diff --git a/source/source_hamilt/module_gint/gint_common.cpp b/source/source_hamilt/module_gint/gint_common.cpp
index 4f346576406..c466a817b0e 100644
--- a/source/source_hamilt/module_gint/gint_common.cpp
+++ b/source/source_hamilt/module_gint/gint_common.cpp
@@ -210,10 +210,10 @@ void merge_hr_part_to_hR(const std::vector<hamilt::HContainer<double>>& hr_gint_
                             + std::complex<double>(clx_i[is], clx_j[is]) * mat_nspin2->get_value(irow, icol);
                         }
                     }
-                    //fill the lower triangle matrix at -R by conjugate transpose of upper at R
-                    // This ensures H(-R) = H(R)^dagger, required for Hermiticity of H(k).
-                    // For real matrices (is=0,3), conj has no effect.
-                    // For complex matrices (is=1,2), conj is essential.
+                    // Transpose only the spatial indices within this fixed spin block.
+                    // The real Pauli-channel matrices obey V_mu(-R) = V_mu(R)^T.
+                    // Full spinor Hermiticity also exchanges the spin indices; the
+                    // conjugate coefficients are already in the opposite spin block.
                     if (iat1 < iat2)
                     {
                         auto lower_mat = lower_ap->find_matrix(-R_index);
@@ -221,7 +221,7 @@ void merge_hr_part_to_hR(const std::vector<hamilt::HContainer<double>>& hr_gint_
                         {
                             for (int icol = 0; icol < upper_mat->get_col_size(); ++icol)
                             {
-                                lower_mat->get_value(icol, irow) = std::conj(upper_mat->get_value(irow, icol));
+                                lower_mat->get_value(icol, irow) = upper_mat->get_value(irow, icol);
                             }
                         }
                     }
diff --git a/tests/03_NAO_multik/CASES_CPU.txt b/tests/03_NAO_multik/CASES_CPU.txt
index a85a0938ec3..c1a91cb6ced 100644
--- a/tests/03_NAO_multik/CASES_CPU.txt
+++ b/tests/03_NAO_multik/CASES_CPU.txt
@@ -69,3 +69,5 @@ get_pchg_k
 get_s
 scf_deltaspin2
 scf_deltaspin4
+scf_nspin4_nsoc_hr
+scf_nspin4_soc_hr
diff --git a/tests/03_NAO_multik/scf_nspin4_nsoc_hr/INPUT b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/INPUT
new file mode 100644
index 00000000000..12e01009f07
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/INPUT
@@ -0,0 +1,22 @@
+INPUT_PARAMETERS
+suffix            autotest
+calculation       scf
+symmetry          0
+pseudo_dir        ../../PP_ORB
+orbital_dir       ../../PP_ORB
+gamma_only        0
+ecutwfc           15
+scf_thr           1e-7
+scf_nmax          50
+noncolin          1
+nspin             4
+lspinorb          0
+basis_type        lcao
+ks_solver         scalapack_gvx
+dft_functional    pbe
+smearing_method   gauss
+smearing_sigma    0.02
+mixing_type       pulay
+mixing_beta       0.4
+mixing_gg0        3.0
+out_hsr           1 12
diff --git a/tests/03_NAO_multik/scf_nspin4_nsoc_hr/KPT b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/KPT
new file mode 100644
index 00000000000..c289c0158aa
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/KPT
@@ -0,0 +1,4 @@
+K_POINTS
+0
+Gamma
+1 1 1 0 0 0
diff --git a/tests/03_NAO_multik/scf_nspin4_nsoc_hr/README b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/README
new file mode 100644
index 00000000000..53f536b8ced
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/README
@@ -0,0 +1,7 @@
+Self-consistent nspin=4 O2 H(R)/S(R) regression, without SOC.
+O-O distance 2.3 bohr in an 18-bohr cubic cell; existing O FR PP and minimal 6-bohr 1s1p orbitals.
+PBE/LCAO, 15 Ry, Gamma 1x1x1; initial moments (1/sqrt(2),1/sqrt(2),0) muB per atom.
+Full SCF uses scf_thr=1e-7, scf_nmax=50, Pulay 0.4 and mixing_gg0=3; no DeltaSpin.
+References are complete raw 16x16 complex matrices with only R=0 and finite y magnetization.
+Repeated local GNU/Intel runs pass the unchanged matrix comparator; the unfixed parent fails H.
+These compact parameters test the assembly regression rather than material-property convergence.
diff --git a/tests/03_NAO_multik/scf_nspin4_nsoc_hr/STRU b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/STRU
new file mode 100644
index 00000000000..8a2d235ee6d
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/STRU
@@ -0,0 +1,22 @@
+ATOMIC_SPECIES
+O 15.999 O_ONCV_PBE_FR-1.0.upf
+
+NUMERICAL_ORBITAL
+O_gga_6au_60Ry_1s1p.orb
+
+LATTICE_CONSTANT
+18
+
+LATTICE_VECTORS
+1 0 0
+0 1 0
+0 0 1
+
+ATOMIC_POSITIONS
+Direct
+
+O
+0.0
+2
+0.5 0.5 0.43611111111111112 0 0 0 mag 0.7071067811865475 0.7071067811865475 0
+0.5 0.5 0.56388888888888888 0 0 0 mag 0.7071067811865475 0.7071067811865475 0
diff --git a/tests/03_NAO_multik/scf_nspin4_nsoc_hr/hrs1_nao.csr.ref b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/hrs1_nao.csr.ref
new file mode 100644
index 00000000000..91015c95b30
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/hrs1_nao.csr.ref
@@ -0,0 +1,56 @@
+ --- Ionic Step 1 ---
+ # print H matrix in real space H(R)
+ 1 # number of spin directions
+ 1 # spin index, E_Fermi = -6.68506 eV
+ 16 # number of localized basis
+ 1 # number of Bravais lattice vector R
+
+ user_defined_lattice
+ 9.52519
+ 1 0 0
+ 0 1 0
+ 0 0 1
+ O
+ 2
+ Direct
+ 0.5 0.5 0.436111111110222
+ 0.5 0.5 0.563888888889778
+
+ #----------------------------------------------------------------------#
+ #                               CSR Format                             #
+ # The outer loop corresponds to the number of Bravais lattice vectors. #
+ # The first line contains the index of the Bravais lattice vector      #
+ # (Rx, Ry, Rz), followed by the number of non-zero elements.           #
+ # The subsequent lines consist of three blocks of data, which are      #
+ # values, column indices, row pointers.                                #
+ #----------------------------------------------------------------------#
+
+ 0 0 0 96
+ # CSR values
+ (-2.004383330761e+00,0.000000000000e+00) (-4.378766338937e-02,4.378766338938e-02) (-4.154791598051e-01,0.000000000000e+00) (9.594409771163e-04,-9.594409771162e-04) (-1.004647828998e+00,0.000000000000e+00) (-8.828573496782e-03,8.828573496782e-03)
+ (1.093397372134e+00,0.000000000000e+00) (9.833707841451e-03,-9.833707841451e-03) (-4.378766338937e-02,-4.378766338938e-02) (-2.004383330761e+00,0.000000000000e+00) (9.594409771163e-04,9.594409771162e-04) (-4.154791598051e-01,0.000000000000e+00)
+ (-8.828573496782e-03,-8.828573496782e-03) (-1.004647828998e+00,0.000000000000e+00) (9.833707841451e-03,9.833707841451e-03) (1.093397372134e+00,0.000000000000e+00) (-4.154791598051e-01,0.000000000000e+00) (9.594409771163e-04,-9.594409771162e-04)
+ (-1.157395052414e+00,0.000000000000e+00) (-1.903117650352e-02,1.903117650352e-02) (-1.093397372134e+00,0.000000000000e+00) (-9.833707841514e-03,9.833707841515e-03) (8.102031397564e-01,0.000000000000e+00) (1.899269201656e-03,-1.899269201656e-03)
+ (9.594409771163e-04,9.594409771162e-04) (-4.154791598051e-01,0.000000000000e+00) (-1.903117650352e-02,-1.903117650352e-02) (-1.157395052414e+00,0.000000000000e+00) (-9.833707841514e-03,-9.833707841515e-03) (-1.093397372134e+00,0.000000000000e+00)
+ (1.899269201656e-03,1.899269201656e-03) (8.102031397564e-01,0.000000000000e+00) (-7.974273329164e-01,0.000000000000e+00) (-5.630637244357e-02,5.630637244358e-02) (-4.148020119102e-01,0.000000000000e+00) (-6.023892705908e-03,6.023892705908e-03)
+ (-5.630637244357e-02,-5.630637244358e-02) (-7.974273329164e-01,0.000000000000e+00) (-6.023892705908e-03,-6.023892705908e-03) (-4.148020119102e-01,0.000000000000e+00) (-7.974273329164e-01,0.000000000000e+00) (-5.630637244357e-02,5.630637244358e-02)
+ (-4.148020119102e-01,0.000000000000e+00) (-6.023892705908e-03,6.023892705908e-03) (-5.630637244357e-02,-5.630637244358e-02) (-7.974273329164e-01,0.000000000000e+00) (-6.023892705908e-03,-6.023892705908e-03) (-4.148020119102e-01,0.000000000000e+00)
+ (-1.004647828998e+00,0.000000000000e+00) (-8.828573496782e-03,8.828573496782e-03) (-1.093397372134e+00,0.000000000000e+00) (-9.833707841514e-03,9.833707841515e-03) (-2.004383330761e+00,0.000000000000e+00) (-4.378766338969e-02,4.378766338969e-02)
+ (4.154791598051e-01,0.000000000000e+00) (-9.594409771072e-04,9.594409771072e-04) (-8.828573496782e-03,-8.828573496782e-03) (-1.004647828998e+00,0.000000000000e+00) (-9.833707841514e-03,-9.833707841515e-03) (-1.093397372134e+00,0.000000000000e+00)
+ (-4.378766338969e-02,-4.378766338969e-02) (-2.004383330761e+00,0.000000000000e+00) (-9.594409771072e-04,-9.594409771072e-04) (4.154791598051e-01,0.000000000000e+00) (1.093397372134e+00,0.000000000000e+00) (9.833707841451e-03,-9.833707841451e-03)
+ (8.102031397564e-01,0.000000000000e+00) (1.899269201656e-03,-1.899269201656e-03) (4.154791598051e-01,0.000000000000e+00) (-9.594409771072e-04,9.594409771072e-04) (-1.157395052414e+00,0.000000000000e+00) (-1.903117650366e-02,1.903117650366e-02)
+ (9.833707841451e-03,9.833707841451e-03) (1.093397372134e+00,0.000000000000e+00) (1.899269201656e-03,1.899269201656e-03) (8.102031397564e-01,0.000000000000e+00) (-9.594409771072e-04,-9.594409771072e-04) (4.154791598051e-01,0.000000000000e+00)
+ (-1.903117650366e-02,-1.903117650366e-02) (-1.157395052414e+00,0.000000000000e+00) (-4.148020119102e-01,0.000000000000e+00) (-6.023892705908e-03,6.023892705908e-03) (-7.974273329162e-01,0.000000000000e+00) (-5.630637244396e-02,5.630637244396e-02)
+ (-6.023892705908e-03,-6.023892705908e-03) (-4.148020119102e-01,0.000000000000e+00) (-5.630637244396e-02,-5.630637244396e-02) (-7.974273329162e-01,0.000000000000e+00) (-4.148020119102e-01,0.000000000000e+00) (-6.023892705908e-03,6.023892705908e-03)
+ (-7.974273329162e-01,0.000000000000e+00) (-5.630637244396e-02,5.630637244396e-02) (-6.023892705908e-03,-6.023892705908e-03) (-4.148020119102e-01,0.000000000000e+00) (-5.630637244396e-02,-5.630637244396e-02) (-7.974273329162e-01,0.000000000000e+00)
+ # CSR column indices
+ 0 1 2 3 8 9 10 11 0 1 2 3 8 9 10 11
+ 0 1 2 3 8 9 10 11 0 1 2 3 8 9 10 11
+ 4 5 12 13 4 5 12 13 6 7 14 15 6 7 14 15
+ 0 1 2 3 8 9 10 11 0 1 2 3 8 9 10 11
+ 0 1 2 3 8 9 10 11 0 1 2 3 8 9 10 11
+ 4 5 12 13 4 5 12 13 6 7 14 15 6 7 14 15
+ # CSR row pointers
+ 0 8 16 24 32 36 40 44 48 56 64 72 80 84 88 92
+ 96
+
diff --git a/tests/03_NAO_multik/scf_nspin4_nsoc_hr/result.ref b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/result.ref
new file mode 100644
index 00000000000..01a2940f4e9
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/result.ref
@@ -0,0 +1,5 @@
+etotref -861.8592868019903
+etotperatomref -430.9296434010
+CompareHR_pass 0
+CompareSR_pass 0
+totaltimeref 1.52
diff --git a/tests/03_NAO_multik/scf_nspin4_nsoc_hr/sr_nao.csr.ref b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/sr_nao.csr.ref
new file mode 100644
index 00000000000..0ce0e2fe378
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_nsoc_hr/sr_nao.csr.ref
@@ -0,0 +1,44 @@
+ --- Ionic Step 1 ---
+ # print S matrix in real space S(R)
+ 1 # number of spin directions
+ 1 # spin index
+ 16 # number of localized basis
+ 1 # number of Bravais lattice vector R
+
+ user_defined_lattice
+ 9.52519
+ 1 0 0
+ 0 1 0
+ 0 0 1
+ O
+ 2
+ Direct
+ 0.5 0.5 0.436111111110222
+ 0.5 0.5 0.563888888889778
+
+ #----------------------------------------------------------------------#
+ #                               CSR Format                             #
+ # The outer loop corresponds to the number of Bravais lattice vectors. #
+ # The first line contains the index of the Bravais lattice vector      #
+ # (Rx, Ry, Rz), followed by the number of non-zero elements.           #
+ # The subsequent lines consist of three blocks of data, which are      #
+ # values, column indices, row pointers.                                #
+ #----------------------------------------------------------------------#
+
+ 0 0 0 40
+ # CSR values
+ (1.000000000000e+00,0.000000000000e+00) (3.122772551440e-01,0.000000000000e+00) (-3.769119232454e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (3.122772551440e-01,0.000000000000e+00) (-3.769119232454e-01,0.000000000000e+00)
+ (1.000000000000e+00,0.000000000000e+00) (3.769119232454e-01,0.000000000000e+00) (-2.556149750289e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (3.769119232454e-01,0.000000000000e+00) (-2.556149750289e-01,0.000000000000e+00)
+ (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00)
+ (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (3.122772551440e-01,0.000000000000e+00) (3.769119232454e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (3.122772551440e-01,0.000000000000e+00)
+ (3.769119232454e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (-3.769119232454e-01,0.000000000000e+00) (-2.556149750289e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (-3.769119232454e-01,0.000000000000e+00)
+ (-2.556149750289e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00)
+ (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00)
+ # CSR column indices
+ 0 8 10 1 9 11 2 8 10 3 9 11 4 12 5 13
+ 6 14 7 15 0 2 8 1 3 9 0 2 10 1 3 11
+ 4 12 5 13 6 14 7 15
+ # CSR row pointers
+ 0 3 6 9 12 14 16 18 20 23 26 29 32 34 36 38
+ 40
+
diff --git a/tests/03_NAO_multik/scf_nspin4_soc_hr/INPUT b/tests/03_NAO_multik/scf_nspin4_soc_hr/INPUT
new file mode 100644
index 00000000000..ece38d58a21
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_soc_hr/INPUT
@@ -0,0 +1,22 @@
+INPUT_PARAMETERS
+suffix            autotest
+calculation       scf
+symmetry          0
+pseudo_dir        ../../PP_ORB
+orbital_dir       ../../PP_ORB
+gamma_only        0
+ecutwfc           15
+scf_thr           1e-7
+scf_nmax          50
+noncolin          1
+nspin             4
+lspinorb          1
+basis_type        lcao
+ks_solver         scalapack_gvx
+dft_functional    pbe
+smearing_method   gauss
+smearing_sigma    0.02
+mixing_type       pulay
+mixing_beta       0.4
+mixing_gg0        3.0
+out_hsr           1 12
diff --git a/tests/03_NAO_multik/scf_nspin4_soc_hr/KPT b/tests/03_NAO_multik/scf_nspin4_soc_hr/KPT
new file mode 100644
index 00000000000..c289c0158aa
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_soc_hr/KPT
@@ -0,0 +1,4 @@
+K_POINTS
+0
+Gamma
+1 1 1 0 0 0
diff --git a/tests/03_NAO_multik/scf_nspin4_soc_hr/README b/tests/03_NAO_multik/scf_nspin4_soc_hr/README
new file mode 100644
index 00000000000..57a5befabff
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_soc_hr/README
@@ -0,0 +1,7 @@
+Self-consistent nspin=4 O2 H(R)/S(R) regression, with SOC.
+O-O distance 2.3 bohr in an 18-bohr cubic cell; existing O FR PP and minimal 6-bohr 1s1p orbitals.
+PBE/LCAO, 15 Ry, Gamma 1x1x1; initial moments (1/sqrt(2),1/sqrt(2),0) muB per atom.
+Full SCF uses scf_thr=1e-7, scf_nmax=50, Pulay 0.4 and mixing_gg0=3; no DeltaSpin.
+References are complete raw 16x16 complex matrices with only R=0 and finite y magnetization.
+Repeated local GNU/Intel runs pass the unchanged matrix comparator; the unfixed parent fails H.
+These compact parameters test the assembly regression rather than material-property convergence.
diff --git a/tests/03_NAO_multik/scf_nspin4_soc_hr/STRU b/tests/03_NAO_multik/scf_nspin4_soc_hr/STRU
new file mode 100644
index 00000000000..8a2d235ee6d
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_soc_hr/STRU
@@ -0,0 +1,22 @@
+ATOMIC_SPECIES
+O 15.999 O_ONCV_PBE_FR-1.0.upf
+
+NUMERICAL_ORBITAL
+O_gga_6au_60Ry_1s1p.orb
+
+LATTICE_CONSTANT
+18
+
+LATTICE_VECTORS
+1 0 0
+0 1 0
+0 0 1
+
+ATOMIC_POSITIONS
+Direct
+
+O
+0.0
+2
+0.5 0.5 0.43611111111111112 0 0 0 mag 0.7071067811865475 0.7071067811865475 0
+0.5 0.5 0.56388888888888888 0 0 0 mag 0.7071067811865475 0.7071067811865475 0
diff --git a/tests/03_NAO_multik/scf_nspin4_soc_hr/hrs1_nao.csr.ref b/tests/03_NAO_multik/scf_nspin4_soc_hr/hrs1_nao.csr.ref
new file mode 100644
index 00000000000..edd4d50eb7a
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_soc_hr/hrs1_nao.csr.ref
@@ -0,0 +1,93 @@
+ --- Ionic Step 1 ---
+ # print H matrix in real space H(R)
+ 1 # number of spin directions
+ 1 # spin index, E_Fermi = -6.68505 eV
+ 16 # number of localized basis
+ 1 # number of Bravais lattice vector R
+
+ user_defined_lattice
+ 9.52519
+ 1 0 0
+ 0 1 0
+ 0 0 1
+ O
+ 2
+ Direct
+ 0.5 0.5 0.436111111110222
+ 0.5 0.5 0.563888888889778
+
+ #----------------------------------------------------------------------#
+ #                               CSR Format                             #
+ # The outer loop corresponds to the number of Bravais lattice vectors. #
+ # The first line contains the index of the Bravais lattice vector      #
+ # (Rx, Ry, Rz), followed by the number of non-zero elements.           #
+ # The subsequent lines consist of three blocks of data, which are      #
+ # values, column indices, row pointers.                                #
+ #----------------------------------------------------------------------#
+
+ 0 0 0 256
+ # CSR values
+ (-2.004383498629e+00,0.000000000000e+00) (-4.378436323814e-02,4.378436323814e-02) (-4.154786008256e-01,0.000000000000e+00) (9.594375732042e-04,-9.594375732042e-04) (2.188274841787e-05,0.000000000000e+00) (-1.654355282713e-06,-1.108420439992e-16)
+ (2.188274841745e-05,0.000000000000e+00) (1.132320665990e-16,1.654355282663e-06) (-1.004647685204e+00,0.000000000000e+00) (-8.827922966156e-03,8.827922966156e-03) (1.093397158138e+00,0.000000000000e+00) (9.832992212382e-03,-9.832992212382e-03)
+ (-1.790536517864e-07,0.000000000000e+00) (-3.550782390893e-05,-3.185158553823e-18) (-1.790536519514e-07,0.000000000000e+00) (1.234122545089e-17,3.550782390893e-05) (-4.378436323814e-02,-4.378436323814e-02) (-2.004383498629e+00,0.000000000000e+00)
+ (9.594375732042e-04,9.594375732042e-04) (-4.154786008256e-01,0.000000000000e+00) (1.654355282982e-06,1.108420439992e-16) (-2.188274842218e-05,0.000000000000e+00) (1.132320665990e-16,1.654355283032e-06) (-2.188274842260e-05,0.000000000000e+00)
+ (-8.827922966156e-03,-8.827922966156e-03) (-1.004647685204e+00,0.000000000000e+00) (9.832992212382e-03,9.832992212382e-03) (1.093397158138e+00,0.000000000000e+00) (3.550782390896e-05,3.185158553823e-18) (1.790536512762e-07,0.000000000000e+00)
+ (1.234122545089e-17,3.550782390897e-05) (1.790536511122e-07,0.000000000000e+00) (-4.154786008256e-01,0.000000000000e+00) (9.594375732042e-04,-9.594375732042e-04) (-1.157395266026e+00,0.000000000000e+00) (-1.903000544585e-02,1.903000544585e-02)
+ (-2.853326133376e-05,0.000000000000e+00) (8.973061401638e-04,2.098216053230e-16) (-2.853326133407e-05,0.000000000000e+00) (-4.455501767514e-17,-8.973061401640e-04) (-1.093397158138e+00,0.000000000000e+00) (-9.832992212353e-03,9.832992212353e-03)
+ (8.102027314692e-01,0.000000000000e+00) (1.899165610565e-03,-1.899165610565e-03) (-3.264529049088e-06,0.000000000000e+00) (-3.947658458326e-05,-4.530762104793e-18) (-3.264529049261e-06,0.000000000000e+00) (1.966384542183e-17,3.947658458327e-05)
+ (9.594375732042e-04,9.594375732042e-04) (-4.154786008256e-01,0.000000000000e+00) (-1.903000544585e-02,-1.903000544585e-02) (-1.157395266026e+00,0.000000000000e+00) (-8.973061401642e-04,-2.098216053230e-16) (2.853326133379e-05,0.000000000000e+00)
+ (-4.455501767514e-17,-8.973061401641e-04) (2.853326133345e-05,0.000000000000e+00) (-9.832992212353e-03,-9.832992212353e-03) (-1.093397158138e+00,0.000000000000e+00) (1.899165610565e-03,1.899165610565e-03) (8.102027314692e-01,0.000000000000e+00)
+ (3.947658458332e-05,4.530762104793e-18) (3.264529048291e-06,0.000000000000e+00) (1.966384542183e-17,3.947658458331e-05) (3.264529048126e-06,0.000000000000e+00) (2.188274841788e-05,0.000000000000e+00) (1.654355282982e-06,-1.108330402869e-16)
+ (-2.853326133376e-05,0.000000000000e+00) (-8.973061401642e-04,2.092659055830e-16) (-7.974284579056e-01,0.000000000000e+00) (-5.630200558565e-02,5.630198117140e-02) (-4.806746675944e-09,-9.008532559618e-04) (-3.786486325673e-08,3.786486315883e-08)
+ (1.790536511522e-07,0.000000000000e+00) (-3.550782390894e-05,-7.114150526696e-19) (-3.264529048043e-06,0.000000000000e+00) (3.947658458329e-05,-3.857801124959e-18) (-4.148020091756e-01,0.000000000000e+00) (-6.023432244226e-03,6.023428942609e-03)
+ (-5.745379495785e-10,-4.122872387800e-05) (-9.266915070238e-09,9.266915063236e-09) (-1.654355282713e-06,1.108330402869e-16) (-2.188274842218e-05,0.000000000000e+00) (8.973061401638e-04,-2.092659055830e-16) (2.853326133379e-05,0.000000000000e+00)
+ (-5.630200558565e-02,-5.630198117140e-02) (-7.974284579056e-01,0.000000000000e+00) (-3.786486325673e-08,-3.786486315883e-08) (-4.806746667032e-09,9.008532559618e-04) (3.550782390896e-05,7.114150526696e-19) (-1.790536519126e-07,0.000000000000e+00)
+ (-3.947658458329e-05,3.857801124959e-18) (3.264529049355e-06,0.000000000000e+00) (-6.023432244226e-03,-6.023428942609e-03) (-4.148020091756e-01,0.000000000000e+00) (-9.266915070238e-09,-9.266915063236e-09) (-5.745379638231e-10,4.122872387800e-05)
+ (2.188274841744e-05,0.000000000000e+00) (1.128110050133e-16,-1.654355283032e-06) (-2.853326133392e-05,0.000000000000e+00) (-4.412980109130e-17,8.973061401641e-04) (-4.806746620283e-09,9.008532559618e-04) (-3.786486325632e-08,3.786486315879e-08)
+ (-7.974284579056e-01,0.000000000000e+00) (-5.630198117140e-02,5.630200558565e-02) (1.790536510459e-07,0.000000000000e+00) (1.179101934482e-17,3.550782390892e-05) (-3.264529048050e-06,0.000000000000e+00) (-1.542578151500e-17,-3.947658458326e-05)
+ (-5.745379395628e-10,4.122872387800e-05) (-9.266915070151e-09,9.266915063253e-09) (-4.148020091756e-01,0.000000000000e+00) (-6.023428942609e-03,6.023432244226e-03) (1.128110050133e-16,-1.654355282663e-06) (-2.188274842261e-05,0.000000000000e+00)
+ (-4.412980109130e-17,8.973061401640e-04) (2.853326133360e-05,0.000000000000e+00) (-3.786486325632e-08,-3.786486315879e-08) (-4.806746611373e-09,-9.008532559618e-04) (-5.630198117140e-02,-5.630200558565e-02) (-7.974284579056e-01,0.000000000000e+00)
+ (1.179101934482e-17,3.550782390897e-05) (-1.790536520225e-07,0.000000000000e+00) (-1.542578151500e-17,-3.947658458332e-05) (3.264529049361e-06,0.000000000000e+00) (-9.266915070151e-09,-9.266915063253e-09) (-5.745379538074e-10,-4.122872387800e-05)
+ (-6.023428942609e-03,-6.023432244226e-03) (-4.148020091756e-01,0.000000000000e+00) (-1.004647685204e+00,0.000000000000e+00) (-8.827922966156e-03,8.827922966156e-03) (-1.093397158138e+00,0.000000000000e+00) (-9.832992212353e-03,9.832992212353e-03)
+ (1.790536511522e-07,0.000000000000e+00) (3.550782390896e-05,-7.114150526696e-19) (1.790536510459e-07,0.000000000000e+00) (1.179101934482e-17,-3.550782390897e-05) (-2.004383498630e+00,0.000000000000e+00) (-4.378436323800e-02,4.378436323800e-02)
+ (4.154786008255e-01,0.000000000000e+00) (-9.594375731979e-04,9.594375731979e-04) (-2.188274842248e-05,0.000000000000e+00) (1.654355283112e-06,-1.692661408515e-16) (-2.188274842149e-05,0.000000000000e+00) (1.334640479212e-16,-1.654355282982e-06)
+ (-8.827922966156e-03,-8.827922966156e-03) (-1.004647685204e+00,0.000000000000e+00) (-9.832992212353e-03,-9.832992212353e-03) (-1.093397158138e+00,0.000000000000e+00) (-3.550782390894e-05,7.114150526696e-19) (-1.790536519126e-07,0.000000000000e+00)
+ (1.179101934482e-17,-3.550782390892e-05) (-1.790536520225e-07,0.000000000000e+00) (-4.378436323800e-02,-4.378436323800e-02) (-2.004383498630e+00,0.000000000000e+00) (-9.594375731979e-04,-9.594375731979e-04) (4.154786008255e-01,0.000000000000e+00)
+ (-1.654355282583e-06,1.692661408515e-16) (2.188274841728e-05,0.000000000000e+00) (1.334640479212e-16,-1.654355282713e-06) (2.188274841837e-05,0.000000000000e+00) (1.093397158138e+00,0.000000000000e+00) (9.832992212382e-03,-9.832992212382e-03)
+ (8.102027314692e-01,0.000000000000e+00) (1.899165610565e-03,-1.899165610565e-03) (-3.264529048043e-06,0.000000000000e+00) (-3.947658458329e-05,-3.857801124959e-18) (-3.264529048050e-06,0.000000000000e+00) (-1.542578151500e-17,3.947658458332e-05)
+ (4.154786008255e-01,0.000000000000e+00) (-9.594375731979e-04,9.594375731979e-04) (-1.157395266026e+00,0.000000000000e+00) (-1.903000544581e-02,1.903000544581e-02) (-2.853326133362e-05,0.000000000000e+00) (8.973061401643e-04,-2.317996996394e-16)
+ (-2.853326133347e-05,0.000000000000e+00) (7.380830795484e-17,-8.973061401641e-04) (9.832992212382e-03,9.832992212382e-03) (1.093397158138e+00,0.000000000000e+00) (1.899165610565e-03,1.899165610565e-03) (8.102027314692e-01,0.000000000000e+00)
+ (3.947658458329e-05,3.857801124959e-18) (3.264529049355e-06,0.000000000000e+00) (-1.542578151500e-17,3.947658458326e-05) (3.264529049361e-06,0.000000000000e+00) (-9.594375731979e-04,-9.594375731979e-04) (4.154786008255e-01,0.000000000000e+00)
+ (-1.903000544581e-02,-1.903000544581e-02) (-1.157395266026e+00,0.000000000000e+00) (-8.973061401638e-04,2.317996996394e-16) (2.853326133399e-05,0.000000000000e+00) (7.380830795484e-17,-8.973061401640e-04) (2.853326133419e-05,0.000000000000e+00)
+ (-1.790536517864e-07,0.000000000000e+00) (3.550782390896e-05,-3.185158553823e-18) (-3.264529049088e-06,0.000000000000e+00) (3.947658458332e-05,-4.530762104793e-18) (-4.148020091756e-01,0.000000000000e+00) (-6.023432244226e-03,6.023428942609e-03)
+ (-5.745379395628e-10,-4.122872387800e-05) (-9.266915070151e-09,9.266915063253e-09) (-2.188274842259e-05,0.000000000000e+00) (-1.654355282583e-06,-1.711007811797e-16) (-2.853326133360e-05,0.000000000000e+00) (-8.973061401638e-04,-2.317345988062e-16)
+ (-7.974284579061e-01,0.000000000000e+00) (-5.630200558547e-02,5.630198117122e-02) (-4.806746364336e-09,-9.008532559618e-04) (-3.786486327102e-08,3.786486327571e-08) (-3.550782390893e-05,3.185158553823e-18) (1.790536512762e-07,0.000000000000e+00)
+ (-3.947658458326e-05,4.530762104793e-18) (3.264529048291e-06,0.000000000000e+00) (-6.023432244226e-03,-6.023428942609e-03) (-4.148020091756e-01,0.000000000000e+00) (-9.266915070151e-09,-9.266915063253e-09) (-5.745379538074e-10,4.122872387800e-05)
+ (1.654355283112e-06,1.711007811797e-16) (2.188274841717e-05,0.000000000000e+00) (8.973061401643e-04,2.317345988062e-16) (2.853326133401e-05,0.000000000000e+00) (-5.630200558547e-02,-5.630198117122e-02) (-7.974284579061e-01,0.000000000000e+00)
+ (-3.786486327102e-08,-3.786486327571e-08) (-4.806746666688e-09,9.008532559618e-04) (-1.790536519514e-07,0.000000000000e+00) (1.234122545089e-17,-3.550782390897e-05) (-3.264529049261e-06,0.000000000000e+00) (1.966384542183e-17,-3.947658458331e-05)
+ (-5.745379495785e-10,4.122872387800e-05) (-9.266915070238e-09,9.266915063236e-09) (-4.148020091756e-01,0.000000000000e+00) (-6.023428942609e-03,6.023432244226e-03) (-2.188274842177e-05,0.000000000000e+00) (1.337552221918e-16,1.654355282715e-06)
+ (-2.853326133347e-05,0.000000000000e+00) (7.346774771233e-17,8.973061401640e-04) (-4.806746307552e-09,9.008532559618e-04) (-3.786486327143e-08,3.786486327919e-08) (-7.974284579061e-01,0.000000000000e+00) (-5.630198117122e-02,5.630200558547e-02)
+ (1.234122545089e-17,-3.550782390893e-05) (1.790536511122e-07,0.000000000000e+00) (1.966384542183e-17,-3.947658458327e-05) (3.264529048126e-06,0.000000000000e+00) (-9.266915070238e-09,-9.266915063236e-09) (-5.745379638231e-10,-4.122872387800e-05)
+ (-6.023428942609e-03,-6.023432244226e-03) (-4.148020091756e-01,0.000000000000e+00) (1.337552221918e-16,1.654355282979e-06) (2.188274841809e-05,0.000000000000e+00) (7.346774771233e-17,8.973061401641e-04) (2.853326133419e-05,0.000000000000e+00)
+ (-3.786486327143e-08,-3.786486327919e-08) (-4.806746609904e-09,-9.008532559618e-04) (-5.630198117122e-02,-5.630200558547e-02) (-7.974284579061e-01,0.000000000000e+00)
+ # CSR column indices
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+ # CSR row pointers
+ 0 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240
+ 256
+
diff --git a/tests/03_NAO_multik/scf_nspin4_soc_hr/result.ref b/tests/03_NAO_multik/scf_nspin4_soc_hr/result.ref
new file mode 100644
index 00000000000..99a1c78f02c
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_soc_hr/result.ref
@@ -0,0 +1,5 @@
+etotref -861.8596226171877
+etotperatomref -430.9298113086
+CompareHR_pass 0
+CompareSR_pass 0
+totaltimeref 1.65
diff --git a/tests/03_NAO_multik/scf_nspin4_soc_hr/sr_nao.csr.ref b/tests/03_NAO_multik/scf_nspin4_soc_hr/sr_nao.csr.ref
new file mode 100644
index 00000000000..0ce0e2fe378
--- /dev/null
+++ b/tests/03_NAO_multik/scf_nspin4_soc_hr/sr_nao.csr.ref
@@ -0,0 +1,44 @@
+ --- Ionic Step 1 ---
+ # print S matrix in real space S(R)
+ 1 # number of spin directions
+ 1 # spin index
+ 16 # number of localized basis
+ 1 # number of Bravais lattice vector R
+
+ user_defined_lattice
+ 9.52519
+ 1 0 0
+ 0 1 0
+ 0 0 1
+ O
+ 2
+ Direct
+ 0.5 0.5 0.436111111110222
+ 0.5 0.5 0.563888888889778
+
+ #----------------------------------------------------------------------#
+ #                               CSR Format                             #
+ # The outer loop corresponds to the number of Bravais lattice vectors. #
+ # The first line contains the index of the Bravais lattice vector      #
+ # (Rx, Ry, Rz), followed by the number of non-zero elements.           #
+ # The subsequent lines consist of three blocks of data, which are      #
+ # values, column indices, row pointers.                                #
+ #----------------------------------------------------------------------#
+
+ 0 0 0 40
+ # CSR values
+ (1.000000000000e+00,0.000000000000e+00) (3.122772551440e-01,0.000000000000e+00) (-3.769119232454e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (3.122772551440e-01,0.000000000000e+00) (-3.769119232454e-01,0.000000000000e+00)
+ (1.000000000000e+00,0.000000000000e+00) (3.769119232454e-01,0.000000000000e+00) (-2.556149750289e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (3.769119232454e-01,0.000000000000e+00) (-2.556149750289e-01,0.000000000000e+00)
+ (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00)
+ (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (3.122772551440e-01,0.000000000000e+00) (3.769119232454e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (3.122772551440e-01,0.000000000000e+00)
+ (3.769119232454e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (-3.769119232454e-01,0.000000000000e+00) (-2.556149750289e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (-3.769119232454e-01,0.000000000000e+00)
+ (-2.556149750289e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00)
+ (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00) (2.212660810968e-01,0.000000000000e+00) (1.000000000000e+00,0.000000000000e+00)
+ # CSR column indices
+ 0 8 10 1 9 11 2 8 10 3 9 11 4 12 5 13
+ 6 14 7 15 0 2 8 1 3 9 0 2 10 1 3 11
+ 4 12 5 13 6 14 7 15
+ # CSR row pointers
+ 0 3 6 9 12 14 16 18 20 23 26 29 32 34 36 38
+ 40
+