Abstract
The interaction of nitric oxide (NO) in its ground state X^{2}Π and the first excited Rydberg state A^{2}Σ^{+} with an argon (Ar) atom has been studied using density functional theory. A number of exchange-correlation functionals that account for dispersion interactions have been considered, including functionals with both empirical and non-empirical treatments of dispersion. To study NO in the excited state, the recently developed maximum overlap method was used. Potential energy surfaces for interaction of NO with Ar have been constructed and parameters describing their minima, such as NO-Ar distance, orientation angle, and binding energy, have been determined. A comparison with combined experimental and accurate theoretical data has been made in terms of these parameters and the overall shape of the surfaces. For the ground state, several of the functionals give very good results. Treatment of the excited state is more problematic. None of the functionals considered provides completely satisfactory results. Several reasons for this failure have been identified: an incorrect description of the non-dispersion component of the interaction and the damping of the dispersion interaction at small interatomic distances.
This work is supported by the Engineering and Physical Sciences Research Council (EP/H004815). We are grateful for access to the University of Nottingham High Performance Computer and to Professor Tim Wright and Dr. Richard Wheatley for useful discussions.
I. INTRODUCTION
II. COMPUTATIONAL DETAILS
III. RESULTS AND DISCUSSION
A. NO(X^{2}Π)-Ar interaction
B. NO(A^{2}Σ^{+})-Ar interaction
IV. CONCLUSIONS
Key Topics
- Density functional theory
- 33.0
- Ground states
- 23.0
- Excited states
- 22.0
- Exchange correlation functionals
- 12.0
- Dispersion
- 10.0
Figures
Jacobian coordinates r, R, and θ of the NO-Ar complex. r is the distance between N and O, R is the distance between the NO center of mass M and Ar, and θ is the angle between the NO molecular axis and the line connecting the Ar nucleus with the NO center of mass.
Jacobian coordinates r, R, and θ of the NO-Ar complex. r is the distance between N and O, R is the distance between the NO center of mass M and Ar, and θ is the angle between the NO molecular axis and the line connecting the Ar nucleus with the NO center of mass.
The best PESs [in cm^{−1}] of NO(X^{2}Π)-Ar interaction obtained with DFT/aug-cc-pvtz: (a) and (b) PW86-PBE-XDM; (c) and (d) HF-PW92-VV09; (e) and (f) ωB97X; (g) and (h) B3LYP-D3; and (i) and (j) CCSD(T)/aug-cc-pvtz+BFs. (a), (c), (e), (g), and (i) A^{′} components; (b), (d), (f), (h), and (j) – A^{″} components. Equipotential lines are separated by 5 cm^{−1} from each other.
The best PESs [in cm^{−1}] of NO(X^{2}Π)-Ar interaction obtained with DFT/aug-cc-pvtz: (a) and (b) PW86-PBE-XDM; (c) and (d) HF-PW92-VV09; (e) and (f) ωB97X; (g) and (h) B3LYP-D3; and (i) and (j) CCSD(T)/aug-cc-pvtz+BFs. (a), (c), (e), (g), and (i) A^{′} components; (b), (d), (f), (h), and (j) – A^{″} components. Equipotential lines are separated by 5 cm^{−1} from each other.
Dispersion energy E _{ d } [in cm^{−1}] dependence on NO-Ar separation R [bohr] and orientation angle θ [°] computed using various DFT functionals: (a) PW86-PBE-XDM, (b) HF-PW92-VV09, and (c) B3LYP-D3. For PW86-PBE-XDM and HF-PW92-VV09, the dispersion energy of the A^{′} component is presented, the difference between A^{′} and A^{″} dispersion in the region of PESs’ minima does not exceed few cm^{−1}.
Dispersion energy E _{ d } [in cm^{−1}] dependence on NO-Ar separation R [bohr] and orientation angle θ [°] computed using various DFT functionals: (a) PW86-PBE-XDM, (b) HF-PW92-VV09, and (c) B3LYP-D3. For PW86-PBE-XDM and HF-PW92-VV09, the dispersion energy of the A^{′} component is presented, the difference between A^{′} and A^{″} dispersion in the region of PESs’ minima does not exceed few cm^{−1}.
Difference between CCSD(T) and DFT without dispersion PESs [in cm^{−1}]: (a) and (b) for the A^{′} component of the ground state; (c) and (d) for the excited state of the NO-Ar complex. (a) CCSD(T)/aug-cc-pvtz+BFs − HF-PW92/aug-cc-pvtz difference, (b) CCSD(T)/aug-cc-pvtz+BFs − B3LYP/aug-cc-pvtz difference, (c) CCSD(T)/d-aug-cc-pvtz+BFs − HF-PW92/aug-cc-pvtz difference, and (d) CCSD(T)/d-aug-cc-pvtz+BFs − B3LYP/d-aug-cc-pvtz difference. Equipotential lines are separated by 5 cm^{−1} from each other.
Difference between CCSD(T) and DFT without dispersion PESs [in cm^{−1}]: (a) and (b) for the A^{′} component of the ground state; (c) and (d) for the excited state of the NO-Ar complex. (a) CCSD(T)/aug-cc-pvtz+BFs − HF-PW92/aug-cc-pvtz difference, (b) CCSD(T)/aug-cc-pvtz+BFs − B3LYP/aug-cc-pvtz difference, (c) CCSD(T)/d-aug-cc-pvtz+BFs − HF-PW92/aug-cc-pvtz difference, and (d) CCSD(T)/d-aug-cc-pvtz+BFs − B3LYP/d-aug-cc-pvtz difference. Equipotential lines are separated by 5 cm^{−1} from each other.
(a)–(c) Interaction energy [in cm^{−1}] of NO(A^{2}Σ^{+}) and Ar obtained with MOM-DFT: (a) PW86-PBE-XDM/aug-cc-pvtz, (b) HF-PW92-VV09/aug-cc-pvtz, and (c) B3LYP-D3/d-aug-cc-pvtz. (d), (e), and (g) Corresponding dispersion energy contribution [in cm^{−1}] to the (a), (b), and (c) PESs, respectively. Equipotential lines are separated by 5 cm^{−1} from each other.
(a)–(c) Interaction energy [in cm^{−1}] of NO(A^{2}Σ^{+}) and Ar obtained with MOM-DFT: (a) PW86-PBE-XDM/aug-cc-pvtz, (b) HF-PW92-VV09/aug-cc-pvtz, and (c) B3LYP-D3/d-aug-cc-pvtz. (d), (e), and (g) Corresponding dispersion energy contribution [in cm^{−1}] to the (a), (b), and (c) PESs, respectively. Equipotential lines are separated by 5 cm^{−1} from each other.
(a) and (b) – ΔT _{ e }(R,θ) [in cm^{−1}] calculated with CCSD(T): (a) using non-scaled NO(A^{2}Σ^{+})-Ar PES; (b) using scaled NO(A^{2}Σ^{+})-Ar PES (see text). Equipotential lines are separated by 50 cm^{−1} from each other. Zero levels are depicted with thicker lines; (c) Zero levels of ΔT _{ e }(R,θ) calculated with coupled cluster and DFT: blue – CCSD(T) from (b), red – PW86-PBE-XDM/aug-cc-pvtz, yellow – HF-PW92-VV09/aug-cc-pvtz, green – B3LYP-D3/aug-cc-pvtz(X^{2}Π), d-aug-cc-pvtz(A^{2}Σ^{+}). All excited state calculations have been done using MOM.
(a) and (b) – ΔT _{ e }(R,θ) [in cm^{−1}] calculated with CCSD(T): (a) using non-scaled NO(A^{2}Σ^{+})-Ar PES; (b) using scaled NO(A^{2}Σ^{+})-Ar PES (see text). Equipotential lines are separated by 50 cm^{−1} from each other. Zero levels are depicted with thicker lines; (c) Zero levels of ΔT _{ e }(R,θ) calculated with coupled cluster and DFT: blue – CCSD(T) from (b), red – PW86-PBE-XDM/aug-cc-pvtz, yellow – HF-PW92-VV09/aug-cc-pvtz, green – B3LYP-D3/aug-cc-pvtz(X^{2}Π), d-aug-cc-pvtz(A^{2}Σ^{+}). All excited state calculations have been done using MOM.
Tables
Summary of literature results for NO(X^{2}Π)-Ar interaction and its PESs minima. Values of D _{ e } marked with * are obtained from D _{0} of the same reference and the ratios D _{ e }/D _{0} from Ref. 20.
Summary of literature results for NO(X^{2}Π)-Ar interaction and its PESs minima. Values of D _{ e } marked with * are obtained from D _{0} of the same reference and the ratios D _{ e }/D _{0} from Ref. 20.
Parameters of the minima of the NO(X^{2}Π)-Ar interaction PESs obtained with DFT/aug-cc-pvtz and CCSD(T)/aug-cc-pvtz, aug-cc-pvtz+BFs. The errors compared to literature data are given in brackets, for D _{ e } the errors are computed relative to the average across the data given in Table I, for R _{ e } and θ _{ e } the errors are computed relative to the data from Ref. 11. The uncertainty of our results related to the numerical fitting is about 1 cm^{−1}, 0.1 bohr, and 2° for D _{ e }, R _{ e }, and θ _{ e } values, respectively.
Parameters of the minima of the NO(X^{2}Π)-Ar interaction PESs obtained with DFT/aug-cc-pvtz and CCSD(T)/aug-cc-pvtz, aug-cc-pvtz+BFs. The errors compared to literature data are given in brackets, for D _{ e } the errors are computed relative to the average across the data given in Table I, for R _{ e } and θ _{ e } the errors are computed relative to the data from Ref. 11. The uncertainty of our results related to the numerical fitting is about 1 cm^{−1}, 0.1 bohr, and 2° for D _{ e }, R _{ e }, and θ _{ e } values, respectively.
Summary of literature results for NO(A^{2}Σ^{+})-Ar interaction and parameters of its PES minima. D _{ e } ^{ g } is the depth of the global minimum, D _{ e } ^{ l } is the depth of the local minimum, and D _{0} is the dissociation energy. “Experimental” values of D _{ e } marked with * are obtained from D _{0} from the same experimental work and ratios D _{ e }/D _{0} from Ref. 12.
Summary of literature results for NO(A^{2}Σ^{+})-Ar interaction and parameters of its PES minima. D _{ e } ^{ g } is the depth of the global minimum, D _{ e } ^{ l } is the depth of the local minimum, and D _{0} is the dissociation energy. “Experimental” values of D _{ e } marked with * are obtained from D _{0} from the same experimental work and ratios D _{ e }/D _{0} from Ref. 12.
NO(A^{2}Σ^{+})-Ar interaction PES depth D _{ e } [in cm^{−1}] calculated using MOM with DFT and CCSD(T). Values marked with * are an estimate of D _{ e } calculated at R = 8 bohrs and θ = 0° which is close to the equilibrium geometry of the complex. Negative values of D _{ e } indicate a lack of binding in the region close to the true equilibrium. The most accurate estimate of D _{ e } derived from experiments^{16,23} and high-level theory^{12} is 105–120 cm^{−1} (see Table III).
NO(A^{2}Σ^{+})-Ar interaction PES depth D _{ e } [in cm^{−1}] calculated using MOM with DFT and CCSD(T). Values marked with * are an estimate of D _{ e } calculated at R = 8 bohrs and θ = 0° which is close to the equilibrium geometry of the complex. Negative values of D _{ e } indicate a lack of binding in the region close to the true equilibrium. The most accurate estimate of D _{ e } derived from experiments^{16,23} and high-level theory^{12} is 105–120 cm^{−1} (see Table III).
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