Density Functional Computations on Noble Gas Atoms
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Density functional theory (DFT) has now become the predominant technique in computational quantum chemistry, having displaced wavefunction-based computations for atoms, molecules and solids. The key reason is that QFT deals with a single electron density function for an -electron system, rather than a complicated combination of orbital functions . The fundamental validity of DFT and its practical implementation by a variational principle are expressed in two theorems of Hohenberg and Kohn. For all necessary background on DFT, refer to the definitive monograph of Parr and Yang [1]. For more recent advances, see also [2].
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Contributed by: S. M. Blinder (June 2019)
Open content licensed under CC BY-NC-SA
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The total electron density is approximated by a sum of shells (one to five shells for He to Xe):
, where ,
which is suggested by Slater's rules for atomic orbitals.
The DFT functional takes the form
,
with
,
, ,
,
,
, ,
.
The last formula is a conjecture by the author based on computations of atomic correlation energies.
It is most convenient to carry out all the integrals numerically. The energy functional , based on the selected shielding parameters , is computed and compared with the exact (nonrelativistic) energy of the atom. By the second Hohenberg–Kohn theorem, the optimized energy for the functional form of is a minimum, although short of the exact energy.
References
[1] R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, New York: Oxford University Press, 1989.
[2] J. Sun, J. W. Furness and Y. Zhang, "Density Functional Theory," Mathematical Physics in Theoretical Chemistry (S. M. Blinder and J. E. House, eds.), Amsterdam: Elsevier, 2018 Chapter 4. doi:10.1016/B978-0-12-813651-5.00004-8.
[3] W.-P. Wang and R. G. Parr, "Statistical Atomic Models with Piecewise Exponentially Decaying Electron Densities," Physical Review A, 16(3), 1977 pp. 891–902. doi:10.1103/PhysRevA.16.891.
[4] S. M. Blinder. "Shell Structure of Noble Gas Atoms" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ShellStructureOfNobleGasAtoms.
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