Kepler Predictor–Corrector Algorithm - Yale Chemistry - Yale ...

Journal of Chemical Theory and Computation
M and e, where each Halley’s iteration is followed by a Newton
Raphson optimization. The iterative scheme typically converges
to machine accuracy in about three iterations. 37 The initial guess
ε (0) a is obtained, as follows:
Elliptic Kepler Equation. For elliptic motion (0 e M e π and
0 e e e 1) and small M, we expand the sin function in eq 56 to
third order, and the resulting approximation of M is substituted
in eq 55, giving a cubic approximation of the elliptic Kepler
equation:
0 ¼ ε 3 00 þ 3qε 00 2r ð58Þ
where q = 2(1 e)/e and r =3M/e. Solving eq 58, we obtain
2r
ε 00 ¼
c 2 v þ q ð59Þ
þðq=wÞ2
where c v =((r 2 + q 3 ) 1/2 + r) 1/3 . On the other hand, for large M,a
good initial guess is
ε 01 ¼ M
Therefore, we define an initial guess that is valid for intermediate
values of M, asanM-weighted average of the small M guess ε 00
and the large M guess ε 01 :
ε ð0Þ
a ¼ 1 π ðM ε 01 þðπ MÞε 00 Þ ð60Þ
¼ 1 π ðM2 þðπ MÞε 00 Þ ð61Þ
Hyperbolic Kepler Equation. For hyperbolic motion (M >0
and 1 e e) with small M, the sinh function in eq 56 is expanded to
third order, and the resulting approximation of M is introduced
into eq 55 to give the cubic approximation to the hyperbolic
Kepler equation:
0 ¼ ε 3 00 þ 3qε 00 2r ð62Þ
where q =2(e 1)/e and r =3M/e, with the same formal solution
introduced by eq 59, although there are different values of q and r.
Bounded coefficients are obtained through an iterative procedure
based on the hyperbolic equation divided by e, as follows:
0 ¼ M ε þ e sinh½εŠ ð63Þ
0 ¼ L gε þ sinh½εŠ ð64Þ
with L = M/e and g =1/e. For large L, a good initial guess is
ε 01 ¼ sinh 1 L
ð65Þ
which is again mixed with ε 00 , as follows:
ε ð0Þ
a ¼ M ε 01 þ 1 ε 00
M þ 1
¼ M sinh 1 L þ ε 00
M þ 1
3. ELECTRON SCATTERING
ð66Þ
ð67Þ
3.1. Time-Dependent Wigner Transform. This section
illustrates the capabilities of the KPC algorithm, introduced in
section 2, as applied to semiclassical dynamics simulations of
electron proton scattering processes in the Wigner-transform
ARTICLE
time-dependent picture. Simulations consider the problem of
electron scattering from stationary protons, as described within
the Born Oppenheimer approximation (i.e., with m = m e and
the Coulombic parameter γ = 1 au, so that μ = γ/m = 1 au).
The initial state for the scattering electron is defined by the
three-dimensional Gaussian:
!
ψ 0 ðxÞ ¼ð2πσ 2 Þ 3=4 ðx x 0 Þ 2
exp
4σ 2 þ i p p 0 ðx x 0Þ ð68Þ
with average position x 0 and momentum p 0 . The corresponding
Wigner transform: 39
P 0 ðx, pÞ ¼ 1
Z ∞
ð2πpÞ 3 ∞ ψ 0 x þ s
s
ψ
2 0 x e ip 3 s ds ð69Þ
2
" #
1
¼
ð2πpσ x σ p Þ 3 exp ðx x 0 Þ 2 ðp p 0 Þ 2
2σ 2 x 2σ 2 ð70Þ
p
defines the initial phase-space distribution function, where σ x = σ
and σ p is defined by the uncertainty relation σ x σ p = p/2.
The full quantum-mechanical Wigner distribution P QM t (x,p)is
computed as
Pt
QM 1
Z ∞
ðx, pÞ ¼
ð2πpÞ 3 ∞ ei=pp 3 s ψ t x þ s
ψ
2 t x
s
2
ds
ð71Þ
where ψ t is the solution of the time-dependent Schr€odinger
equation
!
ip ∂ ∂t ψ tðxÞ ¼ ^p2 þ VðxÞ
2m e
ψ t ðxÞ
ð72Þ
with
VðxÞ ¼
q j
jx R j j
ð73Þ
∑
j
where the sum is over all protons j, with charge q j =+e, and
coordinates R j . ψ t (x) is represented on a three-dimensional grid
and propagated according to the standard Split Operator Fourier
Transform (SOFT) method. 40,41 The grid is defined as follows:
x αk = x α0 + kΔx, k = 1, 2, ..., 128, where α =1 3 enumerates the
Cartesian directions, x 10 = 4 Å, and x 20 = x 30 = 5 Å. The grid
spacings Δx = 10/128 Å and Δt =10 4 fs define a sufficiently fine
space time grid that ensures an accurate representation of the
oscillatory structure of ψ t (x), even during high-energy collisions
(e.g., collisions with tens of electronvolts). The full-quantum
propagation is based on the short-time Trotter approximation of
the time-evolution operator:
ψ tþΔt ðxÞ ¼Uðt, t þ ΔtÞ ψ t ðxÞ
ð74aÞ
≈ e i=pVΔt=2 e i=p^p2 =2m eΔt e i=pVΔt=2 ψ t ðxÞ
ð74Þ
The time-evolved semiclassical Wigner distribution P SC t (x,p) is
computed as follows:
Z ∞
Pt
SC ðx, pÞ ¼ð2πÞ 3 ds
∞
Z ∞
∞
Z ∞
dp 0 dx 0 e iðp
∞
p t Þs δðx t xÞ P 0 ðx 0 , p 0 Þ
ð75aÞ
¼ 1 N ∑N δðp p t ðjÞÞ δðx t ðjÞ xÞ ð75Þ
j ¼ 1
where x t (j) andp t (j) are coordinates and momenta, obtained by
classical KPC propagation. The initial coordinates and momenta
29 dx.doi.org/10.1021/ct200452h |J. Chem. Theory Comput. 2012, 8, 24–35