**1-F. Practical analysis methods**

##### EDA | FZOA

**EDA**

Chemical understanding of the results of quantum chemistry calculations is extremely important in elucidating various chemical, physical, and biological phenomena, and in designing new materials. Nakai’s Group has developed **energy density analysis (EDA)**, which divides total energy into constituent atoms as a method for analyzing quantum chemistry calculations. We have also developed a **bond energy density analysis (BEDA)** that estimates the bond energy between atoms. **EDA** estimates the contribution of constituent atoms using atomic orbitals, similar to Mulliken’s electron density analysis. Therefore, the use of a diffuse basis function might give a non-physical value. Therefore, we could reduce the basis-function dependence by using the **natural atomic orbital (NAO)** basis (**NAO-EDA**).

Based on the original concept of **EDA** that energy density on a spatial grid is estimated rather than that for an atom, we formulated **Grid-EDA**. Furthermore, Grid-EDA estimates the partial sum of the energy density for the contributions of the constituent atoms using Prof. Becke’s partition function, which can also avoide the problem of basis-set dependence. **Grid-EDA** is also applicable to first-principles calculation using plane wave basis, and was implemented in the first-principles calculation program **PHASE** developed by Dr. Ohno’s Group.

Various **EDA**-related analysis methods have been used in many application studies. It is also used for purposes other than the analysis, although that was not envisioned at the time of the proposal. In the **DC-correlation method**, **EDA** was used to estimate the correlation energy of only the central region excluding the buffer region of the subsystem, and the theoretical problem was solved. In **ML-EC**, **Grid-EDA** was used to evaluate the CCSD(T)/CBS correlation energy density for each spatial grid, which is the objective function. Such unexpected theoretical development gives us one of the pleasures of research activity.

** **

**Key Literature**

**＜****EDA****＞**

- H. Nakai, “Energy density analysis with Kohn-Sham orbitals”,
*Chem. Phys. Lett.*,**363**, 73 (2002). - Y. Kawamura, H. Nakai, “A hybrid approach combining energy density analysis with the interaction energy decomposition method”, J
*. Comput. Chem.*,**25**, 1882 (2004).

**＜****BEDA****＞**

- H. Nakai, Y. Kikuchi, “Extension of energy density analysis to treating chemical bonds in molecules”,
*J. Theor. Comput. Chem.*,**4**, 317 (2005). - H. Nakai, H. Ohashi, Y. Imamura, Y. Kikuchi, “Bond energy analysis revisited and designed toward a rigorous methodology”,
*J. Chem. Phys.*,**135**, 124105 (2011).

**＜****NAO-EDA****＞**

- T. Baba, M. Takeuchi, H. Nakai, “Natural atomic orbital based energy density analysis: implementation and applications”,
*Chem. Phys. Lett.*,**424**, 193 (2006).

**＜****Grid-EDA****＞**

- Y. Imamura, A. Takahashi, H. Nakai, “Grid-based energy density analysis: Implementation and assessment”,
*J. Chem. Phys.*,**126**, 034103 (2007). - Y. Imamura, A. Takahashi, T. Okada, T. Ohno, H. Nakai, “Extension of energy density analysis to periodic-boundary-condition calculations with plane-wave basis functions”,
*Phys. Rev. B*,**81**, 115136 (2010).

**＜****Review****＞**

- H. Nakai, Y. Kikuchi, Y. Imamura, “One-body energy decomposition schemes revisited: assessment of Mulliken-, grid-, and conventional energy density analyses”,
*Int. J. Quant. Chem.*,**109**, 2464 (2009).

**＜Review (in Japanese)＞**

- T. Baba, Y. Yamauchi, Y. Kikuchi, Y. Kurabayashi, H. Nakai, “Kohn-Sham DFT法に対するエネルギー密度解析”,
*Bull. Soc. Discrete Variational Xα*,**18**, 7 (2005).

**FZOA**

To improve the accuracy of quantum chemistry calculations, many basis functions are used for flexible description of one-electron functions, and multi-configurational wavefunction is used for descriptions beyond the mean-field theory. However, we often encounter situations where the essence can be seen from a qualitative understanding of the phenomenon. The frontier orbital theory by Prof. Kenichi Fukui can be said to be a representative example. Nakai’s Group has proposed **frozen orbit analysis (FZOA)**, which is an analysis method that considers only the transition between HOMO and LUMO, for the purpose of intuitive understanding of excited states. This led to the discovery/elucidation of the **symmetry rules of degenerate excitations**, the **π*-σ* hyperconjugation**, and the **governing factors of the conical intersection structure**.

**Key Literature**

**＜****FZOA****＞**

- H. Nakai, H. Morita, H. Nakatsuji, “Frozen-orbital analysis for the excited states of metal complexes in high symmetry: O
_{h}case”,*J. Phys. Chem.*,**100**, 15753 (1996). - T. Baba, Y. Imamura, M. Okamoto, H. Nakai, “Analysis on excitations of molecules with Ih symmetry: frozen orbital analysis and general rules”,
*Chem. Lett.*,**37**, 322 (2008).