Page 76 - Advanced Organic Chemistry Part A - Structure and Mechanisms, 5th ed (2007) - Carey _ Sundberg

n r n r 55

F n r = T n r +1/2 drdr +E n r (1.19)
xc
s

r −r
SECTION 1.3
The energy of the system is given by integration: Electron Density
Functionals
0
1 n r n r

E = − drdr − r n r dr +E n r (1.20)
xc
xc
j

1 2 r −r
In principle, these equations provide an exact description of the energy, but the value
E n r is not known. Although E n r cannot be formulated exactly, Kohn and
xc xc
Sham developed equations that isolate this term:
KS
h =
i i i i
1 Z A " r

2

h KS =− ! − + dr +V "
i i XC
2 r −R r −r

i
i
i
A
Various approximations have been developed and calibrated by comparison with
experimental data and MO calculations. The strategy used is to collect the terms that
can be calculated exactly and parameterize the remaining terms. Parameters in the
proposed functionals are generally selected by optimizing the method’s description of
properties of a training set of molecular data. The methods used most frequently in
organic chemistry were developed by A. D. Becke. 69 Lee, Yang, and Parr 70 (LYP)
developed a correlation functional by a fit to exact helium atom results. Combining
such “pure DFT” functionals with the Hartree-Fock form of the exchange is the basis
for the hybrid methods. Becke’s hybrid exchange functional called “B3,” combined
with the LYP correlation functional, is the most widely applied of the many possible
choices of exchange and correlation functionals. This is called the B3LYP method.
Much of the mechanics for solution of the Kohn-Sham equations is analogous to what
is used for solution of the SCF (Hartree-Fock) equations and employs the same basis
sets. That is, a guess is made at the orbitals; an approximation to the Kohn-Sham
Hamiltonian h KS is then reconstructed using the guess. Revised orbitals are recovered
i
from the Kohn-Sham equations, the Hamiltonian is reconstructed, and the process
continued until it converges.
DFT computations can be done with less computer time than the most advanced
ab initio MO methods. As a result, there has been extensive use of B3LYP and other
DFT methods in organic chemistry. As with MO calculations, the minimum energy
geometry and total energy are calculated. Other properties that depend on electronic
distribution, such as molecular dipoles, can also be calculated.
A number of DFT methods and basis sets have been evaluated for their ability to
71
calculate bond distances in hydrocarbons. With the use of the B3LYP functionals, the
∗∗
commonly employed basis sets such as 6-31G and 6-31G gave excellent correlations
∗
with experimental values but overestimated C–H bond lengths by about 0.1 Å, while
C–C bond lengths generally were within 0.01 Å. Because of the systematic variation, it
is possible to apply a scaling factor that predicts C–H bond lengths with high accuracy.
Ground state geometries have also been calculated (B3LYP) for molecules such as
formaldehyde, acetaldehyde, and acetone.
69
A. D. Becke, Phys. Rev. A, 38, 3098 (1988); A. D. Becke, J. Chem. Phys., 96, 2155 (1992); A. D. Becke,
J. Chem. Phys., 97, 9173 (1992); A. D. Becke, J. Chem. Phys., 98, 5648 (1993).
70 C. Lee, W. Yang and R. G. Parr, Phys. Rev. B, 37, 785 (1988).
71
A Neugebauer and G. Häflinger, Theochem, 578, 229 (2002).