Atomic Structure Theory

40 2 Central-Field Schrödinger Equation
This is the basic algorithm used to advance the solution to the radial
Schrödinger equation from one point to the next. Using this equation, we
achieve the accuracy of the predict-correct method without the necessity of
separate predict and correct steps. To start the integration using (2.57), we
must give initial values of the two-component function f(t) atthepoints
1, 2, ··· ,k. The subroutine adams is designed to implement (2.57) for values
of k ranging from 0 to 8.
2.3.2 Starting the Outward Integration (outsch)
The k initial values of y[j] required to start the outward integration using the
k + 1 point Adams method are obtained using a scheme based on Lagrangian
differentiation formulas. These formulas are easily obtained from the basic
finite difference interpolation formula (2.45). Differentiating this expression,
we find
dy
[n − j] =− log(1 −∇)(1−∇)
dx
j y[n] . (2.58)
If (2.58) is expanded to k termsinapowerseriesin∇, and evaluated at
the k + 1 points, j =0, 1, 2, ··· ,k, we obtain the k + 1 point Lagrangian
differentiation formulas. For example, with k = 3 and n = 3, we obtain the
formulas:
dy
[0] =
dt
1
1
(−11y[0] + 18y[1] − 9y[2]+2y[3]) −
6h 4 h3y (4) , (2.59)
dy
[1] =
dt
1
1
(−2y[0] − 3y[1]+6y[2] − y[3]) +
6h 12 h3y (4) , (2.60)
dy
[2] =
dt
1
1
(y[0] − 6y[1] + 3y[2]+2y[3]) −
6h 12 h3y (4) , (2.61)
dy
[3] =
dt
1
1
(−2y[0] + 9y[1] − 18y[2]+11y[3]) +
6h 4 h3y (4) . (2.62)
The error terms in (2.59-2.62) are found by retaining the next higher-order
differences in the expansion of (2.58) and using the approximation (2.51).
Ignoring the error terms, we write the general k + 1 point Lagrangian differentiation
formula as
k
dy
[i] = m[ij] y[j] , (2.63)
dt
j=0
where i =0, 1, ··· ,k, and where the coefficients m[ij] are determined from
(2.58).
To find the values of y[j] at the first few points along the radial grid, we
first use the differentiation formulas (2.63) to eliminate the derivative terms
from the differential equations at the points j =1, ··· ,kand we then solve the
resulting linear algebraic equations using standard methods. Factoring r ℓ+1
from the radial wave function P (r) ,