The inclusion of the correlation effect in many electron system was shown in the self-consistent density functional theory to yield the virial theorem, an important relation between the kinetic and the potential energies. This provides a correspondence principle and an accuracy that could rule out mechanisms not begetting results within the error limit. The coherence effect of exchange antisymmetry for fermions can readily couple with the correlation effect to result in rich physics. We show from a variational quantum Monte Carlo code that in the excited states of helium atom, the triplet state of like spins tends to have less repulsive dispersion than the singlet state of opposite spins.
The inclusion of the correlation effect in many electron system was shown in the self-consistent density functional theory to yield the virial theorem, an important relation between the kinetic and the potential energies. This provides a correspondence principle and an accuracy that could rule out mechanisms not begetting results within the error limit. The coherence effect of exchange antisymmetry for fermions can readily couple with the correlation effect to result in rich physics. We show from a variational quantum Monte Carlo code that in the excited states of helium atom, the triplet state of like spins tends to have less repulsive dispersion than the singlet state of opposite spins.
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.
Within density functional theory, a coordinate-scaling relation for the coupling-constant dependence of the exchange-correlation kernel ${f}_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\omega})$ is utilized to express the correlation energy of a many-electron system in terms of fxc. As a test of several of the available approximations for the exchange-correlation kernel, or equivalently the local-field factor, we calculate the uniform-gas correlation energy. While the random phase approximation (fxc= 0) makes the correlation energy per electron too negative by about 0.5 eV, the adiabatic local-density approximation [fxc=fxc(q= $0,\ensuremath{\omega}$ = 0)] makes a comparable error in the opposite direction. The adiabatic nonlocal approximation [fxc= ${f}_{\mathrm{xc}}(q,\ensuremath{\omega}$ = 0)] reduces this error to about 0.1 eV, and inclusion of the full frequency dependence [fxc= ${f}_{\mathrm{xc}}(q,\ensuremath{\omega})]$ in an approximate parametrization reduces it further to less than 0.02 eV. We also report the wave-vector analysis and the imaginary-frequency analysis of the correlation energy for each choice of kernel.
In no wave function of the type (1) [product of single determinants for each spin] is there a statistical correlation between the positions of the electrons with antiparallel spin. The purpose of the aforementioned generalization of (1) is to allow for such correlations. This will lead to an improvement of the wave function and, therefore, to a lowering of the energy value. E. Wigner, Phys. Rev. 46, 1002 (1934) Summary Although the exact many-body wavefunction cannot be written down in general, we do know of some of its properties. For example, there are differences between the wavefunctions of insulators and metals and the cusp condition gives the behavior as any two charges approach each other. In this chapter we also discuss approximate wavefunctions, ways to judge their accuracy and how to include electronic correlation. Examples of many-body wavefunctions are the Slater–Jastrow (pair product) wavefunction and its generalization to pairing and backflow wavefunctions. In other places in this book, we argue that it is not necessary to look at the many-body wavefunctions explicitly because they are unwieldy; the one- and two-body correlation functions discussed in Ch. 5 are sufficient to determine the energy and give information on the excitation spectra. However, these correlation functions do not always contain all information of interest. In principle, the ground-state energy of a many-electron system is a functional of the density, but the very derivation of the theorem invokes the many-body wavefunction, as expressed explicitly in Eq. (4.16). The effects of antisymmetry are manifest in the correlation functions, but antisymmetry is most simply viewed as a property of the wavefunction; electronic correlation is fundamentally a result of properties of the many-body wavefunction. Studying many-body wavefunctions provides a very useful, different point of view of many-body physics from the approaches based on correlation functions. Many of the most important discoveries in physics have come about by understanding the nature of wavefunctions, such as the Laughlin wavefunction for the fractional quantum Hall effect, the BCS wavefunction for superconductors, p-wave pairing in superfluid 3 He, and the Heitler– London approach for molecular binding. The role of Berry's phases has brought out the importance of the phase of the wavefunction in determining properties of quantum systems. This has led to new understanding of the classification of insulators, metals, superconductors, vortices, and other states of condensed matter.
The exact time-dependent correlation energy in time-dependent density-functional theory is shown to sometimes become positive, which is impossible with most present TDDFT approximations. Both the correlation potential and energy can be strongly nonlocal in time. A new inequality is derived for the time-dependent exchange-correlation energy. The correlation energy appears to scale to a constant function of scaled time in the high-density limit. In the linear response regime, the correlation energy is shown to become purely adiabatic, but the correlation potential is generally nonadiabatic. The usefulness of the virial theorem as a test of numerical accuracy is demonstrated. All results are found or inspired by exact numerical solution of a simple model system (Hooke’s atom), and inversion of the corresponding Kohn–Sham equations.
Abstract It is difficult to imagine that density functional theory (DFT) could be useful in improving quantum Monte Carlo methods, but it is easy to imagine that QMC could be useful in improving density functional methods. Since QMC can produce accurate wavefunctions and since wavefunctions lead to all other properties one should be able, in principle, to use QMC to predict the exchangecorrelation energy functional Exc[n(r)], the key (unknown) quantity required in DFT. In practice, that is not so easy, but this paper shows how QMC can take steps in that direction by providing insights gained from comparisons of results from QMC and DFT. The authors considered the case of the strongly inhomogeneous electron gas using variational Q:tvIC to obtain wavefunctions 1/J>. and energies E[7/J>.] for the ground state of the Hamiltonian iI>- = T +AVee+V >. associated with the coupling constant A in the range 0 to 1. Optimization of a Slater-Jastrow many-body wavefunction was made by minimizing a penalty function given by a weighted sum of the variance in local energies and a sum of squares of electron density differences. The variational parameters in 7/J>. and v>. were repeatedly varied to reach convergence followed by additional calculations to obtain the local values of the exchange-correlation energies e c and the exchange-correlation holes n c from which Exe and nxc could be determined.