原子的解析波函数

我们设计了一套变分波函数,用来计算了周期表中前面十个原子的能量。我们设计的单电子试探波函数具有下列形式:1s:ψ₁(r)=N₁e^-μαr1+(μbr)²], 2s:ψ₂(r)=N₂(μr)e^-μr-Ne^-μcr], 2p:ψ₃(r)=N₃(μdr)cosθe^-μdr, ψ₄(r)=N₄(μdr)sinθe^iφ-μdr, ψ₅(r)=N₅(μdr)sinθe^-iφ-μdr。式中的a,b,c,d及μ为五个变分参数。N₁,N₂,N₃,N₄与N₅为归一化因子;N由ψ₁与ψ₂的正交条件来决定。用这种波函数来计算原子的能量,所得的结果比莫尔斯等人(P.M.Morse,L.A.Young and E.S.Haurwitz)用他们设计的四参数波函数所算得的结果为好,更接近实验值,同时也接近于由自洽场所算出的结果。若我们的波函数中固定c等于1不变,这时就变为只有四个参数的波函数,结果仍比莫尔斯等人的好。

THE ANALYTIC WAVE FUNCTIONS FOR ATOMS

The energies of the first ten atoms in the periodic table are calculated with a set of new variational wave functions. The form of the wave functions used is as follows: 1s:ψ₁(r)=N₁e^-μαr1+(μbr)²], 2s:ψ₂(r)=N₂(μr)e^-μr-Ne^-μcr], 2p:ψ₃(r)=N₃(μdr)cosθe^-μdr, ψ₄(r)=N₄(μdr)sinθe^iφ-μdr, ψ₅(r)=N₅(μdr)sinθe^-iφ-μdr. There are five parameters in all, two for the ls function, two for the 2s and one for the 2p functions. The parameter μ is a scale factor, the best value of which can be determined analytically, leaving but four parameters to be determined numerically. N₁,N₂,N₃,N₄ and N₅ are normalization factors. The constant N is fixed so that ψ₂ is orthogonal to ψ₁. The energies and the parameters of the various states are determined by the variational method. The results of this calculation are better than that calculated by Morse, Young and Haurwitz with their wave functions containing four parameters. If we put c=1 in our wave functions, then there are four parameters only in all and it will be found that the results are still better than that found by Morse et al.