Correlation-kinetic Effects Research Articles

Electron correlations in an excited state of a quantum dot in a uniform magnetic field

Electron correlations in a two-electron two-dimensional ‘artificial atom’ or quantum dot (with harmonic confining potential) in the presence of a uniform magnetic field in an excited singlet state are studied via quantal density functional theory (QDFT). QDFT allows for the separation of the electron correlations due to the Pauli exclusion principle and Coulomb repulsion, as well as the determination of the contribution of these correlations to the kinetic energy. The QDFT mapping is from the excited state of the quantum dot to one of noninteracting fermions in their ground state possessing the same basic variables of the density and physical current density, and the same orbital and spin angular momentum. A detailed analysis of these correlations in terms of their quantal sources, the corresponding ‘classical’ fields, and resulting potentials and energies is presented. The key conclusions are that as in natural atoms, the contributions of the Pauli and Coulomb correlations relative to the total energy for the excited state, are less than but of the same order of magnitude as those for the ground state of a quantum dot. However, in contrast, the correlation-kinetic contributions are an order of magnitude greater than those for a quantum dot in its ground state. These correlations constitute nearly 75% of the kinetic and 25% of the total energy. This result is consistent with prior work on low electron density Wigner systems in three-dimensions in which correlation-kinetic effects too play a significant role. The significance of these correlations to traditional excited state density functional theory is noted.

Electron Correlations in Local Effective Potential Theory

Local effective potential theory, both stationary-state and time-dependent, constitutes the mapping from a system of electrons in an external field to one of the noninteracting fermions possessing the same basic variable such as the density, thereby enabling the determination of the energy and other properties of the electronic system. This paper is a description via Quantal Density Functional Theory (QDFT) of the electron correlations that must be accounted for in such a mapping. It is proved through QDFT that independent of the form of external field, (a) it is possible to map to a model system possessing all the basic variables; and that (b) with the requirement that the model fermions are subject to the same external fields, the only correlations that must be considered are those due to the Pauli exclusion principle, Coulomb repulsion, and Correlation–Kinetic effects. The cases of both a static and time-dependent electromagnetic field, for which the basic variables are the density and physical current density, are considered. The examples of solely an external electrostatic or time-dependent electric field constitute special cases. An efficacious unification in terms of electron correlations, independent of the type of external field, is thereby achieved. The mapping is explicated for the example of a quantum dot in a magnetostatic field, and for a quantum dot in a magnetostatic and time-dependent electric field.

Wigner high-electron-correlation regime of nonuniform density systems: A quantal-density-functional-theory study.

The Wigner regime of a system of electrons in an external field is characterized by a low electron density and a high electron-interaction energy relative to the kinetic energy. The low correlation regime is in turn described by a high electron density and an electron-interaction energy smaller than the kinetic energy. The Wigner regime of a nonuniform electron density system is investigated via quantal density functional theory (QDFT). Within QDFT, the contributions of electron correlations due to the Pauli exclusion principle, Coulomb repulsion, and correlation-kinetic effects are separately delineated and explicitly defined. The nonuniform electron density system studied is that of Hooke's atom in the Wigner regime for which the exact wave function is derived. As such the results of the QDFT analysis are exact. It is observed that in comparison to the low correlation case, not only is the electron-interaction energy greater than the kinetic energy as a fraction of the total energy, but so are its individual Hartree, Pauli, and Coulomb components. The ionization potential as a fraction of the total energy too is greater. But most significantly, in the Wigner regime, the correlation-kinetic energy as a fraction of both the electron-interaction and total energy is substantially greater than in the low correlation case. Hence, we propose that the Wigner regime now also be characterized by a high correlation-kinetic energy. The kinetic energy as a fraction of the total energy, however, is less than in the low correlation case. This fact and the high correlation-kinetic energy value in the Wigner regime is explained by the new concept of 'quantal compression' of the kinetic energy density derived from QDFT. The corresponding results for the low correlation case are in turn a consequence of a 'quantal decompression' of the kinetic energy density. From the QDFT analysis, the exact values for the Kohn-Sham theory 'exchange-correlation' and 'correlation' energy functionals of the density, and their respective functional derivatives are also obtained. These results ought to be of value in the construction and testing of new approximate energy functionals valid for the Wigner regime.

Wigner high electron correlation regime in nonuniform electron density systems: Kinetic and correlation-kinetic aspects

The Wigner high electron correlation regime is characterized in the literature by an electron-interaction energy much greater than the kinetic energy. Via the ‘quantal Newtonian’ first law, we discover that for a nonuniform electron density system in this regime, there is a ‘quantal compression’ of the kinetic energy density. The explanation of this compression provides a fundamental understanding for why the kinetic energy is a smaller fraction of the total energy relative to the same ratio in the low correlation regime. We also discover by application of quantal density functional theory, that the contribution of electron correlations to the kinetic energy – the correlation-kinetic effects – and to the total energy is very significant. We propose that in addition to a high electron-interaction energy, the Wigner regime must thus also be characterized by a high correlation-kinetic energy.

Quantal density-functional theory in the presence of a magnetic field

Quantal density functional theory (Q–DFT) of electrons in an external electrostatic field is generalized to the added presence of a magnetostatic field. This Q–DFT constitutes the mapping from the interacting system of electrons in an external electrostatic and magnetostatic field in any state as described by Schrodinger theory to one of noninteracting fermions with the same density, physical current density, electron number, and canonical orbital and spin angular momentum. To formulate this Q–DFT, Schrodinger theory from the perspective of the individual electron via the corresponding ‘Quantal Newtonian’ first law is developed. It is shown that in addition to the external fields, each electron experiences an internal field which is comprised of components representative of electron correlations due to the Pauli exclusion principle and Coulomb interaction, the density, the kinetic effects, and a contribution due to the external magnetic field. These fields are derived from quantal sources that are expectations of Hermitian operators taken with respect to the system wave function. As such the intrinsic self-consistent nature of the Schrodinger equation is demonstrated. With the Schrodinger equation written in self-consistent form, the magnetic field, (in addition to the vector potential of the field component of the momentum), now appears explicitly in it. The ’Quantal Newtonian’ first law for the model system of noninteracting fermions is derived. It is shown that if the model fermions are subject to the same external potentials, then the only electron correlations that must be accounted for in the Q–DFT mapping are those of the Pauli principle, Coulomb repulsion and Correlation-Kinetic effects. The resulting local electron-interaction potential within Q–DFT is the work done in an effective field that is the sum of fields representative of these correlations. The corresponding many-body components of the total energy can be expressed in integral virial form in terms of the separate fields. To explicate this Q–DFT, it is applied to a quantum dot as represented by the exactly solvable two-dimensional Hooke’s atom in a magnetic field. A key observation is that as a result of the reduction in dimensionality due to the presence of the magnetic field, Correlation-Kinetic effects are significant.

Quantal density-functional theory of the density amplitude: Application to atoms

Local effective potential theory of electronic structure is the mapping from a system of electrons in an external field to one of noninteracting fermions or bosons with the same electronic density. The energy and ionization potential are also thereby determined. The mappings may be achieved via either quantal density-functional theory (QDFT) or Hohenberg-Kohn-Sham density-functional theory (HKS-DFT). The wave function for the model fermionic system is a Slater determinant of spin orbitals, whereas that for the model bosons is the density amplitude. In the QDFT mappings, the contributions of the electron correlations due to the Pauli exclusion principle, Coulomb repulsion and correlation-kinetic effects are separately delineated. It has been proved via QDFT that the contribution of Pauli and Coulomb correlations to these model systems is the same; the difference lies solely in their correlation-kinetic component. In this paper, we apply the QDFT of the density amplitude to study the mapping to the bosonic model. The application is to atoms and performed at the Hartree-Fock theory level of electron correlations. A principal conclusion is that correlation-kinetic effects play a significant role in the mapping to the bosonic model, whereas they are negligible in the mapping to the model fermions. For the bosonic model, this contribution increases with electron number, becoming nearly as significant as those due to the corresponding electron-interaction (the sum of the Hartree and Pauli) term. The significance of the correlation-kinetic effects will be further enhanced on the inclusion of Coulomb correlations and the corresponding correlation-kinetic contributions. The consequences of these conclusions for the HKS-DFT of the density amplitude are discussed, as are directions for future work.

Local effective potential theory: Nonuniqueness of potential and wave function

In local effective potential energy theories such as the Hohenberg-Kohn-Sham density functional theory (HKS-DFT) and quantal density functional theory (Q-DFT), electronic systems in their ground or excited states are mapped to model systems of noninteracting fermions with equivalent density. From these models, the equivalent total energy and ionization potential are also obtained. This paper concerns (i) the nonuniqueness of the local effective potential energy function of the model system in the mapping from a nondegenerate ground state, (ii) the nonuniqueness of the local effective potential energy function in the mapping from a nondegenerate excited state, and (iii) in the mapping to a model system in an excited state, the nonuniqueness of the model system wave function. According to nondegenerate ground state HKS-DFT, there exists only one local effective potential energy function, obtained as the functional derivative of the unique ground state energy functional, that can generate the ground state density. Since the theorems of ground state HKS-DFT cannot be generalized to nondegenerate excited states, there could exist different local potential energy functions that generate the excited state density. The constrained-search version of HKS-DFT selects one of these functions as the functional derivative of a bidensity energy functional. In this paper, the authors show via Q-DFT that there exist an infinite number of local potential energy functions that can generate both the nondegenerate ground and excited state densities of an interacting system. This is accomplished by constructing model systems in configurations different from those of the interacting system. Further, they prove that the difference between the various potential energy functions lies solely in their correlation-kinetic contributions. The component of these functions due to the Pauli exclusion principle and Coulomb repulsion remains the same. The existence of the different potential energy functions as viewed from the perspective of Q-DFT reaffirms that there can be no equivalent to the ground state HKS-DFT theorems for excited states. Additionally, the lack of such theorems for excited states is attributable to correlation-kinetic effects. Finally, they show that in the mapping to a model system in an excited state, there is a nonuniqueness of the model system wave function. Different wave functions lead to the same density, each thereby satisfying the sole requirement of reproducing the interacting system density. Examples of the nonuniqueness of the potential energy functions for the mapping from both ground and excited states and the nonuniqueness of the wave function are provided for the exactly solvable Hooke's atom. The work of others is also discussed.

Asymptotic near-nucleus structure of the electron-interaction potential in local effective potential theories

In local effective potential theories of electronic structure, the electron correlations due to the Pauli exclusion principle, Coulomb repulsion, and correlation-kinetic effects, are all incorporated in the local electron-interaction potential ${v}_{ee}(\mathbf{r})$. In previous work, it has been shown that for spherically symmetric or sphericalized systems, the asymptotic near-nucleus expansion of this potential is ${v}_{ee}(r)={v}_{ee}(0)+\ensuremath{\beta}r+O({r}^{2})$, with ${v}_{ee}(0)$ being finite. By assuming that the Schr\odinger and local effective potential theory wave functions are analytic near the nucleus of atoms, we prove the following via quantal density functional theory (QDFT): (i) Correlations due to the Pauli principle and Coulomb correlations do not contribute to the linear structure; (ii) these Pauli and Coulomb correlations contribute quadratically; (iii) the linear structure is solely due to correlation-kinetic effects, the contributions of these effects being determined analytically. We also derive by application of adiabatic coupling constant perturbation theory via QDFT (iv) the asymptotic near-nucleus expansion of the Hohenberg-Kohn-Sham theory exchange ${v}_{x}(\mathbf{r})$ and correlation ${v}_{c}(\mathbf{r})$ potentials. These functions also approach the nucleus linearly with the linear term of ${v}_{x}(\mathbf{r})$ being solely due to the lowest-order correlation kinetic effects, and the linear term of ${v}_{c}(\mathbf{r})$ being due solely to the higher-order correlation kinetic contributions. The above conclusions are equally valid for systems of arbitrary symmetry, provided spherical averages of the properties are employed.

Quantal density-functional theory of excited states: The state arbitrariness of the model noninteracting system

The quantal density-functional theory (Q-DFT) of nondegenerate excited-states maps the pure state of the Schr\"odinger equation to one of noninteracting fermions such that the equivalent excited state density, energy, and ionization potential are obtained. The state of the model S system is arbitrary in that it may be in a ground or excited state. The potential energy of the model fermions differs as a function of this state. The contribution of correlations due to the Pauli exclusion principle and Coulomb repulsion to the potential and total energy of these fermions is independent of the state of the S system. The differences are solely a consequence of correlation-kinetic effects. Irrespective of the state of the S system, the highest occupied eigenvalue of the model fermions is the negative of the ionization potential. In this paper we demonstrate the state arbitrariness of the model system by application of Q-DFT to the first excited singlet state of the exactly solvable Hookean atom. We construct two model S systems: one in a singlet ground state ${(1s}^{2}),$ and the other in a singlet first excited state $(1s2s).$ In each case, the density and energy determined are equivalent to those of the excited state of the atom, with the highest occupied eigenvalues being the negative of the ionization potential. From these results we determine the corresponding Kohn-Sham density-functional theory (KS-DFT) ``exchange-correlation'' potential energy for the two S systems. Further, based on the results of the model calculations, suggestions for the KS-DFT of excited states are made.

Origin of the derivative discontinuity in density functional theory

The local electron-interaction potential of Kohn-Sham density functional theory is representative of electron correlations due to the Pauli exclusion principle, Coulomb repulsion, and correlation-kinetic effects which are a consequence of the correlation contribution to the kinetic energy. This potential exhibits a discontinuity as the electron number passes through an integer value. We show, via quantal density functional theory, that the physical origin of the discontinuity is the correlation-kinetic effect. The magnitude of the discontinuity is derived to be the work done to move an electron in a conservative field representative of this effect. An explanation is given as to how these correlation-kinetic effects give rise to such a discontinuity. Further, how this understanding relates to that within traditional Kohn-Sham theory and other previous explanations of the discontinuity is also discussed.

Analytical properties of the Kohn–Sham theory exchange and correlation energy and potential via quantal density functional theory

In this study we derive analytical properties of the Kohn–Sham (KS) theory exchange Ex[ρ] and correlation \documentclass{article}\pagestyle{empty}\begin{document}$E_{c}^{\mathrm{KS}}[\rho]$\end{document} energy functionals of the density ρ(r) and of their respective functional derivatives vx(r) and vc(r). These properties are derived via quantal-density functional theory (Q-DFT) in terms of the different electron correlations present by application of adiabatic coupling constant (λ) perturbation theory. The results are: (i) The exchange energy Ex[ρ] and potential vx(r) are representative of electron correlations due to the Pauli exclusion principle as well as the lowest order O(λ) correlation-kinetic effects. While the contribution of the latter to vx(r) is explicit, their contribution to the energy Ex[ρ] is indirect via the orbitals of the noninteracting fermion or s-system. (ii) To leading order O(λ2), the correlation potential vc,2(r) has contributions from both Coulomb correlations and correlation-kinetic effects. However, at this order, the energy \documentclass{article}\pagestyle{empty}\begin{document}$E_{c,2}^{\mathrm{KS}}[\rho]$\end{document} is due entirely to correlation-kinetic effects. At higher order, both these correlations contribute to the correlation energy and potential. In a second component to this work we derive properties of the asymptotic structure of vx(r) and vc(r) via Q-DFT for systems for which the N-electron atom and (N−1)-electron ion are orbitally nondegenerate. The results are: (iii) There is no correlation-kinetic contribution to the potential vx(r) asymptotically. The asymptotic structure of vx(r), to exponential accuracy, is due entirely to Pauli correlations and is the work done Wx(r) to move an electron in the field of the Fermi hole, decaying as −1/r. (iv) The lowest order O(λ2) Coulomb correlations do not contribute asymptotically to vc(r). (v) The leading order O(λ2) correlation potential vc,2(r) is due entirely to correlation-kinetic effects decaying as 8κ0χs/5r5, where \documentclass{article}\pagestyle{empty}\begin{document}$\kappa_{0}^{2}/2$\end{document} is the ionization potential, and χs an expectation value of the s-system ion. (vi) To O(1/r5), the O(λ3) correlation potential vc,3(r) to leading order is entirely due to second-order Coulomb correlations and decays as −αs/2r4, where αs is the polarizability of the s-system ion. There are no Coulomb correlation contributions to vc,3(r) and thus vc(r) to O(1/r5). (vii) For systems for which the N-electron atom is orbitally degenerate, the asymptotic structure of vx(r) contains in addition terms of O(1/r3), O(1/r5), etc. The other remaining asymptotic properties of vc(r), however, remain unchanged. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 555–566, 2000

Analytical properties of the Kohn-Sham theory exchange and correlation energy and potential via quantal density functional theory

In this study we derive analytical properties of the Kohn–Sham (KS) theory exchange Ex[ρ] and correlation \documentclass{article}\pagestyle{empty}\begin{document}$E_{c}^{\mathrm{KS}}[\rho]$\end{document} energy functionals of the density ρ(r) and of their respective functional derivatives vx(r) and vc(r). These properties are derived via quantal-density functional theory (Q-DFT) in terms of the different electron correlations present by application of adiabatic coupling constant (λ) perturbation theory. The results are: (i) The exchange energy Ex[ρ] and potential vx(r) are representative of electron correlations due to the Pauli exclusion principle as well as the lowest order O(λ) correlation-kinetic effects. While the contribution of the latter to vx(r) is explicit, their contribution to the energy Ex[ρ] is indirect via the orbitals of the noninteracting fermion or s-system. (ii) To leading order O(λ2), the correlation potential vc,2(r) has contributions from both Coulomb correlations and correlation-kinetic effects. However, at this order, the energy \documentclass{article}\pagestyle{empty}\begin{document}$E_{c,2}^{\mathrm{KS}}[\rho]$\end{document} is due entirely to correlation-kinetic effects. At higher order, both these correlations contribute to the correlation energy and potential. In a second component to this work we derive properties of the asymptotic structure of vx(r) and vc(r) via Q-DFT for systems for which the N-electron atom and (N−1)-electron ion are orbitally nondegenerate. The results are: (iii) There is no correlation-kinetic contribution to the potential vx(r) asymptotically. The asymptotic structure of vx(r), to exponential accuracy, is due entirely to Pauli correlations and is the work done Wx(r) to move an electron in the field of the Fermi hole, decaying as −1/r. (iv) The lowest order O(λ2) Coulomb correlations do not contribute asymptotically to vc(r). (v) The leading order O(λ2) correlation potential vc,2(r) is due entirely to correlation-kinetic effects decaying as 8κ0χs/5r5, where \documentclass{article}\pagestyle{empty}\begin{document}$\kappa_{0}^{2}/2$\end{document} is the ionization potential, and χs an expectation value of the s-system ion. (vi) To O(1/r5), the O(λ3) correlation potential vc,3(r) to leading order is entirely due to second-order Coulomb correlations and decays as −αs/2r4, where αs is the polarizability of the s-system ion. There are no Coulomb correlation contributions to vc,3(r) and thus vc(r) to O(1/r5). (vii) For systems for which the N-electron atom is orbitally degenerate, the asymptotic structure of vx(r) contains in addition terms of O(1/r3), O(1/r5), etc. The other remaining asymptotic properties of vc(r), however, remain unchanged. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 555–566, 2000

Physical origin of the discontinuity of the Kohn-Sham theory effective potential

We derive a formal line-integral expression for the discontinuity that occurs in the electron-interaction potential of the Kohn-Sham (KS) theory as the electron number passes through an integer. The discontinuity may thus be interpreted as the work done to move an electron in the force of a conservative field. This result is derived via the rigorous description of the KS theory in terms of density matrices, and as such we provide an explanation for the physical origin of the discontinuity from a distinctly different perspective. The derivation further shows that the different electron correlations represented in the KS potentials, viz. those due to the Pauli exclusion principle, Coulomb repulsion, and correlation-kinetic effects, all contribute to the discontinuity, and the contribution of each to the conservative field is given explicitly. The same physics shows that the KS representation of Hartree-Fock and Hartree theories as well as other approximations must also exhibit the discontinuity, and we demonstrate this for the work formalism Hartree-Fock approximation.

Quantum-mechanical interpretation of time-dependent density-functional theory

In this paper we derive the differential virial theorem for both time-dependent (TD) Schrödinger theory and Kohn-Sham (KS) density-functional theory. As such we obtain an exact integral expression for the TD electron-interaction potential of KS theory that is independent of the choice of action. The expression, valid for general TD phenomena other than the ionization or scattering process, is afforded the physical interpretation at each instant of time as being the work done to move an electron in a conservative field. The field is a sum of four component fields representative of Pauli and Coulomb correlations, correlation-kinetic effects, and as discovered here, correlations due to the difference in current densities of the KS and Schrödinger systems. The interpretation further reduces to that for the corresponding electron-interaction potential of stationary-state KS theory for time-independent external potentials.

Physics of transformation from Schrödinger theory to Kohn-Sham density-functional theory: Application to an exactly solvable model

According to Hohenberg-Kohn-Sham density-functional theory (DFT), and its constrained search formulation, the Schr\odinger ground-state wave function \ensuremath{\Psi} is a functional of the ground-state electronic density \ensuremath{\rho}(r). But the explicit functional dependence of \ensuremath{\Psi} on \ensuremath{\rho} is unknown. It is, however, possible to describe Kohn-Sham (KS) DFT and its electron-interaction energy functional and functional derivative rigorously in terms of the wave function \ensuremath{\Psi}. This description involves a conservative field which is a sum of two fields, the first representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, and the second of correlation-kinetic effects. The sources of these fields are expectations of Hermitian operators with respect to \ensuremath{\Psi}. The energy functional is expressed in integral virial form in terms of these fields, whereas the functional derivative is the work done to move an electron in the conservative field of their sum. In this paper we illustrate the physics of transformation from Schr\odinger to KS theory by application of this description to a ground state of the exactly solvable Hooke's atom. As such we determine properties such as the pair-correlation density, the Fermi and Coulomb holes, the Schr\odinger and KS kinetic-energy-density tensors and kinetic fields, and the electron-interaction and correlation-kinetic fields, potentials, and energies, the majority of these constituent properties of the transformation being obtained analytically. In this manner we demonstrate the separate contributions and significance of each type of electron correlation to the KS electron-interaction energy and its functional derivative. Based on this study and previous work, it is proposed that in the construction of approximate energy functionals and their derivatives for application to more complex systems, it is the fields that be directly approximated.