An Accurate Calculation of Potential Energy Curves and Transition Dipole Moment for Low-Lying Electronic States of CO

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1 Commun. Theor. Phys. 59 (2013) Vol. 59, No. 2, February 15, 2013 An Accurate Calculation of Potential Energy Curves and Transition Dipole Moment for Low-Lying Electronic States of CO LU Peng-Fei ( ì), 1, YAN Lei ( ), 1 YU Zhong-Yuan ( ), 1 GAO Yu-Feng (Ô ô), 2 and GAO Tao (Ô ) 2 1 Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), Ministry of Education, P.O. Box 49 (BUPT), Beijing , China 2 Institute of Atomic and Molecular Physics, Sichuan University, Chengdu , China (Received September 3, 2012; revised manuscript received November 26, 2012) Abstract In this paper, potential energy curves for the X 1 Σ +, a 3 Π, a 3 Σ +, d 3, A 1 Π and I 1 Σ states of CO have been calculated using complete active space self-consistent field and multi-reference configuration interaction methods. The calculations have been performed at 108 nuclear separations from 0.7 to 4.0 Å by the aug-cc-pv5z basis set. Spectroscopic constants for the six low-lying electronic states are found in good agreement with experimental data. The vibrational states of the X 1 Σ + and A 1 Π states are also calculated, which are reliable and accurate by comparison with the experimental data and the other theoretical values. The transition dipole moment (TDM) shows that the TDM of the two states (X 1 Σ + A 1 Π) are reduced strongly with increase of bond length. PACS numbers: Vn, jp, Bc Key words: potential energy curves, transition dipole moment, electronic state 1 Introduction Carbon monoxide (CO) is one of the most abundant diatomic molecules in the universe. It has been observed in comets, planetary atmospheres, and the photospheres of the sun and cooler stars. The electronic spectrum of carbon monoxide is one of the best experimentally studied systems. [1] A number of ab initio calculations and experimental investigations have been performed on carbon monoxide. [2 6] In 1988, Deleon [7] obtained the internuclear dependence of CO (A 1 Π X 1 Σ + ) transition moment for the range 1.0 < r < 1.8 Å by making laser induced fluorescence measurements on highly vibrationally excited CO. The ab initio calculations of Kirby and Cooper are in good agreement with the experimental work of Chan. [8] In 1993, Marcel Drabbels et al. [9] studied the transitions from the X 1 Σ + (v = 0) ground state of the CO to the electronically excited A 1 Π (v = 0), B 1 Σ + (v = 0), and C 1 Σ + (v = 0) states by 2-photon laser induced fluorescence spectroscopy. They also obtained accurate molecular constants for the B and C states. In 1994, Morton and Noreau, in their compilation of fundamental spectroscopic data for all CO transitions between 100 and 155 nm, provided a critically evaluated summary of the A 1 Π X 1 Σ + database. [10] In 1999, Spielfiedel et al. [11] calculated the band-integrated oscillator strengths of the CO A 1 Π X 1 Σ + transition for 0 v 23 and v = 0 1. In 2007, Varandas [12] carried out a detailed study to account for the small barrier in the potential energy curve, located at about 2.25 Å. The calculated height of the barrier is 594 cm 1, and the experimental data is 950±150 cm 1. [13] Most recently, Shi et al. [14] obtained eight accurate lowlying electronic states of CO by ab initio quantum chemical method by including the core-valence correlation and relativistic corrections. These advanced options result in more accurate spectroscopic constants for CO. Nevertheless Shi et al. [14] did not present transition dipole moments for the transition between electronic state A 1 Π and X 1 Σ +. The main goal of the present paper is to extend the previous studies by using multireference singles and doubles configuration interaction plus Davison correction (MR- CISD+Q). The potential energy curves (PECs) and the spectroscopic constants of the ground and low-lying excited states of CO molecule are calculated in detail. The transition dipole moments (TDMs) of the transitions from A 1 Π bound excited state to the ground X 1 Σ + state were also calculated. This paper is organized as follows. The computational methods are given in Sec. 2. In Sec. 3, the calculated results and discussions are presented. Our conclusions are summarized in Sec. 4 finally. 2 Computational Methods In the calculations, two types of basis set were employed. The correlation consistent polarized valence basis sets of Dunning and co-workers, [15 17] denoted by ccpv5z, had been used in the present work. In order to Project Supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China under Grant No Corresponding author, photon.bupt@gmail.com c 2013 Chinese Physical Society and IOP Publishing Ltd

2 194 Communications in Theoretical Physics Vol. 59 assess the effect of additional diffuse functions, the augmented correlation consistent set, [16] denoted by aug-ccpv5z, was also used. The augmented correlation consistent polarized valence quintuple zeta (aug-cc-pv5z) basis set was employed to obtain accurate potential energy curves (PECs) for the six bound states (X 1 Σ +, a 3 Π, a 3 Σ +, d 3, A 1 Π, I 1 Σ ) of CO. For low-lying electronic states of CO, complete active space self-consistent field (CASSCF) and multireference configuration interaction (MRCI) method with Davison size-extensivity correction (+Q) [18 20] were performed. Due to the limitation of symmetry of the MOLPRO package, [21] the calculations are performed under C 2v symmetry, where the Σ + is represented by A 1, Σ is represented by A 2, degenerate states and Π are represented by A 1 +A 2 and B 1 +B 2, respectively. The ground-state molecular orbitals (MOs) were calculated first using the Hartree-Fock self-consistent field (HF-SCF) method. The ground X 1 Σ + state of CO is characterized mainly by the close-shell electronic configuration 1σ 2 2σ 2 3σ 2 4σ 2 1π 4 5σ 2. In the CASSCF and subsequent MRCI calculations, the reference configurations were all electronic configurations generated from [1σ 2, 2σ 2, 3σ 0 2, 4σ 0 2, 1π 0 4, 5σ 0 2, 2π 0 4, 6σ 0 2 ]. The eight outermost MOs were selected as the active space, including four A1, two B1 and two B2 symmetry MOs. The 1s-like core orbitals were not correlated (frozen core approximation). The ten valence electrons of CO were placed in the active space and did not restrict the excitation type. Single point energies of different states calculated with the MOLPRO ab initio program. [21] The PECs of the low-lying bond states of CO are constructed. The vibrational energies and wave functions on each adiabatic potential energy curves are calculated by solving the one-dimensional nuclear Schrödinger equation with LEVEL8.0 program of Le Roy. [22] Assuming that the Morse potential is a good approximation at the bottom of the curves, we obtain the equilibrium internuclear distance (R e ), the rotational constant (B e ), the harmonic and anharmonic vibrational constants (ω e and ω e χ e ), and the adiabatic relative electronic energy referred to the ground state (T e ). 3 Results and Discussions 3.1 Potential Energy Curves and Spectroscopic Constants Potential energy curves for the X 1 Σ +, a 3 Π, a 3 Σ +, d 3, A 1 Π, I 1 Σ states of CO were calculated for 108 nuclear separations from 0.7 to 4.0 Å by the AV5Z basis set and are shown in Fig. 1. The six low-lying electronic states of CO are dissociated from the ground state C( 3 P, 2s 2 2p 2 ) with the ground state O( 3 P, 2s 2 2p 4 ). All the calculated potential energy curves properly converge to the correct dissociation limit. Equilibrium bond length (R e ), excitation energy (T e ), rotation constant (B e ), dissociation energy (D e ), and vibrational constants (ω e and ω e χ e ) are also calculated from the potential energy curves, and are summarized in Table 1. Fig. 1 The potential energy curves of the X 1 Σ +, a 3 Π, a 3 Σ +, d 3, A 1 Π, and I 1 Σ states of CO. The spectroscopic labels are shown for each of the potential wells. The CO (X 1 Σ + ) state is described at the SCF level of approximation by (1a 1 ) 2 (2a 1 ) 2 (3a 1 ) 2 (4a 1 ) 2 (5a 1 ) 2 (1b 1 ) 2 (1b 2 ) 2 in C 2v symmetry and by 1σ 2 2σ 2 3σ 2 4σ 2 5σ 2 1π 4 in C v symmetry. The primary electronic configuration for the A 1 Π and a 3 Π states are ( ) 3σ 2 4σ 2 5σ1π 4 2π, that is, the valence states A 1 Π and a 3 Π results from the 5σ 2π excitation. The a 3 Σ +, d 3, and I 1 Σ states are described by the configuration ( ) 4σ 2 5σ 2 1π 3 2π. In all of our CASSCF and CI calculations, the 1σ and 2σ orbitals (K shells of C and O) remain doubly occupied. For the X 1 Σ + state, the computed dissociation energy is ev compared with the experimental value [23] of ev and the theoretical result of Cooper and Kirby [4] of ev. Our calculated value of R e for the X 1 Σ + state is in error by Å. A fit to the first twenty vibrational levels in the potential curve gives ω e = cm 1 and ω e χ e = cm 1 compared with the experimental data [23] of and cm 1, respectively. For the a 3 Π state, the computed dissociation energy D e is ev, which is found in good agreement with the experimental value [1] of ev. Our computed T e for the state, i.e., cm 1 is also in good agreement with the experimental value, cm 1. The best calculated value is that of Shi et al. [14] of cm 1. Our computed value of R e for the a 3 Π state is in error by Å. From Table 1, it is apparent that the calculated values for ω e, ω e χ e, and B e are in good agreement with the experimental values too. Our computed T e for the a 3 Σ + state is cm 1 compared with the experimental value of cm 1

3 No. 2 Communications in Theoretical Physics 195 and the theoretical result of Shi et al. [14] of cm 1. The computed dissociation energy D e is 4.3 ev compared with the experimental value [1] of ev. Shi et al. [14] obtained a D e of ev, which is smaller than the experimental data. From Table 1, our calculated values for R e, ω e, ω e χ e, and B e are also in good agreement with the experimental values. For the d 3 state, the calculated and measured values of T e are in excellent agreement. The other spectroscopic constants (R e, D e, ω e, ω e χ e, and B e ) are compared with the corresponding experimental data in Table 1. Our calculated values for R e, D e, ω e, ω e χ e, and B e for the d 3 state are Å, ev, cm 1, cm 1, and cm 1, where the relative errors are 0.3%, 1.9%, 1.1%, 14%, 0.8%, respectively. The calculated T e value for the A 1 Π state in Table 1 is cm 1, compared with an experimental value of cm 1. A lower relative energy position ( cm 1 ) of the minima of the A 1 Π adiabatic curve was obtained by Vázquez et al. [24] From Table 1, it is apparent that our computed values for R e, D e, ω e, ω e χ e, and B e for the A 1 Π state are all in good agreement with the experimental data. Our calculated R e for the I 1 Σ state is Å compared with the experimental value of Å and the theoretical result of Å of Vázquez et al. [24] The computed vibrational constants (ω e and ω e χ e ) for the I 1 Σ state are cm 1 and cm 1 compared with the experimental values of cm 1 and cm 1, respectively. A fit to the potential curve of the I 1 Σ state gives B e = cm 1, D e = ev, and T e = cm 1 compared with the experimental values of cm 1, ev, and cm 1, respectively. Table 1 Comparison of experimental and theoretical spectroscopic constants for CO. States R e/å ω e/cm 1 ω eχ e/cm 1 B e/cm 1 D e/ev T e/cm 1 X 1 Σ Ref. [23] Ref. [4] Ref. [25] Ref. [11] Ref. [14] Expt. [1] a 3 Σ Ref. [23] Ref. [14] Expt. [1] A 1 Π Ref. [4] Ref. [25] Ref. [11] Ref. [14] Expt. [1] a 3 Π Ref. [14] Expt. [1] d Ref. [14] Expt. [1] I 1 Σ Ref. [14] Expt. [1] Vibrational and Transition Dipole Moment Figure 2 shows the vibration term energy for the X 1 Σ + state, plotted against the vibrational quantum number v. The quality of computed results is assessed in Table 2 for a number of vibrational energy levels for the X 1 Σ + state. The energy of the vibrational ground state (v = 0) is cm 1, the first vibrationally excited state lies above the ground state by cm 1 having an energy of cm 1, while the last bound vibrational level (v = 74) below the dissociation limit has an energy of ev. Black and white circles are the experimental and calculated results, respectively. As Fig. 2 shows, the computed vibrational levels G(v) are in good agreement with the experimental values.

4 196 Communications in Theoretical Physics Vol. 59 Figure 3 shows the v dependence of vibrational level G(v) of the A 1 Π state. The quality of computed results is assessed in Table 3 for a number of vibrational energy levels for the A 1 Π state. The energy of the first vibrational level (v = 0) is cm 1, and the corresponding experimental value is cm 1. [27] The calculated last bound vibrational level (v = 23) below the dissociation limit has an energy of 3.15 ev, which is very close to the dissociation energy (D e = ev) of the A 1 Π state. In our calculations, we computed all the 24 possible vibrational levels (v = 0, 1, 2,..., 23) of CO (A 1 Π). Obviously, the present G(v) result is in good agreement with these experimental data. [27] in Table 4 and compared with the theoretical results of Kirby & Cooper (1989), Chantranupong et al. (1992), and Spielfiedel et al. (1999) in Fig. 4. Table 2 Comparison of the present G v results with the experimental data for the X 1 Σ + state of CO. G v/cm 1 G v/ev v Pw Exp. [2] Pw Exp. [2] The configuration interaction wave functions for the ground state X 1 Σ + and the low-lying excited electronic state A 1 Π of CO were constructed from a common set of valence and virtual orbitals (four A1, two B1 and two B2 symmetry molecular orbitals) in order to complete the calculation of electric dipole transition moments between the two states. A very small step in R (i.e Å for 1.12 R Å) has been used in order to represent correctly the variation of the transition moment near the equilibrium distance of the ground state X 1 Σ + and A 1 Π state. The calculated values are summarized Fig. 2 The vibrational levels of the ground state X 1 Σ +, plotted against the vibrational quantum number v. Black and white circles are the experimental and calculated results, respectively. Expt: experimental data. Pw: present work. Fig. 3 The vibrational levels of the A 1 Π state, plotted against the vibrational quantum number v. Black and white circles are the experimental and calculated results, respectively. Expt: experimental data. Pw: present work. Figure 4 shows the transition dipole moment (TDM) between the X 1 Σ + and A 1 Π states. As shown in Fig. 4, the ab initio dipole transition moment remains positive throughout, but tends asymptotically to zero at large R. The results by Kirby & Cooper, and Chantranupong et al. are larger than our calculated values and those obtained by Spielfiedel et al. Within the calculational error, our calculations give very similar results with those in Ref. [14] in most bond length region, but in detail the bond length dependence of the functions is different. The agreements imply that the calculated TDM for the transition (X 1 Σ + A 1 Π) is very accurate to a certain extent.

5 No. 2 Communications in Theoretical Physics 197 Table 3 Comparison of the present G v results with the experimental data for the A 1 Π state of CO. G v/cm 1 G v/ev T e + V /cm 1 v Pw Expt. [2] Pw Expt. [2] Pw Expt. [2] Table 4 Dipole transition a (in atomic units) as function of the internuclear distance R (in Å). R D(R) R D(R) R D(R) a To avoid congestion, we only present these data that between the corresponding theoretical equilibrium bond length. 4 Conclusion To summarize, the CASSCF and MRCI methods with aug-cc-pv5z basis set have been performed on the six lowlying states (X 1 Σ +, a 3 Π, a 3 Σ +, d 3, A 1 Π, and I 1 Σ ) of CO. The calculated potential energy curves and spectroscopic constants are generally in good agreement with experimental data. The vibrational states of the X 1 Σ + Fig. 4 Transition dipole moment (in atomic units) for the transition (A 1 Π X 1 Σ + ). and A 1 Π states have also been calculated. The comparison demonstrates the present vibrational level G(v) values agree well with the corresponding experimental data. The calculations on the transition dipole moment (TDM) show that the TDM of the two states (X 1 Σ + A 1 Π) are reduced strongly with increase of bond length.

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