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The relation in history part:

Quantum chemistry influences all  branches of chemistry by applying quantum mechanic to chemistry problems. It can be considered a subarea of physical chemistry. In brief, physical chemistry studies the macroscopic, atomic and particular phenomena of the chemical systems. And in particular, the quantum chemistry part of it adopts the concept of quantum mechanics to understand, for example, molecular states and properties (for example bond length and rotation), and thermodynamic properties (for example heat capacity, entropy).

Organic chemistry adopts quantum mechanics to investigate mechanisms of organic reactions, to estimate the stability of molecules, and to analyze NMR spectrums. Analytical chemistry uses quantum mechanics to determine the frequency and intensity in the spectroscopy. Inorganic chemistry approximates quantum mechanical methods and properties of transitional-metal ion with ligand field theory. And bio-chemical method uses quantum chemistry for bio-molecular calculation.

Density functional theory:

Density functional theory (DFT) is one of the computational quantum mechanical methods used in quantum chemistry. It uses the modeling method to investigate the many-body-system electronic structures in particular atoms, molecules, and the condensed phases. The pioneer DFT model, Thomas–Fermi model, was developed independently by Thomas and Fermi in 1927. This was the first attempt to describe many-electron systems on the basis of electronic density instead of wave function, although not very successful in the treatment of entire molecules. The method did provide the basis for what is now known as density functional theory, which is given by,

$T_{tf}(n)=C_{F}\int n({\overrightarrow {r}})n^{(2/3)}(r)d^{3}r=C_{F}\int n^{5/3}({\overrightarrow {r}})d^{3}r.$

Hartree-Fock method came later as a method to approximate the wave function and the energy of a many-body system in the stationery state. It majorly made the band energy calculation practical for crystalline solids, surfaces and molecules. Density functional in quantum chemistry was then lead by Johnson and his coworkers in the Slater’s group in 1960 with the invention of finite system Scatter wave computer system to simplify the Hartree-Fock method. The system assumes an approximation with a single Slater determinant, in the cases of fermion particles, or with a single bosons particle permanent of N orbitals.

Kohn and Sham laid the basis of modern DFT method by proving that electron density could be used as the fundamental property to develop the many-body system. The important difference from previous method was that the equations were no more approximation of Hartree-Fock method as in Slater equation, but an original theory in the same logical base level as Hartree-Fock method. In contrast to the traditional approximation method, modern DFT systematically treats the many-body system with ${\hat {U}}$ to be a single-body system without ${\hat {U}}$. The normalized $\,\!\Psi$ is adopts the electronic density $n({\vec {r}})$ and is given by,

$n({\vec {r}})=N\int {\rm {d}}^{3}r_{2}\cdots \int {\rm {d}}^{3}r_{N}\Psi ^{*}({\vec {r}},{\vec {r}}_{2},\dots ,{\vec {r}}_{N})\Psi ({\vec {r}},{\vec {r}}_{2},\dots ,{\vec {r}}_{N}).$

Density functional is split into four terms; the Kohn-Sham kinetic energy, an external potential, exchange and correlation energies. A large part of the focus on developing DFT is on improving the exchange and correlation terms. Though this method is less developed than post Hartree–Fock methods, its significantly lower computational requirements (scaling typically no worse than basis functions, for the pure functionals) allow it to tackle larger polyatomic molecules and even macromolecules. This computational affordability and often comparable accuracy to MP2 and CCSD(T) (post-Hartree–Fock methods) has made it one of the most popular methods in computational chemistry at present.

1. Levine, Iran.  “Chap 1: The Schrodinger Equation.” Quantum Chemistry. 5th ed. Prentice Hall, New Jersey 07458. Retrieved Nov. 29, 2016.
2. Jan K. Labanowski, Jan W. Andzelm, Density functional Theory method on Chemistry. 1st ed. Electronic. Retrieved Nov. 29, 2016.