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We report here an <i>R</i>-matrix study of electron collision with the BeO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>+</mo></msup></semantics></math></inline-formula> molecular ion in its X <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mn>2</mn></msup><mi mathvariant="normal">Π</mi></mrow></semantics></math></inline-formula> ground state and at a single bond length, namely its equilibrium <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>e</mi></msub><mo>=</mo><mn>2.7023</mn></mrow></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mn>0</mn></msub></semantics></math></inline-formula>. Firstly, a good quality configuration interaction calculation is performed for the BeO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>+</mo></msup></semantics></math></inline-formula> ground and excited states. We then perform scattering calculations using the <i>R</i>-matrix method to yield the cross-section for electronic excitation to several of its excited states. The electron impact dissociation of BeO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>+</mo></msup></semantics></math></inline-formula> through the two lowest dissociation channels, namely the Be<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>+</mo></msup><mrow><msup><mo>(</mo><mn>2</mn></msup><msub><mi>S</mi><mi>g</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> + O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>(</mo><mn>3</mn></msup><msub><mi>P</mi><mi>g</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> and Be<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>+</mo></msup><mrow><msup><mo>(</mo><mn>2</mn></msup><msub><mi>S</mi><mi>g</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> + O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>(</mo><mn>1</mn></msup><msub><mi>D</mi><mi>g</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> dissociation channels, is estimated using the electronic excitation cross-sections. Rotational excitation cross-sections are provided for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mrow><mo>(</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>→</mo><msup><mi>j</mi><mo>′</mo></msup><mrow><mo>(</mo><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> rotational transitions. Our calculations also yield e + BeO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>+</mo></msup></semantics></math></inline-formula> neutral Feshbach resonances and their widths which we present systematically categorized by their symmetry and quantum defects, and BeO-bound Rydberg states at the BeO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>+</mo></msup></semantics></math></inline-formula> equilibrium. The full potential energy curves for the resonant states, their widths and the bound Rydberg states, whose details we propose to give in a subsequent work, can be the starting point of other collision calculations.