Ground-state potential and dipole moment of carbon monoxide: Contributions from electronic correlation, relativistic effects, QED, adiabatic correction, and nonadiabatic correction

The ground ${\mathrm{X}}^1{\mathrm{\Sigma }}^{+}$ state potential energy curve (PEC) and dipole moment curve (DMC) of CO molecule have been revisited within the framework of the relativistic coupled-cluster approach, which incorporates nonperturbative single, double, and triple cluster amplitudes (CCSDT) in conjunction with a finite-field methodology. The generalized relativistic pseudopotential model was used for the effective introducing the relativity in all-electron correlation treatment and accounting the quantum-electrodynamics (QED) corrections within the model-QED-operator approach. The diagonal Born-Oppenheimer correction to PEC has been evaluated using the CCSD approach. The sensitivity of resulting PEC and DMC to variations in basis set parameters and regular intramolecular perturbations were considered as well. The present ab initio results are in a reasonable agreement with their most accurate semiempirical counterparts.

Figure 1

The disparities in the PECs computed for the X2C Hamiltonian (including the Gaunt contribution) and the GRPP Hamiltonian. These calculations have been systematically conducted within both the DHF and CCSD(T) approximations.

Figure 2

Basis set convergence for the ground-state PEC. The results are presented concerning the $\mathrm{aug}\text{−}\mathrm{cc}\text{−}\mathrm{pV}N\mathrm{Z}$ basis set family relative to the CBS values within the CCSD(T) approximation: $\mathrm{\Delta }E=E\left(\text{aug-cc-pV}N\text{Z}\right)-E\left(\text{CBS}\right)$. Additionally, for comparative purposes, the CBS values obtained for the Dyall basis sets are also displayed.

Figure 3

Contribution of diffuse orbital basis functions to the PEC of the ground state. The results are presented within the $\mathrm{cc}\text{−}\mathrm{pV}N\mathrm{Z}$ basis set family, considering both the DHF and CCSD(T) approximations. The “Correlation” notation denotes the $(E_{\text{CCSD(T)}}-E_{\text{DHF}})$ contribution.

Figure 4

Basis set convergence of the perturbative triple $\mathrm{\Delta }{E}^{\text{(T)}}=E_{\text{CCSD(T)}}-E_{\text{CCSD}}$ and residual triple $\mathrm{\Delta }{E}^{\text{T-(T)}}=E_{\text{CCSDT}}-E_{\text{CCSD(T)}}$ cluster amplitudes contributions to the PEC of the ground state. The $\mathrm{\Delta }{E}^{\text{(T)}}$ results are presented relative to the CBS values. These data are obtained under the GRPP approximation and consider explicit correlation of ten electrons.

Figure 5

Relativistic corrections to the PEC of the ground state. The dashed and dotted lines delineate the contributions arising from the DC Hamiltonian and Gaunt corrections, respectively. The solid line represents the cumulative effect. These data are acquired using the CCSD(T) approximation. The results obtained in Ref. [19] are presented with a shift to align them with the DC curve for the purpose of a clear comparison.

Figure 6

QED corrections to the PEC of the ground state. The calculations are performed in the DHF and CCSD approximations. The data obtained by Konovalova et al. [19] and Dulaev et al. [24] are also presented.

Figure 7

The PEC of the ground state obtained at various levels of accounting for correlation effects: DHF, CCSD, CCSD(T), CCSDT, while including all other relevant corrections. The semiempirical potential curve is extracted from Ref. [47]. The minima of the potential curves are normalized to zero.

Figure 8

Deviation of the CCSD(T) and CCSDT PECs from the semiempirical potential [47] for the ground state. The results are showcased for calculations conducted using the aug-$\mathrm{cc}\text{−}\mathrm{pV}6\mathrm{Z}$ basis set and CBS calculations. The relativistic and QED corrections are taken into account. Additionally, for comparative purposes, the data obtained by Mehskov et al. Ref. [23] obtained within the MR-ACPF approach are also displayed.

Figure 9

Comparison of the mass-dependent DBOC functions for the ground state of ${}^{12}\mathrm{C}{}^{16}\mathrm{O}$ molecule: the HF and CCSD methods versus empirical data from Ref. [47].

Figure 10

(a) Comparison of the dimensionless mass-dependent $q$ correction, evaluated according to Eq. (11), to rotational energy of the ${}^{12}\mathrm{C}{}^{16}\mathrm{O}$ ground state with its empirical counterpart [47]. (b) The $J$-dependent contribution of the nonadiabatic effect into the permanent dipole function for the CO ground state.

Figure 11

Basis set convergence for the DMC of the ground state. The results are showcased concerning the $\mathrm{aug}\text{−}\mathrm{cc}\text{−}\mathrm{pV}N\mathrm{Z}$ basis set family relative to the CBS values within the CCSD(T) approximation: $\mathrm{\Delta }\mu =\mu \left(\text{aug-cc-pV}N\text{Z}\right)-\mu \left(\text{CBS}\right)$. For comparative purposes, the CBS values obtained for the Dyall basis sets are also presented.

Figure 12

Contribution of diffuse orbital basis functions to the ground-state DMC. The results are illustrated for the cc-pV5Z basis set family within both DHF and CCSD(T) approximations.

Figure 13

Relativistic correction to the ground-state DMC. The calculations are executed within the DHF, CCSD, and CCSD(T) approximations. Additionally, for comparative purposes, the data obtained by Konovalova et al. Ref. [19] are also displayed.

Figure 14

The ground-state DMC obtained at various levels of accounting for correlation effects: DHF, CCSD, CCSD(T), CCSDT, while incorporating all other relevant corrections. The semiempirical curve is extracted from Ref. [22].

Figure 15

Deviation of the DMC obtained using the CCSD, CCSD(T), and CCSDT methods from the semiempirical curve [22] for the ground state. The results of MR-ACPF and CCSD(T) calculations by Meshkov et al. [22], MR-CISD($+\text{Q}$)-FF by Balashov et al. [11], and MR-ACPF by Chen et al. [17] are also presented.