Complete Single-Center Basis Sets in Atomic Calculations. Scott I. Young. 1. , Kyle Rollin. 1. , Michael W.J. Bromley. 1,2. , Jim Mitroy. 3. , Kurunathan Ratnavelu.
Complete Single-Center Basis Sets in Atomic Calculations 1
1
1,2
3
Scott I. Young , Kyle Rollin , Michael W.J. Bromley , Jim Mitroy , Kurunathan Ratnavelu
4
1
3
Department of Physics and 2 Computational Science Research Center, San Diego State University, San Diego, California Faculty of Technology, Charles Darwin University, Darwin, Australia, 4 Faculty of Science, Universiti Malaya, Kuala Lumpur, Malaysia
Abstract
Expectation Values of the He Ground State
CI-Kohn Variational Scattering Method
Single particle orbitals centered on the nucleus are the most commonly used basis in large-scale calculations of atomic structure. The convergence towards a complete basis set, with respect to both the number and angular momenta of the orbitals included in a configuration interaction (CI) expansion, has been investigated using the ground and excited states of the helium atom. This enabled energies to be determined to within 10−8 Hartree, whilst the convergence of the electron-electron δ-function and other relativistic corrections have been examined in detail. Unusual convergence patterns in the CI-Kohn variational scattering method have been observed and, however, high-precision calculations of elastic positron scattering from atoms are achievable.
The groud state energy and the δ-function for Helium are determined by finding expectation values using a series of N and ` dimensional basis sets. Extrapolation techniques are then used to estimate the expectation values at the limit that N and ` go to infinity.
The scattering basis adds two continuum functions to the Hamiltonian,
-2
He (1Se) 〈δ〉∞ Extrapolations λ10 log10(|〈δ〉∞-0.1557637174|)
〈E〉N
-6
4-term 3-term 2-term
-8 -10
5-term 6-term
〈δ〉N -6
-8
3-term
20
30
40
50
60
10
20
30
4-term
40
N
L Norb 0 44 1 80 2 115 11 430 12 465 `→∞ Exact
There are many approaches to solving the three-body problem numerically. Some include: Hartree-Fock, Many-Body Perturbation Theory, Hylleraas Method, and Configuration Interaction (CI). CI is not the most efficient method for exploring atoms because the method is based on single-center orbital expansions which converge slowly. Configurations are a set of LS coupled radial orbitals (Laguerres). cI |ΦI ; LSi.
50
60
N
Configuration Interaction (CI) NCI 990 1656 2286 7956 8586
hEiL (Calc) -2.879 028 760 -2.900 516 228 -2.902 766 823 -2.903 709 652 -2.903 712 786 -2.903 723 252
hEiL (N → inf) -2.879 028 766 -2.900 516 246 -2.902 766 852 -2.903 709 741 -2.903 712 882 -2.903 724 372 -2.903 724 377
CI is fundamentally the solution of a variational eigenproblem: c1 c1 H11 H12 . . . H1N c2 c2 H21 H22 H 2N = En . .. . . . .. .. .. .. . HN 1
HN 2
HN N
cN
cN
Finding p-values involves calculating expectation values with a finite set of orbitals (N)
It is known that the energy increments with respect to N , behave as ∆EN ≈
AE Np
+
BE N p+t
+
CE N p+2t
10
∆hEiL =
∆hEiN =
+
αE N 3.5
BE (L+ 21 )5
+
βE N4
+
+
γE N 4.5
+...
14 8
0.1
-1.04 -1.06
0.01
-1.08 0.001 8
10 12 N (#)
14
16
18
20
2
4
6
8 10 12 14 16 18 20 N (#)
12 7
10 6 8 He3Se 5
Positron-Hydrogen Threshold Scattering
6 He3Po He1Se
4
He1Po 3
2 20
25
30
35
40
45
50
10
12
N
14
16
18
20
22
24
26
N
Once the p-values are known extrapolation to the limit that N goes to infinity can be performed for the energy: Po 3 o P 3 e P 3 e S
N = 50 (Hartree) -2.122 596 765 537 -2.132 371 380 985 -0.704 662 331 634 -2.174 264 856 186
p-value 3.5 5.0 5.8 5.0
N → ∞ (Hartree) -2.122 596 772 963 -2.132 371 380 373 -0.704 662 331 637 -2.174 264 856 384
The lowest-order relativistic correction to the Hamiltonian is: 4 p ˆ H0 = − 3 2 . 8m c
Using first order perturbation theory the correction is extrapolated out to N → ∞ with a finite radial basis set.
+...
For L it is known that the t-value is 1, and that the power series has a p-value of 4 or 6. When looking at the convergence with respect to N it has been established that the t-value is 0.5 with a p-value of 3.5.
-1.02
Calculations converge independently for Nodd and Neven . The extrapolation methods used are able to fit a power series to each curve.
16
+ . . ..
CE (L+ 21 )6
He1Po He3Pe He3Po He3Se
18
9
The p-values and t-values are then determined by the calculated series. AE (L+ 21 )4
1
20 He1Po He3Po He3Se He1Se
1
Convergence with Respect to N and L
-1
6
4
Since CI is a variational method each calculation will produce an upper bound which will decrease as the calculation size increases. This upper bound behaves as a power series that converges to the actual value, as the calculation is allowed to grow to infinite size. By studying this convergence we are able to extrapolate very accurate values from fairly small sized calculations.
AN - AN+1
Scattering Length vs. Orbitals
-1.1 11
3
P
Hmn = hΦm |H|Φn i = hΦm |Te + Vpe |Φn i + hΦm |Ve1 e2 |Φn i
... ...
I1α + Iα1 c1 −(I10 + I01 ) 6 c2 7 6 −(I20 + I02 ) 7 I2α + Iα2 7 76 7 6 7 76 . 7 6 7 .. . .. 7 6 .. 7 = 6 7 . 76 7 6 7 IN α + IαN 5 4cN 5 4 −(IN 0 + I0N ) 5 1 2Iαα αt − (Iα0 + I0α ) 2 2
As the ` = 1 set of basis functions was added, the series showed dependence on whether the total number of basis functions was even or odd.
Energies and Relativistic Corrections for Restricted He Excited States
P
The Hamiltonian matrix is then populated by computing the matrix elements in terms of these orbitals:
... ...
3
CI-Kohn Convergence
The L = 1 wavefunction looks like: ΨL=1 = ΨL=0 + c4 ψ2p ψ2p + c5 ψ3p ψ3p + . . . .
I21 + I12 2I22 .. . I2N + IN 2 I2α + Iα2
32
Where Iab = hΨa |H −E|Ψb i and α is the combination of the scattering coefficients.
A (a0)
I
2I11 6 I12 + I21 6 6 .. 6 . 6 4I1N + IN 1 I1α + Iα1 2
-10 10
|Ψ; LSi =
giving us this set of linear equations to solve:
2-term
5-term
-14
∂ ∂αv =0=2 hΨt ; LS|H − E|Ψt ; LSi, ∂cn ∂cn
-4
A (a0)
log10(|E∞-Eexact |)
-4
-12
PNCI
The Kohn functional must be stationary with P respect to each variaN tional parameter, |Ψt ; LSi = α0 |φs i + α1 |φc i + cn |φn i.
-2 He (1Se) Energy Extrapolations λ10
√ cos(kr) −βr χc = k 1 − exp , kr
√ sin(kr) , χs = k kr
1
Po 3 o P 3 e P 3 e S
N = 50 hp4 i 40.140 115 838 396 39.597 557 890 054 00.580 323 564 781 41.851 571 085 622
p-value 2.0 2.0 4.0 4.0
N → ∞ hp4 i 40.140 117 030 730 39.597 558 193 327 00.580 323 565 010 41.851 570 929 099
Up to 20 orbitals were used to describe each ` value, plus the 2 scattering states. The calculations were extrapolated with respect to N , then Lmax . The scattering lengths are reported in units of a0 . L 0 1 0 1 2 10 ∞
No 45 45 20 40 60 220
Acalc 0.536 845 -1.114 599 0.536 848 -1.097 006 -1.539 204 -2.018 080
A∞ 0.536 845 -1.123 334 0.536 845 -1.130 388 -1.571 638 -2.072 717 -2.112 691
Zef f −calc 0.474 699 1.746 484 0.474 564 1.714 721 2.906 867 6.163 172
Zef f −∞ 0.474 755 1.758 479 0.474 782 1.766 809 3.022 185 6.924 971 8.795 777
Take Home Lesson Accurate estimates of expectation values for three-body systems can be determined with finite basis sets. These values, extrapolated to infinity, produce far better numbers than calculations with up to twice the amount of basis functions. Future calculations will include other three-body systems. We also plan to use the same techniques on the four-body problem, both in the bound and scattering cases.
