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evaluation
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The evaluation of exchange integrals which occur in the quantum theory of molecules is reduced to the solution of Poisson's equation.

用量子力学解释分子构造时产生一种对换积分。此文将求此种积分之问题变为解Poisson偏微分方程式之问题。

A new approximation method is proposed in this article for the discussion of molecular structures,and this new method includes the two well-known theories,molecular orbital theory and electron-pair bond theory as two special cases.Let a molecule have n bonds and let the ith bond be described by the anti-symmetrical two-electron bond function ψ_i(v_(2i-1),v_(2i)).(If there exist one- electron,three-electron or many-electron bonds,they can be similarly described by the corresponding one-electron,three-electron...

A new approximation method is proposed in this article for the discussion of molecular structures,and this new method includes the two well-known theories,molecular orbital theory and electron-pair bond theory as two special cases.Let a molecule have n bonds and let the ith bond be described by the anti-symmetrical two-electron bond function ψ_i(v_(2i-1),v_(2i)).(If there exist one- electron,three-electron or many-electron bonds,they can be similarly described by the corresponding one-electron,three-electron or many-electron bond func- tions.) Then the stationary state of the molecule is represented by the follow- ing wave function Ψ, where the summation is over all permutations of 1,2,……,2n except those within the interior of the functions,since each ψ_i is already anti-symmetrical.Obviously (2~n/((2n)/!))~(1/2) is the normalization factor. By quantum mechanics the energy of the molecule equals (1) here H_i,T_(ij) and S_(11)' are respectively the following three kinds of operators, (2) (3) (4) The third term of equation (1) is the exchange integral of electrons 1 and 1', while (1,2') is that of electrons 1 and 2'.According to the definition of bond functions,ψ_i may be written as (5) Substituting equation (5) into equation (1) and carrying out the integration over spin coordinates,we obtain (6) It can be easily seen from equation (6) that the combining energy of a mole- cule consists of two parts,one being the binding energy of the bonds represent- ed by the first term of equation (6),and the other being the interaction energy of the bonds denoted by the second term of that equation. If we choose certain functions φ_i~('s) involving several parameters and substi- tute them into equation (6),we may determine the values of those parameters by means of the variation principle. For the discussion of bond interaction energies,we develop a new method for the evaluation of certain types of three-center and four-center integrals.The interaction energy of a unit positive charge and an electron cloud of cylindrical- symmetry distribution may be written as (7) where (8) and R_0~2=a~2+b~2+c~2 The interaction energy of two electron clouds both of cylindrical-symmetry distributions with respect to their own respective axes is evaluated to be (9) (10) where is to sum over j from zero to the lesser value of n-2i and m, is to sum over i from zero to the integral one of n/2 and (n-1)/2,and is to sum over all cases satisfying the relation =m-j,while b_(n,n-2i) represents the coefficient of x~(n-2i) in the n th Legendre polynomial.

本文在分子结构理论方面,作了下列两点贡献:首先建议了用双电子或多电子键函数作为近似基础,来计算分子的近似能量和近似电子云分布。这样计算得来的结果,一定会比用分子轨道理论或电子配对理论好,因为它更真实的反映了分子的化学性质,同时它也包括了后两者,而以它们为特例。我们得到了分子结合能的表示式,用表示式证明了分子结合能由两部分组成:一部分是键的结合能,另一部分是键与键间的作用能。其次是建议了一种新方法,把在计算化学键相互间的作用能中遇到的一些三中心和四中心积分,还原为容易计算的二中心积分。这方法比以往所用的好,因为它计算比较简单,同时限制性也小。

A new method is proposed in this paper for the evaluation of the following three important kinds of three-center and four-center integrals needed in molecular quantum mechanics: (1) (2) (3) Obviously, integral (2) is a special case of integral (3), so we need only to evaluate the first and the third. The following equations have been obtained for the evaluation of the above-mentioned two kinds of integrals (4) (5) where (6) (7) (8) (9) R_0 in equation (4) is the distance of point a apart from the...

A new method is proposed in this paper for the evaluation of the following three important kinds of three-center and four-center integrals needed in molecular quantum mechanics: (1) (2) (3) Obviously, integral (2) is a special case of integral (3), so we need only to evaluate the first and the third. The following equations have been obtained for the evaluation of the above-mentioned two kinds of integrals (4) (5) where (6) (7) (8) (9) R_0 in equation (4) is the distance of point a apart from the origin located in bc line, while that in equation (5) is the distance between the two chosen origins separately located in ab and cd lines. θ_0 is the angle made byand, and θ_(10), θ_(20) and θ_(12) are respectively the angles made by and,and,and, The fourth term of equation (5) is, in general, negligible except in the case of R_0, which is less than two Bohr units. We propose two methods for the evaluation of and : the first one is to choose the origin at the end-point of the bond and evaluate the integral strictly inside and outside the sphere of radius, while the second one is to choose the origin at the mid-point of the bond and evaluate its value inside and outside the ellipsoid passing through the end-point of. The calculation involved in the second method is quite simple and, of course, a small error is introduced in changing the region of integration from the sphere to the ellipsoid, but it is quite negligible in comparison with the result of our first method. Equation (4) is exact in all cases, while equation (5) is exact in many cases but also involves certain errors in some other cases. From our actual calculations, we draw the conclusion that equation (5) is almost exact in the evaluation of integrals L_(aa, bc) and L_(ab, cd) and that certain error is involved in the evaluation of integral L_(ab, bc), but the error introduced does not exceed ten per cent.

在本篇文章中,我們建議了一種新方法來計算量子力學中的三中心和四中心積分;這方法此以往的好,因為計算簡單,應用廣闊,結果也比較可靠。我們用來計算三中心吸引能的公式[方程(5)]是在任何情况下都是正確的,而用來計算三中心和四中心的排斥能積分公式[方程(18)]在某些情况下是正確的,在另一些情况却能引進一些誤差。在計算非相隣鍵的積分時引進的誤差很小,可以忽略不計;在計算相隣鍵的積分時引進的誤差此較大,但不超過百分之十。我們建議兩種計算A_u和B_u的方法,一種方法是以鍵的一個端點爲原點,嚴格按照球內外的區域積分;另一種方法是以鍵的中點為原點,按照橢圓體的內外區域積分。前一種方法理論上嚴密,然而後一種方法計算簡單,收斂性快,引進的誤差也不大;尤其在計算相隣鍵的三中心排斥能的積分時,似乎後一方法得到的結果還比前一方法好。在本文中,為了容易說明起見,常常引用吸引能和排斥能這兩個名詞,實際我們的方法,是用來計算下列三類積分:它們不僅包括吸引能和排斥能積分,也把交换積分包括在內,甚至可以在更廣泛的意義上看待上列積分。若σ_1,σ_2也是Φ_1和Φ_2的函數時,仍可以用我們的一般展開理論處理,不過要此本文複雜。

 
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