Chinese Journal of Chemical Physics  2016, Vol. 29 Issue (6): 657-662

The article information

Wei-xiu Pang, Yun-bin Sun, Jian-jun Zhao, Yi Lu
庞伟秀, 孙运斌, 赵建军, 鲁毅
Ab initio Study of Anharmonic Force Field and Spectroscopic Constants for Germanium Dichloride
Chinese Journal of Chemical Physics, 2016, 29(6): 657-662
化学物理学报, 2016, 29(6): 657-662

Article history

Received on: April 14, 2016
Accepted on: June 23, 2016
Ab initio Study of Anharmonic Force Field and Spectroscopic Constants for Germanium Dichloride
Wei-xiu Pang, Yun-bin Sun, Jian-jun Zhao, Yi Lu     
Dated: Received on April 14, 2016; Accepted on June 23, 2016
Key Laboratory of Magnetism and Magnetic Materials at Universities of Inner Mongolia Autonomous Region, Department of Physics, Baotou Normal University, Baotou 014030, China
*Author to whom correspondence should be addressed. Yun-bin Sun,
Abstract: Ab initio study of the equilibrium structure, spectroscopy constants, and anharmonic force field for several isotopomers of germanium dichloride (70GeCl2, 72GeCl2, and 76GeCl2) have been carried out at the MP2 and CCSD (T) levels of theory using cc-pVTZ basis set. The calculated geometries, rotational constants, vibration-rotation interaction constants, harmonic frequencies, anharmonic constants, quartic and sextic centrifugal distortion constants, cubic and quartic force constants are compared with experimental data. For small mass differences of the Ge isotopes, the isotopic effects for germanium dichloride are much weaker. The agreements are satisfactory for these two methods, but the deviations of CCSD (T) results are slightly larger than that of MP2, because of CCSD (T)'s inadequate treatment of electron correlation in hypervalent Cl atom.
Key words: Anharmonic force field    Spectroscopic constants    Germanium dichloride    

Transient molecules containing silicon and germanium have attracted considerable attention recently because of their roles as important intermediates in the technologically important production of semiconductors. In order to optimize the manufacturing processes, it is crucial to establish sensitive methods for detecting and characterizing such intermediates in theory and experiment. High resolution infrared and microwave spectroscopy are the most important sources of information for molecular force fields. Given the anharmonic potential, the molecular vibrational and rotational energy levels are easily obtained using variational or perturbation theories. The most useful analytical methods for studying semiconductor growth processes are likely to be spectroscopic in nature. Since its first observation in the late 1960s, the electronic absorption spectrum [1, 2] of germanium dichloride has attracted several experimental [3-6] and theoretical [6-9] attention.

The potential function under the influence of which the nuclei are moving is the same for isotopic molecules, since these isotopologues have the same electronic structure. But because of the difference in the masses the vibrational frequencies are different. A study of isotope shifts in spectra of germanium dichloride may be helpful in ascertaining which molecule (or radical) is responsible for a given spectrum and it may also aid in the spectrum. In addition, an accurate measurement of the isotope effect can be used to obtain a precise value for the ratio of the masses of the two kinds of isotopic atoms concerned. Under favorable conditions, the accuracy of the ratio of the masses so obtained is comparable with the accuracy of mass-spectrographic values. Apart from that, the study of the isotope effect in electronic band spectra has led to the discovery of new isotopes and an unambiguous confirmation of the quantum mechanical formula for the energy levels of the oscillator.

Employing the given force field, one can determine all spectroscopic constants of molecule, such as harmonic constants, anharmonic constants, rotational constants, centrifugal distortion constants, rotation-vibration interaction constants, etc. [10]. Recently, with the development of the method of analytic second derivatives of molecular energy, it has become possible to calculate the rovibrational spectra, harmonic or anharmonic force field of small or middle molecules by ab initio method [11-15]. It has been shown that spectroscopic constants from accurate, purely ab initio anharmonic force field are reliable [12-16].

As far as we know, there have been very few reports about the anharmonic force field and the isotopic effects of GeCl2. In 1995, Masaki et al. observed the millimeter-wave spectrum of GeC12 and its isotopic species in the ground and vibrationally excited states, $\nu_1$ , $\nu_2$ , $2\nu_2$ , and $\nu_3$ [17]. In 2000, Vast et al. presented a theoretical method to investigate the effects of isotopic disorder on the lattice dynamics of crystals [18]. Employing the method, they calculated the isotopic effect of Ge. In 2007, Pang et al. calculated spectroscopic constants of 74GeCl2 using three levels of theory [19]. To complete this series of germanium dichloride and the isotopic effects, it would be interesting to determine the anharmonic force field and spectroscopic constants of 70GeCl2, 72GeCl2, and 76GeCl2.

In this work, we determined the equilibrium structure, and calculated the ab initio anharmonic force field at several levels of theories. Accuracy was checked by comparing the ab initio spectroscopic constants with their corresponding experimental values. We also discussed the isotopic effects on anharmonic force field and spectroscopic constants for four isotopologues.


The geometry optimizations calculations presented below have been carried out with the Gaussian 03 program [20], and the second-order M∅ller-Plesset perturbation theory [21] (MP2) has been used. At the computed equilibrium geometries, harmonic force fields were evaluated analytically. No orbital has been kept frozen during these calculations. Cubic and semidiagonal quartic normal coordinate force constants have been evaluated using the ACESII [22] program package for the coupled cluster (CC) theory with single and double excitations augmented by a perturbational estimate of the effects of connected triple excitations (CCSD (T)) method [23], while the Gaussian 03 program for the MP2 calculations. Correlation consistent basis set has been used in the geometry optimizations and anharmonic force field, the standard valence cc-pVTZ basis has been employed for these isotopomers of germanium dichloride [24]. The electronic configuration describes the distribution of electrons of an atom in atomic orbitals. Considering the ground electronic configuration of Ge and Cl can be written as 1s22s22p63s23p63d104s24p2 and 1s22s22p63s23p5, respectively, the cc-pVTZ basis set is employed. The basis of Ge is the [6s5p3d1f] contraction of a (20s13p9d1f) primitive set [24]. The basis of Cl is the [5s4p2d1f] contraction of a (15s9p2d1f) primitive set [24]. This basis set has also been chosen as it was felt that the basis set enhancement beyond cc-pVTZ does not lead to major improvements in the computed anharmonic properties and that polarized triple-zeta basis sets should therefore be sufficient in such applications. For each isotopomer the cubic force field has been used to compute spectroscopic parameters using the usual second-order perturbation theory.


For the main isotopic species the theoretical results for the molecular geometries, the spectroscopic constants, and the full quartic force fields are calculated. They are compared with the corresponding experimental or empirical data [6, 9, 17, 25] whenever these are available. While comparison of anharmonic force constants is possibly the most meaningful way to compare different force field representations of the PES of a molecule, it is more usual to compare the anharmonic force fields by their ability to predict standard rovibrational spectroscopic constants [26]. The isotopic composition for chlorine are 75.8% of 35Cl and 24.2% of 37Cl, and the major isotopic species for germanium have a more complex composition with 20.5% of 70Ge, 27.4% of 72Ge, 36.5% of 74Ge, 7.8% of 76Ge [27].

We calculated the ab initio structures and total energy for the optimized geometries of these molecules, at the MP2 and CCSD (T) level of theory using cc-pVTZ basis set. The computed equilibrium structures of GeCl2 and experimentally derived [17] equilibrium structure are shown in Table Ⅰ. The MP2 results are almost the same values for these isotopomers, due to the fact that the equilibrium internuclear distances are the same in these isotopic molecules but not the effective internuclear distances [10]. So we listed the equilibrium structures of 74GeCl2 instead of other isotopomers. The previous computed results are also listed in Table Ⅰ [6, 9, 25]. From Table Ⅰ we can see that the total energy for the optimized geometries of GeCl2 calculated by CCSD (T) method is slightly lower than that of MP2 method (about 0.13 Hartree), which means that the optimized structure by CCSD (T) method is more stable.

Table Ⅰ Molecular equilibrium geometries including bond lengths R, bond angles A, and total energies ET of GeCl2.

The refined rotational parameters can be used to determine an improved geometry for germanium dichloride. Table Ⅱ lists the computed rotational constants for 70GeCl2, 72GeCl2, 74GeCl2, and 76GeCl2 at MP2 and CCSD (T) methods with cc-pVTZ basis set, along with the available experimental data [17]. The theoretical ground-state rotational constants (A0, B0, C0) have been obtained from the associated equilibrium constants ( $A_\textrm{e}$ , $B_\textrm{e}$ , $C_\textrm{e}$ ) by taking into account of the effects of vibration-rotation coupling via perturbation theory [28, 29]. As far as it goes, no experimental data of equilibrium rotational constants have been reported for 70GeCl2, 72GeCl2, and 76GeCl2. The deviations between the experimental values and the computed values at the equilibrium geometry are within 0.02%-0.25% (in absolute value).

Table Ⅱ Rotational constants of equilibrium and ground states for GeCl2.

The calculated vibration-rotation interaction constants $\alpha_i^X$ (with X=A, B, C and i=1, 2, 3) for these major isotopes are collected in Table Ⅲ, together with the corresponding experimental data for 74GeCl2 [17]. Comparing the experimental data with the calculated data, the MP2 results generally yield vibration-rotation constants with the correct sign and with reasonable magnitude. It is easy to find that the error between the calculated value of $\alpha_i^\textrm{C}$ and the corresponding experimental data is very big. The reason is that there is an important Coriolis interaction between the vibrational states $\nu_1$ and $\nu_3$ in GeCl2. The calculated value of $\alpha_i^\textrm{C}$ in Table Ⅲ does not take this interaction into account, However, the experimental data have contained it [17]. For this reason, the direct comparison with the experimental α constants is not immediate. It is more informative to compare the values of $B_0^X - B_{\rm{e}}^X $ = $\displaystyle{\frac{1}{2}\sum {\alpha _{\rm{i}}^X }}$ because the interaction was mostly cancelled out. As shown in Table Ⅲ, good agreement with experimental values is obtained for the B and C rotational constants. However, for A0- $A_{\rm{e}}$ , the experimental value, -15.665 MHz, is smaller than the computed values (2.06 and 2.29 MHz). The deviations between CCSD (T) results and experimental values are somewhat larger than MP2 results, especially for $\alpha _1^\textrm{A}$ and $\alpha _1^\textrm{C}$ . A possible reason is that CCSD (T) theory has difficulties in describing electron correlation in the hypervalent Cl atom [30, 31].

Table Ⅲ Vibration-rotation interaction constants of GeCl2

Table Ⅳ presents the calculated harmonic and fundamental vibrational wavenumbers of the major isotopes of GeCl2 without any scaling factor. The heavier isotopic molecule has the smaller frequency. For small mass differences the frequencies of the isotopic molecules are close to those of the ordinary molecule. The isotope shifts of $\omega_3$ and $\nu_3$ are very small because the atom Ge which is replaced by its isotope moves very little in the particular normal mode, whereas the shifts of others are more larger because the atom Ge in question has a large amplitude in that normal mode. So the investigation of the isotope effects is invaluable in correlating the observed vibrational frequencies with the theoretical normal modes of GeCl2.

Table Ⅳ Harmonic and fundamental vibrational wavenumbers of GeCl2 (in cm-1)

Table Ⅴ presents the anharmonic constants $X_{ij}$ of the major isotopes of GeCl2. The theoretical fundamental wave numbers were derived from the corresponding harmonic wave numbers and anharmonicity constants. For molecules containing two equivalent bonds of the general formula XY2, the relations [32-34] relate some of the vibrational anharmonicity constants as follows:

${{X}}_{11} \approx {X}_{{\rm{33}}} \approx \frac{1}{4}{X}_{{\rm{13}}}$ (1)
Table Ⅴ Vibrational anharmonicity constants Xij of GeCl2

These relations are based on an approximate model, but they provide a useful check on the calculations because it has been observed experimentally that they hold well for some of the XY2 type molecule. As the appropriate entries of Table Ⅴ testify, the interrelations presented in the Eq.(1). hold very well for the directly determined ab initio spectroscopic constants. So far, both the harmonic wave numbers and anharmonic constants of these molecules have not been deduced experimentally. In view of the limited experimental data on these constants, ab initio predictions of them are expected to be useful for future experimental work.

The experimental ground-state values and computed equilibrium quartic centrifugal distortion constants (A reduction [35]) of 70GeCl2, 72GeCl2, 74GeCl2, and 76GeCl2 are compared in Table Ⅵ. This permits to check the quality of the harmonic force field. The deviations between the experimental and ab initio are only a few percent, the largest deviation ( $>$ 10%) being for $d_{\rm{K}}$ as usual. It is worth noting that the deviations are almost the same (the percentage) as for 74GeCl2. It is likely that a large part of these deviations could be explained by the difference between ground state and equilibrium constants which values are known to be a few percent. The effects for these quartic centrifugal constants are much weaker, and the shift between our results and experiment is still very satisfactory.

Table Ⅵ Equilibrium quartic centrifugal distortion constants of GeCl2

The equilibrium sextic centrifugal distortion constants (A reduction [35]) for these isotopmers of GeCl2 calculated from the MP2/cc-pVTZ cubic force fields are shown in Table Ⅶ. It should be pointed out that the ACESII program used for the CCSD (T) calculations does not permit to calculate sextic centrifugal distortion constants and the comparison is thus restricted to the MP2/cc-pVTZ results for these isotopmers. The calculations generally yield the sextic centrifugal distortion constants with the correct sign and with reasonable magnitude. Although no experimental sextic centrifugal distortion constants are found in references, we believe the results of MP2/cc-pVTZ are reliable.

Table Ⅶ Sextic centrifugal distortion constants of GeCl2 (10-3 Hz) by MP2 method

The investigation of the vibrational frequencies of molecules isotopic with the one considered gives additional equations for the force constants. While the number of constants in the most general quadratic potential function is usually larger than the number of fundamental frequencies. Thus if only one isotopic species is observed not all the force constants can be evaluated unless simplifying assumptions are made. But with the help of the fundamental frequencies of one or more isotopic molecules a sufficient number of additional equations is in general obtained to determine all constants in the most general (quadratic) potential function. Table Ⅷ lists complete ab initio quartic force fields in normal coordinates for GeCl2. It includes the results of MP2 and CCSD (T) with cc-pVTZ basis set. From group theoretical arguments we find that, in its C $_{\textrm{2V}}$ ground electronic state, the germanium dichloride molecule has two geometric variables, and 4 quadratic, 6 cubic, 9 quartic, 12 quintic, and 16 sextic force constants. By definition, the quadratic normal coordinate constants correspond to the harmonic wave numbers. All cubic force constants are predicted to be negative by MP2 level of theory. Therefore, the fact that $f_{211}$ and $f_{222}$ have been predicted to be positive in CCSD (T) results should be regarded as weaknesses in the CCSD (T) calculation rather than indications that there were problems in the MP2 calculations. It is also noteworthy that the cubic constants $f_{221}$ and $f_{233}$ are determined to be much smaller in absolute magnitude than other force constants. So, only the constants higher than 1 cm-1 are reported in this Table. In the quartic part of the force field pronounced deviations exist for some of the constants. First of all, it is noteworthy how different the calculated $f_{3333}$ constants are. This is probably due not only to theory effects but also to higher-order contamination and truncation effects during the determination. Finally, it is interesting to note that the CCSD (T) results do not seem follow any simple rule.

Table Ⅷ Quadratic, cubic, and quartic force constants of GeCl2 in normal coordinates (cm-1)

In conclusion, we present some results to the isotopic effects on GeCl2. The isotopic effects for germanium dichloride are much weaker. All spectroscopic constants calculated with this approach are in good agreement with the available experimental results, proving the high quality of the underlying anharmonic force field and the viability of the perturbation-resonance approach at the same time. The MP2 results are in excellent agreement with the available experimental data for main isotopomers, so that the MP2 predictions for spectroscopic constants yet unknown are expected to be reliable.

Those isotopic molecules are also symmetrical, because the central atom is replaced by an isotope. As an asymmetric top molecule, the effective Hamiltonian in A reduction [35] can be written as

$\begin{align} & {{{\hat{H}}}^{\text{A}}}={{A}_{K}}J_{z}^{2}+{{A}_{J}}{{J}^{2}}+\frac{1}{2}a(J_{+}^{2}+J_{-}^{2})-{{D}_{K}}J_{z}^{4} \\ & -{{D}_{JK}}{{J}^{2}}J_{z}^{2}-{{D}_{J}}{{J}^{4}}\text{+}{{H}_{K}}J_{z}^{6}+{{H}_{KJ}}{{J}^{2}}J_{z}^{4}+ \\ & {{H}_{JK}}{{J}^{4}}J_{z}^{2}+{{H}_{J}}{{J}^{6}}+\frac{\text{1}}{2}[-{{d}_{K}}J_{z}^{2}-{{d}_{J}}{{J}^{2}}+ \\ & {{h}_{K}}J_{z}^{4}+{{h}_{JK}}{{J}^{2}}J_{z}^{2}+{{h}_{J}}{{J}^{4}}, J_{+}^{2}+J_{-}^{2}{{]}_{+}} \\ \end{align}$ (2)
${{A}_{K}}=\frac{A-(B+C)}{2}$ (3)
${{A}_{J}}=\frac{B+C}{2}$ (4)
$a=\frac{B-C}{2}$ (5)

here $[, ]_ +$ represents the positive commutator. Submitting the parameters in Table Ⅵ and Table Ⅶ into the above equation, one can obtain the computed rotational energy of molecule in the study, which is necessary to assign the observed rotational energy of molecule.


This work was supported by the National Natural Science Foundation of China (No.51562032 and No.61565013) and the Inner Mongolia College Research Project (No.NJZZ13220).

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庞伟秀, 孙运斌, 赵建军, 鲁毅     
包头师范学院物理科学与技术学院, 内蒙古自治区磁学与磁性材料重点实验室, 包头 014030
摘要: 用从头算方法的MP2和CCSD(T)方法结合cc-pVTZ基组计算了二氯化锗同位素(70GeCl272GeCl276GeCl2)分子的平衡结构、光谱常数和非谐振力场.二氯化锗的几何结构、转动常数、振转相互作用常数、谐频、非谐振常数、四次和六次离心畸变常数、三次和四次力常数的计算结果与实验结果符合较好,二氯化锗分子的同位素效应较小,可能的原因是Ge同位素的质量变化相对较小.两种方法计算的结果均与实验结果符合,但CCSD(T)方法比MP2计算结果的偏差稍大一些,可能的原因是CCSD(T)方法在描述过共价Cl原子的电子相关时不够充分.
关键词: 非谐振力场    光谱常数    GeCl2分子