fix(rt): normalize LCAO wavefunctions with arrival-endpoint overlap

source/source_lcao/module_rt/evolve_psi.cpp

@@ -104,9 +104,9 @@ void evolve_psi(const int nband,
     // (4)->>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
     /// @brief normalize psi_k
-    /// @input Stmp, psi_not_norm, psi_k, print_matrix
+    /// @input S(t+dt), psi_not_norm, psi_k, print_matrix
     /// @output psi_k
-    norm_psi(pv, nband, nlocal, Stmp, psi_k, ofs_running, print_matrix);
+    norm_psi(pv, nband, nlocal, s_mat.p, psi_k, ofs_running, print_matrix);
 
     // (5)->>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -284,7 +284,11 @@ void evolve_psi_tensor(const int nband,
         upsi_tensor_lapack<Device>(pv, nband, nlocal, U_operator, psi_k_laststep, psi_k, ofs_running, print_matrix);
     }
 
-    // (4) Normalize psi_k
+    // (4) Normalize in the endpoint metric, not the averaged propagation metric.
+    if (s_src != nullptr && (!use_lapack || myid == root_proc))
+    {
+        syncmem_complex_h2d_op()(Stmp.data<std::complex<double>>(), s_src, len_HS);
+    }
     if (!use_lapack)
     {
         norm_psi_tensor(pv, nband, nlocal, Stmp, psi_k, ofs_running, print_matrix, cublas_res);

source/source_lcao/module_rt/test/CMakeLists.txt

@@ -9,6 +9,19 @@ add_library(tddft_test_lib tddft_test.cpp)
 target_link_libraries(tddft_test_lib PRIVATE Threads::Threads GTest::gtest_main GTest::gmock_main)
 #target_include_directories(tddft_test_lib PUBLIC $<$<COMPILE_LANGUAGE:CXX>:${GTEST_INCLUDE_DIRS}>)
 
+# This regression exercises CPU and CPU Tensor-LAPACK propagation. A cuBLASMp
+# target also needs its GPU kernels and distributed-library link dependencies.
+if(NOT ENABLE_CUBLASMP)
+  AddTest(
+    TARGET MODULE_LCAO_tddft_evolve_psi_test
+    LIBS parameter base device tddft_test_lib
+    SOURCES test_evolve_psi.cpp ../evolve_psi.cpp ../middle_hamilt.cpp ../norm_psi.cpp
+            ../band_energy.cpp ../upsi.cpp ../solve_propagation.cpp ../propagator.cpp
+            ../propagator_cn2.cpp ../propagator_taylor.cpp ../propagator_etrs.cpp
+            ../../../source_basis/module_ao/parallel_orbitals.cpp
+  )
+endif()
+
 AddTest(
   TARGET MODULE_LCAO_tddft_middle_hamilt_test
   LIBS parameter base device tddft_test_lib

source/source_lcao/module_rt/test/norm_psi_test.cpp

@@ -121,3 +121,42 @@ TEST(NormPsiTest, testNormPsi)
     delete[] psi_k;
     delete[] Stmp;
 }
+
+// A wavefunction arriving at t+dt must be normalized with S(t+dt).
+// Here the averaged overlap gives a unit norm in the wrong time layer.
+TEST(NormPsiTest, EndpointMetricDiffersFromPropagationMetric)
+{
+    Parallel_Orbitals pv;
+    const int nlocal = 2;
+    const int nband = 1;
+    const int block = 1;
+    const int zero = 0;
+    pv.nloc = 4;
+    pv.nloc_wfc = 2;
+    pv.nrow = pv.ncol = 2;
+    pv.ncol_bands = 1;
+    pv.dim0 = pv.dim1 = pv.nb = 1;
+    pv.blacs_ctxt = ictxt;
+    pv.set_coord(0, 0);
+    int info = 0;
+    int lld = 2;
+    descinit_(pv.desc, &nlocal, &nlocal, &block, &block, &zero, &zero, &ictxt, &lld, &info);
+    ASSERT_EQ(info, 0);
+    descinit_(pv.desc_wfc, &nlocal, &nband, &block, &block, &zero, &zero, &ictxt, &lld, &info);
+    ASSERT_EQ(info, 0);
+    descinit_(pv.desc_Eij, &nband, &nband, &block, &block, &zero, &zero, &ictxt, &lld, &info);
+    ASSERT_EQ(info, 0);
+    std::complex<double> endpoint[4] = {2.0, {0.0, -0.4}, {0.0, 0.4}, 1.0};
+    std::complex<double> midpoint[4] = {1.5, {0.0, -0.2}, {0.0, 0.2}, 1.0};
+    std::complex<double> right[2] = {1.0, {0.0, 0.5}};
+    std::complex<double> wrong[2] = {right[0], right[1]};
+    auto norm = [&](const std::complex<double>* psi) {
+        return (std::conj(psi[0]) * (endpoint[0]*psi[0] + endpoint[2]*psi[1])
+              + std::conj(psi[1]) * (endpoint[1]*psi[0] + endpoint[3]*psi[1])).real();
+    };
+    std::ofstream log;
+    module_rt::norm_psi(&pv, nband, nlocal, endpoint, right, log, 0);
+    module_rt::norm_psi(&pv, nband, nlocal, midpoint, wrong, log, 0);
+    EXPECT_NEAR(norm(right), 1.0, 1e-12);
+    EXPECT_GT(std::abs(norm(wrong)-1.0), 0.1);
+}

source/source_lcao/module_rt/test/test_evolve_psi.cpp

@@ -0,0 +1,107 @@
+#include "source_lcao/module_rt/evolve_psi.h"
+#include "source_base/module_external/scalapack_connector.h"
+#include "tddft_test.h"
+
+#include <algorithm>
+#include <array>
+#include <complex>
+#include <fstream>
+
+namespace
+{
+using Complex = std::complex<double>;
+
+// A two-orbital Hamiltonian with changing overlap and zero energy. This keeps
+// the normalization oracle independent of the propagator's phase convention.
+class EndpointHamiltonian : public hamilt::Hamilt<Complex>
+{
+  public:
+    std::array<Complex, 4> h = {{0.0, 0.0, 0.0, 0.0}};
+    std::array<Complex, 4> s = {{2.0, Complex(0.0, -0.4), Complex(0.0, 0.4), 1.0}};
+
+    void matrix(hamilt::MatrixBlock<Complex>& hk, hamilt::MatrixBlock<Complex>& sk) override
+    {
+        hk.p = h.data();
+        sk.p = s.data();
+        hk.row = sk.row = 2;
+        hk.col = sk.col = 2;
+    }
+};
+
+class EvolvePsiTest : public testing::Test
+{
+  protected:
+    Parallel_Orbitals pv;
+    EndpointHamiltonian hamiltonian;
+    std::array<Complex, 4> previous_s = {{1.0, 0.0, 0.0, 1.0}};
+    std::array<Complex, 2> previous_psi = {{1.0, Complex(0.0, 0.5)}};
+    std::ofstream log;
+
+    void SetUp() override
+    {
+        const int nlocal = 2;
+        const int nband = 1;
+        const int block = 1;
+        const int zero = 0;
+        const int lld = 2;
+        int info = 0;
+        pv.nloc = 4;
+        pv.nloc_wfc = 2;
+        pv.nrow = pv.ncol = 2;
+        pv.nrow_bands = pv.ncol_bands = 1;
+        pv.dim0 = pv.dim1 = pv.nb = 1;
+        pv.blacs_ctxt = ictxt;
+        pv.set_coord(0, 0);
+        descinit_(pv.desc, &nlocal, &nlocal, &block, &block, &zero, &zero, &ictxt, &lld, &info);
+        ASSERT_EQ(info, 0);
+        descinit_(pv.desc_wfc, &nlocal, &nband, &block, &block, &zero, &zero, &ictxt, &lld, &info);
+        ASSERT_EQ(info, 0);
+        descinit_(pv.desc_Eij, &nband, &nband, &block, &block, &zero, &zero, &ictxt, &lld, &info);
+        ASSERT_EQ(info, 0);
+    }
+
+    void expect_endpoint_norm(const Complex* psi)
+    {
+        const auto& s = hamiltonian.s;
+        const Complex norm = std::conj(psi[0]) * (s[0] * psi[0] + s[2] * psi[1])
+                           + std::conj(psi[1]) * (s[1] * psi[0] + s[3] * psi[1]);
+        EXPECT_NEAR(norm.real(), 1.0, 1e-12);
+        EXPECT_NEAR(norm.imag(), 0.0, 1e-12);
+    }
+
+    void run_tensor_lapack()
+    {
+        ct::Tensor psi(ct::DataType::DT_COMPLEX_DOUBLE, ct::TensorShape({1, 2}));
+        ct::Tensor last_psi(ct::DataType::DT_COMPLEX_DOUBLE, ct::TensorShape({1, 2}));
+        ct::Tensor last_h(ct::DataType::DT_COMPLEX_DOUBLE, ct::TensorShape({4}));
+        ct::Tensor last_s(ct::DataType::DT_COMPLEX_DOUBLE, ct::TensorShape({4}));
+        ct::Tensor energy(ct::DataType::DT_DOUBLE, ct::TensorShape({1}));
+        last_h.zero();
+        std::copy(previous_psi.begin(), previous_psi.end(), last_psi.data<Complex>());
+        std::copy(previous_s.begin(), previous_s.end(), last_s.data<Complex>());
+        CublasMpResources resources;
+        const bool use_lapack = true;
+        module_rt::evolve_psi_tensor<base_device::DEVICE_CPU>(
+            1, 2, &pv, &hamiltonian, psi, last_psi, last_h, last_s, energy, 0, log, 0,
+            use_lapack, resources);
+        expect_endpoint_norm(psi.data<Complex>());
+        EXPECT_NEAR(energy.data<double>()[0], 0.0, 1e-12);
+    }
+};
+
+TEST_F(EvolvePsiTest, CpuUsesEndpointOverlap)
+{
+    std::array<Complex, 2> psi;
+    std::array<Complex, 4> last_h = {{0.0, 0.0, 0.0, 0.0}};
+    double energy = 0.0;
+    module_rt::evolve_psi(1, 2, &pv, &hamiltonian, psi.data(), previous_psi.data(),
+                          last_h.data(), previous_s.data(), nullptr, false, &energy, 0, log, 0);
+    expect_endpoint_norm(psi.data());
+    EXPECT_NEAR(energy, 0.0, 1e-12);
+}
+
+TEST_F(EvolvePsiTest, LapackUsesEndpointOverlap)
+{
+    run_tensor_lapack();
+}
+} // namespace

tests/05_rtTDDFT/01_NO_KP_ocp_TDDFT/result.ref

@@ -1,5 +1,5 @@
-etotref -18.06593168523521
-etotperatomref -9.0329658426
-totalforceref 40.752890
-totalstressref 72.113950
+etotref -18.06317131077989
+etotperatomref -9.0315856554
+totalforceref 40.778000
+totalstressref 72.075825
 totaltimeref 1.51

tests/05_rtTDDFT/02_NO_CH_OW_TDDFT/result.ref

@@ -1,6 +1,6 @@
-etotref -323.5398409807095
-etotperatomref -107.8466136602
-totalforceref 19.449586
-totalstressref 329.291622
+etotref -323.5414256083637
+etotperatomref -107.8471418695
+totalforceref 19.479505
+totalstressref 329.532786
 Compare_wfc_lcao_pass 0
 totaltimeref 2.08

tests/05_rtTDDFT/02_NO_CH_OW_TDDFT/wfk1g3_nao_mod.txt.ref

@@ -3,102 +3,102 @@
 10 (number of bands)
 31 (number of orbitals)
 1 (band)
--1.61755597e+00 (Ry)
+-1.61956674e+00 (Ry)
 1.00000000e+00 (Occupations)
-1.7973043938e-02 5.6365832482e-03 4.0292662011e-03 9.3736555060e-07 3.1049873027e-04
-4.4881398078e-01 6.1981640010e-02 2.1866535770e-03 5.3897313978e-05 1.3443943213e-02
-2.4004970606e-03 5.6471009836e-05 2.1371977997e-03 1.1018497715e-02 3.0196749177e-08
-3.4646045381e-05 2.2873667193e-03 1.7872146624e-05 4.7006669445e-01 4.4280513019e-02
-2.3897983686e-03 1.0158791017e-04 1.3497044181e-02 2.1842110247e-03 4.9927225942e-05
-2.2484070449e-03 8.2944340647e-03 5.4571952020e-08 2.2603779928e-05 2.5122887235e-03
-1.9522588113e-05
+1.7978898550e-02 5.6366910412e-03 4.0322435374e-03 9.3798766189e-07 3.1016773183e-04
+4.4883757431e-01 6.1966744236e-02 2.1800265035e-03 5.3946088255e-05 1.3439980727e-02
+2.4036051496e-03 5.6499392644e-05 2.1412949922e-03 1.1024172420e-02 2.9436045423e-08
+3.3815912900e-05 2.2873845642e-03 1.7885120830e-05 4.7001888827e-01 4.4305874063e-02
+2.3831696068e-03 1.0164371922e-04 1.3493438927e-02 2.1869387626e-03 4.9968190799e-05
+2.2523402527e-03 8.3005989975e-03 5.2732435653e-08 2.1805528903e-05 2.5123858069e-03
+1.9536387830e-05
 2 (band)
--7.07747373e-01 (Ry)
+-7.08529790e-01 (Ry)
 1.00000000e+00 (Occupations)
-2.0653069044e-01 1.2922438214e-03 5.5721552369e-03 7.2107890104e-07 1.1945264266e-04
-2.4049797032e-02 9.0534178503e-03 4.1353051803e-01 7.6243598584e-08 1.0850406734e-03
-8.5129373380e-03 2.6137874323e-06 2.3615760670e-03 6.7179031525e-03 6.4628270727e-06
-2.8515439786e-03 7.4067280152e-05 1.8547996963e-06 1.8685696660e-02 1.1546954703e-02
-4.1059383480e-01 1.0508242989e-05 2.0026414555e-03 5.0069936802e-03 4.5255783501e-06
-3.1904695199e-04 6.5641412277e-03 6.4629453133e-06 2.5762247539e-03 6.2136578048e-04
-2.3450104732e-07
+2.0646047614e-01 1.2861261803e-03 5.5751873309e-03 7.2012205818e-07 1.1938872104e-04
+2.4034276362e-02 9.0459619354e-03 4.1340319593e-01 7.3654303492e-08 1.0847619613e-03
+8.5213061349e-03 2.6142104594e-06 2.3609822930e-03 6.7029906478e-03 6.4599215041e-06
+2.8505552253e-03 7.3822408962e-05 1.8538671872e-06 1.8673791495e-02 1.1537276794e-02
+4.1046337053e-01 1.0507171096e-05 2.0014259820e-03 5.0198316820e-03 4.5265044545e-06
+3.1926187220e-04 6.5493908957e-03 6.4594988163e-06 2.5753374295e-03 6.2099617840e-04
+2.3598489551e-07
 3 (band)
--4.79113206e-01 (Ry)
+-4.78693920e-01 (Ry)
 1.00000000e+00 (Occupations)
-7.0949991528e-01 8.1390263005e-02 4.9575118700e-03 2.7483923543e-06 8.1834463522e-05
-5.7799082474e-02 3.6790649728e-03 2.2770637352e-01 2.2541428598e-05 7.3952402289e-03
-1.1241010814e-03 1.1728087063e-05 2.4012336056e-03 1.2402328734e-02 3.1370968456e-06
-5.0148802325e-04 2.4026880519e-04 3.6789407726e-07 4.8045716271e-02 8.0553214099e-03
-2.2283829965e-01 4.5750575711e-05 1.5504092995e-02 6.2851811344e-04 3.2441273217e-06
-2.8429220454e-03 1.2036492083e-02 4.3292811855e-06 2.8188480320e-04 1.5086954993e-03
-5.1152329172e-06
+7.1004946661e-01 8.1450938358e-02 4.9395597457e-03 2.7487224323e-06 8.1631705931e-05
+5.7829797909e-02 3.7031756176e-03 2.2789404977e-01 2.2560792678e-05 7.4008498450e-03
+1.0972358283e-03 1.1737822389e-05 2.4036715824e-03 1.2424733070e-02 3.1366832795e-06
+5.0097504967e-04 2.3981595345e-04 3.7540901618e-07 4.8105627823e-02 8.0655095605e-03
+2.2302561565e-01 4.5797451701e-05 1.5514591641e-02 6.1487470643e-04 3.2526419501e-06
+2.8433201707e-03 1.2058467539e-02 4.3304170759e-06 2.8106838918e-04 1.5095021321e-03
+5.1139697124e-06
 4 (band)
--4.50651893e-01 (Ry)
+-4.52882734e-01 (Ry)
 1.00000000e+00 (Occupations)
-1.3217097755e-02 7.3806064880e-04 1.2039734432e-03 1.7927330311e-06 6.7026358167e-04
-6.9718730997e-03 2.5739269810e-03 2.8140856028e-03 3.8117027304e-03 6.3187344806e-01
-3.1905229613e-04 8.0840443344e-04 1.2861276004e-01 3.1525074076e-03 6.1157600920e-06
-1.0276743126e-03 6.6002886306e-02 3.2860482696e-04 8.4197612364e-03 3.4694185400e-03
-2.9154496332e-03 4.0958343738e-03 6.7963777219e-01 2.1227339874e-04 6.8936701349e-04
-1.1232882734e-01 3.2277104175e-03 6.4194817440e-06 1.0253157697e-03 7.2324878450e-02
-3.6946345590e-04
+1.3217335093e-02 7.3727750351e-04 1.2050544312e-03 1.7935518214e-06 6.6976661385e-04
+6.9736470048e-03 2.5739450331e-03 2.8143333276e-03 3.8113004019e-03 6.3180443143e-01
+3.1973138837e-04 8.0882191415e-04 1.2868341372e-01 3.1523729444e-03 5.9666640386e-06
+1.0014479353e-03 6.6010032458e-02 3.2864614901e-04 8.4188749587e-03 3.4680039854e-03
+2.9157132747e-03 4.0956112998e-03 6.7960033523e-01 2.1269290536e-04 6.8967193189e-04
+1.1237966029e-01 3.2276262110e-03 6.2640547783e-06 9.9842154060e-04 7.2340702498e-02
+3.6955196688e-04
 5 (band)
--4.29244918e-01 (Ry)
+-4.31480781e-01 (Ry)
 1.00000000e+00 (Occupations)
-1.2546345097e-04 1.1549793258e-05 5.4857705532e-06 1.7615067526e-04 4.0243258957e-06
-2.4556284209e-05 1.3132013087e-05 2.6969529589e-05 6.3651794916e-01 3.8379303257e-03
-1.6110581304e-06 1.3046916020e-01 7.5108455649e-04 1.7032402131e-05 1.0434769506e-03
-6.3770696327e-06 4.0366253454e-04 5.5551345877e-02 1.4437473875e-04 7.4710347480e-05
-2.7465612737e-05 6.8689845606e-01 4.1425969972e-03 4.1990382723e-07 1.1228617889e-01
-6.6490946210e-04 1.4347980412e-05 1.0445399518e-03 6.1202194759e-06 4.4273751465e-04
-6.2664386483e-02
+1.2546300088e-04 1.1544419848e-05 5.4942274942e-06 1.7619214274e-04 4.0224364989e-06
+2.4597950718e-05 1.3167764352e-05 2.6971741725e-05 6.3644973778e-01 3.8375039171e-03
+1.6162250031e-06 1.3053917376e-01 7.5151573396e-04 1.7031749016e-05 1.0186590414e-03
+6.2186970867e-06 4.0370503645e-04 5.5558102720e-02 1.4442834930e-04 7.4732362682e-05
+2.7466693884e-05 6.8686156956e-01 4.1423733968e-03 4.2015005442e-07 1.1233643502e-01
+6.6521290261e-04 1.4348155985e-05 1.0184597013e-03 5.9558219401e-06 4.4283742447e-04
+6.2679678678e-02
 6 (band)
-4.03555478e-01 (Ry)
+4.01355569e-01 (Ry)
 0.00000000e+00 (Occupations)
-8.1756853804e-06 1.4726002623e-06 1.9926827054e-06 1.3205847861e-04 2.4037628504e-06
-2.4248674591e-04 1.6038440115e-06 6.9047060430e-06 9.7310772497e-01 7.7983002859e-03
-6.0669465978e-06 5.6609682874e-02 4.9199867740e-04 9.7677240136e-06 1.3864011770e-03
-1.0077715105e-05 1.1828854219e-03 1.6453331700e-01 2.3320409245e-04 1.7959969054e-05
-2.1113749459e-06 9.7075043110e-01 7.7724078957e-03 2.2137732366e-06 6.3595395340e-02
-5.5419599527e-04 6.1531296898e-06 1.3430501429e-03 9.4413885659e-06 1.2904066196e-03
-1.7669219340e-01
+8.1765382256e-06 1.4723424388e-06 1.9944574791e-06 1.2724638244e-04 2.4008924861e-06
+2.4247430396e-04 1.5964249387e-06 6.9211473871e-06 9.7295037212e-01 7.7971327194e-03
+6.0543349782e-06 5.6493656157e-02 4.9112907363e-04 9.7752723629e-06 1.3441238962e-03
+9.7322333870e-06 1.1831228230e-03 1.6456535257e-01 2.3319234580e-04 1.7960741032e-05
+2.1244262132e-06 9.7060145833e-01 7.7713173116e-03 2.2001820834e-06 6.3488033235e-02
+5.5340508093e-04 6.1443976916e-06 1.3010619773e-03 9.0953302969e-06 1.2906710664e-03
+1.7672719027e-01
 7 (band)
-4.29134758e-01 (Ry)
+4.26938573e-01 (Ry)
 0.00000000e+00 (Occupations)
-2.0480946395e-03 2.1273300953e-04 4.7719244715e-04 1.4696326594e-06 1.0470311603e-04
-9.8666670088e-02 1.1656518985e-02 8.5216242340e-04 7.7101178400e-03 9.9362664003e-01
-1.1296243997e-03 4.8820825804e-04 6.7644381447e-02 6.7123713358e-04 1.0840575556e-05
-1.2451732319e-03 1.5673281830e-01 1.3488478325e-03 9.5609650162e-02 1.6674853442e-02
-4.1258058382e-04 7.6978947909e-03 9.9115501373e-01 2.8455247362e-04 5.3144399462e-04
-7.4105554472e-02 2.4095607344e-03 1.0772987703e-05 1.2234007472e-03 1.6933162690e-01
-1.4384419805e-03
+2.0482188045e-03 2.1266023376e-04 4.7721139922e-04 1.4408763205e-06 1.0384483808e-04
+9.8678865855e-02 1.1661715110e-02 8.5538571549e-04 7.7088253246e-03 9.9347173328e-01
+1.1265399113e-03 4.8726654476e-04 6.7530487736e-02 6.7290331423e-04 1.0502159071e-05
+1.2005997230e-03 1.5676439980e-01 1.3491122042e-03 9.5621893424e-02 1.6679939202e-02
+4.0847675357e-04 7.6966572631e-03 9.9100868342e-01 2.8114151531e-04 5.3055963429e-04
+7.4000446053e-02 2.4076022261e-03 1.0436801716e-05 1.1788837476e-03 1.6936625727e-01
+1.4387257564e-03
 8 (band)
-4.92284898e-01 (Ry)
+4.91352987e-01 (Ry)
 0.00000000e+00 (Occupations)
-2.5847686475e-01 3.5257767479e-02 5.4429817444e-02 4.2800479963e-08 1.1707003177e-04
-2.5391693038e-01 3.0906458744e-01 1.0309380927e-01 1.5690058903e-05 1.6756080117e-02
-7.4477563029e-02 7.1711876929e-06 2.7626743177e-03 2.1035597043e-01 4.3173134636e-06
-6.4592447180e-04 1.0156619157e-03 3.3288592277e-05 1.5848550144e-01 3.3347684867e-01
-9.6590964216e-02 9.0011058768e-06 2.2792423066e-03 7.3586649880e-02 5.0701595243e-05
-3.0483400107e-03 2.0961787776e-01 5.5391856269e-06 5.4738559843e-04 2.0894231013e-03
-7.0353203665e-06
+2.5865076697e-01 3.5282240483e-02 5.4427194935e-02 4.2213880655e-08 1.1740832263e-04
+2.5409279287e-01 3.0923887154e-01 1.0318117800e-01 1.5758941746e-05 1.6769661826e-02
+7.4532904324e-02 7.1408186138e-06 2.7601769549e-03 2.1049405581e-01 4.3191791524e-06
+6.4674087229e-04 1.0176402543e-03 3.3302987436e-05 1.5854547576e-01 3.3369622370e-01
+9.6670427626e-02 8.9782364529e-06 2.2862611332e-03 7.3645039878e-02 5.0756339340e-05
+3.0441385650e-03 2.0975437711e-01 5.5385640142e-06 5.4872294175e-04 2.0900052094e-03
+7.0115138556e-06
 9 (band)
-7.32222119e-01 (Ry)
+7.30101827e-01 (Ry)
 0.00000000e+00 (Occupations)
-2.4578208099e-03 8.5847556868e-04 6.2451396786e-04 2.5249221788e-07 1.2684874430e-03
-2.2778095437e+00 3.7725640644e-01 4.7386691353e-02 2.8868075795e-04 3.8786189795e-02
-1.0678281716e-02 8.5395923953e-05 4.7206230460e-03 1.4010781725e-01 7.5926982369e-07
-1.1083169862e-04 7.8127696803e-03 5.1037265705e-05 2.2812749546e+00 3.7121473632e-01
-4.2697424407e-02 3.5567803722e-04 3.9611025367e-02 1.5859297322e-02 1.6143615843e-04
-5.7494623163e-03 1.3199606167e-01 6.0533859920e-07 1.5523468806e-04 8.4274553739e-03
-5.9662172579e-05
+2.4579625212e-03 8.5825303449e-04 6.2440566243e-04 2.4784434719e-07 1.2669747411e-03
+2.2779971233e+00 3.7732774019e-01 4.7287372854e-02 2.8871195305e-04 3.8781817532e-02
+1.0588051098e-02 8.5408224465e-05 4.7172122108e-03 1.4007101459e-01 7.4740623959e-07
+1.1011622396e-04 7.8135869103e-03 5.1022566447e-05 2.2814624594e+00 3.7128603378e-01
+4.2597509895e-02 3.5572258423e-04 3.9607059321e-02 1.5769172146e-02 1.6146222806e-04
+5.7464823043e-03 1.3195913833e-01 5.9323812941e-07 1.5418730122e-04 8.4283669410e-03
+5.9651499967e-05
 10 (band)
-9.17560989e-01 (Ry)
+9.15500991e-01 (Ry)
 0.00000000e+00 (Occupations)
-1.4606023528e-02 3.2672698707e-03 3.5047317472e-03 3.2795571456e-07 1.4170211301e-03
-6.0584573025e-02 1.2710721285e-02 1.4913817050e+00 7.1935525183e-06 2.9201081715e-03
-3.1644405045e-01 2.5005563054e-06 1.5440308394e-03 5.6773861367e-03 9.5814964427e-07
-3.3827412551e-03 3.7440304557e-04 8.0088113958e-07 6.2704186789e-02 3.8211457165e-03
-1.4740767749e+00 1.0480089133e-05 3.1784992560e-03 3.4460940625e-01 6.4683415396e-06
-1.4877866802e-03 1.8028478460e-03 8.3408664858e-08 2.5473380709e-03 3.9643628566e-04
-1.7675027510e-06
+1.4607034345e-02 3.2673316864e-03 3.5029959111e-03 3.2659368534e-07 1.4105684635e-03
+6.0469334046e-02 1.2696158914e-02 1.4912704230e+00 7.1899750411e-06 2.9204853580e-03
+3.1633885213e-01 2.5004628352e-06 1.5438677657e-03 5.7288438809e-03 9.5736870196e-07
+3.3823395333e-03 3.7406983686e-04 8.0336813907e-07 6.2588224078e-02 3.8077941067e-03
+1.4739673442e+00 1.0473590075e-05 3.1798001184e-03 3.4450386526e-01 6.4653602240e-06
+1.4886344286e-03 1.7490552225e-03 8.4800326322e-08 2.5470449473e-03 3.9636150391e-04
+1.7685095749e-06