Nuclear quantum effects
Introduction: NOMO method

Born-Oppenheimer (BO) approximation separates the motions of electrons and nuclei. This is adopted in electronic-state theories: such as molecular orbital (MO) method and density functional theory (DFT). In the BO approximation, the electronic wavefunction for specific nuclear placement is determined by solving the time-independent Schrodinger equation for electronic Hamiltonian. Accuracy and efficiency of the electronic-state calculation have been improved within the BO framework, and its target is expanding not only for the small molecules but for large systems such as biomolecules and nanomaterials. However, it is difficult for the BO approximation to describe the phenomena derived from the quantum nature of nuclei, e.g., zero-point vibration, tunneling, resonance, scattering, interference, etc. To treat these phenomena, we need to deal with nuclei quantum-mechanically.

Fig. 1. Proton tunneling in Malonaldehyde.

Our group has worked on the non-BO theory. Our strategy is to introduce the nuclear orbital (NO), which permits the conventional MO method to be extended to the non-BO theory. The nuclear orbital plus molecular orbital (NOMO) method determines the wavefunction of nuclei and electrons simultaneously. Similar approaches for the non-BO problem were reported by other groups and gradually noticed. As a practical development of NOMO method, our group implemented the Hartree-Fock (HF) method, which determines the wavefunction based on the one-body approximation. The configuration interaction-singles (CIS) and second-order Moller-Plesset (MP2) methods were then combined with the NOMO method. In this page, the theory and result of NOMO/HF and NOMO/CIS methods are described.

Theory

The NOMO method expresses the HF wavefunction as a direct product of electronic and nuclear wavefunctions
$\Phi_0^\text{NOMO/HF} = \Phi_0^\text{e} \cdot \Phi_0^\text{n}$
As a one particle wavefunction, MO
$\{ \varphi_i, \varphi_j, \varphi_k, \cdots \}$
and NO
$\{ \varphi_I, \varphi_J, \varphi_K, \cdots \}$
are introduced. The electronic and nuclear wavefunctions are represented as an antisymmetric (symmetric for bosonic nuclei) product of MOs and NOs, respectively.
$\Phi_0^\text{e} = \lVert \varphi_i \varphi_j \cdots \varphi_k \rVert$
$\Phi_0^\text{n} = \lVert \varphi_I \varphi_J \cdots \varphi_K \rVert$
The NOMO/HF equation for each MO and NO is obtained by applying the variational method to the total energy.
$f^\text{e} \varphi_i = \varepsilon_i \varphi_i$
$f^\text{n} \varphi_I = \varepsilon_I \varphi_I$
The Fock operator includes the mean-field interaction of NO and MO. As for the unrestricted treatment, the NOMO/UHF method is formulated in the same manner. The NOMO/HF wavefunction satisfies the Brillouin and Koopmans theorems. Correlation methods developed basis on the BO approximation can be applied to the NOMO method

Extension to excited-state calculation: NOMO/CIS method

The NOMO/HF method determines the ground-state wavefunction of nuclei and electrons simultaneously. This section extends the NOMO method to excited-state calculations. The NOMO/full-CI method with complete basis functions gives the exact wavefunction in principle. In the conventional MO method, the CIS method, which exploits one electron excitation operator, is a simple approach. By extending the CIS method to the NOMO framework, the excited states of nuclei and electrons are expected to be obtained.

Fig. 2. Schematic pictures of (a) MO theories and (b) NOMO theories.

NOMO/CIS wavefunction is given as follows.
$\Psi^\text{NOMO/CIS} = \left( 1 + \sum_{i,a}^{\{\text{elec.}\}} c_i^a a_a^\dagger a_i + \sum_{I,A}^{\{\text{nuc.}\}} c_I^A a_a^\dagger a_I \right) \Phi_0^\text{NOMO/HF}$
CI coefficients are determined variationally, that is, by the diagonalization of the CI matrix.

Example 1: Electronic and vibrational excited states

Table 1 shows the excitation energies of vibrational (ν = 0→1 and 0→2) and electronic (S0→S1) transitions obtained by the NOMO/CIS method. The cc-pVTZ was used as the electron basis function (EBF). Experimental values are listed for comparison. In addition to the electronic counterparts, vibrational excitation energies were obtained reasonably.

Table 1. Vibrational and electronic excitation energies (in cm-1) of H2, D2, and T2 calculated by the NOMO/CIS method, comparing with experiments.
Example 2: Nuclear basis function dependence

In Figure 3, the energy gaps between ground and vibrationally excited (ν = 0, 1, and 2) states by the NOMO/HF and CIS calculations are summarized with different nuclear basis functions (NBFs). The cc-pVTZ set was used as EBF. The total energy for these states decrease with increasing NBF, but the dependence of ground-state energy is small. The total energy of the first excited state (ν = 1) seems to converge by applying p type functions. The second excited state (ν = 2) does not converge until adding d type functions. This trend corresponds to the exact wavefunction of harmonic oscillator in the first and second excited states.

Fig. 3. NBF dependence of the total energies of H2 in the ground and vibrational excited sates (ν = 0, 1, and 2) calculated by the NOMO/HF and CIS methods.
Example 3：Difference nuclear density map

Figure 4 shows difference nuclear density maps between vibrational ground and excited (ν = 1 and 2) states of H2, HeH+, and LiH obtained by the NOMO/CIS method. The (3s3p3d) and cc-pVTZ set were used as the NBF and EBF. The importance of pz and dzz type functions is seen from the density map. As the nuclear mass increases from H to He, and Li, the nuclear density shrinks. This means that the quantum effect is decreased with the increase of nuclear mass. Density of proton in HeH+ is more delocalized than the others, indicating that hydrogen atom vibrates loosely according to the weak chemical bond.

Fig. 4. Nuclear density difference maps for the vibrational excitations (ν = 0→1 and 0→2) of H2, HeH+, and LiH.

Next, vibrational excited states of H3+ were estimated by the NOMO/CIS method using the NBF of (4s4p4d) type. The point group of H3+ in the ground state geometry is D3h. The lowest two states are degenerated with the E′ symmetry. Third excited state has the A1′ symmetry. These states are attributed to the excited states of the totally symmetric and antisymmetric stretching vibration. Figure 5 (a), (b), and (c) show difference nuclear density maps between the ground and excited states, indicating the direction of the p type function in each excited state. These density difference correspond to vibration modes in Figure 5 (d), (e), and (f). Thus, the NOMO/CIS method was found to describe the vibrational wavefunction properly.

Fig. 5. (a)-(c) Nuclear density difference maps for the three vibrational excitations of H3+ and (d)-(f) schematic illustration for the three normal modes of H3+.