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The energy levels arising from the electronic orbital <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>5</mn><mi>p</mi><mo>−</mo><mn>4</mn><mi>f</mi></mrow></semantics></math></inline-formula> crossing between the ground <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>5</mn><msup><mi>p</mi><mn>2</mn></msup><mspace width="3.33333pt"></mspace><mn>4</mn><mi>f</mi></mrow></semantics></math></inline-formula> and excited <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>5</mn><mi>p</mi><mspace width="3.33333pt"></mspace><mn>4</mn><msup><mi>f</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> configurations in the Nd<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mn>9</mn><mo>+</mo></mrow></msup></semantics></math></inline-formula> ion are investigated by using high-accuracy relativistic ab initio calculations. The accurate atomic data of the lifetime, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>g</mi><mi>J</mi></msub></semantics></math></inline-formula> factor, electric quadrupole moment, and hyperfine structure of the magnetic dipole are also presented. The long-lived states that are suitable for making narrow-linewidth (milli-Hz) clock lines are found. Dominant systematics caused by stray electromagnetic interactions in an experiment and the coefficients of the relativistic sensitivityto variation of the fine-structure constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and of the Lorentz invariance violation are evaluated, thus validating that the Nd<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mn>9</mn><mo>+</mo></mrow></msup></semantics></math></inline-formula> ion can be a new candidate for high-resolution spectroscopy and precision fundamental studies for probing new physics beyond the Standard Model.