Diagonal Born–Oppenheimer Correction (DBOC)

When the nuclear kinetic energy operator

\[\hat{T}_\mathrm{n} = -\sum_{I} \frac{\hbar^2}{2 M_I} \nabla_{\mathbf{R}_I}^2\]

acts on the adiabatic electronic wavefunctions both diagonal and off-diagonal non-adiabatic couplings arise. [16] The diagonal term behaves like a potential energy correction and modifies the adiabatic PESs, \(V_\alpha(\mathbf{R})\), as

\[\tilde{V}_{\alpha}(\mathbf{R}) = V_\alpha(\mathbf{R}) + V_{\mathrm{DBOC}}^{(\alpha)}(\mathbf{R}).\]

Here, \(\mathbf{R} \equiv \{\mathbf{R}_1, \mathbf{R}_2, \dots, \mathbf{R}_n\}\) denotes the ring-polymer coordinates containing n beads. The correction term is

\[V_{\mathrm{DBOC}}^{(\alpha)}(\mathbf{R}) = \sum_{I=1}^N \sum_{k=1}^n \frac{\hbar^2}{2M_{I,k}} \sum_{\gamma \neq \alpha} \left| \mathbf{d}_{I,k,\alpha\gamma}(\mathbf{R}_k) \right|^2.\]

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Total force on bead k for adiabatic state \(\alpha\) is

\[\mathbf{F}_{I,k} = -\nabla \tilde{V_\alpha}(\mathbf{R}_k) = -\nabla V_\alpha(\mathbf{R}_k) - \nabla V_{\mathrm{DBOC}}^{(\alpha)}(\mathbf{R}_k) = \mathbf{F}_{I,k}^{(0)} + \Delta \mathbf{F}_{I,k},\]

where the DBOC force contribution is

\[\Delta \mathbf{F}_{I,k} = \sum_{\alpha \ne \gamma} \frac{1}{M_{I,k}} \mathbf{d}_{I,k,\alpha \gamma}(\mathbf{R}_k) \cdot \nabla \mathbf{d}_{I,k,\alpha \gamma}(\mathbf{R}_k).\]

The non-adiabatic coupling vectors (NACVs) are evaluated at each bead k:

\[\mathbf{d}_{I,k,\alpha \gamma} = \frac{\langle\psi_{k,\alpha}| \nabla_{R_{I,k}} H(\mathbf{R}_k)|\psi_{k,\gamma}\rangle} {E_\gamma(\mathbf{R}_k) - E_\alpha(\mathbf{R}_k)}.\]

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DBOC-corrected forces are computed numerically via finite differences:

\[\nabla \mathbf{d}_{I,k,\alpha \gamma} = \frac{ \mathbf{d}_{I,k,\alpha \gamma}(\mathbf{R}_k+\delta) - \mathbf{d}_{I,k,\alpha \gamma}(\mathbf{R}_k-\delta) }{2\delta}.\]

Warning

The displacement parameter \(\delta\) must be chosen to ensure numerical stability and convergence of the DBOC forces during dynamics simulations.