The article information
 Wenyan Zhang, Feiwu Chen
 张文彦, 陈飞武
 Iterative Multireference Configuration Interaction
 近邻层对相邻层界面能的影响
 Chinese Journal of Chemical Physics, 2019, 32(6): 701707
 化学物理学报, 2019, 32(6): 701707
 http://dx.doi.org/10.1063/16740068/cjcp1905094

Article history
 Received on: May 15, 2019
 Accepted on: July 22, 2019
Single reference wavefunction theories beyond HartreeFock approximation such as single reference perturbation theory [111] and coupled cluster theory [1215] provide accurate ways to calculate the molecular ground state energy as well as the molecular properties [1618] at the geometry near equilibrium. However, these single reference theories will break down if the molecule of interest is at quasidegenerate excited states or far from its equilibrium geometry such as cases of the formation or breaking of chemical bonds. For these situations, the wavefunction used to describe the molecular electronic structure must have at least some multireference characteristics. Multireference configuration interaction theories (MRCI) [1934] are developed along this line.
Allelectron full configuration interaction (FCI) is size extensive and orbitalinvariant [35, 36]. Its computational results are often regarded as a calibration for other wavefunction models. However, due to its large computational cost, its application is limited to small molecules. Werner and Knowles proposed internal contracted MRCI to reduce the computation cost with the accuracy loss almost negligible [21]. Feller proposed an iterative approach to estimate the FCI energies at complete basis set limit in 1993 [22]. Recently Liu and Hoffmann [26, 27] discussed possible ways to obtain the electronic correlation energy in strong correlated systems and proposed a novel iterative configuration interaction approach (iCI) whose energy converges quickly and monotonically from above to the FCI. Liu and Hoffmann [27] also discussed the relationship of iCI with the ICI (iterative configuration interaction) theory proposed by Nakatsuji et al. [3739]. The heatbath CI method of Umrigar and coworkers [3133] is also similar to these approaches. There are two key stages in heatbath CI method: generating the variational wave function and energy, and computing the perturbative energy correction [3133].
Despite the huge progress made in multireference theories, one often ignored problem is the correct guess of initial reference states, which is in general unknown in advance except the reference state at equilibrium. A correct guess of the reference states is essential for the multireferene theoretical models such as MRCI to get results with a high accuracy. Due to this reason and also encouraged by the promising results presented by Liu and Hoffmann [26, 27], an iterative multireference configuration interaction (IMRCI) with a different updating procedure is proposed in this work. Because of huge computational cost of FCI, the purpose here is to investigate how to achieve certain target accuracy such as 10
Multireference configuration interaction starts with a set of
$\begin{eqnarray} \{\Phi_1, \Phi_2, \Phi_3, \ldots, \Phi_p\}\nonumber \end{eqnarray} $ 
These reference functions may be a combination of Slater determinants with the expected electronic multiplicity or the socalled configuration state functions. They are supposed to have important contributions to the true wavefunction of an atomic or molecular system under consideration. All possible single and double excitations from these reference functions are carried out to construct the rest of the configuration functions. The wavefunction of the system is supposed to be a linear combination of these configuration functions, the combination coefficients and the corresponding electronic energy are then determined variationally. This is the socalled MRCI single and doubles. For simplicity, it is abbreviated as MRCI.
The effectiveness of MRCI depends on the choice of
In order to improve the efficiency and accuracy, IMRCI is proposed. The iteration procedure is expressed in three steps as follows:
(ⅰ) Choose an initial set of preference functions with scheme (Ⅰ):
$ \begin{eqnarray} \{\Phi_1^{(i)}, \Phi_2^{(i)}, \ldots, \Phi_p^{(i)}\}\nonumber \end{eqnarray} $ 
where
(ⅱ) Perform an MRCI calculation and determine expansion coefficients of configuration functions variationally.
(ⅲ) Select a new set of
If a better accuracy is required, a relatively larger size of the reference space should be considered and the procedure starting from the step (ⅰ) to (ⅲ) has to be repeated until the selfconsistency about the reference functions is reached.
Ⅲ. RESULTS AND DISCUSSIONIn this preliminary test, H
Three types of H
Computational results of MRCI and IMRCI for H
The results of H
The relative errors of MRCI with CAS(4, 4) and CAS(6, 6), i.e.
The results of H
In comparison with the results listed in Tables Ⅰ and Ⅱ, the accuracy between the results of MRCI and IMRCI in Table Ⅲ is very large. It is found that the leading electron configurations in MRCI calculations are the same, i.e., 10 electrons of H
Since the change of the leading configuration on the potential energy surface is in general unknown in advance, the reference state of the single reference models such as MP2, MP3, CCSD and CCSD(T) on the potential energy surface is always the leading electron configuration of the ground state at equilibrium. Therefore, once the leading electron configuration on the potential energy surface was changed, there would be a corresponding jump on the potential energy surface computed with MP2, MP3, CCSD and CCSD(T). The potential energy curves of H
Methylene is a wellknown model molecule tested by various theoretical approaches [9, 28, 48]. The geometries of the singlet and triplet states of CH
The potential energy curves of the singlet and triplet states of CH
N
MRCI is a wellestablished and reliable configuration interaction approach applicable for the electronic ground state as well as the excited states. It is especially useful for molecules at equilibrium although its slow convergence to the FCI results is often observed. The MRCI accuracy depends strongly on the initial guess of the configuration functions in the reference space. A wrong initial guess which often occurs for molecules far from its equilibrium geometry will lead to very low MRCI accuracy. Similar to iCI [26, 27] and ICI [3739], preliminary test results presented here demonstrated that IMRCI can improve the accuracy of MRCI for molecules both at equilibrium and nonequlibrium geometries with a few iterations due to the fact that the most important electronic configurations can be selected into the reference space through the iterative procedure presented here. It is often thought that the problems of the single reference theoretical models on the potential energy surfaces can be solved with the multireference theoretical methods. However, this is not true since the multireference models are strongly dependent on the guess of initial multireference wavefunctions which, on the other hand, is in general unknown in advance. As we discussed for H
This work was supported by the National Natural Science Foundation of China (No.21473008 and No.21873011).
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