Since we represent the time evolution operator Sl ≡ exp(−iH0l∆t/2) in complete eigensets of atomic Hamiltonian H0l, solving the eigenvalue equation Eq. (2.18) is the central part of the P-space TDSE. Before proceeding, let’s examine the atomic potential first. In the single-active-electron approximation, only the motion of the outer most electron is con-sidered. The interaction between this electron and other inner shell electrons and atomic core can be modeled as a model potential [14, 60, 61]. The form of a model potential is not unique. We list one for example [61]:
V (r) =−1
r −a1e−a2r+ a3re−a4r+ a5e−a6r
r (2.33)
When r → ∞, V (r) → −1/r which is the long-range Coulomb potential between the outer most electron and the ion core. The second short-range potential term account for the screened effect caused by other inner shell electrons where a′is are variation parame-ters to be optimized by fitting the energy levels from experiment data. This can be done by, for example, density functional calculations. Fourier transformation of this model potential is where k =|p − q|. Except the first long-range Coulomb potential term have a singularity at k = 0, others are well-behaved functions in the p-space and are easy to handle. The singularity in long-range Coulomb potential make it difficult to calculate eigenstates as well as eigenvalues accurately. To remove this singularity, Land´e subtraction technique has been proposed and the potential part of Eq. (2.18) is rewritten as [55]:
∫ basis, see Eq. (2.15), and has the following form:
vl(p, q) =− 1
πpqQl(p2 + q2
2pq ) (2.37)
Ql is Legendre function of the second kind and is defined as: and add it back. By this trick, the terms in the square bracket tend to zero faster than the Coulomb potential vl(p, q) tends to infinity when q→ p and singularity is thus removed.
Hence we can redefine vl(p, q) for convenience [43]
vl(p, q) =
0 p = q
Vl(p, q) p̸= q (2.39)
Then, the eigenvalue equation Eq. (2.18) can be rewritten as:
[p2
and the Legendre function of the second kind can be expressed alternatively as [62]
Ql(z) = 1
We should integral the two terms on the right hand side of Eq. (2.46) from 0 to∞ and these two terms can be calculated analytically. The first term is
1
l Il l Il
Table 2.1: Numerical values of Ilfor l=0-19. [56].
We neglect the detail calculation of the second term and just define its as Il, so Jl(p) = π
2 − Il (2.48)
Numerical values of Ilfor l = 0−19 are listed in Table 1 [56]. However, in the numerical calculation, we can only set a finite upper bound pmax rather than ∞. Since Jl(p) is obtained by integrating momentum q from 0 to∞, setting a larger upper bound pmax as well as a large number of grid points for obtaining accurate eigenstates and eigenvalues is expected. As a result, the calculation is inefficient.
This problem can be cured by setting the integration upper bound of Jl(p) to a finite value pmax. We denote the modified Jlby Jlf inite(p). That is where s = p/pmax. The second term of Jlf inite(p) can be calculated numerically. Together with the numerical integration of ql(p) from 0 to pmax, the eigenvalue equation Eq. (2.40) is ready to solve [63]. where L is mapping parameter and xm = 2L/pmax. The grid points are denser at small p for smaller L. x is discretized by using Gauss-Lobatto quadrature where the N grid points {xi} are the roots of the derivative of Legendre polynomial PN +1′ (x) [64].
State E(nl)− exact ∆Φ Table 2.2: Comparison of eigenvalues and wave functions of a hydrogen atom between
”Present” Land´e substraction method method with finite limits and the ”Ordinary” Land´e substraction method. [E(nl)-exact] is the deviation of energy levels for the first few low-lying states. ∆Φ is the root-mean-square deviation of the wave function. 2048 grid points and pmax = 100 a.u. are used. 1.76(−6) ≡ 1.76 × 10−6[44]. Table 2.3: The same as Table 2.3, but 2048 grid points and pmax = 2000 a.u. are used [44].
In the Table 2.2 and 2.3, we present the energy deviation and root-mean-square de-viation of the radial wave functions between our numerical results and exact ones of a hydrogen atom. Root-mean-square deviation of the wave functions is defined as:
∆Φ =
√ 1 N
∫
p2dp|Φ(p) − Φexact(p)|2 (2.52) Φexact is the exact wave function of a hydrogen atom in the P-space. and first few low-lying states are listed in the following [57, 58]:
F10(p) = 25/2
F21(p) = 128
√3π p
(4p2+ 1)3 (2.55)
”Ordinary” means the results of using typical Land´e subtraction method while ”Present”
means the results of using Land´e subtraction method with finite integration limits. In Table 2.2, 2048 grid points and pmax = 100 a.u. are used. In Table 2.3, 2048 grid points and pmax = 2000 a.u. are used. As mentioned above, typical Land´e subtraction method needs a larger pmax to ensure the accuracy. Therefore, the ”Ordinary” results for pmax = 2000 a.u. are more accurate than those for pmax = 100 a.u.. We can find the ”Present” results improve the accuracy to several order than ”Ordinary” results at pmax = 2000 a.u. as well as pmax = 100 a.u.. The results are more accurate at lager pmax for both methods, especially for the s states. This is because P-space radial wave function tends to 1/pl+4 when p → ∞. So, s states (l = 0) are the most diffusive states, a larger pmax is needed to reach higher accuracy. The improvement of the results to an acceptable accuracy at small pmax by the ”Present” method will make the time propagation of TDSE more efficient, as we will see in the next section.
In Fig. 2.1, we compare the numerical wave function for the ”Present” Land´e sub-straction method method with finite limits to exact ones for the first few low-lying states of a hydrogen atom. 2048 grid points are used and left column for pmax = 100 a.u. while right column for pmax = 2000 a.u.
Figure 2.1: Comparison of the numerical wave functions of ”Present” Land´e substraction method method with finite limits with the exact ones for the first few low-lying states of a hydrogen atom. 2048 grid points are used and left column for pmax = 100 a.u. while right column for pmax = 2000 a.u..