History of Several Generalized Gradient Approximations

In 1968, Ma and Brueckner [69] derived the second-order gradient expansion for the correlation energy in the high-density limit, (1.188) and (1.191). In numerical tests, they found that it led to improperly positive correlation energies for atoms, because of the large size of the positive gradient term. As a remedy, they proposed the first GGA, 

In 1980, Langreth and Perdew [83] explained the failure of the second-order gradient expansion (GEA) for E. They made a complete wavevector analysis of be., they replaced the Coulomb interaction /u in (1.100) by its Fourier transform and found 

The sum rule of (1.102) should emerge from (1.204) in the A — 0 limit (since sin(rr)/x — 1 as x —0), and does so for the exchange energy at the GEA level. But the A —0 limit of n (A ) turns out to be a positive number proportional to and not zero. The reason seems to be that the GEA correlation hole is only a truncated expansion, and not the exact hole for any physical system, so it can and does violate the sum rule. 

Langreth and Mehl [11] (1983) proposed a GGA based upon the wavevector analysis of (1.203). They introduced a sharp cutoff of the spurious small-k contributions to E All contributions were set to zero for k kr = 

The errors of the GEA for the exchange energy are best revealed in real space (see (1.100)), not in wavevector space (see (1.203)). In 1985, Perdew [12] showed that the GEA for the exchange hole density - - u) contains 

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