"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Pollock, Mary Elizabeth"@en . "2010-04-16T16:49:35Z"@en . "1930"@en . "Master of Arts - MA"@en . "University of British Columbia"@en . "[No abstract available]"@en . "https://circle.library.ubc.ca/rest/handle/2429/23730?expand=metadata"@en . "CAT LE3%7-^CC. MO. AN ILLUSTRATIVE EXAMPLE OF THE INTER-PENETRATING ORBITS OF THE NORMAL HYDROGEN MOLECULE. by MARY ELIZABETH POLLOCK A Thesis submitted for the Degree of Master of Arts, in the Department of Mathematics. THE UNIVERSITY OF BRITISH COLUMBIA April, 1930 (i) CONTENTS. 1. Introduction 1 2. The Differential Equations of Motion 2 3. A Particular Solution 3 4. The Equations of Variation .. 7 5. Initial Conditions..... 17 6. Integration of and 18 7. Induction to the General Term..... 32 8. The Final Solution 35 9. The Convergence of the Solutions 36 10. An Illustrative Orbit 38 Table I 38 Table II 39 Figure I 41 Figure II 42 THE INTER-PENETRATING ORBITS OF THE NORMAL HYDROGEN MOLECULE. 1. Introduction.^ The normal hydrogen molecule consists of two nuclei having eoual positive charges of electricity and of two electrons with equal negative charges. If we suppose that the electrons are infinitesimal in mass as compared with the nuclei, then there are two types of orbits for the electrons, viz.y the pendulum and the inter-penetrating. In the present paper the inter-penetrating orbits will be considered. The electrons are constrained to move in a plane, viz.% the right bisector plane of the join of the nuclei. They are further constrained to move so that they are symmetrically situated with respect to the point at which the plane of the motion is cut by the join 01 the nuclei. ^ The theoretical work given here is contained in a paper by Dean D. Buchanan, head of the Department of Mathema-tics in the University of British Columbia. The evaluation o the constants, and the algebraic computation and the drawing of the graphs in the numerical example are the work of the candidate, who wishes to take this opportunity to express her thanks to Dean Buchanan for suggesting the topic. - 2 -2. The Differential Equations of Motion. z. \ -A'. \ \ \ \ ^ 1 \u00E2\u0080\u0094 \ \ \ \ Let us take a system of rectangular coordinates with the origin at the mid-point of the join of the nuclei, the x-axis along the join and the yz-plane in the plane of motion. The force of attraction between the nuclei and the elect-rons and the force of repulsion between the electrons are assumed according to the Newtonian law of the inverse square. Cnly the classical mechanics is used and no application of Lprmor's theorem is made. Let JT denote the ratio of the repulsion to the attraction. Let the distance from the origin to a nucleus be chosen as the unit distance and let the unit of time be chosen so that the gravitational constant is unity. Let the coordinates of one electron, E, , be ( ^ ^ ^ ^ 1 /b^* ^ ^ l^y? -. - 21 -/y/y,^ v ^ X J jL i J??? - ^ j . y/y^ _ /? ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ y * ^ / J - 22 -On integrating (18,b) we find , -When we substitute in (18,a) the values for from (17) and for & from (19) we obtain 7 \u00E2\u0080\u0094 L 7<* = ^ ^ ^ _ ^ ^ -7' f* ) .-/J =- 25 -where ^ j )( J j f^ ^ ^ * ^ < i r ^ ^ ^ \u00E2\u0080\u00941/3 Ht ^ 3 3 3 / ^ / - 24 -^ ' / / / ^ - - - - 7 yj / - ^^.yj-a-jr^ -^y<7f3TT 3 ^ ^ ii - 25 -f / ^ ^ ^ ^ y ^ - 26 -- ffjL? r ^ J r ^ ^ ^ ) - 27 -) J / In order that shall be periodic we must put and the solution for (20) then becomes Substitute for in (19). It then follows that We must annul the constants in order that shall be peri-odic. Hence - \u00E2\u0080\u0094 * - o or - 2 8 -From the initial conditions (16) we obtain ^ -The periodic solutions for and ^ therefore, which satisfy the initial conditions are y -or where /-V. r ' c<) ) 7 - 29 -^ ^ ^^ ^ ^ ^ , j / ^ ^ ^ ^ ^ ^ ^ / r ^ -L J ^ .3/3.? ^ ^ ^ ^ , ^ V3 ^ ^ ^ - 50 -^ ^ ^ f.s , \"A K = - - 3 \u00E2\u0080\u0094 / ^ - A f yir ^ f ^ % ^ ) y'* ^ ^ ^ ^ ^ ^ / ^ ^ / Sj 3T X? - 31 -/ -- ^ ^ ^ ^ ^ ^ ^ / J ^ ^ ^ ^ n ) ^ ^ 33-7/ ^ . ^ r ^ ^ 1 7 ^ ^ ^ ' ^ ^ ^ ^ / Jj 7 ^ / \u00E2\u0080\u0094 ^ ^ ^ ^ ^ / ^ ^ - 32 -7. Induction to the General Term. The remaining steps of the integration are similar to the preceding step and an induction will show that the inte-gration can be carried to any desired degree of accuracy. Let us suppose that ^ ^ ^ ^ have been de-termined for and that fr) L ; 2L /*/ ^ L (3 3) ' ^ <7 f where AA , /I- , = are known c "Thesis/Dissertation"@en . "10.14288/1.0080146"@en . "eng"@en . "Mathematics"@en . 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