Determination of a Wave Function Functional

Pan, Xiao-Yin; Sahni, Viraht; Massa, Lou

doi:10.1103/physrevlett.93.130401

Cited by 22 publications

(29 citation statements)

References 16 publications

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“…After the completion of these calculations, we became aware of the work of Pan et al [29], who have developed a formulation of the Hylleraas integrals that changes the order of integration and thereby reduces equation (6) The restricted Hartree-Fock energies were calculated through a method employing a zeroth order spherical Bessel function representation of the Hartree-Fock orbitals. This method and its implementation has been described in detail in [25].…”

Section: Computational Detailsmentioning

confidence: 99%

Electron correlation energy in confined two-electron systems

et al. 2010

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Radial, angular and total correlation energies are calculated for four two-electron systems with atomic numbers Z = 0 -3 confined within an impenetrable sphere of radius R. We report accurate results for the non-relativistic, restricted Hartree-Fock and radial limit energies over a range of confinement radii from 0.05 -10 a 0 . At small R, the correlation energies approach limiting values that are independent of Z while at intermediate R, systems with Z  1 exhibit a characteristic maximum in the correlation energy resulting from an increase in the angular correlation energy which is offset by a decrease in the radial correlation energy.

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Section: Computational Detailsmentioning

confidence: 99%

Electron correlation energy in confined two-electron systems

et al. 2010

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“…Some of these theories seek to use not only the electron density but also additional information ͑values of certain properties, [58][59][60] information about the pair density, 20,[61][62][63][64] information about the local kinetic energy ͑or equivalently, the local temperature͒, 23,37,65 etc.͒ That class of generalized densityfunctional theories seeks to achieve greater accuracy at greater computational cost than the conventional Kohn-Sham approach. Other sorts of generalized density-functional theories are designed for cases where less accuracy would be acceptable, provided that the method was less computationally expensive than an ordinary Kohn-Sham calculation.…”

Section: Discussionmentioning

confidence: 99%

Alternatives to the electron density for describing Coulomb systems

Ayers

Nagy

2007

The Journal of Chemical Physics

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Stimulated by the difficulty of deriving effective kinetic energy functionals of the electron density, the authors consider using the local kinetic energy as the fundamental descriptor for molecular systems. In this ansatz, the electron density must be expressed as a functional of the local kinetic energy. There are similar results for other quantities, including the local temperature and the Kohn-Sham potential. One potential advantage of these approaches-and especially the approach based on the local temperature-is the chemical relevance of the fundamental descriptor. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2718950͔ I. MOTIVATIONAlthough the Kohn-Sham approach to density-functional theory ͑DFT͒ is now well established it bears remembering that almost 40 years separate the seminal papers of Thomas and Fermi from the breakthrough of Kohn and Sham. [1][2][3] Even with the rise of the Kohn-Sham approach, "orbitalfree" computational techniques have never entirely vanished from the scene because orbital-free DFT is much faster than conventional Kohn-Sham calculations. In orbital-free DFT one only varies a single function of the three spatial coordinates ͑the electron density͒ instead of using N threecoordinate functions ͑orbitals͒ or one six-coordinate function ͑the density matrix͒. ͑N is the number of electrons.͒ At present, however, orbital-free calculations are not really more "efficient" than Kohn-Sham calculations because while orbital-free calculations are computationally inexpensive, they are also very inaccurate. The problem is that the approximate kinetic energy functionals are ordinarily inadequate for describing bond breaking and other chemical processes in molecular systems. ͓Orbital-free methods have had more success in solid state materials, although seemingly only in cases where the electron density is low ͑most of the electrons are treated with a pseudopotential͒.4 ͔ Kohn-Sham theory circumvents the problem of evaluating the kinetic energy directly: instead the electron density is used to evaluate the Kohn-Sham potential, which is used to evaluate the Kohn-Sham orbitals, which are then used to evaluate the kinetic energy of the reference system of noninteracting electrons,Although T s ͓͔ is usually slightly smaller than the true kinetic energy, the Kohn-Sham kinetic energy functional is N representable by construction and, as such, avoids the "variational catastrophes" that afflict ordinary orbital-free DFT. 5-7The 80 years that has elapsed since the original paper of Thomas testifies to the fact that expressing the kinetic energy as a functional of the electron density is extraordinarily difficult. This raises the question: might it be easier to express the electron density as a functional of the local kinetic energy or another similar quantity-like the local temperature or the Kohn-Sham potential-that determines the kinetic energy? We do not yet know whether or not it is easier to express the electron density as a functional of these descriptors, but we can establish that it is possib...

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“…The extension to the time-dependent case via the "Quantal Newtonian" second law with examples is discussed in Section 5. Concluding remarks are made in Section 6 together with a comparison of the self-consistent method and the variational [23] and constrained-search variational methods [24][25][26] for the determination of the wave function.…”

Section: Introductionmentioning

confidence: 99%

Schrödinger Theory of Electrons in Electromagnetic Fields: New Perspectives

Sahni

Pan

2017

Computation

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The Schrödinger theory of electrons in an external electromagnetic field is described from the new perspective of the individual electron. The perspective is arrived at via the time-dependent "Quantal Newtonian" law (or differential virial theorem). (The time-independent law, a special case, provides a similar description of stationary-state theory). These laws are in terms of "classical" fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. The laws reveal the following physics: (a) in addition to the external field, each electron experiences an internal field whose components are representative of a specific property of the system such as the correlations due to the Pauli exclusion principle and Coulomb repulsion, the electron density, kinetic effects, and an internal magnetic field component. The response of the electron is described by the current density field; (b) the scalar potential energy of an electron is the work done in a conservative field. It is thus path-independent. The conservative field is the sum of the internal and Lorentz fields. Hence, the potential is inherently related to the properties of the system, and its constituent property-related components known. As the sources of the fields are functionals of the wave function, so are the respective fields, and, therefore, the scalar potential is a known functional of the wave function; (c) as such, the system Hamiltonian is a known functional of the wave function. This reveals the intrinsic self-consistent nature of the Schrödinger equation, thereby providing a path for the determination of the exact wave functions and energies of the system; (d) with the Schrödinger equation written in self-consistent form, the Hamiltonian now admits via the Lorentz field a new term that explicitly involves the external magnetic field. The new understandings are explicated for the stationary state case by application to two quantum dots in a magnetostatic field, one in a ground state and the other in an excited state. For the time-dependent case, the evolution of the same states of the quantum dots in both a magnetostatic and a time-dependent electric field is described. In each case, the satisfaction of the corresponding "Quantal Newtonian" law is demonstrated.

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Determination of a Wave Function Functional

Cited by 22 publications

References 16 publications

Electron correlation energy in confined two-electron systems

Electron correlation energy in confined two-electron systems

Alternatives to the electron density for describing Coulomb systems

Schrödinger Theory of Electrons in Electromagnetic Fields: New Perspectives

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