The article information
 Jiang Yi, Zhongfu Xie, Feiwu Chen
 易江, 谢忠甫, 陈飞武
 Theoretical Study on Divergence Problems of Single Reference Perturbation Theories
 单参考态微扰理论发散问题的理论研究
 Chinese Journal of Chemical Physics, 2019, 32(5): 597602
 化学物理学报, 2019, 32(5): 597602
 http://dx.doi.org/10.1063/16740068/cjcp1904083

Article history
 Received on: April 28, 2019
 Accepted on: June 6, 2019
Beijing Key Laboratory for Science and Application of Functional Molecular and Crystalline Materials, Beijing 100083, China
Since the work of Møll and Plesset (MP) in 1934 [1], MP perturbation theory has been widely used in quantum chemistry for closedshell atomic and molecular systems. Due to its great success, the single reference RayleighSchrödinger perturbation theory has also been extended to openshell systems [213]. Among them, to list a few, there are unrestricted MP (UMP), the single reference openshell version of CASPT2 [2], the restricted openshell MøllerPlesset theory (ROMP) of Amos et al. [3], OPT1 and OPT2 of Murray and Davidson [4], the restricted MøllerPlesset theory (RMP) of Knowles et al. [5], the Zaveraged perturbation theory (ZAPT) of Lee and Jayatilaka [7], and the openshell perturbation theory (OSPT) of Chen [10]. Lee et al. [11] compared the performances of OPT1, OPT2, RMP, ROMP and ZAPT. Formaldehyde (H
Divergence problems are often unavoidable in perturbation theories [14] as well as coupled cluster theories [15, 16]. For both closedshell and openshell systems, if the eigenvalues of zeroth order Hamiltonian become degenerate or quasidegenerate, the perturbation series will be divergent even at lower orders. It is usually thought these problems may be solved with the socalled multireference perturbation theories [1731] such as the CASPT2 [2], the blockcorrelated secondorder perturbation theory of Li et al. [26], the valence bond perturbation theory of Wu et al. [27], and the multireference RayleighSchrödinger perturbation theory of Yi and Chen [31].
In 1996, Olsen et al. found a surprising divergent case of MP perturbation theory due to the addition of diffuse functions [32], in which the MP series are divergent at high orders. Olsen et al. attempted to explain this divergent problem with a simple twostate model [33]. Larsen et al. [34] reported subsequently the divergent behaviors of molecular electric dipole moments of HF calculated with MP perturbation theory at a stretched geometry in the ccpVDZ basis set. Similar highorder divergence cases without addition of diffuse functions for molecules at stretched geometries were also investigated by Chen et al. [13], Murray et al. [35], Wheel et al. [36], and Leininger et al. [37]. Finley proposed the socalled maximum radius of convergence perturbation theory to yield rapid convergence of RayleighSchrödinger perturbation series [38]. Yokoyama et al. tested Finley's theory with Be, Ne, H
Except the work on the doublet states of NO and CN, and the triplet state of O
In RayleighSchrödinger perturbation theory [45], the Hamiltonian
$\begin{eqnarray} \hat H = \hat H_0 + \hat V \end{eqnarray} $  (1) 
where
$ \begin{eqnarray} \hat H_0 \psi _i^{(0)} = E_i^{(0)} \psi _i^{(0)} \end{eqnarray} $  (2) 
where
$ \begin{eqnarray} {\rm{ }}E_i^{(n)} = \left\langle {\psi _i^{(0)} \left {\hat V} \right\psi _i^{(n  1)} } \right\rangle \end{eqnarray} $  (3) 
with
$\begin{eqnarray} \left(\hat H_0  E_i^{(0)} \right)\left {\psi _i^{(n)} } \right\rangle \hspace{0.1cm} = \hspace{0.1cm} \sum\limits_{k = 1}^n {E_i^{(k)} \left {\psi _i^{(n  k)} } \right\rangle } \hspace{0.1cm} \hspace{0.1cm} \hat V\left {\psi _i^{(n  k)} } \right\rangle \end{eqnarray} $  (4) 
where
In order to calculate the perturbation energy as shown in Eq.(3),
$ \begin{eqnarray} \hat H_0 = \sum\limits_{i = 1}^N {\hat F_i } \end{eqnarray} $  (5) 
where
$\begin{eqnarray}  \frac{{\hat F_\alpha }}{2} + \frac{{3\hat F_\beta }}{2}, \frac{{\hat F_\alpha }}{2} + \frac{{\hat F_\beta }}{2}, \frac{{3\hat F_\alpha }}{2}  \frac{{\hat F_\beta }}{2} \nonumber \end{eqnarray} $ 
where
Feenberg transformation [40, 41] is one of effective techniques to treat the divergence problems for perturbation series. In comparison with Eq.(1), the Hamiltonian
$ \begin{eqnarray} \hat H = \frac{1}{{1  \lambda }}\hat H_0 + \left(\hat V  \frac{\lambda }{{1  \lambda }}\hat H_0\right) \end{eqnarray} $  (6) 
where
$ \begin{eqnarray} E_\lambda ^{(n)} \hspace{0.15cm} &=&\hspace{0.15cm} (1  \lambda )\sum\limits_{k = 0}^{n  2} {\left( {\begin{array}{*{20}c} {n  2} \\ k \\ \end{array}} \right)} \lambda ^{n  2  k} (1  \lambda )^k E_i^{(k + 2)}, \nonumber\\ && \hspace{0.15cm}(n\geq2) \end{eqnarray} $  (7) 
Eq.(3) and (7) will be exploited to discuss the convergence behaviors of series
In this work, two types of basis sets are used, which are 631+G [47] and augccpVDZ [48]. The molecules under study are the singlet states of N
Olsen et al. reported that the MP perturbation series of Ne and HF systems were divergent due to the addition of diffuse functions. The number of electrons in these two systems is 10. We found numerically for small systems that the MP perturbation series is divergent only for systems with 10 electrons if diffuse functions are added. For examples, the MP perturbation series of Li
In contrast to the closedshell systems discussed in the subsection above, it seems that there are more divergence cases for openshell systems. The open systems are the doublet states of F, OH and NH
The convergence behaviors of the quadruplet state of N are shown in FIG. 5. In comparison with the doublet state of NH
The triplet state of C is chosen for the present study on the basis set effect. In this case 631+G and augccpVDZ are employed for investigation. The convergence behaviors of perturbation series of OSPT and Feenberg for the two basis sets are presented in FIG. 6 and FIG. 7, respectively. As can be seen from these two figures, it seems that the basis set effect on the OSPT series can be ignored. However, the basis set effect on the Feenberg series is more significant. For the same value of parameter
As discussed in the subsections above, the Feenberg transformation plays an important role in improving the convergence of MP and OSPT series. For various values of the parameter
In this work, two closedshell systems (N
Feenberg transformation is one of effective techniques to accelerate the convergence speed of MP and OSPT series. It is found for both closedshell and openshell systems that there always exists a minimum perturbation order
This work was supported by the National Natural Science Foundation of China (No.21473008 and No.21873011).
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