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An illustrative example of the interpenetrating orbits of the normal hydrogen molecule Pollock, Mary Elizabeth 1930

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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  5.  Initial Conditions.....  17  6.  Integration of  18  7.  Induction to the General Term.....  32  8.  The Final Solution  35  9.  The Convergence of the Solutions  36  An Illustrative Orbit  38  10.  and  ..  7  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 Mathematics 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. 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  z. \  the yz-plane in the plane of  -A'. \ \  motion. The force of attraction  \  \  ^  1 —  \  between the nuclei and the electrons 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 (<?,-y,  Then according  to the restrictions of motion the coordinates of the other electron E^, are (<?,-y,-^). If E< only is considered, the differential equations of motion are  7  *  (!) *  = 3 Now choose a system of rotating coordinates,  ^  which rotate in the yz-plane about the origin with the uniform angular velocity unity. The necessary transformations are y ^ 7 r  7  -  and when the equations (1) are thus transformed we obtain  (3J  ^  3.  7 'J  A Particular Solution. Let  where  /  (1J  ^ and 6 are parameters. Then equations (3) become  - 4 We generalize the parameters ^ ering that ^  and 6 by consid-  is fixed and that 6 varies. We first obtain a  solution of (5) when  and then make the analytic continu-  ation of this solution for  The final results hold for  the physical problem only when ^^ and  e have values which  satisfy (4). For  where  a particular solution of (5) is  satisfies the equation  Let Then  n  %  ^  [  / t^  ^*,  - 5 -  +  ^  -  7  ^ =  %  ^  ^  <2, 7/"  Z^ 4,4, - 33 a, /  ^ <3,  Ms  t 7$  73  * 73  ^ ^ ^ ^ A / ^ ^  ^  ^  *% 1  ) )  - 6 -  ^  3,"  a-" <?,*  )  d,'*  ^  '  ^^ < \ )  - 33 F,  %  Equating coefficients of equal powers ofy*^we obtain  a,  % = = <7  <2. ^  a,''  . 7 4.  The Equations of Variation. Let f ) -r  where  , ^  J  are the displacements from the particular sol-  ution, ^ is an arbitrary parameter depending upon  and  ^  is the new independent variable. When equations (7) are substituted in (5) we obtain, the dots denoting derivation with respect to  Where  ^ ^  y/v^/Q^Y/t^  Then equation (5',a) becomes  . 8 -  ^ 9 -  - 10 Where  f ^  % 2/  /  -- % y/^  ^ ^ ^ ^  ?  ;  =  ^ -  -  ^L 7-  f)  -  - F/.  =r J2 _  — Y?"  - 11 -  JO  7*  - i X  ^  =  -  ^c  1^  ir""'*  -  A?'  V 3 -  J/  ^  /A ^ ^ ^  - 12 -  <2)  13 Equation (5',B) becomes  -f-  - 14 -  which has the form (8,b) i.e.  where "6/ * 7*  y*  y  - 15 o; , =r -3 ^ J? _ A? _ T^f ,1% Mr  <?3  = - APj  ^  We shall show that equations (8) can be integrated as power series in g and that  f can be determined also as a power  series in 6 so that ^ and  j will be periodic. Let us there*  fore suppose that  -  .  -  Let these substitutions be made in (8) and let the resulting equations be cited as (8'). On equating the coefficients of the various powers of e in (8s) we obtain sets of differential equations which define the various  and  The  and  the constants of integration at each step can be determined, as we shall show, so that each ^  and  is separately peri-  odic and satisfies certain initial conditions.  -  16  -  If we consider only terms of (8') which are independent of g we obtain the equations of variation, viz.,  A -  ^ - #  ^  ^o)  = O Integrating equation (10,b) we obtain ^ where  ^ -  ^^  OiJ  is arbitrary, and on substituting for ^  in (10,a)  we have =  -  (/x)  The general solution of this equation is  where  and ^  are constants of integration. When this value  for jo, is substituted in (11) we obtain 17  ^  V"  As we desire a solution for ^ which will be periodic we must  put  or On substituting for  <2 r -  2-.  in (13) and integrating (14) the peri"  odic solutions of the equations of variation are found to be  17  where  is a constant of integration. The period of these  solutions is ^ ^ y ^ i n  7* and this will be chosen as the period  for the solutions at the succeeding steps^  5.  Initial Conditions. In the solutions of the equations of variation  four arbitrary constants arose, viz.,  ,  ,  , Z) .  At each succeeding step of the Integration four constants will arise similarly and they will be denoted by  3 (j= 1,2,3,,..........). It will be seen that J is determined so that ^ will be periodic but that , j  y  ^ , d  ID' , will otherwise remain undertermined. We may therefore d Impose three initial conditions upon ^ and  ^ , and we shall  choose Inasmuch as ^ carries the factor <s in (7) there will be no loss of generality in putting  p(0)=l. When these conditions  are imposed upon (9) it follows that  18 As a consequence of (16) the solutions (15) become  C7) "  /y^* L  ^^ /  or.  *  /V  - _ -L-  /I-/-'.  6.  Integration of  and  When we equate the terms in 6 in (8'), that is in (8) after (9) have been substituted, we obtain  19 (17) have  substituted, (18,b) becomes  t  .  where  Q'*'  _  / A .  g/^yl*  X  ^  )  - 20 -  ^  ^ ^  —  ^  /b^* ^ ^  .  /  -  ^  l^y?  J  ^  j>  ^ 1  -  - 21 -  /y/y,^  jL i J??? -  v  ^ ^  ^  ^  y *  ^ ^  ^  X  ^ j  J  . 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 — L  =  ^ ^  ^  _  7'  ^  ^  -  f*  ) .-/J  7<*  =- 25 -  ^ j  )(  J j f^ ^  ^  where *  ^ < i r  ^  ^  ^  —1/3  3 33  /  ^  /  Ht  ^  - 24 -  ^ '  /  /  ^  -  - --  yj  /  -  /  ^^.yj-a-jr^ -^y<7f3TT  3  ^ ^ ii  7  - 25 -  f / ^  ^  ^  ^  ^ y  - 26 -  - ffjL?  r  ^  r  ^  J ^  ^  )  - 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 odic. Hence or  — *  -o  shall be peri-  -  28  -  From the initial conditions (16) we obtain  ^ The periodic solutions for  and  -  ^ therefore, which  satisfy the initial conditions are  y  -  or  r '  where c<) /-V.  ) 7  - 29 -  ^  ^  ^  ^^  ^  ^  , j  /  ^  ^  ^  ^ ^  ^  r -L J  ,^  ^  ^  ^  V3  /  ^ ^  ^ .3/3.?  ^  ^  ^  ^  - 50 -  ^  f.s K =  ^  , - - 3 — /  ^  "A ^  -  A  f  yir  f  ^  ^  y'*  ^ ^  /  %  ^ ^  ^  3T  ^  ^  X?  ^  ^  /  Sj  )  - 31 -  /-  ^  ^^ ^  ^^ ^  ^^  /  ^  J  ^  n  )  ^ ^ 33-7/ ^ .  ^  r  7  ^  ^  ^  ^  ' ^ ^  7^/  —  ^  ^ 1  ^  ^ ^  ^ ^  /  ^  ^ ^  /  Jj  - 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 integration can be carried to any desired degree of accuracy. Let us suppose that  ^ ^ ^ ^  termined for  and that  L '  where  have been de-  fr) /*/ ^  ; 2L  AA , /I- , c <?  ^  <7  L f  =  (3 3)  are known  constants. We desire to show that the solutions have the form (23) when -<=n. When we equate the coefficients of <g in (8') we obtain  -  , ^ J"  -- -  ,  -  ^  The only undetermined constant which enters (24) is enters only as shown. The terms in (v=0,l,2, sum of sines.  and it  carry either ^ or  .,.,n-l) as a factor and hence consist of a  - 33 Equation (24,b) has therefore the form  --ty  Integration of this equation yields r  ^  ci  Equation (24,a) involves terms in to even degrees, (v= 0,1,2,.  , ^  ,n-l). Hence H  , and  ^  consists  of a sum of cosines and when ^ is eliminated by means of (25) equation (24,a) becomes  (7  If  M  is to be periodic, ^Jr^and the general solution of (26)  becomes  When this equation is substituted in (25) the constants must be annulled if  is to be periodic, hence  or .  ^ A  ^ 34 Integrating the remaining terms in (25) we obtain  y  /7  <7  .  -tZ  ^  By imposing the initial conditions upon the resulting integral and uoon (27) we find that  tl) ^  J J  - 35 Then Z—  A fr 2/  7 -  ——t - _  )  7*  ^  or r*)  ^^  i.e. the desired solutions for ^ as (23) when  ^ is replaced by  and ^ have the same form  n . This completes the  induction.  8.  The Final Solution, On substituting (25) in (9) and the result in (7)  we obtain i? -  ^ Z Z  ^K - Z^ Z,-/  A/,  /V  ^  ^  ^  7  - 36 The period of these solutions is  in rT and in  it is  / The solutions (r8) are in terms of the rotating coordinates. With respect to the fixed axes, the solutions for the orbits of the electron J-, are  y where  ^ ,  ^ , have the values in (28). The orbits for the  electron ^ can be obtained from those for signs of  9*  y  and  z  by changing the  in (29).  The Convergence of the Solutions. Only the formal construction was considered in the  preceding sections and it remains now to show that the solutions there obtained converge. An existence proof in which Poinca^re's extension to Cauchy's theorem may be used to establish the convergence of the solutions for sufficiently small values of je/, but such a proof is cumbersome and generally more difficult than the actual construction. Instead of such a proof we make use of MacMillan's theorem in which he has shown that if the constants of integration in a system of differential equations to which (8) are reducible can be determinedso as to make the solution formally periodic  ^ 57  -  the solutions will converge for all finite values of the time provided a parameter corresponding to e is sufficiently small numerically. As  <s is arbitrary we may, therefore,  assign to it such values that the solutions will converge.  UBC Scanned by UBC Library  - 38 10.  An Illustrative Orbit. Recall equation (4)  For the normal hydrogen molecule ^  the value  Hence if we assign to  .2, the parameter <s must take the value  =.05,  The numerical values of certain cinstants entering the solutions, together with the number of the equation in which these constants first arise, are listed in Table I.  Tablel.- Numerical Values of Certain Constants, (  Constant  mr Hr Kr m* Hr Hf K7 Kr H. H. H, K. K;  Equation in which constant first appears.  .2 ) -.05)  Numerical Value.  (6) (8) (9) (13) (17)' (17)' (17)' (22) (22) (22) (22) (22) (28) (28) (28) (28) (28)  These values then give rise to the solutions  1.2214 -3.8436 13.7321 0.3955 -6.4409 7.4409 -37.63 -10.7118 13.59 -7.5726 286.34 1.9906 1.5166 -0.3381 -0.0189 2.5974 0.00498  y* which are expressed in terms of the rotating coordinates. The period of these solutions is  / = Table II. contains the values of the solutions (  Table 11.- Solutions for ^ T t x.0268  0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180  .3381 x cos T  0.3381 0.3330 0.3177 0.2928 0.2590 0.2173 0.1691 0.1156 0.0587 0.00 -0.0587 - 0.1156 -0.1691 -0.2173 -0.2590 -0.2928 -0.3177 -0.3330 -0.3381  .0189 x cos 2T  0.0189 0.0178 0.0145 0.0095 0.0033 -0.0033 -0.0095 -0.0145 -0.0178 -0.0189 -0.0178 -0.0145 -0.0095 -0.0033 0.0033 0.0095 0.0145 0.0178 0.0189  and  j  2.5974 x sin T  0.00 0.4511 0.8884 1.2987 1.6696 1.9897 2.2494 2.4408 2.5579 2.5974 2.5579 2.4408 2.2494 1.9897 1.6696 1.2987 0.8884 0.4511 0.00  .00498 X sin 2T  0.00 0.0017 0.0032 0.0043 0.0049 0.0049 0.0043 0.0032 0.0017 0.00 -0.0017 -0.0032 -0.0043 -0.0049 -0.0049 -0.0043 -0.0032 -0.0017 -0.00  1  1 .1596 1 .1658 1 .1844 1 .2143 1 .2543 1 .3026 1 .3570 1 .4155 1 .4757 1 .5355 1 .5931 1 .6467 1..6952 1 .7372 1 .7723 1 .7999 1 .8198 1 .8318 1 .8358  0.00 0.4528 0.8916 1.3030 1.6745 1.9946 2.2537 2.4440 2.5596 2.5974 2.5562 2.4376 2.2451 1.9848 1.6647 1.2944 0.8852 0.4494 0.00  -  40  -  In Figure I. the orbits of the two electrons, F, and E^ in the  ^ - p l a n e are shown, while Figure II. represents  the orbit of JE, with respect to the fixed axes.  Scanned by UBC Library  T;  

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