Hartree Wave Functions Research Articles

Symmetries in the multi-configurational time-dependent Hartree wavefunction representation and propagation.

In multi-configurational time-dependent Hartree (MCTDH) approaches, different multi-layered wavefunction representations can be used to represent the same physical wavefunction. Transformations between different equivalent representations of a physical wavefunction that alter the tree structure used in the multi-layer MCTDH wavefunction representation interchange the role of single-particle functions (SPFs) and single-hole functions (SHFs) in the MCTDH formalism. While the physical wavefunction is invariant under these transformations, this invariance does not hold for the standard multi-layer MCTDH equations of motion. Introducing transformed SPFs, which obey normalization conditions typically associated with SHFs, revised equations of motion are derived. These equations do not show the singularities resulting from the inverse single-particle density matrix and are invariant under tree transformations. Based on the revised equations of motion, a new integration scheme is introduced. The scheme combines the advantages of the constant mean-field approach of Beck and Meyer [Z. Phys. D 42, 113 (1997)] and the singularity-free integrator suggested by Lubich [Appl. Math. Res. Express 2015, 311]. Numerical calculations studying the spin boson model in high dimensionality confirm the favorable properties of the new integration scheme.

Open system dynamics using Gaussian-based multiconfigurational time-dependent Hartree wavefunctions: Application to environment-modulated tunneling.

A variational approach for the quantum dynamics of statistical mixtures is developed, which is based upon the representation of the natural states of the mixture in terms of hybrid Gaussian-based Multiconfiguration Time-Dependent Hartree (G-MCTDH) wavefunctions. The method, termed ρG-MCTDH, is combined with a treatment of dissipation and decoherence based on the nonstochastic open-system Schrödinger equations. The performance and the convergence properties of the approach are illustrated for a two-dimensional tunneling system, where the primary tunneling coordinate, represented by flexible single-particle functions, is resonantly coupled to a second harmonic mode, represented by Gaussian wave packets. The harmonic coordinate is coupled to the environment and two different processes are studied: (i) vibrational relaxation at zero temperature described by a master equation in the Lindblad form and (ii) thermalization induced by the Caldeira-Leggett master equation. In the second case, the evolution from a quantum tunneling regime to a quasistationary classical-limit distribution, driven by the heat bath, is visualized using a flux analysis.

The 't Hooft Coupling and Baryon Mass Splitting in the Large-Nc Quark Model**Supported by the National Natural Science Foundation of China under Grant No 11265014.

We study the 't Hooft coupling gt and the mass splitting of the ground-state baryons in terms of the large Nc-inspired quark model, by which the Hartree wavefunctions of light quarks are obtained. By fitting the spectra of decuplet and octet baryons, we obtain the 't Hooft coupling gt to be around 1.57. We generalize the scenario to the case of heavy baryons, such as ΛAc, gt values which does not deviate much from 1.57, as well as to the case of mesons with gt far from that for baryons. The consequence is discussed.

Heavy baryons in the large [formula omitted] limit

It is shown that in the large Nc limit heavy baryon masses can be estimated quantitatively in a 1/Nc expansion using the Hartree approximation. The results are compared with available lattice calculations for different values of the ratio between the square root of the string tension and the heavy quark mass σ/mQ. These estimates implement important 1/Nc corrections and assume a string tension independent of Nc. Using a potential adjusted to agree with the one obtained in lattice QCD, a variational analysis of the ground state spin averaged baryon mass is performed using Gaussian Hartree wave functions. Relativistic corrections through the quark kinetic energy are included. The results provide good estimates for the first sub-leading in 1/Nc corrections.

Quantum Solitary Wave Solutions in a Ferromagnetic Chain with Nearest- and Next-Nearest-Neighbor Interaction

The quantum solitary wave solutions in a one-dimensional ferromagnetic chain is investigated by using the Hartree–Fock approach and the multiple-scale method. It is shown that quantum solitary wave solutions can exist in a ferromagnetic system with nearest- and next-nearest-neighbor exchange interaction, and at the certain value of the first Brillouin zone, the solitary wave solution of the Hartree wave function becomes the intrinsic localized mode.

Magnetic field dependence of the low-energy spectrum of a two-electron quantum dot

The low-energy eigenstates of two interacting electrons in a square quantum dot in a magnetic field are determined by numerical diagonalization. In the strong correlation regime, the low-energy eigenstates show Aharonov-Bohm-type oscillations, which decrease in amplitude as the field increases. These oscillations, including the decrease in amplitude, may be reproduced to good accuracy by an extended Hubbard model in a basis of localized one-electron Hartree states. The hopping matrix element t comprises the usual kinetic energy term plus a term derived from the Coulomb interaction. The latter is essential to get good agreement with exact results. The phase of t gives rise to the usual Peierls factor, related to the flux through a square defined by the peaks of the Hartree wave functions. The magnitude of t decreases slowly with magnetic field as the Hartree functions become more localized, giving rise to the decreasing amplitude of the Aharonov-Bohm oscillations.

A Gaussian wavepacket approach for including correlation in the time-dependent Hartree wavefunction for short times

A hybrid Gaussian wavepacket time-dependent Hartree approach to quantum dynamics is presented. The time-dependent Hartree method yields an efficient methodology for solving the multi-dimensional time-dependent Schrödinger equation, but has the deficiency of enforcing a factorized form of the wavepacket for all time. We present a strategy for including the correlations in the time-dependent Hartree wavefunction by including a time-dependent correction to the Hartree wavefunction. This correction has the form of a time-dependent Gaussian wavepacket. We test the hybrid Gaussian wavepacket-time-dependent Hartree approach and find that it is able to predict correlations in the wavefunctions for at least a single vibrational period.

Two-center p-space integrals: diatomic molecular moments

Analytical expressions for the calculation of both orbital and total moments of diatomic molecules using Gaussian‐type orbitals are formulated. Moments \(\left\langle {p^t } \right\rangle \) with \( - 2 \leqslant t \leqslant 4\) are computed for the ground state of 35 diatomic molecules at equilibrium bond length using 6‐311G(d,p) basis sets. In order to test our expressions, these expectation momentum values are compared with the values calculated using self‐consistent‐field wave functions of Hartree–Fock quality, which give Hartree–Fock limit energies.

Time-dependent calculation of reactive flux employing complex absorbing potentials: General aspects and application within the multiconfiguration time-dependent Hartree wave approach

A novel modification of the flux operator formalism is introduced that combines the merits of the flux operator approach with those of complex absorbing potentials. The method is used to determine initial-state selected reaction probabilities for a broad energy range from a single appropriately chosen time-dependent wave packet. The propagation may be performed solely in the coordinates of the reagents arrangement channel. State-to-state transition probabilities can also be obtained when appropriate projectors are included. In contrast to similar methods the present one does not require the calculation of derivatives with respect to the reaction coordinate. More importantly, it avoids the need to (E,t)-Fourier transform the wave packet at every grid point on a dividing surface. The proposed formula, though completely general, is especially well suited to handle multiconfiguration time-dependent Hartree wave functions. As a check of the reliability initial-state selected reaction probabilities for the collinear H+H2→H2+H reaction are calculated and compared with (numerically) exact results. We also show that the initial wave packet may be placed close to the interaction region when its energy distribution is corrected for the mean potential energy.

Semiclassical molecular dynamics for strongly coupled Coulomb systems

Based on the time-dependent variational principle for a Hartree wave function, a semiclassical approximation for a strongly correlated plasma is derived. This procedure maps the quantum dynamical problem to a classical one with one additional degree of freedom per particle and is considered the natural extension of classical molecular dynamics. As a test case, a single electron in a Coulomb potential is studied. For the full many-body problem the pair distribution function, the velocity autocorrelation function, the conductivity, and the diffusion constants are calculated for a plasma. We also consider liquid molecular hydrogen and lithium as a further test case which is sensitive to the treatment of the Pauli exclusion principle.

Quasielastic 40Ca(e,e'p) cross sections in a many-particle self-consistent Hartree model.

A fully distorted partial-wave calculation is presented for the reaction $^{40}\mathrm{Ca}$(e,e'p${)}^{39}$K which uses relativistic self-consistent Hartree wave functions for both the target nucleus and the residual nucleus. The final state interaction of the knocked-out nucleon is described by an optical potential and the Coulomb distortion of the electron wave functions is also included. We assume standard values for meson coupling constants and masses in the Hartree calculations and the optical potential used comes from global fits to elastic proton scattering; hence, this calculation is a description of (e,e'p) reactions with no free parameters. We examine the proton knocked-out from the ${\mathit{d}}_{3/2}$ state of $^{40}\mathrm{Ca}$ which leaves the residual nucleus $^{39}\mathrm{K}$ in its ground state. Agreement with the experimental data of NIKHEF is good and the calculated spectroscopic factor agrees very well with the one extracted from experiment. There seems to be no need to make the assumption that the ${\mathit{d}}_{3/2}$ state is appreciably depleted. We also calculate the (e,e'p) cross section from the 2${\mathit{s}}_{1/2}$ and 1${\mathit{f}}_{7/2}$ orbitals. For the 2${\mathit{s}}_{1/2}$ orbital the shape does not agree so well with the experimental data. The ${\mathit{f}}_{7/2}$ orbital leads to approximately the correct shape and we extract a spectroscopic factor of 3.7%.

Semileptonic weak interactions in nuclei: A relativistic analysis

We use a consistent relativistic formalism, with relativistic Hartree wave functions and a covariant effective current operator with all the properties of the standard model, to study semileptonic weak processes in nuclei. Results are obtained for β-decay, μ- capture, charge-changing v-reactions and neutral v-scattering involving the isodoublet systems 17F 17O and 3H 3He.

On the effect of the helium ground state wave function on the electron impact triple differential ionization cross section in asymmetric geometry

The sensitivity of the triple differential cross sections for the electron impact ionization of helium to the target ground state wave function is investigated at high energies in the asymmetric Ehrhardt geometry. The calculation is done in the modified Glauber approximation using (i) one-parameter Hartree wave function and (ii) three-parameter analytical fit of Byron and Joachain to the Hartree–Fock wave function for the helium ground state. The latter choice leads to somewhat better agreement with the recent absolute data of Jung et al. at 600 eV incident energy. The results are however not very sensitive to this choice.

Local density approximations to moments of momentum of diatomic molecules with Hartree–Fock–Roothaan quality electron distributions

By generalizing local density arguments for atoms it is found that the mth moment of momentum for molecules 〈pm〉 is proportional to ∫ρ(r)1+(1/3)mdr, where ρ(r) is the electron density. This relation is tested for m=−1, 1, and 2 using wave functions of Hartree–Fock–Roothaan quality: 〈pm〉 is computed directly and the density integrals are evaluated using ρ(r) generated from these wave functions. Eleven neutral diatomic systems and two singly charged positive ions are considered. Apart from changes in the proportionality constants, the local density approximation is vindicated; for m=−1, 1, and 2 the magnitudes of the changes in the predicted constants are less than ∼10%. Finally, 〈p〉, which is proportional to the Dirac–Slater exchange energy, is compared with the appropriate self-consistent Hartree–Fock–Roothaan energy terms and, again, a linear relationship emerges.

The electric dipole sum rule in a relativistic self-consistent theory of finite nuclei

Using a relativistic self-consistent theory of finite nuclei including meson degrees of freedom, the energy weighted electric dipole sum rule is evaluated for 4He, 16O, 40Ca, and 208Pb. Due primarily to the presence of scalar meson exchange, the lower effective nucleon mass results in a substantial increase over the Thomas-Reiche-Kuhn result. The importance of self-consistency is stressed and comparison is made with previous non-relativistic calculations of the sum rule as well as with experimental data. Allowing modifications from the filled Dirac sea in the Hartree wavefunctions significantly alters the predicted sum rule value for all nuclei considered.

Correlation hole for an exactly solvable problem

AbstractThe probability distribution of the interparticle distance is obtained for the N‐body boson problem where the potential is an attractive delta function. This is done for both the exact wave function and the Hartree wave function allowing the calculation of the correlation hole.

Effect of the electron-phonon interaction on the subband structure of inversion layers in compound semiconductors

Recently electron inversion layers have been fabricated on a number of compound (principally III–V) semiconductors. Because of the polar nature of these materials, the effect of the electron-optical phonon interaction is expected to be significant. This has certainly been true in recent calculations of the ground state energy and equilibrium density of electron-hole liquids in a number of compound semiconductors [1]. We have calculated the subband structure of inversion layers in polar semiconductors. The starting point of our many-body calculation is the Hartree subband structure determined using a variational method described in ref. [2]. Using the Hartree wavefunctions as basis functions we have calculated the self-energy of the ground and first excited subbands to lowest order in the effective interaction. This interaction is the sum of Coulomb (direct and image) and electron-phonon terms which are screened by a dielectric function that includes both e-e and e-ph effects. The screening by inversion layer electrons has been treated in the plasmon-pole approximation. We have obtained the electron self-energy, the quasi-particle energy, and the subband separations as a function of inversion layer concentration for parameters corresponding to GaAs. The coupling of electrons to bulk phonons is considered. Our results are compared to those obtained using the simpler ϵ 0 ∗ approximation [1]. Here, explicit phonon terms are neglected, but the bare band mass is replaced by the polaron mass, and the high frequency dielectric constant (ϵ ∞) is replaced by the static dielectric constant (ϵ 0). This approximation has been applied to the electron-hole liquid for situations in which the optical phonon frequency is larger than the electron plasma frequency. The idea behind the ϵ 0 ∗ approximation is to account in a very rough way for vertex corrections to the electron-phonon interaction. A principal criticism of our dynamic RPA calculations is the neglect of vertex corrections in the effective interaction. In three-dimensional metallic systems (high density), Migdal's theorem applies and vertex corrections are small. However, it is not clear that we may neglect such corrections in inversion layers. This point will be discussed further.

Time-dependent Hartree-Fock formalism for the dielectric function

In this paper we derive a formula for the time-dependent Hartree-Fock dielectric function by using Green's-function theory. We develop a set of self-consistent equations for the self energy, polarization, and Green's function which can be iterated to obtain successively more accurate approximations for the polarization. In this procedure the Hartree and Hartree-Fock approximations occur naturally as the first and second steps in the iterations. As the iterations are carried out, we obtain not only the expressions for the polarization, but also the one-electron Hartree-Fock equations, so that there is no question as to which polarization formula corresponds to which equation. In this way we obtain an expression for the dielectric function which clearly and directly corresponds to the Hartree-Fock equation and which is therefore called the time-dependent Hartree-Fock (TDHF) dielectric function. In this TDHF formula for the dielectric function, the first term is the time-dependent Hartree or random-phase approximation dielectric function. We show that in order to be consistent with the theory, one should use Hartree wave functions and energies to compute random-phase approximation dielectric functions and Hartree-Fock wave functions and energies to compute TDHF dielectric functions. Numerical results are presented for the free-electron gas.

On the Fermi hole in atoms

A recent proposal for the evaluation of the shape of the Fermi hole in atoms is investigated. It is pointed out that the arbitrariness inherent in the Hartree-Fock orbitals introduces an ambiguity in the definition, which can be overcome by using the unorthogonalized Hartree orbitals. This is still not quite satisfactory, the use of a proper Hartree-Fock as well as a proper Hartree wavefunction, each independently optimized, being essential for obtaining the shape of the Fermi hole. Calculations demonstrating these points are presented for the 23S state of the helium isoelectronic sequence.

The Fermi hole in atoms

A new definition of the Fermi hole in many-electron systems is provided in terms of the distribution function falpha alpha (r12) of the interelectronic distance for electrons with parallel spins. By analogy with the Coulomb hole, the Fermi hole is defined as the difference between the values of f22(r12) derived from the Hartree-Fock and the Hartree wavefunctions. this definition, unlike previous ones, provides a simple picture of the Fermi hole as a function of r12. By assuming that the Hartree and Hartree-Fock orbitals are identical, an analytical formula is derived for the Fermi hole. Explicit calculations are presented for the 23S state of He and the ground state of Be. It is observed that the Fermi hole is remarkably similar in these two cases; and that the effects of Coulomb correlation are more long-ranged than those of Fermi correlation.