Kohn Theorem Research Articles

Distribution of oscillator strength in Gaussian quantum dots: An energy flow from center-of-mass mode to internal modes

The energy spectra and oscillator strengths of two, three, and four electrons confined by a quasi-two-dimensional attractive Gaussian-type potential have been calculated for different strength of confinement $\ensuremath{\omega}$ and potential depth $D$ by using the quantum chemical configuration interaction (CI) method employing a Cartesian anisotropic Gaussian basis set. A substantial redshift has been observed for the transitions corresponding to the excitation into the center-of-mass mode. The oscillator strengths, concentrated exclusively in the center-of-mass excitation in the harmonic limit, are distributed among the near-lying transitions as a result of the breakdown of the generalized Kohn theorem. The distribution of the oscillator strengths is limited to the transitions located in the lower-energy region when $\ensuremath{\omega}$ is large but it extends towards the higher-energy region when $\ensuremath{\omega}$ becomes small. The analysis of the CI wave functions shows that all states in the energy range covered by the present study can be classified according to the polyad quantum nnumber ${v}_{p}$. It is shown that the distribution of the oscillator strengths for large $\ensuremath{\omega}$ occurs among transitions involving excited states with the same value of ${v}_{p}$ as the center-of-mass excited state, ${v}_{p,\mathit{cm}}$, while it occurs among transitions involving the excited states with ${v}_{p}={v}_{p,\mathit{cm}}$ and ${v}_{p}={v}_{p,\mathit{cm}}+2$ for small $\ensuremath{\omega}$.

Just how good is DFT?

Density functional theory (DFT) has become a ‘‘household tool,’’ so to speak, for the average chemist. We now take it for granted: we expect DFT to provide a reasonably good description of almost any molecule we can make—or rather, we can think of—and justifiably so. For organic molecules, this is a perfectly valid expectation. However, for transition metal complexes and transition states, the performance of DFT, while generally very good, is less certain, while van der Waals complexes are largely beyond the range of applicability of current DFT methods. Given DFT’s broad and increasing popularity among bioinorganic chemists, a need has been felt for a critical evaluation of the performance of DFT, from a bioinorganic perspective. This series of four Commentaries is intended to address that need. Modern DFT began in 1964, with a famous paper by Hohenberg and Kohn [1] which contained the following two rigorous theoretical results. First, the groundstate electron density uniquely determines the electronic wavefunction and hence all ground-state properties of an electronic system. Second, the energy of an electron distribution is a function—more technically, a functional—of the electron density and this functional is a minimum for ground-state densities. The Hohenberg–Kohn theorem assures us that the density functional is a universal quantity, but it does not specify its form. These results reduce the problem

The Hellmann–Feynman theorem and its relation with the universal density functional

The integrated Hellmann–Feynman theorem provides an expression for the energy as a functional of the density. However, this result is completely defined only for the densities corresponding to the eigen functions of the Schrödinger equation. Since this density functional is obtained by integrating the Hellmann–Feynman theorem, it is not possible to obtain bounds to the true energy by using the variation method together an approximate density. This point is discussed through some elementary examples. The relation between the integrated Hellmann–Feynman theorem and the Hohenberg and Kohn theorem, is also stressed in this Letter.

Magnetoelasticity theory of incompressible quantum Hall liquids

A simple and physically transparent magnetoelasticity theory is proposed to describe linear dynamics of incompressible fractional quantum Hall states. The theory manifestly satisfies the Kohn theorem and the $f$-sum rule, and predicts a gaped intra-Landau level collective mode with a roton minimum. In the limit of vanishing bare mass $m$ the correct form of the static structure factor, $s(q)\sim q^4$, is recovered. We establish a connection of the present approach to the fermionic Chern-Simons theory, and discuss further extensions and applications. We also make an interesting analogy of the present theory to the theory of visco-elastic fluids.

Density functional theory in terms of a Legendre transformation for beginners

The derivation of the Hohenberg–Kohn (HK) theorem by means of a simplified version of Lieb's Legendre transformation is presented, with the stress on physically problematic aspects, and caring about the definition of the domains of the respective functionals only to the extent, that this is absolutely necessary. The take-home lesson consists in a critical analysis of some statements often found in the DFT literature. As a simple illustration for a Legendre transformation we discuss that in parameter space for the family of n-electron isoelectronic atomic ions, with the nuclear charge as parameter.

On the proof by reductio ad absurdum of the Hohenberg–Kohn theorem for ensembles of fractionally occupied states of Coulomb systems

AbstractIt is demonstrated that the original reductio ad absurdum proof of the generalization of the Hohenberg–Kohn theorem for ensembles of fractionally occupied states for isolated many‐electron Coulomb systems with Coulomb‐type external potentials by Gross and colleagues is self‐contradictory, since the to‐be‐refuted assumption (negation) regarding the ensemble one‐electron densities and the assumption regarding the external potentials are logically incompatible to each other due to the Kato electron‐nuclear cusp theorem. It is proved, however, that the Kato theorem itself provides a satisfactory proof of this theorem. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Collective motions of a quantum gas confined in a harmonic trap

Single-component quantum gas confined in a harmonic potential, but otherwise isolated, is considered. From the invariance of the system of the gas under a displacement-type transformation, it is shown that the center of mass oscillates along a classical trajectory of a harmonic oscillator. It is also shown that this harmonic motion of the center has, in fact, been implied by Kohn's theorem. If there is no interaction between the atoms of the gas, the system in a time-independent isotropic potential of frequency ${\ensuremath{\nu}}_{c}$ is invariant under a squeeze-type unitary transformation, which gives collective radial breathing motion of frequency $2{\ensuremath{\nu}}_{c}$ to the gas. The amplitudes of the oscillating and breathing motions from the exact invariances could be arbitrarily large. For a Fermi system, appearance of $2{\ensuremath{\nu}}_{c}$ mode of the large breathing motion indicates that there is no interaction between the atoms, except for a possible long-range interaction through the inverse-square-type potential.

Charge-density response function of two-dimensional electron systems in a magnetic field

We examine charge density excitations in two-dimensional electron systems in a strong perpendicular magnetic field. Our theoretical framework is the extended random-phase approximation (extended RPA, ERPA), which takes into account all the second-order self-energies of particle-hole propagators. Our results are different from the RPA response functions qualitatively, and are consistent with recent empirical findings. We explain our results in detail in connection with two-particle-two-hole degrees of freedom and the Kohn theorem.

On the original proof by reductio ad absurdum of the Hohenberg–Kohn theorem for many‐electron Coulomb systems

AbstractIt is shown that, for isolated many‐electron Coulomb systems with Coulombic external potentials, the usual reductio ad absurdum proof of the Hohenberg–Kohn theorem is unsatisfactory since the to‐be‐refuted assumption made about the one‐electron densities and the assumption about the external potentials are not compatible with the Kato cusp condition. The theorem is, however, provable by more sophisticated means, and it is shown here that the Kato cusp condition actually leads to a satisfactory proof. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Kohn’s theorem in a superfluid Fermi gas with a Feshbach resonance

We investigate the dipole mode in a superfluid gas of Fermi atoms trapped in a harmonic potential. According to Kohn's theorem, the frequency of this collective mode is not affected by an interaction between the atoms and is always equal to the trap frequency. This remarkable property, however, does not necessarily hold in an approximate theory. We explicitly prove that the Hartree-Fock-Bogoliubov generalized random phase approximation (HFB-GRPA), including a coupling between fluctuations in the density and Cooper channels, is consistent with both Kohn's theorem as well as Goldstone's theorem. This proof can be immediately extended to the strong-coupling superfluid theory developed by Nozi\'eres and Schmitt-Rink (NSR), where the effect of superfluid fluctuations is included within the Gaussian level. As a result, the NSR-GRPA formalism can be used to study collective modes in the BCS-BEC crossover region in a manner which is consistent with Kohn's theorem. We also include the effect of a Feshbach resonance and a condensate of the associated molecular bound states. A detailed discussion is given of the unusual nature of the Kohn mode eigenfunctions in a Fermi superfluid, in the presence and absence of a Feshbach resonance. When the molecular bosons feel a different trap frequency from the Fermi atoms, the dipole frequency is shown to depend on the strength of effective interaction associated with the Feshbach resonance.

Parametric resonance of a two-dimensional electron gas under bichromatic irradiation

In an ultrahigh mobility 2D electron gas, even a weak nonparabolicity of the electron dispersion, by violating Kohn's theorem, can have a drastic effect on dc magnetotransport under ac drive. We study theoretically the manifestation of this effect in the dc response to the combined action of two driving ac-fields (bichromatic irradiation). Compared to the case of monochromatic irradiation, which is currently intensively studied both experimentally and theoretically, the presence of a second microwave source provides additional insight into the properties of an ac-driven 2D electron gas. In particular, we find that nonparabolicity, being the simplest cause for a violation of Kohn's theorem, gives rise to new qualitative effects specific to bichromatic irradiation. Namely, when the frequencies $\omega_1$ and $\omega_2$ are well away from the cyclotron frequency, $\omega_c$, our simple classical considerations demonstrate that the system becomes parametrically unstable with respect to fluctuations with frequency $(\omega_1+\omega_2)/2$. As an additional effect of nonparabolicity, this parametric instability can manifest itself in the dc properties of the system. This happens when $\omega_1$, $\omega_2$ and $\omega_c$ are related as 3:1:2, respectively. Even for weak detuning between $\omega_1$ and $\omega_2$, the effect of the bichromatic irradiation on the dc response in the presence of nonparabolicity can differ dramatically from the monochromatic case. In particular, the equations of motion can acquire multistable solutions. As a result, the diagonal dc-conductivity can assume several stable negative values at the same magnetic field.

Linear response at the 4-component relativistic density-functional level: application to the frequency-dependent dipole polarizability of Hg, AuH and PtH2

Abstract We report the implementation and application of linear response density-functional theory (DFT) based on the 4-component relativistic Dirac–Coulomb Hamiltonian. The theory is cast in the language of second quantization and is based on the quasienergy formalism (Floquet theory), replacing the initial state dependence of the Runge–Gross theorem by periodic boundary conditions. Contradictions in causality and symmetry of the time arguments are thereby avoided and the exchange-correlation potential and kernel can be expressed as functional derivatives of the quasienergy. We critically review the derivation of the quasienergy analogues of the Hohenberg–Kohn theorem and the Kohn–Sham formalism and discuss the nature of the quasienergy exchange-correlation functional. Structure is imposed on the response equations in terms of Hermiticity and time-reversal symmetry. It is observed that functionals of spin and current densities, corresponding to time-antisymmetric operators, contribute to frequency-dependent and not static electric properties. Physically, this follows from the fact that only a time-dependent electric field creates a magnetic field. It is furthermore observed that hybrid functionals enhance spin polarization since only exact exchange contributes to anti-Hermitian trial vectors. We apply 4-component relativistic linear response DFT to the calculation of the frequency-dependent polarizability of the isoelectronic series Hg, AuH and PtH2. Unlike for the molecules, the effect of electron correlation on the polarizability of the mercury atom is very large, about 25%. We observe a remarkable performance of the local-density approximation (LDA) functional in reproducing the experimental frequency-dependent polarizability of this atom, clearly superior to that of the BLYP and B3LYP functionals. This allows us to extract Cauchy moments (S(−4) = 382.82 and S(−6) = 6090.89 a.u.) that we believe are superior to experiment since we go to higher order in the Cauchy moment expansion.

Is the Hohenberg–Kohn–Sham version of DFT a semi-empirical theory?

The question of whether the Hohenberg–Kohn–Sham version of density functional theory (HKS-DFT), a semi-empirical theory is treated by resorting to an analysis of its foundations. In particular, it is shown that the construction of the universal functional F[ ρ] in HKS-DFT is equivalent to solving the N representability problem for the 2-matrix. Also, the relationship between the Hohenberg–Kohn theorem and the Legendre transformation is discussed and some of the positions concerning the N-dependence (or N-independence) of the universal functional are presented. Finally, some comments are made concerning the reasons why in many cases HKS-DFT turns out to be so highly successful.

Magneto-optics of three-dimensional quantum dots: a real time, time-dependent local spin–density approach

We perform adiabatic time-dependent local spin–density approximation (TDLSDA) calculations in real time of the excitation spectrum of three-dimensional quantum dots (QD's) in magnetic fields of arbitrary direction. In the case of parabolic confinement and electric dipole modes, the calculations reproduce exactly the generalized Kohn theorem, which is a stringent test of the numerical accuracy achieved by our practical implementation of TDLSDA. We apply the method to the study of spin dipole modes in a QD. Real time TDLSDA can be more efficient than Green's function methods to compute the dynamical properties of confined electrons, especially when the finite thickness of the system has to be taken into account. As an illustration, we obtain the dipole spin modes and the acoustic modes of vertical diatomic artificial quantum molecules at zero magnetic field.

Collective oscillations of a confined Bose gas at finite temperature in the random-phase approximation

We present a theory for the linear dynamics of a weakly interacting Bose gas confined inside a harmonic trap at finite temperature. The theory treats the motions of the condensate and of the non-condensate on an equal footing within a generalized random-phase approximation, which ({\it i}) extends the second-order Beliaev-Popov approach by allowing for the dynamical coupling between fluctuations in the thermal cloud, and ({\it ii}) reduces to an earlier random-phase scheme when the anomalous density fluctuations are omitted. Numerical calculations of the low-lying spectra in the case of isotropic confinement show that the present theory obeys with high accuracy the generalized Kohn theorem for the dipolar excitations and demonstrate that combined normal and anomalous density fluctuations play an important role in the monopolar excitations of the condensate. Mean-field theory is instead found to yield accurate results for the quadrupolar modes of the condensate. Although the restriction to spherical confinement prevents quantitative comparisons with measured spectra, it appears that the non-mean field effects that we examine may be relevant to explain the features exhibited by the breathing mode as a function of temperature in the experiments carried out at JILA on a gas of $^{87}$Rb atoms.

Response of Bose-Einstein condensates to external perturbations at finite temperature

We present a theory of the linear response of a Bose-condensed gas to external perturbations at finite temperature. The theory developed here is the basis of a recent quantitative explanation of the measurements of condensate excitations and decay rates made at JILA [D. S. Jin et al., Phys. Rev. Lett. 78, 764 (1997)]. The formalism is based on a dynamic, number-conserving, mean-field scheme and is valid in the collisionless limit of well-defined quasiparticles. The theory is gapless, consistent with the generalized Kohn theorem for the dipole modes, and includes the time-dependent normal and anomalous averages, Beliaev and Landau processes, and all relevant finite size effects. The important physical process where the thermal cloud is driven directly by the external perturbation is explicitly included. This is required for consistency with the dipole modes and is also needed to explain the JILA observations.

Atomic and molecular energies as functionals of the electrostatic potential

Starting from the Hohenberg–Kohn theorem, atomic and molecular energies have been expressed rigorously as functionals of the electronic electrostatic potential, Velec(r). Explicit formulations have been derived for the functionals representing the kinetic energy and electronic interaction contributions to the total energies.

Density functional theory with arbitrary basic variables

By modifying the Levy constrained-search formulation, the Hohenberg–Kohn theorem of the density functional theory is extended. The new theorem allows us to choose arbitrary physical quantities as basic variables, which uniquely determine the ground-state properties of the system. The theorem also establishes the variational principle with respect to the basic variables chosen. By using this theorem, self-consistent single-particle equations are derived. If the occupation matrix of localized orbitals is chosen as a basic variable, these equations are equivalent to those of the LDA+U method.

Spin Effects of Low-dimensional Electron Gases Studied by Far-infrared Photoconductivity Experiments

AbstractWe review our recent work on spin effects in low-dimensional electron gases studied using far-infrared photoconductivity technique. We measure the spin-orbit coupling parameter α via spectroscopy by detecting the combined resonance. Detailed filling-factor dependent study shows the collective nature of this excitation, in accordance to theoretical predictions that both Kohn and Larmor theorem are broken for long-wavelength excitations that changes both the Landau and spin quantum numbers. We find that the long spin-relaxation time of a two-dimensional electron gas results in a novel bolometric spin effect, which gives rise to a substantial photo resistance change by reversing the spin polarization of electrons at the Fermi-level.

Finite-temperature excitations of a trapped Bose-Fermi mixture

We present a detailed study of the low-lying collective excitations of a spherically trapped Bose-Fermi mixture at finite temperature in the collisionless regime. The excitation frequencies of the condensate are calculated self-consistently using the static Hartree-Fock-Bogoliubov theory within the Popov approximation. The frequency shifts and damping rates due to the coupled dynamics of the condensate, noncondensate, and degenerate Fermi gas are also taken into account by means of the random-phase approximation and linear-response theory. In our treatment, the dipole excitation remains close to the bare trapping frequency for all temperatures considered, and thus is consistent with the generalized Kohn theorem. We discuss in some detail the behavior of monopole and quadrupole excitations as a function of the Bose-Fermi coupling. At nonzero temperatures we find that, as the mixture moves towards spatial separation with increasing Bose-Fermi coupling, the damping rate of the monopole (quadrupole) excitation increases (decreases). This provides us a useful signature to identify the phase transition of spatial separation.