Magnetic Properties#

Calculate principal g values for isolated Kramers doublets, field-dependent electronic levels, magnetization and differential magnetic susceptibility. Results use the retained spin–orbit states of the CASCI/selected-CI or MPPT2 Hamiltonian chosen by mppt.heffso.iorder (1 or 2, respectively). Magnetic properties use the magnetic-moment operator at zeroth order in the MPPT perturbation, projected into those retained states.

Request a calculation#

Add this block to a working YAML calculation:

mppt:
  heffso:
    enabled: true
    properties: true
    magnetic_properties:
      fields_tesla: [[0.0, 0.0, 0.0], [0.0, 0.0, 1.0]]
      temperatures_kelvin: [2.0, 10.0, 300.0]
      susceptibility: true

This requests response at zero field and at 1 T along the input z axis, at each of three temperatures. For zero-field operators and eligible g tensors only, replace the nested request with magnetic_properties: {}. See all magnetic input controls for defaults, dependencies and optional field-matrix output.

Workflow reuse checks fields, temperatures, spin g and the effective doublet-grouping tolerance against the native artifact. A changed control needs a matching HEFFSO result; an old response grid is not silently reused.

The Ce³⁺, Pr³⁺ and OH examples provide complete inputs, calculation commands and comparisons with literature or analytic limits.

Magnetic moment and field Hamiltonian#

Zeroth-order magnetic-moment operator#

The operator order refers to the electronic MPPT perturbation. In the MPPT2 calculation, states come from the second-order effective Hamiltonian, while the magnetic-moment operator remains at zeroth order. First- and second-order MPPT corrections to that operator are not included.

The dimensionless angular-momentum matrices L and S use units of ℏ. The physical electronic moment is \(\mu_B\) times the numerical moment matrix m:

\[\begin{split}\begin{aligned} m_a&=-(L_a+g_eS_a),\\ H(\mathbf B)&=H(0)-\mu_B\sum_a B_a m_a. \end{aligned}\end{split}\]

Here \(g_e\) is spin_g (default 2.00231930436092) and B is in T. H(0) is the zero-field Hamiltonian in the retained SOC basis, with its lowest energy subtracted. The absolute reference is stored separately. Original zero-field gaps, including small pair splittings, are preserved.

The stored Zeeman matrices are derivatives of H with respect to the Cartesian field components, in hartree/T. The constants are \(\mu_B\) = 2.127191078656686 × 10⁻⁶ hartree/T and \(k_B\) = 3.166811563455608 × 10⁻⁶ hartree/K. Cartesian axes follow the input geometry; the angular-momentum origin is recorded in model/origin_bohr.

Operator order versus field response#

Zeroth order in the MPPT perturbation does not mean zero field or zero response. The Zeeman interaction is linear in B. Diagonalizing H(B) mixes retained states and can produce nonlinear magnetization and the off-diagonal Van Vleck susceptibility. These field effects are included with the same zeroth-order magnetic-moment operator.

Principal g values and their applicability#

g measures the linear Zeeman splitting of a doublet. In an anisotropic system, the splitting depends on field direction, so one scalar g is insufficient. HEFFSO reports three nonnegative principal values and their spatial axes.

The input spin_g is the electron-spin coefficient in the moment operator. The calculated doublet g includes orbital and spin matrix elements and their SOC mixing. An atomic Landé factor instead describes an entire angular-momentum multiplet; these three quantities should not be interchanged.

For an eligible two-state manifold, project each moment component into the pair and remove its scalar trace. The resulting metric is independent of the chosen basis within that pair:

\[\begin{split}\begin{aligned} \bar m_a&=m_a^{\mathrm{pair}} -\tfrac12\operatorname{Tr}(m_a^{\mathrm{pair}})I_2,\\ G_{ab}&=2\operatorname{Re}\operatorname{Tr}(\bar m_a\bar m_b),\\ g_i&=\sqrt{\lambda_i(G)}. \end{aligned}\end{split}\]

In the degenerate-pair, weak-field limit, the field-induced splitting is

\[\Delta E(\mathbf B)=\mu_B\sqrt{\mathbf B^{\mathsf T}G\mathbf B}.\]

The reported dimensionless values are ordered \(g_1 \le g_2 \le g_3\). Principal axes are columns in input xyz coordinates. Their signs are arbitrary; axes within repeated principal values are nonunique. A pair g tensor is not the Landé factor of an entire atomic multiplet.

Requirements for a Kramers-doublet g tensor#

The implementation requires an odd-electron, isolated two-state manifold closed under time reversal, with the correct time-reversal square, a time-odd moment and a negligible scalar moment trace. Higher degenerate multiplets are not split into arbitrary pairs.

The energy grouping threshold τ uses the configured tolerance and the largest actual returned eigensolver residual \(r_{\max}\):

\[\tau=\max(\tau_{\mathrm{input}},4r_{\max}).\]

Separation from neighboring manifolds must exceed 10τ. For a pair at the top of a truncated spectrum, the upper gap is unknown and g is withheld. If the entire represented model consists of that pair, the gap is known and stored as zero, meaning no external states in that model.

The per-state status codes explain omitted g values. Missing g does not prevent field levels or thermal response within the represented model. Inspect the time-reversal diagnostics when interpreting such a result.

Which states receive g values?#

Retained states	Automatic doublet g	Field levels, M and χ
Complete isolated odd-electron Kramers pair	Yes, if all applicability checks pass	Yes
Even-electron ion or molecule, including Pr³⁺	Withheld by this implementation	Yes
Fourfold or larger degenerate manifold	Withheld; no arbitrary pairing	Yes
Pair with missing time-reversed weight	Withheld	Available, but inspect the broken-closure diagnostics
Highest pair with an unknown omitted-state gap	Withheld	Available within the retained model

Even-electron systems can have magnetic response and effective spin-Hamiltonian parameters. Their general spin Hamiltonians and non-Kramers pseudospin fits are outside this automatic two-state analysis. A missing g row does not mean zero magnetic moment or zero susceptibility.

For a free atom, a numerically isolated pair created by an approximate multiplet splitting is a property of that computed model. It is not automatically an experimental crystal-field doublet. Inspect the complete multiplet and its convergence before assigning that interpretation.

Magnetization and susceptibility#

At each requested B, diagonalize H(B) and use its eigenvalues and eigenvectors to form the equilibrium density matrix at T > 0:

\[\begin{split}\begin{aligned} \beta&=(k_BT)^{-1},\\ Z&=\operatorname{Tr}e^{-\beta H(\mathbf B)},\\ \rho&=Z^{-1}e^{-\beta H(\mathbf B)},\\ M_a&=\operatorname{Tr}(\rho m_a),\\ \chi_{ab}&=\left.\frac{\partial M_a}{\partial B_b}\right|_T. \end{aligned}\end{split}\]

M is the numerical moment in \(\mu_B\) per ion or molecule; χ is in \(\mu_B/\mathrm{T}\) per ion or molecule. This is response to B, not a volume or molar susceptibility. No density, Avogadro-factor or macroscopic unit conversion is applied.

The derivative is evaluated analytically in the retained space. In the field eigenbasis, populations \(p_n\) and the divided-difference weights are

\[\begin{split}\begin{aligned} p_n&=e^{-\beta E_n}/Z,\\ D_{nm}&=\frac{p_n-p_m}{E_m-E_n} \quad(E_n\ne E_m),\\ D_{nm}&\longrightarrow\beta p_n \quad(E_m\to E_n),\\ \chi_{ab}&=\mu_B\Bigl[ \sum_{nm}D_{nm}\operatorname{Re} \bigl((m_a)_{nm}(m_b)_{mn}\bigr)\\ &\hspace{3em}-\beta M_aM_b\Bigr]. \end{aligned}\end{split}\]

This includes diagonal thermal fluctuations, off-diagonal Van Vleck terms and the degenerate limit. It is not a finite difference between the supplied field points: a single field is sufficient for χ there. All field–temperature combinations are evaluated. Principal susceptibilities are the ascending eigenvalues of the full Cartesian χ tensor, with axes stored as columns.

Isotropic response and effective moment#

For zero-field comparisons, the orientationally averaged susceptibility and the pointwise Curie-inferred effective moment are

\[\begin{split}\begin{aligned} \chi_{\mathrm{iso}}&=\tfrac13\operatorname{Tr}\chi,\\ \frac{\mu_{\mathrm{eff}}(T)}{\mu_B} &=\sqrt{\frac{3T\chi_{\mathrm{iso}}}{\mu_B/k_B}}. \end{aligned}\end{split}\]

Use the numerical χ values in \(\mu_B/\mathrm{T}\), temperature T in K and \(\mu_B/k_B \approx 0.671714\ \mathrm{K/T}\). This effective moment is a derived comparison quantity, not an additional native response array. The examples state the appropriate complete-multiplet or isolated-doublet Curie reference and its temperature range for each system.

Analytic Curie references#

For an unsplit atomic LS multiplet with Landé factor \(g_J\), or an isolated effective spin-½ pair with principal values \(g_1, g_2, g_3\), define a dimensionless factor C:

\[\begin{split}\begin{aligned} C_{\mathrm{atom}}&=g_J^2J(J+1),\\ C_{\mathrm{pair}}&=\tfrac14(g_1^2+g_2^2+g_3^2),\\ \chi_{\mathrm{iso}}^{\mathrm{Curie}}(T)&=\frac{\mu_B}{k_B}\frac{C}{3T},\\ \frac{\mu_{\mathrm{eff}}^{\mathrm{Curie}}}{\mu_B}&=\sqrt C. \end{aligned}\end{split}\]

These references assume the zero-field limit and positive temperatures well below the excited manifolds. They describe the chosen ground manifold alone; the full retained-state response also includes inter-manifold contributions. They are analytic comparisons, not measured susceptibilities.

From a molecular response to bulk permeability#

HEFFSO does not output the magnetic permeability of a material. Its M and χ refer to one fixed-nuclei ion or molecule in a specified B field. In SI, macroscopic magnetization is a moment per volume and susceptibility is commonly defined with respect to H:

\[\begin{split}\begin{aligned} \mathbf B&=\mu_0(\mathbf H+\mathbf M_{\mathrm{vol}}),\\ \boldsymbol\chi_H^{\mathrm{vol}} &=\frac{\partial\mathbf M_{\mathrm{vol}}}{\partial\mathbf H},\\ \boldsymbol\mu_r&=I+\boldsymbol\chi_H^{\mathrm{vol}}. \end{aligned}\end{split}\]

The last relation is the linear-response relation, or the differential relation about a specified field. The numerical HEFFSO χ in μB/T cannot simply be added to one: it has different units and a different field variable.

For a dilute, weakly magnetic ensemble of noninteracting identical centers, neglecting local-field corrections and using B ≈ μ₀H, the conversion is

\[\begin{split}\begin{aligned} \mathbf M_{\mathrm{vol}}&=n\mu_B^{\mathrm{SI}}\mathbf M^{\mathrm{code}},\\ \boldsymbol\chi_H^{\mathrm{vol}} &\simeq\mu_0 n\mu_B^{\mathrm{SI}}\boldsymbol\chi_B^{\mathrm{code}}. \end{aligned}\end{split}\]

Here n is the number of centers per m³, μB in SI is in A·m², and the code symbols mean the numerical values printed in μB and μB/T per center. Replacing n by Avogadro’s constant gives the corresponding dilute-limit molar susceptibility in m³/mol. Orientational averaging must match the sample.

Materials may additionally need exchange and dipolar interactions, local fields, sample demagnetization and the omitted diamagnetic response. The current equilibrium calculation also does not give hysteresis, relaxation rates or frequency-dependent AC susceptibility.

Read the results#

The managed run stores the complete result in data/<prefix>heffso_properties.h5. The MAGNETIC PROPERTIES text section shows totals and up to 20 eligible doublets and 20 thermal-response rows. Full grids, moment matrices, g metrics, axes and time-reversal diagnostics remain in the HDF5 magnetic contract.

With Caspy installed, export a Ce run using:

caspy-heffso-report --run-dir runs/ce --prefix Ce --magnetic

This writes the following files under runs/ce/reports/heffso/ from stored native values:

File	Contents
`Ce_levels.csv`	Zero-field energies, angular values and per-state magnetic status
`Ce_doublets.csv`	Eligible principal g and axes; header only if none qualify
`Ce_field_levels.csv`	Levels for each requested Cartesian field
`Ce_magnetic_response.csv`	M and, when requested, all nine χ components for every field–temperature point
`Ce_magnetic.png`	Overview of g, field levels and available thermal response

New fields or temperatures require a HEFFSO run with the revised input; the export command does not recompute thermal response. Current readers require mppt_heffso_properties_v4; regenerate older properties artifacts.

Model scope and convergence#

The current model uses the magnetic-moment operator at zeroth order in the MPPT perturbation, projected into the determinant catalog and retained SOC states, with real spatial orbitals. First- and second-order MPPT corrections to the moment operator are not included. The model has no separate quadratic orbital-diamagnetic term and no field relaxation of orbitals. It describes fixed-nuclei electronic response, not molecular rotation, vibrational averaging or hyperfine structure.

The selected determinant space and the number of retained scalar/SOC states affect response. States with small thermal populations can still contribute through off-diagonal susceptibility terms. Establish convergence for the requested fields and temperatures; thermal populations alone do not bound the omitted response.

For a doublet-only description, both thermal excitation and field mixing with neighboring manifolds must be small. The multistate field calculation can include mixing among retained states beyond that limit, but not mixing with discarded states. Increase the retained-state window and check the actual M and χ curves; there is no universal field or temperature cutoff valid for every system.

Zero-field Kramers protection requires odd electron count, time-reversal closure and a time-reversal-preserving approximation. Atomic multiplet completeness in S and applicable L/J is a separate question. Rebuilding MPPT with changed input functions or selection can change non-Kramers splittings; increasing selection need not reduce them monotonically.

Magnetic analysis preserves these gaps. A passive transformation of already constructed Hamiltonian and moment matrices must preserve spectra and principal g. See MPPT Symmetry and Convergence for the distinction and the evidence needed to interpret splittings.