energy. While the correlation correction is negligible for certain pairs, it is quite sizable in others, in one case lowering the long-range dipole-dipole interaction energy by 27%. On the other hand, the poor convergence of the multipole expansion at equilibrium separation leads to unsatisfactory results at this geometry. It is suggested that a distributed multipole approach would yield superior results over a wide range of intermolecular distance.

First- and second-order contributions to the effective Hamiltonian parameters for Gd 3+( 8S) and Mn 2+( 6S)

The effective operator formalism is used to calculate the values of effective Hamiltonian parameters which arise from higher order perturbation corrections due to the relativistic crystal field and spin-orbit interactions. The method is illustrated on two examples: the Gd 3+ ion in the C 3h-symmetry and the Mn 2+ ion in the C 3v-symmetry. Transitions to all excited orbitals for l = s, p, d, f and n = l+1,…, 10 are included in calculations. The role of second-order perturbation corrections is significant for both excitations within the ground configuration and outside it.

Ground-state energetics of helium and deuterium fermion fluids

The method of correlated basis functions (CBF) is applied to the evaluation of the ground-state energy of atomic fermion fluids as a function of density. As a first step, liquid $^{3}\mathrm{He}$ in both unpolarized and fully polarized spin configurations is considered variationally, using Slater-Jastrow trial wave functions. Results are reported for a conventional analytic choice of the state-independent two-body correlation function $f(r)$ and for the optimal $f(r)$ determined by the solution of a suitable Euler equation. The Jastrow treatment is found to be inadequate in that (i) the energy expectation value lies above the experimental equilibrium energy by some 1.5 K, and (ii) the polarized phase is predicted to be more stable than the unpolarized one. For a given polarization, a correlated basis is formed by application of the assumed Jastrow correlation factor to the elements of a complete set of noninteracting-Fermi-gas Slater determinants. The exact ground-state energy may be developed in a perturbation expansion in the correlated basis, the leading term being the Jastrow energy expectation value. Considerable improvement on the Jastrow description of the unpolarized phase is achieved upon inclusion of the correlated two-particle\char22{}two-hole component of the second-order CBF perturbation correction. At the experimental equilibrium density, this contribution, which incorporates important momentum- and spin-dependent correlations, can amount to some 0.6-1.1 K [depending on the choice of $f(r)$]. The required correlated-basis matrix elements are calculated by Fermi hypernetted-chain (FHNC) techniques, crucial Pauli effects of the elementary diagrams being introduced through the FHNC/C algorithm. The Euler equation is approximated within the same framework. The momentum-space integrations in the second-order perturbation correction are evaluated by a Monte Carlo procedure. One may reasonably expect that further refinements of the CBF method will lead to an accurate microscopic description of the ground-state energetics of liquid $^{3}\mathrm{He}$. Bulk atomic deuterium with all electronic spins aligned is treated at the same level of approximation as applied to helium. Three choices of nuclear-spin distribution are examined, with a single spin state present, or two or three equally populated nuclear spin states. The finite-density energy minimum is found to lie very close to zero energy in all three examples; a very precise many-body calculation will thus be needed to decide their liquid or gaseous nature at zero temperature under zero external pressure.

The care and handling of CBF corrections

We report calculations of second-order perturbation corrections within the method of correlated basis functions (CBF) for the OMY nuclear interaction. For nuclear matter, Monte Carlo integration enables us to evaluate independently the accuracy of approximations made in previous calculations. We suggest a prescription which permits substantial improvements within the framework of those calculations.

A correlated-basis-functions approach to realistic nuclear matter

The method of correlated basis functions is adapted to the nuclear-matter problem with two-nucleon potentials containing tensor as well as central components. Procedures are described for evaluating through three-body cluster order the energy expectation value with respect to a constrained trial ground-state wave function incorporating tensor and central correlations, and for calculating in two-body cluster approximation the second-order perturbation correction in a basis of likewise-correlated functions. Results for the 5100, 5200, Gammel-Christian-Thaler and Hamada-Johnston potentials are presented and dissected.

Atomic correlation energies. III. Second-order corrections to the Hartree-Fock ground state of B, C, N, O and F

For pt.II see ibid., vol.11, no.12, p.2069 (1978). The authors have calculated the second-order perturbative corrections to the Hartree-Fock ground state energies of B, C, N, O and F. The basis used was sufficiently large to assure truncation errors less than 10-3 au in all cases. They found EB(2)=-0.1113 au, EC(2)=-0.1476 au, EN(2)=-0.1840 au, EO(2)=-0.2540 au and EF(2)=-0.3206 au which represent 89.2%, 94.3%, 97.6%, 98.5% and 99.6% respectively of the empirical correlation energies.

Atomic correlation energies. II. Converged E(2)values for Ne and Ne+

For pt.I see ibid., vol.11, p.1157 (1978). The second-order perturbative corrections to the Hartree-Fock ground-state energies of Ne and Ne+ are calculated using a Hartree-Fock-Roothaan approach with practically complete basis sets. The results are ENe(2)=-0.3855 au and ENe+(2)=-0.3214 au. It was found necessary to include virtual orbitals with relatively high angular momenta in the basis sets in order to saturate the E(2) values. The Ne result represents 98.8% of the empirical correlation energy. From ENe(2), ENe+(2) and the Hartree-Fock ground-state energies the authors find an ionisation potential, IPNe,calc=0.7934 au or 21.59 eV, in excellent agreement with the measured value, IPNe=21.56 eV.

Atomic correlation energies. I. Rigorous evaluation of E(2)for He, Li and Be

The authors have calculated the second-order perturbative corrections to the ground-state energy of He, Li and Be with the Hartree-Fock Hamiltonian as H0. Increasingly large basis sets were used, up to the point where results stabilised. The converged basis-independent second-order corrections are Ehe(2)=-0.0372 au, ELi(2)=-0.416 au and EBe(2)=-0.757 au, which amount to 88%, 92% and 80% respectively, of the experimental correlation energies.

Perturbation corrections to the Jastrow energy for simple models of nuclear matter

The second-order perturbation correction to the Jastrow energy expectation value is calculated in the leading cluster approximation for two models of nuclear matter in which the nucleons are assumed to interact via “semi-realistic” or “test” central potentials containing hard cores. This negative correction, which acts to compensate for the state independence of the Jastrow correlation factor, is reasonably small in magnitude. Nevertheless it significantly aggravates the already serious disagreement of Jastrow and Brueckner results, i.e., the corrected Jastrow energy is depressed still further below the lowest-order Brueckner energy, the net discrepancy increasing to some 10 or 15 MeV at the Jastrow equilibrium. Higher-order cluster contributions to the Jastrow energy expectation value are considered, and some of the implications for nuclear many-body theory of this and other known examples of striking confict between Brueckner and Jastrow results are discussed.

The tensor J8 contribution to the rotational energy of a tetrahedral molecule

Two methods are outlined for calculating the tensor distortion energy of a tetrahedral molecule to the eighth degree in the rotational angular momentum. In the first, the entire distortion Hamiltonian including the eighth-rank distortion tensor Ω 8 is diagonalized. In the second, the basis which diagonalizes only the fourth rank tensor Ω 4 is used to generate tables which allow the direct determination of the octic energy terms. For J ≤ 20, numerical values are given of the diagonal matrix elements of Ω 8 and of the second-order perturbation correction due to offdiagonal matrix elements of the sixth-rank tensor Ω 6 . Explicit expressions in terms of angular momentum operators are given for Ω 8 and a selection of the operator equivalents T 8, m .

Second-order perturbation correction to the Bogoliubov excitation spectrum

Two substantially different perturbation methods are used to obtain two equivalent exact results for the leading correction to the Bogoliubov formula for the energy spectrum of an elementary excitation in a boson fluid.

Many-Body Green's Functions for Finite, Nonuniform Systems: Applications to Closed Shell Atoms

Ab initio calculations of natural orbitals, ionization potentials, and total ground state energies for the closed shell systems helium and beryllium are given using the technique of many-body Green's functions. The necessary formalism for application of the Green's function theory to finite, nonuniform, many-body systems is developed following the work of Layzer; connections with standard perturbation theory are made. The natural orbitals obtained were of high quality. Analysis of the diagrammatic expansion of the Green's function led to the surprising result that the main effect of the infinite-order summations implicit in the solution of Dyson's equation was to ``renormalize'' the second-order perturbation corrections.

Corrections to the Variational Ground State of a Charged-Boson Gas

The variational ground-state energy of a charged-boson gas, obtained previously for an intermediate density range using a Bijl-Dingle-Jastrow (BDJ) type of correlated trial wave function, is improved in two steps: (i) application of the paired-phonon analysis for optimization of the variational description of the ground state in the BDJ function space, using the hypernetted-chain equation for the correlation function, and (ii) use of the improved liquid-structure function to evaluate the second-order perturbation correction generated by three-phonon components. The paired-phonon analysis yields quite small corrections to the ground-state properties including the energy and the liquid-structure function, and the perturbation corrections to the energy are also found to be very small. It is demonstrated that in the high-density limit the procedure yields the ground-state energy exact through second order in the density expansion.

Theory of the interaction of the lattice vibrations with the vibrational excitations in solid hydrogen

The theory of the interaction of the vibrational excitations (vibrons) with the lattice vibrations in solid hydrogen is developed. The phonons are treated in the self-consistent harmonic (SCH) approximation appropriate to quantum crystals. The vibron–phonon interaction is expanded in terms of the SCH phonon operators rather than in powers of the displacements of the molecules from their equilibrium positions. First- and second-order perturbation corrections to the vibron energies arising from the vibron–phonon coupling are calculated. The effect of the anharmonicity of the intramolecular vibrations in conjunction with the vibron–phonon coupling is also discussed.