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We consider some remarkable central elements of the universal enveloping algebraU(gl(n)) which we call quantum immanants.


They result in many nontrivial properties of quantum immanants.


Quantum integrable systems and differential Galois theory


This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry.


We study the tensor category of tilting modules over a quantum groupUq with divided powers.

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 The main difficulty of the oneelectron theory of optical activity lies in the calculation of the socalled vicinal actions. This may be overcome, as suggested in the present investiga tion by assuming that the vicinal actions are due to the interactions of various chemical bonds in the molecule with the chromophoric electron. For the calculation of these interactions the charge distribution of a single covalent bond is assumed to consist of two nuclei, each of unit positive charge, separated by the bond distance... The main difficulty of the oneelectron theory of optical activity lies in the calculation of the socalled vicinal actions. This may be overcome, as suggested in the present investiga tion by assuming that the vicinal actions are due to the interactions of various chemical bonds in the molecule with the chromophoric electron. For the calculation of these interactions the charge distribution of a single covalent bond is assumed to consist of two nuclei, each of unit positive charge, separated by the bond distance and two electrons with opposite spins and a spherically symmetrical orbital eigenfunction represented by (?) where r is the distance measured from a certain point on the bond axis and near the center of the bond. The location of this point is so determined that the model will give the proper dipole moment of the bond. The exponent α, which determines the extent of diffusivity of the electron cloud, is obtained by an approximate variational treatment. The quadruple moment of this model, calculated for the CH bond, agrees very well with that estimated by Lassettre and Dean from a study of the potential barrier hindering the internal rotation of the ethane molecule. To test the theory and also to illustrate the procedure which is followed in calculating optical rotations using the new model of vicinal actions, the optical rotations of several methyl derivatives of cyclopentanone have been calculated and the following conclusions have been reached: (1) The proposed model gives rise to optical rotations in agreement with the observed values. (2) The effect of the internal rotation of the methyl group on the molecular rotation [M]_D of 3methylcyclopentanone is large, [M]_D being +44° and 130° for the cis transconformations, respectively. (3) The observed [M]_D may be regarded as composed of two parts, namely, the contribution from the multipole terms and that from the orbital overlapping. The latter corresponds to the "incomplete screening of atomic nuclei", and has been regarded as the most important vicinal action in earlier calculations. In the present work, it is shown, however, that this effect contributes only a few percent of the total [M]_D, unless the perturbing group is very near to the chromophoric group as it is in the case of 2methyl cyclopentanone. The failure of the previous calculations is, therefore, due mainly to the omission of the multipole (especially the quadrupole) terms. (4) The absolute configuration of 3methylcyclopentanone, indicated by the present calculation, is the opposite of the one suggested by Eyring, since he based on the overlapping contribution alone, which bears a sign opposite to that of the total [M]_D (5) That the observed [M]_D of 2,4,5,5tetramethylcyclopentanone is considerably smaller than that of 2,4dimethylcyclopentanone may be due to one or both of the following possibilities: (a) that the conformation of the 2 and 4methyl groups slightly deviates from the transposition in the tetramethyl compound, due to the presence of the two additional methyl groups; (b) that the two additional methyl groups in 5,5positions are not exactly symmetrical with respect to the ring so that they make contributions to [M]_D.  本文在量子力學的單電子旋光理論的鄰近作用問題上,作了如下的貢獻: 1.指出旋光度應由分子中各化學鍵,而不是分子中各原子(如像前人所假定的)對於生色團電子的微擾作用來計算,兩者的主要不同點在於是否考慮鍵的多極矩。 2.建議在旋光度的計算中,共價單鍵可以看作是由兩個處於鍵端的正電荷和一個以單中心狀態函數,表示出來的電子雲所組成。根據這個假定計算了環戊酮的甲基衍生物的旋光度,其結果與實驗值甚爲一致(詳見結果討論)。 3.計算結果證明甲基的內旋轉對於旋光度的影響很大,例如順式和反式構型的3甲基環戊酮的旋光度,應分別為+44°和30°。 4.指定了3甲基環戊酮的絕對構型,其結果Eyring所指定者相反。  A new method is proposed in this paper for the evaluation of the following three important kinds of threecenter and fourcenter 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 abovementioned 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... A new method is proposed in this paper for the evaluation of the following three important kinds of threecenter and fourcenter 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 abovementioned 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 endpoint 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 midpoint of the bond and evaluate its value inside and outside the ellipsoid passing through the endpoint 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。  A new set of values of the effective principal quantum numbers and screening constants in  本文考虑了 s 和 p 电子的区别以及半充满前的 p 电子和半充满后的 p′电子的区别,并在此基础上改进了 Slater 的原子能级近似计算法中有效主量子数和屏蔽常数的数值。用改进的数据计算了各原子的各极电离能和 X 光光谱线系的极限,其结果与实验的近似程度一般都比 Slater 法好。   << 更多相关文摘 
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