Exchange and Correlation Energy Functionals

Since the quality of any DFT calculation is limited by the quality its approximation to the true XC energy funaional, , c[p( )] surprising 

The heart of the LSDA is the approximation that a particular point in space of an inhomogeneous distribution of electrons (such as an atom or molecule) with densities p (r) and pP(r) has the same values 6xc v c nd v f. as any point in a homogeneous distribution of electrons of the exact same densi- 

Though this may seem to be a crude approximation, theoretical consider-ations s do justify the LSDA s ability to provide quantitatively accurate geometries, charge distributions, and vibrational speara on a wide variety sys-tems.2.8- It should not be surprising to see the LSDA results surpass HF and even challenge correlated post-HF methods. One must remember that correlation effects are included within the LSDA XC functional. Systems that prove to be extremely problematic in FIF theory, where correlation effects are by definition absent, are therefore not necessarily systems that offer any great challenge to the LSDA. A simple molecule like ozone is one such classic example.22 

HF theory does not do particularly well, as the hydrogen bond is 0.2 A too long. However, the MP2 calculation cleans up the geometry very nicely, with the hydrogen bond now in error by less than 0.02 A. Within the LSDA, malonaldehyde is very poorly described. An overestimation of the strength of the hydrogen bond essentially drives the system towards a Civ structure, with the 

H hydrogen bond almost 0.5 A shorter than experiment and only 0.02 A 

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Approximate gradient-corrected exchange and correlation energy functionals are developed using theoretical considerations such as the known behavior of the true (but unknown) functionals and E in various limiting situations as a guide, with, per-... 

From the above, it is conclnded that because the ground states have different spin, searching for density fnnctionals without taking into account the spin is a failure. The same holds true for the exchange and correlation energy functional that has a stronger spin dependence. 

Given the fact that the exchange and correlation energy functional is not known, approximations have to be made, e.g. the local density approximation and/or the approximation or another suitable scheme. The... 

Exchange and correlation energy functional. The KS approach exactly transforms a many-body problem into a simple non-interacting problem, thanks to the introduction of the XC energy functional xc[p]-However, the functional form of Ey f p] is not known and it must be... 

The total density is the sum of die a and /3 contributions, p = Pa + Pp, and for a closed-shell singlet these are identical (p, = pp). Functionals for the exchange and correlation energies may be formulated in terms of separate spin-densities however, they are often given instead as functions of the spin polarization C, (normalized difference between p and pp), and the radius of the effective volume containing one electron, rs-... 

Gritsenko, O. V., Schipper, P. R. T., Baerends, E. J., 1997, Exchange and Correlation Energy in Density Functional Theory. Comparison of Accurate DFT Quantities With Traditional Hartree-Fock Based Ones and Generalized Gradient Approximations for the Molecules Li2, N2, F2 , J. Chem. Phys., 107, 5007. 

The term Exc[p] is called the exchange-correlation energy functional and represents the main problem in the DFT approach. The exact form of the functional is unknown, and one must resort to approximations. The local density approximation (LDA), the first to be introduced, assumed that the exchange and correlation energy of an electron at a point r depends on the density at that point, instead of the density at all points in space. The LDA was not well accepted by the chemistry community, mainly because of the difficulty in correctly describing the chemical bond. Other approaches to Exc[p] were then proposed and enable satisfactory prediction of a variety of observables [9]. 

Since the Coulomb, exchange, and correlation energies are all consequences of the interelectronic 1 /r12 operator in the Hamiltonian, one can define the exchange energy functional Ex [p] in the same manner as... 

The simplest way to gain a better appreciation for tlie hole function is to consider the case of a one-electron system. Obviously, the Lh.s. of Eq. (8.6) must be zero in that case. However, just as obviously, the first term on the r.h.s. of Eq. (8.6) is not zero, since p must be greater than or equal to zero throughout space. In die one-electron case, it should be clear that h is simply the negative of the density, but in die many-electron case, the exact form of the hole function can rarely be established. Besides die self-interaction error, hole functions in many-electron systems account for exchange and correlation energy as well. 

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