Time-dependent Hartree–Fock frequency-dependent polarizability calculation applied to divide-and-conquer electronic structure method
Graphical abstract
DC–TDHF method reduces the computational cost of dynamic polarizability calculation with fairly good accuracy.
Introduction
By virtue of recent developments of quantum chemical methodologies, it becomes easy to compute the energy of a large system in ab initio quality. Various schemes making large-scale quantum chemical calculations possible have been proposed at the levels of the independent particle [Hartree–Fock (HF) or density functional theory (DFT)] models and more sophisticated post-HF correlation methods so far [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. These methods usually reduce the order of the computational time which scales O(N3) or more in the traditional formalisms, where N represents the system-size. The energies obtained by these methods have been also used to compute static response properties [25], [26].
Although it is doubtless that the total energy of the system and its derivatives are the most important properties that relate to stability, reaction, structure, vibrational modes, and other static properties of the system, the frequency-dependent response properties are also valuable ones that implicitly have the information of the electronic excited states. To obtain these dynamic properties, one has to solve the response equations such as the time-dependent HF (TDHF) equation [or equivalent random phase approximation (RPA) equation] at the level of the independent particle model [27], [28]. Thus far, low-scaling methodologies which orient toward large-scale dynamic (hyper)polarizability calculations have been proposed [29], [30], [31], [32], [33].
We have developed the divide-and-conquer (DC) quantum chemical methods which ensure asymptotic linear-scaling costs with respect to the system-size and implemented DC-based HF [5], DFT [5], and post-HF [7], [8], [9], [10], [11], [12] codes into the Gamess package [34], [35]. Although the linear-scaling static polarizability calculations based on the DC methods are available by the finite differentiation of the energy with respect to the electric field, its numerical assessments have never been reported, to the best knowledge of the authors. Moreover, no dynamic response calculation scheme based on the DC method has been proposed to date. Unlike the simple fragmentation methods, the DC-based formulation has the potential to effectively calculate dynamic polarizabilities, which relate to the electronic excitation energies of the entire system, because the DC method can construct the density response of the entire system, not of each subsystem.
In this Letter, we propose a scheme to evaluate dynamic polarizability by solving TDHF equation based on the linear-scaling DC strategy. The organization of this Letter is as follows. Section 2 presents a short summary of the DC electronic structure method and the theoretical aspects of the present DC–TDHF method. Numerical results are shown in Section 3. Finally we give concluding remarks in Section 4.
Section snippets
TDHF method for polarizability calculation
The definition of the polarizability follows from an expansion of the total dipole moment d with respect to the external electric field E:where d0 is the dipole moment of the unperturbed system, α is the polarizability and β is the first hyperpolarizability. The polarizability for an oscillating electric field is denoted as α(ω), and that for a static field E0(t) = A is α(0).
We consider the interaction of an N-electron closed-shell
Numerical assessments
The DC–TDHF method was assessed in the dynamic polarizability calculations of n-alkanes, CnH2n+2 (Fig. 1, C2h symmetry), by comparing the findings with the conventional TDHF results. The following results were obtained by the modified version of the Gamess program [34], [35]. All C–C and C–H distances were fixed to 1.534 and 1.100 Å, respectively. All C–C–C and H–C–H angles were fixed to 113.6° and 105.9°, respectively. All calculations were performed with the 6-31G** basis set [36] unless
Concluding remarks
In the present study, we have proposed and implemented the DC–TDHF method for the calculation of dynamic polarizability. By comparing the polarizability of the n-alkane system obtained by the DC–TDHF method with those by the conventional TDHF method, it was found that the DC–TDHF method gives accurate static and dynamic polarizabilities. In addition, we confirmed that the DC–TDHF calculations are faster than conventional TDHF ones, possessing the lower order computational cost with respect to
Acknowledgments
This study was supported in part by a Grant-in-Aid for Scientific Research on Priority Areas ‘Molecular Theory for Real Systems’ KAKENHI 18066016 from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan; by the Next Generation Integrated Nanoscience Simulation Software Project of the MEXT; by the Global Center Of Excellence (COE) ‘Practical Chemical Wisdom’ from the MEXT; and by a Project Research Grant for ‘Development of High-performance Computational Environment
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