By Carlos Fiolhais, Fernando Nogueira, Miguel A.L. Marques
Density useful thought (DFT) is by way of now a well-established process for tackling the quantum mechanics of many-body platforms. initially utilized to compute homes of atoms and easy molecules, DFT has speedy develop into a piece horse for extra complicated purposes within the chemical and fabrics sciences. the current set of lectures, spanning the complete variety from easy ideas to relativistic and time-dependent extensions of the idea, is the precise advent for graduate scholars or nonspecialist researchers wishing to familiarize themselves with either the elemental and such a lot complex strategies during this box.
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Extra info for A Primer in Density Functional Theory
213) in the Z → ∞ limit have also be found  for ions with 3, 9, 10, and 11 electrons. Now we turn to the construction of a GGA for the exchange energy. 205). To recover the good LSD description of the linear response of the uniform gas (Sect. 194) to write s → 0 : Fx (s) = 1 + µs2 . 215) Then the gradient coeﬃcients for exchange and correlation will cancel for all rs and ζ, apart from small ∇ζ contributions to ExGGA , as discussed in the next section. 196), the proper gradient coeﬃcient for exchange in the slowly-varying limit.
138) 1 Density Functionals for Non-relativistic Coulomb Systems 27 In other notation, ex (n) = − 1/3 3 3 (9π/4) (3π 2 n)1/3 = − 4π 4π rs . 139) Since the self-interaction correction vanishes for the diﬀuse orbitals of the uniform gas, all of this exchange energy is due to the Pauli exclusion principle. 3 Correlation Energy Exact analytic expressions for ec (n), the correlation energy per electron of the uniform gas, are known only in extreme limits. The high-density (rs → 0) limit is also the weak-coupling limit, in which ec (n) = c0 ln rs − c1 + c2 rs ln rs − c3 rs + .
Note that s and p are small not only for a slow density variation but also for a density variation of small amplitude (as in Sect. 4). The slowly-varying limit is one in which p/s is also small . 40), s(r) → sγ (r) = s(γr). 106), so their gradient expansions are Ts [n] = As d3 r n5/3 [1 + αs2 + . 184) Ex [n] = Ax d3 r n4/3 [1 + µs2 + . 185) Because there is no special direction in the uniform electron gas, there can be no term linear in ∇n. 186) ∂n via integration by parts. 185), which are fourth or higher-order in ∇, amounts to the second-order gradient expansion, which we call the gradient expansion approximation (GEA).
A Primer in Density Functional Theory by Carlos Fiolhais, Fernando Nogueira, Miguel A.L. Marques