References are as follows LYP 67 B3PW91 97 PBE 59 B88 63 KS 4 HK 3 Paper Web of

References are as follows lyp 67 b3pw91 97 pbe 59 b88

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References are as follows: LYP, 67 B3PW91, 97 PBE, 59 B88, 63 KS, 4 HK. 3 Paper Web of Science Google Scholar LYP 43 123 49 703 B3PW91 42 642 52 028 PBE 30 575 37 771 B88 24 766 28 529 KS 21 670 31 251 HK 15 222 27 317 densities from other sources, TFD energies have errors of around 10%, too large for computational purposes. Further- more, Teller deduced 13 that Thomas-Fermi theory cannot bind molecules. TFD is useful for rough estimates of atomic prop- erties only. Shell structure is a consequence of the Pauli exclusion principle and, as such, arises from electron pairs in orthonor- mal orbitals arranged in Slater determinants. Hartree-Fock (HF) theory is the simplest realization. Given the Hamiltonian operator, H , for N electrons in an external (nuclear) potential v ext : H = − 1 2 i 2 i + i v ext ( r i ) + 1 2 j = i i 1 | r j r i | , (1) we minimize the energy of a Slater determinant with respect to variations in the occupied spin orbitals ψ i σ . The result is the famous HF orbital equation 1 2 2 ψ (1) + v ext (1) ψ (1) + v el (1) ψ (1) j ψ (2) ψ (2) r 12 d 2 ψ (1) = ε ψ (1) , (2) v el (1) = ρ (2) r 12 d 2 , ρ = σ ρ σ , ρ σ = i | ψ | 2 , with the summation over j in the exchange term over orbitals of parallel spin only. The total HF energy is given by E HF = − 1 2 σ i ψ 2 ψ + v ext ρ + 1 2 ρ (1) ρ (2) r 12 d 1 d 2 + E X , (3) E X = − 1 2 σ | i ψ (1) ψ (2) | 2 r 12 d 1 d 2 , where E X is the Hartree-Fock exchange energy. The first three terms are the total kinetic energy, the interaction energy with the external potential, and the classical Coulomb self-interaction energy, respectively. HF theory, while immensely more useful than TFD, is still not accurate enough for energy predictions in chem- istry. Bond energies are significantly underestimated. Some molecules, F 2 , for example, are not even bound at the Hartree- Fock level. Thus post -HF methods, adding to Hartree-Fock numerous other determinants involving excited or “virtual” orbitals, are generally required for viable chemical com- putations. Post-HF technology is well developed (see, e.g., Refs. 14 and 15 for overviews) and capable of very high accu- racy, but the development and computational costs are severe. Simply put, it is complicated , and the computer-time scaling with system size N is several orders larger than for Hartree- Fock depending on the method (i.e., the formal scaling 16 of HF is N 4 , while post-HF methods scale like N 5 and higher as their sophistication increases). I will refer to HF and post-HF methods as wave-function methods, and the theory as wave- function theory (WFT). The more common terminology, ab initio theory, is regrettable as it denigrates density-functional theory. As we shall see in Secs. III and IV , DFT is as ab initio ” as WFT. In the mid 1900s, computation of HF orbitals in condensed-matter systems was intractable. The problem is the orbital-dependent, nonlocal, exchange operator in Eq. (2) .
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