Efficient vibrational analysis for unrestricted Hartree–Fock based on the fragment molecular orbital method
Graphical abstract
Introduction
Vibrational frequency calculations are very useful to simulate vibrational infrared (IR) and Raman spectra, estimate thermodynamic properties such as entropy and free energy, and perform stationary point search. In order to obtain vibrational frequencies, one has to calculate the matrix of the second derivatives of the energy and diagonalize it [1], [2], [3].
For large systems Hessian calculations are very expensive even at the level of Hartree–Fock. One can apply the combined quantum-mechanical and molecular-mechanical (QM/MM) method [4], [5], [6], our own n-layered integrated molecular orbital and molecular mechanics (ONIOM) [7], and density functional tight binding method [8]. There are also various partial approaches where the Hessian is computed usually numerically only for a subset of atoms [9], [10], [11].
Alternatively, there is a variety of fragment-based methods [12], [13], [14], [15], for some of which the analytic second derivative has been developed for closed shell molecules: molecular tailoring approach [16], generalized energy-based fragmentation method [17], integrated multi-center molecular orbital (MO) approach[18], and the fragment molecular orbital (FMO) method [19]. There is a need for a normal mode analysis applicable to large radical systems.
In FMO [20], [21], [22], one divides the system into fragments (also called monomers), and calculates each of them in the embedding electrostatic potential (ESP) due to the remaining fragments. After the fragment electronic states converge with respect to ESP, fragment pair calculations are performed. FMO has been applied to proteins [23], [24], DNA [25], and inorganic systems [26], [27]. Geometry optimizations [28], [29] and molecular dynamic simulations [30], [31] can be conducted using the fully analytic energy gradients in both restricted [32] and unrestricted [33] formulations. An important by-product of fragment-based calculations is the pair interaction energies (PIEs), which can be decomposed into physically meaningful components [34], [35].
In this Letter, we develop the analytic second derivative of the energy for unrestricted Hartree–Fock (UHF) based on FMO (FMO-UHF), using coupled perturbed Hartree–Fock (CPHF) [36] to obtain the response terms arising from the coupling of the electronic state of fragment pairs and the embedding ESP. The FMO Hessian as originally formulated for restricted Hartree–Fock (RHF)[19] is very expensive, because a quadratic number of Hartree–Fock calculations for fragment pairs has to be done. To address this problem, in this Letter we develop the second derivative for the electrostatic dimer approximation (ES-DIM) [37], where the number of Hartree–Fock dimers scales linearly with the system size, while the rest is treated with a very fast and accurate approximation. The accuracy and efficiency of FMO-UHF Hessian are evaluated in comparison with the ab initio calculation and experimental results.
Section snippets
Second derivative of the FMO-UHF energy
In FMO-UHF [33] some fragments are calculated with UHF, and the rest with RHF. A dimer calculation is performed with UHF, only when at least one fragment in the dimer is UHF, otherwise RHF dimers are computed. Although we derive the equations for any number of UHF fragments, our current implementation of FMO-UHF is limited to one UHF fragment.
The second derivative of the total energy E in FMO-UHF with respect to nuclear coordinates a and b in the two-body expansion is
Computational details
Because production version of GAMESS [39], [40] did not have the analytic ab initio UHF Hessian, we implemented both conventional and FMO-based UHF Hessians. The latter was parallelized with the generalized distributed data interface (GDDI) [41].
The accuracy of the FMO-UHF Hessian is evaluated in comparison to ab initio for stable organic radicals [42] (see the structures in Figure S1 in the Supplementary materials) and a small polypeptide NH2-IYPIG-COO, which we denote in short as IYPIG (IYPIG
Accuracy of the FMO-UHF Hessian
We evaluated the accuracy of FMO-UHF vibrational frequencies and IR intensities for solvated stable organic radical molecules, 2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPO) and dimethyl-amino-nitronyl-nitroxide (DMANN). Because the many-body effects in water are significant, DMANN, solvated in water, shows somewhat larger errors than TEMPO, solvated in DMF.
The results are shown in Figure 1 and Table 1. Ab initio and FMO frequencies are very similar, while the IR intensities show some deviations.
Conclusions
We have derived the analytic second derivative of the energy for open-shell systems using the FMO-based UHF method and parallelized in GAMESS. In addition, we have derived the second derivative of the energy for the separated dimer approximation (ES-DIM) for both restricted and unrestricted FMO methods, which is crucial to reduce the cost of FMO Hessians. The accuracy of vibrational frequencies and thermodynamical properties is demonstrated for solvated organic radicals and the method is
Acknowledgement
This work has been supported by the Next Generation Super Computing Project, Nanoscience Program (MEXT, Japan) and Computational Materials Science Initiative (CMSI, Japan) and JSPS KAKENHI Grant No. 24540443. Calculations were performed on TSUBAME2.0 at the Global Scientific Information and Computing Center of Tokyo Institute of Technology, RIKEN Integrated Cluster of Clusters (RICC) at RIKEN and Research Center for Computational Science (Okazaki, Japan) for the computer resources.
References (50)
- et al.
Chem. Phys. Lett.
(2002) - et al.
J. Mol. Struct. THEOCHEM
(1999) - et al.
Biophys. J.
(2005) - et al.
Theor. Chem. Acc.
(2002) - et al.
Chem. Phys.
(1975) - et al.
Chem. Phys. Lett.
(1999) Chem. Phys. Lett.
(2003)- et al.
Chem. Phys. Lett.
(2002) - et al.
Chem. Phys. Lett.
(2009) - et al.
Comput. Phys. Commun.
(2010)
J. Comput. Chem.
Mol. Phys.
J. Am. Chem. Soc.
J. Chem. Phys.
Ann. Rev. Phys. Chem.
J. Chem. Phys.
J. Chem. Phys.
Chem. Rev.
Phys. Rev. Lett.
J. Phys. Chem. B
J. Chem. Phys.
J. Phys. Chem. A
J. Phys. Chem. A
J. Chem. Phys.
J. Phys. Chem. A.
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