The magnetic moments becomes increasingly large due to small (1e-11) numerical fluctuations

[hongriTianqi]

Describe the bug

As can be seen below in an example, the small magnetic moment in the order of 1e-12 will induce larger and larger magnetic moments in y direction, resulting in difficulty of convergence in the lambda loop

Step (Outer -- Inner) =  6 -- 24          RMS = 3.7238680504e-06
Step (Outer -- Inner) =  6 -- 25          RMS = 7.5230684743e-07
Step (Outer -- Inner) =  6 -- 26          RMS = 5.2305459901e-08
Meet convergence criterion ( < 1.0000000000e-07 ), exit.
after-optimization spin: (print in the inner loop): 
ATOM 1   -4.4782397511e-08 -1.2412521673e-12 1.6999999730e+00
ATOM 2   1.4722431406e+00 -1.2428208289e-12 -8.5000002527e-01
after-optimization lambda: (print in the inner loop): 
ATOM 1   -1.4648089740e-02 0.0000000000e+00 -8.5237087421e-03
ATOM 2   -1.4705793350e-02 0.0000000000e+00 -8.4237631640e-03
Inner optimization for lambda ends.
===============================================================================
 GE7    -6.440357e+03  -1.215134e-03  3.030e-03  1.425e+02  
===============================================================================
Inner optimization for lambda begins ...
Covergence criterion for the iteration: 1.000e-07
initial lambda: 
ATOM 1   -1.4648089139e-02 -1.1764815605e-08 -8.5237089776e-03
ATOM 2   -1.4705793253e-02 1.1765657717e-08 -8.4237625258e-03
initial spin: 
ATOM 1   3.4261262311e-07 1.7520288126e-05 1.6999993302e+00
ATOM 2   1.4722427777e+00 -1.7520323655e-05 -8.4999936841e-01
target spin: 
ATOM 1   0.0000000000e+00 0.0000000000e+00 1.7000000000e+00
ATOM 2   1.4722431864e+00 0.0000000000e+00 -8.5000000000e-01
Step (Outer -- Inner) =  7 -- 1           RMS = 1.7536450148e-05
Step (Outer -- Inner) =  7 -- 2           RMS = 3.3443089017e-02
Step (Outer -- Inner) =  7 -- 3           RMS = 2.5955056083e-02
Step (Outer -- Inner) =  7 -- 4           RMS = 2.5927924763e-02
...
Step (Outer -- Inner) =  9 -- 93          RMS = 4.4377218400e-04
Step (Outer -- Inner) =  9 -- 94          RMS = 4.4377218201e-04
Step (Outer -- Inner) =  9 -- 95          RMS = 4.4377219502e-04
Step (Outer -- Inner) =  9 -- 96          RMS = 4.4377218185e-04
Step (Outer -- Inner) =  9 -- 97          RMS = 4.4377219201e-04
Step (Outer -- Inner) =  9 -- 98          RMS = 4.4377218491e-04
Step (Outer -- Inner) =  9 -- 99          RMS = 4.4377218733e-04
Step (Outer -- Inner) =  9 -- 100         RMS = 4.4377218912e-04
Reach maximum number of steps ( 100 ), exit.
after-optimization spin: (print in the inner loop): 
ATOM 1   3.6933035125e-04 2.3908847039e-04 1.6999417588e+00
ATOM 2   1.4723771494e+00 -2.3916070965e-04 -8.4965105393e-01
after-optimization lambda: (print in the inner loop): 
ATOM 1   -1.5843811389e-02 -1.3502661244e-07 -9.1852863177e-03
ATOM 2   -1.5876420578e-02 1.3625511696e-07 -9.1288001577e-03
Inner optimization for lambda ends.
===============================================================================
 GE10   -6.440358e+03  -9.363864e-05  4.719e-04  5.262e+02  
===============================================================================
Inner optimization for lambda begins ...
Covergence criterion for the iteration: 1.000e-07
initial lambda: 
ATOM 1   -1.5843811430e-02 -1.3505272075e-07 -9.1852863122e-03
ATOM 2   -1.5876420594e-02 1.3628123314e-07 -9.1288001966e-03
initial spin: 
ATOM 1   3.6931028468e-04 2.3912735290e-04 1.6999417913e+00
ATOM 2   1.4723771675e+00 -2.3919959239e-04 -8.4965108754e-01

Expected behavior

fix the bug

To Reproduce

180.tar.gz

Environment

No response

Additional Context

No response

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[hongriTianqi]

Still met numerical instability when increasing the value to 1e-10

Meet convergence criterion ( < 1.0000000000e-07 ), exit.
after-optimization spin: (print in the inner loop): 
ATOM 1   2.1135377358e-08 1.5681893429e-10 1.7000000074e+00
ATOM 2   1.4722432074e+00 1.4857703488e-10 8.5000000750e-01
after-optimization lambda: (print in the inner loop): 
ATOM 1   -1.2705585560e-02 0.0000000000e+00 -2.1193206280e-02
ATOM 2   -1.2001810852e-02 0.0000000000e+00 -2.1599489650e-02
Inner optimization for lambda ends.
===============================================================================
 GE16   -6.440739e+03  -1.210098e-03  1.932e-03  2.029e+02  
===============================================================================
Inner optimization for lambda begins ...
Covergence criterion for the iteration: 1.000e-07
initial lambda: 
ATOM 1   -1.2705585489e-02 -9.7075849115e-09 -2.1193206404e-02
ATOM 2   -1.2001810982e-02 9.6032862428e-09 -2.1599489659e-02
initial spin: 
ATOM 1   1.6988781714e-07 1.4383571552e-05 1.6999999200e+00
ATOM 2   1.4722430579e+00 -1.4379694521e-05 8.5000009254e-01
target spin: 
ATOM 1   0.0000000000e+00 0.0000000000e+00 1.7000000000e+00
ATOM 2   1.4722431864e+00 0.0000000000e+00 8.5000000000e-01
Step (Outer -- Inner) =  16 -- 1           RMS = 1.4382682084e-05
Step (Outer -- Inner) =  16 -- 2           RMS = 1.1216229030e-01
Step (Outer -- Inner) =  16 -- 3           RMS = 5.5398041356e-03
Step (Outer -- Inner) =  16 -- 4           RMS = 5.3719204620e-03

[hongriTianqi]

I have found the reason behind this issue is the numerical error accumulation in the quantity orbMulP.

Ok, how do I define the numerical error?

Answer: In the above attached example, I set target mag to [0,0,1.7], and constrain to [1,1,1], which means that I want to constrain atomic magnetic moments in three directions to 0, 0, and 1.7, respectively. During the iterations, it is expected that the atomic magnetic moments in x and y direction should keep zero as no reason should result in moments in x and y directions. However, due to numerical error, they are actually not zero.

Ok, another question, how much will the numerical error reach?

I have tested the above example, and see numerical error as large as 1e-11 happened.