Openmp result differs in LCAO cases

[Qianruipku]

Describe the Bug

When we use openmp (OMP_NUM_THREADS not equal to 1), the LCAO results are different.
For example of 830_NO_GO_LT_fcc:



Worse still, sometimes segmental fault may occur in 830_NO_GO_LT_fcc!

Expected behavior

A clear and concise description of what you expected to happen.

To Reproduce

Steps to reproduce the behavior:

1. [e.g. clone the source code from ...]
2. [e.g. configure CMake with options ...]
3. [e.g. build ABACUS with ...]
4. [e.g. run ABACUS with ...]
   It is recommended to attach your calculation case here for the developers to reproduce the bug.

Environment

• OS: [e.g. Ubuntu 20.04]
• Compiler: [e.g. gcc 5.1.0]
• Dependencies: [e.g. Intel MKL, OpenBLAS]

Additional Context

Add any other context about the problem here.

[Qianruipku]

For test 211_NO_S2_elec_add，if we use "export OMP_NUM_THREADS=1" or "export OMP_NUM_THREADS=2".
The test cannot be passed.

OMP_NUM_THREADS = 4 can pass the test.

It means the results of different OMP_NUM_THREADS are different.

[dyzheng]

Test the Gamma_only cases:
modify: tests/integrate/Autotest.sh: line12 ' case="^[^#].*_GO_ .*$" ' and set correct abacus path in line4,
run Autotest.sh dozens of times and check if any case failed.

[dyzheng]

Can you give more information about your top three graphs? @Qianruipku

[Qianruipku]

cd 830_NO_GO_LT_fcc
export OMP_NUM_THREADS=4
../../../build/abacus
(get figure 1)
../../../build/abacus
(get figure 2)
../../../build/abacus
(get figure 3)

[caic99]

Using OMP_NUM_THREADS=4 mpirun -n 4 abacus, I got the following result:

 ITER   TMAG      AMAG      ETOT(eV)       EDIFF(eV)      DRHO       TIME(s)    
 GE1    -4.77e-14 4.78e-14  -2.037124e+02  0.000000e+00   1.247e-01  2.164e-01  
 GE2    -2.19e-13 2.20e-13  -2.037347e+02  -2.231557e-02  3.101e-02  2.096e-01  
 GE3    -1.15e-12 1.15e-12  -2.037348e+02  -3.447545e-05  8.795e-03  2.082e-01  
 GE4    -5.22e-12 5.22e-12  -2.037348e+02  1.445857e-05   2.591e-03  2.071e-01  
···
 GE25   -2.50e-01 2.55e-01  -2.037509e+02  -3.757284e-06  3.298e-04  2.144e-01  
···(TMAG & AMAG converged)
 GE42   -2.50e-01 2.55e-01  -2.037509e+02  1.691800e-13   6.333e-10  2.079e-01  

For OMP_NUM_THREADS=1 mpirun -n 4 abacus, TMAG & AMAG output is always 0.
@dyzheng Could you take a look at this?

[dyzheng]

In collinear spin calculation for LCAO base, initial magnetization should not be 0, which would cause Hamiltonian of spin-up and spin-down always be same. The convergent state is not the real fundamental state.
In this issue, OMP_NUM_THREADS>1 would introduce a little digital error in "Gint" procedure, which lead to real fundamental state.
I will fix this test case and add warnings in ABACUS.