From 16cbaefa64c3097d7a67258ec25d5d0fb8e49874 Mon Sep 17 00:00:00 2001
From: Mark Dean <mpmdean@gmail.com>
Date: Fri, 2 Aug 2024 13:18:12 -0400
Subject: [PATCH 1/9] update examples

---
 examples/sphinx/example_3_AIM_XAS.py          | 17 ++--
 examples/sphinx/example_5_charge_transfer.py  | 98 +++++++++++++++++++
 ...ransitions.py => example_7_transitions.py} |  0
 ...ons.py => example_8_Hunds_interactions.py} |  0
 ...mple_8_Coulomb.py => example_9_Coulomb.py} |  0
 5 files changed, 108 insertions(+), 7 deletions(-)
 create mode 100644 examples/sphinx/example_5_charge_transfer.py
 rename examples/sphinx/{example_5_transitions.py => example_7_transitions.py} (100%)
 rename examples/sphinx/{example_7_Hunds_interactions.py => example_8_Hunds_interactions.py} (100%)
 rename examples/sphinx/{example_8_Coulomb.py => example_9_Coulomb.py} (100%)

diff --git a/examples/sphinx/example_3_AIM_XAS.py b/examples/sphinx/example_3_AIM_XAS.py
index 6a2b9240..37033b93 100644
--- a/examples/sphinx/example_3_AIM_XAS.py
+++ b/examples/sphinx/example_3_AIM_XAS.py
@@ -91,12 +91,15 @@
           [F0_dd, F2_dd, F4_dd, F0_dp, F2_dp, G1_dp, G3_dp])  # with core hole
 
 ################################################################################
-# Charge-transfer energy scales
+# Energy of the bath states
 # ------------------------------------------------------------------------------
-# The charge-transfer :math:`\Delta` and Coulomb :math:`U_{dd}` :math:`U_{dp}`
-# parameters determine the centers of the different electronic configurations
-# before they are split. Note that as electrons are moved one has to pay energy
-# costs associated with both charge-transfer and Coulomb interactions. The
+# In the notation used in EDRIXS, :math:`\Delta` sets the energy difference
+# between the bath and impurity states. :math:`\Delta` is defined in the atomic
+# limit without crystal field (i.e. in terms of the centers of the impurity and
+# bath states before hybridization is considered) as the energy for a
+# :math:`d^{n_d} \rightarrow d^{n_d + 1} \underline{L}` transition.
+# Note that as electrons are moved one has to pay energy
+# costs associated with the Coulomb interactions. The
 # energy splitting between the bath and impurity is consequently not simply
 # :math:`\Delta`. One must therefore determine the energies by solving
 # a set of linear equations. See the :ref:`edrixs.utils functions <utils>`
@@ -336,6 +339,6 @@
 # .. [1] Maurits Haverkort et al
 #        `Phys. Rev. B 85, 165113 (2012) <https://doi.org/10.1103/PhysRevB.85.165113>`_.
 # .. [2] A. E. Bocquet et al.,
-#        `Phys. Rev. B 53, 1161 (1996) <https://doi.org/10.1103/PhysRevB.53.1161>`_
+#        `Phys. Rev. B 53, 1161 (1996) <https://doi.org/10.1103/PhysRevB.53.1161>`_.
 # .. [3] Arata Tanaka, and Takeo Jo,
-#        `J. Phys. Soc. Jpn. 63, 2788-2807(1994) <https://doi.org/10.1143/JPSJ.63.2788>`_
+#        `J. Phys. Soc. Jpn. 63, 2788-2807(1994) <https://doi.org/10.1143/JPSJ.63.2788>`_.
diff --git a/examples/sphinx/example_5_charge_transfer.py b/examples/sphinx/example_5_charge_transfer.py
new file mode 100644
index 00000000..8a7a1558
--- /dev/null
+++ b/examples/sphinx/example_5_charge_transfer.py
@@ -0,0 +1,98 @@
+#!/usr/bin/env python
+"""
+Charge-transfer energy for NiO
+================================================================================
+This example follows the :ref:`sphx_glr_auto_examples_example_3_AIM_XAS.py`
+example and considers the same model. This time we outline how to determine
+the charge transfer energy in the sense defined by Zaanen, Sawatzky, and Allen
+[1]_. That is, a :math:`d^{n_d} \\rightarrow d^{n_d + 1} \\underline{L}` transition
+in the atomic limit, after considering crystal field. Although this can be
+determined analytically, the easiest way is often just to calculate it, as we will
+do here.
+"""
+import edrixs
+import numpy as np
+import matplotlib.pyplot as plt
+import scipy
+import example_3_AIM_XAS
+import importlib
+_ = importlib.reload(example_3_AIM_XAS)
+
+################################################################################
+# Determine eigenvectors and occupations
+# ------------------------------------------------------------------------------
+# The first step repeats what was done in
+# :ref:`sphx_glr_auto_examples_example_4_GS_analysis.py` but it does not apply
+# the hybridization between the impurity and both states.
+
+from example_3_AIM_XAS import (F0_dd, F2_dd, F4_dd,
+                               nd, norb_d, norb_bath, v_noccu,
+                               imp_mat, CF, bath_level,
+                               hyb, ext_B, trans_c2n)
+ntot = 20
+umat_delectrons = edrixs.get_umat_slater('d', F0_dd, F2_dd, F4_dd)
+umat = np.zeros((ntot, ntot, ntot, ntot), dtype=complex)
+umat[:norb_d, :norb_d, :norb_d, :norb_d] += umat_delectrons
+emat_rhb = np.zeros((ntot, ntot), dtype='complex')
+emat_rhb[0:norb_d, 0:norb_d] += imp_mat
+indx = np.arange(norb_d, norb_d*2)
+emat_rhb[indx, indx] += bath_level[0]
+tmat = np.eye(ntot, dtype=complex)
+for i in range(2):
+    off = i * norb_d
+    tmat[off:off+norb_d, off:off+norb_d] = np.conj(np.transpose(trans_c2n))
+
+emat_chb = edrixs.cb_op(emat_rhb, tmat)
+v_orbl = 2
+sx = edrixs.get_sx(v_orbl)
+sy = edrixs.get_sy(v_orbl)
+sz = edrixs.get_sz(v_orbl)
+zeeman = ext_B[0] * (2 * sx) + ext_B[1] * (2 * sy) + ext_B[2] * (2 * sz)
+emat_chb[0:norb_d, 0:norb_d] += zeeman
+basis = np.array(edrixs.get_fock_bin_by_N(ntot, v_noccu))
+H = (edrixs.build_opers(2, emat_chb, basis)
+  + edrixs.build_opers(4, umat, basis))
+
+e, v = scipy.linalg.eigh(H)
+e -= e[0]
+
+num_d_electrons = basis[:, :norb_d].sum(1)
+alphas = np.sum(np.abs(v[num_d_electrons==8, :])**2, axis=0)
+betas = np.sum(np.abs(v[num_d_electrons==9, :])**2, axis=0)
+
+################################################################################
+# Plot
+# ------------------------------------------------------------------------------
+# Let's vizualize :math:`\alpha` and :math:`\beta`.
+
+fig, ax = plt.subplots()
+
+ax.plot(e, alphas, '.-', label=r'$\alpha$ $d^8L^{10}$')
+ax.plot(e, betas, '.-', label=r'$\beta$ $d^9L^{9}$')
+
+ax.set_xlabel('Energy (eV)')
+ax.set_ylabel('Population')
+ax.set_title('NiO')
+ax.legend()
+plt.show()
+
+################################################################################
+# One can see that the mixing between impurity and bath states has disappered
+# because we have turned off the hybridization. The charge transfer energy can
+# now we straighforwardly computed
+
+GS_energy = min(e[np.isclose(alphas, 1)])
+lowest_energy_to_transfer_electron = min(e[np.isclose(alphas, 0)])
+Delta_ZSA = lowest_energy_to_transfer_electron - GS_energy
+print(f"Charge transfer energy is {Delta_ZSA:.3f} eV")
+
+################################################################################
+# where we have used :code:`np.isclose` to avoid errors from finite numerical
+# precision.
+
+##############################################################################
+#
+# .. rubric:: Footnotes
+#
+# .. [1] J. Zaanen, G. A. Sawatzky, and J. W. Allen,
+#        `Phys. Rev. Lett. 55, 418 (1985) <https://doi.org/10.1103/PhysRevLett.55.418>`_.
diff --git a/examples/sphinx/example_5_transitions.py b/examples/sphinx/example_7_transitions.py
similarity index 100%
rename from examples/sphinx/example_5_transitions.py
rename to examples/sphinx/example_7_transitions.py
diff --git a/examples/sphinx/example_7_Hunds_interactions.py b/examples/sphinx/example_8_Hunds_interactions.py
similarity index 100%
rename from examples/sphinx/example_7_Hunds_interactions.py
rename to examples/sphinx/example_8_Hunds_interactions.py
diff --git a/examples/sphinx/example_8_Coulomb.py b/examples/sphinx/example_9_Coulomb.py
similarity index 100%
rename from examples/sphinx/example_8_Coulomb.py
rename to examples/sphinx/example_9_Coulomb.py

From f36fda074e47c392227a7731697bd8be13412e83 Mon Sep 17 00:00:00 2001
From: Mark <mpmdean@gmail.com>
Date: Fri, 2 Aug 2024 15:18:12 -0400
Subject: [PATCH 2/9] Update examples/sphinx/example_5_charge_transfer.py

Co-authored-by: thomas-jinu <98235678+thomas-jinu@users.noreply.github.com>
---
 examples/sphinx/example_5_charge_transfer.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/examples/sphinx/example_5_charge_transfer.py b/examples/sphinx/example_5_charge_transfer.py
index 8a7a1558..ca4eaf31 100644
--- a/examples/sphinx/example_5_charge_transfer.py
+++ b/examples/sphinx/example_5_charge_transfer.py
@@ -82,7 +82,7 @@
 # now we straighforwardly computed
 
 GS_energy = min(e[np.isclose(alphas, 1)])
-lowest_energy_to_transfer_electron = min(e[np.isclose(alphas, 0)])
+lowest_energy_to_transfer_electron = min(e[np.isclose(betas, 1)])
 Delta_ZSA = lowest_energy_to_transfer_electron - GS_energy
 print(f"Charge transfer energy is {Delta_ZSA:.3f} eV")
 

From ddd8be6edd1d244286f05c184d5e5194c563999b Mon Sep 17 00:00:00 2001
From: Mark <mpmdean@gmail.com>
Date: Fri, 2 Aug 2024 15:18:18 -0400
Subject: [PATCH 3/9] Update examples/sphinx/example_5_charge_transfer.py

Co-authored-by: thomas-jinu <98235678+thomas-jinu@users.noreply.github.com>
---
 examples/sphinx/example_5_charge_transfer.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/examples/sphinx/example_5_charge_transfer.py b/examples/sphinx/example_5_charge_transfer.py
index ca4eaf31..f744afa2 100644
--- a/examples/sphinx/example_5_charge_transfer.py
+++ b/examples/sphinx/example_5_charge_transfer.py
@@ -27,7 +27,7 @@
 
 from example_3_AIM_XAS import (F0_dd, F2_dd, F4_dd,
                                nd, norb_d, norb_bath, v_noccu,
-                               imp_mat, CF, bath_level,
+                               imp_mat, bath_level,
                                hyb, ext_B, trans_c2n)
 ntot = 20
 umat_delectrons = edrixs.get_umat_slater('d', F0_dd, F2_dd, F4_dd)

From d95f0de175357766f736637f02da2087ec4c7788 Mon Sep 17 00:00:00 2001
From: Mark Dean <mpmdean@gmail.com>
Date: Sat, 10 Aug 2024 14:40:55 -0400
Subject: [PATCH 4/9] Add note about block diagonalization

---
 examples/sphinx/example_5_charge_transfer.py | 32 ++++++++++++++++++--
 1 file changed, 29 insertions(+), 3 deletions(-)

diff --git a/examples/sphinx/example_5_charge_transfer.py b/examples/sphinx/example_5_charge_transfer.py
index f744afa2..6b7078b6 100644
--- a/examples/sphinx/example_5_charge_transfer.py
+++ b/examples/sphinx/example_5_charge_transfer.py
@@ -51,14 +51,14 @@
 emat_chb[0:norb_d, 0:norb_d] += zeeman
 basis = np.array(edrixs.get_fock_bin_by_N(ntot, v_noccu))
 H = (edrixs.build_opers(2, emat_chb, basis)
-  + edrixs.build_opers(4, umat, basis))
+     + edrixs.build_opers(4, umat, basis))
 
 e, v = scipy.linalg.eigh(H)
 e -= e[0]
 
 num_d_electrons = basis[:, :norb_d].sum(1)
-alphas = np.sum(np.abs(v[num_d_electrons==8, :])**2, axis=0)
-betas = np.sum(np.abs(v[num_d_electrons==9, :])**2, axis=0)
+alphas = np.sum(np.abs(v[num_d_electrons == 8, :])**2, axis=0)
+betas = np.sum(np.abs(v[num_d_electrons == 9, :])**2, axis=0)
 
 ################################################################################
 # Plot
@@ -90,6 +90,32 @@
 # where we have used :code:`np.isclose` to avoid errors from finite numerical
 # precision.
 
+################################################################################
+# Diagonalizing by blocks
+# ------------------------------------------------------------------------------
+# When working on a problem with a large basis, one can take advantage of the
+# lack of hybridization and separately diagonalize the impurity and bath
+# states
+
+energies = []
+
+for n_ligand_holes in [0, 1]:
+    basis_d = edrixs.get_fock_bin_by_N(10, nd + n_ligand_holes)
+    Hd = (edrixs.build_opers(2, emat_chb[:10, :10], basis_d)
+          + edrixs.build_opers(4, umat[:10, :10, :10, :10], basis_d))
+    ed = scipy.linalg.eigh(Hd)[0][0]
+
+    basis_L = edrixs.get_fock_bin_by_N(10, 10 - n_ligand_holes)
+    HL = (edrixs.build_opers(2, emat_chb[10:, 10:], basis_L)
+          + edrixs.build_opers(4, umat[10:, 10:, 10:, 10:], basis_L))
+    eL = scipy.linalg.eigh(HL)[0][0]
+    energies.append(ed + eL)
+
+print(f"Charge transfer energy is {energies[1] - energies[0]:.3f} eV")
+
+################################################################################
+# which yields the same result.
+
 ##############################################################################
 #
 # .. rubric:: Footnotes

From 8c0bdf6f7347915578e439cd4bb39626f9961c33 Mon Sep 17 00:00:00 2001
From: Mark Dean <mpmdean@gmail.com>
Date: Tue, 13 Aug 2024 11:56:30 -0400
Subject: [PATCH 5/9] consider hopping

---
 docs/source/_static/energy_level.png         | Bin 0 -> 15460 bytes
 examples/sphinx/example_5_charge_transfer.py |  33 ++++++++++++++-----
 2 files changed, 24 insertions(+), 9 deletions(-)
 create mode 100644 docs/source/_static/energy_level.png

diff --git a/docs/source/_static/energy_level.png b/docs/source/_static/energy_level.png
new file mode 100644
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diff --git a/examples/sphinx/example_5_charge_transfer.py b/examples/sphinx/example_5_charge_transfer.py
index 6b7078b6..a7947a19 100644
--- a/examples/sphinx/example_5_charge_transfer.py
+++ b/examples/sphinx/example_5_charge_transfer.py
@@ -6,9 +6,9 @@
 example and considers the same model. This time we outline how to determine
 the charge transfer energy in the sense defined by Zaanen, Sawatzky, and Allen
 [1]_. That is, a :math:`d^{n_d} \\rightarrow d^{n_d + 1} \\underline{L}` transition
-in the atomic limit, after considering crystal field. Although this can be
-determined analytically, the easiest way is often just to calculate it, as we will
-do here.
+in the atomic limit, after considering Coulomb interactins and crystal field. Although
+this can be determined analytically in some cases, the easiest way is often just to
+calculate it, as we will do here.
 """
 import edrixs
 import numpy as np
@@ -61,7 +61,7 @@
 betas = np.sum(np.abs(v[num_d_electrons == 9, :])**2, axis=0)
 
 ################################################################################
-# Plot
+# Energy to lowest energy ligand orbital
 # ------------------------------------------------------------------------------
 # Let's vizualize :math:`\alpha` and :math:`\beta`.
 
@@ -78,13 +78,12 @@
 
 ################################################################################
 # One can see that the mixing between impurity and bath states has disappered
-# because we have turned off the hybridization. The charge transfer energy can
-# now we straighforwardly computed
+# because we have turned off the hybridization. The energy required to
 
 GS_energy = min(e[np.isclose(alphas, 1)])
 lowest_energy_to_transfer_electron = min(e[np.isclose(betas, 1)])
-Delta_ZSA = lowest_energy_to_transfer_electron - GS_energy
-print(f"Charge transfer energy is {Delta_ZSA:.3f} eV")
+E_to_ligand = lowest_energy_to_transfer_electron - GS_energy
+print(f"Energy to lowest energy ligand state is {E_to_ligand:.3f} eV")
 
 ################################################################################
 # where we have used :code:`np.isclose` to avoid errors from finite numerical
@@ -111,11 +110,27 @@
     eL = scipy.linalg.eigh(HL)[0][0]
     energies.append(ed + eL)
 
-print(f"Charge transfer energy is {energies[1] - energies[0]:.3f} eV")
+print(f"Energy to lowest energy ligand state is {energies[1] - energies[0]:.3f} eV")
 
 ################################################################################
 # which yields the same result.
 
+################################################################################
+# Energy splitting in ligand states
+# ------------------------------------------------------------------------------
+# The last thing to consider is that our definition of the charge transfer
+# energy refers to the atomic limit with all hopping terms switch off, whereas
+# the ligand states in the model are already split by the oxygen-oxygen hopping
+# term :math:`T_{pp}` as illustrated below. So the final charge transer energy
+# needs to account for this.
+#
+#     .. image:: /_static/energy_level.png
+#
+
+T_pp = 1
+print(f"Charge transfer is {energies[1] - energies[0] + T_pp:.3f} eV")
+
+
 ##############################################################################
 #
 # .. rubric:: Footnotes

From 3e32647da062341bc533924d9b38850409c24c49 Mon Sep 17 00:00:00 2001
From: Mark Dean <mpmdean@gmail.com>
Date: Tue, 13 Aug 2024 13:29:39 -0400
Subject: [PATCH 6/9] just get one eigenvalue

---
 examples/sphinx/example_5_charge_transfer.py | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/examples/sphinx/example_5_charge_transfer.py b/examples/sphinx/example_5_charge_transfer.py
index a7947a19..4f97dc51 100644
--- a/examples/sphinx/example_5_charge_transfer.py
+++ b/examples/sphinx/example_5_charge_transfer.py
@@ -102,12 +102,13 @@
     basis_d = edrixs.get_fock_bin_by_N(10, nd + n_ligand_holes)
     Hd = (edrixs.build_opers(2, emat_chb[:10, :10], basis_d)
           + edrixs.build_opers(4, umat[:10, :10, :10, :10], basis_d))
-    ed = scipy.linalg.eigh(Hd)[0][0]
+    ed = scipy.linalg.eigh(Hd, eigvals_only=True, subset_by_index=[0, 0])[0]
 
     basis_L = edrixs.get_fock_bin_by_N(10, 10 - n_ligand_holes)
     HL = (edrixs.build_opers(2, emat_chb[10:, 10:], basis_L)
           + edrixs.build_opers(4, umat[10:, 10:, 10:, 10:], basis_L))
-    eL = scipy.linalg.eigh(HL)[0][0]
+    eL = scipy.linalg.eigh(HL, eigvals_only=True, subset_by_index=[0, 0])[0]
+
     energies.append(ed + eL)
 
 print(f"Energy to lowest energy ligand state is {energies[1] - energies[0]:.3f} eV")

From 1eaa2efffcd53efacc3f08784f1759acb7d10f51 Mon Sep 17 00:00:00 2001
From: Mark Dean <mpmdean@gmail.com>
Date: Tue, 13 Aug 2024 13:54:43 -0400
Subject: [PATCH 7/9] update diag

---
 docs/source/_static/energy_level.png | Bin 15460 -> 14301 bytes
 1 file changed, 0 insertions(+), 0 deletions(-)

diff --git a/docs/source/_static/energy_level.png b/docs/source/_static/energy_level.png
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From 0359e27169b79dbf4db3336d677fbc4b7172cc41 Mon Sep 17 00:00:00 2001
From: Mark Dean <mpmdean@gmail.com>
Date: Wed, 14 Aug 2024 13:40:52 -0400
Subject: [PATCH 8/9] final typo

---
 examples/sphinx/example_5_charge_transfer.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/examples/sphinx/example_5_charge_transfer.py b/examples/sphinx/example_5_charge_transfer.py
index 4f97dc51..78d98cc8 100644
--- a/examples/sphinx/example_5_charge_transfer.py
+++ b/examples/sphinx/example_5_charge_transfer.py
@@ -120,7 +120,7 @@
 # Energy splitting in ligand states
 # ------------------------------------------------------------------------------
 # The last thing to consider is that our definition of the charge transfer
-# energy refers to the atomic limit with all hopping terms switch off, whereas
+# energy refers to the atomic limit with all hopping terms switched off, whereas
 # the ligand states in the model are already split by the oxygen-oxygen hopping
 # term :math:`T_{pp}` as illustrated below. So the final charge transer energy
 # needs to account for this.

From 47ded2b493f783444cb20243f026777ba4c6837a Mon Sep 17 00:00:00 2001
From: Mark Dean <mpmdean@gmail.com>
Date: Wed, 14 Aug 2024 13:44:53 -0400
Subject: [PATCH 9/9] final typo

---
 examples/sphinx/example_5_charge_transfer.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/examples/sphinx/example_5_charge_transfer.py b/examples/sphinx/example_5_charge_transfer.py
index 78d98cc8..ca259206 100644
--- a/examples/sphinx/example_5_charge_transfer.py
+++ b/examples/sphinx/example_5_charge_transfer.py
@@ -6,7 +6,7 @@
 example and considers the same model. This time we outline how to determine
 the charge transfer energy in the sense defined by Zaanen, Sawatzky, and Allen
 [1]_. That is, a :math:`d^{n_d} \\rightarrow d^{n_d + 1} \\underline{L}` transition
-in the atomic limit, after considering Coulomb interactins and crystal field. Although
+in the atomic limit, after considering Coulomb interactions and crystal field. Although
 this can be determined analytically in some cases, the easiest way is often just to
 calculate it, as we will do here.
 """