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13.5 Nuclear–Electronic Orbital Method

13.5.5 NEO-DFT(V)

(July 14, 2022)

Within the NEO framework, select nuclei are treated quantum mechanically at the same level as the electrons. This removes the Born-Oppenheimer separation between the quantum nuclei and the electrons and naturally includes nonadiabatic effects between the quantum nuclei and the electrons. At the same time, quantizing the select nuclei gives rise to a potential energy surface with fewer nuclear degrees of freedom, which prevents a direct calculation of the vibrational frequencies of the entire molecule. Consequently, diagonalization of a coordinate Hessian in the NEO framework yields vibrational frequencies and accompanying normal modes of only the classical nuclei, with the quantum nuclei responding instantaneously to the motion of the classical nuclei. 1066 Schneider P. E. et al.
J. Chem. Phys.
(2021), 154, pp. 054108.
Link
Although the fundamental anharmonic vibrational frequencies of the quantum nuclei can be accurately obtained through NEO-TDDFT, 248 Culpitt T. et al.
J. Chem. Phys.
(2019), 150, pp. 201101.
Link
the couplings between the vibrations of the classical and quantum nuclei are missing. To obtain the fully coupled molecular vibrations, an effective strategy denoted NEO-DFT(V) was developed. 1324 Yang Y. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 1167.
Link
, 249 Culpitt T. et al.
J. Chem. Theory Comput.
(2019), 15, pp. 6840.
Link
The NEO-DFT(V) method has been shown to incorporate key anharmonic effects in full molecular vibrational analyses and to produce accurate molecular vibrational frequencies compared to experiments. 1324 Yang Y. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 1167.
Link
, 249 Culpitt T. et al.
J. Chem. Theory Comput.
(2019), 15, pp. 6840.
Link

The NEO-DFT(V) method involves diagonalization of an extended NEO Hessian composed of partial second derivatives of the coordinates of the classical nuclei (𝐫c) and the expectation values of the quantum nuclei (𝐫q). This extended Hessian matrix is composed of three sub-matrices: 𝐇0=(2E/𝐫c2)|𝐫c=𝐫q, 𝐇1=2E/𝐫q𝐫c, and 𝐇2=2E/𝐫q2, where in each case, all other coordinates of the classical nuclei and expectation values of the quantum nuclei are fixed. The extended Hessian has the following structure:

𝐇DFT(V)=[𝐇0𝐇1𝐇1𝐇2] (13.51)

where

𝐇2 =𝐔𝛀𝐌𝐔 (13.52a)
𝐇1 =-𝐇2𝐑 (13.52b)
𝐇0 =𝐇NEO+𝐑𝐇2𝐑. (13.52c)

The quantity 𝐑=d𝐫q/d𝐫c and the NEO Hessian matrix is 𝐇NEO=d2E/d𝐫c2 (without the constraint that the expectation values of the quantum nuclei are fixed). In the expression for the 𝐇2 matrix, 𝐌 is the diagonal mass matrix, and 𝛀 is the diagonal matrix with elements corresponding to the squares of the NEO-TDDFT fundamental vibrational frequencies. 1324 Yang Y. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 1167.
Link
𝐔 is a unitary matrix that transforms 𝛀 to the target coordinate system and is approximated with the transition dipole moment vectors afforded by a NEO-TDDFT calculation. 249 Culpitt T. et al.
J. Chem. Theory Comput.
(2019), 15, pp. 6840.
Link

Diagonalization of 𝐇DFT(V) produces the fully coupled molecular vibrational frequencies including anharmonic effects associated with the quantum protons. The NEO-DFT(V) method is available for use with the epc17-2 functional or when no electron-proton correlation functional is used. The NEO-HF(V) method, which involves building the extended NEO-Hessian based on the NEO-HF Hessian and inputs from NEO-TDHF, is also available.