Density Functional Theory
Introduction

Fundamental theories for quantum chemical calculations are classified into the wave function theory (WFT) and density functional theory (DFT). DFT expresses the electronic energy as a functional of electron density. In Kohn-Sham DFT (KS-DFT) [1], which is currently used as standard, kinetic and electron-electron repulsion energies are given analytically like the Hartree-Fock (HF) method. Instead of the HF exchange, an approximated exchange-correlation functional is adopted. The exchange-correlation functionals have been sophisticated by including density gradient, kinetic energy density, and HF exchange. KS-DFT was successful both in numerical accuracy required for applications to molecular systems and in realistic computational cost. To expand the applicability of KS-DFT, a number of exchange-correlation functionals and correction methods have been developed. In this page, we introduce functionals and correction methods developed in our laboratory.

Density functional tuned for different types of excitation

Time-dependent density functional theory (TDDFT) [2] has been utilized to obtain excitation energies and response properties. In the TDDFT calculation of excitation energy, the valence excitation is calculated with reasonable accuracy, but core and Rydberg excitations are not. We developed the Core-Valence-Rydberg B3LYP (CVR-B3LYP) functional [3,4], which accurately describes core and Rydberg excitations. This functional adjusts the ratio of the HF exchange for different types of Kohn-Sham orbitals to follow the behavior of following methods: the BHHLYP functional for core orbitals, the HF method for Rydberg orbitals, and the B3LYP functional for valence orbitals. Though different Fock operators are obtained, the invariance of unitary transformation is ensured by the Roothaan coupling operator [5].

As shown in Figure 1, mean absolute errors (MAEs) of excitation energies by the CVR-B3LYP functional were 0.3 eV (core-valence), 0.8 eV (core-Rydberg), and 0.28 eV (valence-valence), respectively. We confirmed that all kinds of excitations are calculated with the same accuracy.

Fig. 1. Mean absolute errors of core-valence, core-Rydberg, valence-valence, valence-Rydberg excitations (in eV).

Toward a minimum empiricism

The accuracy of KS-DFT was dramatically improved by mixing the HF exchange with the DFT exchange. The ratio of the HF exchange is usually determined based on numerical assessments. The improvement of reliability and the expansion of coverage are expected by determining the HF exchange ratio non-empirically.

Janak's theorem [6] is known as a basic theorem of Kohn-Sham DFT. The differentiation of total electronic energy E with respect to the number of the electron occupation fi in orbital i is equal to the corresponding orbital energy εi.
$\cfrac{\partial E}{\partial f_i} = \varepsilon_i \ \ \left( 0 \leq f_i \leq 1 \right)$
If the exchange-correlation functional is exact, the right side is constant. Therefore, the occupied orbital energy should be equal to ionization energy. If the above equation is differentiated by the occupation number, the linearity condition for exchange-correlation functionals is derived.
$\cfrac{\partial^2 E}{\partial f_i^2} = 0 \ \ \left( 0 \leq f_i \leq 1 \right)$
Many approximated functionals do not satisfy this condition. Orbital-specific (OS) functional [7,8] determines the ratio of the HF exchange to satisfy the linearity condition.

MAEs of ionization energies based on the orbital energy are shown in Figure 2. Compared with the B3LYP functional, which is widely used at present, and the LC-BLYP functional, which describes the valence orbital energy with good accuracy, the OS functional provides a small error in any types of orbitals.

Fig. 2. Mean absolute errors of ionization energies of core and valence orbitals (in eV).

A density-dependent dispersion correction method

Many correlation functionals in DFT include the electron correlation effect locally, but cannot treat the dispersion force originated in long-range correlation. Since the dispersion force is a dominant component of van der Waals interaction, many exchange-correlation functionals cannot evaluate the intermolecular interaction energy quantitatively. Our group developed the local response dispersion (LRD) method [9,10] as a method of dispersion correction for DFT.

To evaluate non-empirical dispersion energy with small additional cost, the LRD method was derived by the multipole expansion of dispersion energy at each atomic center and the local response approximation for the density response function. As a result, following simple formula was obtained.
$E_\text{disp} = - \sum_{a>b} \sum_{n \geq 6} C_n^{ab} R_{ab}^{-n} f_\text{damp}^{(n)}$
Cn in the above equation is called the n-th order dispersion coefficient. Though the use of predetermined values is possible, the LRD method calculates Cn using the electron density obtained by the DFT calculation.

Figure 3 is the potential energy surface of parallel-displaced benzene dimer. Compared with the result of the CCSD(T) method, which is called the "gold standard" method in quantum chemistry, the LC-BOP functional combined with the LRD method provided very accurate curves, especially if multicenter interactions are included.

Fig. 3. Potential energy surface of parallel-displaced benzene dimer.
LRD[m,n] means the cutoff of dispersion correction energy up to n-th two-center and m-th multicenter interactions.

The LRD method was implemented in the public version of the GAMESS program package developed in the laboratory of M. S. Gordon, Iowa State University. The combination with the LC-BOP functional was found to provide accurate results for various intermolecular interactions.

References
1. “Self-Consistent Equations Including Exchange and Correlation Effects”
W. Kohn and L. J. Sham, Phys. Rev., 140, A1133 (1965).
2. M. E. Casida, in Recent Advances in Density Functional Methods, edited by D. P. Chong (World Scientific, Singapore, 1995), Vol. 1, p. 155.
3. “Hybrid exchange-correlation functional for core, valence, and Rydberg excitations: Core-valence-Rydberg B3LYP”
A. Nakata, Y. Imamura, and H. Nakai, J. Chem. Phys., 125, 064109 (2006).
4. “Extension of the Core-Valence-Rydberg B3LYP Functional to Core-Excited-State Calculations of Third-Row Atoms”
A. Nakata, Y. Imamura, and H. Nakai, J. Chem. Theory Comput., 3, 1295 (2007).
5. “Self-Consistent Field Theory for Open Shells of Electronic Systems”
C. C. J. Roothaan, Rev. Mod. Phys., 32, 179 (1960).
6. “Proof that ∂E⁄∂ni = ε in density-functional theory ”
J. F. Janak, Phys. Rev. B, 18, 7165 (1978).
7. “Linearity condition for orbital energies in density functional theory: Construction of orbital-specific hybrid functional”
Y. Imamura, R. Kobayashi, and H. Nakai, J. Chem. Phys., 134, 124113 (2011).
8. “Linearity condition for orbital energies in density functional theory (II): Application to global hybrid functional”
Y. Imamura, R. Kobayashi, and H. Nakai, Chem. Phys. Lett., 513, 130 (2011).
9. “Density Functional Method Including Weak Interactions: Dispersion Coefficients Based on the Local Response Approximation”
T. Sato and H. Nakai, J. Chem. Phys., 131, 224104 (2009).
10. “Local Response Dispersion Method II. Generalized Multicenter Interactions”
T. Sato and H. Nakai, J. Chem. Phys., 133, 194101 (2010).