Automatic Determination of Buffer Region in Divide-anc-Conquer Quantum Chemical Calculations

Abstract

本稿では，著者らが開発してきた大規模量子化学計算手法である分割統治（DC）法において，バッファ領域を自動的に決定する手法について述べる．バッファ領域は，DC法の近似に伴い導入される誤差に直接関係し，その選択はエネルギー精度を決める要である．繰り返し計算であるDC Hartree-Fock法では，二層の階層構造を持つバッファ領域を用いる．外側バッファ領域の各原子からのエネルギー寄与を概算し，エネルギー閾値に基づいてバッファ領域をその方向に拡大すべきか否かを判断することで，バッファ領域を徐々に拡大していく．一方，繰り返し計算ではない2次Møller-Plesset摂動計算に対するDC法では，元のバッファ領域内の各原子に対してエネルギー寄与を概算し，与えられたエネルギー閾値以上の寄与を持つ原子のみをバッファ領域に残す．いずれの手法も，エネルギーに基づく1つの閾値だけをパラメータとして，ほぼ一定の精度でエネルギーを計算できることを実証した．

Translated Abstract

A scheme to automatically determine the buffer region in the divide-and-conquer (DC) large-scale quantum chemical method is introduced. The buffer region directly relates to the error introduced by the DC method. In the iterative DC Hartree-Fock procedure, the automatic scheme adopts two-layered buffer region and gradually enlarges the buffer region by evaluating the energy contribution from the outer buffer region and determining whether the buffer region should be enlarged or not based on the energy-based threshold. On the other hand, in the non-iterative DC second-order Møller-Plesset perturbation calculation, the energy contribution is approximately estimated for the atoms in the buffer region and only those atoms that contribute more than an energy-based threshold are left in the buffer region. We demonstrated that both methods achieve almost constant accuracy in the energy using only one energy-based threshold as a parameter.

Figures

Fig. 1.

 Structure of central, inner buffer, and outer buffer regions in the two-layer DC method.

Fig. 2.

 Expansion of the buffer region for Gly10 subsystem in Trp-cage system. (a) initial buffer region for $r_{\text{b}}^{\text{in}}=3.5\text{Å}$; (b) final DC-HF buffer region with $e_{\text{thresh}}^{\text{SCF}}=0.1\text{ μ}{E}_{\text{h}}{\text{atom}}^{-1}$; (c) final DC-MP2 buffer region with $e_{\text{thresh}}^{\text{corr}}=10\text{ μ}{E}_{\text{h}}{\text{atom}}^{-1}$.

Fig. 3.

 System-size dependence of the CPU time of the evaluation of $e_B^{\alpha }$ for polyacetylene chain system containing 2n carbon atoms C_2nH_2n+2. The initial inner and outer buffer sizes for the DC-HF calculation were fixed at $r_{\text{b}}^{\text{in}}$=5.0 Å and $r_{\text{b}}^{\text{out}}$= 6.5 Å, respectively.

Fig. 4.

  cis-trans isomerization of polyacetylene C₁₀₀H₁₀₂.

Tables

Table 1.  Initial buffer-size dependence of the automated DC-HF energy for the Trp-cage system with 6-31G* basis set.

$r_{\text{b}}^{\text{in}}/\text{Å}$	$r_{\text{b}}^{\text{out}}/\text{Å}$	Energy /Eh	(Diff.)/μEh atom−1	$⟨l_{\text{local}}^{\text{HF,}\alpha }⟩$ /Å
4.0	5.0	−7439.551778	(+0.04)	9.79
4.5	5.5	−7439.551761	(+0.09)	9.65
5.0	6.0	−7439.551722	(+0.23)	9.64
Standard HF		−7439.551790

Table 2.  Buffer-size dependence of the actual and estimated errors in the one-electron part of the Hellmann-Feynman term of the HF energy gradient of α-helix glycine oligomer Gly₁₀.

$r_{\text{b}}^{\text{in}}$	$r_{\text{b}}^{\text{out}}$	MaxAD/Eh bohr−1			MAD /Eh bohr−1
/Å	/Å	actual	estimated		actual	estimated
3.5	4.5	0.5021	0.2697		0.1032	0.0208
4.0	5.0	0.5795	0.0998		0.0808	0.0072
4.5	5.5	0.0664	0.0076		0.0107	0.0007
5.0	6.0	0.0201	0.0010		0.0037	0.0001
5.5	6.5	0.0160	0.0002		0.0031	0.0000

Table 3.  Buffer-size dependence of the actual and estimated errors in the HF energy and the Pulay term of the HF energy gradient ofα-helix glycine oligomer Gly₁₀. The standard HF energy is −2142.6879 E_h.

$r_{\text{b}}^{\text{in}}/\text{Å}$	$r_{\text{b}}^{\text{out}}/\text{Å}$	Energy error /Eh			MaxAD of Pulay term/Eh bohr−1			MAD of Pulay term/Eh bohr−1
		Actual	estimated		actual	Estimated		actual	estimated
3.5	4.5	+0.0553	+0.0540		0.0265	0.0317		0.0043	0.0027
4.0	5.0	−0.0397	−0.0511		0.0188	0.0209		0.0026	0.0014
4.5	5.5	+0.0027	+0.0022		0.0021	0.0007		0.0003	0.0002
5.0	6.0	+0.0002	−0.0007		0.0007	0.0007		0.0001	0.0001
5.5	6.5	+0.0001	−0.0003		0.0006	0.0003		0.0001	0.0000

Table 4.   cis-trans isomerization energy of polyacetylene C₁₀₀H₁₀₂ evaluated at (DC-)MP2/6-31G* level.

DC-MP2 method	Total energy /Eh		Isomerization energy
	trans	cis	/ kcal mol−1
$r_{\text{b}}^{\text{MP2}}$= 6 Å	−3858.088	−3857.721	229.9
$r_{\text{b}}^{\text{MP2}}$= 8 Å	−3858.094	−3857.721	234.0
$r_{\text{b}}^{\text{MP2}}$= 10 Å	−3858.098	−3857.722	235.8
$e_{\text{thresh}}^{\text{corr}}$= 10 μEh	−3858.094	−3857.721	234.1
$e_{\text{thresh}}^{\text{corr}}$= 0.1 μEh	−3858.098	−3857.722	235.9
Standard MP2	−3858.100	−3857.722	237.2

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