Overlap integration among basis functions about fac HOT 23 OPEN

To be clear, what I want is this matrix

which may be a function (tensor) of initial and final basis indexes (mu and nu) as well as the multipole L.

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you can use SetCILevel(-1) to disable CI, then calculate the multipole matrix emements with the TRTable function. the levels directly correspond to basis functions in this case.

However, if you are using mixing coefficients from another code, it's unlikely that the basis functions in that code would be exactly the same as those used in FAC, as the radial wavefunctions most likely differ.

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Hi, @mfgu

Thanks for the reply.

you can use SetCILevel(-1) to disable CI, then calculate the multipole matrix emements with the TRTable function.

Yes, I thought this, but the sign of the element will be gone?

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the last column in the output is the matrix element with the sign when a specific multipole moment is specified in the TRTable function. e.g., TRTable(fn, lower, upper, -1) for E1.

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Ah, nice. It is what I exactly want. Thanks.

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Hi @mfgu

I am working on this, but still have some trouble.

I tried to reproduce the multipole moment (line strength) based on the following equation,

I computed
1 Hamiltonian with configuration interaction
2 multipole moment without configuration interaction

From 1, I could compute b_fu and b_iv in the above equation; they are just eigenvectors of Hamiltonian.

From 2, in principle, I could calculate the rest of the part

However, I couldn't reproduce the original multipole element.

• For the parity-changing transitions (E1, M2, E3), the line strength is almost reproduced (but not perfect)
• For the parity-conserved transitions (M1, E2, M3), the reproduced line strengths are completely different from that by FAC.
  For example, if I calculate naively, the finite line-strength is computed for the diagonal (S_ff > 0).

I think something is omitted in equation (24).
I saw the source and this function

I would be grateful if you could give me some advice.

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I used the following for testing

SetAtom('Ni')
Config('lev_3d9_4s1', '1s2 2s2 2p6 3s2 3p6 3d9 4s1')
Config('lev_3d9_4p1', '1s2 2s2 2p6 3s2 3p6 3d9 4p1')
ConfigEnergy(0)
OptimizeRadial('lev_3d9_4s1')
ConfigEnergy(1)
Structure('Ni_small.en', 'Ni_small.ham', ['lev_3d9_4s1', 'lev_3d9_4p1'])
BasisTable('Ni_small.basis')
MemENTable('Ni_small.en')
SetTransitionCut(1e-6)
TransitionTable('Ni_small.tr', ['lev_3d9_4s1', 'lev_3d9_4p1'], ['lev_3d9_4s1', 'lev_3d9_4p1'], -3)
TransitionTable('Ni_small.tr', ['lev_3d9_4s1', 'lev_3d9_4p1'], ['lev_3d9_4s1', 'lev_3d9_4p1'], -2)
TransitionTable('Ni_small.tr', ['lev_3d9_4s1', 'lev_3d9_4p1'], ['lev_3d9_4s1', 'lev_3d9_4p1'], -1)
TransitionTable('Ni_small.tr', ['lev_3d9_4s1', 'lev_3d9_4p1'], ['lev_3d9_4s1', 'lev_3d9_4p1'], 1)
TransitionTable('Ni_small.tr', ['lev_3d9_4s1', 'lev_3d9_4p1'], ['lev_3d9_4s1', 'lev_3d9_4p1'], 2)
TransitionTable('Ni_small.tr', ['lev_3d9_4s1', 'lev_3d9_4p1'], ['lev_3d9_4s1', 'lev_3d9_4p1'], 3)
FinalizeMPI()

and I got the following,

For parity-conserved transition, there is a huge difference, not only in the diagonal.
This is neutral nickel.
I am not sure how the relativistic effect is important for this case...

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Energy file looks like this

FAC 1.1.5
Endian	= 0
TSess	= 1576536733
Type	= 1
Verbose	= 1
Ni Z	=  28.0
NBlocks	= 1
E0	= 0, -4.13122011E+04

NELE	= 28
NLEV	= 16
  ILEV  IBASE    ENERGY       P   VNL   2J
     0     -1  0.00000000E+00 0   400    6 1*2.2*8.3*17.4*1                 3d9.4s1                                          3d+5(5)5.4s+1(1)6
     1     -1  1.11532151E-01 0   400    4 1*2.2*8.3*17.4*1                 3d9.4s1                                          3d+5(5)5.4s+1(1)4
     2     -1  2.18253137E-01 0   400    2 1*2.2*8.3*17.4*1                 3d9.4s1                                          3d-3(3)3.4s+1(1)2
     3     -1  6.87719716E-01 0   400    4 1*2.2*8.3*17.4*1                 3d9.4s1                                          3d-3(3)3.4s+1(1)4
     4     -1  2.99682911E+00 1   401    4 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d+5(5)5.4p-1(1)4
     5     -1  3.13381309E+00 1   401    2 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d+5(5)5.4p+1(3)2
     6     -1  3.17369793E+00 1   401    8 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d+5(5)5.4p+1(3)8
     7     -1  3.21177697E+00 1   401    0 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d-3(3)3.4p+1(3)0
     8     -1  3.21200115E+00 1   401    6 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d+5(5)5.4p-1(1)6
     9     -1  3.33965544E+00 1   401    4 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d-3(3)3.4p-1(1)4
    10     -1  3.43886347E+00 1   401    6 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d+5(5)5.4p+1(3)6
    11     -1  3.50782720E+00 1   401    4 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d+5(5)5.4p+1(3)4
    12     -1  3.53238109E+00 1   401    6 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d-3(3)3.4p+1(3)6
    13     -1  3.66456060E+00 1   401    2 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d-3(3)3.4p+1(3)2
    14     -1  3.67396000E+00 1   401    4 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d-3(3)3.4p+1(3)4
    15     -1  4.00078372E+00 1   401    2 1*2.2*8.3*17.4*1                 3d9.4p1                                          3d-3(3)3.4p-1(1)2

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Thanks.
I am studying some statistics of the hamiltonian, eigenvector and line strength and therefore I need these matrices rather than the results...

I modified fac source code and enabled the line strength among the levels with the same energy.
However, the discrepancy increased and I have no idea...

Do you have any comments?

I changed

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@mfgu
Thank you for your help!

Do you have any idea about dni.py?
I will try to make it, but it would be nice if I could hear your advice...

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Nice! Thank you so much!!

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I think this option does not reproduce the transition rate or even worsen the result.
Probably something is missing...

For me, it is ok as E1 transitions are dominant and they are reproduced relatively well from the computation without the options.
But I am a little wondering if it is an indication of some bugs inside...

With the output with the options

SetOption('transition:aw', 0)
SetOption('transition:ls_all', 1)

Without the options

# SetOption('transition:aw', 0)
# SetOption('transition:ls_all', 1)

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Ah, you are right.
Something should be wrong in MY script.
Thank you for taking time. I'll try to fix.

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I found the reason.
With options

SetOption('transition:aw', 0)
SetOption('transition:ls_all', 1)

the transition matrix is not symmetric.

With the options

Without the options

I assumed it is symmetric in my script.
Probably the upper diagonal is only correct?

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the line strength matrix is only symmetric up to a phase factor. depending on the angular momenta, S_ij=S_ji or S_ij=-S_ji

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