推广的Eckart定理与原子激发态的新变分法
Extended Eckart Theorem and New Variation Method for Excited States of Atoms
摘要：
我们将Eckart定理从基态推广到激发态，在激发态的变分判据中引入了能量增加。结果表明，一个非常好的激发态试探函数的能量可以略低于精确的本征值。此外，由试验激发态波函数计算的能量，这是最接近的确切本征态通过Gram-Schmidt正交归一化到一个基态近似，也低于确切的本征值。为了避免基于Hylleraas，Undheim，McDonald [HUM]和Eckart定理的上限变分理论中固有的变分限制，我们提出了一种新的变分泛函Omega-n，并证明了它在本征态处具有局部极小值，从而允许通过改变试探波函数无限逼近本征态.作为一个例子，我们用新的变分泛函计算了三重态S（偶）氦原子的能量和径向期望值，并与用HUM和Eckart定理计算的结果进行了比较。初步的数值计算结果表明，计算得到的激发态第三S（偶）和第四S（偶）激发态的能量可能略低于本文所证明的广义Eckart定理的精确本征值（HUM理论无法达到），而近似的波函数则优于HUM理论.
英文摘要：
We extend the Eckart theorem, from the ground state to excited statew, which introduces an energy augmentation to the variation criterion for excited states. It is shown that the energy of a very good excited state trial function can be slightly lower than the exact eigenvalue. Further, the energy calculated by the trial excited state wave function, which is the closest to the exact eigenstate through Gram-Schmidt orthonormalization to a ground state approximant, is lower than the exact eigenvalue as well. In order to avoid the variation restrictions inherent in the upper bound variation theory based on Hylleraas, Undheim, and McDonald [HUM] and Eckart Theorem, we have proposed a new variation functional Omega-n and proved that it has a local minimum at the eigenstates, which allows approaching the eigenstate unlimitedly by variation of the trial wave function. As an example, we calculated the energy and the radial expectation values of Triplet-S(even) Helium atom by the new variation functional, and by HUM and Eckart theorem, respectively, for comparison. Our preliminary numerical results reveal that the energy of the calculated excited states 3rd Triplet-S(even) and 4th Triplet-S(even) may be slightly lower than the exact eigenvalue (inaccessible by HUM theory) according to the General Eckart Theorem proved here, while the approximate wave function is better than HUM.
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