docs: update SR docs (#5621)

* docs: update SR docs

* Refactor: remove useless var in get_s

* docs: update docs

* Docs: polish docs

Author: YuLiu98

docs/advanced/elec_properties/hs_matrix.md

@@ -61,6 +61,11 @@ We also offer the option of only calculating the overlap matrix without running
 
 A file named `SR.csr` will be generated in the working directory, which contains the overlap matrix.
 
+> When `nspin` is set to 1 or 2, the dimension of the overlap matrix is nlocal $\times$ nlocal, where nlocal is the total number of numerical atomic orbitals. 
+These numerical atomic orbitals are ordered from outer to inner loop as atom, angular quantum number $l$, zeta (multiple radial orbitals corresponding to each $l$), and magnetic quantum number $m$. 
+When `nspin` is set to 4, the dimension of the overlap matrix is (2 $\times$ nlocal) $\times$ (2 $\times$ nlocal). In this case, the numerical atomic orbitals are ordered from outer to inner loop as atom, angular quantum number $l$, zeta (multiple radial orbitals corresponding to each $l$), magnetic quantum number $m$, and npol (index of spin, ranges from 0 to 1).
+
+
 ## examples
 We provide [examples](https://github.com/deepmodeling/abacus-develop/tree/develop/examples/matrix_hs) of outputting the matrices. There are four examples:
 

docs/advanced/elec_properties/position_matrix.md

@@ -18,4 +18,7 @@ Each file or each section of the appended file starts with "STEP: " followed by
 
 Each block here contains the matrix for the corresponding cell. There are three columns in each block, giving the matrix elements in x, y, z directions, respectively. There are altogether nbasis * nbasis lines in each block, which emulates the matrix elements.
 
-In molecular dynamics (MD) calculations, if [out_app_flag](../input_files/input-main.md#out_app_flag) is set to true, then `data-rR-tr` is written in an append manner. Otherwise, output files will be put in a separate directory, `matrix`, and named as `$x`_data-rR-tr, where `$x` is the number of MD step. In addition, The output frequency is controlled by [out_interval](../input_files/input-main.md#out_interval). For example, if we are running a 10-step MD with out_interval = 3, then `$x` will be 0, 3, 6, and 9.
+In molecular dynamics (MD) calculations, if [out_app_flag](../input_files/input-main.md#out_app_flag) is set to true, then `data-rR-tr` is written in an append manner. Otherwise, output files will be put in a separate directory, `matrix`, and named as `$x`_data-rR-tr, where `$x` is the number of MD step. In addition, the output frequency is controlled by [out_interval](../input_files/input-main.md#out_interval). For example, if we are running a 10-step MD with out_interval = 3, then `$x` will be 0, 3, 6, and 9.
+
+## get_S
+We also offer the option of only calculating the position matrix without running SCF. For that purpose, in `INPUT` file we need to set the keyword [calculation](../input_files/input-main.md#calculation) to `get_S`, and [out_mat_r](../input_files/input-main.md#out_mat_r) to `true`.

docs/advanced/input_files/input-main.md

@@ -1720,7 +1720,7 @@ These variables are used to control the output of properties.
 
 - **Type**: Boolean
 - **Availability**: Numerical atomic orbital basis (not gamma-only algorithm)
-- **Description**: Whether to print the matrix representation of the position matrix (in Bohr) into a file named `data-rR-tr` in the directory `OUT.${suffix}`. For more information, please refer to [position_matrix.md](../elec_properties/position_matrix.md#extracting-position-matrices).
+- **Description**: Whether to print the matrix representation of the position matrix (in Bohr) into a file named `data-rR-tr` in the directory `OUT.${suffix}`. If [calculation](#calculation) is set to `get_S`, the position matrix can be obtained without scf iterations. For more information, please refer to [position_matrix.md](../elec_properties/position_matrix.md#extracting-position-matrices).
 - **Default**: False
 
 ### out_mat_hs2

source/module_esolver/esolver_gets.cpp

@@ -52,8 +52,8 @@ void ESolver_GetS::before_all_runners(UnitCell& ucell, const Input_para& inp)
         this->pelec = new elecstate::ElecStateLCAO<std::complex<double>>(&(this->chr), // use which parameter?
                                                                          &(this->kv),
                                                                          this->kv.get_nks(),
-                                                                         &(this->GG), // mohan add 2024-04-01
-                                                                         &(this->GK), // mohan add 2024-04-01
+                                                                         nullptr, // mohan add 2024-04-01
+                                                                         nullptr, // mohan add 2024-04-01
                                                                          this->pw_rho,
                                                                          this->pw_big);
     }
@@ -100,7 +100,8 @@ void ESolver_GetS::runner(UnitCell& ucell, const int istep)
                          search_radius,
                          PARAM.inp.test_atom_input);
 
-    this->RA.for_2d(this->pv, PARAM.globalv.gamma_only_local, orb_.cutoffs());
+    Record_adj RA;
+    RA.for_2d(this->pv, PARAM.globalv.gamma_only_local, orb_.cutoffs());
 
     if (this->p_hamilt == nullptr)
     {

source/module_esolver/esolver_gets.h

@@ -4,8 +4,6 @@
 #include "module_basis/module_nao/two_center_bundle.h"
 #include "module_cell/unitcell.h"
 #include "module_esolver/esolver_ks.h"
-#include "module_hamilt_lcao/module_gint/gint_gamma.h"
-#include "module_hamilt_lcao/module_gint/gint_k.h"
 
 #include <memory>
 
@@ -34,18 +32,9 @@ class ESolver_GetS : public ESolver_KS<std::complex<double>>
     void cal_stress(UnitCell& ucell, ModuleBase::matrix& stress) {};
 
   protected:
-    // we will get rid of this class soon, don't use it, mohan 2024-03-28
-    Record_adj RA;
-
     // 2d block - cyclic distribution info
     Parallel_Orbitals pv;
 
-    // used for k-dependent grid integration.
-    Gint_k GK;
-
-    // used for gamma only algorithms.
-    Gint_Gamma GG;
-
     TwoCenterBundle two_center_bundle_;
 
     // temporary introduced during removing GlobalC::ORB