**1-D. Practical nuclear wavefunction calculation method**

**NOMO**

The **Born-Oppenheimer (BO) approximation**, which separates the motions of electrons and nuclei, has been accepted as the basis of quantum chemistry. In quantum chemistry calculation based on BO approximation, the electronic state is calculated when fixing the positions of point-charge nuclei. However, various phenomena based on the wave nature of nucleus have been observed. Nakai’s Group has developed the **nuclear orbital plus molecular orbital (NOMO) method**, which firstly introduces a one-particle nuclear orbital similar to the electronic ones, and succeeded in determining the nuclear and electronic wavefunctions simultaneously. So far, the **NOMO method** has been applied to the evaluation of isotope effects in hydrogen bonds, the calculation of vibrationally excited states, and the efficient calculation of proton binding energy. We also reported that the **NOMO method** is also effective in calculating electron-positron annihilation γ-ray spectra.

We have also been working on improving the accuracy of the **NOMO method** from various perspectives. The **NOMO method** starts with the NOMO/HF method, which treats electron-nucleus interactions as mean fields. Therefore, in addition to the usual electron correlation (electron-electron correlation), it is necessary to consider electron-nucleus correlation and nucleus-nucleus correlation. In this regard, we firstly appliedy many-body perturbation theory and coupled cluster methods. Another problem is originated from the basis function that describes the nuclear wavefunction. Since most degrees of freedom for the nuclear motion correspond to the vibrational modes, we proposed to use the Gaussian function, which is the exact solution of the harmonic oscillator. However, plane waves and spherical harmonics should be used for translational and rotational motions, respectively. Because Gaussian basis functions are not suitable for the translational and rotational motions, we proposed a method (**TRF-NOMO method**) to remove the effect of translational and rotational motions, and improved the accuracy. We also developed a method (**ECG-NOMO method**) that combines the NOMO method with the ECG method by Prof. Adamowicz, which gives a highly accurate solution although the applicability is very limited.

Nakai’s Group has been working on many theoretical improvements in this way for more than 20 years, but the **NOMO method** (and equivalent methods around the world) still has problems to be solved theoretically. We believe that a breakthrough exists to achieve the standard theory.

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**Key Literature**

**＜****NOMO/HF****＞**

- M. Tachikawa, K. Mori, H. Nakai, and K. Iguchi, “An extension of ab initio molecular orbital theory to nuclear motion”,
*Chem. Phys. Lett.*,**290**, 437 (1998). - H. Nakai, “Simultaneous determination of nuclear and electronic wave functions without Born-Oppenheimer approximation: Ab initio NO+MO/HF theory”,
*Int. J. Quantum Chem.*,**86**, 511 (2002).

**＜****NOMO/CIS****＞**

- “Non-Born-Oppenheimer theory for simultaneous determination of vibrational and electronic excited states: ab initio NO+MO/CIS theory”, H. Nakai, K. Sodeyama, M. Hoshino,
*Phys. Lett.*,**345**, 118 (2001).

**＜****NOMO/MBPT&CC****＞**

- H. Nakai, K. Sodeyama, “Many-body effects in nonadiabatic molecular theory for simultaneous determination of nuclear and electronic wave functions: ab initio NOMO/MBPT and CC methods”,
*J. Chem. Phys.*,**118**, 1119 (2003).

**＜****TRF-NOMO****＞**

- H. Nakai, M. Hoshino, K. Miyamoto, S. Hyodo, “Elimination of translational and rotational motions in nuclear orbital plus molecular orbital theory”,
*J. Chem. Phys.*,**122**, 164101 (2005). - M. Hoshino, H. Nakai, “Elimination of translational and rotational motions in nuclear orbital plus molecular orbital theory: application of Møller-Plesset perturbation theory”,
*J. Chem. Phys.*,**124**, 194110 (2006). - K. Miyamoto, M. Hoshino, H. Nakai, “Elimination of translational and rotational motions in nuclear orbital plus molecular orbital theory: contribution of the first-order rovibration coupling”,
*J. Chem. Theory Comp.*,**2**, 1544 (2006).

**＜****NOMO/DFT****＞**

- Y. Imamura, H. Kiryu, H. Nakai, “Colle-Salvetti-type correction for electron-nucleus correlation in the nuclear orbital plus molecular orbital theory”,
*J. Comput. Chem.*,**29**, 735 (2008).

**＜****ECG-NOMO****＞**

- M. Hoshino, H. Nishizawa, H. Nakai, “Rigorous non-Born-Oppenheimer theory: combination of explicitly correlated Gaussian method and nuclear orbital plus molecular orbital theory”,
*J. Chem. Phys.*,**135**, 024111 (2011). - H. Nishizawa, M. Hoshino, Y. Imamura, H. Nakai, “Evaluation of electron-repulsion integral of the explicitly correlated Gaussian-nuclear orbital plus molecular orbital theory”,
*Chem. Phys. Lett.*,**521**, 142 (2012). - H. Nishizawa, Y. Imamura, Y. Ikabata, H. Nakai, “Development of the explicitly correlated Gaussian-nuclear orbital plus molecular orbital theory: incorporation of electron-electron correlation”,
*Chem. Phys. Lett.*,**533**, 100 (2012).

**＜****NOMO/qFCI****＞**

- L. Veis, J. Višňák, H. Nishizawa, H. Nakai, J. Pittner, “Quantum chemistry beyond Born-Oppenheimer approximation on a quantum computer: a simulated phase estimation study”,
*Int. J. Quantum Chem.*,**116**, 1328 (2016).

**＜****NOMO/DCPP2****＞**

- Y. Tsukamoto, Y. Ikabata, J. Romero, A. Reyes, H. Nakai, “The divide-and-conquer second-order proton propagator method based on nuclear orbital plus molecular orbital theory for the efficient computation of proton binding energies”,
*Phys. Chem. Chem. Phys.*,**18**, 27422 (2016).

**＜****Positron Annihilation Spectra****＞**

- Y. Ikabata, R. Aiba, T. Iwanade, H. Nishizawa, F. Wang, H. Nakai, “Quantum chemical approach for positron annihilation spectra of atoms and molecules”,
*J. Chem. Phys.*,**148**, 184110 (2018).

**＜****Review****＞**

- H. Nakai, “Nuclear orbital plus molecular orbital theory: Simultaneous determination of nuclear and electronic wave functions without Born-Oppenheimer approximation”,
*Int. J. Quantum Chem.*,**107**, 2849 (2007).