Density Functional Theory Density functional theory or DFT is a method of computing the energy of a molecular structure. It is an ab-initio method in that the only information needed is the coordinates and types of the atoms in the molecule. The energy is calculated by solving SchrĂśdinger’s equation: đ??ťđ??ťđ??ťđ??ť = đ??¸đ??¸đ??¸đ??¸
where H is the Hamiltonian operator and E are the energy eigenvalues. ď ™ď€ is a wavefunction describing the positions of the electrons in the molecule. There are two main methods of solving this equation: choose a model wavefunction (molecular orbital theory) or choose the electron density (density functional theory).
But what does the term “functionalâ€? mean? Normally we think of a function as an operation that maps one number to another; i.e. đ?‘“đ?‘“(đ?‘Ľđ?‘Ľ ) = đ?‘Śđ?‘Ś. A functional maps a set of functions to a number; i.e. {đ?‘“đ?‘“(đ?‘Ľđ?‘Ľ)} → đ?‘…đ?‘…. This is essentially a function of functions.
The Hamiltonian operator is simply the potential and kinetic energy operator of the electrons and nuclei of the molecule but it is the interactions of all of the particles that complicate the form. Mathematically the Hamiltonian is given by: đ??ťđ??ť = đ?‘‡đ?‘‡đ?‘’đ?‘’ + đ?‘‡đ?‘‡đ?‘›đ?‘› + đ?‘‰đ?‘‰đ?‘’đ?‘’−đ?‘’đ?‘’ + đ?‘‰đ?‘‰đ?‘›đ?‘›âˆ’đ?‘›đ?‘› + đ?‘‰đ?‘‰đ?‘’đ?‘’−đ?‘›đ?‘›
where the T are the kinetic energy operators of the electrons and nuclei, respectively. V are the potential energy operators of the electron – electron interactions, the nucleus – nucleus interactions, and the electron – nucleus interactions.
When constructing the Hamiltonian using the Born-Oppenheimer approximation keeps the atomic nuclei stationary so Tn is zero and Vn-n is constant. Ve-n can be considered as an external field of the nucleus charge acting on the electron. What is left, Te + Ve-e is approximated in the various DFT methods. The approximations are given by functions of the electron density, đ?œŒđ?œŒ(đ?‘Ľđ?‘Ľâƒ— ).
In DFT, functionals are chosen that fit calculated energies of a test set of molecules with known energies. Density functional theory uses functionals of electron density to approximate the two-electron interaction when constructing SchrÜdinger’s equation. Wavefunctions comprised of a linear combination of basis sets are used to solve SchrÜdinger’s equation based on the Kohn-Sham method to calculate the energy of a molecule given the coordinates of and types of atoms.
The most popular functional is designated the B3LYP functional and is popular because it gives good energy calculations in a reasonable computational time. B3LYP combines a correlation functional with an exchange functional. The B3 is a 3 - 39 VOLUME 83, NUMBER 3