All-order package

All-order programs

The all-order part of the package calculates corrections to the bare Hamiltonian due to the core shells for the CI code pconf. The all-order label refers to inclusion of the large number of terms (second-, third-, fourth-order, etc.) in order-by-order many-body perturbation theory expansion using the iterative solutions until the sufficient numerical convergence is achieved. This code version implements a variant of the linearized single double coupled-cluster (CC) method. This CC version has been developed specifically for atoms fully utilizing atomic symmetries and capable of being efficiently ran with very large basis sets (over 1000 orbitals), reaching negligible numerical uncertainty associated with the choice of basis set.

The all-order package consists of four programs: allcore-ci, valsd-ci, sdvw-ci, and second-cis. The first three compute core, core-valence, and valence-valence all-order excitations, respectively, and store resulting data in SGC.CON (small file) and SCRC.CON (up to a few GB file). The fourth program, second-cis, is described below. The all-order codes can be omitted if high precision is not required, in which case only second-cis is run, leading to a method referred to as CI+MBPT.

The following code executions read the files hfspl.1 and hfspl.2 and generate the files SGC.CON and SCRC.CON. It also takes in the input file inf.aov and writes the results to the respective out. files.

./allcore-rle-ci <inf.aov >out.core      # computes Sigma_ma, Sigma_mnab -> writes pair.3 file
./valsd-rle-cis <inf.aov >out.val        # computes Sigma_mv (and thus Sigma_1, the one-body correcton to the Hamiltonian), Sigma_mnva -> writes val2 and sigma files
./sdvw-rle-cis <inf.aov >out.sdvw        # computes Sigma_mnvw (Sigma_2, the two-body correction to the Hamiltonian) -> writes pair.vw and sigma1 files

Click here to see a description of inf.aov

12               # ncore - number of core shells
1 -1             # n and kappa of the core shells
2 -1
2  1
2 -2
3 -1
3  1
3 -2
3  2
3 -3
4 -1
4  1
4 -2
35  5                # nmax and lmax in correlation diagram summations
0   0                # internal parameters, do not change
30                   # max number of iterations for core
2 7 1            # Stabilizer code parameters, see below
0.d0                 # damping factor, not active not zero
1                # kval (key_en) - key for energies,  see explanation below
24               # nval - number of valence orbitals in list to follow
 5  -1 30        # n and kappa of the valence orbitals, max number of iteration
 6  -1 30
 7  -1 30
 8  -1 30
 5   1 30
 6   1 30
 7   1 30
 8   1 30
 5  -2 30
 6  -2 30
 7  -2 30
 8  -2 30
 4   2 30
 5   2 30
 6   2 30
 7   2 30
 4  -3 30
 5  -3 30
 6  -3 30
 7  -3 30
 4   3 30
 5   3 30
 4  -4 30
 5  -4 30

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Additional explanations:
2 7 1  # Stabilizer code parameters
2 - DIIS method (change to 1 for RLE method)
7 - number in iterations before stabilizer code runs
1 - do not change (controls the type of linear algebra code)

Note

It is important to always check all output files. In out.core and out.val, check that the core and valence orbitals have converged, respectively. If any cases completed 30 iterations (max) with values slowly drifting up, you can reduce the number of interactions to 3 or 5, and rerun the respective codes. Here, it’s important to look at how much energies fluctuated during the first few iterations. In cases of severe divergence, check all inputs for errors. If nothing is found, and the atom/ion with closed d shell, but not p shell, set kval=2.

Note

kval controls the values of \(\tilde{\epsilon}_v\) in denominators [1]. kval=1 is the default choice, where \(\tilde{\epsilon}_v\) for all \(nl_j\) is set to the DHF energy of the lowest valence \(n\) for the particular partial wave. For example for Sr, \(\tilde{\epsilon}_v\) for \(v=ns\) is set with the DHF energy of the \(5s\) state, \(v=np_{1/2}\) is set with the DHF energy of the \(5p_{1/2}\) state, \(v=nd_{3/2}\) is set with the DHF energy of the \(5d_{3/2}\) state, and so on. kval=2 is only used when the all-order valence energies are severely divergent. So far, this was observed with highly-charged ions with filling \(p\)-shell (e.g. Sn-like). By “severe”, we mean that the energies begin to diverge after the first or second iteration, immediately driving the correlation energy to be very large. Such a divergence cannot be fixed by a stabilizer. In this case, we have to identify which partial waves diverge and manually change the energies for these orbitals - we set them to the lowest DHF energy of the partial wave for which the all-order converged. For instance, if \(s\) diverges, but \(p\) does not, then set the \(ns\) energies to the lowest \(np_{1/2}\) DHF energies. The rest are left as DHF as in kval=1. The format would be as follows:

2                               # kval
3                               # lmax for the input to follow
0  -0.28000                         # l=0 energies
1  -0.22000  -0.22000           # l=1 energies p1/2 p3/2
2  -0.31000  -0.31000           # l=2 energies d3/2 d5/2
3  -0.13000  -0.13000           # l=3 energies f5/2 f7/2

MBPT program

second-cis is the fourth program of the all-order package. It calculates second-order MBPT corrections to the bare Hamiltonian for a much larger set of orbitals than the all-order codes, since high accuracy is not required for higher orbitals. It can be run on its own (CI+MBPT) or after the all-order codes, in which case it reads the all-order results and overlays them with MBPT coverage for the remaining orbitals. Such overlay of the all-order and MBPT corrections drastically improves the efficiency of the method.

The following code execution reads the files hfspl.1, hfspl.2 and HFD.DAT (to read the list of orbitals), and writes the files SGC.CON and SCRC.CON used by pconf. It also takes in the input file inf.vw and writes the results to the respective out.sdvw files.

./second-cis <inf.vw >out.second.vw   # writes SGC.CON and SCRC.CON files

Click here to see a description of inf.vw

12               # ncore - number of core shells
1 -1             # n and kappa of the core shells
2 -1
2  1
2 -2
3 -1
3  1
3 -2
3  2
3 -3
4 -1
4  1
4 -2
35  5                    # nmax and lmax in correlation diagram summations
195                              # Nmax  (max_orb) # of orbitals (from BASS.INPT) for how many Sigmas to calculate
3                                # Lmax (lvmax) for how many Sigmas to calculate, supersedes Nmax. Maybe best to set to 4 when 4f is    important
 9                               # Kmax (kvmax) for higest partial wave for Sigma2 S_Kmax(ijkl). May increase for f shell cases
 195 100                 # nav1 nav - maximum average of orbitals to calculate Sigma for nav1=(i+j)/2 for Sigma(ij) and nav=(i+j        +k+l)/4 for Sigma_K(ijkl)
0                        # if just second order, 1 to read all-order input from PAIR.VW and sigma1
1                        # kval, set the same as in the all-order, see explanations of inf.aov

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Additional explanations about Sigma_1 and Sigma_2 restrictions:
(1) Put the first number (Sigma 1) to be the last orbital you plan to list in ADD.INP.
Look up the number in BASS.INP and add the number of core shells.
Here, the last orbital to be included in pconf is 21d5/2, so the number is 195.
Example: Last orbital in BASS.INP:
183  2.1201  3  2.1201      # Here, the last orbital to be included in pconf is 21d5/2, so the number is 183+12 = 195
Note: this is inputted twice, keep it the same.
(2) Set 100 as the second value. This was tested a while ago, can increase.

References

[1]

M. S. Safronova, M. G. Kozlov, W. R. Johnson, and D. Jiang. Development of a configuration-interaction plus all-order method for atomic calculations. Phys. Rev. A, 2009.