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The Nature of the Chemical Bond of Hydrogen-Type Molecules and Molecular Ions q n 0 4 Defining x, y, z in terms of we put 0 , it may be easily verified that
0
395
(11.40)
x2 y2 z 2 ( 0) (11.41) a 4 b 4 c 4 a 2b 2 c 2 Consequently, the charge density in rectangular coordinates is q 1 (11.42) 2 4 abc x y2 z2 a 4 b4 c4 (The mass-density function of an MO is equivalent to its charge-density function where m replaces q of Eq. (11.42)). The equation of the plane tangent to the ellipsoid at the point x0 , y0 , z0 is x y z (11.43) X 20 Y 20 Z 20 1 a b c where X , Y , Z are running coordinates in the plane. After dividing through by the square root of the sum of the squares of the coefficients of X , Y , and Z , the right member is the distance D from the origin to the tangent plane. That is, 1 D (11.44) 2 x y2 z2 a 4 b4 c4 so that for an electron MO e (11.45) D 4 abc In other words, the surface density at any point on a charged ellipsoidal conductor is proportional to the perpendicular distance from the center of the ellipsoid to the plane tangent to the ellipsoid at the point. The charge is thus greater on the more sharply rounded ends farther away from the origin. In the case of hydrogen-type molecules and molecular ions, rotational symmetry about the internuclear axis requires that two of the axes be equal. Thus, the MO is a spheroid, and Eq. (11.37) can be integrated in terms of elementary functions. If a b c , the spheroid is prolate, and the potential is given by
1
e
8 0
a 2 b2
ln
a 2 a 2 b2 a 2 a 2 b2
(11.46)
SPHEROIDAL FORCE EQUATIONS
ELECTRIC FORCE The spheroidal MO is a two-dimensional surface of constant potential given by Eq. (11.46) for 0 . For an isolated electron MO the electric field inside is zero as given by Gauss’ Law
S
EdA V
dV 0
(11.47)
where the charge density, , inside the MO is zero. Gauss’ Law at a two-dimensional surface with continuity of the potential across the surface according to Faraday’s law in the electrostatic limit [4-6] is n E1 E2
0
(11.48)
E2 is the electric field inside which is zero. The electric field of an ellipsoidal MO with semimajor and semiminor axes a and b c , respectively, is given by substituting given by Eq. (11.38-11.42) into Eq. (11.48). e 1 e e 1 (11.49) Di E i i i 4 0 ( 4 0 abc 4 o abc x 2 y 2 z 2 0 a 4 b4 c4 wherein the ellipsoidal-coordinate parameter 0 at the surface of the MO and D is the distance from the origin to the tangent plane given by Eq. (11.44). The electric field and thus the force and potential energy between the protons and the electron MO can be solved based on three principles: (1) Maxwell’s equations require that the electron MO is a equipotential energy surface that is a function of alone; thus, it is a prolate spheroid, (2) stability to radiation, and conservation first principles require that the angular velocity is constant and given in polar coordinates with respect to the origin by Eq. (11.24), and (3) the equations of motion due to the central force of each proton (Eqs. (11.5-11.19) and Eqs. (11.68-11.70)) also determine that the current is ellipsoidal, and based on symmetry, the current is a prolate spheroid. Thus, based on Maxwell’s equations, conservation principles, and Newton’s Laws for the equations of motion, the electron MO constraints and the motion under the force of the